MARIANO GIAQUINTA GIUSEPPE MODICA ji SOUdEK
Volume 37
Ergebnisse der Mathematik and ihrer Grenzgebiete 3. Folge
Cartesian Currents in the Calculus of Variations I
A Series
of Modern Surveys in Mathematics
Cartesian Currents
" Springer 4:s"
Mariano Giaquinta Giuseppe Modica Jir i Soucek
Cartesian Currents in the Calculus of Variations I Cartesian Currents
Springer
Mariano Giaquinta Dipartimento di Matematica University di Pisa Via F. Buonarroti, 2 1-56127 Pisa
Italy Giuseppe Modica
Dipartimento di Matematica Applicata University di Firenze Via S. Marta, 3 1-50139 Firenze
Italy Jiii Soucek Faculty of Mathematics and Physics Charles University Sokolovska, 83 i86oo Praha 8 Czech Republic Library of Congress Cara log tng-,n-Publication Data Giaquinta. Marlano. 1947Cartesian currents in the calculus of variations G,aqu,nta, Giuseppe Modica, Juui Soucek. o.
Mar, and
cm. -- iErgemnsse der Mathematlk and ,hrer Grenzgeb,ete
3. Fdlge, v. 37-38) Includes b,bl,ograph)ca I references and index. ISBN 3-540-64009-6 (v hardcover alk. paper). -- ISBN 3-540-64010-X in 2 hardcover alk. paper) I. Calculus of variations. I. Medica. Giuseppe. II. Soucek. Jlli. III Title. IV. Series Ergebnisse der Mathemat)k and hrer Grenzgeblete 3. Folge. Bd. 37-38QA316.G53 1998 515'.64--dc21 98-18195 CIP 1
,
Mathematics Subject Classification (1991): 49Q15, 49Q20, 49Q25, 26B30, 58E20, 73C50, 76A15
ISSN 0071-1136
ISBN 3-540-64009-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
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To
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Preface
Non-scalar variational problems appear in different fields. In geometry, for instance, we encounter the basic problems of harmonic maps between Riemannian manifolds and of minimal immersions; related questions appear in physics, for example in the classical theory of o-models. Non linear elasticity is another example in continuum mechanics, while Oseen-Frank theory of liquid crystals and Ginzburg-Landau theory of superconductivity require to treat variational problems in order to model quite complicated phenomena. Typically one is interested in finding energy minimizing representatives in homology or homotopy classes of maps, minimizers with prescribed topological singularities, topological charges, stable deformations i.e. minimizers in classes of diffeomorphisms or extremal fields. In the last two or three decades there has been growing interest, knowledge, and understanding of the general theory for this kind of problems, often referred to as geometric variational problems. Due to the lack of a regularity theory in the non scalar case, in contrast to the scalar one - or in other words to the occurrence of singularities in vector valued minimizers, often related with concentration phenomena for the energy density - and because of the particular relevance of those singularities for the problem being considered the question of singling out a weak formulation, or completely understanding the significance of various weak formulations becames non trivial. Keeping in mind the direct methods of Calculus of Variations, this amounts roughly to the question of identifying weak maps or fields, and weak limits of sequences of weak maps with equibounded energies. As we shall see, the choice of the notions of weak maps and weak convergence is very relevant, and different choices often lead to different answers concerning equilibrium points.
The aim of this monograph is twofold: discussing a homological theory of weak maps, and in this context treating several typical and relevant variational problems. The basic idea in defining a weak notion of vector valued maps is to think of them not componentlywise but globally, i.e. as graphs. In other words we define
weak maps between two oriented and boundaryless Riemannian manifolds X and y similarly to distributions or Sobolev functions, using a standard duality approach, but testing with functions which live in the product space X x Y. Thus one is naturally forced to move from the context of Sobolev maps, for instance, to that of currents and to allow vertical parts, and one is naturally led
viii
Preface
to the basic notion in this monograph of Cartesian currents. One should think of Cartesian currents as weak limits in the sense of currents of graphs of smooth maps, although this is not always true. In particular Cartesian currents satisfy the homological condition of having zero boundary in the cylinder X x y, and they induce a homology map which is continuous for the weak convergence of currents. As the natural context for those notions is geometric measure theory, we first develop an elementary introduction to the theory of currents of Federer and Fleming to provide all needed information. In particular we prove the deformation and closure theorems of Federer and Fleming, that, as we shall see, play a relevant role not only to study parametric but also non parametric integrals. In the first part of our monograph, after a preliminary chapter about measure theory and the phenomenology of weak convergence, we discuss integer rectifiable and normal currents, differentiability properties of the graphs of maps, continuity of Jacobian minors, and how those notions are related in terms of approximate tangent planes, area or mass, and the homological notion of boundary. We also discuss related topics as, for instance, higher integrability of determinants, functions of bounded variations and degree theory, and of course closure, compactness and structure properties of special classes of Cartesian currents. Finally in Vol. I Ch. 5 we deal with the homology theory for currents. There we present classical topics as for instance Hodge theory, Poincare-Lefschetz and de Rham dualities and intersection numbers, and we conclude by discussing the homology map associated to a Cartesian current in terms of periods and cycles. In doing this we have tried to keep our treatment elementary, illustrating with simple examples the results, their meaning and their typical use, and we always give detailed proofs. Also, at the cost of some repetition we have tried to make each chapter, and sometimes even sections, readable as far as possible indipendently of the general context, so that parts of this monograph can be easily used separately for example for graduate courses. This we hope justifies the size of our monograph. Open questions are often mentioned and in the final section of each chapter we discuss references to the literature and sometimes supplementary results. In the second part of our monograph we deal with variational problems. In Vol. II Ch. 1 we discuss general variational problems, their parametric and non parametric formulations, and in connection with them, different notions of ellipticity (parametric ellipticity, polyconvexity, and quasiconvexity). The rest of the second part of this monograph is then dedicated to specific variational problems in the setting of Cartesian currents: in Vol. II Ch. 2 we deal with weak diffeomorphisms and non linear elasticity, in Vol. II Ch. 3, Vol. II Ch. 4, and Vol. II Ch. 5 we discuss some issues of the harmonic mapping problem and of related questions; and, finally, we shortly deal in Vol. II Ch. 6 with the non parametric area problem. For further information about the content of this monograph we refer the reader to the introductions to each chapter, to the detailed table of contents, and to the index.
Preface
ix
In preparing this monograph we have taken advantage from discussions with
many friends and colleagues. Among them it is a pleasure for us to thank G. Anzellotti, J. Ball, F. Bethuel, H. Brezis, F. Helein, S. Hildebrandt, J. Jost, H. Kuwert, F. Lazzeri, D. Mucci, J. Necas, K. Steffen, M. Struwe, V. Sverak and B. White. We thank also Mirta Stampella for her invaluable help in the typing and retyping of our manuscript. We would like to aknowledge supports during the last years from the Ministero dell'Universita e della Ricerea Scientifica, Consiglio Nazionale delle Ricerche, EC European Research Project GADGET I, II, III, and Czech Academy of Sciences. Parts of this work have been written while the authors were visiting different Universities. We want to extend our thanks to Alexander von Humboldt Foundation, Sonderforschungsbereich 256 of Bonn University, to the Departments of Mathematics of the Universities of Bonn, Cachan, Keio, Paris VI, Shizuoka, and to the Forschungsinstitut ETH, Zurich, to the Center for Mathematics and its Applications, Canberra and to the Academia Sinica, Taiwan. Firenze, October 1997
Mariano Giaquinta Giuseppe Modica
Jiri Soucek
Contents Cartesian Currents in the Calculus of Variations I and II
Volume I. Cartesian Currents 1
General Measure Theory ....................................
2
Integer Multiplicity Rectifiable Currents ..................... 69
3
Cartesian Maps .............................................. 175
4
Cartesian Currents in Euclidean Spaces ...................... 323 Cartesian Currents in Riemannian Manifolds ................ 493
5
1
Bibliography ..................................................... 667 Index ............................................................ 697
Symbols ......................................................... 709
Volume II. Variational Integrals 1
Regular Variational Integrals ................................
1
5
Finite Elasticity and Weak Diffeomorphisms ................. 137 The Dirichlet Integral in Sobolev Spaces ..................... 281 The Dirichlet Energy for Maps into S2 ...................... 353 Some Regular and Non Regular Variational Problems ....... 467
6
The Non Parametric Area Functional ........................ 563
2 3
4
Bibliography .....................................................
653 Index ............................................................ 683 Symbols ......................................................... 695
Contents Volume I. Cartesian Currents
1.
General Measure Theory .................................... 1
General Measure Theory ................................... 1.1 Measures and Integrals ...............................
1 1
1
(Q-measures and outer measures, measurable sets, Caratheodory mea-
sures. Measurable functions. Lebesgue's integral. Egoroff's, Beppo Levi's, Fatou's and Lebesgue's theorems. Product measures and Fubini-Tonelli theorem)
1.2
....................
10
Hausdorff Measures ..................................
12
Borel Regular and Radon Measures
(Borel functions and measures, Radon measures. Caratheodory's criterion in metric spaces. Vitali-Caratheodory theorem. Lusin's theorem)
1.3
(Hausdorff measures and Hausdorff dimension, spherical measures. Isodiametric inequality. Cantor sets and Cantor-Vitali functions. Caratheodory construction. Hausdorff measure of a product)
1.4
.....
23
Covering Theorems, Differentiation and Densities ........
29
Lebesgue's, Radon-Nikodym's and Riesz's Theorems
(Lebesgue's decomposition theorem, Radon-Nikodym differentiation theorem. Vector valued measures and Riesz representation theorem)
1.5
(Vitali's and Besicovitch's covering theorems. Symmetric differentiation and Radon-Nikodym theorem. Densities. Approximate limits and measurability. Densities and Hausdorff measure)
2
Weak Convergence ......................................... 36 2.1
Weak Convergence of Vector Valued Measures .......... (Definitions. Banach-Steinhaus theorem and the compactness theorem. Convergence as measures and L1-weak convergence. Lebesgue's theorem
36
about weak convergence in L1)
2.2
Typical Behaviours of Weakly Converging Sequences ..... 40
(Oscillation, concentration. distribution, and concentration-distribution. Nonlinearity destroys the weak convergence) 2.3
Weak Convergence in L4, q > 1 ....................... 44
2.4
spaces. Riemann-Lebesgue's lemma. Radon-Riesz theorem and variants) Weak Convergence in L' ............................. 50 (The proof of Lebesgue theorem on weak convergence in L'. Weak con-
(Weak convergence in L' and weak and weak" convergence on Banach
vergence of the product)
2.5
2.6
2.7 3
Concentration: Weak Convergence of Measures ..........
(The universal character of the concentration-distribution phenomenon. The concentration-compactness lemma)
55
Oscillations: Young Measures ......................... 58 More on Weak Convergence in Ll ..................... 64 (Equivalent definitions of Young measures. Examples)
(Convergence in the sense of biting and convergence of the absolutely continuous parts) Notes .................................................... 67
Contents Volume I. Cartesian Currents
Integer Multiplicity Rectifiable Currents ..................... 69 Area and Coarea. Countably n-Rectifiable Sets ................ 69 Area and Coarea Formulas for Linear Maps ............. 69 1.1
1
(Polar decomposition theorem. Area and change of variable formula for linear maps. Coarea formula for linear maps. Cauchy-Binet formula)
1.2
Area Formula for Lipschitz Maps ...................... 74
(Area formula for smooth and for Lipschitz maps: curves, graphs of codimension one, parametric hypersurfaces, submanifolds, and graphs of higher codimension)
1.3
Coarea Formula for Lipschitz Maps ....................
82
(Coarea formula for smooth and for Lipschitz maps. C'-Sard type theorem)
1.4
1.5 2
Rectifiable Sets and the Structure Theorem ............. 90
(Countably n-rectifiable and n-rectifiable sets. The approximate tangent space of sets and measures. The rectifiability theorem for Radon measures. Besicovitch-Federer structure theorem)
The General Area and Coarea Formulas ................ 99
(Area and coarea formulas for sets on manifolds and for rectifiable sets. The divergence theorem)
Currents ................................................. 103 2.1
Multivectors and Covectors ........................... 104
(k-vectors, exterior product of multivectors, simple k-vectors. Duality between k-vectors and covectors. Inner product of multivectors. Sim-
ple k-vectors and oriented k-planes. Simple n-vectors in the Cartesian product 1R" x 1RN. Characterization of simple n-vectors in IR" x RN. Induced linear transformations)
2.2
Differential Forms ..
... .......
................. 118
(Exterior differentiation, pullback, forms in a Cartesian product. Integration of differential forms)
2.3
Currents: Basic Facts ................................ 122
(Currents and weak convergence of currents. Boundary and support of currents. Product of currents. Currents with finite mass, currents which are representable by integration. Lower semicontinuity of the mass. Compactness-closure theorem for currents with locally finite mass. ndimensional currents in R" and BV functions. The constancy theorem. Examples. Image of a current under a Lipschitz map. The homotopy formula)
2.4
Integer Multiplicity Rectifiable Currents ................ 136
2.5
Slicing ............................................. 151 (Slices of codimension one. Slices of codimension larger than one)
2.6
(Currents carried by smooth graphs. Rectifiable and integer multiplicity rectifiable currents. The closure theorem for integer multiplicity 0dimensional and 1-dimensional rectifiable currents. Examples. Image of a rectifiable current under a Lipschitz map. The Cartesian product of rectifiable currents)
The Deformation Theorem and Approximations ......... 157
(The deformation theorem. Isoperimetric inequality. Weak and strong polyhedral approximations. The strong approximation theorem for normal currents)
2.7 3
The Closure Theorem
................................ 161
(The classical proof: slicing lemma, the boundary rectifiability theorem, the rectifiability theorem. White's proof) Notes .................................................... 173
Cartesian Maps .............................................. 175 1
Differentiability of Non Smooth Functions
.................... 179
Contents Volume I. Cartesian Currents 1.1
xv
The Maximal Function and Lebesgue's Differentiation Theorem
........................................... 180
(The distribution function. Hardy-Littlewood maximal function and inequality. Lebesgue's differentiation theorem. Lebesgue's points. The Hausdorff dimension of the set of non Lebesgue points. Calderon-Zygmund decomposition argument. The class L log L)
1.2
Differentiability Properties of W 1,P Functions
........... 192
(Differentiability in the LP sense. Calderon-Zygmund theorem. MorreySobolev theorem)
1.3
Lusin Type Properties of W''P Functions ............... 202
(Kirszbraun and Rademacher theorems. Lusin type theorems for Sobolev functions. Whitney extension theorem. Liu's theorem)
1.4 1.5
Approximate Differential and Lusin Type Properties ..... 210
(Approximate continuity and approximate differentiability. Lusin type properties are equivalent to the approximate differentiability)
Area Formulas, Degree, and Graphs of Non Smooth Maps 218
(Area formula, graphs and degree for non smooth maps. Rado'-Reichelderfer theorem) 2
Maps with Jacobian Minors in L' ........................... 228 2.1 2.2
2.3 2.4
The Class AI (.(l, RN) , Graphs and Boundaries .......... 229 (The class A'(0, 1RN ). The current integration over the graph. Convergence of graphs and minors) Examples .......................................... 233 (The map x/Ixl. Homogeneous maps: boundary and degree)
Boundaries and Integration by Parts ................... 238
(Approximate differential and distributional derivatives. Analytic formulas for boundaries. Boundary and pull-back )
More on the Jacobian Determinant .................... 247
(Maps in Wl,"-1 The distributional determinant. Isoperimetric in-
equality for the determinant. The class A,,,,, and Higher integrability of the determinant. BMO and Hardy space 1 (]R"))
2.5
Boundaries and Traces ............................... 265 (The boundary of a current integration over a graph and the trace in
3
the sense of Sobolev. Weak and strong anchorage) Cartesian Maps ........................................... 276
3.1
Weak Continuity of Minors ........................... 277 (Weak convergence of minors in L1, as measures, and convergence of
graphs. Examples Reshetnyak's theorem)
3.2
The Class cart' (.fl, RN): Closure and Compactness ...... 285
(The class of Cartesian maps. The closure theorem. A compactness theorem)
3.3
The Classes cartP(f?, R'`'), p > 1 ....................... 293 (The class cart P (0,1RN) closure and compactness theorems.)
4
Approximability of Cartesian Maps .......................... 296 The Transfinite Inductive Process ..................... 299 4.1 (Ordinal numbers. The transfinite inductive process and the weak sequential closure of a set)
4.2
Weak and Strong Approximation of Minors ............. 303
(Sequential weak closure and strong closure of smooth maps in the class of Cartesian maps. Cart1(D,RN) = CART' (f2, RN) C cart' (S2, RN))
4.3 5
The Join of Cartesian Maps .......................... 313 (Composition and join of Cartesian maps. Weak continuity of the join)
Notes .................................................... 318
Contents Volume I. Cartesian Currents
xvi
4.
Cartesian Currents in Euclidean Spaces ...................... 323 Functions of Bounded Variation ............................. 327 1 1.1
1.2
The Space BV (Q, R) ................................ 329
(The total variation. Semicontinuity. Approximation by smooth functions A variational characterization of By. Sobolev and Poincare inequalities. Compactness. Fleming-Rishel formula)
Caccioppoli Sets .................................... 340 (Sets of finite perimeter. Isoperimetric inequality. Approximation by
smooth open sets)
1.3
1.4
1.5
De Giorgi's Rectifiability Theorem ..................... 346
(Reduced boundary. De Giorgi's rectifiability theorem. Measure theoretic boundary. Federer's characterization of Caccioppoli sets)
The Structure Theorem for BV Functions .............. 354
(Jump points and jump sets. Regular and singular points. The structure of the measure total variation. Lebesgue's points and approximate differentiability of BV-functions)
Subgraphs of BV Functions .......................... 371
(Characterization of BV functions in terms of their subgraphs; jump and Cantor part of Du in terms of the reduced boundary of the subgraph of u)
2
Cartesian Currents in Euclidean Spaces ...................... 379 Limit Currents of Smooth Graphs ..................... 380 2.1 (Toward the definition of Cartesian currents)
2.2 2.3
The Classes cart (,fl x IRN) and graph(.f2 x ISBN) .......... 384
(Cartesian currents and graph-currents)
The Structure Theorem .............................. 391
(The structure theorem. The map associated to a Cartesian current. Weak convergence of Cartesian currents)
2.4 2.5
2.6
Cartesian Currents in Codimension One ................ 403
(BV functions as Cartesian currents and representation formulas. Cantor-Vitali functions. SBV functions)
Examples of Cartesian Currents ....................... 411
(Bubbling off of circles, spheres, tori. Attaching cylinders. Examples of vector-valued BV functions. A Cartesian current with a Cantor mass on minors. A Cartesian current which cannot be approximated weakly by smooth graphs)
Radial Currents ..................................... 439
(Currents associated to radial maps u(x) = U(IxI)x/Ixi. A closure theorem)
3
Degree Theory ............................................ 450 3.1
n-Dimensional Currents and BV Functions ............. 451 (Representation formula and decomposition theorem for n-dimensional
currents in F. Constancy theorem and linear projections of normal currents)
4
3.2
Degree Mapping and Degree of Cartesian Currents ...... 460
3.3
The Degree of Continuous Maps ....................... 471
3.4
h-Connected Components and the Degree .............. 474
(Definition and properties of the degree for Cartesian currents and maps)
(The degree of continuous Cartesian maps agrees with the classical degree for continuous maps)
(Homologically connected components of a Caccioppoli set) Notes .................................................... 479
Contents Volume I. Cartesian Currents
5.
xvii
Cartesian Currents in Riemannian Manifolds ................ 493 1
More About Currents ...................................... 494
1.1
The Deformation Theorem ........................... 494
(Proof of Federer and Fleming deformation theorem and of the strong approximation theorem)
1.2 1.3
Mollifying Currents .................................. 505
(e-mollified of forms and currents. A representation formula for normal currents)
Flat Chains .........................................512
(Integral flat and flat chains and norms. Image of a flat chain. Federer's flatness theorem. Mollification and a representation formula for flat chains. Federer's support theorem. Cochains) 2
Differential Forms and Cohomology .......................... 527 2.1
Forms on Manifolds .................................. 528
(Tangent and cotangent bundle. Null forms to a submanifold)
2.2 2.3
Hodge Operator ........... ........ ................531
(Interior multiplication of vectors and covectors. Hodge operator. The L2 inner product for forms)
Sobolev Spaces of Forms ............................. 536 (The classes Lr2,(X) and WP'2(X))
2.4 2.5
Harmonic Forms .................................... 538
(The codifferential S. Laplace-Beltrami operator on forms and the Dirichlet integral. Their expressions in local coordinates)
Hodge and Hodge-Kodaira-Morrey Theorems ........... 543
(Gaffney Lemma. Hodge-Kodaira-Morrey decomposition theorem. De Rham cohomology groups. Hodge representation theorem for cohomology classes)
2.6
2.7
2.8
3
Relative Cohomology: Hodge-Morrey Decomposition ..... 549
(Collar theorem. Tangential and normal part of a form. Coboundary operator. The lemma of Gaffney at the boundary. Hodge-Morrey decomposition. Hodge representation theorem of relative cohomology classes)
Weitzenbock Formula ................................ 559
(Connections and covariant derivatives. Levi-Civita connection. Second covariant derivatives. Curvature tensor and Laplace-Beltrami operator on forms)
Poincare and Poincare-Lefschetz Dualities in Cohomology 565
(De Rham cohomology groups Poincare duality. Relative cohomology groups on manifolds with boundary. Cohomology long sequence. Poincare Lefschetz duality)
Currents and Real Homology of Compact Manifolds ........... 570 3.1 3.2
Currents on Manifolds ............................... 572
(Currents in X. Flatness and constancy theorems)
Poincare and de Rham Dualities ...................... 574
(Isomorphism of (n - k) cohomology groups and k homology groups. Integration along fibers Poincare dual form. de Rham duality between cohomology and homology. Periods. Normal currents and classical real homology)
3.3
Poincare-Lefschetz and de Rham Dualities .............. 589 (Relative homology. Homology long sequence. Poincare-Lefschetz duality
theorem. De Rham theorem for manifolds with boundary. Relative real homology classes are represented by minimal cycles)
3.4
Intersection of Currents and Kronecker Index ........... 599
(Intersection of normal currents in R' and on submanifolds of R'. Intersection of cycles is the wedge product of Poincare duals. Kronecker index. Intersection index)
Contents Volume I. Cartesian Currents
xviii
3.5
4
Relative Homology and Cohomology Groups ............ 608
(Homology and cohomology in the Lipschitz category. Closure of cosets. Generalized de Rham theorem)
Integral Homology ......................................... 615 4.1
Integral Homology Groups ............................ 615
(Integral homology groups. Integral relative homology groups. Isoperimetric inequalities and weak closure. Torsion groups. Integral and real homology)
4.2
5
Intersection in Integral Homology ..................... 624
(Intersection of cycles on boundaryless manifolds. Intersection of cycles on manifolds with boundary. Intersection index in integral homology. An algebraic view of integral homology)
Maps Between Manifolds ................................... 631 5.1 5.2
Sobolev Classes of Maps Between Riemannian Manifolds . 632
(Density results of Schoen-Uhlenbeck and Bethuel. d-homotopy White's results)
Cartesian Currents Between Manifolds ................. 640
(Approximate differentiability. Area formula. Graphs. Cartesian currents. The class carta,l(!2 x y))
5.3 5.4 6
Homology Induced Maps: Manifolds Without Boundary .. 648
(Homology and cohomology maps associated to a Cartesian current)
Homology Induced Maps: Manifolds with Boundary...... 658
(Homology and cohomology maps associated to flat chains)
Notes .................................................... 663
Bibliography ..................................................... 667 Index ............................................................ 697
Symbols ......................................................... 709
Contents Volume II. Variational Integrals
1.
Regular Variational Integrals ................................ 1 The Direct Methods .......................................
1.1
2
The Abstract Setting ................................
3
Some Classical Lower Semicontinuity Theorems .........
10
(Lower semicontinuous and coercive functionals. Weierstrass theorem. r-relaxed functional)
1.2
1
(Lower semicontinuity with respect to the weak convergence in L1. The
role of Banach-Saks's theorem and Jensen's inequality. Regular and smooth integrands in the Calculus of Variations)
1.3
A General Semicontinuity Theorem ....................
19
(Lower semicontinuity with respect to the weak' convergence in L1) 2
Polyconvex Envelops and Regular Parametric Integrals ......... 23 2.1
2.2 2.3
Polyconvexity and Polyconvex Envelops ................ 26
(A few facts from convex analysis. n-vectors associated to the tangent planes to graphs. Polyconvex functions and polyconvex envelopes)
Parametric Polyconvex Envelops of Integrands .......... 37
(The parametric polyconvex l.s.c. envelop of an integrand. Mass and comass with respect to the integrand f)
The Parametric Extension of Regular Integrals .......... 44 (The parametric extension of an integral as integral of the parametric polyconvex l.s.c. envelop)
2.4 3
The Polyconvex l.s.c. Extension of Some Lagrangians .... 45
(Area of graphs. The total variation of the gradient. The Dirichlet integral. The p-energy functional. The liquid crystal integrand)
Regular Integrals in the Class of Cartesian Currents ........... 74 Parametric Integrands and Lower Semicontinuity ........ 75 3.1 (Parametric integrands and lower semicontinuity of parametric integrals)
3.2
Existence of Minimizers in Classes of Cartesian Currents . 82
(Lower semicontinuity of the parametric extension of regular integrals. Existence of minimizers in subclasses of Cartesian currents)
3.3
Relaxed Energies in the Setting of Cartesian Currents .... 88
3.4
Relaxed Energies in the Parametric Case ...............
(The relaxed functional in classes of Cartesian currents and maps)
(The approximation problem for parametric integrals. A theorem of Reshetnyak. Flat integrands and Federer's approximation theorem. In-
90
teger multiplicity rectifiable and real minimizing currents)
4
Regular Integrals and Quasiconvexity ........................ 106 4.1 4.2
Quasiconvexity
...................................... 107
(Quasiconvexity as necessary condition for semicontinuity. Rank-one convexity and Legendre-Hadamard condition)
Quasiconvexity and Lower Semicontinuity
.............. 117
(Quasiconvexity as sufficient condition for semicontinuity in classes of Sobolev maps or Cartesian currents)
Contents Volume II. Variational Integrals
roc
4.3 5
2.
Ellipticity and Quasiconvexity ........................ 127 (Ellipticity, quasiconvexity and lower semicontinuity)
Notes .................................................... 131
Finite Elasticity and Weak Diffeomorphisms ................. 137 1
State Space and Stored Energies in Elasticity ................. 139 1.1
Fields and Transformations ........................... 139
1.2
Kinematics ......................................... 140 (Bodies, states, and deformations of a body. Deformations as 3-surfaces
(Non local and non linear structure of transformations) in R6, or as graphs of diffeomorphisms)
1.3
Local Deformations .................................. 143
1.4
Perfectly Elastic Bodies: Stored Energy, Convexity and
(Infinitesimal deformations as simple tangent vectors to the deformation surface)
Coercivity .......................................... 147
(Stored energy: different forms and constitutive conditions. Polyconvexity and coercivity)
1.5
Variations and Stress ................................ 152 (Infinitesimal variations and the notion of stress. Piola-Kirchhoff and
Cauchy stress tensors. Energy-momentum tensor.) 2
Physical Implications on Kinematics and Stored Energies ....... 155 Kinematical Principles in Elasticity: Weak Deformations . 156 2.1 (Material body and its parts. Impenetrability of matter. Weakly invertible maps. Weak one-to-one transformations. Existence of local deformations. Weak deformations. Elastic bodies and absence of fractures. Elastic deformations)
2.2
Frame Indifference and Isotropy ....................... 169 (Frame indifference principle. Energies associated to isotropic materials)
2.3
2.4 2.5 3
Convexity-like Conditions ............................ 170
(Convexity is not compatible with elasticity. Noll's condition. Polyconvexity and Diff-quasiconvexity)
Coercivity Conditions ................................ 175
(A discussion of the coercivity conditions)
Examples of Stored Energies .......................... 179 (Ogden-type stored energies for isotropic materials)
Weak Diffeomorphisms .....:; .............................. 182 3.1
3.2
3.3 3.4 3.5
3.6
The Classes dif p'q (,f2, ,fl) ............................. 183
(The class of (p, q)-weak diffeomorphisms. Weak convergence. Closure' and compactness properties. An example of a discontinuous weak diffeomorphism)
The Classes dif
p 4 ,
(0, Rn) ............................ 191
(Weak diffeomorphisms with non-prescribed range. Closure and compactness properties. Elastic deformations as weak diffeomorphisms)
Convergence Theorems for the Inverse Maps ............ 199 (Convergence of the ranges and of the inverse maps)
General Weak Diffeomorphisms ....................... 204
(Weak diffeomorphisms with vertical and horizontal parts: structure, closure, and compactness theorems)
The Dif-classes ...................................... 213
(The approximation problem for weak diffeomorphisms)
Volume Preserving Diffeomorphisms ................... 214 (The Jacobian determinant of weak one-to-one maps and of weak deformations)
4
Connectivity Properties of the Range of Weak Diffeomorphisms . 216
Contents Volume II. Variational Integrals 4.1
Connectivity of the Range of Sobolev Maps
(Connected sets and d,_1-connected sets.
............. 217
"mapped" into connected sets by Sobolev maps)
4.2
xxi
sets are
Connectivity of the Range of Weak Diffeomorphisms
..... 219
(Weak diffeomorphisms "map" connected sets into essentially connected sets. Weak diffeomorphisms do not produce cavitation. Examples)
4.3
Regularity Properties of Locally Weak Invertible Maps ... 229
(Weak local diffeomorphisms. Local properties of weak local diffeomorphisms. Vodopianov-Goldstein's theorem. Courant-Lebesgue lemma)
4.4
Global Invertibility of Weak Maps ..................... 238
(Conditions ensuring that weak local diffeomorphisms be one-to-one and homeomorphisms)
4.5 5
(A.e. open sets and a.e. continuous maps. Weak diffeomorphisms in W1' ', p > n - 1, "are" a.e. open maps)
Composition .............................................. 247 5.1
6
An a.e. Open Map Theorem .......................... 243 Composition of weak deformations ..................... 248
(Composition of one-to-one maps and of weak diffeomorphisms)
5.2
On the Summability of Compositions .................. 250
5.3
Composition of Weak Diffeomorphisms ................. 254
(Binet's formula and the summability of the composition)
(The action of weak diffeomorphisms on Cartesian maps and the pseudogroup structure of weak diffeomorphisms. Weak convergence of compositions)
Existence of Equilibrium Configurations ...................... 260 6.1
Existence Theorems ................................. 261
(The displacement pressure problem. Deformations with fractures)
6.2 6.3 7
3.
Equilibrium and Conservation Equations ............... 264
(Energy-momentum conservation law and Cauchy's equilibrium equation)
The Cavitation Problem .............................. 268
(Elastic deformations do not cavitate)
Notes .................................................... 278
The Dirichlet Integral in Sobolev Spaces ..................... 281 1
Harmonic Maps Between Manifolds .......................... 281 1.1 First Variation and Inner Variations ................... 283 (Euler variations and Euler-Lagrange equation of the energy integral. Inner variations, energy-momentum tensor, inner and strong extremals. Conformality relations. Stationary points. Parametric minimal surfaces)
1.2
Finding Harmonic Maps by Variational Methods ........ 293
(Existence and the regularity problem. Mappings from B" into the upper hemisphere of S". Mappings from B" into S"-1)
2
Energy Minimizing Weak Harmonic Maps: Regularity Theory ... 296 2.1 Some Preliminaries. Reverse Holder Inequalities ......... 297 (Some algebraic lemmas. The Dirichlet growth theorem of Morrey. Reverse Holder inequalities with increasing supports)
2.2
Classical Regularity Results .......................... 303 An Optimal Regularity Theorem ...................... 307 (Morrey's regularity theorem for 2-dimensional weak harmonic maps)
2.3
2.4
(A partial regularity theorem and the existence and regularity of energy minimizing harmonic maps with range in a regular ball: results by Hildebrandt, Kaul, Widman, and Giaquinta, Giusti.)
The Partial Regularity Theorem ....................... 319
(The partial regularity theorem for energy minimizing weak harmonic maps: Schoen-Uhlenbeck result)
Contents Volume II. Variational Integrals
)odi
3
Harmonic Maps in Homotopy Classes ......................... 333 The Action of Wl'2-maps on Loops .................... 334 3.1 (Courant-Lebesgue lemma. In the two-dimensional case the action on loops is well defined for maps in W122)
3.2 3.3 4
4.2
4.
(Energy minimizing maps with prescribed action on loops. Schoen-Yau, Saks-Uhlenbeck, Lemaire, Eells-Sampson and Hamilton theorems)
Local Replacement by Harmonic Mappings: Bubbling .... 337 (Jost's replacement method and existence of minimal immersions of S2)
Weak and Stationary Harmonic Maps with Values into S2 ...... 339 4.1
5
Minimizing Energy with Homotopic Constraints ......... 336
The Partial Regularity Theory ........................ 339 (An alternative proof. More on the singular set) Stationary Harmonic Maps ........................... 345 (Partial regularity results for stationary harmonic maps)
Notes .................................................... 350
The Dirichlet Energy for Maps into S2 ...................... 353 1
Variational Problems for Maps from a Domain of R2 into S2 .... 354 1.1
Harmonic Maps with Prescribed Degree ................ 354
(Homotopic equivalent maps and degree. Bubbling off of spheres. The stereographic and the modified stereographic projection. e-conformal maps)
1.2
The Structure Theorem in cart2'1(Q x S2), .f2 C JR2 ...... 362 (The structure and approximation theorems in cart2.1(1? X S2))
1.3
2
Existence and Regularity of Minimizers ................ 366
(The relaxed energy and existence of minimizers. Energy minimizing maps with constant boundary value: Lemaire's theorem. The simplest chiral model and instantons. Large solution for harmonic maps: BrezisCoron and Jost result. A global regularity result)
Variational Problems from a Domain of JR3 into S2 ............ 383 2.1
The Class cart2'1(,f? X S2), ,f2 C R3 .................... 385 (The D-field and homological singularities)
2.2
Density Results in W1'2(B3, S2) ....................... 392 (Approximation by maps which are smooth except at a discrete set of points)
2.3
2.4
Dipoles and Gap Phenomenon ........................ 400 The Structure Theorem in cart2'1(.f2 x S2), Q C J 3 ...... 409
(Dipoles and the approximate dipoles. Lavrentiev or gap phenomenon)
(Structure of the vertical part of Cartesian currents in cart2'1(S2 x S2))
2.5
Approximation by Smooth Graphs: Dirichlet Data ....... 412
(Weak approximation in energy by smooth graphs. The minimal connection and its continuity properties with respect to the W1,2-weak convergence. Cart2,1(1 X S2) = cart2 1(12 x S2). Weak approximability by smooth maps in W,P'2(S2 X S2))
2.6
Approximation by Smooth Graphs: No Boundary Data... 419 (T belongs to carte"1(S2 x S2) if and only if it can be approximated weakly and in energy by smooth graphs Guk possibly with uk = UT on
an) 2.7
The Dirichlet Integral in cart2'1(S? X S2), 0 C JR3 ....... 423 (The parametric polyconvex extension of the Dirichlet integral is its relaxed or Lebesgue's extension. The relaxed of V(u, f2) in W 1,2 (D, S2))
2.8
Minimizers of Variational Problems .................... 429 (Variational problems and existence of minimizers)
2.9
Contents Volume II. Variational Integrals
xxiii
A Partial Regularity Result ...........................
433
(The absolutely continuous part uT of minimizers T is regular except on a closed set whose Hausdorff dimension is not greater than 1. Tangent cones)
2.10
The General Dipole Problem .......................... 449
(The coarea formula and the minimum energy of dipoles)
2.11
Singular Perturbations
............................... 452
(Trying to solve Dirichlet problem by approximating by singularly perturbed functionals of the type of Ginzburg-Landau) 3
5.
Notes .................................................... 458
Some Regular and Non Regular Variational Problems ....... 467 1
The Liquid Crystal Energy ................................. 467 1.1 The Sobolev Space Approach ......................... 470
(Existence and regularity of equilibrium configuration)
1.2
1.3
The Relaxed Energy ................................. 470
(Existence of equilibrium configurations for the relaxed energy. The dipole problem. Relaxed energies in Sobolev spaces and Cartesian currents. Equilibrium configurations with fractures)
The Dipole Problem ................................. 477
(Approximation in energy: irrotational and solenoidal dipoles. The general dipole problem) 2
The Dirichlet Integral in the Regular Case: Maps into S2 ....... 485 2.1
Maps with Values in S2 .............................. 485
(Maps from a n-dimensional space into S2 and the class carte' 1(S2 x S2).
The (n - 2)-D-field) 2.2
The Dipole Problem ................................. 489 (Degree with respect to a (n - 3)-curve and the dipole problem)
2.3
The Structure Theorem .............................. 494
(Structure theorem for currents in cart2'1(0 x S2S2 C LQ") 3
The Dirichlet Integral in the Regular Case: Maps into a Manifold 496 3.1
The Class cart2" (.(2 x y) ............................. 497
(The structure theorem for currents in cart2'1(D x y), 0 C E2)
3.2
Spherical Vertical Parts and a Closure Theorem ......... 501
(Reduced Cartesian currents. Closure theorem. Vertical parts of currents
in Cart2.1(!2 x y) are of the type S2)
3.3
4
The Dirichlet Integral and Minimizers .................. 506
The Dirichlet Integral in the Non Regular Case: a Homological Theory ................................................... 508
4.1
(n,p)-Currents ...................................... 509 ((n, p)-graphs and the classes A(P),'. rectifiable (r, p)-currents, (r, p)mass, (r, p)-boundary. Vertical (r, p)-currents and cohomology. Integer multiplicity rectifiable vector-valued currents)
4.2
Graphs of Sobolev Maps ............................. 516 (Singularities of Sobolev maps and the currents P(u; a) and D(u; a). The class me - W1'p (S2, y))
4.3
p-Dirichlet Graphs and Cartesian Currents ............. 525 (The classes Vp-graph(S2 x y), red-Dp-graph(S2 x y), and carte" '(Ox y): closure theorems)
4.4 4.5 5
The Dirichlet Integral ................................ 534
(Representation. Minimizers and homological minimizers of the Dirichlet integral)
Prescribing Homological Singularities .................. 543
(s-degree. Lower bounds for the dipole energy) Notes .................................................... 546
Contents Volume II. Variational Integrals
xxiv
6.
The Non Parametric Area Functional ........................ 563 Area Minimizing Hypersurfaces ............................. 564 1 1.1
1.2
2
Non Parametric Minimal Surfaces of Codimension One ... 579
(Solvability of the Dirichlet problem. Bombieri-De Giorgi-Miranda a priori estimate. The variational approach. Removable singularities. Liouville type theorems. Bernstein theorem. Bombieri-De Giorgi-Giusti theorem on minimal cones)
Problems for Maps of Bounded Variation with Values in Sl ..... 590 2.1
2.2 2.3
3
Parametric Surfaces of Least Area ..................... 564
(Hypersurfaces as Caccioppoli's boundaries: De Giorgi,s regularity theorem, monotonicity, Federer's regularity theorem. Surfaces as rectifiable currents: Almgren's regularity theorem. Minimal surfaces as stationary varifolds: a survey of Allard theory. Boundary regularity: Allard's and Hardt Simon's results)
Preliminaries ....................................... 594
(Forms and currents in .fl x S1. BV(fl,E): a survey of results)
The Class cart(,fl x Sl) .............................. 600
(The structure and the approximation theorems)
Relaxed Energies and Existence of Minimizers .......... 610 (The area integral for maps into S1: the relaxed area. Minimizers in cart(Q x Sl). Dipole-type problems)
Two Dimensional Minimal Surfaces .......................... 619 3.1
3.2
Plateau's Problem ................................... 619
(Morrey's e-conformality theorem. Douglas-Rado existence theorem. Hildebrandt's boundary regularity theorem. Branch points and embedded minimal surfaces: Fleming, Meeks-Yau, and Chang results)
Existence of Two Dimensional Non Parametric Minimal
Surfaces ............................................ 625 (Rado's theorem and existence, uniqueness, and regularity of two-dimensional graphs of any codimension)
3.3
4
The Minimal Surface System ......................... 627
(Stationary graphs are not necessarily area minimizing. Existence and non existence of stationary Lipschitz graphs. Isolated singularities are not removable in high codimension. A Bernstein type result of Hildebrandt, Jost, and Widman)
Least Area Mappings and Least Mass Currents ................ 632 4.1 4.2
Topological Results .................................. 633
(Representation and homology of Lipschitz chains) Main Results .......................................635
(Least area mapping u : B" - W' and least mass currents "agree" if n > 3. If n > 3 the homotopy least area problem reduces to the
homology problem) 5
The Non-parametric Area Integral ........................... 639 5.1
5.2
The Mass of Cartesian Currents and the Relaxed Area ... 641
(Graphs of finite mass which cannot be approximated in area by smooth graphs)
Lebesgue's Area ..................................... 649
(The mass of 2-dimensional continuous Cartesian maps is Lebesgue's area of their graphs) 6
Notes .................................................... 651
Bibliography ..................................................... 653
Index ............................................................ 683 Symbols ......................................................... 695
1. General Measure Theory
This chapter deals with general measure theory. In Sec. 1.1 we collect definitions
and results from general measure theory that will be freely used later on; in Sec. 1.2, due to its relevance for the sequel, we shall discuss in more details weak convergence of functions and of measures and we shall illustrate some of its main features. In Ch. 2 we shall then develop some of the basic theory of n-rectifiable sets and integer multiplicity rectifiable currents. Of course we do not aim to completeness, for instance Sec. 1.1 of this chapter contains no proof; and in principle, we supply proofs essentially when claims or
their proofs are especially relevant for the sequel; sometimes, proofs are postponed to later chapters. Our goal in these first two chapters is to state precisely results and notations (though we shall usually adopt standard notations) and to illustrate them mainly by examples. In some sense, the first two chapters may be regarded as a simple, and in some regard, rough introduction to the elementary part of geometric measure theory, the right context in which the content of the following chapters lives.
At first lecture the reader can start from Ch. 3 and use Ch. 1 and Ch. 2 as reference chapters.
1 General Measure Theory In this section we collect some basic definitions and results from general measure
theory
1.1 Measures and Integrals Let X be a set and let 2X denote the collection of all subsets of X.
Definition 1. A collection. of subsets of X, F C 2X, is said to be a o--algebra in X if F has the following three properties
(i) X E F,
(ii) IfE,FEF, then E\FE.F,
1. General Measure Theory
2
(iii) If {Ek} C F, k = 1, 2, ... , then U' 1Ek E Y.
Definition 2. Let.F be a a-algebra in X and let µ be a function defined on .F, whose range is [0, +oo],
µ :.F , [0, +oo]. We say that µ, or better (µ, F), is a a measure on X if µ is countably additive on F, that is, if {Ek} is a disjoint countable collection of members of F, then 00
iµ(Ek)
µ U Ek)
(1)
k=1
k=1
If (µ,.F) is a a-measure on X, we evidently have (i) (ii)
µ(O) = 0,
p is in particular finitely additive, i.e., if El, ... , E,, is a finite collection of disjoint sets in.F, then µ(E1 U ... U E,) = 4(E1) + ... + p(E.."), (iii) if E1, E2 E.F, E1 D E2, then µ(E1 \ E2) = µ(E1) - µ(E2) > 0, hence µ is monotone µ(E1) D µ(E2). Given a a-measure µ on X, we may define the measure of any subset of X by trying to measure it in the best possible way by means of the elements of .F, i.e., by setting 00
00
µ* (E) := inf { E µ(Fk)
(2)
I
Fk E .F , U Fk D E}
k=1
k=1
and, if there is no countable collection {Fk } in F covering E, by setting µ* (E) _ +oo. This way we define a new set function µ*
:
2X -> [0, +oo]
which satisfies, as it is not difficult to see, the following properties (i)
(0) = 0,
µ* is monotone, i.e., if A C B then µ*(A) < µ*(B), (iii) p* is countably sub-additive, i.e., if {Ek} is a countable collection of subsets of X, then (n)
(3)
(iv) (v)
µ( UEk <_00p(Ek), k=1 k=1
µ* is an extension of µ, i.e., p* (E) = p(E) whenever E E F, µ* can also be computed as (4)
p* (E) = inf{µ(F)
F E F F D E} ,
Notice that in general µ* is not countably additive.
.
1.1 Measures and Integrals
3
Definition 3. A set function A : 2X --3 [0, +oo] is said to be an outer measure, or simply a measure on X if A(O) = 0 and A is monotone and countably subadditive.
Therefore we can say that every o--measure (µ, F) can be extended by (2), or by (4), to a measure u* on X. In a sense to be made precise, also the converse is true. To see this, we introduce, following Caratheodory, the important notion of measurable sets. Very roughly they are those sets which divide well every subset of X, more precisely
Definition 4 (Caratheodory). Let A be a measure on X. A subset E of X is called A-measurable if for every subset F of X we have
A(F) = A(F-E)+A(FnE)
(5)
The collection of all A-measurable subsets of X is denoted by MA,
MA :_ {E C X I E is A-measurable} Observe that, for any set E and any set F, we have
A(F) < A(F - E) + A(E n F);
also X and 0 are measurable and every A-null set E, i.e., every set of zero measure, A(E) = 0, is measurable. Actually we have
Theorem 1. Let A be a measure on X. Then Ma is a o--algebra; moreover (AIMS, Ma) is a a-measure. In particular A is countably additive on MA. As an immediate consequence of this we get
Proposition 1. Let p be a measure on X and let {Ek} be a sequence of measurable sets. (i) If El C E2 C E3 C ..., then µ(Ek) T u (Uk iEk) (ii) If El D E2 D E3 D ... and µ(E1) < oo, then lc(Ek)
µ (nk 1Ek). The process (2) of generating an outer measure from a o--measure is essentially due to Caratheodory and it is known as Caratheodory's method; for this .
reason outer measures are often called Caratheodory's measures. Such a method in fact works starting from any set function.
Let g be any sub-family of 2X, which contains the empty set, and let a -p [0, +oo] be any set function with a(0) = 0. Define 00
a* (E) := infa(Gk) 0" I Gk E k=1
,
U Gk D E} k=1
if there exists {Gk} c 9 such that UGk D E, or a* (E) :_ +oo otherwise. Then it is not difficult to verify that a* is a measure on X; but, this time, in general we have 9 0 Ma., a*(G) < a(G) and (4) does not hold in general.
4
1. General Measure Theory
Definition 5. A measure A on X is said to be c-regular, where 9 is a subcollection of 2X, if for every E C X we have
A(E) = inf{A(G) I E C G, G E 9}
.
If 9 = M), we say A regular instead of Ma-regular.
As we have seen, if a* is the extension of the v-measure (p, F), then µ* is ,F-regular and even regular, but in general a measure on X need not be regular. However to any measure p we can associate a regular measure j) defined by
µ(E) := inf{µ(F)l In this case one has (t)
If A E Mv, then A E Mµ; and µ(A) = µ(A),
(ii) if A E Mr, and A (A) < +oo, then A E M.
Regular measures are very important; later we shall return on that concept. For the moment we just mention the following:
Proposition 2. If A is a regular measure, then (i) and (ii) of Proposition 1 hold for all sequences {Ek} not necessarily in Ma. Moreover the following measurability criterion holds: if A U B is measurable and A(A U B) = A(A) + A(B), then A and B are A-measurable
Finally, we observe that, if 9 is a v-algebra, then 9-regularity of A. is equivalent to the existence, for every E C X, of an element F E 9 such that A(F) = A(E). Observe that if A is regular and E ¢ Ma, then there is F E Ma with A(F) = A(E), but A(F - E) > 0, otherwise F - E would be measurable, so that also E = F - (F - E) is measurable. We can now conclude our general discussion on the definition of outer measure
by saying: If (p, .F) is a o -measure, then µ* defined by (2) is a measure on X, which is .F-regular and regular; moreover the elements of F are measurable sets and u* = p. Conversely, if A is an (outer) measure on X, which is F-regular with respect to a o--algebra F contained in Ma, then (A 1,F, .F) is a v-measure on X and (A1 )* = A.
Definition 6. Let p be a measure on X and let A C X. The restriction of p on A is defined as the measure
(pLA)(B) := µ(An B)
dB C X.
It is easily seen that p-measurable sets are also p L. A-measurable; moreover, if p is regular and A is measurable, then p L A is regular.
We recall that a property P(x) is said to hold for p-a.e. x c X, p-almost every x in X, if there is a null set A C X, p(A) = 0, such that P(x) holds for
each xEX\A.
1.1 Measures and Integrals
5
Finally let us recall that Lebesgue's outer measure in RI is the result of Caratheodory's construction by taking for instance as family G the family of intervals in R' and as set function a the function which associates to each interval its standard measure. Let us now introduce the integral with respect to a measure A. Here the key point is Lebesgue's point of view which in some sense reverses the classical approach of Cauchy and Riemann. The integral is defined not as limit of sums of terms of the type p((x1,x2))f(X1), but instead as limit of sums of terms of the type y1 µ(f-1((yi,y2))) Therefore the notion of measurable functions plays a fundamental role.
Definition 7. Let-p be a measure on X, and Y a topological space. A function f : X -* Y is said to be p -measurable if and only if f -1(U) is µ-measurable whenever U is an open set in Y.
If Y = [-oo, +00] := ft, and fk, k = 1, 2, 3, ..., are µ-measurable functions from X into [-oo,+oo], then f, +f2, fife, f1j, min(fl, f2), max(fl, f2), infk fk, SUPk fk, liminfk_,,,, fk, limsupk-OO fk are all µ-measurable functions whenever they are defined.
Definition 8. Let µ be a measure in X. A function g : X -+ [-oo, +oo] is called a it p -step function if g is µ-measurable and its image g(X) {g(x) I x E X} is a countable set. A µ-step function is said to be µ-integrable, respectively µ-summable, if the sum
E yµ(g-1(y))
I (g; µ)
yEg(X)
exists in [-oo, +oo], respectively in (-oo, +oo). Here we use standard conventions on 12, in particular 0.o0 = 0, while oo - 00 is undefined. It is not difficult to prove
Proposition 3. Let f be a non-negative p-measurable function. Then f is the pointwise limit of a non decreasing sequence of steps functions each of which takes only a finite number of values.
Let f : X -+ [-oo, +oc] be any function and let Sµ denote the set of all µ-integrable step functions. We define
ffdµ := inf{I(g; µ) I g E SN, f< g µ -a.e}
6
1. General Measure Theory
ffdµ := sup{I(g; µ) 19 E Sµ
g < f µ -a.e}
(as usual we set inf 0 :_ +oc, sup O:= -oo).
Definition 9. The function f : X --+ A is said to be µ-integrable if and only if
r
f
Jfdµ := J
fdµ=
f fdµ;
f is called µ-summable if and only if f f dx E R. Evidently, for every µ-integrable step function g : X -* Il8 we have
I(9;µ) =
fgd#.
One also verifies that every non-negative µ-measurable function is µ-integrable and that every µ-summable function is µ-measurable. It is usual to identify functions which are equal for µ-almost every x E X. The space of all (classes of equivalence of) functions which are µ-summable is denoted by L' (X ; p) . We have (i) L' (X; µ) is a linear space; moreover If I E L' (X ; µ) if f E Ll (X; µ) . (ii) The operator
f E L1(X; µ) ->
ffdµ
E
is linear and continuous in the sense that
ff dµ < f IfI dµ. (iii) Also, the integral is monotone, i. e., f f dp > 0 whenever f > 0 a. e. in X.
Next theorem collects some relevant results concerning Lebesgue's integration.
Theorem 2. Let µ be a measure in X and let { fk}, fk : X - [-oc, +oc], be a sequence of µ-measurable functions. Then we have (i)
Egoroff's theorem. Suppose that the measurable functions fk are finite, i.e., fk : X -* R, and let e > 0 and A C X with µ(A) < +oo. If fk(x) ----p f (x)
for a. e. x E A,
then there exists a µ-measurable set B C X such that µ(A \ B) < E and
fk(x) -> f (x)
uniformly in B
.
1.1 Measures and Integrals
7
(ii) Beppo Levi theorem. If 0 < fl < f2 < ... and f (x) := limk-,,. fk(x), then
ffkdP T
Jfd.
(iii) Fatou's lemma. If fk > 0, k = 1, 2,..., then
f (iv)
lim
r
f fk dµ < lim n f fk dµ
.
Lebesgue's dominated convergence theorem. If fk(x) -+ f (x) for pa. e. x E X and if there exists a p-summable function g such that
then
bk=1,2,...
9,
Ifkl
f lfk - fIdtt f fkdµ --> ffd. 0,
consequently
(v)
Absolute continuity theorem. If f : X -* [-oo, +oo] is p-summable, then for every e > 0 there exists b > 0 such that
Jfdi < C A
for all p-measurable sets A with p(A) < J.
Definition 10. We say that fk converge to f in L1(X; p) if and only if
f f-fd
0.
We say that fk converge to f in measure p if and only if for every e > 0 p{x E X I Ifk(x) - f(x) I > E} -> 0
as k -> 00
Since for every E > 0 we have
p{x E X I I fk (x) - f(X)I> E} < E-1
f
l fk - f I dp ,
we deduce that fk - f in measure p provided f I fk - f I dp -+ 0, i.e., Llconvergence implies convergence in measure. Also, because of Egoroff's theorem, convergence a.e. in a set of finite measure implies convergence in measure.
1. General Measure Theory
8
Proposition 4. Let {fk} be a sequence of p-measurable functions, which converges to f in measure p. Then there exists a subsequence { fk, } such that fk. (x) -+ f (x) for p-a. e. x E X
Notice that in general the entire sequence fk does not converge to f a.e., even if fk converge to f in L1. 1 For instance let f (x) = 0 for x E R. For each k E N, write k = 21 + m, where 0 < m < 2n, k = 0,1.... ; then n and m are uniquely determined by k. Let
_ A (x)
'
r SI
1
if 2 < x< 2±1
0
otherwise .
Then
f fk(x)-f(x)dx = 2-n ->0
k->oo,
but
lim fk(x) koo exists for no x E [0, 1]. If we instead set
_ irk (x)
2"`
{ 0
if 2 < x < 2±1 otherwise
.
we see that for every positive e
meas{xER I but
fk(x)- f(x)I >e}<2-"-+0
ask -->oo
f Ifk(x) - f(x)ldx = 1,
i.e. the convergence of fk to f in measure does not imply Ll convergence.
Every function f : X -> Y induces a map f# which associates to each measure p on X the image measure f#1L
on Y by the formula
f#p(B) := p(f -1(B)) One readily verifies that f -1(B) is p-measurable if and only if B is f# (ii L. A) -
measurable for every A C X. One also sees that if µ(X) < +oo, p is regular, and B C Y is f#p-measurable, then f -1(B) is p-measurable. To state our next results we need the concept of countably p-measurable sets and functions.
Definition 11. Let p be a measure on X.
1.1 Measures and Integrals
9
(i) A set A C X is called countably p-measurable or µ-o -finite if and only if it is expressible as the union of some countable family of p-measurable sets Ak, A = Uk=1 Ak, with µ(A k) < +00(ii) A µ-measurable function f with values in a topological vector space is called countably ,u-measurable or µ-a-finite if and only if the set {x c X I f (x)
0} is µ-o finite. Observe that every p-measurable function is µ-o--finite.
Definition 12 (Product measures). Let µ be a measure on X and let v be a measure on Y. For any M C X x Y define
(µ x v)(M) := inf { >µ(Ak) x v(Bk)} k
where the infimum is taken over all sequences of µ-measurable sets Ak C X and v-measurable sets Bk C Y such that M C UkAk x Bk. The set function µ x v is called the product measure of p and v.
Theorem 3 (Fubini). Let µ and v are respectively measures on X and Y. (i) µ x v is a regular measure in X x Y (ii) If A C X is p-measurable and B C Y is u-measurable, then A x B is
µ x v-measurable and (µ x v)(A x B) = p(A)v(B). (iii) If M C X x Y is countably it x v-measurable, then the set
M. :_ {y E Y I (x, y) E MI,
XEX
is v-measurable for p-a.e. x E X and (µ x v) (M) =
fv(Mx)d(x) x
(iv)
If f: X X Y --p [-oo, +oo] is a µ x v-integrable and countably µ x vmeasurable function, then f (x, ) : y E Y -+ f (x, y) is v-integrable for p-a.e. x E X, the function h(x) := ff(x,Y)dv(Y)
Y
%s p-integrable and
f f(x,y)dµxv = XxY
fhd µ = X
J
X
I
f (x, y) dv(y)
dµ(x)
Y
However finiteness of iterated integrals does not imply in general that of the multiple integral. But for non-negative f and more generally we have
10
1. General Measure Theory
Theorem 4 (Tonelli). Let f : X X Y -+ [0, +oo] be a (p x v) -measurable and countably p x v-measurable function. Then the existence of one of the two integrals
f f(x,y)dy x v XXY
1 X
(ff(xv)()) dp(x) Y
implies the existence of the other and equality as well.
1.2 Borel Regular and Radon Measures In this subsection we assume that X is a metric space with distance d and the induced topology.
Definition 1. Let p be a measure on the metric space X. (i) The smallest o-algebra containing all open sets in X is called the Borel o-
algebra and is denoted by B(X). A function f : X -> Y, with values in a topological space Y is called a Borel function if, for every open set U C Y,
f-1(u) E B(X). (ii) p is called a Borel measure if every Borel set is p-measurable, i.e., B(X) C
Mµ. (iii) W is called Borel-regular if p, is Borel, i.e., B(X) C MN,, and for every A C X, there exists B E B(X) such that B A and µ(B) = µ(A). Definition 2. A measure p over a locally compact and separable space X is called a Radon measure if p is Borel-regular and for every compact subset K of X we have µ(K) < +oo. An important example of Radon measure in JRY is given by Lebesgue's ndimensional outer measure Cn-.
One easily verifies that: If p is Borel-regular and A is p-measurable, then p L A is Borel-regular.
If µ is Borel-regular and A is a p-measurable subset of a locally compact and separable space X with µ(A) < +co, then p L A is a Radon measure. Borel measures, in contrast to measures on a generic set, have a quite rich family of measurable sets. The following theorem gives a criterion ensuring that a measure over a metric space is a Borel measure.
Theorem 1 (Caratheodory's criterion). Let p be a measure over a metric space X. If
µ(A U B) = µ(A) + µ(B) for all A and B with positive distance, i. e.,
dist (A, B) := inf{d(x, y) I x E A, y E B} > 0, then p is a Borel measure, i.e., B(X) C M,,.
1.2 Borel Regular and Radon Measures
11
Next theorem states the basic approximation property in terms of Borel measures
Theorem 2. Let p be a Borel measure on the metric space X and let B C X be a Borel set.
If µ(B) < +oo and e > 0, then there exists a closed set C C B such that µ(B \ C) < e. (ii) If B C U' iUk, where Uk are open sets with p(Uk) < +oc, and e > 0, then (i)
there exists an open set U D B such that lc(U \ B) < E.
Moreover, if p is Borel-regular, then (i) and (ii) hold for all p-measurable sets
BCX.
The following two propositions follow at once from the previous theorem.
Proposition 1. Let p be a Borel-regular measure on the metric space X. Suppose that X C Uk 1Uk where Uk are open sets with p(Uk) < +oo. Then
p(A) = inf{µ(U) U D A, U open} for every subset A C X, and
µ(A) = sup{ µ(C) Cc A, C closed} for every p-measurable subset A C X.
Proposition 2. Let p be a Radon measure on a locally compact and separable space X. Then
p(A) = inf{p(U) U D A, U open} for every subset A C X, and
µ(A) = sup{µ(K) I K C A, K compact} for every p-measurable A with µ(A) < +oc. Let µ be a Borel-regular measure in X. Of course continuous or semicontin-
uous functions f : X -> R are p-measurable and in fact Borel functions; it is also easily seen that the inverse image of a Lebesgue measurable set in ]I8 by a semicontinuous map is measurable. Notice instead that the direct image of a measurable set by a continuous map f in general is not measurable. For instance Cantor-Vitali function in Sec. 1.1.3 below maps the Cantor set, which is a null set, into a set of positive measure, therefore we can obtain non-measurable sets as continuous image of a null set.
Definition 3. We say that a µ-measurable map f : X -> Rm, m > 1, satisfies Lusin's property (N) if p-null sets in X are mapped into null sets in Rm.
1. General Measure Theory
12
The following is then easily proved taking into account Lusin's theorem below: Let µ be a Borel-regular measure in X and let f : X -, ]R'n be a Borel map which has Lusin property (N). Then f (E) is Lebesgue measurable provided E is p-measurable, compare Step 1 of the proof of Theorem 1 in Sec. 2.1.2. Recall that f : X R is a Borel function if the inverse image of an open set is a Borel set.
Also recall that not every Lebesgue measurable function f : Rn -, JR can be obtained as a monotone limit of continuous functions, but we have
Theorem 3 (Vitali-Caratheodory). Every Lebesgue-measurable function f
:
I[Pn -, Ifs is the a.e. pointwise limit of a non-decreasing sequence of functions cpk which are upper semicontinuous and bounded from above. If moreover f is integrable, we have
lim
f
(f - Wok) dx = 0 .
Finally next theorem shows an important relation between measurable and continuous function.
Theorem 4 (Lusin). Let p be a Borel-regular measure on a metric space X and let f : X -+ III be a p-measurable function.
If A C X is p-measurable with u(A) < +oo and e > 0, then there exists a closed set C C A such that µ(A \ C) < e and fic is continuous. (ii) If X is moreover p-o--finite, then there exists a Borel function g : X -, JIB such that f = g p-a.e. in X. (iii) Let X = Ian, let A C Rn be p-measurable with µ(A) < +oo, and let e > 0. (i)
Then there exists a continuous function g : IRn ---, R such that
p{x E A I f (x)
g(x)} < e.
Notice that (iii) follows from (i) by means of Tietze's extension theorem for continuous functions on closed sets. Remark 1. Observing that RI is the union of a countable family of closed cubes with disjoint interiors and with side one, one easily sees that Lusin's theorem holds for any measurable set A of R', not necessarily of finite measure, if p is the Lebesgue measure C. This remark will be used later.
1.3 Hausdorff Measures Hausdorff measures are among the most important measures. They allow us to define the dimension of sets in IRn and provide us with s-dimensional measures in JR'1 for any s, 0 < s < n and actually in any metric space. They will be continuously used in the sequel. Let s be a nonnegative real number. We denote by w9 the volume of the unit ball in IRS f o r s = 1, 2, 3, ..., we set wo := 1 and we let w9 any convenient fixed constant for non-integers s. Since the measure of the unit n- ball is given by
1.3 Hausdorff Measures
rn/2
13
n=1,2,...
(n/2)I'(n/2) where F(t) is Euler's gamma function 00
j
xt-1 exp(-x) dx
,
we may and do take as ws
WS :_ (s/2)F(s/2) .
Fig. 1.1. k (E) - 2, f' (E) - 4. Definition 1. Let X be a metric space. For 6 > 0 and A C X we set (1)
'H'(A) := inf { ws (diamCk)8 2
k
UkCkJA, diamCk<6, CkCX} The Hausdor$ measure ?i is then defined by 7-ls(A)
(2)
Observe that 71
bola(A),
AC X
> 7-1o2 for 61 < b2i thus 7-1 is well defined and
'H'(A) = supW (A)
.
5>0
Notice that it is essential to use the approximating measures 7-lb in order to define H'. The naive definition
ECUCk}
7ls(E) k
k
1. General Measure Theory
14
in fact would have very unpleasant features. In (1) we may also require that the sets Ck be closed or open without changing the value (2). By the same procedure we can define the so-called s-dimensional spherical measure Ss by requiring in (1) that the sets Ck be balls. Then we obviously have
Hs < S8 <
25'Hs
but there exist sets A for which 'H' (A) < 88(A). For X = Rn, one easily verifies (i) 71s(.A) = As71s(A), \ > 0, where AA :_ {Ax I x E Al (ii) 7{° is the so-called counting measure , i. e.,
7H° (A) _ #A := number of elements of A (iii) 7{s = 0 on Rn for s > n (iv) 7{s is not a Radon measure on IIBn for 0 < s < n. (v) If 7l6(A) = 0 for some b > 0, then 'H" (A) = 0.
Finally, observe that for 0 < s < r
xb (A) <
(8)r8718(A)
thus 7{s (A) = +oo if 7lr (A) > 0 and ?ir (A) = 0 if 'Hs(A) < oo. This in particular shows that 71s (A) can be positive and finite for at most one (possibly none) s > 0. This motivates the following definition
Definition 2. The Hausdorff dimension of a set A C X is defined by
dim%(A) := inf{s > 0
:
1s(A) = 0}
From the previous remark it follows that if dim HA > 0, then diin (A) = sup{s > 0 1 7-ts(A) _ +oo}
and that, if 0 <7{s(A) < oc, then dimHA = s. It is easily seen that 7{a and 7{s are measures over X. Borel sets are not in general 71b-measurable for b > 0: for instance the half-space {(x, y) E R2 1 x > 0} c R2 is not 7{b-measurable. However, observe that 7-I = xn on III and we have
7{a (A U B) = li (A) + 7{b (B) if dist (A, B) > 26; therefore Borel sets are 7{s-measurable by Caratheodory criterion. Actually one proves
Proposition 1. 7-a and 7{s are measures on X; moreover 1s is Borel-regular
for alls>0. If X = RI and A is a Hs-measurable subset of X with 7-ls(A) < oc, then Hs L A is a Radon measure on X.
1.3 Hausdorff Measures
15
Using the so-called isodiametric inequality
Ln (A) <(diarn w2 A '
VA C R,
which says that among sets A C ][fin with a given diameter p, the ball with diameter p has the largest Lebesgue measure, one also proves the following important theorem
Theorem 1. The Hausdorff measure H-ln in R' coincides with the Lebesgue measure Ln* on IEBn. Moreover if M is an s-dimensional smooth submanifold in 1R , 0 < s < n, then ?-l9 L M is the standard volume measure in M, induced by the Euclidean metric in M.
0 Cantor sets. It is often difficult to determine the Hausdorff dimension of a set and even harder to find its Hausdorff measure. Of course the most difficult part is to give a lower estimate. The most familiar sets of real numbers of non-integral Hausdorff dimensions are Cantor's sets. These are obtained by the following construction. Fix 8 E (0, 1/2), begin with Eo :_ [0, 1] and in the first step remove an open interval of length 1 - 25 in the middle. Inductively then define Ek+1 by removing in each interval of Ek a centered open interval of length dk (1 - 26). Of course Ek D Ek+i for all k and each Ek is the union of 2k intervals of length b. The Cantor set associated to b is then defined as the closed, non denumerable, and dense in itself set cc C :=
I
IEk.
k=0
Classically one chooses b = 1/3 and the resulting set C is known as Cantor's middle thirds set.
{
F-]
F-]
F-I
F- -4
F--I
H-I
7--1
Fig. 1.2. Cantor's middle thirds set.
To be more precise and for future references we can describe Cantor set C as follows. For each k = 0, 1, ... and j = 1, ... , 2k define the base points bk,j of the Cantor set C inductively by bo,1
:= 0
16
1. General Measure Theory
and
Then
if j = 1, ... , 2k
8 bk,9
bk+l,j
t (1- 8) + 8 bk,j
if j = 2k + 1, ... , 2k+1
'k-l,j := bk_1, j + 8k-1(6,1 - 8)
j = 1, ....
2k-1
are the intervals removed from Ek_1 to get Ek at the step k, and Jk,j
bk,j + 8k [0, 1]
j = 1, ... , 2k
are the intervals whose union yields Ek. Consequently 00
C
2k
= k=0 n uj=1Ji,k
From this expression it is easily seen that Cantor set C is self-similar in the
Fig. 1.3. Self-similarity of Cantor sets in two dimensions.
sense that certain homothetic expansions of C are locally identical to the original set; more precisely we have
bk,j + 8kC = C n Jk,j .
(3)
In fact it is readily seen that x E Jh, j
if
bkX E Jh+k,j
BkEh
= Eh+k n [0,
for all h, k, j
hence Sk]
1.3 Hausdorff Measures
17
bk,j + BkEh = Eh+k n Jk,j for all h, k > 0, j = 1, ... , 2k, so that (3) follows by taking the intersection in h. By introducing the map rr : P([0,1]) --, P([0,1]) defined by
-r(A) := (8A) U (1- 8 + 8A)
,
we also see that rrk([0,1]) = Ek, rk({0}) _ {bk,l, ... , bk,2k }. Therefore the selfsimilarity of C can be also expressed by cc
c = n rk([O,1]) ,
7- (C) = C .
i.e.,
k=0
Proposition 2. The Hausdorff dimension of the Cantor set C, actually C5, is s := log 2/ log(1/8). Moreover
fs(C) = 2-8ws
.
Finally for all x, y E [0,1], x < y RI (C n [x, y]) = k iin £1(Ek n [x, y] ) Proof. Since C c Ek and Ek is the union of 2k intervals of length 8k we see that
lbk(C) < 2-sws2kdks = 2-sws(28s)k = 2-sws Letting k --> oc, 7-ls(C) < 2-sws. To prove the opposite inequality we consider
an open covering F of C. As C is compact we may assume that F is finite F = {K1, ... , KK} and we have Kl U ... U KK D Ek for some k E N. For any e > 0 choose k large enough so that 48k$ < e, and let K := (a, b) denotes one of the intervals in T. For convenience denote by R(I) and L(I) respectively the right and left end points of any interval I. We have 2k
[0, 1] _
k -1
2 "`
(U Jk,j) U ( U U j=1
m=0 i=1
where the J-type intervals yield Ek and the I-type intervals are the ones which are taken out up to step k. First we slightly change K as follows
if a E I,,,,,i, we set a' = R(I00,i); a' > a
if aEJk,j, we set a'=L(Jk,j); a'>a-8k Similarly
if b E I,,,,,i, we set b' = L(I,,i); b' < b if b E Jk,j, we set b' = R(Jk,j ); b' < b + 8k
.
1. General Measure Theory
18
This way for K' _ (a', b') we find
8K' C aEk
I JK'j - IKI I < 2bk ,
and of course K' covers K fl C. From now on we can therefore assume that for each Kr in F
aKr C aEk ,
R(Kr) = R(Jk,i)
L(Kr) = L(Jk,j),
for some j < 2k and j 2k Next we note that, by the construction of Cantor set, each Jk,j is preceded and followed by one of the I-type intervals, one taken out at the k-step and one at least at step k - 1. More precisely there are (m1, i1) and (m2i i2) with
ml, M2:5 k - 1, it < 211, i2 < 212 such that
R(Iml,il) = L(Jk,j) ,
R(Jk,.i) = L(Im2,i2)
and moreover either ml = k - 1, m2 < k - 2, or m1 < k - 2, m2 = k - 1. Consider now the largest I-interval Im,i in Kr = [ar, br]. We claim that (4)
ar > L(Im i) - bm+1
In fact we have the disjoint decomposition bm,i + bm [0, 1] = Jrn+1,.7 U Im,i U J,,,,+1,j+1
for some j, where
= bm,i + [0, Sm+1] Jm+l,j+1 = bm,i + (1- b)81 + [0,6-+'] Jm+1,.7
Suppose that ar < brn,i = L(I,,,,,i) - 51+1 Then there exists I,,,,',i' such that
R(Im',i') = L(Im+1,.7) C Kr. By the previous remark we then have m' < m, i.e., Im',i' I > IIm,i 1, that is Im,i is not the largest I-interval in Kr, a contradiction. Similar we proceed to show that br > R(Im,i) + 8m+1 On account of (4) we can define hence
Jl
Kr fl [0,L(Irn,i)]
J2
,
Kr n [R(Im,i),1]
and find the disjoint decomposition Kr = J1 U Im,,i U J2 with IJ1 1,
1 J21 <
1
SZb IIm,iI
1.3 Hausdorff Measures
19
Therefore using the concavity of t5 and the fact that 285 = 1 we deduce
Kris =
(IJlI + IIm,iI + IJzI)5 > [IJII + 21 b2a(IJII + IJ2l) +
IJ2I]5
-5(21J1I + 2IJ2I)5 ? 2a-5(IJlls + IJ2I5) IJl is + 1-72I5
.
Thus, replacing Kr by the two intervals Jl and J2 which still cover Ek fl Kr and satisfy aJl, 9J2 c aEk, we do not increase the sum IKl
5+...+IKKI8
.
Proceeding this way we obtain in a finite number of steps IKrls
>_
IJk,jls Jk,jCK,
and eventually after a finite number of steps IEhI
for a suitable h. This yields ls(C) > 28 w5.
2 Cantor-Vitali functions. A procedure similar to the one which leads to Cantor set with parameter 8 allows to construct a non decreasing continuous function V, which we call Cantor- Vitali function, whose derivative vanishes almost everywhere. For that, with the notation of O1 we consider for each k the two functions Vk, gk : [0,1] -+ lR defined by 9k(x)
2-k8-kXEk(x)
,
Vk(x)
IYktdt
=
2-ks-(Ekn[0,x])
Clearly Vk(0) = 0, Vk(1) = 1, Vis nondecreasing and
f 9k (t) dt = 2-k
Jk,j
For x E [0, 1] we denote by j (x, k) the largest integer between 0 and 2k such that Jk,j(y,k) is contained in [0, x]; if such an interval does not exist we set j (x, k) = 0. We then see that for x Ek
1. General Measure Theory
20
f
k)
Vk(x)
=
9k (t) dt = 2-kj (x, k)
j=1 Jk,j j(x , k)
fgk+l(t)dt = 2-kj (x, k)
Vk+1(x)
j=1 Jk,j
so that
if x
Vk(x) = Vk+1(x) = 2-k.7 (x, k) Vk+1(x) =
Ek
if x E Ik, j
2k+1
while, if x E Ek, so that x E Jk,j for some j, Vk(x) is linear on each Jk,j, V,' (x) = (2b)-k in Ek and we have Vk+1(x) - Vk (x) I
(5)
G f I9k+1(t) - 9k (t) I dt < 2-k+l Jk,j
Equivalently the approximate Cantor-Vitali functions Vk can be inductively defined by
2 Vk(b-lx) 1
2
if x E [0, 6]
ifxE[6,1-b] +2Vk(d-l(x-(1-b))) ifxE [1-6,1]
Then again we readily see that Vk(0) = 0, Vk(1) = 1, Vk(x) = (26)-k in Ek. Moreover, being Jk,j = [bj,k, bj,k + 6k], we have
Vk(bj,k) _
2k
Vk(bj,k + bk) = '
j 2k
and
2j-1
Vk(x) = 2m+1
for x E 1,,,j
m = 0, ... , k - 1, j = 1,... ,
,
2,m
.
By induction we immediately see that for each k we have (6)
IVk(x)-Vk(y)I < CIx-2,/Is
where
c := (1 - 2b) -1
and
s=
log 2 log(1/b)
From (4) and (6) we infer that Vk converge uniformly to a Holder-continuous and non decreasing function V, called the Cantor- Vitali function.
1.3 Hausdorff Measures
21
Fig. 1.4. An approximation of Vitali's
+--
function.
Notice that Vk(x) = Vk+1(x)
for x E [0, 1] \ Ek
V (x)
and that, by Proposition 2
V(x) _ Hs (c n [o, x])
V x E [0,1]1
xs(C) self-similarity of V is expressed by
V(x) =
V(bkJ)+2-kV(b-k(x-bk,j))
forxEJk,.7:_[bk,9,bk,3+8kl
Finally it is easily seen, compare for more details Ch. 4, that (i)
V has bounded variation in (0, 1), V E BV(0,1), since it is non-decreasing,
andV(0)=0,V(1)=1. (ii)
On each interval in [0,1] \ Ek we have V (x) = Vk (x) = constant. In particular, V (x) is differentiable on each x E [0, 1] \ E, and (approximately) differentiable almost everywhere in [0, 1] since El = 0, and
apV'(x) = 0 (iii)
for a.e. x E [0,1]
.
Since V is continuous, V' has no jump part but only a Cantor part
V' = Va(c). and V' has support in E. (iv) Obviously V([0,1]) = [0, 1], V maps [0, 1] \ E into the denumerable dyadic
set
D :_ {y E lI
I
y=
2k
, j E [0, 2k], j, k E N}
.
Therefore V maps E onto [0, 1] \ D. In particular V has not Lusin property (N).
22
1. General Measure Theory
Let f : Rn --> R'm', m > 1, be a map and let A be a subset of ln. The graph off over A is given by
cf,A := {(x, f(x)) I x E Al
.
Recall also that f is said to be Lipschitz continuous if
Lip f := sup{ If (x)
- f(Y) I
Ix - yI
I x, y E Rn , x
y} < co
.
Then we have the following theorem
Theorem 2. Let f : ][8n -. R' and let A C lien (i) If 7-ln (A) > 0, then dim -h9 f,A >_ n (ii) If f is Lipschitz-continuous, then for s > 0
7{s(f(A)) 5 (Lip f)s1s(A) (iii) If f is Lipschitz-continuous and 7-Ln(A) > 0, then
dim%gf,A = n
.
Remark 1. Hausdorf measure 7i is the result of a special case of a construction known as Caratheodory's construction,conpare Sec. 1.1.1. Given any collection F of parts of X with 0 E F and any function V) :.7= --> [0, +ooJ with 0(0) = 0, we define a measure p = pV,,.F on X as follows. First, for each S > 0, we set
ps(A) := inf{Ezb(Ak) I Uk Ak D A , Ak E .7= diarn Ak < 6} ,
,
ACX
,
and then
µ(A) := sup pi(A)
,
b>0
AcX
Of course Hausdorff measure 7-[s corresponds to the choice F = 2X and This way one may construct several interesting geometric measures, but for our purposes it suffices to work with Hausdorff measure. 0 (A) = w9 (2 diam A)
Having in mind Lebesgue's measure, a natural question is whether 7-ls+t(A x
B) and 7-ls(A) x 7dt(B) are equal, if A C R' and B C R"n. It turns out that such a result is far from being true and in general we only have
Theorem 3. Let A C Wt and B C III". Then 7-Ls+t(A x B) > b H" (A) x 7-lt(B) for some constan b > 0. In particular dim %(A x B) > dim HA + dim HB.
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems
23
Theorem 3 is a special case of the following inequality
(7)
Jfl8(Aflf_1(y))dt
< Wswt(Lipf)s7-[t+s(A) Ws+t
which holds for any Borel set A, any Lipschitz function f, and non negative real numbers s and t. Its proof is presented in Ch. 2, Theorem 2 in Sec. 2.1.3, step 2, in case s, t are integers, but it works as well for general s, t. The opposite inequality in (7) does not holds. For instance in Federer [226] one finds for any s general Cantor-type sets C1 and C2, 7-l8 (Cz) = 1, i = 1, 2, for which 7.12, (Cl x C2) = +oo. However, it is easily seen that 7-128(C x C) < CH, (C) x R, (C)
if C is the Cantor set in 1 A better result holds if s = n Theorem 4. For any B C JR'n with 7{t(B) < oc there exists a number c such that Wn 1 < cwt Lo-1
< 2-'(n+ 1)(t+n)/2
and
,Hn+t (A X B) = c7.n (A)7.1t (B) for every Lebesgue measurable set A.
In general the constant c in Theorem 4 is different from 1 but it is 1 if B has good tangential properties, i.e. if B is t-rectifiable, compare Theorem 3 in Sec. 2.1.5.
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems In this subsection we shall assume that X is a locally compact separable metric space. We shall denote by B,(X) the collection of all relatively compact Borel sets of X. If X is not compact the collection B, (X) is not a a-algebra, it is only a local a-algebra in the sense that
BC(X) L A :_ {B E Bc(X) I B C Al is a a-algebra in A for each A E Bc(X). A set function v : Bc(X) -- JR is called a a-additive measure if 00
v
(UA) k= 1
00
= k=1
for all sequences {Ak} of disjoint sets in Bc(X) such that Uk 1Ak E Bc(X), i.e., if v restricted over BC(X) LA is a-additive on BC(X) LA for all A E B, (X). For any f : X --* [-oo, +oc] and A C X we set
1. General Measure Theory
24
f inA 0 inX\A
fLA
f fdp :=
and
ffL.AdP. X
A
Let lc be a Radon measure over X. We denote by L ,:(X; i) the collection of all locally µ-summable functions f : X --+ [-oo, +00], i.e.,
L o_(X;,u) := { f I f L K E L1(X; A) VK C X, K compact}. For any f E L' Lc (X; p), the expression
(µL f)(A) :=
(1)
r
JA
f da
A E 13,(X)
defines a o--additive measure A L f .
Definition 1. Let p and v be two Radon measures over X. We say that v is absolutely continuous with respect to p, and write v << u, if v(A) = 0 for every A for which µ(A) = 0.
The measures µ and v are called mutually singular, and we write v1µ, if there exists a Borel set A C X such that lU L (X \ A) = 0 and v L A = 0. One of the most important result of measure theory is contained in the next theorem.
Theorem 1. Let i and v be two Radon measures over X (i)
Lebesgue's decomposition theorem. There exist unique Radon measures v(') and v(9) on X such that v = v(a) .+ U(s)
v(s) l
(a)
,
U(s) j. tL
,
U(a) << P.
v(a) is called the absolutely continuous part of v with respect to µ, and v(8) the singular part of v with respect to p. (ii) Radon-Nikodym's theorem. If v << µ, then there exists a unique Borel
function f in Lioc (X; u) such that v = it L f , i.e.,
v(A)=Jfdy
`dAEt3a(X).
A
The function f is called the Radon-Nikodym derivative of v with respect to p and is denoted also by dµ .
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems
25
Obviously, we can write the claim of Theorem 1, as
V = b IL
(2)
dy (a)
dµ
+v
or, equivalently, for any Borel function f with compact support
f f dv =
(3)
r
J
f dy(-a) dµ + ffdi/8).
Moreover, we have forA E 8, (X) (a) (A)
(4)
:= inf{v(B) I BE ]3,(X), µ(A \ B) = O}
Definition 2. We say that the set function v : 13,(X) -4R is a signed measure if there exists a Radon measure µ on X and f E L o,_(X; µ) such that
v = µLf
onB,,(X).
The set function v : Ba(X) -> R n, m > 1 is called a vector valued measure if there exists a Radon measure µ and a vector valued function f = (f 1, ... , f m) E L1 a (X, Rm; µ), i.e., with f k E L1(X ; µ) for k = 1, ... , m, such that v = u L f, i. e.,
(v1 ...,Um) _ (µL f1,...,pLfm) We define the total variation vI : B,(X) -+ [0, +oo) of the vector valued mea-
sure v:Ba(X) ->R"n, v=µL f, as IuI
where I f I = (>2k
:= µLIfI
IfkI2)1/2
We define the positive and negative parts v+ and v- of a signed measure
v= p L f,f :X-,R,as
v+ = ILL f+,
v
= fLLf
where f+=max(f,0), f- =min(f,0). The collection of Radon, signed, and vector valued measures on X will be denoted respectively by M(X), M(X), and M (X, R'n) . A few remarks are now in order. Trivially we have p({x I f (x)
0}) = 0 if (µ L f) (A) = 0 for all A. We also remark that vI, v+ and v- do not depend on the decomposition v = µ L f. For instance, if v = µ1 L f1 = 42 L f2i we have µl,µ2 << µ := Al + µ2 Hence by Radon-Nikodym's theorem we have, for k = 1, 2,
Pk = {1,L
Ak, µkLfk = (µL aµ)Lfk = pL(fk dµ
Thus f 1 t' = f 2 dµ =: f p-a.e. in X and
1. General Measure Theory
26
UkLlfkI = JLL(Ifk W k) = ULifl The same argument yields the result for v+ and v-. If v is a signed measured, we obviously have v = v+ - v-, I vl = v+ + vthus v+
Therefore if we define
, v- <<
'_
dv d-Ivl
Iv.1
dv+
dv-
dlvl
dIvl '
then we have
v = Ivl L
dv
dlvl
and by definition dv
+
( dJvJ )
dv+
du
dv-
d-lvl
(dlvl)
dI vI
'
If vk, k = 1, ... , m, are signed measures, vk = Ak L fk, defining p ;_ Ek µk
we
find
/k = /LL (fk
dd/G
).
thus we conclude
Proposition 1. If vI, ... , v' are signed measures, then v = (vl, ... , vm) is a vector valued measure in the sense of Definition 2. Therefore µ = (41,.. ., Ak) is a vector valued measure if and only if each µk is a signed measure. We also have
Proposition 2. The total variation I vl of a vector valued measure v can by equivalently defined as
I vI (A) = sup{E00I v(Ak) I
I
Ak E B ,(X) , Ak disjoint , UkAk C A} ,
k=1
for A E 15,(X).
If v is a signed measure, we have, for A E Bc(X):
v7'(A)=sup{±v(B) I BCA, BE13,(X)} I vl (A) = sup{ lv(AI) I + I v(A2) I I AI n A2 = 0, AI UA2 C A, Al i A2 E 5,(X)}-
1.4 Lebesgue's, Radon-Nikodym's and Riesz's Theorems
27
Let v = (v' , ... , v') and µ be respectively a vector valued measure and a Radon measure on X. We say that v is absolutely continuous with respect to p, V << µ, if lvl
<< µ .
or equivalently if l vi l << µ for all i. For two vector valued measures v and A we define
vIA
iff
vI I IAL
Then Theorem 1 readily extends to vector valued measures. In particular we see that, if p is a Radon measure on X, the set of all vector valued measures v E M (X; IE8") such that v << ,v is equal to { p L_ f I f E Llc (X, RI; by definition v << l v I, we also get
Since
Proposition 3. Let v be a vector valued measure. Then v = lvI L i7, where V
dv
E Lloc(X,Pm; lvl)
and lvl = 1 IvI-a.e. in X. Given a vector valued measure v = (vl, ... , v') and a vector valued function g = (gl,... , gm) in Ll (X, lRm; vl ), we define the integral of g with respect to v by
f
:= E J
gk dvk
k
where
J gk dvk :=
J
gk dvk+
-
fokdvk_
.
By Proposition 3 we then get
f
(5)
=
fdivi ,
Denoting by Cc (X, II8m) the space of all compactly supported and continuous functions on X with values in Rm, evidently
A. : f E Cc (X, III') - fdivI defines a linear functional on Cc (X, R"'') for every v E M (X, RI). Next theorem roughly inverts this observation.
Theorem 2 (Riesz). We have (i) Let A : CC(X, Ii') -+ R be a linear functional which is continuous in the sense that it has locally finite norm, i.e., (6)
Sup{) (f) l f E Cc(X, IRm) , If 1 < 1
,
spt f C K} < oo
1. General Measure Theory
28
for each compact K C X. Then there exists a unique vector valued measure v= IV I L W E M (X, ]I8m) such that
A(f) =
f = fdH
df E CC(X,Rm)
(ii) Let A : C, (X, R) -, ][8 be a linear functional which is positive, i.e., A(f) > 0
whenever f > 0. Then there exists a Radon measure µ on X, z E Nl+(X), such that
A(.f)=
ffdy
VfECc(X)
Notice that in (ii) there is no condition on the norm of A, the positivity of A suffices. This expresses the well-known fact that every positive distribution is a measure. A consequence of Theorem 2 is that M (X, Rm) is the dual space of C°(X, ]IPm). We moreover remark that a distribution A : C°° (X, ]1800) -> II8 is a measure (or better extends to a measure) if and only if we have sup{.1(f)
I
f EC°°(X,R''), IfI<1,sptf CK} < 00
for all compact K C X. In this case it is common to identify A with p and write A(A) for fA dµ.
We also readily deduce that, for all open sets U in 13,(X), we have (7)
IvI(U) = sup{
f < f, dv> I f E C°(X, l[8m) ,
If 1
<_ 1, spt f C U}
.
This can in fact be used as an equivalent definition of I vI . Given v E M (X, R-) and an open set U in X, one defines the mass of v in U by (8)
Mu(v) := sup{
f
< f, dv> I f E CC(X, Rm) , If I < 1, spt f c U}
and one simply writes M(v) for Mx(v). Of course, if we extend the set function
M(v) : U --+ Mu(v)
U open
to the Radon measure on X given by
MA (V) := inf{Mu(v) I A C U, U open} we find
MA (11) = I vI (A) .
Actually, we should call Mu(v) in (8) the Euclidean mass, because we used the Euclidean norm in If I < 1. If we use another norm I in R00, the resulting mass M,(v) can be expressed as II
Mu(v) =
II v II'(U)
,
II v II'(A)
f
II
dvl
II' dI vI
1.5 Covering Theorems, Differentiation and Densities
where I
II' is the dual norm of II
29
II, i.e.,
IIy1I' := sup{ 111 X 11 <_ 1, x Exam}
Finally we remark that any linear functional A : CC (X, Rm) -> R satisfying
(6), being a vector valued measure A = IvI L v, v = 1 J vI-a.e., extends to a linear functional on the space LI(X,RI;IvI). In particular if X = UkFk, Fk compact and I vI (X) = M(v) < oc, then constant functions are v-summable and A extends to all Borel bounded functions X ---+ 112m. Such an extension can be computed by Lebesgue's theorem.
1.5 Covering Theorems, Differentiation and Densities A classical theorem of Vitali, actually in dimension n = 1, states the following
Theorem 1 (Vitali). Let E be a subset of R which is covered by a collection of closed balls 13 with the property that for each x E E and E > 0 there exists B E 13 such that x E B and diam B < E. Then we can choose a disjoint sequence {Bk} of balls in 13 such that
meas (E - UkBk) = 0 . It turns out that the property of covering a set in the sense of the previous theorem is the key point for differentiating a measure v with respect to another measure p. To state precisely such a theorem we need a few preliminaries. In this subsection X will denote , if not otherwise states, a locally compact
separable metric space. We denote, as usual, the ball of radius r and center x E X by B(x,r). Given a collection 13 of balls in X, the set of the centers of balls in 13 is denoted by C(B)
C(B) := {x E X 12r > 0 so that
B(x, r) E 13}
.
The collection B covers a set A C X if A C UBEBB.
Definition 1. We say that 13 covers A C X finely if for every x E A and e > 0 there exists B(x, r) E 13 with r < E, i.e., if for all x E A
inf{r > 0
1
B(x,r) E 13} = 0
.
Definition 2. Let µ be a Radon measure over X. We say that X has the symmetric Vitali property relative to µ if for every collection B of balls in X satisfying
(i) 13 covers finely the set of its centers C(13) (ii) µ(C(13)) < +oo there exists a sub-collection 13' of B satisfying
30
1. General Measure Theory
(i) the elements of B' are disjoint (ii) 13' covers p-almost all of C(B), i.e.,
h (C(B) \ UBES'B) = 0 . Next theorem provide us with two important cases in which X has the symmetric Vitali property relative to µ.
Theorem 2. We have (i) Let p be a Radon measure over X. If there exists a constant c such that
E.c(B(x, 5r)) < cy(B(x, r)) for all balls B(x, r) in X, then X has the symmetric Vitali property relative to p. (ii) The Euclidean space X = Rn has the symmetric Vitali property relative to p for any Radon measure p on 1E$n. The proof of Theorem 2 uses the following
Lemma 1 (Vitali's covering theorem). Let B be a collection of closed balls in X with equi-bounded radii, i.e., sup{r a sub-collection B' of B such that
I
B(x, r) e 13} < oo. Then there exists
(i) the elements of B' are disjoint, therefore B' is numerable (ii) for all B E B there exists a ball B' E B' such that B n B' 0 0 and f3' D B,
where for B = B(x, r) we have set B = B(x, 5r). In particular
UB C UB BES
BES'
From Lemma 1 one deduces that, if A is covered finely by B, then one has
A \ U{B I BEBill C U{B I BEB'\Bill for each finite sub-collection B" of B', and this yields Theorem 2 (i). The proof of Theorem 2 (ii) uses instead the following
Lemma 2 (Besicovitch's covering theorem). Let E be a subset of 1R
and
let r : E --a R be a positive bounded function defined on E (for instance, let E = C(B) where B is a collection of closed balls with center in each point of E and radius r(x)). Then one can choose an at most countable family of points A := {xi}iEN in E such that
(i) E C UiB(xi, r(xi)) (ii) The balls B(xi, Ir(xi)), xi E A, are mutually disjoint (iii) Moreover, the balls B(xi, r(xi)), xi E A, can be distributed in e(n) families Bk of disjoint closed balls, where l; (n) is a constant depending only on n (iv) No point of E belongs to more than C(n) balls of the family {B(xi, r(xi)) xi E Al I
1.5 Covering Theorems, Differentiation and Densities
31
We notice that Lemma 1 with X = R' is a consequence of Lemma 2. Later we shall refer to Lemma 1 and Lemma 2 as to Besicovitch's covering theorem. Let now µ and v be Radon measures over X. For x E X we set 17µv(x) := limsupµ(B(x,T)) r-O inf " a x,r Dµv(x) :_ lim r-O M(B(x,r))
whenever p(B(x, r)) > 0 for all r > 0, otherwise, i. e., if A(B(x, r)) = 0 for some r > 0, we set
Dµv(x) = Dµv(x) = +00. If Dµv(x) = Dµv(x) E [0, +oo], we simply write Dv(x) for Dµv(x) =Dµv(x). Definition 3. We say that v is differentiable with respect to A at x if and only if
Dµv(x) = Dµv(x) E [0, +00] In this case, Dµv is called the (symmetric) derivative of v with respect to µ or the density of v with respect to A.
The basic facts concerning the differentiation of a Radon measure v with respect to a Radon measure u are collected in the next theorem.
Theorem 3 (Radon-Nikodym). Let y and v be two Radon measures over X. Suppose that X has the symmetric Vitali property with respect to A. Then (i) Dµv(x) exists for p-a.e. x E X and the function Dµv(x) is u-measurable. (ii) There exists a Borel set Z with µ(Z) = 0 such that
v(A) = fDvdp + (VLZ)(A) A
holds for all Borel sets A C X. (iii) Taking into account Radon-Nikodym's theorem and Lebesgue's decomposition theorem in Sec. 1.1.4 we have v(a) = /.L L Dµv ,
v(s) = V L Z
thus
Dµv =
dv(a)
µ-a.e. in X
dµ
Moreover, if v is absolutely continuous with respect to /1, we may take Z = 0 and we have
Dµv = dv
.z
-a.e. in X
1. General Measure Theory
32
(iv) If X has also the symmetric Vitali property with respect to v, then Dµv(x)
exists also for v-a.e. x E X and we may take
Z = {x E X I Dµv(x) = +oo}
.
... (iv) of Theorem 3 hold if X = Rn. The proof of Theorem 3 is based on the following lemma whose proof uses in turn Vitali's property. In view of Theorem 2 (ii), the claims (i)
Lemma 3. Let A be any subset of X. (i) If Dµv(x) < a in A, then v(A) < ay(A) (ii) If D,,v(x) > a in A, then v(A) > ap(A)
A simple consequence of Theorem 3 is the following theorem to which we shall return in Sec. 3.1.1.
Theorem 4 (Lebesgue-Besicovitch's differentiation theorem). Let a be a Radon measure over R' and let f be a function in L oc (R 2; µ) with p > 1. Then
lim
r-D (B(x, r))
for µ-a.e. x
f f dµ = f (x)
B(x,r)
and even lim
1 r)) r-.O p,(B(x,
J B(x,r)
If
-f(x)
rdµ = 0
for p-a.e. x
If we apply Theorem 4 to the characteristic function XE of a p-measurable subset E of Rn, we obtain that lim
r-o
p(B(x, r) n E) µ(B(x, r))
(B
lim
µ(B(x, r))
r-O
=
d(ti L E) dµ (x)
equals 1 for p-a.e. x c E and equals 0 for µ-a.e. x E Rn \ E. The sets IntA(E)
dµL
{x
(x) = 1}
I
Extµ(E)
{x
1
d( E (x) = 0}
might be referred as to the measure-theoretic interior and exterior of E with respect top and we may refer to
am(E) := Rn \ (Int (E) U Ext (E)) as to the measure-theoretic boundary of E. Of particular relevance is the case in which µ is the Lebesgue measure in R', or equivalently the n-dimensional Hausdorff measure in Rn. We then set
1.5 Covering Theorems, Differentiation and Densities
33
Definition 4. Let E be a Lebesgue measurable set in R'. The upper and lower densities of E at x E E are defined by limsup7-ln(Ef1B(x,r))
9*(E,x) B* (E, x)
_ Dy{n(,HnLE)(x)
x (B (x, r))
r- O
:= lim inf r-.O
7-ln (E fl B(x, r)) 1 (B (x, r))
x
_ D n (q-ln L E) (x)
and the density of E at x by 8(E, x) = 0* (E, x) = 0* (E, x) if the upper and lower densities are equal at x. Of course 7{n-a.e. x E E is a point of density 1 and 7-ln-a.e. x E IISn \ E is a point of density 0 for E. In terms of densities we can recover interesting relations between measurable and continuous functions
Definition 5. Let f : R --> R be a Lebesgue measurable function. The approximate upper, respectively lower, limit of f at x is defined as
aplimsup f (y) := inf{t E R
y-x
I
0*({z E IRn I f (z) > t}, x) = 0}
respectively
apliminf f (y) y-ix
sup{t E R 10* ({z E IRn I f (z) < t}, x) = 0}
.
We speak of approximate limit of f at x in case aplim f (y) := aplimsup f (y) = apliminf f (y) y-+x
'Y-+x
and we say f to be approximately continuous at x if
aplim f (y) = f (x) y-x We then have, compare Sec. 3.1.4 (i)
f is approximately continuous at x if for every open set U C we have
R
with f (x) E U
0(f-1(u), x) = 1 (ii) f is approximately continuous at x if there exists a measurable set E with
x E E such that 0(E,x) = 1 and fj E is continuous at x, (iii) if f E Lioc(R ; µ) and aplimy-x f (y) = f (x), then lim.
1
o Gn(B(x, r))
f If (y)-f(x)Idµ = U. B(x,r)
1. General Measure Theory
34
and finally
Proposition 1. Let f : I[8n - IR be a function which is a.e. defined in RI. Then f is Lebesgue measurable if and only if it is approximately continuous at a.e.
xEIR'. The previous results do not apply, at least directly, to lower dimensional Hausdorff measures, as fl9 is not a Radon measure in IRn for 0 < s < n. For this reason one introduces lower-dimensional densities which in some sense extend the notion of Radon-Nikodym derivative to Hausdorff measures.
Definition 6. Let p be a Borel measure over a metric space X. For any subset A C X, and any point x E X, we define the n-dimensional upper and lower densities by On* (p, A, x)
lim sup
9*(p,A,x)
liminf
r-.0 r--.0
it (A n B(x, r)) Wnrn
p(AnB(x,r)) Wnrn
In case A = X we simply write Bn* (p, x) :=
9n*
(p, X, x) ,
9 (p, X, x)
9 (A, x)
In case On* (p, A, x) = 0, (p, A, x) we write on (p, A, x) for the common value. Clearly we have
Bn*(p,A,x) = Bn*(pL A, x)
On (p, A, x)
=
on
(pLA,x)
and, since p(A n B(x, r)) is an upper semicontinuous function in x for r fixed, On* (p, A, x) and 9* (p, A, x) are both Borel functions. Notice moreover that the same definition of densities can be used for any measure p. Next theorem provides useful connections between p and fn based on information about the upper density of p.
Theorem 5. Let p be a Borel regular measure on a metric space X and let
t>0.
(i) If A C B C X and 9n* (p, B, x) > t for all x E A, then p(B) > t7in (A). The
-
case A = B is important, we have: if 0n* (p, A, x) > t for all x E A, then p(A) > t7-ln(A).
(ii) If A C X and On* (p, A, x) < t for all x c A, then p(A) < 2ntfn(A).
A consequence of Theorem 5 is
Theorem 6. Let p be a Borel regular measure on a metric space X, for instance let p be the n-dimensional Hausdorff measure fn. If A is a p-measurable subset of X with p(A) < oo, then 9n* (p, A, x) = 0 for HI-a. e. x E X \ A.
1.5 Covering Theorems, Differentiation and Densities
35
Finally we have the following important bounds for densities with respect to Hausdorff measure
Theorem 7. Let A be any subset of X. (i) If In (A) < oc, then on* (7{n, A, x) _< 1 for Hn-a. e. x E A. (ii) I f 7lb (A) < oo for each 8 > 0, then Bn* (7 h, A, x) > 2-n for 7-ln -a. e. X E A.
Since 7-1 > 7-i > 7-(n, we deduce in particular
2-n < On* (7ln, A, x) < 1
for 7-Cn -a. e. x E A
if Nn (A) < oc.
Even in the case that X is an Euclidean space we cannot infer in general that the n-dimensional density of A is 1 at In - a.e. point of A. This in fact depends upon the rectifiability of A, compare Theorem 5 in Sec. 2.1.4
Remark 1. Consider the measures 1-tk and 7{1, k _< n, in R n. Although they can be very different, it is easily seen that 7dk (A) = 0 if and only if Hk 00 (A) = 0. Moreover, as we have seen,
lim sup r-k Hk 00 (E n Br) _> 2-kWk r-O for 7-lk-a.e. x E E, and one can show that, if Q, Qv v = 1, 2.... are compact se and if every open set A D Q contains Q for v sufficiently large, then
7{k (Q) > lim sup 7{1 (Qv) . v-oo
Notice that the last inequality is in general false for the Hausdorff measure 7-Ck. The properties of 7l"0 illustrated above sometimes make the use of 7-lOO more convenient than the use of 7{k. A consequence of the differentiation theory of measures is the following theorem
Theorem 8 (Disintegration of measures). Let p be a Radon measure on X x Y. Then there exists a measure A on X and, for A-a.e. x E X, a measure vx on Y such that for any M C X x Y we have
p(M) =
fL'(M)dA(x) X
{y E Y I (x, y) E M}. For any µ-summable function f on X x Y, the function f (x, ) is vx-summable for A-a.e. x E X and where Mx
f f dp = f f f (x, y) dvx XxY
X
(y)
da(x)
Y
Moreover, if µ(X xY) < oo, we can take. = 7r#Fp where 7r is the linear projection
rr(x, y) := x. In this case vx turns out to be a probability measure for A-a.e. x, i.e., vx (X) = 1.
1. General Measure Theory
36
2 Weak Convergence Due to its relevance for the sequel, in this section we would like to discuss weak convergence of functions and measures, with the aim of illustrating some of its main features.
2.1 Weak Convergence of Vector Valued Measures Let X be a locally compact separable metric space.
Definition 1. Let v, vk E M(X, R') be vector valued measures over X, k =
1,2,... (i)
We say that vconverge
to v, vk weaklyf
v, if
f --> (ii)
cp E C,(X,R n)
We say that vk converge to v in the mass norm , vk vk converge locally in the mass norm, vk + v, if
MU(vk-v)--p0
if M(vk - v) -+ 0;
VUCCX, U open.
(iii) We say that vk converge to v weakly with the mass iff vk M(v) < oo.
v and M(vk)
We have
Theorem 1. Let {vk} be a sequence of vector valued measures over X satisfying
supMu(vk) < oo
(1)
d U CC X
open.
k
where Mu(v) = IvI(ll) is the mass of v in U. Then there exist a subsequence {vk, } of {vk} and a vector valued measure v such that vk, - v. The converse of Theorem 1 is also valid, i.e., if vk - v then (1) holds. This is consequence of the following
Theorem 2 (Banach-Steinhaus). Let {vk} be a sequence of vector valued measures over X. If
SUP I J j < oo
V cp c CC(X,Rn)
k
then the masses of vk are locally equi-bounded, i.e.,
sup Mu(vk) < oc k
We also have
V U CC X , U open
2.1 Weak Convergence of Vector Valued Measures
37
Theorem 3. Let p and µk, k = 1, 2,..., be Radon measures over X. Then the following three statements are equivalent (i) Ak-'/1 (ii) pk(B) -* p(B) for all relatively compact Borel sets B E B0(X) satisfying
pI(aB) = 0 (izi) for all open subsets U C X u(U) < liminfk-.oopk(U), and for all compact
subsets K C X limsupk_,,,. pk(K) < µ(K). Notice that pk(B) does not converge in general to µ(B) if p(tB) # 0. For example if {xk} is a sequence in B(0,1) C RT converging to x". E 3B(0,1), we have 81k
8,_, but 8xk(B(0,1)) = 1 74 6 ,(B(0,1)) = 0
.
Notice also that in Theorem 3 (ii) we can in fact state pk(B) --> µ(B) for all open set B with p(aB) = 0. In fact every Borel subset can be decomposed as 0
0
0
B =B U(B n 8B) with p(B) = µ(B), and a BC aB. Theorem 3 does not extend immediately to vector valued measures. For example consider the sequence of signed measures
An := 81/n - 6_1/n E M(IR) . p, p = pI = 0, thus, for B = [0, 1), we have p(aB) = JAI(aB) _ 0, but A, (B) = 81/n(B) = 1, so p (B) 74 µ(B). In order to extend Theorem 3 we need a stronger condition. Obviously µn
Definition 2. We say that a sequence of vector valued measures µk E M(X, lRn)
quasi-converges to (µ, a), µk - (µ, a), where p E M(X, Rn) and a e M+(X) if and only if µk
In general a
p and µkl - a. pI and in fact by lower semicontinuity IAI << a. We have
Theorem 4. Let {µk} be a sequence of vector valued measures over X. The following two claims are equivalent (i) Pk q' (A, a) (ii) pk(B) - µ(B) and Iµkl (B) - a(B) for all relatively compact Borel sets B E 130(X) satisfying a(aB) = 0.
Notice that, just by definition, the mass is lower semi continuous with respect to the weak convergence of measures, that is,
Mu(v) < liminf Mu(vk) k-oo
if vk - v
.
Let {µk} be a sequence of vector valued measures such that Pk -s p and pk I (X) --> Jp (X) where Jp (X) < oo. Using the lower semicontinuity of the total variation on open sets we have I p I (A) < 1iim of µk (A) I k-oo
38
1. General Measure Theory
On the other hand for any closed set F we have lim SUP IIik I (F) = lim SUP (I Irk i (X) - Iuk I (X \ F) ) k-+oo
k--+oo
= I a K (X) -
likm
-oo f l /1k I (X \ F) >- l µ l (X) -III (X \ F) = I µ l (F)
Thus by Theorem 3 we infer
Proposition 1. Let {µk} be a sequence of vector valued measures which converges with the masses to y, i.e., ILk
IA,/LkI(X) --p III (X),
Iul(X) < oo
Then Ilk quasi-converges to (µ, WD-
The next remark will be used later. O
Proposition 2. Suppose that X = U' 1Fh, Fh compact, Fh+1» Fh If µk
y with the mass then Lk (cp) ---> u(cp), 1 tck I (f) -> l pl (f) for all continuous and bounded functions cp, f on X.
Proof. In fact, let Xh E CC(X, IR), 0 < Xh < 1, Xh = 0 on X\ Fh, Xh T 1. Since
limsuplµlk(X\ Fh) < IILI(X\ Fh) k-+oo
,
we have for any h lim sup (Ilk - I%v)l <_ HM SUP I(Ilk -A)((1 - Xh)'P)I < 211 W IIILI(X\ Fh) k- oo
k-oo
limsup I(lPkl - IILI)(f)I <- 211 f IIIILI(X\ Fh). k-oo Letting h --> oo the result follows.
Often it is convenient to check the condition (2)
>
f<,du>
not for all cp E C, (X, RI), but for cp in suitable subsets A of CC (X, IRm). In this respect we can state (i)
Suppose that A has the following property. For every cp E CC(X, R m) there
exists a compact set K such that for every E > 0 there exists b E A with spt 4' C K and lcp-0I < E on X. Then vk - v if (2) holds for all cc A and (ii)
supk MU(vk) < oo for all open sets U CC X, Suppose that A is dense in C,(X, IR') in the C-norm topology. If (2) holds for all cp E A and supk M(vk) < oo, then vk - v.
2.1 Weak Convergence of Vector Valued Measures
39
The weak convergence in M(X, R') is not induced in general by a metric. But we have the following. Consider for any positive constant c the bounded set in M(X, Rm)
Kc :_ {v E M(X,
)
I
M(ii)<e}.
Since Cc (X, R') is separable in the C-norm, then there exists a metric p on Kc
such that vk -s v (vk and v in Kc) if and only if p(vk, v) , 0, i.e., the weak convergence on Kc is metrizable. A metric p which does it is for instance r_2=12-e
P(µ1,µ2)
f < - f <
1+f <We,dµ1>-f
where Wj, $ = 1, 2.... is a dense set in C0(X, ][gym).
Let µ be a Radon measure on X and let p be a real number with 1 < p < oc. The space of measurable functions which are p-summable with respect to µ, i.e., for which
f
If Ipdµ < oo
is denoted by LP(X; µ). As it is well-known LP(X; µ) is a Banach space with norm
1/p
(J If I P dµ)
II f II LP (X;N,) = II f Il p
(3)
moreover Cc (X) is dense in LP (X; µ), and the dual space of LP (X; µ) is LP' (X; µ)
where p' is the dual exponent of p, P + -, = 1 if p > 1 or p' = oc if p = 1.
Definition 3. Let µ be a Radon measure on X and let f, fl, f2.... belong to LP(X; µ). (i)
We say that fk converge weakly in LP(X; µ) to f, fk -s f in LP, iff (4)
f fk9 dµ - f f
g dµ
V g E Lp (X ; µ)
We say that fk converge weakly in the sense of measures, or as measures,
to f,fk - f,iff (5)
(ii)
ffk9dµ
'
ffd
V 9 E Cc(X )
µ
We say that the function fk, k = 1, 2,..., are equi-summable or equiintegrable if for every e > 0 there exists a 8 > 0 such that
fIfkd/L<
e
holds for all k and all Borel sets B C X with µ(B) < S. We have
1. General Measure Theory
40
Theorem 5. Let p be a Radon measure over X and let fi, f2,... be functions in LP (X; p).
(i) Suppose p > 1. If supk II fk II LP(x;u) < oo, then there exist f E LP(X; p.) and a subsequence {fk;} such that fk, - f weakly in LP(X; p). (ii) Suppose p = 1. Then
f weakly in L'(X;lc) iff fk -1 f as measures and the a) We have fk fk's are equi-integrable in L'(X;{i) b) If SUN II fk IIL'(x;,t) < oo and the fk's are equi-integrable, then there exist f c Li (X, p) and a subsequence { fk; } such that f k, f weakly in Li(X; µ).
While bounded sequences { fk } in Li (X; A) converge as measures to a mea-
sure modulo passing to subsequences, they in general do not converge, even passing to subsequences, weakly in Li (X; µ), compare Sec. 1.2.2 and Sec. 1.2.4 below. Finally, concerning the equi-integrability condition we mention the following
Proposition 3. A sequence { fk} C L' (X; µ) is equi-integrable if and only if there exists a continuous function 0 : [0, oc) -> [0, oo) such that sup
f
0(I fk1) dy < oo and 0(t) It
oo as t --> oo.
In the remaining part of this section we would like to discuss in more details weak convergence of measures and of functions.
2.2 Typical Behaviours of Weakly Converging Sequences We begin by illustrating some typical features of the behaviour of sequences which are weakly, but not strongly converging.
Oscillations. Sequences of rapidly oscillating functions provide examples of weakly, but not strongly, converging sequences.
0
Consider the sequence
uk (x) := sin kx ,
x E (0, 2ir) C R , k E N
.
then, by Riemann-Lebesgue's lemma, see e.g. Proposition 1 in Sec. 1.2.3 uk u := 0 weakly in Lq(0,27r) V q > 1 but Iluk IIL4(0,2,) = c(q) > 0 so that {uk} does not converge to u in any LQ(0, 2a), q > 1, and II U IIL4(o,2ir) < likm of II Uk IIL9(0,2,)
If Uk - u weakly in LQ, q > 1, by the weak lower semicontinuity of the L9-norm we have
2.2 Typical Behaviours of Weakly Converging Sequences
41
IIuIILI <
(1)
If Uk -> u strongly in L9, we instead trivially have equality in (1), but we may as well have equality under mere weak convergence. Let uk(x) := 1 + sin kx, x E (0, 2ir), ic E N. Then Uk - u 1 weakly in L9 for all q > 1. Clearly 27r
277r
fIUkdx =
J0
0
dk
I u l dx = 27r
fIuk_uI dx = J Isinkxldx = 4 0
0
so that uk - u weakly in L', Uk 74 u strongly in L' but II uk IILI(0,27r)
--;
II U IIL1(0,27r)
Notice that instead 27r
lim inf k-.c
27r
I uk I4 dx >
J 0
Vq>1
fiu l4
,
0
because otherwise, passing to a subsequence, we would have Uk
u weakly in
LI, II Uk II Ly - 11 U II L9 , q > 1 which implies Uk --> u strongly in L9, see Sec. 1.2.3.
1 + t2 is strictly convex, for k = 1, 2, .. .
Notice also that, since 27r
2
27r
1+ukdx = fVFI+U2dx >
1+ I J ul I
=f
0
so that
21r
277r
liminfJ k--oo
1+ukdx > J
1+u2dx
0
0
Concentrations. This is for instance the way Dirac mass appears. But concentration phenomena appear also just in the context of functions.
2 Let Uk : (-1,1) C R , R be defined by uk (x) ['hen
1 k ifO<x<1/k 0
otherwise.
1. General Measure Theory
42
uk
(2)
8o
weakly as measures in (-1, 1)
.
Of course, we may smooth Uk and still have (2). Notice that uk measures in (0,1) while Uk 7 0 weakly in L1(0, 1). By considering instead vk : (-1, 1) , JR given by
0 weakly as
1-k if - 1/k < x < 0 k if 0 < x < 1/k
Vk :=
otherwise
0
we clearly have Vk -s v := 0 weakly as measures in (-1, 1) while no subsequence of {vk} does converge weakly in L'. In fact for cp := X(o,1) E L°°(-1, 1) we have 1
1
J vkcpdx=1 74 fvdxU. This shows that {vk} is not pre-compact in the weak convergence of L', though f 11 IvkI dx = 2
`d k. Notice also that we have a loss in energy: IIVIIL1
while I
<
lira inf 11 vk 11L1
k-.oc
vk41-+ooifq>1.
3 A similar in certain respect, and different for others, concentration phenomena shows also in Lq-spaces for q > 1.
Letq>1,andletuk : (-1,1)-+R be defined by u k (x) :_
We have II uk IIL9 = 1 V k, uk strongly in Lq, since the norms LP(0,1) for any p < q. In fact 11
J
Iu)
k1/9 0
if 0 < x < 1/k ot herw i se
0 weakly in Lq but {uk} does not converge concentrate. But uk --f 0 strongly in
uk
asP-1<0.
dx = k9-1 --+ 0
q
In particular the norms II uk I ILr I p < q, do not concentrate.
Distributions. Concentrated measures may distribute producing a distributed measure in the weak limit. But also functions may concentrate and produce in the limit a function. 4
Consider the Radon measures in (0, 1) C ]R given by 1
Ak
k
ai/k
4.
i=1
2.2 Typical Behaviours of Weakly Converging Sequences
43
ft = An opposite phenomenon to concentration then appears, as clearly uk L1 L (0, 1) weakly as measures, in fact for any smooth ¢ with support in (0, 1) k
1
ki=1E0() -i
f
Odx
0
Replacing each Dirac mass Ji/k with an "almost Dirac mass" a similar phenomenon may appear in the context of functions. Consider for instance the sequence of functions in L1(0, 1) 1
Uk := k
k
k2X(i/k_i/k2 ilk) i=1
Then we again have Uk - u := X(o,1) weakly as measures. This is a typical concentration-distribution phenomenon. In fact L1(spt uk)
k
so that, outside a small set the uk's are zero. One could expect then that L1(spt u) = 0, but this is not true. The uk's concentrate in sets of small measure which are almost densely distributed in such a way that the weak limit is distributed. Note also that Uk, u > 0, and 11 Uk IILI(o,1) = 11 u IILl(0,1) = 1. Notice also that the sequence {uk} is not equi-absolutely integrable, therefore by Theorem 5 in Sec. 1.2.1, {uk} (as well as any of its subsequence) does not converge weakly in
.
L1.
Nonlinearity destroys the weak convergence. If {uk} converges weakly but not strongly, then for a generic non linear function f the sequence M(uk)} U01 does not converge weakly to f (u) 5 Let uk (x) = sin kx, x E (0, 2ir) and let f (t) = t2. Then we have Uk - u := 0 in L4, q > 1 while
f (uk) =sin2 kx = 2 (1 - cos 2kx) - 2
Au) = 0 .
Notice that {uk} is equibounded in L°°. Similarly, if we let f (t) := max(0, t), we have f (uk)
or, if we let f (t) = Jtl, jukI
f (0)
1
weakly in L1(0, 27r)
Jul weakly in L1(0, 2ir).
1. General Measure Theory
44
We notice instead that, if for instance f (t) is a continuous function satisfying
0 < f (t) < t1r,
p>1
and uk -* u strongly in LP(Sl), then
f (uk) - f (U) strongly inLp(.f2). This is readily seen taking into account Egoroff theorem and the absolute continuity of Lebesgue integral.
2.3 Weak Convergence in L4, q > 1 Let Sl be an open set in R' and let 1 < q < oc. For Lebesgue summable functions weak convergence just amounts to: (i) Let uk, u E Lq(Sl). We say that uk --* u weakly in Lq iff
J uk cp dx
(1)
for all cp E
Lq'(Sl), q'
fucodx
= q/(q - 1).
(ii) Let uk,u E L°°(Q). We say that Uk -+ u weakly* in LO°, Uk
-
U, if (1)
holds for all cp E L' (Sl). (iii)
Let uk, u E L1(0). We say that a) Uk ---* u weakly in Ll if (1) holds for all cp E L°O (Sl),
b) Uk -- u as measures in ( if (1) holds for all cp E C°(Q), c) Uk - u as distributions in Q iff (1) holds for all cp E C°O(Q). Clearly
(ii)
=>
(i)
(iii)a ' (111)6
(iii),
.
As it is known the previous definitions can be subsumed to general definitions in the context of Banach spaces. Let B be a Banach space and let B* denotes its dual with duality <x*, x > _ x*(x) for x E B, x* E B*. (a) Let xk, x E B. One says that xk _ x weakly iff x* (xk) --> x* (x) V x* E B*. (b) Let xk, x* E B*. One says that x* - x* weakly* if x* (xk) --4 x* (x) V x E B. Thus the weak convergence is defined in every Banach space, while the weak* convergence is defined only on dual Banach spaces. Of course B** B, hence the weak convergence in B* implies the weak* convergence. In particular, if B is reflexive, B** = B, then weak and weak* convergence in B* (and in B) agree. Banach-Steinhaus theorem ensures that weakly (weakly*) converging sequences are equibounded in B (in B*). The norm in B (B*) is lower semicontinuous with respect to the weak (weak*) convergence. Moreover we recall that bounded
2.3 Weak Convergence in Lq, q > 1
45
sequences in a reflexive Banach space B (in the dual B* of a separable Banach space B) admit weakly (weakly*) converging subsequences. Finally let us mention Banach-Saks or Mazur lemma which states that for any e > 0, if xk - x in some Banach space B, then there exist an integer n and real al, ... , a,,, with
a1 +... + a,,, = 1 such that n
Il X - E ajxi 11 < E . i=1
A similar result is instead false in general for the weak* convergence. The definitions in the beginning, as well as many results on weak convergence of functions may then be seen as applications of the previous abstract facts to LP-spaces, taking into account that
I. If 1 < q < oo, then Lq(Q) is reflexive and (Lq(Q))* = Lq' (Q), q' = q/(q-1). II. (L1(.(2))* = L°°(Q) and (LOO (Q))* # L1(,f2).
This explains why one defines only the weak* convergence in L°°. In particular we see that we can apply Banach-Steinhaus theorem to all L1'spaces, we can infer a simple weak compactness theorem in Lq, 1 < q < cc, and
also in LO° = LI*; while a compactness theorem in L1 turns out to be more delicate.
Finally we mention that in order to have xk - x, (xk - x*) it suffices to check that supk 11 Xk II < CC (supk II xk II < oo) and z*(xk) --+ z*(x) (xk(z) x*(z)) for a dense set z* E B* (of z E B).
-i
From the previous remark or directly we can now easily deduce
Theorem 1. Let 1 < q < oc, and let uk, u E L" (D). Suppose that
supJ IukIgdx < oc k S2
then the following statements are equivalent (i) Uk u weakly in Lq(Q). (ii) Uk -s u as measures in 0. (iii) Uk -s u as distributions. (iv) For any Borel set B C ,(2 with 1 B I > 0 f-B Uk dx -1 f -B u dx. (v)
For any cube Q C .f2 fQ uk dx
fQ u dx.
Moreover one sees that a similar equivalence holds also if q = oe, replacing in (i) the weak convergence with the weak* convergence in L. We can then say that the weak convergence in L' , q > 1, is characterized by the equi-boundedness of the Lq-norms and the convergence in the sense of distributions. Condition (v) expresses precisely the intuitive idea of weak convergence (in Lq, q > 1) as convergence of the mean values.
46
1. General Measure Theory
The meaning of the weak convergence as the process of smoothing oscillations is expressed by the next proposition which extends Riemann-Lebesgue's lemma that we have already used in Sec. 1.2.2. fT (ai, bi) C Rn and let u E Lq(,fl), where 1 < q < oc.
Proposition 1. Let 1?
i=1
Extend u by periodicity from .fl to Rn and let
x E Rn, k E N.
uk (x) := u(kx) Then Uk
U:= l u dx weakly in LQ (.fl), if 1 < p < oo
uk
U:= fu dx weakly* in Loo (Q)
92
n
and uJluxdx.
Proof. Trivially we have 11 uk(x)
f
=
Next we show that (2)
f uk(x) dx
fu(x)dx=ii.
Q
Q
This of course concludes the proof in the case 1 < q < oo. Of course it suffices to show (2) in the case Q has faces parallel to those of .f2. In this case we can find a,,3 E 1n such that n
Q=a+Q0=fl(ai+Oiai,ai+Oibi) i=1
and we have ([x] = integral part of x), using the periodicity,
f(nk_I)d= J
(u(kx) - v,) dx = kn
ka+kpn
a+ 3Q
Q
kn
ka+(k3-[k3])12
([kll)' f (u(y) - u) dy + k,n n 1
kn
(u(y) - u) dy
(u(y) - u) dy +n
J ka+[ka]Q
=
1 (u(y) - u) dy
f ka+(k0- [k0] )
f k«+(k3-[k3])S7
(u(y) - u)dy
(u(y) - i) dy
2.3 Weak Convergence in Lq, q > 1
47
therefore
f(uk - i) dy
kn
f I u(y)
- uI dy
which implies (2).
In order to conclude the proof in the case q = 1, we need to show that the sequence Uk is equi-absolutely integrable, compare Sec. 1.2.1 and the next subsection. For 8 > 0 and x E (1 define vb(x) := max(min(u, 8), -8),
W'5 (X) := u(x) - vb(x).
Of course for any E > 0 we can find 8 = 8(E) such that fn I wb I dx < 2 . By periodicity we extend v6 and wa to ]R'.
J
w8 (kx) I dx
=
Twe find for k = 1, 2,
J I wb (x) I dx <
2
Therefore
f Iuk(x)Idx < f Ivs(kx)I dx + J lwb(kx)l dx < 8IEI + 2 E
E
E
and we find f Iuk(x)Idx < E E
if we choose meas E < E/28.
Notice finally that Theorem 1 does not hold for q = 1. In fact as we have seen in Sec. 1.2.2 there exist {uk} with sup II uk IILI < oo ,
uk - 0 as measures , uk f' 0 weakly in L'
k
The simple remark contained in the next proposition is often quite useful.
Proposition 2. Let 1 < q < oc, and let uk, u E L9 (Q), vk, v E Lq' (Q), q' _ q/(q - 1). Suppose that
Uk -k u weakly in Lq(Q) and vk -k v strongly in L9 (.f2). Then ukvk
uv weakly in L1(.fl).
The next proposition shows that if a weakly converging sequence in Lq, q > 1, does not converge strongly in Lq, then there must be a loss in the q-norm energy.
Proposition 3 (Radon-Riesz). Let uk,u E Lq(Q), 1 < q < oo, be such that
Uk - u weakly in Lq(Q), Then Uk -> U strongly in Lq(.Q).
II Uk IILQ((2) -> II U L9(12).
1. General Measure Theory
48
We notice that Proposition 3, is false for q = 1, compare 01 in Sec. 1.2.2. It is usually stated as a consequence of the so-called Clarkson's inequalities, which express the uniform convexity of the La-spaces for q > 1. We refer for instance to Riesz and Sz.-Nagy [556, n. 37] for a proof. u in L2(12) we have The case q = 2 is /parfticularly simple. In fact if Uk limes I
(3)
k
J
I uk 12 dx -
J
auk - ul2 dx)
=
f
I ul2 dx
u12 Iu12 weakly in L1(Q). And clearly (3) = -Iu12 + 2uku since IukI2 - I uk implies II uk - U IIL2 (O) --+ 0 under the further assumption II uk II --> 11 u II.
As corollary of Proposition 3 or by a direct proof we can also state the following weaker form of Proposition 3
Proposition 4. Let uk,u E Lq(.Q), 1 < q < oo, be such that Uk --> u
a.e. and II uk IIL(Q) - II U II L9 (n)
Then Uk --* u strongly in L4(.f2).
Proof. In fact passing to subsequences, Uk, converge weakly to some v, and v must agree a.e. with u. More directly, by Egoroff theorem we infer that uk _4 u uniformly except on an open set of small measure b. Choosing S so that by the absolute continuity of the integral
Jn
Iul9 dx < e
for sets A with meas A < 5, and observing that also fA auk I9 dx < e V k, we easily conclude.
Notice that we have in fact proved that Proposition 4 holds also for q = 1. Since the convergence in measure implies a.e. convergence passing to a subsequence, it is also easily seen that Proposition 4 remains true replacing the a.e. convergence by the convergence in measure. However, under this assumption, it does not hold anymore for q = 1. There is an analogous of (3) also for q 54 2.
Proposition 5. Let 1 < q < oo. Suppose that (4)
Uk - u a.e. in 12 and Uk
u weakly in L4(,2).
Then
kliM
(.(2).
oo(IIukIILr(57)-
2.3 Weak Convergence in L', q > 1
49
Proof. Fix e > 0 and let vk := sup(0, I Iuklq - Iuk - uNq - Iulq I - eluk - uIq). Clearly vk -4 0 a.e.. From the elementary inequality < elalq + c(e, q)Iblq
I Ia + blq - I aI q I
we also infer vk < Sup(0, I Iuklq - Iuk - ul q I + I uI q - eI uk - uIq) < (c(e, q) + 1) Iulq
Thus the dominated convergence theorem yields limk-,,,, f v' dx = 0, and we n conclude
lim sup k-boo
f
- Iuk - ulq - Iul'Idx
IIuklq
< e lim sup k--oo
f
Iuk - uI°) dx = O(e)
D 71
It is easily seen, compare the proof of Proposition 4, that, if 1 < q < oo, we can replace the assumptions (4) in Proposition 5 by (5)
u a.e.
Uk
sup II Uk IIL9(Q) < o0
,
and the conclusion still holds. However (5) does not suffices, if q = 1. For example consider for A > 0 and for k > A vk (x)
k
-k
if x E (0,1/k) if x E (-1/k, 0)
0
otherwise
and set uk(x) :_ A+vk(x). We have uk(x) --> A a.e. and II Uk IILl = II Uk - A IIL1 = 2. Finally we remark that a key ingredient in the previous statement is convexity, as we shall see better in Vol. II Ch. 2.
We conclude this subsection by restating the compactness property of Lqspaces, 1 < q < oo.
Theorem 2. Let 1 < q < oo. Then (i) If uk - u weakly in Lq(,f2) (uk - u if q = oo), then SUP II tL IIL4(Q) < oo .
(ii)
If SUP I I k
u I I La (.a) < 00
then there exist a subsequence {uk;} of {uk} and u E Lq(.f2) such that
uk; - u uki
u
weakly in Lq(s2), if q < oo weakly* in L°°(.Q), if q = oo
.
50
1. General Measure Theory
2.4 Weak Convergence in L' Most interesting for us will be the notions of weak convergence and convergence as measures in L'(fl) where Si is a bounded open set in RI. As we have seen, compare in particular Sec. 1.2.2, bounded sets in Ll (.fl) are not compact with respect to the weak convergence of L'. This is the reason for regarding L'(0) as embedded in the Banach space of Radon measures with bounded total variation in .f2
L'(Q) - M(Q)
.
Since A4 (0) is the dual space of Co ((7), we can then consider in L' (Q) the weak* convergence of M = (CO ((l))*, which is nothing else than the convergence as measures we have already considered. The important outcome is then
Theorem 1. Let {uk} be a bounded sequence in LI(Q). Then there exist a subsequence {uk; } and a Radon measure a E M (.f2) such that Uk - µ as measures in 0 or, equivalently, weakly* in M((). Next theorem, due essentially to Lebesgue and which we have already stated in Sec. 1.2.1, yields a necessary and sufficient condition for the more delicate compactness with respect to the weak-L'-convergence.
Theorem 2. Let {uk} be a sequence in LI(Q) such that (i) the Uk's are equibounded in L1(S?), supk 11 uk IIL1((l) < 00 (ii) the uk's are equi-integrable, or in other words the set functions E E C (2 are equi-absolutely continuous, i.e., for any e > 0 fE ukI dx ,
there is b > 0 such that fE Iuk I dx < e for all k provided meas E < J. Then there exists a subsequence of {uk} which converges weakly in L'(f2). Conversely, if {uk} converges weakly in L'(0), then (i) and (ii) hold true. Proof. Since step functions are dense in L°°(Q), and taking also into account
the Radon-Nikodym theorem, it is easy to see that it suffices to show that for a subsequence uk, the limit
lim f uk; dx
i
00
B
exists for all measurable set B C Q. Suppose (i) holds. From Theorem 1 we infer that there is a subsequence {uk; } and a measure p such that
fukdx --4
fd:)
E C° (Q)
Using (ii) we shall now show that for such a subsequence and all measurable B C .fl { f e uk; dx} is a Cauchy sequence.
2.4 Weak Convergence in L1
51
By Lusin theorem we can find a sequence {cph } C C° (,f2) with cOh
---+ XB
a.e. in (2 ,
IIWh IILOO(Q) < 1
In correspondence of S = b(e) in (ii) we find an open set Bb C (2 such that IB8I < 6 and {cph} converges uniformly in (2 \ Bb to XB. Now we have
f(uk: - uk,) dx
f(uki - uk,)XB dx
B
fu ki - ukj)(XB - Wh) dx B6
J (uk;-uk;)(XB-Wh)dx + <2
f B6
+
S-AB6
(uki - uk; )Wh dx
12
(Iuk I + Iuk; I) dx + Sup IXB - OhI \B6
f
f f
(uk1 - ukj) cOh dx
(Iuk I + Iuk, I) dx
12
.
S2
In dependence of e we now find some ho such that
Sup IXB- ohI < E
.n\B6
dh>ho.
Since cphc, E C°(0) for all i, j larger than some ko depending on ho and e we have
f(zLk-uk3)cohadx < 6Therefore we infer
f(uk -uk)dxi < B
This concludes the proof of the first part of the theorem. Suppose now that Uk -k u weakly in L' (Q) and assume for simplicity u = 0 (otherwise we would consider the sequence {uk - u}). Then (i) follows from Banach-Steinhaus's theorem. In order to verify (ii), we first prove the following claim:
If (ii) does not hold, then there exist a positive number z, a sequence of disjoint measurable sets Ei C .f2 and an increasing sequence of integers vi such that
Jlttvidx > E;
z
foralliEN.
52
1. General Measure Theory
By assumption, there exists an & > 0 such that, for all b > 0, we can find some set F C ,(2 with meas F < 8, and some v E N which may be taken arbitrarily large such that
fkidx > e. F
Since u, E L' (,fl), for any or > 0 there is some 77 > 0 such that
J
I
B
holds true for all measurable sets B with meas meas B < 77. Choose now o = of = e/2, b = b1 = mess ,f2; then we find 77i > 0, F1 C .f2 and v1 E N such that I dx < o1 for all B with meal B < 771. I dx > e and fB meal F1 < 51, fF, Next we choose 0'2 = z , b2 = Sl /2; then we find 772, F2 C f2 1/2 E N such that dx > e and fB Iu, 2 I dx < o2 for all B with meas B < 772. meas F2 < b2, fF2 , bi = min{ , ... , '7121 }; then Similarly for all i > 2, we choose of = e2
2 meas Fi < bi, fF; u,,, J dx > e, we find 77i > 0, Fi c .f2, vi > vi_1 such that
fB u,,, I dx < of for all B with meas B < 77i. Set now
Ei := Fi - UFq. q>i
We have
meas I U Fq I < q>i
/
meas F. < q>i
24-ti = 77i q>i
and also
J
E;
lu,,,Idx=
lu,,Idx>e-2i>E/2
J
for alli>1.
Fi\Uq>iFq
Since the Ei are disjoint, the claim is proved with z = 6/2. We now observe that replacing Ei by Ei fl {x : u,,, (x) > 0} or by Ei n {x u,,, (x) < 0} and z by z/2 we can modify the claim as follows: There exist Ei C ,f2 and vi such that
f
u,,,dxI
>z
for alliENN.
Ei
We shall finally construct a function cp E L°° (,(2) for which the sequence
f
ukcp dx
does not converge to zero. This will conclude the proof. The function cp will be defined as 1 on the union of suitable Ei and zero otherwise. We set E(1) := E1 and v(1) := vl, and we choose 6(1) so that
2.4 Weak Convergence in L'
JB
for all B with meas B < E(l)
iu (1) I dx < z/3
53
.
If fE(1) uk dx does not converge to zero, the proof is complete as we can take (p = XE(1). Otherwise we choose E(2) as the first E2 for which the remaining ones, i.e., the Ej with j > i, have total measure less than E(1) (this is possible since the Ei are disjoint) and the corresponding index vi is such that
< z/3
for all k > vi
E(1)
(this is possible since fE(1) Uk dx -> 0). Denoting by A(2) the index corresponding to E(2), we choose E(2) > 0 such that
J lu, (2) 1 dx < z/3
for all B with meas B < E(2)
B
If fE(1)uE(2) Uk dx does not converge to zero, the proof is complete. Otherwise we proceed as before and in general we find E(k), v(k), 6(kl) such that
meas U Ei < E(k-1)
J Ui>kE(')
i>k
uh dx
> z/3
for all h > v(k)
and fB I uv(k) I dx < z/3 for all B with meas B < E(k). Finally, set (p := Xuk>1E(k). Then we find U,(k)
dx
=
J
U,(k) dx +
>
,I
U,(k)
U,(k) dx +
E(k)
Ui
S2
J
dX
- 3 - 3> 3 Z
Z
J
U,(k) dx
Ui>kE(')
for all k E N
E(k)
and this gives a contradiction to the weak convergence of Uk to zero.
Often Theorem 2 is referred to as the Dunford-Pettis theorem, although it was first proved by Lebesgue [425].
It is not difficult to show that paralleling the case q > 1, in L'(0) we have
Theorem 3. Let Uk, u E L1(Q). Suppose that SUPk 11 uk IIL1(s2) < oc and {uk} is equi-integrable. Then the following statements are equivalent (i) Uk - u weakly in L'(,R) (ii) Uk -k u as measures (iii) Uk - u as distributions (iv) for any Borel set B C ,R, f-B uk dx -> f-B u dx
54
1. General Measure Theory
(v) for any cube Q C .R, fQ uk dx ---> fQ u dx.
Next proposition, though simple, is very useful
Proposition 1. Let ak, a, bk, b be measurable functions on a measurable set
0CRn. (i) If ak(x) -* a(x) a.e., ak bk
(ii)
A V k, and bk
II ak
ab weakly in L' (.(2) .
If ak -* a in L', II ak I.,n 5 A `d k, and bk ak bk
b weakly in L1(,(2), then
b weakly in L'(s7), then
ab weakly in L'(.Q).
Proof. From the weak convergence of bk to b, we deduce that {bk} is equiabsolutely continuous, i.e., for any e > 0 there exists SE > 0 such that if mess E <
bE then fE Ibk I dx < e d k. In correspondence of SE, Egoroff's theorem yields a closed set F such that meas (.(2 \ F) < bf and ak --, a uniformly in F. Thus for W E L°°(,(2) we get
f cp(ak-a)bkdx <
f Ibkldx < 2AIIcpIIe S2\F
,(2\F
fco(ak-a)bkdx ---> 0 ask---*oo. F
0 weakly in L'(Q). On the other hand, From this we conclude (ak - a)bk 0 weakly in Ll ((2). Thus (i) is proved. since a E LOO, a(bk - b) Part (ii) now follows from (i). In fact from any subsequence of {ak} we can extract a subsequence which converges a.e. to a. Part (i) then yields that any subsequence of akbk has a subsequence which converges weakly in L' to ab. Since the limit is independent from the chosen subsequence, (ii) follows.
Of course (iv) of Theorem 3 does not hold if we merely have Uk -s u as measures in 0. However one can prove, compare Sec. 1.2.1,
Theorem 4. Let uk, u E L' (Q) be such that supk 11 uk 41(fl) < oo and uk - k u as measures in ,(2. Assume moreover that there exists a function v E Ll (12) such that uk > v d k E N. Then (i) for any compact K C 12, fK u dx > lim supk-. fK uk dx, (ii) for any open set U c 12, fu u dx < lim infk-,,. fu Uk dx, (iii) for any Borel set B C .(2 with I aB I = 0, fB Uk dx ---+ fB u dx.
The second example in 2 in Sec. 1.2.2 shows that the assumption Uk > v is essential. The example in 4 in Sec. 1.2.2 shows that strict inequality may occur in (i) and (ii). As we have seen in the previous subsection Uk -i u a.e.
and
11 uk IILl(S2) - 11 U IILl(12)
2.5 Concentration: Weak Convergence of Measures
55
imply strong convergence in Ll (.rl) of Uk to u. Examples in Sec. 1.2.2 show that one cannot replace the a.e. convergence with the weak convergence in L1. However we shall prove in Proposition 1 in Vol. II Sec. 1.3.4, as consequence of a theorem due to Reshetnyak the following interesting result Proposition 2. Let Uk, u E Ll (.(l). Suppose that Uk - u as measures in IZ and
f/1+u2dx.
f/1+udx -+ n
.a
Then uk converge strongly in Ll (Si) to u.
2.5 Concentration: Weak Convergence of Measures Here we would like to show that concentration-distribution phenomena associated to weak convergence of measures are in some sense universal, and shortly discuss weak convergence of measures in all of R''2.
0 Every (concentrated) Radon measure p E M(Rn) can be obtained as weak limit (in the sense of measures) of a sequence of smooth functions with equibounded L1-norm. In fact regularizing A.c, i.e., setting
uE(x) := A*,-)
(x)
where 0,,,,(y) := En o , 0 being a standard mollifying kernel, we have uE E C°° (Rn) and Ln L uE - p weakly in M (W'). 2 Let u E C° ([0, 1] ). Then the distributed measure Ln L u is the weak limit of the concentrated measures k
/ ku
Pk
kJilk
E .M((0,1))
Similarly the sequence of functions 1
Uk
k
2 k RL 2=1
W X(k-ky k
converge in the sense of measure to u. Notice that the measures Ll L Uk are 'almost-concentrated' in fact L1(spt µk) < 1/k. If .11 is not bounded, C°(.(l) is not a Banach space and the space of Radon
measures with bounded total variation in P turns out to be the dual of the separable space
56
1. General Measure Theory
Co (f2) :_ {u E C°(Q) I V e> 0 there exists a compact K C Q
such that I u l < e in 0 \ K I
.
Independently of the boundedness or the unboundedness of .f2 one says that µ locally in M(n) 1ff µk
fWdµk ->
r J
dµ
V
EC°(Q).
n
.a
The Banach-Steinhaus theorem then says that
sup1µkI(V) < 00
(1)
VVCCf2,
k
while, as consequence of the compactness theorem, every sequence {µk} satisfying (1) has a subsequence which converges locally to some µ with locally finite total variation. Consider now a sequence of probability measures µk in IE81, i.e.,
µk?0,
fdk=1.
Then there is a subsequence µki such that µk; - µ locally in M(Rn) with µ(I[8') < 1 and, in general, µ(R'2) < 1. The following result, due to P.L. Lions, describes modulo subsequences what may happen
Theorem 1 (Concentration-compactness lemma). Let {µk} be a sequence of probability measures in R1. Then there exists a subsequence {µk;} such that one of the following three conditions holds (i) (Compactness) There exists a sequence {x,} C R such that for any e > 0
there is a radius R with the property that
µki (B(xi, R)) > 1 - e
`d i
(ii) (Vanishing) For all R > 0 we have s up
µki (B ( x, R))
0
as i -* oo
xEII2n
(iii) (Dichotomy) There exists A E (0, 1) such that for any e > 0 there is a radius
R > 0 and a sequence {xi} satisfying the following property: for any R' > R there exist non-negative measures µk; and µk; such that
0 < /4, + µk; , spt µk; C B(xi, R) , spt µk. C R' \ B(xi, R') limsup(IA-µk;(IRn)l+ (1-A)-µk.(Rn)l <e i- 00
2.5 Concentration: Weak Convergence of Measures
57
Proof. The proof is based on the notion of concentration function
Q(r)
sup p(B(x, r)) XER-
of a non-negative measure, introduced by Levy [431]. Let Qk denote the concentration function of µk. Clearly {Qk} is a sequence of non-decreasing, non-negative, bounded functions on [0, oc) with
lim Qk(R) = 1
R-oc
.
In particular {Qk} is a sequence which is locally bounded in BV(0, oc). Thus there exist a subsequence uki and a bounded, non-negative, non-decreasing function Q with
Qki(R) ---> Q(R)
as i -+ oo and for a.e. R R.
Let
A := lim Q(R).
R
Clearly 0 < A < 1. If A = 0, we have vanishing, case (ii). Suppose A = 1. Then for some Ro > 0 we have Q(Ro) > 1/2. Choose xi such that Qki (Ro) <- µki (B(xi, Ro)) +
a
Then for 0 < e < 1/2 fix R such that Q(R) > 1 - e > 1/2 and let yi satisfy Qki (R)
Akj (B(yi, R)) + 1
For large i we have
l1k<(B(yi,R))+µk<(B(yi,Ro)) > 1=/.Lki(R") It follows that for such i B(yi, R) fl B(xi, Ro)
0
.
That is B(yi, R) C B(xi, 2R + Ro), and hence 1 - E <- NJki (B(xi, 2R + Ro) for large i. This proves (i). If 0 < A < 1, given e > 0 we choose R and a sequence {xi}, depending on e and R, such that
µki (B (xi, R)) > A - e
for all i > io (e)
.
Enlarging io (e), if necessary, we can also find Ri --> oe such that
58
1. General Measure Theory
Qki (R) < Qk. (Ri) < A + e
for all i > io (e)
.
Now let µki
µk; L B(xi, R) ,
{ski := /lki L (lR' \ B(xi, Ri,))
Obviously 0 < ck. + µki < µki and, given R' > R, for large i we also have spt FLki C R"' \ B(xi, Ri) C ]lgn
spt Ak. C B(xi, R) ,
B(xi,
Finally, for i > io(e) we get
I), -A',(Wn)I+I1-A-µki (Rn) = IA - JLki(B(xi,R))I + l-Lk; (B(xi, Rj) - Al < 2e which concludes the proof.
2.6 Oscillations: Young Measures A very important and fruitful idea, due to L.C. Young in the context of Calculus
of Variations and revived more recently among others by L. Tartar, is that one can understand better oscillation phenomena by looking at graphs, or more precisely at the associated Young measures. For instance those measures will give information on the weak limit of the square of a weakly converging sequence. A similar idea is the starting point in the sequel of this monograph for studying currents associated to graphs of maps u. In fact Young measures may be seen as the first component of currents associated to graphs, compare Vol. II Ch. 5.
The main difference is that in the context of currents one regards graphs as 1-graphs, i.e., as sets which carry at (almost) all of their points tangent planes, while in our present context of measures one regards graphs just as sets.
Definition 1. Let Q be a bounded open set in Rn and let u E L°° (Q, RN). The measure Yu E M+(Q X RN) defined for each continuous and bounded function, cp : ,(lXRN --->R, cpEC°(nXRN), by
f
o dYu
nxRN
f(x,u(x))dx Q
is called the Young measure associated to u. Clearly
7r#Yu = G' L 0
if 7 : (x, y) E .fl x Rn --p x E 12 denotes the standard projection and given by 7r#A(E)
µ(E x RN) .
7r#µ is
2.6 Oscillations: Young Measures
O(x) we have
In fact for O(x) E Cb (Q) and cp(x, y)
J O(x) d7r#yu = J S2
59
7r#0 dy,
=J
W(x, u(x)) dx = f co(x) dx
S2XRN
S2XRN
.
S2
This motivates the following
Definition 2. A Young measure in Q x RN is a non-negative Radon measure µ E .M+(fl x RN) satisfying 7r#p = Ln L .fl i.e.
µ(E x RN) = L (E)
for all Borel set E C .fl
The disintegration theorem for measures then yields at once
Proposition 1. Let p E JV1+(.f2 x RN) be a Young measure. Then for Lna.e. x E 12 there exists a (Radon) probability measure vx on RN, i.e., vx > 0, vx(RN) = 1, such that for all bounded and continuous cp E C°(Q x RN) we have
(i) the map
f
W (x, y) dvx (y)
RN
is Lebesgue-measurable (ii) the following equality holds w (x, y) dµ (x, y) =
fdxJco(x)dvx() RN
f2
This way to every Young measure we can associate a collection of probability measures on RN (1)
{1-'x}a.e.xG0
.
Conversely, to every collection (1) we can clearly associate the Young measure defined by (ii), provided (i) holds. Thus one can also define Young measures as a collection (1) for which (i) holds.
Let {uk} be an equibounded sequence in L°°(.l x RN), and let yuk be the Young measures associated to uk. We have
/ = .Cn(,f2) n / 0 < yuk (D X RN) = 7r#yuk (n) Therefore for a subsequence we have yuk. - p in M+ (!2 x RN). Moreover it is easily seen that p is a Young measure too. It is often called the Young measure associated to the sequence {uk;} In fact for any open set V we have 7r#A(V) = p,(V x RN) < limt yuk. (V X ]RN) = 7r#Y,
.
(V) = Gn(V)
and for any compact K C (2 and R such that spt Yuk, C (1 x B (0, R) we have
1. General Measure Theory
60
7r#µ(K) =,u(K x RN) = µ(K x B(0, R)) > lim sup Y,,,,. (K x B (O, R)) = lim sup YU,ki (K X RN ) z-00 i-00
= lr#Yuk (K) = Ln(K). Let {vx} be the collection of probability measures associated to p, i.e. let {v,} be the disintegration of p. For any co(x, y) of the type cp(x, y) = p(x)V(y), cp E E C° (/RN ), we then get from Proposition 1(ii) C° (fl ),
il J
(x)) dx
S?
=lim
-' O J
co(x, y) dYuki (x, y)
J
fl xRf'N
= lim
00
/
P(x, y) dp(x, y) =
f(x)( fb(y) dv(y))1dx 1
1RN
We c an therefore state
Theorem 1. Let {uk} C L°°(.(2, IRN) be such that SUPk 11 uk 11,,. < oo. Suppose
that the Young measures Yuk associated to uk converge weakly Yuk -i p in M+(0 x RN). Then p is a Young measure. Moreover for any b E C' (RN) we have
o uk -
weakly* in L°° (0)
v
where for almost every x E .(2
fb(Y)dvx(Y),
v(x)
RN
{vim} being the disintegration of p.
The probability measures {vv} can be thought of as giving the limiting probability distribution as k -+ oc of the values of Uk near x. This shows that all information on the weak* limit of 0(uk), 0 E C°, are just contained in the Young measure p ' {vx} associated to the sequence {uk}: any information on {vv} yields some information about the weak* limit of 1'(uk).
0
Suppose for instance that for the limiting Young measure {vx} we know that
vu is a Dirac mass, say v. = Su(x) for a.e. x E (2. Then we can infer Uk -4 u strongly in L2(.(2). In fact for b(y) := y2 we get
fuk2dx
--->
n and for V) (y) := y, p E C°(!2)
f f dx
n
RN
Iy12dvx(y) =
fu(x)2dx
2.6 Oscillations: Young Measures
fp(x)(f y dvx (y)) dx = fp(x)u(x)dx.
fPuk dx -> f2
61
RN
S?
D
Let us discuss now oscillations and associated Young measures in the simple
casen=N=1, Q=(0,1).
Proposition 2. Let u E L°°(0,1). Extend u by periodicity, u(x + m) := u(x) for x E (0, 1), m E Z, and set
uk(x) := u(kx)
k E N, x E (0,1) .
,
Then we have YUk - L1 L (0, 1) x A E M+((0,1) X IR) where A is the measure defined by 1
fdA
(2)
:= J 0(u(x)) dx
R
E C°(R)
b'
o
Proof. For cp E C°(0,1) we have
f
f k
cO(x)V) (uk(x)) dx =
0
k
dl;
0
f
k_1
k_1 Tn+1
=k
1:
cP( /k) (u(C))
m=0 m
1
= k E f W(x km)
(u(x)) dx.
k=0 0
The result then easily follows, since k
for x E (0, 1).
Lim o co ("`k
x)
d uniformly
- fo
The next proposition allows us to compute the measure in (2) for special classes of functions u
Proposition 3. We have (i) Suppose u E C1(0,1) and u'# 0 a.e. in (0,1). Then the measure A in (1) is given by
A = £f,
f (y)
E XEu-l (y)
1
u (y)
(ii) Suppose that u E L°°(0,1) takes only a finite number of values
u(x) := E yiXA; (x)
y1,
, ys E
IIR
i=1
where Ui=1Ai is a decomposition of (0,1). Then A in (1) is given by
62
1. General Measure Theory
A= E'C' (U-' i=1
yE]R
Proof. (i) Assume for instance u increasing, u(0) = 0, u(1) = I. Changing variable, x = u(y), u o u = id, we get 1
f
0da = f(u(x))dx
J and u'(y) =
u
1
I
=
0
(y)u (y) dy
0
The result then easily follows.
(ma(y))
(ii) We have
f J
1
dA = /(u(x))dx V)
J
')(u(x)) dx
i=1 A,
0
1R
f
=
s
i=1
s
(yi)L' (Ai) = f d( G1(Ai)by ) i=1
0 Let u(x) :=x, x E (0,1) uk (x) = kx - [kx] = non integral part of kx
.
By Proposition 3 (i) we get Yu, a := G1 L (0,1)2. Notice that the disintegration of Yu, is given by {bu, (x) } while the disintegration of the limit Young measure µ is given by vv := G1 L (0, 1). As consequence of Theorem 1 we obtain uk
1/3 weakly* in L'; in fact
1
v(x)
fY2dJ:1L(0,1) _ f y2 dy 1
Notice that Uk
3
0
1/2 as f ydC1 L (0, 1) = 2.
0
3 Let u(x) := sin2-7rx, x E (0,1), so that uk(x) = sin2k-7rx in (0,1). We have G1 L (0,1) x A
Yu,
in M+((0,1) X I[8)
and by Proposition 1 (i) we have A
C1L f
1
f (y)
1
y2X(-"')(y)
In fact lu'(x) I = 27r V/1 - y2, y = sin27rx and the multiplicity is 2 if Iy
0 if M > 1. Of course
< 1 and
2.6 Oscillations: Young Measures
63
1
1
1
1-
7r_1
y2
dy = 1.
As f is even we have f y- dvv. = f y'n f (y) dy = 0, then we find again
(sin 2irkx)' -k 0
for m odd.
For zb(y) := y2 we instead get as we know sin 2 2kirx
-
J0
sin2 27rx dx
in fact 1
y
1-y 2
-1 4
dy =
sine x dx (-ir/2 7r/2)
Let 0
fa if0<x
ifa<x<1
a, b E
IR
and, as previously, uk(x) := u(kx) x E (0,1). In this case it is easily seen that yv, - Ll L (0, 1) x (o-b,, + (1 - a)bb) the disintegration of the Young measure associated to Uk is constant in x and given by vx := aSa + (1 - U)bb
As consequence, for instance, of Theorem 1 we instead have Uk
ca + (1 - U)b
weakly* in L°° (0, 1)
.
The weak* convergence of Uk takes into account the mean value distribution of the values, while the convergence of Young measures informs us about values, a and b, and weights, o- and (o- - 1), which indicate how much time the graph of Uk spends at a and b. . We conclude this subsection by stating without proof a quite general theorem on Young measures.
Theorem 2 (Ball). Let .(l be a bounded measurable set in IIR' and let K C RI be closed. Let {Uk}, Uk
:
(2 -+ RN, be a sequence of measurable maps satisfying
for any open set U D K
Ln ({xE(2I Uk(x)V U}) -> 0
ask --oo.
Then there exists a subsequence {uk;} of {Uk} and a family of positive measures {vx} on RN such that
64
1. General Measure Theory
(i) vx(RN) < 1 for a.e. x E 0 (ii) spt V, (RN) C K for a.e. x E 11 (iii) for any V E Co(RN) with zb(y) --* 0 for IyI -> oo we have
? o uk: - v
weakly* in L°°
where
v(x)
f(Y)dv(Y). RN
If moreover we assume that supk fn h(Iukl) dx < oo for some continuous non decreasing function h : (0, oc) --+ 1R with h(t) - + oc for t --+ oo (for instance, for h(t) = t, supk fn Iukl dx < oo), then vx is a probability measure, i.e., vx(RN) = 1.
Finally, if furthermore we assume that for some subset E C Q and some Ill E CO (RN) the sequence {g o uki LE} is weakly relatively compact in Ll(E), then g o ukt L E vLE weakly in L' (E) .
2.7 More on Weak Convergence in L1 A bounded sequence {uk} in L1(Q) is not weakly relatively compact in Ll(Q), if {uk} is not equi-integrable. To overcome such a difficulty we have discussed two possible ways (i)
regard L1((2) as embedded in the space of Radon signed measures M(1) and work with the weak convergence of measures
(ii)
associate to Uk its Young measure Yuk and work with the weak convergence of the measures Yuk .
Of course both ways force us to allow limits which do not belong to Ll (2) anymore.
Here we would like to mention two other possibilities which do not require to go outside L'(Q). (i) (ii)
the so called biting convergence the weak convergence in the sense of absolutely continuous parts
Definition 1. Let .(2 C R be a bounded domain and let {uk} be a bounded sequence in L'(Q) Sup jI Uk IIL1(S2) < 00k
We say that Uk converge in the biting sense to u E L'(.fl), Uk - u in 12 if for every e > 0 there exists a measurable set E C 12 such that C' (E) < e and Uk - u weakly in L1(dl \ E). The following compactness result turns out to hold
2.7 More on Weak Convergence in L1
65
Theorem 1 (Biting lemma). Let {uk} C Ll be such that supk II uk IILI(c) < oc. Then there exist a subsequence {uki} of {uk } and u E L1(Q) such that uk -6' u in Q. This means that removing (bitten) sets of possible concentrations (of small measure) we have weak convergence.
Theorem 1 can be easily inferred from the following lemma due to Acerbi and Fusco [2]
Lemma 1. Let {Oh} C L'(Q) be a bounded sequence. Then, for any E > 0 we can find a measurable set CE, a positive number 8 and an infinite set S C N such that meas (CE) < E and moreover for any measurable C in (1 \ CE with meas (C) < 8 we have
fhIdx
<E
`dh E S
C
Proof.
Suppose that the claim is not true. Then there exists e: > o such that
for every (C5, S, S) as in the statement we may choose a measurable set B with B n CE = 0, meas B < 8, and k E S such that
fkdx
>E
B
This implies that for every set C with meas C < E, an every infinite set S C N, there exists a set A with A n C = 0, meas (A U C) < e, and an infinite subset T of S such that
VkET.
P 7kIdx > E
A
We postpone the proof of the claim, and we prove that this leads to a contradiction. Set C = 0, S = N and let Al and T1 be as above. Starting from C = Al and S = T1 we pass to A2 and T2 where Al n A2 = 0 and
f lOkldx=IAldx+IIOkldx>2E A1UA2
Al
`dkET2
A2
Since meas (A1 U A2) < e, we may set C = Al U A2, S = T2 and continue with the same argument. If
N>
E_1
SUP IIcbkIILI k
after N iterations we obtain a contradiction. Finally, let us prove the claim. Let C and S be as stated, set Sl = S and take 81 <
e - measC 2
66
1. General Measure Theory
Then there exist B1 disjoint from C, with meas B1 < 61 i and k1 E S1 such that J , I'k I dx > e. By induction, setting 1
Si
Si :_ {k E Si-1
16i-1,
I
k > ki-1}
T={kiIiEN}
A=UhBh,
it is easily seen that A and T fulfill all requirements. The first example in 02 in Sec. 1.2.2 shows that Uk -k So as measures and 0. But uk L 0; in the second example we have Uk - 0 as measures and Uk the example in ® in Sec. 1.2.2 shows that
-
b
Uk
v:= 0
Uk - u := 1
and
as measures,
i.e., in general, there is no connection between the weak limit in the sense of measures and the biting limit. More information can be instead obtained if those limits agree. We state without proof one such a result due to Muller [501]
Proposition 1. Let .(2 be a bounded open set and let {uk} be a non-negative sequence such that b Uk - u as measures in .(Z and Uk -
1.
U.
Then for any compact set K C S? we have Uk - u weakly in L1 (K). The second example in [4] in Sec. 1.2.2 shows that the assumption uk > 0 is essentially necessary.
Definition 2. Let .(2 be a bounded open set, {uk} C L1(.(2), supk II uk IL1(n) < oo, and u c L' (0) . We say that ACP Uk
-
'
U
if for every subsequence {uk,} Of {uk} satisfying uki -k it as measures, where p E
M(Q), the absolutely continuous part of p (with respect to Lebesgue measure) agrees with u, i.e., pa = U. Remark 1. The rather complicated form of Definition 2 is necessary. In fact we cannot define
ACP- lim uk := (weak*- lim uk) a In this case in fact, by considering a sequence uk ACP Uk
But if Vk = 0, the sequence
0
ACP Uk
-'
.
Jo as measure we have 0.
3 Notes
67
{wk} = {u1, v1, v2, u3, v3, ...}
would not converge in the ACP sense. Instead, trivially we have wk
0.
Of course, if uk -k u E M(f2) as measures then Uk 2P µ(°') , and, if Uk u E L1(12) as measures then uk ASP
U.
Also a compactness theorem is trivial
Theorem 2. Let {uk} C L1(Q) with supk 11 uk IIL'(n) < oc. Then there exist a subsequence {uk=} of {uk} and u E L1(Q) such that uk; we can also assume that
uki -1 p E M(17)
A2.P
u in P. Moreover
as measures
with pa = U.
Notice that in the first example of 2 in Sec. 1.2.2 we have
Uk - 8o as measures in (-1, 1) uk
0
ACP Uk
0
in (-1,1) in (-1,1)
in the second example in 2 in Sec. 1.2.2 uk
Uk Uk
as measures in (-1, 1)
0
0
ASP
0
in (-1,1) in (-1,1)
and finally in the example 4 in Sec. 1.2.2 as measures in (0, 1)
Uk -1 1 Uk Uk
in (0,1)
- 0 ASP
1
in (0,1)
3 Notes 1 There are many books which deal with general measure theory, among them we mention e.g. Caratheodory [135], Federer [226], Halmos [338], Hewitt and Stromberg [375], Munroe [505], Rudin [566], Saks [570], Wheeden and Zygmund [662], see also Caratheodory [134]. Concerning Hausdorff measures the reader is referred to the original paper by Hausdorff [362], and to Rogers [564], Falconer [218], and to Federer [226]. Cantor-Vitali functions appears in Vitali [648], compare also Vitali [649]. Concerning the classical differentiability properties of Cantor-Vitali functions the reader is also referred to Darst [174] and its references.
68
1. General Measure Theory
2 The content of Sec. 1.2.1,..., Sec. 1.2.5 is quite classical. We only mention that Proposition 5 in Sec. 1.2.3 is taken from Brezis and Lieb [110], while Theorem 1 in Sec. 1.2.3 is due to P.L. Lions [435]. In connection with weak convergence we would like to mention the lecture notes by Evans [215]. 3 Concerning Young measures, a part from the original works by Young, see e.g. Young [678] [682], the reader may consult Tartar [626] where Young measures are developed as a tool for analyzing non linear partial differential equations, Ball [69] (and its references)
where Theorem 2 in Sec. 1.2.6 is proved, and Pedregal [530]. The proof of the biting lemma, in Sec. 1.2.7, can be found in Brooks and Chacon [114], compare also Acerbi and Fusco [2] Ball and Murat [72]. Finally, with reference to Proposition 2 in Sec. 1.2.4 the reader may consult e.g. Visintin [647], Amrani, Castaing, and Valadier [40], Balder [59], Kinderleher and Pedregal [412], Brezis [105].
4 Quite interesting results connecting weak convergence in the Hardy space 7{i (Jr) with a.e.-convergence, biting convergence and weak convergence in Ll can be found in Coifman, Lions, Meyer, and Semmes [156].
2. Integer Multiplicity Rectifiable Currents
It is the aim of this chapter to develop and illustrate some of the basic theory of integer multiplicity rectifiable currents. In Sec. 2.1 we shall discuss the important area and coarea formulas and some of the theory of n-rectifiable sets. After two preliminary subsections in which we collect relevant definitions and facts on multivectors, covectors, and differential
forms, we shall present, in Sec. 2.2, some of the basic theory of currents and especially of integer multiplicity rectifiable currents. Because of the relevance of those topics for the sequel we shall give in principle proofs of our claims, though sometimes we refer to later parts or elsewhere; in any case we try to illustrate our claims by simple examples.
1 Area and Coarea. Countably n-Rectifiable Sets In this section we shall present the important area and coarea formulas and some of the basic aspects of the theory of countably n-rectifiable sets.
1.1 Area and Coarea Formulas for Linear Maps Let f be a function from lE8n into RN and let A C R'. We distinguish two cases n < N. In this case we look for an explicit formula, the area formula, which expresses the n-dimensional measure of f (A), Nn (f (A)), (ii) n > N. In this case we consider the (n - N)-dimensional slices f -1(y) f1 A of A for y E f (A) and we want to relate their xn-N-measure with the total measure of A by a formula, called the coarea formula. (i)
Such formulas describe of course transformation properties of volume elements. Therefore we begin by considering first the case in which F is a linear map.
Let us recall a few facts from linear algebra. A linear map U : I[8n -p RN is said to be orthogonal if (Ux) (Uy) = x y for
allx,yER'.
A linear map S : lid' -> R" is said to be symmetric if x (Sy) = Sx y for all x,yElR .
70
2. Integer Multiplicity Rectifiable Currents
A linear map D : R' -> R' is said to be diagonal if there exist d1, ... , do E R such that Dx = (dlxl,... , dx''') for all x E IRn. The adjoint T* of a linear map T : Rn defined
-->
RN is the linear map T* : RN yE
--..
Rn
RN.
For linear maps R and T one easily verifies that T** = T, (R o T)* _
T* o R*, T* = T-1 if and only if T is orthogonal, i.e., T is orthogonal if and only if T *T = id, T* = T if and only if T is symmetric. Identifying the linear map T : Rn -> R" with an n x n- matrix, we also have
detU = ±1 if U is orthogonal. Finally, we recall that for every symmetric linear map S : Rn -+ Rn there exist an orthogonal map U : Rn -+ Rn and a diagonal map D : IRn -> Rn such that
S = UoDoU-1.
The following theorem will be useful
Theorem 1 (Polar decomposition). Let T : R' ---> RN be linear, (i) If n < N, there exist a symmetric map S : ]Rn _ Rn and an orthogonal map
U:Rn ->RN such that T = U o S, (ii) If n > N, there exist a symmetric map S : RN --+ RN and an orthogonal
map U : RN --+ Rn such that T = S o U*
Proof. The claim (ii) follows applying (i) to T*. Thus assume n < N. It is easily seen that C := T* o T : Rn -+ Rn is symmetric, and nonnegative definite, i.e., (Cx) . x > 0. Hence there exist µl, ... , µn > 0 and an orthogonal set {x1, ... , xk} in Rn such that
Cxk = /Lkxk
k = 1, ... , n .
,
Set µk = a', A k > 0. We claim that there exists an orthonormal set {z1, ... , z,.} in RN such that
k=1,...,n.
Txk = Akzk, For that it suffices to define 1
- k Txk
zk
if Ak # 0. As for Ak, At # 0 Zk - ze =
AkAe --
Txk ' Txe =
ae (CXk) - xe =
Ak AkAQ
Xk ' x2 =
Ak SPk Ai
we see that {zk Ak # 0} is orthonormal. Then, for Ak = 0, we define zk to be Rn any vector such that {z1, ... , zn} is orthonormal. We now define S : R' I
by
SXk = Akxk
k=1,...,n
1.1 Area and Coarea Formulas for Linear Maps
71
and U : R' -+ RN by
U2k = Zk Clearly S is symmetric and U orthogonal, and Uo SXk = )tkUxk = Akzk = Txk,
soT=UoS. We can now state
Theorem 2 (Area formula for linear maps). Let T : Rn -* RN be linear, and n < N. Then for any measurable set A C ItSn we have
7-ln(T(A)) = JTLn(A)
(1)
where
JT =
det (T*T)
Proof. By the polar decomposition theorem we can write
T=US and, being S symmetric,
S=VSdV* where Sd is diagonal, and U and V are orthogonal, that is isometries. Thus we deduce, taking into account Binet's formula det AB = det A det B, ,Hn(T(A))
= 7-nr(US(A)) = H(S(A)) =7cn(Sd(A))
J
Sd(A)
dx =
f(det SdI dCn =
Idet SdILn(A) = Idet S(Ln(A)
.
A
It now remain to observe that T*T = S* U* US = S*S, so that
(detS)2 = detT*T. It is usual to refer to JT as to the Jacobian of the map T and denote it also
by J(T). Notice that JT 0 0 if and only if T is injective, i.e., dim T(Rn) = n, therefore we can also write the area formula (1) as
fJTdx = f 7-l°(AnT-1(y))dfn(y)
(2) A
T(A)
A simple consequence of Theorem 2 is the following
.
72
2. Integer Multiplicity Rectifiable Currents
Theorem 3 (Change of variable formula for linear maps). Suppose that T is a linear map from R' into Rn , that for the sake of simplicity we assume invertible and let u : R' -+ R be a Lebesgue summable map. Then we have
fu(x)IdetTldx = f u(T-'y) dy
(3)
.
Rn
Rn
Proof. By writing u as the difference of its positive and negative part, we reduce to prove (3) in the case u > 0. Since in this case u is the limit of a non- decreasing
sequence of step functions, the result easily follows applying the theorem of monotone convergence and taking into account the linearity of the integral and the equality
2 XA(x) = 7-l°(A n T-'(y)) -ET-1(y)
for the characteristic function XA of A.
Suppose now n > N, and let T : Rn -+ RN be linear. In this case the Jacobian of T, denoted again JT or J(T), is defined by
JT =
(4)
det (TT*)
and we have
Theorem 4 (Coarea formula for linear maps). Let T : Rn -+ RN be lin-
ear, and n > N. Then for any Lebesgue measurable set A C Rn the mapping y E RN --: 7-ln-N(A n T-1(y)) is Lebesgue measurable and we have
fNn_N(AnT_1(y))dy = JTLn(A)
(5)
RN or equivalently
fn_N(AflT_1(y))dN(y) = f JT dx
(6) R
.
A
Proof.
If dimT(Rn) < N the claim is trivial. In fact for 1.tN-a.e. y E RN we have A n T-1(y) = 0 and consequently 7-ln-N(A n T-1(y)) = 0. Also, as T = SU* by the polar decomposition theorem, we have T(RI) = S(RN); thus dim S(RN) < N and hence JT = Idet S1 = 0. Suppose therefore dim T(R) = N. By the polar decomposition theorem we have
T = SU* where U : RN -4 Rn is orthogonal and S : RN --+ RN is symmetric; moreover det S = JT 0. By considering any orthogonal map V : Rn -+ Rn such that
1.1 Area and Coarea Formulas for Linear Maps
73
V*(X"...' x N,0,. . .,0) = U(x1,.. . ,xN) and denoting by P* : RN -> R' the linear inclusion
N) = (1
p*( 1
N
)
we also see that U* = PV, i.e., T = SPV. We now observe that for each y E RN P-1(y) as well as V-1P-1(y) and T-1(y) are the translate of the (n-N)-dimensional subspaces of 1Rn P-1(0), V-1P-1(0) and T-1(0), respectively. By Fubini's theorem we then infer that the functions of the variable y
,H--N(V
,Hn N(AnV-1P-1(y))
(A) n P-1(y))
fln N(AnT-1(y))
are Lebesgue measurable and
xn-N (V (A) n P-1(y)) dy
,C" '(A) = Ln (V (A)) RN
= fnn_N(Aflv_1P_1())dy RN
By changing variable z = Sy, the claims follow.
Notice that for n = N the area and coarea formula are just the same, and the Jacobian of T : ]Rn -* R' reduces simply to the determinant of the matrix associated to T (with respect to the standard basis of Rn). We conclude this subsection by stating a formula which expresses the Jacobian of a linear map T : Rn -> RN First assume n < N, and set I(n, N) :_ {A :_ (A1, ... , An)
I
Ai integers, 1 < Al < ... < An < N}
For each A E I (n, N) we denote by Sa the span of {eal , ... , e,\,,} and by Pa the projection of RN into Sa defined by 1
N) = (
al
Theorem 5 (Cauchy-Binet formula). Let T : Rn --> RN be a linear map (i) Ifn < N, then JT2 = (det Pa o T) 2 AEI(n,N)
(ii) Ifn > N, then
JT = JT. thus we can use (i) to compute JT.
;
74
2. Integer Multiplicity Rectifiable Currents
We postpone the proof of Theorem 5 till Sec. 2.2.1, here we only remark that,
analytically, it says that JT is the square root of the sum of the squares of the n x n-subdeterminants of the (N x n)-matrix associated to T, geometrically in view of Theorem 2, it says that the square of the 7f -measure of A equals the sum
of the squares of the ln-measures of the projections of A onto the coordinate planes, thus it may be regarded as a generalization of Pitagora's theorem to which it reduces if n = 1.
1.2 Area Formula for Lipschitz Maps In this subsection we shall generalize Theorem 2 in Sec. 2.1.1 to Lipschitz maps. Let f : Rn -* RN be a Lipschitz map. In Ch. 3 we shall see that
(A) f is almost everywhere differentiable in the classical sense with differential represented by Jacobi's matrix
af'
aft
Ofnfn
axl
Df(x) = afN
afN ..
ax'
Also the partial derivatives derivatives.
afn l
which exist a.e., agree with the distributional
(B) For every E > 0 there exist a closed set FE and a C'-map gE : 1Rn such that
f=gE,
Df=DgE
IAN
in FE
Gn (Rn \ FE) < E
II DgE IIo.,R, < cLip f for some constant c independent of E. Using these results we shall now prove
Theorem 1 (Area formula). Let f : Rn --+ R v be a Lipschitz map, n < N, and let A C Rn be a Lebesgue measurable set. Then (1)
f Jf(x)dx =
f 7H°(An f-I(y))d7-C(y) f(A)
A
where Jf (x) is the Jacobian of f (2)
In particular we have
Jf(x)
det(Df(x)*Df(x))
1.2 Area Formula for Lipschitz Maps
(f (A)) =
(3)
75
fJf(X)dx A
if f is injective on A. The function
y -i f°(An f-1(y)) = #{x I f(x) = y, x E A} =: N(f, A, y) is called the multiplicity function or the Banach indicatrix or the counting function of f over A. Notice that
f(A) = {y I N(f,A,y)
0}
Since a are Borel functions, being limits of continuous functions, the Jacobian J f (x) is a Borel function. Below we shall see that the multiplicity function and f (A) are Lebesgue-measurable. Therefore both integrals in (1) exist: Theorem 1 states equality of those integrals; in particular if one of them is finite, the other one is finite, too. A simple consequence of Theorem 1 will be
Theorem 2 (Change of variable formula). Let f : ][fin --> RN be Lipschitz, n < N, and let u : R" -* [0, +oc] be a Lebesgue measurable map. Then the map
y -' E u(x) xEf -1 (y)
is measurable and (4)
fu(s)Jf(x)dx Rn
f ( E u(x)) dx"(y) RN
xEf-1(y)
In particular, denoting by N(f, A, y) the counting function of f over A,
N(f, A, y) := #{x I f (x) = y, x E A }, for any H'-measurable function v : R' -+ [0, +oo] we have (5)
fv(f(x))Jf(x)dx = fv()N(f,A,y)dfln(y). A
Proof of Theorem 1. We split it into several steps. Step 1. f (A) is measurable. We may assume A bounded otherwise we write A as A = k Ak, Ak bounded, Ak C Ak+1, and we notice that f (A) = f (U Ak) _
f(Ak) k
76
2. Integer Multiplicity Rectifiable Currents
Let {Kh} be a non decreasing sequence of compact sets such that Ln(A) _ suphCn(Kh). Then
Gn(h Kh) = Ln(A)
h Kh C A.
,
Being f continuous, it follows that f (U Kh) = U f (Kh) is a Borel set. Now
f(A) = f(UKh) Uf(A\hKh), and, since f is Lipschitz, it has Lusin property (N)
7T(f(A\hKh)) < (Lipf)nCC(A\UKh) = 0; this shows that f (A) is measurable.
Step 2. H°(An f-1(y)) is H' -measurable and (Lip
f 1t°(An
f)n£72(A) .
RN
We construct a nondecreasing sequence {9k} of 7'ln-measurable functions converging a.e. to 7-l° (An f-1(y)); this shows that N°(An f -1(y)) is 7{n-measurable. Dividing the unit cube of Rn into 2n congruent cubes, iterating the process, and extending by periodicity such a subdivision, we consider a so-called dyadic decomposition of ]1V {Qi } with side Q2 = 2-k. Set now 00
E i=1
9k(Y)
(y)
As f (A n Qk) is 7-ln-measurable for all i, 9k is measurable for each k. Moreover we clearly have 9k (Y) < gk+1(y) Observe now 1. If f -1(y) n A = 0, then 9k(y) = 0 = 7-l°(f -1(y) n A) for all k. 2. If N° (An f -1(y)) < oo, then for k sufficiently large every point in f -1(y) n A belongs to exactly only one cube Qk, hence
9k (Y) =
7-l°(An f(y))
3. If 7-l° (A n f -1(y)) = +oo, we trivially have limk-.oo gk (y) = +00. In conclusion we find 9k (Y)
? 7-l°(An f-1(y))
for each y E RN.
Finally by the Monotone Convergence Theorem f 7-1°(A n f -1(y)) d7-ln (y)
= k i . f 9k (y)
d7-l"'
(y)
RN
RN
= k--.oo lim Tfn(f(AnQk)) < limsupj(Lip f)nfn(AnQk) k-oo
= (Lip f )n
Gn
(A)
1.2 Area Formula for Lipschitz Maps
77
Step 3. It suffices to prove (1) for Lipschitz maps which are also of class
C'. In
(B) above we infer
fact assuming (1) valid for Cl-maps and a
f J(x) dx =
J9(x) dx +
J
=
f
Jf (x) dx
A\Fe
FEnA
A
H°(FEnAn f(y))dxn(y)+ f
1
f (FEJnA)
Jf(x)dx.
A\FE
By the Monotone Convergence Theorem letting e -> 0 we then get (1), as .C (A\ FE) - 0.
Step 4. In view of Step 3 from now on we may assume f of class C'. Denote by B the Borel set, actually the open set
B := {x E JRn
I
Jf(x) > 0} .
We now claims: Let t > 1. There exists a sequence {Ej} of disjoint Borel sets such that
B = U E, ,
f Ej is invertible.
Moreover, there exist linear, symmetric, invertible maps TT ]E8n -+ Wn and Lipschitz invertible maps gj : Tj (Ej) --f f (Ej) with Lip gj < t, Lip gj- 1 < t such that :
fI E; = gj o TI
(6)
In particular (7)
t IdetTjI < Jf(x) < t' IdetTjl
b' x E Ef .
Of course we can split B as union of Borel sets Ej in such a way that f, Ej is invertible. Taking into account the continuity of D f we can also assume that the Ed's be small enough so that, setting
Tj := (Df(xj))* (Df(x5)) where xj is a point in Ej, the map gi := fl Ej 0 T7-1 is as close as we like to the identity. This proves the claim. We are now ready to conclude the proof. For that we distinguish two cases Such a claim could be proved directly for Lipschitz maps. But its proof is quite more delicate, compare Federer [226, 3.2.2], as in this case one cannot use the continuity
of Df.
78
2. Integer Multiplicity Rectifiable Currents
Jf(x) > 0}. Consider the dyadic decomposition {Qk} as in Step 2, and let t > 1. In view of Step 4 we write B = UjEj and we
Step 5. A C B := {x E R' define
I
F j :=AnE; nQk,
Gk(y) => Xf(Fk)(y)
ij
For all k the family {Fj} defines a partition of A, and the map Gk is H-measurable. Moreover X f(F j) = 1 iffy E f (Fkj), that is, if f (y) E F j, and f is invertible on F. It follows, compare Step 2,
Gk(y) T 7-l°(An f-1(y)) as k -> oo, and the Monotone Convergence Theorem yields
fGk(Y)d7-c(Y)
fflo(Aflf1(y))dflfl(y)
T
RN
RN
On the other hand
fGk(Y)dN(Y) =
j:xn(f(Fj)) 2,7
RN
and
xn(f(Fj))=xn(97(Tf(Fj)))
(Lip
9;)"xn(Tj(F
f
< tnG'(Tj(Fj))=tnIdetTj I < t2n
f
))
det7 dx
Jf (x) dx ,
F
if we take into account Theorem 2 in Sec. 2.1.1 and (7) above. Also
xn(Tj(Fj))
xn(9j 1(f(Fj)))
(I' 2p g3- 1)'i1
(f(Fkj))
< tnxn(f(Fj hence, similarly to the above,
xn(f(F )) ?
t-n71n(T
> t-2n
f
F k.
Therefore we conclude
=t-n IdetT;IGn(FZj)
f(.F ))
Jf(x) dx
.
1.2 Area Formula for Lipschitz Maps
t-2n
79
f J f (x) dx < fGk(Y)dn(Y) < t2n f Jf (x) dx A
RN
A
and the result follows first letting k --> oo, and then t -- 1. Step 6. Jf - 0 on A. In this case it suffices to check that
f fl°(An f-1(y)) dxn(y) = 0 RN
or equivalently that fn (f (A)) = 0. For E > 0 we define gE : A C Rn -> RN by (f(x),Ex)
gE(x)
and we factor f as f(x) = ir(gE(x))
where 7r(y, z) := y for y E ii /" and z E ][fin. Now we apply Step 5 to gE to infer
x'y(f (A)) =1-l'(-r(gE(A))) S (Lip 7r)n 7-i'(gE(A)) =
f
Jg. (x) dx .
A
Letting e - 0 we then get
xn(f(A)) <
fg0xx = 0 A
by the continuity of J9E (x) in x, as Jgo (x) = 0. To conclude the proof of the theorem it suffices now to split A as
A = {x E A
I
Jf(x) > 0} U {x E A I Jf(x) = 0}
and apply Step 5 and Step 6. Proof of Theorem 2. If u is the characteristic function of a measurable set (4) simply reduces to (1) as
Y XA (X) = 7-l°(An f-1(y)) x E,f -' (y)
Consequently (4) holds for all step functions, and therefore for all measurable functions u as they are monotone limits of step functions. In order to prove (5) it suffices to prove that u(x) := v(f (x)) is 7-la'-measurable. This follows from the fact that every Lipschitz map has Lusin property (N), therefore maps measurable sets into measurable sets. We shall now state a few important special cases of the area formula, using Cauchy-Binet formula
2. Integer Multiplicity Rectifiable Currents
80
1 Curves in ][8N. Let cp : JR -,. JRN be a smooth curve in RN. Then
Jw (t) = I0(t)I and bb
r[1(,p(a, b)) =
- oo < a < b < +oo ,
fdt a
if cp is a one-to-one parametrization. If f is not one-to-one the multiplicity of the covering of o((a, b)) comes into play. For example, if co (t) _ (cost, sin t) E 1[82, a = 0, and b = 2k-7r + b1i 0 < b1 < 21r, we have
k+1 if 0 < argy < bl
7-l°((a, b) n cp-1(y))
if b1 < arg y < 27r
k
2 Graphs of codimension one. Let u : I[Rn --+ II8 be a Lipschitz function and let P be a bounded open set in Rn. The graph of u over 12 is
Ju,sl = {(x, y) E Rn x R I y = u(x)I and can be regarded as the image of P under the injective map U : Rn --+ Rn x JR given by U(x) (x, u(x)) .
Then one easily computes
Ju(x) =
1 + DDu(x)
hence
xn(gu,.?) = rcn(u(P)) =
f+D2dx. S?
0 Parametric hypersurfaces. Let U : Rn --r Rn+1 be a Lipschitz, and one-toone into its image map. According to Cauchy-Binet formula Ju (x) = sum of the squares of all n x n-subdeterminants of Du(x)
.
Thus the area of the Lipschitz n-dimensional manifold u(Q), fl c l[8n, is given by
7L
=f R
r
n+1 i=1
a(ul
ui-1 ui+ ,...a,
,ufl+l) )21l/2 dx ,
11
1.2 Area Formula for Lipschitz Maps
81
Submanifolds in RN. Let M C RN be an n-dimensional embedded submanifold. Let A C M and let f : Q C Rn -+ M be a parametrization of A over 12, i.e., a smooth one-to-one map with f (.f2) = A. Denote by {g2j } the metric tensor 4
on M
Of gz;
of
axi - axj
and set
g := det (gij) Then we have
(Df)*Df = (gij) so that
volume of A in M = xn (A) =
J n
lg- dx .
5 Graphs of higher codimension. Let u : Rn --* II$N be a Lipschitz map and let f2 a bounded open set in Rn. The graph of u over .f2 is G,,, s-2
:_ {(x, y) E ,f2 X RN I y = u(x)}
and can be regarded as the image of S? under the injective map U : }R given by U(x) := (x, u(x)). Set JM(o) (Du) I
:= 1
and for k = I,-, n, n:= min(n, N) I M(k) (Du) 12
:= sum of the squares of the determinants of all k x k-submatrix of Du .
Applying Cauchy-Binet formula, compare e.g. (17) in Sec. 2.2.1, we then find n
E IM(k)(Du)j2 k=0
Consequently, n
I M(k) (Du) I2 dx S2
V
k=0
.
82
2. Integer Multiplicity Rectifiable Currents
1.3 Coarea Formula for Lipschitz Maps In this subsection we shall generalize Theorem 2 in Sec. 2.1.1 to Lipschitz maps, proving
Theorem 1 (Coarea formula). Let f : RI -+ RN be a Lipschitz map, n >_ N, and let A C Rn be a Lebesgue measurable set. Then
f Jf(x)dx = f Hn-N(An f-1(y))dfN(y)
(1)
A
IAN
where Jf(x) is the Jacobian of f
Jf(x)
(2)
detDf(x) (Df(x))*
A simple consequence of Theorem 1 will be
Theorem 2. Let f : Rn -> RN be a Lipschitz map, n > N, and let u : Rn -4 [0, +oo] be a Lebesgue measurable map. Then
f(x)Jf(x)dx
(3)
J IRN
,/
I
nd?-ln-N
dHN(y) J
f-1(y)
Before proving those results let us make a few remarks. As we saw in Sec. 2.1.2
Jf (x) is a Borel function, below we shall prove that y --> 7.In-N (An f -1(y)) is 7ln-measurable; therefore both integrals in (1) exist. Theorem 1 states equality of those integrals. Also, for every y E ]LAN f -1(y) is clearly a Borel set, thus f -1(y) is 7.ln-Nmeasurable. It is part of the claim in Theorem 2 that
fu
y --->
d7-ln-N
f -'(Y)
is 7-lN-measurable, too.
Finally notice that Theorem 1 and Theorem 2 may be regarded as a curvilinear generalization of Fubini's theorem. Proof of Theorem 1. We split it into several steps. Step 1. f (A) is 7-lN-measurable. This can be proved exactly as in Step 1 of the proof of Theorem 1 in Sec. 2.1.2. Step 2. We have (4)
fn_N(Anf_1(y))dy RN
<
Wn- Nn WN
(Lip f)Nxn(A)
1.3 Coarea Formula for Lipschitz Maps
83
For each j = 1, 2, ... we choose closed balls {Bi } such that A C UO°1B2 , diamBi < 1/j and >°_1 JB?1 < JAI + 1/j. Then we define Wn -N
(y)
C diam Bi
)fl_N Xf(B,)(y)
2
measurable, also for all y E RN we have
By Step 1 the functions 9- 1'
00
,Hn-N (A n f -1(y)) <
gi (y) i=1
Using Fatou's lemma we therefore get
f
I*
if-N(An f-1(y))dy= 1 limb ln-N(Anf-1(y))dy RN
N RN
<
inf f lim J~
00
00
gj3 dy < lim inf
RN
r
gi dy
J
RN
°° =liminfw_N 00
(diamBi )n_N ,HN 2
7= 1
(f (Bf ))
and, using the isodiametric inequality, compare Sec. 1.1.3, we infer
Hn-N(An f-1(y)) dy < RN 00
<1lm inf
Wn_N
diam Bi )
< In-NWN (Lip f)N
In
n-N
2
WN
diam f (Bi )
N
2
00
x n(Bi) i=1
Wn-NWN
Wn
(Iiip
f)N'
n(A)
Step 3. The mapping y -> xn-N(An f-1(y)) is .Cn-measurable, consequently
f Rn-N(An f-1(y))dy < In-NWN (Lip f)Nxn(A) Wn
(5)
RN
Of course (5) follows from (4) once we establish the measurability of the mapping y --r
xn-N(An f-1(y)).
(i)
If A is a compact or an open set, then y -->
map.
In-N (A
n f -1(y)) is a Borel
84
2. Integer Multiplicity Rectifiable Currents
First assume A compact. Fix t > 0, and for each positive integer i, consider the set Ui of all y E RN such that there exist finitely many open sets S1,. .. , SP such that
6Si
Anf -1(y) C
1
diam Sj
,
i
j=1 Wn-N
C
diam Sj 2
) n-N
1
j=1
We now claim that the sets U2 are open sets and 00
Hn-N (An f-1(y)) < t} =
(6)
flu. i=1
This clearly proves (i) in case A is compact. e
To prove that Ui is open, assume y E Ui, A n f -1(y) c U Sj. Then since j=1
f is continuous and A is compact e
Anf-1(z) C U Sj j=1
for all z sufficiently close to y. Let us prove (6). If 7-ln-N (A n f
(y)) < t, then for each 8 > 0
xb-N (An f-1(y))
< t.
Given i, choose 8 E (0, 1/i). Then there exist open sets Sj such that cc
Anf-1(y) C U Sj
diam S. < 8 < 1
j=1
DC
E Wn-N j=1
n-N
( diam Sj 2
<
t+1.
J
Since An f -1(y) is compact, a finite collection {S,. .. , SP} covers An f -1(y), and hence y E U. Thus 00
{y 17-Ln-N(An f-1(y)) 5 t} C n Ui . i=1
i=1
On the other hand, if y E n Ui, then for each i
fl?2 N(Anf-1(y)) < t+a
1.3 Coarea Formula for Lipschitz Maps
85
This proves (6) and therefore (i) when A is
hence 7-ln-1 (A n f-1(y)) compact.
Finally, suppose A is open. Then there exist compact sets K, C K2 C ... C A such that A = U°_1Ki. Thus, for each y E RN, 7..Cn-N(An f-1(y))
= lim 7-ln-N(Kin f-1(y)) i- 00
hence the mapping y --> 7-ln-N (A n f -1(y)) is again Borel measurable. (ii) If A is a Lebesgue measurable set, then y -+ 7-ln-N(A n f-1(y)) is Lnmeasurable. By writing A as union of an increasing sequence of bounded Lnmeasurable sets, we can reduce to the case in which Ln(A) < oc. Then there exist open sets V, D V2 D ... D A such that
lim £(V j\ A) = 0 00
2
Ln (V1) < oo
.
Now
xn-N(Vi n f-1(y))
<7ln N(An f-1(y)) +x"-N((V \ A) n.f-1(y))
and by Step 2
limsup 2-+00
J RN
j7-l"-N(V n f-1(y))
f-1(y)) I dxN(y)
< limsup wn-NWN (Lip f)n7-ln(V \A) = 0 i-00 Wn
.
Consequently
xn-N(V n
f-1(y)) -'
Hn-N(An f-1(y))
LN-a.e. Therefore (ii) follows from (i).2 Step 4. It suffices to prove (1) for Lipschitz maps which are also of class C'. In fact assuming (1) for C' maps and applying (B) in the previous subsection, we infer
f Jf(x) dx = FE nA
and passing to the limit for e
fn_N(FflAflf_1(y))dn(y), RN
--0
f J(x) dx = fn_N(uFflAflf_1(y))dn(y) 2 Notice that, since every measurable set can be exhausted apart from a zero-set by an increasing sequence of compact sets, one could slightly shorten the proof by using only Step 2 and the fact that y -. ?fin-N(An f-1(y)) is a Borel map, if A is compact.
86
2. Integer Multiplicity Rectifiable Currents
The result then follows since by Step 2 1
Ixn-N(EFenAnf-1(y))-xn-N(An f(y)) dxn(y) = 0
RN
as fn-N(A\(UF nA))=0. Step 5. In view of Step 4 from now on we may assume f of class C1. Denote by B the Borel set, actually the open set,
B := {x E R'
Jf (X) > 0}
and assume that A C B. The idea is to apply the coarea formula for linear maps. In order to do that it is convenient to introduce for any A E 1(n-N, n) the map ha : Rn --. RN x Rn-N given by
ha(x) := (f(x),PA(x)) where PA(x) is the orthogonal projection P'\ (x 1,. . ., xn) := (x Al ,...,x an).,
in particular f (x) = 7r o ha(x), if 7r : RN x R71-N -.. RN is the orthogonal projection ir(y, z) = y. Set also
Aa :_ {x E A and we observe that
A=
I
det Dha # 0}
U
A,
AEI(n-N,n)
so we may as well assume that A = Aa for some A E I(n - N, n). Accordingly, we write h instead of ha. By the local invertibility theorem, for any t > 1, we can find disjoint Borel sets {Dk} and linear isomorphisms {Sk}, Sk : ]Rn -> ]R', such that (i) ,Cn (A \ Uk Dk) = 0, 1 (ii)
hl Dk is one-to-one,
iii)
the maps 9k (x) := S 1 o h : Dk --i Rn are one-to-one with Lip 9k <_ t
Lip gk' < t ,
compare Step 4 of the proof of the area formula in Sec. 2.1.2. Since
DflDk = D(7roh) = lroSkoD9k we have
Jf = det((Df)*.Df)=det((Dgk)*.(lroSk)*.(7roSk).D9) 2
J7roSk (detDgk)
2
2 2 = J7roSk J9k
1.3 Coarea Formula for Lipschitz Maps
87
h% (f -'(y))
h;,
y
Fig. 2.1. Coarea: the map ha.
therefore by the area formula, setting
Gk := An Dk
fi
we get (7)
dx = JnoSk t (9k(Gk))
Gk
On the other hand, using the rough estimate for the area we have for any y E
tN-nxn-N(9k(Gk n f-1(y))) < ln'-N(Gk n f-1(y))
(8)
<
tn-Nxn-N
(9k(Gk n f
being Lip gk, Lip g- 1 < t. Integrating (8) in y, taking into account the coarea formula for the linear maps Tr o Sk, compare Theorem 4 in Sec. 2.1.1, using (7) and noting that by construction
9k(Gk n .f-1(y)) = 9k(Gk) n (r o'Sk)-1(y) we then get
tN-n fJf dx = tN-n JlrOSk Hn(9k(Gk)) Gk
tN-n f xn-N(9k(Gk) n (7r o Sk)-1(y)) dy RN
f xn-N
RN
(Gk n .f -1(y)) dy
2. Integer Multiplicity Rectifiable Currents
88
< to-N J hn-N(9k(Gk) n Or o SO -'(Y)) dy RN
to-N f Jf dx. Gk
Now since 7-ln (A
\ Uk 1 Gk) = 0 we can sum over k, use Step 3, and let t -- 1
to conclude that (1) holds in this case.
Step 6. It remains to consider the case in which A C {Jf = 0}, as in the general case we can write A = Al U A2 with Al C {Jf > 0} and A2 C {Jf = 0}. Fix e > 0 and define g : IR" x 1 [ 8 N
f (x) + Ey
g(x, y)
I[8N
?:W XRN _.RN , ?(x,y):=y.
,
x E IR", Y E IR v
Then Dg = (D f, EI), therefore eN < J9 < Ce. Also we observe that for all w
f 'H"-N(An f-1(y))dy= f fn-N(An f-1(y-Ew))dy RN
RN
hence
f ,Hn-N(An f-1(y))dy= RN
N B(0,1) f
f-1(y-ew))dy. RN
A x B(0,1) we have
We now claim that for y E RN, W E RN and B
Bng-1(y)n?-1(w)=S (Aof-1(y-ew))x{w} ifwEB(0,1). In fact (x, z) E B n g-1(y) n F-1(w) if and only if
xEA,
zEB(0,1),
f(x)+ez=y,
z=w
i.e., if and only if
xEA,
z=wEB(0,1),
f(x)=y-ez
or, if and only if
w E B(0,1) ,
(x, z) E (A n f -1 (y - Ew)) x {w}
.
Thus we can continue the computation above using Fubini's theorem and Step 5
1.3 Coarea Formula for Lipschitz Maps
89
f xn-N (A n f-1(y)) dy = IRN
=N
f
dw
f xn-N (B n g-1(y) n Fr-1(w)) dy RN
]RN
Nf
7-ln(Bng-1(y))dy=
1
RN
f J9dxdz B
< 7-ln (A) sup J9 < 7-LT(A) e B
and, for e -> 0, we obtain
I 7-ln-N (An f-1(y)) dy
=0=
f
Jf dx .
A
RN
Proof of Theorem 2. It is exactly the same as the proof of Theorem 2 in Sec. 2.1.2.
E 1 C1-Sard type theorem. A consequence of Theorem 1 is that
7{n-N({Jf = 0} o f-1(y)) = 0
for 7-ln-a.e. Y E
On the other hand if f is of class C', x E A, A open, is such that Jf (x) # 0, then as consequence of the implicit function theorem we can find a neighbourhood V of x such that V n f -1(y) is an (n - N)-dimensional C1-submanifold. Therefore we can state
Corollary 1 (C'-Sard type theorem). Suppose f : A C Rn -> RN is of class C1, where A is open. Then for 7-lN-a.e. y E f (A) we can decompose f -1(y) as
f -I(y) = (f -1(y) \ C) U (f -1(y) n C) where
C := {x E A
I
Jf (x) = 0} = {x
I
rank D f (x) < N}
f-1(y) \ C is an (n - N) -dimensional submanifold and
xn-N (C n f-1(y)) = 0 . Recall that Sard theorem asserts that, if f is of class Cn-N+1, then f -1(y)nC = 0
for HN-a.e. y E RN, so that f -1(y) is itself an (n - N)-dimensional
Cn-N+1
submanifold for 7-HN-a.e. y E RN.
0 Choosing f (x) := lxI in Theorem 2, we infer at once the following well-known formula for polar coordinates
90
2. Integer Multiplicity Rectifiable Currents
J0fdr 0
+00
u dx = J dr J f-1(r)
-oo
II2n
f
u d7{"i-1 = 0
u dHn-1
8B((0,r)
In particular we see that for 7-ll-a.e.. r
d(
f
B(0,r)
udx) = /
In the codimension one case, f :
1R'
r ud7-in-1
J 8B(O,r)
--> 1i8, as Jf = P f f 1, Theorem 1 yields at
once
f IDfIdx =
Jfl1({f =t})dt. -00
Rn
More generally, we have
f IDfldx = ffl-1({ft})dt
{f>t}
t
so that for 7-ll-a.e. t
df
dt
IDfI dx = -7ln 1/{f
t})
{f>t}
1.4 Rectifiable Sets and the Structure Theorem The notion of rectifiable sets we shall discuss in this section turns out to be the appropriate generalization of n-dimensional Cl-submanifolds. Roughly, nrectifiable sets are characterized by the equivalent properties of being countable union of measurable pieces of n-dimensional C1-submanifolds or of possessing approximate tangent space ?-l'-almost everywhere.
Definition 1. A set M C Rn+N is said to be countably n-rectifiable if there exist n-dimensional embedded submanifolds N1iN2,... and No C Rn+N with 7-ln(N0) = 0 such that cc
(1)
M cArouUNk. k=1
Let M be a countably n-rectifiable set which is also 'H-measurable. Then clearly we can write M as disjoint union CO
(2)
M = .MO U U Mk k=1
1.4 Rectifiable Sets and the Structure Theorem
91
where H '(MO) = 0, and each Mk is a Borel subset of an n-dimensional submanifold. If moreover W (M) < oo, we then have 00
,Hn(M) = E7-ln(1Vtk)
(3)
.
k=1
We may also and do assume that at each point x E Mk we have (compare Theorem 6 in Sec. 1.1.5)
O'(M \ Mk, x) = 0
B"(Mk, x) = 1,
.
Definition 2. A countably n-rectifiable set M is said to be n-rectifiable if and only if M is xn-measurable and fn (M) < 00, or equivalently, if it can be written apart for a null 70-set as disjoint union of Borel subsets of n-dimensional C'-submanifolds with finite xn measure. If M is countably ?in-measurable, then M is countably n-rectifiable if and only if it is a countable union of n-rectifiable sets. Notice also that an n-rectifiable
set apart for sets of small 7-ln-measure is a finite union of Borel pieces of C'submanifolds. Despite this nice structure however n-rectifiable sets can be everywhere dense.
Consider for instance a dense set {xk} in Rn+N and denote by Bn(xk, rk) the n-dimensional ball in Rn+N of radius rk and center xk. Choose rk so that
Lark <
00 '
k=1
then clearly 00
M := U Bn(xk,rk) k=1
is a dense n-rectifiable set in Rn+N Taking into account Lusin type property (B) of Sec. 2.1.2 for Lipschitz maps, and also the extension properties of Lipschitz and CI-maps, compare Ch. 3, it is not difficult to prove also that countably rectifiable sets are characterized by the property of being countable union of Lipschitz images of n-dimensional sets
Proposition 1. A set M C R'+N is countably n-rectifiable if and only if M C Ao U (k=1 fk(Ak)) where ?in(Ao) = 0 and fk : Ak -} IIBn+N are Lipschitz maps, AkCR W.
We shall now prove that rectifiable sets are characterized by their tangential properties. First let us define the approximate tangent space in terms of a blow-up procedure
2. Integer Multiplicity Rectifiable Currents
92
Definition 3. Let M C R'+N be an 7-I' -measurable set such that 7-ln (M nK) < 0o for every compact K C IIBn+N. We say that an n-dimensional subspace P of II$n+N is the approximate tangent space for M at x E Rn+N if slim
(4)
f 77T,A(M)
where ix,),(y) :=
Vf
f(y)dxn(y) = f f(y)d7-ln(y) 2
ECc(Rn+N)
P
for x, y E Rn+N, A > 0.
Of course P is unique, if it exists, and will be denoted by TIM. By definition it is characterized by the property
Hn L rlx,a(M) _ 7-ln L P
(5)
.
The condition that M has locally finite H'-measure can be removed; and it is convenient to do it. It suffices to suppose the existence of a positive locally W -integrable function 0 on M. This is equivalent to the requirement that M can be expressed as the countable union of lu-measurable sets with locally finite fn-measure. We then set
Definition 4. Let M C Rn+N be an 7dn-measurable set and let 0 be a positive locally'H' -integrable function on M. We say that an n-dimensional subspace P of Rn+N is the approximate tangent space for M at x E Rn+N with respect to 0 if
f
(6)l
rl.,a(M
f (y)0(x + Ay)d7-Cn(y) = 0(x) f
f(y) dxn(y) df E C.(i18n+N
P
Actually P does not depend on 0 in the sense that the approximate tangent spaces of M with respect to two different positive fl''-integrable functions 0 and
0 coincide fn-a.e. in M. In fact if f1 := H. L 8 and ME := {x E M I 9(x) > e} we have 7-ln (.ME n K) < oo for every compact K C Rn+N and, by Theorem 6 in
Sec. 1.1.5, 0* (p, M \ Me) = 0 for Hn-a.e. X E M. Hence for 7dn-a.e. x E ME the approximate space for M with respect to 8 coincide with TxME as defined in Definition 3 if the latter exists. Let M C Rn+N be an 7-ln-measurable and countably n-rectifiable set. Then we can write M as the disjoint union
M = Mo U U Mk k=1
where 7-C'(Mo) = 0, Mk is l'-measurable and Mk C Mk, where Nk is
an
embedded n-dimensional C'-submanifold of Let A := 07-Cn L M where B is any positive locally R'-integrable function on M, and assume without loss of generality that 7f (Nk) < oo V k. Since Mk is C1 by the differentiation theorem IIBn+N.
we have
1.4 Rectifiable Sets and the Structure Theorem Nl(Akrkn B(B(xx, p
Bn(P,
k,x) = P0 Hn
fl
P)))
= 9(x)
93
7ln a.e. X E Mk
while, by Theorem 6 in Sec. 1.1.5,
9"n(µ,Nk \.Mk,x) = 0 7-l''-a.e. x E Mk 9*n(lc, M \ Mk, x) = 0 7-ln'-a.e. x E Mk From this we easily infer that for a.e. x E Mk and for every f E CC(1Rn+N)
f
f (y) 0(x + Ay) dx"(y)
0
f(y)9(x+Ay)d'h'(y)
0
Thr,A(M\Mk)
f
Th,,A(Nk\Mk)
as A - 0, that is, by Lebesgue theorem,
o
f = 9(x) f
f f (y) 9(x + Ay) dy = 77r,A(M)
o
f (y) 9(x + Ay) dy
77 ,,A(Nk)
TxNk
f (y) dy
as Nk is of class C1. This shows that every 71n-measurable and countably nrectifiable set M admits 7-1' -a. e. on M approximate tangent plane with respect to any positive locally 7-ln-integrable function 0 on M, moreover
TxM = TxNk
(7)
7-ln-a.e. x E Mk
.
The converse is also true and we can state
Theorem 1. Suppose M is 1-ln-measurable. Then M is countably n-rectifiable if and only if there is a positive locally 7-ln-integrable function 0 on M with respect to which the approximate tangent space exists for 7-ln-a.e. x E M. Theorem 1 characterizes countably n-rectifiable sets as sets with good tangential properties: an approximate tangent plane exists at 7-fn a.e. point. We remark that countably n-rectifiable sets can be "fractured". We need to prove the "if' part. For future purposes we shall derive it from a slightly more general result on the rectifiability of Radon measures, applied to the measure µ := 97 C L M. Definition 4 uses the measure 7-ln L 0, but we can easily extend it to any Radon measure
Definition 5. Let y be a Radon measure on ]Rn+N, and for x c Rn+N and A > 0 let flx,a be the measure given by
94
2. Integer Multiplicity Rectifiable Currents
µ.,a(A) = )-nµ(x + AA)
(8)
.
We say that the n-dimensional subspace P C Rn+N is the approximate tangent space for µ at x with multiplicity 9(x) E (0, +oo) if
a o f f (y) d,4.,,\ (y) = 0(x) J f (y) dxn (y)
(9)
P
Clearly 0 E P, and P and O(x) are unique, if they exist. For any Radon measure µ we denote by M. the set of points in Rn+N for which there exists the approximate tangent space for jL at x with multiplicity O(x) E (0, oc).
we have
Then
Theorem 2 (Rectifiability theorem for
Radon measures). Assume that µ has an approximate tangent plane for y-a. e. x E IRn+N with multiplicity 0(x) E (0, oc), i.e.
,U(rn\Mµ)=0. Then M. is countably n-rectifiable, 9 is H -measurable, and p = O7-ln L Mµ,
Proof. Choose any ball B(0, R). For any x E B(0, R) and any bounded Borel
set A we have
(N- L (IRn+N \ B(0, R)))x,a (A) = 0
provided A is sufficiently small. From this we easily deduce that
Mµ n B(0, R) _ .M/,
L B(O,R)
Therefore replacing u by p L B(0, R) we may and do assume that p(Rn+N) < co. Given any N-dimensional subspace 17 C R,,+N and any a E (0,1) let ply denote the orthogonal projection of IRn+N onto 17 and Ca (I7, x) denote the cone
0(17,x) = {yE1R.'n+N I
Ipir(y -
x)I > aly-xI}.
1.4 Rectifiable Sets and the Structure Theorem
95
having "axis" H. Also, recalling that there is a bijection between N-dimensional subspaces and projectors, define the distance between two N-dimensional subspaces 17, IF as the norm of the difference of their projectors dist (17,17') = sup I P17 (x) - P11' (x) I 111=1
This way we obtain a compact metric space. For s e N choose now 0, > 0 and a Borel subset F C 1[Bn+N such that µ(Rn+N \ F)
(10)
µ(Rn+N)
s
and such that for each x E F, µ has an approximate tangent space PP at x with multiplicity 9(x) > 9 . In particular, we therefore have lim P-0
(11)
µ(B(x, p))
> B3
lim
µ(C1/2 (Pj , x) n B(x, p))
P-.0
WnP n
Wn pn
= 0.
Fori=1,2,... andx EFdefine µ(B(x, p))
inf
O
4i(x)
Wnpn
µ(C1 /
sup
,
x) n B(x , P))
WnPn
O
Then
lim fi(x) > 9,
(12)
i- Oc
,
lim qi(x) = 0
VxEF
2-+00
and by Egoroff's Theorem we can choose a µ-measurable set E C F with µ(F \ E) < s /,r,(l[Bn+N)
(13)
and with (12) holding uniformly in E. Therefore for each e > 0 we find S > 0 such that (14)µ(B(x, p)) > 0, - e, µ(C1/2(Py , x) n B(x, p)) WnPn
WnPn
<eifxEE, 0
According to the compactness property of the metric space of N-planes, we now choose a finite number of N-dimensional planes 171 i ... , Hk in Rn+N such that for each N-dimensional plane 17 there is a j E {1, ... , k} such that dist (17, 11j) < 1/16, and let El, ... , Ek be the subsets of E defined by
Ej := {x E E I d(IIj, Pte) < 1/16}
.
Clearly E = Uj=1Ej. We now claim that for a = 0,/16n and 6 so that (14) holds we have
96
(15)
2. Integer Multiplicity Rectifiable Currents
C3/4(Ilj,x) nE; n BS12(x) = {x}
V x E Ej, j = 1,...,k
.
In fact otherwise we find x E Ej and Y E C3/4(J1 , x) n E; n aB(x, p) for some 0 < p < 6/2. From (14) we then have u(C112(PP ,x) nB(x,2p)) < ewn,(2p)' and, since
we have B(y, p/8) C C112 (Py , x) n B(x, 2p) and, again by (14) A(C1/2 (P2 , x) n B(x, 2p)) >_ A(B(y, pl8)) ? (BS - E)wn ( 8)
.
This gives 63 < (1 + 8')e, a contradiction, since E = O /16n. From (15) we also see, taking into account the extension theorem for Lipschitz maps (compare Theorem 1 in Sec. 3.1.3) that for any fixed x0 E Ej Ej n B(xo, S/2) C q(graph f )
(16)
where q is an orthogonal transformation of Rn+N with q(17;) RN and where f = (f 1 .. f N) is Lipschitz. In fact we may assume xo = (0, 0) E Rn X RN, 113 = {0} x RN. Let A denote the projection of Ei nB(xo, 6/4) on Rn x {0}, then for any (x, 0) E A we claim there is a unique (x, y) E Ej n B((0, 0), 6/2). In fact suppose that (xi, yi) E Ej n B((0, 0), S/2), i = 1, 2, then from (15) we infer (x2, y2) 0 C314 (11j, (x1, y1)), hence 3
Y2-Y11 < 4 (Ix2 - X 1
+ Iy2 - y, 12)1/2 <
3 (Ix2
4
- x1I + Iy2 - yll)
i.e.
Iy2-Y1I < 3Ix1-x21. This yields at once uniqueness, and shows that the map
(x, 0) E A -> (x, y) E E; n B((0, 0), 5/2) is Lipschitz. Extending such a map to a Lipschitz map defined on R x {0} D A we then easily get (16). From the above we conclude that E is an n-rectifiable set and
n+N \ E) < 2 /l(R n+N) This shows that
1.4 Rectifiable Sets and the Structure Theorem
97
00
MI, = MoU U.Mk k=1
with µ(Mo) = 0 and Mk C JVk, dVk being n-dimensional Cl-submanifolds of 1Rn+N Since for any x in MN, the existence of the approximate tangent plane implies that (17)
0(x) =
lo
p))
=: Bn(Il,x)
we have
for p - a.e. x E MN, . 0' (µ, x) > 0 Therefore on account of Theorem 5 in Sec. 1.1.5, we infer 7-C(Mo) = 0, hence Mµ is 7ln-measurable and countably n-rectifiable. As B*n(ii, x) is a Borel function, we also infer from (17) that 0(x) is K"-measurable on M. Finally (17) yields
µ(B(x,p))
lim
PoHn(Nk rB(x, p))
6(x)
which in turn implies µ = O7-Cn L M, by the differentiation theorem of measures in Sec. 1.1.5.
We conclude this subsection by stating without proof the important and difficult Besicovitch-Federer structure theorem for purely unrectifiable sets. Let A be an arbitrary set in 1Rn+N which can be written as countable union of sets of finite Nn-measure. Consider the collection of all countably n-rectifiable subsets of A ordered by inclusion. Since the union of families of countably nrectifiable sets in A is again a countably n-rectifiable set, by Hausdorff maximal principle there exists a maximal element R, and we have
A = RUP where R is countably n-rectifiable and P contains no countably n-rectifiable subset of positive 7-ln-measure.
Definition 6. A set P which contains no countably rectifiable subset of positive 7C'-measure is called purely n-unrectifiable.
Purely unrectifiable sets Q of Rn+N are characterized by the fact that they have 7-ln-null projection via almost all orthogonal projection onto n-dimensional subspaces of Rn+N Of course also purely unrectifiable sets Q can be written as countable union of sets of finite 7{'-measure. An example of a purely 1-unrectifiable set in R2 is given by C x C where C is the Cantor set in Sec. 1.1.3, of dimension 1/2. In fact, as we have seen, in this case,
0 < 7-ll(CxC) < oo. Therefore if M C C x C is 1-rectifiable we should have either 7{1(7r(M)) > 0 or 7{1(F(M)) > 0, being it and the projections of ]R x IR into the first and second factor. A contradiction as
98
2. Integer Multiplicity Rectifiable Currents
7 '(ir(M)) <_ x1(C) = 0 ,
x1(ir(M)) <_ x1(C)
= 0.
Theorem 3 (Besicovitch-Federer structure theorem). Let Q be a purely n-unrectifiable subset of ]n+N with 00
Q = UQk,
(Th(Qk) < 00
Vk.
k=1
Then 7-C (p(Q)) = 0 for almost (in the sense of Haar measure for 0(n + N, n)) all orthogonal projections of PJ +N onto n-dimensional subspaces of Rn+N, P E
0(n+N,n).
Notice that only the purely n-unrectifiable subsets could possibly have the null projection property stated in Theorem 3. An immediate consequence of Theorem 3 is
Theorem 4 (Rectifiability theorem). Let A be an arbitrary subset of Rn+N which can be written as countable union cc
A = U Ak
with 7-ln(Ak) < 00
Vk
.
k=1
Suppose that every subset B C A with positive 'H' -measure has the property that
fn(p(B)) > 0 for a set p E 0(n + N, n) of positive Haar measure. Then A is countably n-rectifiable.
Of course the 7-lk-density at lk-a.e. point x of a k-rectifiable set is 1 while it is zero o*k(Hk,.M,x) = 1
1
for Hk-a.e. x E M
0
for Hk-a.e. x 0 M
A natural question is to ask for which sets A of finite 71k-measure k non necessarily integer, there exists for 7-lk-a.e. x E A ok (Hk,
A, x) = ok (Hk, A, x) = B*k (7-lk A, x)
The following theorem gives a complete characterization of such sets
Theorem 5 (Preiss). Let A C Rn, with fk(A) < oo. Then Bk(7-lk, A, x)
exists for fk-a.e. x E A
if and only if k is integer and A is k-rectifiable.
1.5 The General Area and Coarea Formulas
99
1.5 The General Area and Coarea Formulas In this last subsection we would like to state the area and coarea formulas for
Lipschitz maps f : M -* RN, where M is an lu'-measurable and countably n-rectifiable set.
Recall that an n-dimensional C', r > 1, submanifold of 1Rm, m > n, is a set M c RI such that for any y c M there are open sets U and V in Rm with y E U, 0 E V and a C""-diffeomorphism 0: U-* V with 0(y) = 0 and
o(M n U) = V n ]Rn where 1Rn = {(xl, ... , xm) E Rm local representations of M
:VnR' ->Rm, (indeed
I
xn+1 =
... = x' = 0}. In this case we have
v)(VnRT)=MnU,
0(0)=y,
_ ¢ VrR-) such that
ax
(o), ... , ax (°)
are linearly independent vectors in Rm which form a basis of the tangent space
TyM of M at y. The tangent space TM of M at y is the subspace of iRm consisting of those -r E JRm such that r = 'y(0) for some C1 curve y M, y(0) = Y.
Let f : M -> RN be a Cl function on M. The directional derivative DTf E RN in Y, T E TyM is defined by
DTf
dtf('Y(t))It=o
for any Cl curve y : (-1,1) -> M with y(0) = y, ,(o) = T, and it turns out to be independent of the particular curve y. The induced linear map dfy : TM -> RN defined by
dfy(T) := DTf
,
T E TyM
yields the tangent map. For scalar functions of class C', f : M -- ]R, the directional derivatives can be represented in terms of the gradient V' f (y) defined as the element in TyM such that DTf `drETTM. If r1, ... , r,,, is an orthonormal basis of TM, the gradient is also given by the formula n
VMf(y) = E(DTkf)rk k=1
If f is a C1 function defined in a neighbourhood U of M with f = f on M, the gradient VM f is related to the ordinary gradient of f by
100
2. Integer Multiplicity Rectifiable Currents
V M.f (y) = P(V f (y)) where P : Rm --i TyM is the orthogonal projection onto TyM.
The divergence of a vector field X = (X',. .. , X"n) : M similarly
ii
is defined
rn
> ek (VMXc)
div MX
k=1
where e 1, ... , e,n is the standard basis of R. It is easily seen that n
divMX =
7-k
k=1
where r 1 , . .. , Tn
is any orthonormal basis for T. M. We have
Theorem 1 (The divergence theorem). Let M be such that the closure M is a smooth compact manifold with boundary aM = M \ M. If Xy E TyM for each y E M, i.e., X iss a tangent vector field, then
J M
div MX
dW = - JX. rl
dHn-1
am
where rl is the inward pointing unit co-normal of aM, i.e., 1771 = 1, rl is tangent to M, 77 is normal to aM and points into M. In general, i.e., without any assumption on 3M, we have
Jd1VMX = 0 M
for any tangent vector field X such that spt X n M CC M. The mean curvature of a C2 submanifold M C R'n is expressed in terms of its second fundamental form. For y E M we denote by Tl...... n an orthonormal basis of TyM and by vi, ... , V _n any orthonormal basis of (TyM)-L, the orthogonal complement of .,y, that is, vl, ... , v,,,,_n is a basis of the normal space of M at y. The second fundamental form of M at y is the bilinear form
By : TM x TyM --* (TyM)' defined by
m-n B. (-r, rl)
(rl - DTva(y)) va(y) a=1
It is not difficult to infer
Proposition 1. We have
1.5 The General Area and Coarea Formulas (i) Let -y
101
: (-1, 1) -> M be a C2 curve satisfying -y(O) = y, 'y(0) = r, TI = 1.
Then
By(rr,r) = (y(0))1 where v-1- denotes the orthogonal projection of v onto (TM)J-.
: U -+ M is a C2 map
(ii) Let U be a neighbourhood of (0, 0) E R2 and let
with 0(0, 0) = y and
a (0,0) = r'
a
Then
r
By (r, ) =
(
(°'0) - ''
a2 atlat2
(0, 0)
11 I
.
The vector (%y(0))± in (i) is the norm al component of the curvature of -y; in other words By (T, r) measures the normal curvature of M in the direction r. From (ii) it follows that By is a symmetric bilinear form with values in (TyM)1. The mean curvature vector field H of M at y is the trace of By
H(y)
By ("rk, "rk) E (TyM)1 k=1
or equivalently
m-n n
H(y) :=
E E(rk - DTkva(y)) va(y) - U=1 k=1
so that
H :_ -
m-n (div Mva) va a=1
In terms of the mean curvature vector H one can write the divergence theorem for generic vector fields X not necessarily tangent to M. Theorem 2. Let M be as in Theorem 1. Let X : M -> R m be any smooth vector field. Decompose X into its tangent and normal part m-n
X = XT+X1,
X1 = a=1
Then we ha ve
divX1 = and
Xy
f div MX d7-l" = - f X . H d7-C -
M
M
aM
d7ln-1
102
2. Integer Multiplicity Rectifiable Currents
Without any assumption on aM and provided sptX n M cc M, we have
fdivMXd7e2 = _ f X H d7-ln . M
M
Consider now an 7-1n-measurable and countably n-rectifiable set M C R", n < m. As we have seen M can be written as
M = MODUMk k=1
where H (Mo) = 0, Mk is ?in-measurable for all k, and each .Mk lies in some n-dimensional C' submanifold Nk, Mk C Nk. Suppose f : U - R be a locally Lipschitz map from a neighbourhood of M into R. Of course for any k fjvk is also locally Lipschitz, hence 'H'-a. e. differentiable (compare (A) of Sec. 2.1.2 and is differentiable Ch. 3), also it is easy to check that at all points x where affine space L := x + T 14 at the point x, we have fj L is differentiable on the we can set VNk f (x). Thus and grad f L (x) =
Definition 1. The gradient off relative to M is defined for 7fn a.e. x E Mk by
VMf(x) := VNxf(x)
.
We explicitly note that V--"f(x) exists and belongs to T,,M for 7fn-a.e. x E M, and, up to a set of 7{n-measure zero, is independent of the particular decomposition M = UkMk used in the definition. This is easily seen as TxNk = TM for 7ln a.e. x E Mk and T,M is independent of the particular decomposition. For 7-ln-a.e. point in M for which TxM and VM f (x) exist we can then define
the linear map
d'Mfx:TTM- R
by
r ETxM .
dMfx(T) = If f : M --> RN is a Lipschitz map we then set N
dMfx(r) := E(T . VMfe) ee s=1
where e1,.. . , eN is the standard basis in RN, and we define the Jacobian JM (x) for 1f -a.e. x E M as the Jacobian of the linear map dM fx : TxM RN, i.e.
JM(x)
(det (d'p'i fx)*(dM fx))1/2
JM(x)
(det (d'Afx)(dM fx)*)1/2
Often we shall also use the notation
if n < N if n > N .
2 Currents
J. (d' f (x))
JM(x)
for
103
.
Taking into account the approximation property of Lipschitz maps (compare (B) of Sec. 2.1.2) and the fact that sets of zero measure are mapped into sets of zero measure by Lipschitz maps it is not difficult to show that the area and coarea formulas hold for Lipschitz maps f : M -> 1R N, i.e.,
Theorem 3. Let M be an 7-ln-measurable and countably n-rectifiable subset of R', and let A C M. be an H'-measurable set. Then for any non-negative 7-Lnmeasurable function u : M -> R we have
fu(x)J(x)d7-C(x) =
J
ud7-l° I d7-Cn(y) (f-'(y ,I )nA /
if n < N, and
f
udl1 dNN(y) fu(x)Jr(x)dW(x) = f \ RN f-1(y)nA ifn>N. We finally notice that from the decomposition M = UkMk, together with the CI-Sard type theorem in Sec. 2.1.3 and the approximation property (B) in Sec. 2.1.2 it follows
Proposition 2. Let M be an 7-ln-measurable and countably n-rectifiable subset of R'+' and let f : M -. ISBN be a Lipschitz map. Then the slices M n f -1(y) are countably (n - N) -rectifiable subsets of JRt for 7{n -a. e. y E RN.
2 Currents In this section we deal with the general notion of currents and in particular of integer multiplicity rectifiable currents. We do not aim of course to completeness but only to discuss basic definitions and results which are relevant for the sequel. In particular not all proofs are included, and sometimes they are postponed to later chapters.
After the first two subsections on multivectors, covectors and differential forms, where we also fix some notation which will be used later on, basic definitions and results, as the important closure-compactness theorem for integer multiplicity rectifiable currents, are presented in Sec. 2.2.3 and Sec. 2.2.4. In Sec. 2.2.4 we also present some elementary examples, while further developments are discussed in Sec. 2.2.5, Sec. 2.2.6 and Sec. 2.2.7.
2. Integer Multiplicity Rectifiable Currents
104
2.1 Multivectors and Covectors Vectors of a finite dimensional vector space V may be multiplied giving rise to new objects called multivectors; for v1, ... , vk E V the result
C = vln...AVk is called a k-vector. But before describing such a procedure, let us first fix some notations. Usually we shall identify V with Rn with the standard basis e1,. .. , en; its dual V* will be again identified with IRn with dual basis e1, ... , en, so that
<ei,e-1 > = bi
.
We shall use the standard notations for ordered multi-indices
1(k, n) :_ {a = (al, ... ak)
(1)
and for convenience we set
I
ai integers, 1 < al < ... < ak < n}
I(0,n) = {0}
and
if a E I(k,n) . IaI = k If a E 1(k, n), k = 0,1, ... , n, we denote by a the element in 1(n - k, n) which complements a in {1, 2, ... , n} in the natural increasing order; of course we have
a = a, 0 = (1,... , n). If lal =n-1 and 10I = 1, we shall often write i instead of a, i = 1,... , n, and j instead of /3, j = 1,... , n, and i for 1.... ,n). Often in the sequel we shall have to consider maps from Rn into RN. We shall
denote the standard basis of RN by El, ... , EN, so that (el, ... , en, El, ... EN) is the standard basis of Rn x RN; with respect to those bases the coordinates , xn, yl, _ (x, y), and in order to stress this will be denoted by (xi, Y )
we often write 1l
x Ry instead of R x RN.
RN be a linear map, identified, with respect to the standard Let G : Rn basis with the N x n matrix
G=
i = 1,...,N , j = 1,...,n
and let (2)
n := min(n, N)
Given two ordered multi-indices with 1/3 = k, 1 < k < n, la = n - k, we shall denote by (3)
Gea
the k x k-submatrix of G with rows determinant will be denoted by (4)
M« (G)
/3k) and columns (al, ... , ak). Its
det Ga ;
we shall set (5)
MOD (G) := 1
2.1 Multivectors and Covectors
105
k-vectors. Let V be a finite dimensional vectors space over 1R, with basis el,... , en. For k, 1 < k < n, generic k vectors can be multiplied by means of the wedge product 1;
:= v1A ... /nvk
obtaining a new object. This wedge product is characterized by the following properties (i)
it is multilinear vin ... A(avi + bwi) A ... nvk = av1A ... twin... AVk + bv1A ... Awin ... nvk
(ii) it is alternating
vln ... Av2A ... AV3 A ... AVk = -VlA ... AV3 A ... AV A ... AVk
equivalently
v1 A ... nvk = 0 if vi = vi for some i 0 j or
v1A ... nvk = 0 if v1, ... , vk are linearly dependent. In terms of the basis e1,. .. , en, computation of l; yields an object of the form
= E `e« oGI (k,n)
where 6' E 118, and ea is in { ea = eat A ... Ael,_
I
a C 1(k, n) }
.
The set of all linear combinations of the e,,,, a c I(k, n), is the space of k-vectors, denoted AkV . It is a vector space of dimension (k). Every k-vector v1A... AVk defines a k-linear and alternating map with domain the product of k copies of V*, the dual space of V vin ... /nvk
:
V* x ... x V* --> 1[8
defined by
(vin...Avk)(wl,...,wk)
det(<w',vj >)
,
wi EV*
In fact it turns out that AkV can be identified with the linear space of all k-linear
and alternating maps on V*. More precisely, by considering the k-linear and alternating forms e, a E I(k,n) defined by (6)
e"'(wl, ... , w k )
det (wa) ,
O E V*
106
2. Integer Multiplicity Rectifiable Currents
we have for any Q = ()31, ... , Ok) E I (k, n), e1, ... , en denoting the dual basis of V* ea(eQ
...
SO,,
SO := det(Sai)
,
and the ea defined this way form a basis of the linear space of all k-linear alternating forms on V*. This shows that Ak V is defined intrinsically, i.e., independently of any fixed basis.
k-covectors. They are defined similarly to k-vectors replacing V by its dual V*. The linear space of k-covectors is denoted by AkV := AkV*, and is identified with the linear space of k-linear and alternating forms on V, with basis ea, a E I(k, n), given by (7)
ea(v1, ... , vk) := det (va')
,
v1,. .. , vk E V .
Exterior product of multivectors. Given two multivectors e E AkV
,
77 E
AhV, h, k > 0, h + k < n, e _ EaEI(k,n) Saea,'7 = >QEI(h,n),70eQ we define the exterior product of 6 and 77 by (8)
6A77 := E E a77QeaneQ cEI(k,n) /3E1(h,n)
where eaneQ by (i) and (ii) is given by eaAeQ
J0
if ano #0
l a(a,(3)eau0 ifaf1Q=0
where a U Q E 1(h + k, n) is the multi-index obtained reordering the multi-index (a, Q) and a(a, /3)
is the sign of the permutation which reorders (a, 0) in the natural increasing order, and v(0, 0) = 1. This way eA77 E Ah+kV. In order to give a meaning to (8) also in the case h + k > n we set eaneQ = 0
forh+k>n.
It is not difficult to see that the exterior product defined above has the following properties: Let a, 5 E AkV, /3 E AhV, -y E ASV, and c E R. Then (i) (a + S)A/3 = (aA/3) + (SAO) , c (aA/3) = (ca)n/ = aA (co) aA/3 = (-1)hk$Aa
(ii) (iii) (iv) (v)
OA(a + S) = (OAa) + (/3A5)
(aA/3)A'Y = aA(OA-y)
If v1, ... , vk E V _ A1V, and e1, ... , k E V*, then
v1A...Avk(e1,...,Ck) = det()
2.1 Multivectors and Covectors
107
Properties (i),...,(v) in fact characterize the notion of exterior product, so that it is an intrinsic notion independent of any fixed basis. Of course one can proceed similarly in order to define the exterior product of covectors. In this case (v) reads: f o r v1, ... , vk E V we have (9)
C'A...A k(U1i...,vk) = det()
.
Simple k-vectors. A k-vector
E AkV is called simple or decomposable if it can be written as a single wedge product of vectors, = Vi A ... AVk for some V 1 ,- .. , vk E V. Of course An,V - J1 and all n-vectors are simple as they are multiple of the base vector e1A ... Aen. A1V e_- V, hence all 1-vectors (vectors) are simple. The
two-vector := e1Ae2 + e3Ae4
provides an example of a non decomposable 2-vector in A2R4. Assume in fact it
is simple = v1Av2, then n = 0, i.e. e1Ae2Ae3Ae4 + e3Ae4Ae1Ae2 = 0
and, as
e1Ae2Ae3Ae4 = (_1)2 e3Ae4Ae1Ae2
we infer e1Ae2Ae3Ae4 = 0: a contradiction, since e1Ae2Ae3Ae4 is a basis of A4]R4.
Simple k-vectors play a fundamental role since they represent oriented kplanes in V, as we shall see below, and they are one of the main reason to introduce the space of k-vectors. In fact in many respect it is convenient to work with a linear space instead of working with simple k-vectors which do not form a linear space.
Duality between AkV and AkV. Covectors and multivectors are in duality. Given _ ,aEI(k,n) Sae( E AkV, W = EaEI(k,n) waea e AkV the duality between AkV and AkV is defined by (10)
<e,w> := E Eawa aEI (k,n)
It is easily seen that <, > does not depend on the fixed basis. In fact one can show that on simple n-vectors we have
= w(vl,...,vk) and a linear form on AkV is uniquely identified by its values on simple vectors. In particular, from (7) and (11) we infer that the duality product between a simple k-vector and a simple k-covector is given by
2. Integer Multiplicity Rectifiable Currents
108
.
= det() where < v2, w2 > is the duality between V and V*. The duality above of course allows to identify canonically, i.e., independently from the choice of the bases, (AkV)* and AkV* =: AkV. Finally notice that, since the k-vectors ea are simple, a k-vector e is zero if w> = 0 for all simple k-covectors w. The inner product of multivectors. Suppose that V is endowed with a inner product v w, and let e1, ... en be an orthonormal basis. We can then define an inner product in AkV (respectively in AkV) in such a way that ea (respectively ea) be an orthonormal basis of AkV (AkV) by setting
E arla
(12)
w u :_ E waOa aEI(k,n)
aEI(k,n)
where
= EaEI(k,n) Caea, 77 = EaEI(k,n) 17ae. and w = EaEI(k,n) waea, o =
EaEI(k,n) aaea
The inner product defined this way actually depends only on the scalar product in V and not on the fixed orthonormal basis 61, ... , en, and it is invariant for orthogonal changes in V. To see that we consider the duality map
R : V --* V* ,
=v-w Vv,
w -- Rw,
and we define R# AkV --* AkV by setting R#S(viA...AVk) = C(Rv1i...,Rvk)
Since Rei = e, el, ... , en being the dual base in V*, we easily see that
R#ea = ea . As by definition R# depends only on the inner product in V, (13)
C-
77 =
R#'1> ,
and the duality product is independent of the coordinate system, we see that also the inner product is independent of the coordinate system. Given now two simple k-vectors v1 A ... AVk, w1 A ... AWk from (13) we com-
pute
v1A...nvk w1A...nwk = = R# (wl A ... nwk) (vl, ... , Vk) = (W1A ... nwk) (Rvl, ...
= det () = det (v2 wj) hence (14)
(v1A... nvk) (win... nwk) = det (v2 wj)
,
RVk)
2.1 Multivectors and Covectors
109
Consequently the modulus of a simple vector is given by (15)
v1A... AVk
l
=
det (vi vj)
Let us interpret geometrically such a formula in the case V = R. We associate to the simple k-vector v1A... AVk the linear map
T:Rk -* R' ,
Tw := Ewivi.
Such a map is represented by the matrix with columns the components of the vectors v1i...,Vk
T :_ (v1 ...
I vk),
I
i.e., vi := Tei, and the matrix (vi vj) is exactly T*T. Therefore we see that
Iv1A...Avk1 = ITeln...ATekI = detT*T
(16)
One can interpret v1 A ... nvk I as the k-dimensional measure of the k-parallele-
piped generated by the vectors v1,...,vk,i.e. Iv1A... AvkI
= Hk(T[0, 1]k))
In fact from the area formula
hk(T[0, 1]k)) = .JTfk([0,1]k) = detT*T = v1A... nvk Also, if we write
v1A... AVk = E l;aea aEI(k,n)
we have
Sa = (v1A ... Avk)(eal , ... , elk) = det (< vi, ea' >) = det (v'')
and (v'') is the matrix obtained from T choosing the rows al, ... , ak. Hence (17)
detTT* = det(vivj) = IviA...Avk12 = E (ta)2 aEI (k,n)
(detva')2 aEI(k,n)
This proves Cauchy-Binet formula in Sec. 2.1.1. In fact in terms of the projections
Pa : l[8n -> Rk
Pav = (val, ... , vak )
the matrix (v'') is nothing else than Pa o T, hence (17) yields
T*T = E (det (Pa o T)) 2 aEI(k,n)
.
110
2. Integer Multiplicity Rectifiable Currents
Usually we shall work with the Euclidean norm in Rn and the induced norm )1/2
Iei = (
(S-)2 IaI=k
in the class of k-vector t; E AkRn. However in connection with the calculus of variations it is convenient and natural in many instances to consider the so-called mass and comass norms on multivectors and covectors. Here we confine ourselves
to give just the definitions and mention a property, compare Federer [226] and Vol. II Ch. 1. The comass norm of w E AkV is defined by (18)
II w II
:= sup{
E AkV, ICI < 1, e simple}
I
.
Of course
Ilwll<_Iwl. The mass norm of e E AkV is defined by (19)
IVII
sup{<1=, w> I w E AkV, II W II < 1} ,
and of course
11 0 ?
161
.
One can also show that for some c = c(n, k) one has
IIwII ? 1 IwI
,
II
II
<_
cICI
.
However in general
11wII < wl,
Iei
and we have:
II = For example for the 2-vector ICI = vr2.
wl
if and only if 1; is simple if and only if w is simple
:= eine2 + e3Ae4 in A2 R4 we have II e II = 2,
Simple k-vectors and oriented k-planes. Let us come now to one of the main motivation to introduce k-vectors. We shall show that, similarly to vectors which identify oriented one-dimensional lines, unit simple k-vectors identify the oriented k-dimensional planes in V. We shall assume V endowed with an inner product. First we observe that k vectors v1, ... , Vk are linearly dependent if and only if v1A... AVk = 0.
2.1 Multivectors and Covectors
111
In fact if V1, ... , vk are linearly dependent we know that vin ... nvk = 0. Suppose
on the contrary that vi, ... , vk are linearly independent. Then we can complete them in order to form a basis of V and consider k elements w1, ... , wk of the dual basis, so that < vi, wj > = S2 . We then find
= det() = 1 hence v1 A ...
nvk 0 0. Let P be an oriented k-dimensional subspace of V and let r1, ... , Tk be a positively oriented orthonormal basis of P. We then set bP := T1A...ATk . If v1i ... , vk is another basis of P then we have k
vi =
z=l ...,k
aijTj j=1
and it turns out that v1A...AVk = detA7-1A...ATk as one can easily verify checking on simple covectors. Hence, if vi, ... , Vk is another oriented orthonormal basis (so that det A = 1, being det A > 0) we have
P = v1 A ... hvk . This shows that SP is uniquely determined by the oriented k-plane P. In general, if v1i ... , vk is any oriented basis of P, then v1 A ... AVk
P = Iv1A ... AvkI Conversely, let
E AkV be a simple non-zero k-vector. Set
Pg := {v E V I eAv = 0} . Of course PP is a linear subspace of V. The dimension of Pg is k. In fact, being C simple, = v1 A ... nvk, and C 0 0 implies that v1, ... , vk are linearly independent; moreover, the observation above says that v E PP if and only if v is a linear combination of v1, ... , vk. It follows that
PP = span Orienting P in such a way that v1, ... , Vk be positively oriented we then see that APE _
1
2. Integer Multiplicity Rectifiable Currents
112
This proves our claim in the beginning. We would like now to read the previous characterization of simple vectors in a somewhat more algebraic way. Let C _ Eaea E AkV be a simple vector with C 74- 0. Of course a 0 for some a and by a suitable choice of the basis we may assume that a = (1, ... , k). This way we may reduce to a Cartesian situation. As such a situation is especially relevant for us in the sequel, we shall discuss it with some details.
Simple n-vectors in the Cartesian product Rn x RN. With the notation above every n-vector in A,,,Rn+N can be written as
_E
aQ eane,3
Ial+IR1
where a and Q are ordered multi-indices. We shall refer to Caa as to the components of and to Co' as to the 00-components or the first component of C.
Denoting for any k, 0 < k < n := min(n, N) by Vn,k the linear sub-
space of AnRn+N given by the linear combinations of n-vectors of the type v1A... Avn-knw1A... Awk with vi E R', wi E ]IAN, so that Vn,k := An-kin ® AkRN
(20)
,
the n-vector e can be uniquely decomposed as n
_ E (k)
(21)
k=0
where e(k) E Vn,k is given by
as eaAEg
(k) _ HI=k
in particular c(o) _ X00 e1, ... /den .
Similarly every n-covector splits as n
w,,,3 eaAE,3 _ EW(k)
W=
W(k) E Vn,k
=
An-kRn ®AkRN
k=0
la+Ql=n
and the duality <, > reads as
<W, > _ T WaQ aQ Ia+J01=n,
=
<W (k) , (k) > k
C <W(k), C> = <W, e(k) > = <w(k), e(k) >
,
2.1 Multivectors and Covectors
113
Let G := (G)21 ,...,n, i=1,...,N be the N x n-matrix associated to the linear transformation that we still denote by G from R' into RN. The vectors
i=1,...,n
ei+Gei,
yields a basis of the tangent plane to the graph of G in Rn+N. The unit simple n-vector
M(G) M(G)
(22)
where
M(G) := (el + Gel)A ... A(e,,, + Ge,,,)
(23)
identifies the plane graph of G, and in fact orients such a plane. It will be then called the tangent n-vector to the graph cG of G. Notice that if we denote by id m G the map id x G : RI -> Rn+N given by
(id >a G)(x) = (x, Gx)
(24) then
9,3 :_ (id >a G)(lRn)
.
Notice also that l: depends only on the orientation of R and not on the special fixed basis. Conversely, if e is a unit simple n-vector with positive first component, boo > 0, then 1; represents an n-plane with no vertical vector, and therefore there exists
a linear map L : Rn -> RN such that
_ M(L) M(L) More precisely we have
Proposition 1. Let V be an n-dimensional subspace of Rn+N, i.e., an n-plane in 118'+N through the origin oriented by the unitary n-vector V
V=
VaQ eaneQ
,
E (V' )z = 1 Iad+1131=n
laI+IRI=n
and let 7rv be the restriction to V of the orthogonal projection 7r to R'. We have (i) V contains vertical vectors, i.e., non zero vectors z such that 7r(z) = 0, if and only if V00 = 0. (ii) If V00 > 0, then there exists a linear map L : R' -- RN such that V = range (id >a L) and
V
I M(L) I
,
M(L)
(ei + Lel)A ... A(en + Len)
Moreover -7rv is the inverse of id >a L : R' --+ V.
2. Integer Multiplicity Rectifiable Currents
114
(iii) If J., denotes the Jacobian determinant of lrv, then
J,1 =
Voo
Proof. Suppose that V contains at least a vertical vector z1. Let (z1, ... , zn) be a basis for V. We split each zi as vi + wi, where vi E R and wi E RN. Since zl is vertical we have vi = 0, and, as z1A... nzn is a multiple of V
V = kzjA...Azn = k(vl+wl)A...A(vn+wn) Being V00 = k det (v1, ... , vn), we therefore deduce V°° = 0. Suppose V does not contain vertical vectors. Choose a basis z1, . . . , zn in V and split as previously each zi as zi = vi+wi. Clearly the n vectors vi are linearly independent, otherwise we could find Ai such that E Aivi = 0 and therefore > Aizi = > Aiwi would be
vertical. Thus V00 := kdet (v1, ... ,v) # 0. This proves (i). Define the linear map L : R' --, RN by Lvi = wi, where zi = vi + wi is a basis for V. Clearly, vi +Lvi = zi thus the range of id x L is V, moreover irv is the inverse of id x L. Finally, M(L) _ z1A ... nzn zln ... /nzn M(L) I
zjA...Azn Iz1A...AznI
) = 1 Moo( L
oo
>0
we therefore conclude that V = M(L)/IM(L)1. This proves (ii). If V00 > 0, being lrv the inverse of id x L, the Jacobian determinant of 7rV is the inverse of the Jacobian determinant of id < L, thus
_
1
1
IM(L)I
Jidr
=
Voo
If V00 = 0, then 7rv(V) is a lower dimensional subspace of F", thus J,( = 0. In both cases V00 > 0 and V°° = 0, we therefore have J, = V°°. This proves (iii) and concludes the proof of the proposition. In coordinates it is not difficult to compute, compare (31) (33) below, (25)
M(G) _
a) Ma (G) eaAEO
where recall M9 (G) = det Ga denotes the determinant of the (/, a)-minor of G Ga = JEa and MO'(G) = 1. In particular we have M(o) (G) = e1A ... Aen (26)
M(l)(G) _ E(-1)n-ZG2eine, M(k) (G) _
Q(a, a)Ma (G)eaAe'R InI+I
2.1 Multivectors and Covectors
115
and in terms of the components l;aa of := M(G)/CM(G)(
boo _
1
M(G)j (27)
X19
=
CU O
o (a, a)Ma
X00 =
(-1
boo
In conclusion we can then state E Anon+IV with
Proposition 2. An n-vector
non-zero first component is
simple if and only if
/
X00
v(a, a)Ma I
((-1)"-Z X00)
We also see that the map A ,mn+Iv
i ces} M : { N x nmatr -
is injective, and that the map
M(i) : {N x n-matrices} --> Vn,l :=
An-lRn ®Allf81`'
yields an isometry of linear spaces. This allows us to say that El :_ {l= E A,Rn+N
I
t; = M(G) for some N x n-matrix G}
is the graph of an algebraic function over Vn,1.
M(Gg
Fig. 2.3. The map M.
2. Integer Multiplicity Rectifiable Currents
116
For each t; E A.,,R,+N with t; 00 > 0, we can associate to C the N x n-matrix Ge defined by (28)
Ge
l
M(1)(\X00) .
Then t; -- Gg is "linear" and GCI El = M-1, i.e.,
GM(G) = G for all N x n-matrices G and l; = M(Ge) if and only if a is simple and
E Al where COO
Al := {l; E AI TRn+N
= 1} .
Also
GC = 0 if and only if e(1) = 0 .
In terms of Gg Proposition 2 above reads: E fl j n+N with 600 > 0 is simple if and only if M(Ge) (29) Finally we notice that, as dim An]IPn+N = and simple n-vectors can and by 660, simple n-vectors form a "small" be parametrized by e(1) nN) -nN-1 manifold of dimension nN+1; in other words, there are (n conditions for simplicity.
Induced linear transformations. With any linear map L : V -> W between two finite dimensional vector spaces V and W, we can associate the linear maps AkW defined by AkL : AkV
AkL(V1A... Avk) = Lv1A... ALvk
(30)
.
One can show that the family of linear maps AkL is characterized by the following two properties (i) A1L = L (ii)
Ah+kL(aAQ) = AhLaAAkL/.3 ,
a E AhV, 9 E AkV
With respect to bases e1,... , en in V and E1, ... , EN in W, the map L can be Accordingly we then find represented by a matrix (aj') so that Lei = >Z AkLea
ca 6,3 0
where (31)
c« = det (aa')
i.e. the matrix associated to AkL has as entries the k-order minors of the matrix A associated to L. In fact, denoting by e', ... , EN the dual basis of E1, ... , EN we have
2.1 Multivectors and Covectors cO = (AkL)(ea)(en'..., eOk) _ (Leaf A ... ALeak) (EQl
117
... 16 Yk )
= (EQ' A ... AeOk) . (Leaf A ... ALeak ) = det (<eQ=, Lea, >) = det (<ea=, aa, ei >) = det (aaj ) and, by (16)
AkL = det T*T. It is also readily seen that, if = v1 A. - - Avk is a simple vector and P the
(32)
subspace generated by V1, ... , vk, then AkL(e) is again a simple vector, and, in fact it is the k-vector associated to the plane L(P), if L has maximal rank or 0. Returning to n-vectors in A,,Rn+N and linear maps G : Rn --4 RN, we can then write
M(G) = (e1 + Gel)A... A(e, + Gen) = An-(id x G)(eiA ... Aen)
.
The computations to prove (31), then yield a proof of the decomposition (33)
M(G) _ E u(a, a) Ma (G) eanep
.
laI+IAI=n
In fact computing M(G) we get terms of the type
... vein... AGejA... Aehn ... AGekn ... reordering such terms then we get
M(G) = Ea(a,a)ean(A1 G)(e5)
(34)
a
that by (31) is just (33). Similarly, there is a linear induced map A k L: AkW --> AkV characterized by the properties
(i) A1L=L*, L*:W-+V (ii)
Ah+kL(WAS) = AhL(w)AAkL(b) , w E AhW, S E AkV
and again defined by AkL(W1A...AWk)
= L*wln...AL*wk
.
One easily checks that AkL is the dual map of AkL, i.e., (35)
<e,AkLw>
V 6 E AkV, w E AkW
.
118
2. Integer Multiplicity Rectifiable Currents
and (36)
jjAkLjJ = IAkLHH
If G : W --+ Z is another linear map, we have
Ak(G o L) = AkG o AkL .
(37)
Equalities (35) and (37) express the so-called Binet formulas , compare Vol. II Ch. 2, which in the case V = W = Z and k = dim V amount respectively to det L = det LT
det GL = det G det L
,
as one can see using (31).
2.2 Differential Forms Let X be an n-dimensional manifold. A k-vector field C on X is a map
I;:xEX -+ e(x)EAkTXM. Similarly, a k-covector field on X is a map
w:xEX -+ w(x) EAkTXMf-- Ak(TXM)* . A k-covector field is also called a (differential) k-form. In fact it can be generated by applying the exterior differential d to functions and using the wedge product. For the sake of simplicity here we shall only consider k-forms on an open set
U of R', so that w is a map
w:UCRn -+ AkR')we shall discuss the general situation in Ch. 5. Exterior differentiation. As usual, let e1, ... , e,z be the standard basis in Rn. We denote by dxi the 1-form on Rn defined as the constant map (Jn)* dx2:xERn --+ e'EA ell ... , en being the dual basis of Rn, so that dxi(x) = xi. For any smooth function f : U -+ R its differential is defined as the 1-form df : U --+ A'R' n
df : x E U
E ff+ (x)dxi E A i=1
Of course df does not depend of the fixed basis, in fact for any smooth vector
field :xEU--+ 1;(x)
we have
< (x),df,> = =
. dtf(x+tC(x))jt=o
2.2 Differential Forms
119
Notice also that the definition of dxi is consistent with the definition of df ; in fact if f (x) := xk, then n
dfy = > ffi (x) dx2 = bkie2 = ek i=1
For any ordered multi-index a E I(k, n) the k-form
dxa:xEU -f eaEAkRn can be written as
dxa = dxa' A ... Adxak
,
and every k-form w : U C IEFn -+ Ak1n as w(x) = Eal_k wa(x) dxa; so that
w = E wa dxa lal=k
where the product of a k-form w by a function is defined by
(fw)(x) := f(x)w(x), and more generally the exterior product of two forms by
(wAS)(x) := w(x)A5(x)
.
The exterior derivative of a k-form, k > 1, w = k I=k wa dxa with smooth coefficients wa (x) is defined by
dwandxa _
dw
wax= dx''Adxa Ial=k i=1,...,n
lal=k
This way (i) df agrees with the ordinary differential for C' function f (ii) d(w + 77) = dw + dry for all k -forms w and 77 (iii) d(whr7) = dwA'i + (-1)kwndi] if w has degree k (iv) d(dw) = 0, for any form of class C2.
Notice that (iii) generalizes the ordinary differentiation rule for the product of functions, and (iv) expresses in case w is a function the equality of mixed derivatives
DiDj f = DjDi f . Conversely one can easily show that the exterior derivative operator d is characterized by properties (i)...(iv). This in particular implies that dw is intrinsically defined, independently of the coordinate system.
2. Integer Multiplicity Rectifiable Currents
120
The space of infinitely differentiable and compactly supported k-forms in U is defined by wadxa wa E C,°(U)} D"(U) I
lal=k
Similarly we define wadxa
£k(U)
I
wa E Coo(U)}
lal=k
and we speak of rn-differentiable k-forms. We may then regard the exterior derivative as an operator
d : £k (U) -> £k+l(U)
.
Of course it increases of one the degree of a form and decreases of one its regularity property.
Pull-back. Let U C R and V C RN, and let f : U -* V be a smooth map. To every k-form w = > wQ dyQ E Dk (V) we can associate the form f #w E £k (U), called the pull-back of w and defined by
f#W = E w,o of d fa
(1)
1131=k
where
dfQ :=
dfQ'A...Adf,Qk
dfQh =
n
fR! dxi
.
It is not difficult to show that for any x E U we have (2)
f#w(x) = Akdff(w(f(x))
and
f#(dyalA... Ady3k) = df01A...
Adf,ok
so that pulling back is an intrinsic operation, and f:# enjoys the following properties (i)
(ii) (iii) (iv)
f#(w+77) = f#(w)+f#(77), f#(wAr7) = f#(w)nf#(,7),
f#(dg) = d(g o f ), if g : V --> 1R, d(f#w) = f # (dw) if f E C2.
2.2 Differential Forms
121
Differential forms in a Cartesian product. Later we shall be interested in
differential r-forms in U :_ 0 x RN C 112' x RN. As for r-vectors and r-covectors, we can write every r-form in Dr(U) as
w=
Wap dX'Ady" I+1O1=r
I
or
min(r,N) W(k) 1: k=max(O,r-n)
w = where
E
W(k) :_
w,,o dx' Ady 3
.
I-I+IR1=*
PI=k
Integration of differential forms. Let M be a smooth, say C1, k-dimensional manifold that for the sake of simplicity we assume to be embedded in R. Let (Ui, Oi) define local charts in M and cpi : V C Rk --p m f1 Ui, oi = ?Pi 1, be a local parametrization of M with parameter variables u. The tangent space of M at cpi (u) is given by
Tw,(u)M := span {D1cc (u), ... , Dkco (u)} and the unit k-vector
S
D1cpi(u)A... ADkcpi(u) Dlcpi(u)A... ADkco (u) I
!L E Ui
Recall that two parametrizations yield the
defines an orientation of
same orientation if the Jacobian determinant of the transformation which reduces one chart to the other is positive. In this case we have
Dico (u)A... ADkwi(u) Dicpi(u)A... ADk
D1cpj (u)A... ADkcpj(u) I Di.Wj (u)A... ADkWj(u)
for cpi(u) = cOj(u)-
If one can choose charts or parametrizations in such a way that 1; (cpi (u)) defines
a smooth unitary tangent vector field on M, one says that M is an oriented manifold with orientation l;. Suppose now w is a smooth k-form on M (or in a neighbourhood of M in ), and suppose spt w C Ui, then classically the integral of w on M is defined as the integral of the k-form cp#w on Rk
Jw :=
(3)
M given by
Jco w
V
2. Integer Multiplicity Rectifiable Currents
122
f
f<w(u)eiA.. . Aek > dfk(u)
w
M
.
V
One easily computes
=w(co (u))(Dco (u)e1h...ADcpi(u)ek) = = ADkWi(u) (u)) > JW, (u) = (u)),
(4)
In the general case, that is if sptw is not contained in a chart Ui, and M is oriented by 6 E A,, TM, one shows by means of a decomposition of unity {i
that
f w = E ftiw) ,
}
cPi#(eiA ... /yen)
a'2 V
M
is well defined. According to the computation in (4) and taking into account the area formula in Sec. 2.1, J. (u) d7-lk (u) = d7-lk L M, we then find
fw
(5)
=
M
f<e(x)w(x)>d(x). M
This shows that a k-form is a natural object to integrate over an oriented, kdimensional rectifiable set, because it is sensitive only to both the location x E M
and the tangent plane to M at x. Given a k-rectifiable set M C R'ti, we say that M is oriented by Ak]R
(i) (ii)
M ->
if
is Hk-measurable, for 7 1k-a.e. x 1e(x) I = 1 and fi(x) is associated to TIM, in the sense that is the wedge product of k vectors of a basis of TIM.
In this case we define the integral of the k-form w on the oriented rectifiable set M simply by (5). Of special relevance for us will be the case in which M is the graph say of a smooth map u : (1 C R' --, RN. In this case we easily infer from the above (6)
f w = J(idu)#w = f <w(x, u(x)), M(Du(x)) > dx . S2
0
2.3 Currents: Basic Facts Let U C R' be an open set, 0 < k _< n. We denote by Dk (U) the space of all infinitely differentiable and compactly supported k-forms in U topologized by the usual topology which is characterized by the assertion that
2.3 Currents: Basic Facts
w' dxa -p W
wa dxa ,
123
i --* 00
aEI(k,n)
aEI(k,n)
if there is a fixed compact K C U such that (i) (ii)
sptw"CKVaEI(k,n) and V i lim DRw' = DQwa `d a E I(k, n) and every multi-index 0.
Definition 1. A k-dimensional current in U is a continuous linear functional on Dk(U). The space of k-dimensional currents in U is denoted by Dk(U). Given T E Dk(U), we can define for any a E I(k, n) the components of T (with respect to standard basis e1, .
, en in Rn) by setting for any cp E C°O (U)
. .
T'(co) := T(Vdxa) Clearly with a certain abuse of language we have
T = E Taea IaI=k
where we regard ea as the operator
ea (E wQ dxO ) := Wa ea(dxo) = ba = aa . 5a2 ... yak or equivalently
Ta(wa)
T(w)
w=
if
wa dxa . aEI(k,n)
aEI(k,n)
From this point of view a current is just a collection of ordinary distributions indexed by a E I(k, n). Accordingly we have
Definition 2. Let Th,T E Dk(U). We say that the sequence {Th} converges weakly to T, Th - T, if Th(W) -i T(w) for all w E Dk(U). The first notion which distinguish currents from ordinary distributions is that of boundary. It is simply defined as the dual operation of the exterior derivative.
Definition 3. The boundary of T E Dk(U) is defined as the (k - 1) -current (1)
We set
8T(77) := T(dr7)
8T = 0
if
V
rl E Dk-1(U) .
T E Do (U)
.
2. Integer Multiplicity Rectifiable Currents
124
We shall see that the boundary operator
: Dk(U) -f Dk-1(U)
8
comprises two apparently different concepts
(a) that of derivative of a function (b) that of boundary of submanifolds. Before discussing the important subclass of currents with components which are measures in U, let us make a few general remarks. As for distributions the support of a current T E Dk(U) is defined by
sptT = n{K C U
I
K relatively closed in U, T(w) = 0 for all w E Dk (U) with spt w c U \ K} .
One can define the Cartesian product of currents. Let Ti E Dki (Ui), Ui C Rni,
i = 1,2. The current T := T1 X T2 E Dk1+k2 (Ul X U2) is defined as follows. If w E Dk1+k2 (U) has the form
w = E wap (x, y) dxa Adya ICI=k1
=k2
then we set
T(w)
(2)
= Ti I
If Ial
k1i
J(3(
w«R(x, y) dy3
T2 (
\ jal=k1
Ial=k2
k2, IaI + IQI = kl + k2, and
I
/
dxa)
E C°°(U), we set
T(cpdxandy,3) = 0.
(3)
One easily shows that
O(Ti x T2) _ &T1 x T2 + (-1)k1T1 x 0T2
(4)
.
As a special case consider T E Dk (U), U C Rn, and the 1-current Q (0,1) l defined as integration of 1-forms on the oriented segment (0, 1) of R. Let w E Dk+1([0,1] X U). Split w into its horizontal and vertical parts, w = WH +wv where
WH(t,x) = E wHa(t,x)dtndxa Ial=k
wv (t, x) =
wva(t,x)dxa Iaj=k+1
and denote by [w](0,1) the k-form in U obtained by averaging wH with respect to the t variable
2.3 Currents: Basic Facts
125
1
U wxa(t, x) dt dx".
[w](o,l) (x)
0
JJ
From the definition of product of currents we infer, compare (3), that Q (0, 1) x T(wv) = 0, while from (2)
\
1
(5)
(0,1) 11 x T(w) =
JT( o
wx(tx) dxI dt //
Ial=k
T([w](0.1))-
Also (6)
a([(0,1)lxT) _ (81-8o)xT-Q(0,1)}jxaT =61xT-8oxT-Q(0,1)1xaT
where 8°, 81 E Do (R1) are the Dirac masses at 0 and 1, 8o(cP) = W(0)
,
81((P) = W (1)
V c0
E D°(R1)
Currents which are representable by integration. Those are currents with components which are measures with locally finite total variations instead of mere distributions.
Definition 4. Let U C R and V C U be open sets, and let T E Dk(U). The mass of T in V is defined by MV(T) := sup{T(w) I W E Dk(U), sptw C V, Iw(x)I < 1 V X E U} . Actually there are very good reasons for replacing in the previous definition the Euclidean norm I w (x) I by the comass norm defined in Sec. 2.2.1, compare Vol. II Ch. 1.
Definition 5. Let U C 1 and V C U be an open set, and let T E Dk (U) . The mass of T in V is defined by Mv(T) := sup {T(w) I w E D"(U), spt w C V,
IIHII<1VXEU}. The reason for such a definition will be apparent in Vol. II Ch. 1 where dealing with approximation theorems, compare also Sec. 2.2.6. The Euclidean and mass norms are equivalent, in fact
MV(T) < Mv(T) < cMv(T) where c is an absolute positive constant, see Sec. 2.2.1; but in general we have
MV(T) < Mv(T).
2. Integer Multiplicity Rectifiable Currents
126
If V = U we shall simply write M(T) instead of Mv(T), and we set
M(T) < oo} Mk,lo,:(U) :_ {T E Dk(U) Mv(T) < oo d V CC U1.
Mk(U) :_ {T E Dk(U)
I
I
Of course T E Mk,loc(U) (respectively Mk (U)) if and only if the Ta's are Radon measures (respectively Radon measures with finite total variation in U).
Introducing the total variation of T with respect to the comass and the Euclidean norm
:= sup {T (w) := sup{T(w)
II T II (w)
ITIO
II w(x) II <_ o(x)}
I
Iw(x)I <_ W(x)},
I
cp > 0, an immediate consequence of Riesz theorem for measures is
Theorem 1. Let T E Mk,1oc(U). We have (i)
There exists a ITI-measurable function f with 1 < f < c such that IIT 1= ITI L f.
(ii)
There exists a unique I T II -measurable map I
1
U ---* Ak1n with I ' I I =
II T II-a.e. such that for all w E Dk(U)
T(w)=f d1ITII.
(7)
U
Moreover
VVCCU, V open
IITII(V)=Mv(T)
and T is the Radon-Nikodym derivative of T with respect to II T II dT
T
dIITII
(iii) Also
T(w) =
fdITI U
TI(V) = Mv(T)
VVCCU, V open
and I fTI = 1 ITI-a.e.
Notice that conversely, given a non-negative Radon measure µ and a µmeasurable and locally summable map f : U -+ AkRn the current T = p L f defined by T(w)
:
fd u
2.3 Currents: Basic Facts
127
belongs to Mk,loc(U) and we have
IITII = ILLIIf1I
µ a.e.
Let T E Mk,loc(U), i.e., T(w)
= f< T, w> d I i T11. Though in general
MV (T) 54 MV(T), if T is simple, we infer from Theorem 1
1 = f ITI = f II T II = f
ITI-a.e., equivalently I ITI I-a.e..
Therefore we conclude in this case Mv(T) = MV(T). In the sequel we shall always work with the mass and comass norms, but we understood equalities like IITII = ITI,
IITII = ITI
II T II-a.e. or ITI-a.e.
in the case that T is simple. From the definition of mass we readily infer
Proposition 1 (Lower semicontinuity of the mass). Let Tj,T E Vk(U). If Tj T then for any V C U, V open, MV (T) < lim inf j,oo MV (Tj) . and from the compactness theorem for Radon measures
Proposition 2 (Compactness-closure theorem). Let {Tj} C Mk,lo0(U) be a sequence satisfying supj MV (Tj) < cc d V CC U. Then there exists a subsequence {Tj,} and T E Mk,loc(U) such that Tj, - T. Moreover
M(T) < lim inf M(Ty) < cc j -moo
if the masses of Tj, are equibounded.
By the monotone convergence theorem any current with finite mass, T E Mk (U), can be extended to all k-forms with Borel bounded coefficients, and again we have
T(w) := f<,w>d(IT((. U
A similar extension works also for T E Mk,loc(U) and for compactly supported k-forms w in U with bounded Borel coefficients. In particular, if T E Mk(U) and A C U is any Borel set we can define the restriction of T on A, T L A, by (8)
TLA(w) := fdMTM. A
More generally, we can define for any bounded Borel function
128
2. Integer Multiplicity Rectifiable Currents
T L f (w) :=
(9)
f<w>fdIlTIl.
However a certain care is needed when dealing with weak convergence. From {Tj} C ivtk(U), sups M(T?) < oo, Tj - T, i.e.,
Tj (w) -+ T(w)
VwEDk(U)
one can easily infer that TT (w) --* T (w) on k-forms with continuous coefficients with compact support, but in general the Tj (w) do not converge to T (w) if w has only bounded Borel coefficients (even with support contained in U). The reason for that is of course that a sequence of functions in L' ((1) which converges as measures need not converge in L1, compare Ch. 1.
For future purposes we note that for any T E Mk,lo.(U), T := II T II L T, the
current [ (0, 1) x T has locally finite mass and actually II
(0,1) J1 x T II =
G1
Q (0,1). x
x I ITI I
= e1AT.
For the reader's convenience we include the simple proof. With the same notation as above let w E Dk+1(U), W = WH + WV. As clearly II WH II < II w II, we have II [w](0,1) II (x)
f II w(t, x) II dt, hence 0
(0,1)1 x T(w)I
=
IT([w](0,l))1 < II T II(II [W](0,1) II)
IITII
JIIwt,x )I Idt)
C1xIITII(liw(t,x)II),
0
from which we infer II [ (0,1)1 x T II < Ll x II T 11. To prove the opposite inequality, for e > 0 and f E CO ([0, 1] x U), let w E Dk ([0, 1] x U) be such that for any
t
T(w(t,x)) > II T Integrating with respect to t in (0, 1) we get 1
x T II. To conclude the proof of the II claim it suffices to notice the following equality II
1
1
(0,1)1xT(w) = f T(wH)dt= f dt f <WH(t,x),T(x)>d1ITII 0
=
0
f<w(t,x),eiAf>d' x 11 T II.
2.3 Currents: Basic Facts
129
Normal currents. The class of normal currents is defined as Nk(U)
M(T) +M(aT) < oo}
{T E 1)k(U)
aT E NIk-1(U)} ,
{T E Mk(U) and the class of locally normal currents by
Nk,loc(U) := {T E Vk(U) J Mv(T) +Mv(3T) < oc V V CC U}
If T E Nk(U) one sets
Nv(T) := M v(T) +Mv(0T). Of course also for normal currents a closure-compactness theorem holds: If Tj E Nk,l°°(U) with
sup{Mv(Ti) +Mv(aTi)} < oo
d V CC U ,
then there exists a subsequence {Tj, } of {Tj } and T E Nk,lo,(U) such that
Tj, -S T
.
An interesting result for normal currents, which will be proved in Theorem 3
in Sec. 4.3.1, states that they cannot concentrate on small closed sets; more precisely we have
Theorem 2. Let T E Nk,loc(U), U C Rn, and let p,, denotes for any a E I(k, n) the orthogonal projection 1
n
Ck'
ak
Then, for any closed set E C U with Nk(pa(E)) = 0 `d a c 1(k, n), we have
ITII(E) = 0
i.e. TAE=0. n-dimensional currents in Rn. Those currents will be discussed with some details in Ch. 4. Here we shall confine ourselves to state for the reader's convenience a few facts. Let U be an open set in R'. Of course
-D- (U) = {bdx'A... Adxn
I
b E D°(U) = C,° (U)}
.
Thus to any n-current T it corresponds the distribution
b --> T(bdx'A...hdxn) and, conversely, to any distribution S E Do(U) it corresponds the n- current
130
2. Integer Multiplicity Rectifiable Currents
bdx1A... Adxn
S(b)
.
In particular, to any function u E L o, (U) we can associate the n-current
T,(bdx'A...Adx'):= fbudx.
Tv,:D'(U)->R,
U
Obviously
Mv(T,,) =
fluldx
V V CC U.
V
As every w E D'-'(U) can be written as w = Ej' 1(-1)j-'a; dx1A... Adx3A ... ndxn, we compute 09T.)(W)
=
fudivadx
,
a = (ai, ... , a.) .
We therefore see that every function u E BV0,,(U), i.e. every L1-function whose distributional derivatives are Radon measures defines the n-dimensional locally
for any V cc U, we have normal current T, moreoverfudx, MV 09T.) =
MV (T.) =
fdDu. v
V
Conversely we shall see in Ch. 4
Theorem 3. For any locally normal current T E Nn,lo,;(R'n) there is a function
u E BV(U) such that T = T. As for any n-current in U C Rn we have T = T°e1A ... Ae,, and aT = 0 implies DiT° = 0, i = 1, . . . , n, in the sense of distributions, i.e., To =const, we also infer the following characterization of n-currents in U without boundary in U.
Theorem 4 (Constancy theorem). Suppose U is a connected open set in Rn, T E Dn(U) is such that aT = 0, then T is a real multiple of the current integration on U, i.e.,
T = rl[U
for some r E R. 1 For the reader's convenience we compute here the boundary of k-currents in Rn, n = 1, 2, 3.This will show in which sense a means derivative.
1. In R1 we have 0-dimensional currents S which are simply distributions, and for which 8S = 0, and 1-currents. Clearly 1-currents have the form
T = T1e1
2.3 Currents: Basic Facts
131
and according to d27 = 27' dx for 27 E D°(R), we compute
8T(r7) = T(di2) = Tl(7]) = -DT'(r7)
i.e., 8T = -DT'. 2. In 1R2, apart from 0-currents, we have currents of the form S = S' el +S2 e2 E Dl (1R2).
T = To e,Ae2 E D2(R2)
Since for 77 = 97, dx1 + 772 dx2 E D' (IR2) we have d77
771,x2dx2Adx1 +772,xidx1Adx2 = (772,x1 - 77i x2) dx'Adx2
we can compute 8T (77)
= T (d7?) = T° (772,x1 - r71,x2) _ = -DxiT°(r72) + Dx2T°(771) _ -(DxiT°el - Dx2T°el)(77)
i.e.
8T = Dx2T°el - Dx1T°e2 and aT may be regarded as minus the gradient of T°. For 77 E D° (R2) we have d77 = 77x 1 dxl + 77x2 dx2 hence aS(77) = S(d77) = S1 (77x1) + S2 (77x2) = -Dx1 S1(77)
-
Dx2 S2 (77)
i.e.
aS = -div (Sl, S2) E Do (]R2)
.
3. In R3, a part for 0-dimensional currents which are distributions, currents have
the form
T = T°e,Ae2Ae3 E D3(]R3) S = S1e2Ae3 +S2 e3Aej +S3 ejAe2 E D2(1 3) R = R'el + R2e2 + R3e3 E Dl (R3) As in ]R2 we easily computes
8T - - gradient of To
,
aR - -div (R1, R2, R3)
For 77 = r7idxl +772dx2 +773dx3 E D1 (R3) we have d77 = (772,x1
- r7i,x2)dx1Adx2 + (773,x2 - r72,x3)dx2Adx3 + (773,x1 - 71i,x3)dxlhdx3
we therefore see that aS(77) = -curl (S1, S2, S3)(771, 772, 773)
2. Integer Multiplicity Rectifiable Currents
132
Image of a current. For its relevance in the sequel we conclude this subsectioi
by discussing the notion of image or pushing forward of a normal current f#T by a smooth map f . Under mild conditions one can define the pushing forward of a current T E Dk (U), U C R1. Let f : U ---p V, V C R', be a C°° map such that f lspt T is proper, i.e., f -1(K) n spt T is compact in U for any compact set K. Then the current f#T E Dk(V) is defined by (10)
f#T(w)
T ((f#w)
w E Dk(V)
where C E Co (U) with c = 1 in a neighbourhood of spt T n spt f #w. One easily checks that such a definition is independent of ( and (11)
a f# = f#a ,
spt f#T C f (spt T) .
Suppose now that T has finite mass, i.e. T E Mk(U). As we have seen T extends to all differential forms rl with Borel coefficients and f Inj d1J T 11 < oo. In particular for any map f : U -+ V of class Cl with bounded derivatives in U, supu I D f I < oc, the form f # (w) has bounded and continuous coefficients, hence T is well defined on f # (w). Therefore we can conclude that for any f of class C1 and T in Mk(U) f#T is well defined again by
f#T(w)
(12)
T (.f #w) .
In particular if U = V x RN and Tr : U -> V is the orthogonal projection onto the first factor, lr(x, y) = x, ?r# is well defined for any T E Mk(U). However in general f# is not continuous with respect to the weak convergence of currents. For example if U = (0, 1) x R, V = (0, 1) the sequence of currents Tk defined as integration over the line segment from the point (0, k) to (1, k), we have M(aTk) = 0,
M(Tk) = 1,
7r#(Tk) = To
but Tk - 0 as k -s oo which says that 7r# is not continuous. We explicitly note that (12) implies that f#T has finite mass if T has finite mass and actually f#T I I = f#11 T 1 1 ,
f -!'(f (x)) = AkDf(x)(T(x))
Similarly, assuming that M(T), M(8T) < oo and f E C2(U) has bounded second derivatives the pushing forward of aT
f#8T(w) = aT(f#(w)) = T(df#(w)) is well defined, but in general
f#aT 0 of#T. Let us consider now mappings f : U --* V which are just Lipschitz continuous. In this case for any w with smooth coefficients the pull-back f #w is a form with
2.3 Currents: Basic Facts
133
bounded measurable, but not necessarily Borel coefficients. Consequently we can apply T to f #w only if II T II is absolutely continuous with respect to the Lebesgue measure, and in general T (f #w) as in (12) is meaningless. The following example illustrates the situation. Take U = V = 1 and let f : Iii -> lib be defined by
f (x) :=
x+1 forx<-1 for -1<x<1 0 x-1 forx>1.
If w = cp(y) dy E D' (R),
f#w = co(f(x)) f'(x) dx =: zb(x) dx
where O(x) is defined and smooth for x # f1. But b(1) and 0(-1) are not defined. By taking T := 81 E No (IR) we then see that
T(.f#w) = 81(') = 0(1) is meaningless.
However it turns out that if T is a locally normal current in Dk(U) and f is a Lipschitz map which is proper on spt T, then f#T can be well defined. But before doing that let us show why we need to assume f spt T proper.
Let f : U := (0, +oo) y V := ]IP be just the inclusion map, and let T [ (0, b) D, b > 0. We have aT = 8b so that T E N1(U), consequently f#aT = 8b. The push-forward of Q (0, b) by f is instead the current Q (0, b) E Dl (IR), f#T = Q (0, b) , hence of#T = Sb - 8o. Therefore we have f#aT # of#T. As one would like to have the equality f#aT = af#T, what is wrong is the fact that f is not proper f-1([-1, 1]) = (0,1] and (0,1] is not compact in U = (0, +oo). Suppose now f Lipschitz and proper on sptT, T E Nk,loc(U). Let 0 E C° ° (Rn) be a standard mollifier and let r-'O(r-1x)
'f'r(x)
r>0
and
fT
f *0,
be a smoothing of f. We define the image of T by f as
f#T(w) = limo fT#T(w)
w E Dn(V)
i.e. as the weak limit of the well defined currents fT#T, compare Sec. 5.1.2. In fact we have
Proposition 3. If T E
f : U -> V is Lipschitz and fIsptT proper,
then
f#T(w) := o fT#T (w) 11
2. Integer Multiplicity Rectifiable Currents
134
exists and is independent of the mollifier 0 for all w E Dk(V). It agrees with f#T defined by (12) if f is of class C'. Moreover spt f#T C f (spt T) and k
sup JDfj) Mf-1(w)(T) f-1(W)
Mw(f#T) <
for any open set W CC V. Finally
f#8T = of#T . The proof of Proposition 3 is based on the important homotopy formula for currents. Consider two smooth maps f, g : U -> V and a smooth homotopy
h : [0,1] x U -> V,
h(0, ) = f,
h(1, ) = g.
Suppose h is proper in [0, 1] x spt T. Then h#(1(0,1)1 x T) is well defined and we have
8h#(1(0,1)1 x T) = h#a([ (0,1) x T) = h#(51 xT - So x T- Q(0,1) x 8T) = g#T- f#T - h#(Q(0, 1) 1 x9T) . Therefore we can state
Proposition 4 (Homotopy formula). Let U, V be open sets in R', U --+ V be smooth maps, and let h : [0, 11 x U -- V be a smooth homotopy between f and g; finally let T E Dk (U) . Assume that h-1(K) n spt T is compact in [0, 1] x U for any compact set K C V, then we have (13)
g#T - f#T = ah#(1(0,1) ] x T) + h#([ (0,1) fl x aT)
where h#(Q (0,1) l x aT) := 0 if k = 0. Recalling (5) Q (0, 1) JI x T(w) = T([w](o,l))
we can write (13) as (14)
T(g#w
- f#w) =T([h#dw](o,l) +d[h#w](o,l))
where the last summand has to be taken equal to zero if k = 0. It is easy to see that (13) and (14) are also valid even if h is not proper provided T has compact support in U, T E Ek(U). By applying (14) to currents of the type
w -.
J
wnrl
rl E Dn-k (U)
2.3 Currents: Basic Facts
135
we then recover the usual homotopy formula for forms g#w
(15)
- f#w = [h#dw](o,1) + d[h#w](o,l).
In the special case of the affine homotopy
h(t, x) := (1 - t) f (x) + tg(x) we have
Ak+1Dh(t, x)(elAT(x))
h# (t,x)
= (9(x)-f(x))AAkDxh(t,x)(T(x)) hence
h#(Q(0,1)1 x T)(w) =
f <w(h(-)),(9-f)AAkDxh(T)>d(L' x IIT II)
which implies
M(h#(Q(0,1)]1 x T)) :5 cIITII(I9-fI(IDfIk+IDyIk)) and therefore (16)
M(h#Q (0, 1) ] x T) < sup If - 9I sup (ID.f I + IDgI)k M(T) spt T
spt T
Returning to Proposition 3, we consider the affine homotopy
h(t, x) := tf, (x) + (1- t) ff (x) corresponding to two parameters T, Q > 0. The homotopy formula yields for any O
w with spt w c K, K CC V, f,-#T(w) - fv#T(w) = h# (If (0,1) x T)(dw) + h# (E (0, 1) x 0T)(w) therefore, if moreover IwI, Idwl < 1, we infer from (16)
I.fT#T(w) - ff#T(w)I < cM(T)
sup f-' (K) nspt T
IfT -
f01 (Lip f)k
The proof of Proposition 3 can be now easily concluded. We conclude this subsection by stating one more simple but useful consequence of the homotopy formula.
Let T E Dk(U) with spt T C U. Suppose that U is star-shaped relative to the origin, and consider the affine homotopy which joins the null map to the identity map
h(t, x) = tx
.
We define the cone over T as the current in Dk+1 (U) given by
136
2. Integer Multiplicity Rectifiable Currents
OxT := h#(Q(0,1)JJxT). From the homotopy formula we infer
8(OxT) = T-(Oxcxc T) which shows that every boundaryless current T is a boundary, i.e., if T E Dk(Rn)
and aT = 0, then there exists R E Dk+1 such that T = 8R. Another immediate consequence of the homotopy formula is the following
Proposition 5 (Poincare lemma). Let U be a starshaped domain in IEBn and let w be a closed form in U. Then w is exact. In fact, assuming U starshaped with respect to the origin and considering the affine homotopy between 0 and the identity, h(t, x) := tx, (15) yields w = g#w = d[h#w](o,l).
2.4 Integer Multiplicity Rectifiable Currents In the class of k-dimensional currents in some open set U of R' we may identify two typical situations which in some sense are extremal from the point of view of smoothness. (i)
Given a Lebesgue measurable k-vector field k-dimensional current
T := ,Cn L C =
Gn
L:
in U,
Cc'ell the
L COea
oI=k
is a current "distributed" on the n-dimensional set U. Actually by considering a non-negative measure y and a 4-measurable k-vector field C we can realize k-dimensional current
T := 1JLC ii)
which are "distributed" in sets of any "dimension" less than n. Given a smooth k-dimensional oriented submanifold of IPn, M, M C U, the current integration of infinitely differentiable k-forms with compact support in U, Q M , defines a k-current which lives on the k-dimensional set M. Moreover, as we have seen in Sec. 2.2.2, the current
QMI(w) =
J
M
w
w E Dk(U)
2.4 Integer Multiplicity Rectifiable Currents
137
can be represented as
I M I (w) = f<(x),w(x)>d7-Lk M
where fi(x) orients for H' -a-e. x c M the tangent plane TAM of M at x. One also refers to Q M I as to the current carried by M.
Integer multiplicity rectifiable currents arise as the natural extension of the currents of the type Q M 1, in which we allow M to be xk-measurable and countably rectifiable and we also allow integer multiplicity. As we shall see in Sec. 2.2.6
such a class turns out to characterize the class of the weak limits of the kdimensional currents carried by smooth submanifolds. But before turning to precise definitions, let us return to the k-dimensional
currents which are integration over smooth graphs, as they shall be specially important for us in the sequel.
Currents carried by smooth graphs. Let ,(2 be an open set in 1Rn and let u: f2 C 1Rn _+ RN be a smooth map. Its graph is defined by
9u,o := {(x, u(x))
E IRn+N
I
x E .(2}
and the Cartesian current carried by the graph of u is given by the n-dimensional current
G. i.e., by
Cu (w) =
f (id x u)#w = f<w(xu(x)),M(Du(x))>dx
d w E Dn (U)
n
S2
where U := .f2 x 1R`v. By (33) in Sec. 2.2.1 we have
(a, a) f waQ(x, u(x)) Ma (Du(x)) dx
Gu(w) IaI+IQI=n
D
for any n-form w
w =
E waQ(x, y) dxahdya Iat+-IQI=n
where a(a, a) is the sign of the permutation which reorders the multi-index (a, a) in its natural order, and Ma (Du) denotes the determinant of the minor of Du with rows din) and columns (al, ... , an). Alternatively this can be seen as follows. We have
duo' A...hdu8k =
up ;k dxjlA...Adxj'k
and those terms will contribute to waQ dxandyQ only if the differentials
138
2. Integer Multiplicity Rectifiable Currents
and dx31,...,dx31-
dx«l,...,dx«>.-,.
are different, so that (ii,... , jk) must be a permutation of 5. Reordering (ji, ... , jk) to d, and then (a, a) we then get dx«1 A ... Adx«n-k AduQ1 A... Adupk = Mg (Du) dx«ndxa = c- (al &) Ma (Du) dx'A ... Adxn . {E Ma (DU) 211/2 is the element of area
It is worth noticing that I M(Du) on cu,.fl and the n-vector
M(Du) _
a(a, a) Ma (Du)
l
I«I+IQI=n
_ (el + Dlu''ei)A ... A(e,, + D,luiei) or more precisely M(Du)/jM(Du)j is the unitary tangent n-vector orienting gu,,?. Thus we can write Gu (w) _
f <w, M(Du)
> d7-ln L Gu,,f2
compare Sec. 3.1.5 and Sec. 3.2.1. For future purposes we would like to discuss some special cases. This in particular will show in which sense the boundary operator describes the geometric boundary.
0
Dimension and codimension 1. Let n = N = 1, Q = (0, 1), U = (0,1) x R.
As a 1-forms in U have the form w = cpl (x, y) dx + cp2 (x, y) dy, with cpl, W2 E C,c°(.fl x R), we get
G.(w) = J[i(x,u(x)) + cp2(x,u(x))u'(x)] dx . 0
For any 0-form 77 = cp(x, y), we have d77 = cps dx + cpy dy so that
if[cox(x,u)+co(x,u)uu]dx aGu(97) = G(dr7) = 0
dx(x, u(x)) dx = cp(b) - cp(a) 0
where b = (1, u(1)), a = (0, u(0)). From the previous computations we see that
aGu=0
2.4 Integer Multiplicity Rectifiable Currents
139
if we regard Gu as a current in U, as 77 E Do(U), and 8Gu = Sb - Sa E Do (IR x IR)
if we regard G,,, as a current in R x R. Notice that 8b - dd E Mo(R x IR) and is the current carried by the graph of ul arz, so that, regarding Gu as a current in IR x IR
8Gu = Nan,uiej the graph of ul an being oriented in the induced way.
2 Codimension 1. Suppose n > 1, N = 1. Then Gu,f2 has dimension n and an n-form in IRn+1 can be written as w = wo (x, y) dxl A ... Adxn +
Then Gu(w)
f
=
wi (x, y) dx' A ... Adxzn ... Adxn ndy . n.
[wo(x, u(x)) +
j:(-1)n_zwz(x, u(x)) ux, I dx . i=1
By Gauss-Green theorem we can compute Gu (dri) = Q agu,.? (77) = Q guy an J 0/)
so that 8Gu = 0 if we regard Gu as an element of Dn (.fl x ]R) 8G,, = Q 9u1637 1 if we regard Gu as an element of Dn (Rn x IR)
This can be easily seen, for instance, if n = 2 and Q = {(x1,x2)
1
x1 < 0}. If
77 = cot dx1 + 02 dx2 + cp3 dx3 E Dl (IR2 x IR)
one easily computes
(id x u)#d7) _ (-colx2 + 2x1 - Wlx3ux2 + (Q3xiux2 -cp3x2ux1)dx'Adx2 { - D2 [c1(x, u(x))] + D1
l
(x, u(x))] + Dl [co3 (x, u(x) )]ux2
- D2[co3(x, u(x))]uxl } dxlndx2.
Denoting by v = (1, 0) the exterior normal to 812, we then get
J
(id x u)* d77
f[c02(o,x2, u(0, x2)) + c3(0, x2, u(0, x2))ux2] dx2 9uian
(77)
Similarly one proceeds in the general case, compare Sec. 3.2.3.
2. Integer Multiplicity Rectifiable Currents
140
0
R2
Mapping from R2 into R2. Let u E C1(.(l, R2), 7 C I[F2. Any 2-form in R2 X can be written as 2
wij (x, y) dxindyj
w = wo (x, y) dxl ndx2 +
+ w2 (x, y) dy' Ady2
i,j=1
Thus G,,(w) is given by 2
[Ou (x)) + E (-1)z_lwi n
X, u(x))uxi + W2 (X, u(x))det Du(x) I dx.
i,j=1 11
Rectifiable and integer multiplicity rectifiable currents. Let U be an open set in R'. Given (i) an lk-measurable and countably k-rectifiable set M C U (ii) an lk-measurable map 9 : M -> IEi; which is locally 71k L.M-summable (iii) an 71k-measurable map 1; : M - AkR' with I ICI I = 1 xk L M-a.e.
we say that a current T E Dk (U) is of the type 'r(M, 0, ), T = ir(M, 9, 1;) if
T(w) =
T= By changing l; --* l sign 0, 9 -> 9 sign 0, we can and do assume that 0 > 0, and we call 9 the multiplicity of r(.M, 9,1;). Sometime one refers to
set (T) := {x E IR'
9k(kk T 1 1
,
x) > 0}
as to the set of T. Of course T determines set (T), while possible M's are determined only up to sets of Nk-measure zero.
Definition 1. We say that a current T E Dk(U) is rectifiable if and only if T = -r(M, 9, C) for some M, 0, C as previously and moreover l; (x) is a k-vector associated to the tangent plane TTM for 71k L M-a.e. x. If moreover the density 0 is integer-valued, then we say that T = r(Nl, 0, C) is integer multiplicity rectifiable, briefly i.m. rectifiable. The class of integer multiplicity rectifiable k-currents in U will be denoted by Rk(U). For currents of type rr(.M, 9, l;) we have
1ITHH = 97,ckLM
2.4 Integer Multiplicity Rectifiable Currents
141
while, according to our discussion after the definition of the mass and the Euclidean mass in Sec. 2.2.3, for rectifiable currents we have and
IC11=ICI=1
IITII=ITI=BfkL.M.
Of course the class of rectifiable currents in U form a linear space, also
Ti +T2 E 7Zk(U) ,
if
T1, T2 E Rk(U)
but in general AT E Rk(U) only if T E Rk(U) and A is an integer. The following two theorems of Federer and Fleming make the class of i.m. rectifiable k-currents Rk (U), very natural and important especially in connection with the calculus of variations.
Theorem 1 (Closure theorem). Let {T3} C Rk(U) be a sequence of i.m. rectifiable k-currents in some open set U C 118' satisfying
sup[MV(Tj) +MV(aTj)] < oo
(1)
VV open, V CC U
j
and weakly converging to some current T E Dk (U), Tj -s T. Then T E Rk (U)
.
This theorem together with the compactness theorem for general currents yields
Theorem 2 (Compactness theorem). Let {T3} C Rk(U) be a sequence satisfying (1). Then there exists a subsequence {Tj, } of {T3 } and T E Rk (U) such
that Tj' - T. Defining the weak convergence in Nk,ioc(U) as
Tj - T if
weakly in Nk,1oc(U)
Tj - T
in Dk(U)
sup (MV (Tj) + My (aTj)) < 0o
j
bVCCU
we can state Theorem 1 as: Rk (U) n Nk,l0C (U) is (sequentially) weakly closed in Nk,l0C (U)
Notice that not every current in Rk (R') belongs to Nk (JRI) . For instance a collection of 2-dimensional disjoint disks in R3 with radius rj satisfying 00
00
rJ2 j=1
< 00 ,
T3 = 00 j=1
clearly yields a current T in 7 2(IR3) for which M(aT) = oo.
2. Integer Multiplicity Rectifiable Currents
142
(a)
Fig. 2.4. (a) A sequence of density 1 currents converging to a current of density 2, (b) A rectifiable current with boundary of infinite mass.
(b)
We shall return to Theorem 1 later in Sec. 2.2.6 and Sec. 2.2.7. But first we would like to present a few examples with the aim of illustrating Definition 1 and Theorem 1 and Theorem 2. Of course limits of smooth manifolds Mj, identified with the rectifiable currents 7(M (MjQAi ) with density 1, can be in principle rectifiable currents with integer density not necessarily 1, therefore it is natural to allow integer densities in order to identify such limits. As for distributions we cannot expect any regularity of the limits in the sense of currents of smooth manifolds Mj, if there is no control on the mass of the currents Q Mj ][, i.e.,
supM(QMj f) = +oo .
Assuming only Q Mj - T in Dk(U), the components of T need not even be measures. 4
Consider the sequence of smooth maps
uj
:
[0,1] --> R,
uj(x) := sin 2 j7rx
.
For the associated i.m. rectifiable currents Gui carried by the graphs of uj we trivially have
M(Gu,,) -) oo
,
asj ->oc.
Also i
Gu, (cp(x, y)dx) = fca(x,uj)dx 0 1
Gu, (z/)(x, y)dy) = f(x,uj)jdx 0
2.4 Integer Multiplicity Rectifiable Currents
143
Fig. 2.5. Young measure and distributions as zero and first components of the limit of i.m. currents. 1r
J 0
d J V1 (X, T) dr - f(xr) i
(171
[ dx
0
dx
0
f
Ui
1f
- J dx 0
'0. (x, T) d-r
0
Therefore, taking into account Proposition 2 in Sec. 1.2.6 we see that the zero component of Gu,, converge to the Young measure in (0, 1) x 1[P given by
G1L(0,1)xA,
r
J
dA
)n2dx
,
o 1
Gui (co(x, y)dx)
-- f(x,sin2x)dx 0
while 1
Gu; (0(x, y)dy)
sin/27rx
- f dx J 0
Ox (x, T) dr
0
which is not a measure, but just a distribution in R x R.
Next we want to show that the closure theorem does not hold for realmultiplicity rectifiable currents.
5 The closure theorem for i.m. rectifiable 0-currents. Let k = 0, U = R'n. It is readily seen that T E 7Zo(U) if and only if
2. Integer Multiplicity Rectifiable Currents
144
T=
aisz;,
zi E R', ai E Z .
,
i=1
Let {T3 } be a sequence in Ro (JRY) with
N(T) := M(T) +M(aT)
sup N(T3) < oo ,
.
j
Since the coefficients aii) are integers, we can choose a common s and write each Teas
Tj = i=1
with supj,i Ja(') < oc. Therefore passing to a subsequence we have
P)
-+ zi
or
z2j) I
-+ oo ,
and
ai
as j -* oo, from which we infer the (elementary) closure theorem
T=
Tj
aiazt E Ro(U) . i=1
Let us allow now real densities. Assume for instance k = 0, n = 1 and U :_ (0, 1), and consider the sequence of rectifiable currents with real multiplicity 1
Tj
bile
1
3
i=1
We then have, compare Sec. 1.2.5, that T. converge weakly to T = ,C1 L (0,1) which belongs to Mo (1k) but is not rectifiable. Of course similar examples occurs
in any dimension. For instance the sequence of real-multiplicity rectifiable 1dimensional currents in R2
Tj := 1EQ(0,1)x{i/j} 3i=1 with sups N(T3) < oc converges to the 1-dimensional normal, but non rectifiable current T := C2 L (0, 1) x (0, 1) e1. 6 The closure theorem for i.m. rectifiable 1-currents in R. and let T = -r (M, 9,T) R, (R). Of course we can assume
= oe1,
Let k = n = 1,
v:M -+ {tl}.
From the decomposition theorem for normal n-currents in IR', compare Sec. 4.3.1, or by a simpler argument in this case, we see that each T E Ri (J) n Ni (IR) has the form
2.4 Integer Multiplicity Rectifiable Currents
T=
145
giQ(ai,bi)
0. Since aT = Ei=1 qi (Sb - Say) and
where fail f1 {bi} = 0, qi E Z, qi M(aT) = 2 E8=1 IgiI > 2s we get
s < N(T)
M(T) + M(aT)
.
If {Tj } C Rl (R), sups N(Tj) < oo we can therefore write S
Zj
qi
Q (ai
bi
)
i=1
for some fixed s independent of j. If moreover we assume that spt TT in contained in some fixed compact set we then find as previously a(j)
-, ai,
qz.7)
b(j) ""' bi,
--, qi,
Tj -s
qi Q (ai, bi)
i=1
We would like to show now that the assumption
sup M(aTj) < 00 i
is essentially necessary in the closure theorem. We shall see that if the previous bound does not hold, and Tj - T, we can have
Concentration: the support of T may be a lower dimensional set Distribution: T = T(M, 0, T) where T orients M, but 0 is not integer (iii) T = T(M, 0, C), where 0 is integer but 6 is not tangent to M. (iv) combination of the previous phenomena, compare Ch. 1. (i) (ii)
7 Concentration. Consider the sequence of 1-dimensional currents
Tj := j Q (0,1/j) ]j E 7Z1(R) fl N1(IE8) Of course M(TV) = 1, M(aTj) = 2j -, oc, and we have T,j
-s boe1
which belongs to M1([8), but is not rectifiable, its support is a zero-dimensional set. 8
Distribution. Consider the sequence of 1-dimensional currents
i-A i
T, i=1
E 71(R)nN1(R)
2. Integer Multiplicity Rectifiable Currents
146
where 0 < a < 1. Of course supj M(Tj) < oo, M(0Tj) = 2j -- oo. One easily sees that
Tj - T := A
(0, 1)
and, trivially, T is not i.m. rectifiable. 9
Consider the sequence of i.m. rectifiable 1-currents in U = R X R = R2
j (0, 1/j) X {i/j}l E R1(R2) n N1(11 2) .
Tj
i=1
Then Tj = T(Mj,1, e1), Mj = Ui=1(0,1/j) x {i/j}, and M(Tj) = 1, M(aT3)
E Fig. 2.6. One dimensional i.m. rectifiable currents converging to a non rectifiable current.
2j -- oo. Clearly 711
Tj
L Nl3
H' L M,
T = 7-(.A4, 1, el)
M := {0} x (0,1) .
Since the tangent vector to M is M := e2 # e1, the limit current T is not rectifiable. 10 We remark that M(aT) = +oo in the previous example. More generally, if T = 'r (M, 0, C) E Mk (RI) where M is a smooth k-dimensional submanifold, 0 > 0, and C is a smooth vector field which is not tangent to M, then we have
M(aT) = +oo. Let us prove it in the special case T = Q (0, 1) x {0} e2 E .M1(R2) , as the same argument works in general, compare also Lemma 1 in Sec. 3.3.2. We have
T=T1e1+T2e2, consider the 0-form
T1-0,
T2=711 LM
,
.M=(0,1)x{0}.
2.4 Integer Multiplicity Rectifiable Currents
147
77(x1,772)
where
if xI < 2
1
3-xI if2<xj <3 ifxI>3
0
and, for a fixed E > 0,
-1
if x2 < -E
(1/E)x2
if Ix21 < E
1
ifx2>E.
One computes 1
8T(77) = T2(7l2) = fz(x'o)dx =
1 E
0
Being 1771 < 1, we therefore infer M(aT) > 8T(77) = E, i.e., M(aT) = oo. 11 Concentration-distribution. The different phenomena corresponding to (i) (ii) (iii) may appear in a combined form. Let us see two simple cases. Set for j = 2, 3, .. .
Mj:=U i=1
z
\j
z
3
C(0 1).
Clearly Tj E R1 (]R) n Ni (EP); as xi (Mj) = 1/j2, we have
M(Tj) = 1,
while
M(aTj) = 2j3 -* oo.
The hypotheses of the closure theorem are not satisfied, but
Tj - T := r((0,1),1, e1) = Q (0,1)11 E 7Z, (R) n Ni The concentration behaviour is seen from the fact that
1'(Mj) -' 0 ,
>
11(Mj) < 00
3
and for any positive e, we can find a Borel set E with x1 (E) < E such that for large j we have
IITj11((0,1)\E) = 0, i.e., M3 C E. Such a concentration however produces the i.m. rectifiable current Q (0, 1) 1 which is distributed in (0, 1). Consider now the following variant. Set again
148
2. Integer Multiplicity Rectifiable Currents
Fig. 2.7. One dimensional i.m. rectifiable currents converging to a 1-dimensional diffused current. Tj
T(Mj,j2,el) E 7Z1(R2)
where this time
i_
Mj :=
1
,
7-12LQ
,
i) x f h
C R2
i,h=1
Then
j2 7 1LMj
Q= (0,1) x (0,1)
and
Tj
T:= T(Q, 1, el) = 7-12 L Q e1
.
Again we have concentration, with this time a non rectifiable current distributed on the 2-dimensional set Q. Of course in this case we again have M(aTj) = 2j4 -+ 00. 12 Finally we would like to remark that the vanishing of OT in some sense fixes the orientation of T. Let B(0, 1) be the unit ball in 1R2, trivially T =
T(B(0,1), 1, eine2) defines an i.m. rectifiable current in R2 without boundary in B(0,1). Also T1 +T2, where T(B(0, 1/2), 1, eiAe2)
T1 T2
:=
T(B(0,1) \ B(0,1/2),1, -eine2)
,
is an i.m. rectifiable current in R2 ; however we have
a(T1 +T2) L B(0,1) = 2QOB(0,1/2) ]
0.
2.4 Integer Multiplicity Rectifiable Currents
149
Image of a rectifiable current under a Lipschitz map. Let T be an i.m. rectifiable k-current in U C R', T = -r (M, 0, C) E 7Zk(U), U C 1R71, and let f : U C R'n -+ V C RN be a Lipschitz map such that f, spt T is proper.
Then the push-forward or the image of T under f we have already defined in the general context of currents turns out to be an i.m. rectifiable k-current which can be explicitly written as
f#T(w) = f
(2)
<w(f(x)), (AkdMfx)S(x)>0(x) d7-(k(x) .
M
This can be easily seen in the case in which T is the current integration over a smooth submanifold and f of class C1, and actually in the same way in our present situation, as we can write M apart from an 7{k-zero set as union of 7-lk-measurable subset of k-dimensional C'-submanifolds. Recall in fact that
I
an orientation of the rectifiable set f (M). Therefore from the general area formula we have
f#T(w) = f <w(y),
(3)
f(M)
xEf-'(y)rm+ 0(x)
> d7lk(y) I
I
where (4)
Nl+ = {x E M I JM(x) > 0}
.
In fact choosing any orthonormal basis (Tl (y), ... , Tk (y)) in Ty f (M) in a 7-lk-measurable way for 7-lk-a.e. y E f (M), setting 71(y)
Ti
(y)A... ATk(y)
and for Nk-a.e. y E f (M)
N(y) :=
9(x)E(x) xEf -' (y)nM+
where e(x) = ±1 is defined by
(ilkdMfx)e(x) (AkdMfx)t(x)
= 6(x ) 77(f (x))
2. Integer Multiplicity Rectifiable Currents
150
we can write
f#T(w) =
J
Nd7-lk
f(M)
f#T = r(f(M), INS, sign Nil) and clearly
E
IN(y)I =
0(x)
xEf -1(y) nM+
_I
0
I
Fig. 2.8. (a) 7r# Q S' J = 0, 7r# I I Q flI I = 2-L' L (-1,1); (b) -#Q AB -J = Q (0,1) ], =AA.C' L(0,1).
Later we shall be interested to the case in which f is the projection it
:
(x, y) E Rn+N -+ x E 1n and T = -r (M, 0, ) E Rn(R n+N) As a trivial example consider the case n = N = 1 {(x, kx)
Nt and
I
x E (0,1)} , e + kE
T = T Mill 1+k2
k> 0
E R1(R2) .
Then we have 7r#T = T((0,1), 1, e) = Q (0, 1) 1
.
Notice that
T - H1LM 1+k2+711LM
1+k
and, as measures, we have
7r(7-I1L.M) =
l+k2H'L(0,1)
k2e
2.5 Slicing
151
while -ir#e = e, lr#E = 0 so that 7r#T=
1+k2H1L(0,1)
1+k2
e=f1L(0,1) e.
Remark the difference between projecting a measure and projecting a current!
The Cartesian product of rectifiable currents. It is also not difficult to show that the Cartesian product of two i.m. rectifiable currents we have defined in the general context of currents is also a rectifiable current. More precisely, if Ti = T (Mi, 8i, Si) E Rki (Ui) ,
U i E ]E8'
,
i=1,2
then
Ti X T2 = 7-(Ml x M2, 8102, eiA 2) E Rki+k2 (Ul X U2) Notice however that T1 and T1 x T2 rectifiable do not imply that T2 must be rectifiable. For instance take
Ti = kQ(0,1)l , so that
T2 = k Q(0,1)
T1xT2 = [(0,1)x(0,1)1.
2.5 Slicing Let T be a k-dimensional current in Rn and let f : lI , R' be a Lipschitz map. Corresponding to y E R' one often needs to consider the part of T in f -1(y) as a (k - m)-dimensional current k > m, usually denoted by . There is a measure theoretic method to define the slice of a current T by f at least if T has certain regularity property. It has very interesting applications for instance to intersection theory in algebraic geometry and to complex analysis. Later we shall use some elementary properties of the slices. Therefore we shall collect here a few facts concerning slicing. First let us consider codimension-one-slices, i.e. the case m = 1. In order to
define the slice of T by f at y we need T to have some regularity property. We shall now see that it can be defined for locally normal or i.m. rectifiable k-currents, and of course the two definitions agree if T E Rk(Rn) f1 Nk,1oc Il$ n
Definition 1. Let T E Nk,ioc (U) where U is an open set in R', i.e., let
Nv(T) := Mv(T) +Mv(aT) < oo and let f : U
IR be a Lipschitz function. We set
d V cc U
,
2. Integer Multiplicity Rectifiable Currents
152
(b)
(a)
(c)
Fig. 2.9. (a) Q B(0,1)1 n if < t}, (b) -8Q B(0,1) L if < t}, (c) < QB(0,1)
(1)
a(T L { f < t}) - 8T L { f < t}
-a(TL{f >t})+aTL{f >t}
t >.
where
If > t} { f < t}
{x E U I f (x) > t} {x E U f (x) < t} I
Of course we have
:= = if
M(TL{f =t})+M(aTL{f =t}) = 0,
and this happens for all but countably many values of t. Therefore is well defined for 7-11-a.e. value t.
Consider now the case of i.m. rectifiable currents. As a consequence of the theory of rectifiable sets and of the general coarea formula, in particular taking into account Said type lemma in Sec. 2.1.4, we infer Lemma 1. Let .M be an 7-lk-measurable and countably rectifiable k-dimensional
set in R', f : R -+ I[8 be a Lipschitz function, and
M+ := {x E M I Jm(x)
:_ I (AkdMfx)eI > 0}
where 1; (x) is a k-vector orienting TIM . Then for 7-1' -a. e. t E R we have (i) ,Mt := f-1(t) n M+ is countably rectifiable and 7-1k-1-measurable. (ii) For 7 1k-1-a.e. x E .Mt TM t and TxM both exist, TxMt is a (k - 1)dimensional subspace of TxM and
TxM = {T + AVM f (x)
I
T E TxMt, A E IR}
where V- "f (x) is the gradient off projected on TxM.
2.5 Slicing Moreover
fdtfgd71_1
(2)
R
153
= JpvMfIgd7k M
Mt
for all non-negative 7-lk-measurable functions g.
Applying (2) to g X{f
f
r
tt
I V f lg d7-lk =
Mn{f
J -00
ds
J M..,
g
dHk-1
so that the left hand side is an absolutely continuous function of t and for a.e.
tE]l. (3)
dt
f
IVMf Ig dRk
= fgdh/c_1 Mt
Mn{f
Using the scalar product <, > in AkTTM, define now for l; E AkTTM and w E TAM the (k - 1)-vector 6 L w by (4)
:= <e,wA97>
drlEAk-1TM.
Then one sees that for 7-L1-a.e. t E ]l8 and 7-fk-1-a.e. x c Mt
Fig. 2.10. Slice
VMf (x) L I VMf (x) is a unit simple (k - 1)-vector.
E
Ak-1TTMt
154
2. Integer Multiplicity Rectifiable Currents
Definition 2. Let T = -r(M, 0,1;) E Rk(U). For a.e. t E R we define the slice of T by f at t
as the rectifiable current in 7Zk_1(U) given by
:= 7-(JV1t, et, fit) where for 7-1k-1-a.e. x E Mt
et(x)
fi(x)L
et(x)
0+,&t
Vmf(x) I v f (x) I ,
0+
0XM+
We have
Proposition 1. Let T = -r (M, 0, 1;) E Rk(U), U C RT. Then (i) For any open set V C U
f Mv()dt= f IvMflBdxk<Jsup IoMfIMV(T). MnV
Ilt
(ii) If moreoverT E Nk,lOC(U), i.e, Mv(aT) < oo for all V c c U, then for a.e.
tER
= a(TL{f
aTL{f < t}
.
(iii) If also aT is rectifiable, then for a.e. t E R
= -a
.
For locally normal currents in Nk,lo,:(U) we finally have
Proposition 2. Let T E Nk,lo,(U), U C Rn. Then
(i) spt C sptTn{f=t}
.
(ii) For all open sets V C U
Mv() <supIDfI limio hMv(TL{t < f < t+h}) v
Mv() <supjDfl limia Mv(TL{t - h < f
fMv(TL{f
a
Thus bb
fMv()dt< supIDfIMv(TL{a < f
V
2.5 Slicing
155
We shall not prove Proposition 1 and Proposition 2, but we would like to make a few remarks on their proofs. Proposition 1 (i) is a consequence of (2). The proof of Proposition 1 (ii) is done by approximating f with its smoothings fem. The proof of Proposition 1 (iii) is simple. One applies (ii) of Proposition 1 to OT
<3T, f, t> = a(3T L {f < t})
,
while applying a to the identity in (ii)
a< T, f, t > = -a(aT L If < t})
.
In order to prove (2) one first consider the case f E C', and then one gets the result by approximation. For f E C' and -y : IR --* [0, +oo) smooth and increasing we get
[a(TLyo f)(3TLyo f)] (W) =(TL7of)(dW)-aT(yo fW)=T(-yo fdw)-T(d(yofw)) =T(yofdw-d(-yofw)) = -T(-y' c, fdfAw)
.
Choosing -y such that -y = 0 for t < a, -y = 1 for t > b, and
0 < y' <
b+¢ E
fora0,
we infer
a(T Ly o f)(W) - (aT L-y o f)(W) --> if b 1 a and this implies (i). Also
IT(y'ofdfAW)I
SsupIDfI-±-Mv(TL{a< f
which implies (ii) and (iii). Quite more complicated is to deal with slices of codimension larger that 1. To
motivate the procedure let us consider the special case T = r(M,1, ) where M is a smooth k-dimensional submanifold in U C Ilk' oriented by the k-vectorfield 6, with ?-l!` (M) < oo, and let
f:R' -- R',
k>m
be smooth. For x e M we factor
e(x) _ r7(x)AC(x) so that 77(x) is a simple m-vector of TIM, 17(x) = 1
= Jf (x) ((x) is a simple (k - m)-vector of TxM, C(x)I = 1, and
,
156
2. Integer Multiplicity Rectifiable Currents ker d1M f (x) is associated with C(x)
in case J fm (x) > 0. Using the coarea formula one finds that if w E Dk-m (U) and cp is any real valued Borel function on ]gym, then
T L f#(cpdyln...ndym)(w) =T(f#(cpdy1n...ndym)nw)
f
=
f#(cpdy1A...Adym)Aw(x)>d7-lk(x)
M
= Jpj(x))Jf (x)<((x),w(x)>dHk(x) M
= li
=
f f
f
cc(y)
<((x),w(x)>d7-lk-m(x)d7-Cm(y)
Mn f-1(y)
W(y),r (Mn f(y),1,()(w)dxm(y)
R-
It follows from the differentiation theory that for 71 -a.e. y TL
f#(XB(y,p)dyiA...
Adym)/(wnpn) -, T(M n f-1(y),1, C)
in Dk-.,,,.(U) as p - 0. When T E Nk(U) is an arbitrary normal current (or even a flat chain), one introduces for y E U and p > 0 the (k - m)-currents TP :=
1
wmP,m,
TL
f#(XB(y,P)(t)dt1n...ndtm)
E Dk-m(Rn)
and defines the slice of T at y as the weak limit of Tp
:= li oTp whenever it exists. It turns out that exists for 7-lm-a.e. y. In fact for a E Dk-m(U) we have f# (T L a) E N,,,, (Rm), hence, compare Theorem 2 in Sec. 4.3.1,
f#(TLa)(w) = fhaWdX,
w E Dm(Rm)
for some ha E Ll(]Rm). Consequently TT(a) = (_l)m(k-m) f ha(t) d7-Lm B(y,p)
and limp_,o Tp (a) exists for each fixed a and 7-lm-a.e. y by the Lebesgue differentiation theorem. If we choose a denumerable subset C of Dk-m (U) which is dense
2.6 The Deformation Theorem and Approximations
157
with respect to the uniform topology we infer that for 7-lm'-a.e. y limp-o Tp(a) exists for all at E C, and, as
IT( a)1 <M(f#(TLa))
Dk-,,,,(U).
Also one has (i) (ii)
spt c f -1(y) n spt T,
a = (-1)"' ,
k > m.
Moreover (iii) (iv)
E Nk_m(U) for fm'-a.e. y (T L f#dy'A... ndym)(w) = f (w) dxm(y), w E Dk-m(U)
(v) M(TL f#dy'A...Adym) = fM()d?-lm(y), and finally (vi)
is i.m. rectifiable for 7..( a.e. y if T is i.m. rectifiable.
Finally we remark that there is no problem in extending the theory of slicing to the case in which f takes values in an oriented m-dimensional submanifold of class C1.
2.6 The Deformation Theorem and Approximations In this section we shall state without proofs the important deformation theorem and some of its consequences: the isoperimetric inequality and the approximation theorem.
Consider that e-grid or E-skeleton, E > 0, of Rn+N Ln,e
{x E In
I
1x E Zn+N} E
and denote by Ln,E the n-dimensional faces of Ln,E, or equivalently the family of all n-dimensional cubes in lRn+N with side E and vertices in Ln,e
Theorem 1 (The deformation theorem). We have (i) Let T E Nn(Rn+N) and let E be any positive real number. Then there exist a real polyhedral chain P made of n faces in Ln,E, i.e.
P = E aj j Fj I
aj E R
Fj E.Cn,c
and
R E Nn+l(1E2n1N)
such that
S E Mn(In+N)
T = P+aR+S
.
158
2. Integer Multiplicity Rectifiable Currents
Moreover P is controlled as follows
M(P) < cM(T) ,
M(aP) < cM(aT)
and the rest aR + S is small in the sense that
M(R) < ceM(T)
M(S) < cEM(aT)
and
M(3R) < c{M(T) +EM(aT)} , where c is an absolute constant. Finally spit P, spt R C U5 (spt T)
spt aP, spt aR C Ub (spt aT )
where b = b(E) = 2 n _+ NE and U5(A) :_ {x
I
dist (x, A) < b}.
(ii) If moreover T E Rn (Rn+N) is i.m. rectifiable, we can choose P and R i. m.
rectifiable, i.e. R E Rn+1(iR +N) and
P = E aj [ Fj
with aj E 7G
,
FjEG,,,E
and, if also aT is i.m. rectifiable, we can choose S i.m. rectifiable, too.
Such a result is obtained by deforming T into Ln,e, so that the result is automatically a polyhedral chain. The main error term is aR, where R is the surface through which T is deformed; S is a secondary error term due to moving aT into the skeleton Ln,E. It is not difficult to see that there is no real problem to perform such a procedure by projecting from points inside the (n + 1)-cubes defined by Ln,E, if T does not wind about the center projection and has no autointersections. In the general case the key point is choosing the center of projection suitably by observing that in the average the distortion is still controlled. We shall postpone the proof to Sec. 5.1.1. A first consequence of Theorem 1 is the following
Theorem 2 (Isoperimetric inequality). Let T be a boundaryless i.m. rectifiable (n - 1)-dimensional current in Rn+N with compact support. Then there exists an i.m. rectifiable n-current R with compact support such that aR = T and (1)
M(R) < c
M(T)n/(n-1)
where c = c(n, N).
Proof. From the deformation theorem we can write in correspondence of E > 0
T = P+aR+S . The result follows by choosing e, but strangely, E large. From
2.6 The Deformation Theorem and Approximations
159
M(S) < c_-M(OT) = 0 we infer S = 0. Next we observe that if
P= E aj[F,
O,
F3 EGA,E
we have
N(e) integer M(P) = N(e) en-1 , as a. E Z and M(QF, ]]) > en-1. From M(P) < cM(T), and the previous inequality we therefore infer by choosing (2cM(T))1/(n-1)
e that
N(e)en-1 = M(P) < cM(T)
=Zn-1
,
i.e. N(e) = 0. Then P = 0 and T = 8R with spt R compact and
M(R) < csM(T) = c'
(M(T))n/(n-1)
As in the classical case it turns out that the best constant c in (1) is that corresponding to equality in
M(vT) <
cM(T)n/(n-1)
which arises when T is a k-dimensional ball.
The next consequence of the deformation theorem we want to state is the following
Theorem 3 (Weak polyhedral approximation). Let T E Rn(U), U open in Rn+N, be such that aT E Mn_1(U). Then there is a sequence of i.m. polyhedral chains of the form Pj
a(i) QF50) ]I
a(j) E Z, FS(j) E Ln,j,
s
S
such that (2)
sups (M(P,) + M(8PM < oo
PP - T
weakly in Dn(U)
and consequently (3)
aPj -1 aT
weakly in Dn_1(U)
.
-- 0
160
2. Integer Multiplicity Rectifiable Currents
Proof. First assume U = Rn. We apply the deformation theorem with e = ej to obtain
T = P3 +&R;+S3
with
sup(M(Pj) +M(aPj)) < c(M(T) +M(aT)) < oo M(RS) < cep M(T) ,
M(SS) < cep M(aT)
.
Hence S? - 0, Rj - 0, consequently ORS - 0, and we get Pj - T. In the general case we take cp : 118'+N _, ]18 with cp > 0 in U, cp - 0 in n+N \ U such that {x v(x) > Al CC U V A > 0. For x1-a.e. A > 0 we infer from Sec. 2.2.5 that TA := T L {x W(x) > A} is such that M(aTA) < oo. Since spt TA CC U, we can apply the previous argument and conclude the proof by choosing a suitable sequence A, 10. I
I
Notice that in particular Theorem 3 says that every i.m. rectifiable ndimensional current is the weak limit in the sense of currents of a sequence of smooth n-dimensional submanifolds of I18'+N One can also prove, compare Sec. 5.1.1
Theorem 4 (Strong approximation). Let T be an i.m. rectifiable current in Rn+N, T E Rn(Rn+N) with compact support and OT E Rn_1(118'). Let e > 0. Then there exist an i.m. rectifiable polyhedral chain P in Rn+N and a C1diffeomorphism of ][8'+N into ll8'+N such that (4)
M(P - f#T) + M(aP - of#T) < e
.
Moreover
spt P C {x dist (x, spt T) < e} =: Ue (spt T) I
(5)
Lip f-1 < 1+e Lip f < 1+e , f(x) - xI < edxEI[8'+N and
We emphasize that inequality (4) means that P and f#T, as well as aP and f#8T coincide up to a piece of small mass. Usually the class of i.m. polyhedral chains of dimension n in Rn+N is denoted by P,(R,+N) It is defined as the class of currents of the form
where a, . E Z and Zj are n-dimensional simplexes in R +N Finally we would like to state a similar approximation theorem with mass for normal currents. Here it is essential to use the mass norm of n-vectors instead of the Euclidean norm, compare Sec. 2.2.1 and Vol. II Ch. 1. We denote by Pn,(I8'+N) the class of polyhedral chains with real coefficients. We have, compare Vol. II Sec. 1.3.4 and Federer [226] for a complete proof
2.7 The Closure Theorem
161
Theorem 5 (Strong approximation theorem for normal currents). Let T belong to N,,(R'+N) with sptT compact. Then there exists a sequence of polyhedral chains P; with real coefficients, P3 E Pn(Rn+N), such that
T
Pj
weakly in Dn(Rn+N)
aPj - aT weakly in Dn_1(Rn+N) and
M(T)
M(PG)
and
M(aPj) -+M(8T).
2.7 The Closure Theorem In this final subsection we would like to indicate the main steps in the proof of the closure theorem for i.m. rectifiable currents, Theorem 1 in Sec. 2.2.4.
Let U C Rn+N be an open set and let T E Nn,loc(U). For a E U and r < ra, := dist (a, aU) we know from Sec. 2.2.5 that the slice of T over 9B(a, r) is well defined as
:= 8(TLB(a,r))-(8T)LB(a,r) and is an i.m. rectifiable (n-1)-current for ?1-a.e. r < ra, if moreover T E Rn(U). We have
Lemma 1 (Slicing lemma). Let Tk,T E Nn,loc(U) be such that sup N v (Tk) < no
bVCCU
k
Tk - T
weakly in V,(U)
Then, given a e U, for.Cl-a.e. r < ra there exists a subsequence, depending on r, {Tk} of {Tk} such that
supNv() < no
(1)
(2)
VVcc U.
k
If moreover for some Vo CC U we have Nv0 (Tk) -+ 0 then {Tk} can be chosen in such a way that also Nvo () -+ 0.
Proof. Passing to a subsequence we can assume that 11Tk11+I
aTkH - Y
where p is a positive Radon measure on U. If p(OB(a, r)) = 0, (notice that this holds for all but countably many r at most), we then infer
TkLB(a,r)
TLB(a,r)
,
9TkLB(a,r)
aTLB(a,r)
,
162
2. Integer Multiplicity Rectifiable Currents
hence
a(TkLB(a,r)) -s 8(TLB(a,r))
,
consequently (1) holds. For 0 < rl < r2 < rd and V CC U, we also know from Sec. 2.2.5 that r2
f Mv()dr < cMv(TkLB(a,r2)). rl
The same holds for
a = -a(aTk L B(a, r)) _ - hence
r2
f fk(r)dr < c Nv (Tk L B (a, r2)) . rl
where
fk(r) := Nv() > 0.
Fatou's lemma and the equiboundedness of Nv (Tk) then yield r2
r2
lkm inf fk(r) dr < lim of j fk(r) dr < oc
.
rl
rl
Therefore for a.e. r E (rl, r2) we have liminfk_. fk(r) < oo. Choosing those is we can finally find a subsequence such that (2) holds. The rest of the claim is trivial.
A part from the simple remark in Lemma 1, the first key ingredient in the proof of the closure theorem is the following theorem which says that if almost all slices of T by spheres, i.e., by f (x) := Ix - al, are rectifiable then T is rectifiable, the second ingredient is the boundary rectifiability theorem stated below.
Theorem 1. Let T E Mn,lar(U). Suppose OT = 0 and that for all a E U and almost all r < ra, the slices
:= a(T L B(a, r)) are i.m. rectifiable. Then T is i.m. rectifiable.
Let us postpone the proof of Theorem 1 and let us give the Proof of the Closure theorem. We shall prove it by induction on the dimension n of the currents (keeping fixed n + N and U).
Let Tk E Rn(U), supk Nv(Tk) < oo V V CC U, Tk -L T in Dn(U). By the same argument in Theorem 3 in Sec. 2.2.6 we can assume U = Rn+N and supk N(Tk) < oo.
2.7 The Closure Theorem
163
The theorem clearly holds for 0-dimensional currents, compare Sec. 2.2.4. Suppose that it holds for (n - 1)-dimensional currents and let us prove it for n-currents. From the proof of the boundary rectifiability theorem below (where one uses the closure theorem for (n - 1)- currents) we infer that
aTk E Rn_1(U) and of course N(aTk) < oo. Using again the closure theorem for (n-1)-currents we therefore infer aT E Rn-1(U). By the isoperimetric theorem, for instance, we can then find S E Rn_1(U) such that OS = aT. By considering
T := T-S
TK := Tk - S,
we see that it suffices to prove the theorem in the case aT = 0. Fix a. By the slicing lemma, for a.e. r < ra, there is a subsequence {Tk} of {Tk} such that
,
sup N() < oo. k
Again by the proof of the boundary rectifiability theorem / E R,-,(R n +N )
and by the inductive assumption we conclude
E Rn_1(Rn+N) The conclusion now follows from Theorem 1.
Theorem 2 (Boundary rectifiability theorem). Let T E R ,,,(U), U open in I[Bn+N Suppose
VVccU.
MS (T) < oo
Then OT is an i.m. rectifiable current in U, aT E Rn_1(U). Proof. By the polyhedral weak approximation theorem in Sec. 2.2.6 we find a sequence {Pk} of polyhedral chains with integer coefficients with
supN(Pk) < oo
Pk - T .
,
k
Trivially
aPk E Rn-1(U)
,
sup N(aPk) < oc . k
Therefore applying the closure theorem (note: for (n -1)-currents) we infer that aT is i.m. rectifiable, as aPk - aT.
2. Integer Multiplicity Rectifiable Currents
164
Remark 1. We have seen that the closure theorem for boundaryless i.m. rectifiable (k - 1)-currents in lRn implies the boundary rectifiability of i.m. rectifiable k-currents in R. Let {Ti} be a sequence in Rn_1(Rn) satisfying spt T2 C B(0,1) ,
5Ti = 0 ,
Ti - T in Dn,_1(Rn)
We can write
Ri E Rn(ln) ,
Ti = aRi ,
spt Ri C B(0,1) ,
sup N(Ri) < oo . k
Trivially, since the Ri's are n-currents in ll8n, then Ri - R, R E Rn(R'), and
Ti - T=8R By the boundary rectifiability theorem we then conclude T E Rn,_1(R' ). In other words: The closure theorem for boundaryless i.m. rectifiable currents of codimension 1 is equivalent to the boundary rectifiability theorem for i.m. rectifiable currents of maximal dimension.
Let us discuss now the proof of Theorem 1. It relies on the important and difficult Besicovitch-Federer structure theorem, compare Theorem 3 in Sec. 2.1.4. The first step consists in proving the following lower density lemma.
Theorem 3 (A lower density lemma). Let T E V (U), U C Rn+N with
MS(T)+Mv(aT) < oc
VVCCU.
Then
(i) For II T II-a.e. x
A(x, r)
_
T OIITII(B(x,r)) - 1 where
A(x, r) := inf{M(S) 18S = a(T L B(x, r)), S E Dn(U)} . (ii)
If moreover aT = 0 and if a(T L B(x, r)) is i.m. rectifiable for every x and almost every r, then there is S > 0 such that
B*(IIT II,x) > d for II T
II-a.e. x E U.
Proof. Clearly .(x, r) < II T II (B(x, r)), so if (i) is false there is an e > 0 and a set X such that I I T I I (X) > 0 and for each x E X there exist arbitrarily small
balls B (x, r) with
A(x, r) < (1 - e) M(T L B(x, r)) . We assume
2.7 The Closure Theorem
165
XcVccU. By Besicovitch's covering theorem, for every p > 0 there exists a disjoint collection B1, B2 ... C V of such balls covering II T II-almost all of X and having radii < p. Let Si E V,,,(U) be a current such that
M(Si) < (1 - e) M(T L Bi)
9Si = 3(T L Bi) , and set
T. := T->TLBi+I:Si. i
i
Then, denoting by xi the center of the ball Bi, we have
(T-Tp)(w) =1: (TLBi -Si)(w) i
E ,9(5.i x (T L Bi - Si)) (w)
< 1: < hence Tp
(8yi x (T L Bi - Si)) (dw)
pM(TLB2-Si)supIdwl
M(Ox(TLBi-Si))supIdwl
2pM(T L Bi) sup dwl < 2pMv (T) sup Idwl
,
T as p --+ 0 and
Mv(T) < liminfMv(Tp) p-0
(3)
.
On the other hand Mv(TT)
<
Mv(T->TLBi)+>Mv(Si) i
i
Mv(T-TLBi)+(1-E)Mv(TLBi) i
i
Mv(T)-EEMv(TLBi) i
< Mv(T) - EII T 11(X) which contradicts (3). To prove (ii) consider a point x at which (i) holds, and let
f (r) := M(T L B(x, r))
.
Then for r small, r < R,
f (r) < 2A (x, r) By the slicing theory in Sec. 2.2.5, as 8T = 0,
M(8(T L B(x, r))) < f' (r)
for a.e. r
2. Integer Multiplicity Rectifiable Currents
166
and by the isoperimetric inequality A(x
c f'(r)
r)(n-1)/n <
for a.e. r
7
since the slice a(T L B(x, r)) is rectifiable. Thus f(r)(n-1)/n <
c f'(r)
or
dr
f (r)1/n > c'
.
Therefore
f(r)
>_
(c'r)n
for r < R.
Next step is the following rectifiability theorem
Theorem 4 (Rectifiability theorem). Let T E M,(U), U C Rn+N, with aT c M,-,(U). Set
M :_ {x E U I and suppose that
0' (11 T II , x) > 0}
IITII(U\M) = 0
i. e. that
e*n(IITII,x)>0
for IITII
- a.e. x EU.
Then T is a real-multiplicity rectifiable current.
Proof. First we show that Nl is a countable union of sets of finite fn-measure. In fact, setting Mk :_ {x 19*n (II T II, x) > 1/k} by Theorem 5 in Sec. 1.1.5, we obtain
k Hn(Mk) < II T II (U) < oc
M = UMk
fn (.Mk) < 0C.
k
Let us prove now that l;(x) orients TIM, for H'-a.e. x E M. Write 00
M = Mo U U M j=1
with Hn(M0) = 0, M3 pairwise disjoint, and Mj C N N; a C1 submanifold. From
2.7 The Closure Theorem (4)
0*n(JI
0*n(1I T 1j, U Mr, x) = 0,
167
T II,JVj \ Mj, x) = 0
r,-Ej
for Hn-a.e. x E M j, writing as usual 77,,,a (y) := x + A-1(y - x) we have for any w E V(Rn+N)all x E .Mj for which (4) holds, and all A sufficiently small d < ?-ln , #aw+>B6f
yx,a#T (w) = T (rl#Aw) =
(A)
N.
where e(A) -- 0 as A 10, i.e. 77y,A#T(w) =
I
<6(x + Az), w(z) >B(x + Az) dH'(z) + e(A) .
7=a (Nj )
Since Nj is of class C1, this gives limr7x,a#T(w) = 0(x)
r
JP
w(z) > d7-ln(z)
for 'H'-a.e. x E Mj, where
P=TXNj =TxMj On the other hand from M(aT) < oo we get 1ar7y,a#T(w)I_ 1?7x,a#8T(w)I = 18T(71#aw)j
IWlPWn0*n(HHaTHH,x)
A
for H'-a.e. x E Mj. Thus for such x
lim M(aix,a#T) = 0. a-o Finally we clearly have
lim M(r7X,a#T) < oo .
a-.o
Thus we conclude that for fn-a.e. x E M we can find aZ 10 such that
97z,a,#T - Sx
,
where (5)
S. (W) = 0(x)
SS E Dn(Rn+N)
aSx = 0
r <e(x),w(z),>d?-ln(z) J
.
T. M
We finally claim that (5) and OS,, = 0 imply that fi(x) orients TIM. Without loss of generality we can assume
T 0 M= Rn x {0} C Rn+N
2. Integer Multiplicity Rectifiable Currents
168
Choose
w :=
x'cp(x)dx"A...Adx2°-1
where j > n+ 1, cp E and 5Sxo = 0
0 =
C°O(Rn+N).
E
Dn-1(Rn+N)
Since xi = 0 on Rn x {0} we deduce from (5)
asxo(w) = Sxo(dW) _
= B(xo) f w(x)<e(xo), dx'AdxilA...
Adxin-1 > d7{n(x)
ToM 0(xo)
f
e,Aej1A... / \eZn-1 dH'(x)
ToM As W is arbitrary we then get
e(xo) ejAei1A... Aei,,_1 = 0
whenever j > n + 1
This yields at once C = ±e1 A ... Aen as required, since
1.
Proof of Theorem 1. By the lower density lemma, Theorem 3, there is 8 > 0 such that o (MT
'X) > 8
for 11 T 11-a.e. x
.
Thus if we let
M = {x E 1[$n+N
0'(11 T 11, x) >Jj
we have 11 T
(6)
I(Rn+N
\ M) = 0,
and from Theorem 5 in Sec. 1.1.5
Hn(M) < b-1HH T 11 (M) < oc
(7)
.
On the other hand Theorem 3 in Sec. 4.3.1 and (6) imply that
11 T
absolutely continuous with respect to xn L M, thus we can write
11TI1(f):=J fOd7"(n and
(8)
T(w) = f<w(x),T(x)>8(x)d(x). M
One can now apply the rectifiability theorem Theorem 4 and infer that T is a real multiplicity rectifiable current. Therefore it remains to prove that O(x) E Z
7-Lna.e.xEM.
2.7 The Closure Theorem
169
With the notations in the proof of Theorem 4 we have
Si := rlx,a,#T - 0(x) If TXM
for xn-a.e x E M
where QTXM]I is oriented by e(x), and N(S,) < oo. Let d(y) := IyI, y E ]lBn+N Applying the slicing lemma we infer that for a.e. r and for a subsequence Sn3 , that we again denote by S we have
<Sj,d,r> - 9(x)IfTMf18B(0,r)If with sups N(< Si, d, r>) < oc. Choosing now r in such a way that is rectifiable, we infer that all the < S? , d, r > are rectifiable too and the closure theorem for (n - 1)-currents then yields that 9(x)QTTM fl 8B(0, r) If
is rectifiable, i.e. 0(x) E Z.
A direct proof of the closure theorem.. We close this section giving an alternative proof of the closure theorem, in fact of Theorem 1, which does not use the structure theorem nor the rectifiability theorem, but the rectifiability criterion for measures Theorem 2 in Sec. 2.1.4. A key point is again the lower density lemma Theorem 3
The proof proceeds by induction, and similarly to the previous proof it is enough to prove the following claim:
Theorem 5. If T E Dn(lRn+N), 8T = 0, M(T) < oo, and for every Lipschitz function f , the slice = a(T L { f < r}) is i. m. rectifiable, then T is i.m. rectifiable.
In the proof one can and do freely use the closure theorem for (n - 1)-currents in R,+N which is available because of the inductive hypotheses. Proof of Theorem 5. As in the proof of Theorem 4 we infer using the lower density lemma that if we let
M = {x E Rn+N
I
B; (11 T 11, x) > d}
then (9)
(10)
I T 11 (IRn+N \ .M)
= 0,
xn(M) < 5-'j 1TII(.M) < co.
Moreover from Theorem 3 in Sec. 4.3.1 and (9), we then infer that absolutely continuous with respect to xn L M, thus we can write
T(w) = or
T II is
170
2. Integer Multiplicity Rectifiable Currents
f<w(x),r(x)>dm(x)
T(w) =
T(x) := O(x)T(x)
M
and we need again to show that M is rectifiable, for 7-ln-a.e. x E M, T(x) is a
simple vector associated to TAM, and 8(x) is an integer. We then prove the claim using the rectifiability criterion of measures, Theorem 2 in Sec. 2.1.4 instead of the Besicovitch-Federer structure theorem. From general measure theory we know that for Hn almost every a E M we have (11)
0'* (fn, M, a) < 1
(12)
lim
f
7-ln(.M n B(a, r))
I F(x) - F(a) I d?ln(x) = 0
MnB(a,r)
and we fix one such a point a E M. By (12) and the definition of M we have
O (Hn,M,a) = F(a)I-le*(IITII,a) > 0
(13) and
limsup7-l' L77a,A(M)(B(0, R)) = lim sup A-n7{n(M n B(a, AR)) A-0
A-0 wnRn8n* (7 (, M, a) < oc
where as usual
,
A-1(x - a). It easily follows passing to subsequences
7-ln L ?7(M)
(14)
p
a Radon measure, and
J
f<wr(a)>ddb =: sa(w)
<w, (a) >dn L a.\, (M)
for any n-form w E V
,n+N ). Setting
(
Tak = 'la,Ak Yf'T from
(M)) _
MB(o,1) (T)1k - F(a)Rn L
nMB(a,akR) (T - , (a)7-1n L M)
J
.\k n
A
1F(x) - f(a) I d7-ln (x) --'. 0
MnB(a,Akr) we also infer (15)
TAk Tak 11
- Sa
lL L F(a) = S.
2.7 The Closure Theorem
171
Since for every Lipschitz f, almost all slices are rectifiable, from (15) the slicing lemma, and the (n - 1)-dimensional case of the closure theorem we infer that almost all slices <µLT(a), f,r> are i.m. rectifiable. Thus by Theorem 3 we have the lower density bound
j
0*(µ,x) -! T(a)I > 0
(16)
for µ-a.e. x.
We now claim that p is translation invariant in at least n-directions, and, since the set of point where (16) holds has locally finite 7-ln-measure, in exactly
n-directions. Denoting by V the subspace of jn+N of vectors v such that T is invariant under translation in the direction of v, we shall in fact prove that F(a) E AnV. This implies that dim V > n, and by the previous remark, dim V = n and 7(a) is simple. By making an orthogonal change of coordinates, we can assume that V has a basis consisting of vectors of the standard basis of Rn+N, e1, ... , en+N. Of course, in order to prove the claim, it then suffices to show that if el V, then F(a) has no-component of the form e1Aei2A ... Aei,,. By mollification we can also assume that II µ LT(a) jI = 7{,+N L 6 for a smooth 6. Then for all f E (Rn+N) 0
=
LT(a))(fdx2A...Adxn) = (pLT(a))( ax dxindx2A...Adxn)
(0(tZ
ai-xi0(x)dx= f>aif i where
ai := the eiAe2A... nen component of T(a) . Hence
i.e. 0, consequently FL L r(a) is invariant under translation in the direction E aiei. Since el V we conclude that a1 = 0 which means that 7(a) has no component of the type e1Ae2A ... nen. The same argument can be used to prove that T(a) has no component of the type elsei2A ... Aei,,. Returning to the proof of the theorem, we then find a collection PI, ... , Pp of n-planes parallel to the n-dimensional subspace determined by T(a) such that p
jU -
j=1
aj(xn LPj)
By (11) and (14) p
Eai < on*(7{n, M a) < 1 j=1
172
2. Integer Multiplicity Rectifiable Currents
and by the lower bound (16) aj > 8/IT(a)I, sop must be finite. We now claim p = 1. Suppose not and let
3E < min{dist (Pi, Pj)
I
Pj I.
Pi
Fix j and let f (x) := dist (x, Pj ). Also assume without loss of generality that Pi is parallel to ]I8n x {0} C Rn+N and set U:= B(0, R) C I2n and W := U x I[8N C fgn+N By the second of (15) lim Mw (T.\, L {E < f < 2E}) < ?(a)Iµ(W n {E < f < 2e})
koo
so by the slicing lemma we can find a further subsequence and t c (E, 2e) such that
kL{f
(17) Tl
and such that
Mw(8(Ta,, L { f < t})) --> 0. In particular
Mu(87r#(TAkLIf
(18)
Also 7r#Ta, L If < t} is an n-dimensional current in IR with bounded mass and boundary mass, hence it is represented by a BV function Uk
7r#T,\k L If < t}(W) =
JU
UkW
with
MU(87r#(Ta,, L{f < t})) = fIDuk. U
By Poincare inequality for suitable /3k we infer
f IUk - akI dx < C U
JU
IDUkI.
From (17) (18) it follows that (3k
Q = ajIT(a)I,
fInk -pI dx
0.
U
But
QkCn'({XEU
I
Uk(X)=0}) < J *k-ak(dx < c U
so
f U
IDUki
3 Notes
Gn({xEU
I
uk(X)=0})
173
0
Thus using also (14)
lim G'({x E U I G'(U) = k-.oo
uk(x)
0})
urn " (7f(fla,ak (M) n if < t}) n u)
<
lim7{CL7]a,Ak(M)({f
k--.w
µ({ f < t} n w) = aj7-C(U) . It follows that each aj > 1, and by (11), (13) and (14) 0 < 0 (7l", M, a) < E aj < 0'n* (f', M, a) < 1
j
Hence p = 1 and al = 1. Furthermore P1 must pass through the origin since for every r > 0 ,u(B(O, r))
> lim ff'L r)a,ak (M)(B(0, r)) k-oa liminf)\- H'(M n B(a,)\r)) a-.o
r-"0 (HjT j], a) > 0 by (13). Finally note that P1 is determined by T(a) and therefore does not depend on particular subsequences.
This shows that M has an approximate tangent plane for fl -a.e. a E M. Hence M is n-rectifiable by Theorem 2 in Sec. 2.1.4. As in the previous proof, see the proof of Theorem 1, or observing that by the (n -1)-dimensional closure theorem µ LT(a) has integer multiplicity slices and µ = 7{n L P1, we finally get that 0(a) = ft(a) is an integer.
3 Notes 1 The classical references for geometric measure theory are Federer's treatise [226], and Simon's book [592]. In our presentation we mostly followed Simon [592] and its terminology and notation, which slightly differ from those of Federer [226]. The reader will surely find also interesting consulting Morgan [485] Hardt and Simon [357]. We also mention Whitney [674], the foundational paper of Federer and Fleming [230], and De Rham [189].
2 The rectifiability theorem for Radon measures Theorem 2 in Sec. 2.1.4 is essentially due to De Giorgi [177]. In our presentation we have follow Simon [592]. The structure theorem is due to Besicovitch [83] for k = 1 and to Federer [223]. A presentation of Besicovitch's original proof can be found in Falconer [218], the general case is discussed in Federer [226].
174
2. Integer Multiplicity Rectifiable Currents
Concerning rectifiable sets, we would like to mention the following interesting criterion, see e.g. Alberti, Ambrosio, and Cannarsa [8] and Simon [596]. Given any set S C R' and any point x E R', define T(S, x)
J{rp j
r > 0, p = li .
x - xh X
- xh
l xh I x}t xh E S \f{x},
and the tangent space of S at x, Tan(S, x) as the smallest subspace of R' which contains T(S, x). We then have
Theorem 1. If
dimTan(S,x) < k
d x E S,
then S is fk-measurable and 7-lk-countably rectifiable.
3 The content of Sec. 2.2.1 and Sec. 2.2.2 is quite standard. The reader may consult e.g. Federer [226], and for an elementary presentation Fleming [238]. In Sec. 2.2.3, Sec. 2.2.4 we presented basic definitions and facts, as well as some simple examples. The reader is referred to the books quoted above. There one also finds the proof of the general slicing theorem, of the deformation theorem, and of its consequences. For the isoperimetric inequality the reader is referred to Almgren [22], where the optimal inequality in any codimension is proved, and to Almgren [21]. For further information on isoperimetric inequalities the reader is referred to Burago and Zalgaller [117] and to its bibliography.
4 The closure theorem was first proved for codimension one currents, i.e. (n - 1)dimensional currents in R' by De Giorgi [177]. We shall present it in Ch. 4 in the context of Caccioppoli sets. The ideas introduced in De Giorgi [177] play an important role in the general case. The first proof of the closure theorem in Sec. 2.2.7 is essentially the classical Federer-Fleming proof in Federer and Fleming [230] as modified by Simon [592], except for one point where we have used the lower density lemma, Theorem 3 in Sec. 2.2.7. Theorem 3 in Sec. 2.2.7 is taken from White [669], compare also Fleming [240] and Solomon [604]. The second proof, which develops in the same spirit as De Giorgi's proof in codimension 1, is due to White [669].
3. Cartesian Maps
In this chapter, we shall be concerned with the notions of graph, tangent plane to a graph, and weak convergence of graphs for non-smooth maps from IR' into RN, n, N > 1.
We begin in Sec. 3.1 by investigating the differentiability properties of Sobolev maps. As it is well known in this context the basic notion, at least for scalar maps, is that of distributional derivative, and Sobolev maps do not possess smoothness properties in the usual sense. In particular Sobolev maps are not even almost everywhere differentiable in the classical sense, except in the case that they have gradient summable with a power larger than the dimension n. However a theorem of Calderdn and Zygmund states that they are almost everywhere differentiable in the so-called LP-sense, compare Sec. 3.1.2. Essentially as consequence of this result we shall infer, compare Sec. 3.1.3,
that Sobolev maps, u E Wl,r(,f2,IRN), fl C IR', p > 1 possess the following Lusin type properties (i) (ii)
there is a denumerable disjoint family of measurable sets {Fk}, such that ulF, is Lipschitz continuous and meas (( \ UkFk) = 0. except on arbitrarily small open sets, u is of class C'.
It turns out, compare Sec. 3.1.4, that those Lusin type properties are equivalent to a sort of measure-theoretic pointwise differentiability notion called approximate differentiability almost everywhere. Approximate differentiability provides the right way to express local approximability by linear transformations and, just using the equivalence with the Lusin type properties mentioned above, the classical formulas for area and coarea and for the degree can be extended to almost everywhere approximately differentiable maps. All that may be regarded as a special case of the rectifiability theory of Besicovitch, De Giorgi, Federer and Fleming, we have partly described in Ch. 2. For us the important outcoming is that the graph of a map which is almost everywhere approximately differentiable is well defined, and is a countably nrectifiable set. Our definition of graph, CJu,s-i in Sec. 3.1.5, more than to the usual notion of graph of a continuous function defined in 0, is modeled after a kind of 1-graph in which each points z E Gu,SQ carries the tangent plane to CJu,n at z = (x, u(x)). An important consequence of such a definition is Lusin's property (N), i.e., that null sets in .f2 are mapped into 7i -null sets in fix RN, a
176
3. Cartesian Maps
property which is typical of Lipschitz maps, but which does not hold in general for continuous maps. This way we see that the graph and its tangent planes are not really related to the distributional derivative, but to the in some cases weaker and in general different notion of almost everywhere approximate differentiability of the map, which in fact will be the starting point for us in this chapter. As a simple example the real map x/Ixj : R -* R has a rectifiable graph with horizontal tangent plane everywhere except at zero, corresponding to the naive, and correct in the sense of approximate differentials, idea that it has zero (approximate) differential everywhere except at zero. A more complicate example which shows a lot of fractures in the graph is Cantor-Vitali function for which the approximate differential is zero almost everywhere and the graph in the previous sense consists of infinitely many horizontal segments, compare next chapter. Usually the approximate differential is denoted by apDu. As here it is our main notion of derivative, in the sequel we shall simply write Du for apDu if no confusion with the distributional derivative may arise. Approximately differentiable maps provide essentially the natural and correct class of maps whose graphs are ?f''-measurable and countably n-rectifiable
sets in the product Q x RN. Their tangent planes, at points of approximate differentiability, are described by the minors of the Jacobian matrix in terms of the approximate differential exactly as in the classical case. Consequently the n-dimensional area of the graph of an almost everywhere approximately differentiable map is finite if and only if all the minors of the Jacobian matrix are summable.
In Sec. 3.2 we introduce and discuss the class of approximately differentiable maps with summable Jacobian minors, denoted by A' (,(l, RN). Each map u E A'(17,RN) turns out to be identified completely by the integer multiplicity rectifiable n-dimensional current Gu defined as integration of compactly supported differentiable n-forms in 1 x RN over the graph GJu,.fl of u in .fl naturally oriented by the independent variables. Being interested in weak limits of sequences {uk}, there are at least three important elements which turn out to be relevant for a satisfactory geometric description of such weak limits, compare Ch. 1 and Ch. 2. (i)
the tangent plane, that is the existence almost everywhere of the tangent plane to the graphs of the {uk}'s or equivalently the rectifiability of the graphs of the {uk}'s.
(ii)
(iii)
the area of the graph or better the equiboundedness of the areas of the approximating graphs {uk}. In fact limits of graphs may be measures in the product RI x ISBN already in the case n = N = 1, if they do not have equibounded masses. the boundary in ,(2 x RN. The rectifiability of the graphs and the equiboundedness of of their areas are not sufficient to provide a good geometric descrip-
tion of limits of smooth graphs. In fact such limits may again be measures
3. Cartesian Maps
177
distributed in the product .f2 x 1RN, if the control of the size of boundaries of the approximating sequence is lost.
With respect to the first two points, the class Al (f2, RN) yields a satisfactory setting. In the sequel of Sec. 3.2 we discuss the boundary question. In the scalar case, N = 1, the summability of the distributional derivative turns out to be equivalent to the coincidence of the approximate differential and the distributional derivative and to the non existence of boundaries of the current Gu inside the cylinder (2 x R
aGuLQxIR = 0. This is no longer true in the vector valued case, N > 1. It will be true for maps u E Wl,?`(0,1RN), with n = min (n, N), that
3GULQXRN = o.
(1)
However (1) does not hold in general for maps u E Wl,p(.f2, RN) if p < n. For instance the map x/jxj from the unit ball B3 of ]R3 into JR3 belongs to W1'p for all p < 3; as det D(x/I x1) = 0, its graph has finite area, but one has aG
L B3 XR3 = -Jo x a{ B3
Condition (1) is obviously satisfied by smooth maps, and is also closed with respect to the weak convergence of graphs. In fact (1) can be equivalently stated as
G,,,(&,)) = 0
for all (n - 1)-forms w supported in ,f2 x 1R N. Thus it is natural to require (1) also for reasonable weak maps. This way we are naturally led to introduce the class of Cartesian maps as cart l (0, IRN) = {u E Al (.f2, JRN) aGu L !2 x ]RN=01 In Vol. II Ch. 2 we shall see that (1) expresses in the context of finite elasticity
the physical requirement that a perfectly elastic body does not fracture if we deform it. In Sec. 3.2.3 of this chapter we shall see that condition (1) amounts not only to the integration by parts formula for the derivatives (or equivalently to the existence in Ll of the distributional derivatives) but to all formulas of integration by parts, actually in the product space ,f2 x IRN, for all Jacobian minors, corresponding to the classical differential identities which state that all minors are null Lagrangians or divergence free fields. In Sec. 3.2.5 we shall then briefly compare traces in the sense of Sobolev spaces and boundaries. A few significant examples will be discussed in Sec. 3.2.2. In Sec. 3.2.4 we shall introduce special subclasses, denoted Ap,q, p > n - 1, q > n/(n - 1) whose
elements are Cartesian maps and discuss higher integrability of the Jacobian determinant.
178
3. Cartesian Maps
Having always in mind the question of identifying weak limits of smooth maps, we shall discuss in Sec. 3.3 the weak continuity of the minors. More pre-
cisely we ask under which conditions it happens that for a sequence {uk} in A' (fl, RN) such that strongly in L1 weakly in Ll Ma (Duk) - va weakly in Ll Uk --> u
v
Duk
we have
v = Du,
va
= Ma (Du)
.
After discussing a few examples, we shall show that essentially this happens
if aGuk LQ x RN = 0, i.e. if the {uk}'s are Cartesian maps. For the class cart' (Q, RN) we shall provide in Sec. 3.3.2 a closure theorem which essentially relies on Federer-Fleming closure theorem for integer multiplicity rectifiable currents. This way we are able to formulate quite general results concerning the weak continuity of minors which extend results as Reshetnyak's theorem on the weak continuity of determinants in the Sobolev setting. Of course cart' (Q, RN) misses nice compactness properties, it just inherits the compactness property of the L' space. This makes that class not suitable for the calculus of variations, and in fact we postpone to next chapter introducing and discussing the important class of Cartesian currents. Better compactness properties are of course enjoyed by the classes
cartP(fl, RN) :_ {u E AP(.fl, RN)
I
aGu L ,fl x RN = 0}
when p is strictly larger than 1.This will be discussed in Sec. 3.3.3. Starting from a sequence of smooth maps {uk}, for which we clearly have (2)
aGukL.0XIRN = O,
by Stokes's theorem, and assuming for instance that all Jacobian minors of {uk} are equibounded in some LP-norm with p > 1, we may then infer passing to a subsequence that
Guk - Gu where u E cartP(f2, RN). Therefore the non validity of (1) is a homological obstruction for the approximability of a map in AP(1?,RN). However it turns out that this is not the only obstruction. There are also homotopic obstructions to the approximability of maps in carte (0, RN). This follows from a result concerning the strong and the weak approximability of minors in LP. In this respect the known results seem to be still at an initial stage and better understanding seems to be needed. We refer the reader to Sec. 3.4 for precise statements and open questions.
1 Differentiability of Non Smooth Functions
179
1 Differentiability of Non Smooth Functions In this section we shall investigate pointwise differentiability properties of functions in Sobolev spaces. Let £2 be a domain in R', and let p be a real number in [1, oo). The Sobolev space W1'p(Q,118), simply W1"p(Q), we recall, can be defined as the class of all LP-functions in £ with derivatives in the sense of distributions in LP (fl)
W1'p(,fl) := {u E L2(Q)
(1)
j
Du E Lp(,f2,R")}
normed by 1/p
1/p
r (2)
II U II
J n
wl,P(n)
/
J IDuIp dx n
dx
Juj
It turns out that the linear space W1,p(,f2) is complete, i.e., is a Banach space with respect to the norm in (2), and even a Hilbert space for p = 2, with scalar product r / (u, v)w1,2(S?) := (3) u v dx + J Du Dv dx .
JQ
.n
One also shows that W1'p(Q) can be characterized as the closure of smooth functions in Wl,p, i.e., as the closure in W',P(,fl) of
{u
I
u E Cl (f2) f1 W 1'p(f2)} .
In general one interprets u in W1,P as a class of equivalence of functions in LP; in this section we shall instead work with a suitable representative of u and we are interested in its pointwise behaviour. Of course we cannot expect that Sobolev functions possess smoothness properties in the usual classical sense, but we shall show that a Sobolev function u is almost everywhere differentiable in the LP-sense, which in turn implies a Lusin type property, that is Sobolev functions are of class C1 except an arbitrarily small open sets. Of course the situation is
much easier for p > n. In this case it turns out that u is "almost everywhere differentiable" in the classical sense, and in fact we can find a representative of u which is continuous in ,f2 and almost everywhere differentiable: this is the content of the classical Morrey-Sobolev theorem. From our point of view the important outcoming of this will be the possibility of defining the graph 9Ju,.f2 of a Sobolev function u E W1,P(0, RN) and of showing that ccu,A is a countably n-rectifiable set for all p > 1. Actually both Lusin type property and the rectifiability of gu, fl turn out to be equivalent to the almost everywhere approximate differentiability, Sec. 2.1.4. In fact the notion of approximate differentiability, which is no more related to the distributional derivatives, is the right notion in order to define the graphs of a map and its tangent planes and will be one of the basic notions in this book.
180
3. Cartesian Maps
In Sec. 3.1.1 we present a few relevant facts about Hardy-Littlewood maximal operator which will be used later. As a consequence we shall then prove Lebesgue's theorem about the convergence of averages of a function on balls and an estimate of the Hausdorff dimension of the set of points which are not Lebesgue points of Sobolev functions. We conclude Sec. 3.1.1 proving the classical Calderon-Zygmund decomposition argument and giving a characterization of Zygmund class L log L. In Sec. 3.1.2 we shall prove the classical Calderon-Zygmund LP-differentiability theorem, some of its consequences, and Morrey-Sobolev theorem.
In Sec. 3.1.3, after recalling the classical Whitney extension theorem and Rademacher theorem, we shall present Lusin type properties for Sobolev functions. In Sec. 3.1.4 we shall discuss the notions of approximate limits, approximate differentials, and we shall prove equivalence of approximate differentiability and Lusin type properties. In Sec. 3.1.5, after discussing the validity of the area formula, we shall intro-
duce the notions of graph and tangent plane for vector valued maps which are almost everywhere approximately differentiable and we shall see that the area formula holds for such maps. This allows us to define the degree for maps with Jacobian determinant in LI. However we shall see that in such a general case the degree loses homological invariance and it is not stable by weak convergence.
1.1 The Maximal Function and Lebesgue's Differentiation Theorem In order to study the differentiability properties of Sobolev functions it turns out to be very useful to work with Hardy-Littlewood maximal operator which is defined, for every u E L o, (1R) and for all x E I[8"' as (1)
f
M(u) (x) = sup
r>O
I u(y) I dy
,
B(x,r)
where
1 ff()d.y
I f(y) dy
JAI
A
A
We shall now prove a few properties of M(u) which will be used in the sequel. Given a measurable function u(x) in R', the function (2)
X(t) := meas {x E R
u(x)I > t}
I
t E R+
is called the distribution function of u, and, if u E LP(]Wn), p > 1, one has
fudx Rn
= p f 00tP-1X(t) dt . 0
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
181
Moreover, for any u E L' (1R') we have
a(t) <
(3)
t
Estimate (3) is usually referred as a 1-1-weak estimate for u. In fact (3) is weaker than the summability of u, as it is shown for instance by the extension to zero of the function 1/x in (0, 1) which trivially satisfies (3) but is not in Li. where A = II U
11L1.
One can easily see that the maximal function M(u) of a non zero u E Ll (1R.' ) is not in Ll (R'y), in other words the operator u -> M(u) is not bounded in L' ('). Indeed if I u I > 0 on a set of positive measure, then M(u) decreases at infinity no faster than c IxHowever M (u) verifies a weak (1-1)-estimate which makes it very useful.
Proposition 1 (Maximal theorem). Let u E LP(R"), with p > 1. Then for any t > 0 (4)
meas {x E Rn
M(u)(x) > t} <
I
f
s
Iu(y)I dy
.
u >t/2
In particular M(u)(x) is finite almost everywhere.
Proof. Let u E Ll (Rn) and let
At := {x E Rn
M(u)(x) > t}
I
.
For every x E At there exists a ball B(x, r(x)) such that
Iu(y)I dy > t
T
B(x,r(x))
or equivalently
measB(x,r(x)) <
t
f
Iu(y)I dy
.
B(x,r(x))
Using Besicovitch's covering theorem we can choose a denumerable set of points {xi} in At such that UiB(xi, r(xi)) A and the balls B(xi, r(xi)) do not overlap
more than 3' times. Thus measAt
< >measB(xi,r(xi)) i
(5)
<
t
f a
u(y) I dy <
f
8
B(x;,r(x;))
IIfn
Suppose now that u c Lr(R'), with p > 1. The function ii (X)
{ u(x) if I u(x) I > t/2 0
if I u(x) I < t/2
I u(y) I dy
3. Cartesian Maps
182
is easily seen to be in L' (RI). Moreover
M(u)(x) < M(v,)(x) + 2
V x E Rn
hence
{x E lE$n
I
M(u) (x) > t} C {x E Rn
M(u)(x) > t/2} .
I
Inequality (5) applied to u then yields
meas{xEIR
I
M(u)(x)>t}
< 2 t3' 1 I u(y) I dy =
2
tan f
I u(y) I dy
juI>t/2
Len
The second part of the theorem follows easily from (1).
Remark 1. If u E LP(Rn), p > 1, Holder inequality trivially yields
M(u)(x)
[M(UP)(x)I1/P
Therefore we find
meas {x E R' I M(u) (x) > t} < meas {x E Rn I M(uP) (x) > tP}
f
c
f
IuIPdy
dy,
It
ju>t/2
ItLIP>tP/2
i.e., the following (p-p)-weak estimate meas {x E
(6)
R
I
M(u)(x) > t} <
IuVPdy.
Actually we have
Proposition 2. Suppose 1 < p:5 +oo and u E LP. Then M(u) E LP and IIM(u)IILP(R') < A(p) IIuIILP(Jn)
(7)
where A(p) is a constant which depends only on p and on n, A(p) -> 00 as p -+ 1.
Proof. Using Proposition 1 we infer 00
f IM(u) I P dx = p f tP-1meas{ x I M(u) (x) > t} dt Rn
o 2ju(x)P
00
= 6np f
tP-2 ( f
Iu(x)I dx) dt = 6np
Iu>t/2
0
p
=6n2P-1
p
-
f ( f tP-2dt) Iu(x)I dx
JR"
0
IJu(x)IPdx
1
Rn F-I
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
183
Remark 2. In fact we also have u E LP if M(u) E LP, p > 1, as consequence of Lebesgue's differentiation theorem below.
Proposition 1 is a useful tool in many questions which we are not going to touch. For our purposes, we only point out a trivial consequence of (4) contained in the following proposition, and we derive the fundamental differentiation theorem of Lebesgue.
Proposition 3. Let Uk, u E LP (R'), with p > 1. (i) If Uk --i u in LP, then M(Iuk - uIP) -> 0 in measure, i.e., for any e > 0
meas {x E Rn (ii)
I
M(luk - u1P)(x) > s} -4 0 as k --} no
We have tPmeas {x E l[8n
M(u) (x) > t}
I
----->
0
as t - oc
Proof. The weak estimate (6) yields meas {x E
IR
I
M(l uk
- uI P) (x) > E} <
6p II uk - u 111P
I
thus (i) follows; while (6) and the absolute continuity of the integral yield (ii).
0 Theorem 1 (Lebesgue's differentiation theorem). Let u E LP(Rn), p > 1. Then for almost every x we have
lu(x)I < M(u)(x)
(8)
and (9)
to
f
l u(y) - u(x) I P dy = 0;
B(x,r)
in particular lim
(10)
r-.o
-f
u(y) dy = u(x)
11
B(x,r)
holds for almost every x.
Let {uk} be a sequence in C0 (l ) which converges to u in LP(lRn), Passing to a subsequence we can assume that Proof.
(11)
uk(x) -> u(x)
for a.e. x
and, since M(Iuk - uIP)(x) -- 0 in measure, that
184
3. Cartesian Maps
M(luk - uIP)(x) --> 0
(12)
for a.e. x
.
Denote by E the set where (11) and (12) hold
E := {x E R'
(13)
(11) and (12) hold}
1
Clearly meas (llU \ E) = 0. Let us prove that (8) holds for all x E E. Since Uk is smooth, we have
uk (x) = lira f uk (y) dy B(x,P) thus
uk(x)I C O t
IUk(y)I dy :5 M(Uk)(x)
B(x,p)
from this inequality (8) follows taking into account (11) and (12). In order to prove (9) define now VP(u)(x)
:= sup f lu(y) - u(x) 11 dy p
For p < r we have 1/P
Cf
l u(y) - u(x) 11 dy
< oscB(x,P)uk
+lt
dy}1/P
T
[(u(y) - uk(y)) - (u(x) - uk(x))]P
B(x,P)
< oSCB(x,P)uk + [M(Iu - ukVP)(x)]1/P + lu(x) - uk(x)I .
Thus, taking into account (8) for x E E, we find that for all k and x E E (VP(u)(x))1/P < oSCB(x,r)uk + 2[M(lu - ukl')(x)]1/P
(14)
Now for every k we choose bk so that osCB(x,bk)uk < 2-k uniformly in x, we then deduce from (14) that for any x E E
lim (Vak(u)(x))1/P < lim 2-k +2 lim
k-00
k--.oo
k-,oo
ukIP)(x)]1/P = 0
and this implies (9). Finally (10) is a consequence of (9). Remark S. For future uses, we explicitly state that (8), (9) and (10) of Theorem 1 hold for all x E E, where E is the set defined in (13).
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
185
Actually the previous considerations apply to any representative u of a func-
tion in L'(R') since we have used, compare Theorem 1, pointwise values of U.
Definition 1. Let u E
The Lebesgue set Cu of u is the set of points x E R" for which there exists A E IR such that (15)
f
l u(y) - A dy
as r --> 0 .
0
B(x,r)
For x E Lu, the number A = Au(x) is uniquely defined by (15) and it is named the Lebesgue value of u at x. It is clear from the definition that Lu and Au(x) are uniquely defined by the class of equivalence of u E Ll(IRn) and by Theorem 1 meas (W1 \ Lu) = 0 .
(16)
Any function u(x) such that u(x) = Au(x)
dx E Lu
is referred to as a Lebesgue representative of u. For instance we can set
u(x) - f 0Au(x)
if x 0 Lu if x E Lu
,
It is easy to check that Vx E Lu
Iu(x)I < M(u)(x) If moreover u E L oc(R") we also have (17)
-/
u(y) - u(x) Ir dy ---- 0
as r -> 0 .
B(x,r)
for almost every x E W'. We shall denote by Lu,p the set of points where (17) holds.
Points in ,f2\Lu are often called exceptional points, and Q\Lu the exceptional set of u. We shall now prove that the Hausdorff dimension of the exceptional set of a function u E W11 P(Q), 12 open in ll8u, is not greater then n - p. In doing that the next lemma is the key point.
Lemma 1. Let .f2 be an open set of IIRA, v be a function in Ll r(f2) and 0
EaxES-l I limsupp ' p-0
J B (x,p)
Then we have
'H-(E,,) = 0
jv(y)jdy>0
}.
3. Cartesian Maps
186
Proof. It suffices to show that for each compact subset K C 12 we have 7-((F) _ 0 where F = Ea f1 K. Set
P-' f I v(y) I dy > s-1 }.
x E F I lim sup
F(s)
B(x,p)
Obviously F = Us_1F(S), hence it suffices to show that ?-('(F(s))
0 for
all s. Let Q be a bounded open set with K CC Q CC .(1 and let min(1, dist (K, 8Q)). For all e > 0, 0 < e < d, and for all x E F(s)
d :_ there
exists r(x), 0 < r(x) < e, such that
f
r(x)-«
Iv(y)I dy > 2s
B(x,r(x))
Applying Besicovitch's covering theorem we can then choose a denumerable set
of points {xi} in F(s) such that U,B(xi,r(xi)) J F(s) and the balls B(xi,ri), ri = r(xi), do not overlap more than 3' times. Therefore we infer (18)
Eri < 2sE i
i
v(y)I dy = 23ns
I
f
Iv(y)I dy
UiB(xi,ri)
B(xi,ri)
Since a < n, this inequality implies (19)
ri <
meas(UiB(xi,ri)) C w,,,
i
i
< 23'se'-" J Iv(y)I dy
w,,,e'''-aEr"
wn, = meas (B(0, 1))
Q
From (19) and the absolute continuity theorem, applied to (18), it follows that the right hand side of (18) goes to zero as e -+ 0, and therefore that 1-(" (F(s)) = 0.
We can now state
Theorem 2. Let ,(l be an open set in ]Rh and let u E Wlo'cp'(fl), 1 < p < n. Set
El :_ {x E .(L
I
lim sup
pp-n f
P-0
I DuIp dy > 0}
B(x,p)
E2
Then we have
{x E (I lim o
u(y) dy does not exist finite} B(x,p)
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
187
=0
(20)
Wn-P (E1)
(21)
7 fn-P+e (E2) = 0
b e> 0.
Moreover the exceptional set of u has Hausdorff dimension not greater than
n - p, dimH(.fl\Gu,)
(22)
de > 0.
Proof. Lemma 1 yields at once (20). To prove (21), for a fixed xo we consider the function
u(x) dx =
O(P) :=
u(xo + Py) dy
-}
B(0,1)
The function 0(p) is differentiable in (0, oo) and
0' (P) = f Y Diu (xo + py) dy B(o,1)
hence 10' (P) I
1/P
(f
<-
I Du (x) I P dx)
.
B(xo,p)
Suppose now that xo 0 En-P+E, i.e.,
p-n+P-e
f
I Du(x) I P dx -+ 0
B(xo,p)
then
W(P)I <
cp-1+;
,
and therefore P
10(P) -O(s)1 <
which implies that xo
0'(t)I dtj <
IPA/P -SEIPI
E'2. Therefore we conclude that
E2 C E.-p+,
Vc
.
This implies (21), because of Lemma 1. Finally Poincare inequality
f B(x,p)
I u(y) - u.o,P 1 P dy < c pP
f
I Du(y) I P dy
B(x,p)
yields at once that every point in .(1 \ (E1 U E2) is a Lebesgue point for u and consequently (22) holds, if we take into account (20) and (21).
188
3. Cartesian Maps
The function
u(x) = loglog
lxj-1
which belongs to W1"2(B(0,1)), B(0,1) C R2, shows that Theorem 2 is optimal
for p > 1. Later we shall see that the case p = 1 is special. We shall in fact see in Sec. 4.1.4, Remark 3 in Sec. 4.1.4, that the exceptional set of a function u E Wl"l(f2) has zero (n - 1) -dimensional Hausdorf measure. We conclude this subsection by proving a fundamental decomposition theorem which is very useful in many instances from real and complex analysis to partial differential equations. Here we shall use it in order to give a characterization of the class of functions with maximal function in L'. Let Q be a cube in Rn. Dividing Q into 21 congruent cubes and iterating the process one obtain the so-called dyadic decomposition of Q.
Lemma 2 (Calderon-Zygmund decomposition argument). Assume that u is a non negative function in L'(Q) and let t be a positive number such that
I u(x) dx < t
.
Q
Then there exists a countably family {Qi} of cubes in the dyadic decomposition of Q such that -(i)
t< f u(x)dx<2'1t Qi
(ii)
u(x) < t
for a.e. x E Q \ UiQi
Proof. By bisection of the sides of Q, we subdivide Q into 2n congruent and equal subcubes. Those subcubes P which satisfy -1- u(x) dx < t P
are similarly subdivided and the process is repeated indefinitely (or finitely if there is no such cube). Let Q = {Qi} denote the family of subcubes so obtained which satisfy
f u(x) dx > t Q.
and for each Qi denote by Qi the subcube whose subdivision gives Qi. Since
jQil = 2" QjI we get immediately (i) as
f u(x) dx < t . Qi
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
189
If x E Q \ UiQi and is not on the boundary of some Qi then clearly it belongs to infinitely many cubes P of the successive subdivision in 2n subcubes with API --+ 0. Since, on the other hand, at every Lebesgue point x we have i
liim
t
Ju(y) - u(x) I dy < c(n) b(1limo
EP P
Ju(y) - u(x) I dy = 0 ,
B(x,,8(P))
J(P) being the diameter of P, in particular
u(x) = PIlimo + u(y) dy i
,EP JP
we infer (ii) at once.
Remark 4. Actually Calderon-Zygmund decomposition Q = {Qt} depends on the parameter t. Let {Q(t)} and {Q(s)} be the families corresponding to the parameters t and s, t < s. Then each Q(3) is contained in some Qjt). This is clear from the construction. In fact we have
f u(x)dx>s>t, Q2")
therefore in the chain of cubes {P} which are successively divided into 21 equal cubes and produce Q,8) the first cube P for which
f u(x) dx > t P
belongs to Q(t).
We shall now prove that the distribution function of the maximal function M(u)(x), the function
if
t -> t lu(x)I dx uI>t
,
and the measure of the union of the cubes Q,t) of the Calderon-Zygmund decomposition relative to Jul and t are equivalent for large values of t. In fact we have
Proposition 4. Let Q be a cube in W1, let u E L' (Q) and let t be such that
t>
f Q
3. Cartesian Maps
190
Denote by Q(t) the family of subcubes of Q in the Calderon-Zygmund decomposition relative to Jul and t. We have (23)
2_n t-1
J IuI>t
Iu(x)I dx < E
f
2 t-1
i
lu(x)I dx
IuI>t/2
and for a constant -y = 'y(n) (24)
t J
Iu(x) l dx < cl (n) meal { x E Q l M(u) (x) > y(n)t }
IuI>t
<
f J
C2 (n) t
Iu(x)I dx
IuI>7(n)t/2
Proof. Calderon-Zygmund lemma yields
f
f
Iu(x)I dx <
IuI>t
Iu(x) I dx < 2n t
IQzt) I
Q(t)
and
f
t IQzt)I << f Iu(x)I dx =
Iu(x)I dx +
Q,t)
f
Iu(x)I dx
Q;t)n{IuI>t/2} IQit)I
2
f
+
Iu(x)I dx .
Q(t)n{IuI>t/2}
from which (23) follows at once.
If x E Q(t), then
Iu(x)I dx > t
.
Q;
By considering the smallest ball B with center x containing Q?t), we then get M(u) (x) ? f Iu(y)I dy > c(n) B
l u(y) I dy ? y(n) t
.
Qit)
This proves (24) taking also into account (23) and (4).
As first consequence we shall now give a characterization of the class of functions with maximal function in L' ([8n)
1.1 The Maximal Function and Lebesgue's Differentiation Theorem
191
Definition 2. We say that u belongs to L log L in Q if u is measurable and
JQ
Jul log(2 + Jul) dx < oc
.
Notice that
LlogL(R') C L'(IR)
.
Proposition 5. We have (i)
If u E LlogL(R7) then M(u) E Ll c(1R ).
ii)
Let Q a cube in 1R', u a measurable function with M(u) E L'(Q). Then u E LlogL(Q).
Usually Proposition 5(ii) is proved under the additional hypotheses that u = 0 outside of Q. This is not necessary and actually the two claims are equivalent. Proof. Let K be a compact in W. Using Proposition 1 we get
fM(v)(x)dx<2IKH K
J M(u)>2
M(u) (x) dx 00
00
=2IKI+J I{M(u)>t}Idt < 2JKJ +cJ (J t
2
lu(x) I dx) dt I
CI>e
Iu(x)I
= 2 1KI+ c
=2IKI +c
J IiI>1 J
Iu(x)I (
J
t dt) dx
1
Jul log Juldx.
UD>1
This proves (i). Because of Lebesgue's differentiation theorem, lu(x)l < M(u)(x)
(25)
for a.e. x in Q, hence u E L'(Q). Since (26)
J
< log 3
Q
/
J
Q
r Jul dx + J IuI>1
it suffices to prove that the last integral is finite. Set
tQ :_ Q
Jul log Jul dx
3. Cartesian Maps
192
We have
f (27)
J
lu(x) I dx + J
Jul log Jul dx < log tQ J
,
ltI>tQ
lul
lu(x)I>1
Jul log Jul dx
Using Proposition 4 we infer u(x)1
(28)
J ul>tQ
Jul log Jul dx = J lul>tQ
00
f
dt
J
lu(x)l
t
tQ
J
( f dt) dx 1
lu(x)Jdx < c1(n)lJ M(u) IL1(Q)
lul>t
We therefore conclude from (25) (26) (27) (28)
1
ullog luldx < c(n)(log(3tQ) + 1)II M(u) llL1(Q)
which proves the claim (ii).
1.2 Differentiability Properties of W1,P Functions Let u E Wlo'P(R' ), with p > 1. In this subsection we shall be concerned with pointwise differentiability properties of u, thus it is not restrictive to assume that in fact u E W1'1(Rn), or even u E W1,P(lR') and we shall think of u as of a representative of the equivalence class of u. We denote by Let u E Wlo'P
Ru := IunLDun{xlu(x)=A.u(x)} the set of Lebesgue points for both u and the weak derivatives Du where u(x) is the Lebesgue value of u at x. By Lebesgue differentiation theorem,
meas (Rn \ Ru) = 0 . For u
W' (l]), x E R and 8> 0 we define Va (Du) (x) :=
sup J
O
I
Du(y) - Du(x) IP dy
B(x,r)
Notice that, for almost all x E Ru, we have Ua (Du) (x) ---p 0
as 8 -> 0
The following proposition contains the basic estimates in order to study the differentiability properties of functions in W,1'P(IIYn)
1.2 Differentiability Properties of W" P Functions
193
Proposition 1. Let u E Wio (R ), p > 1, x E A. and u(x) be the Lebesgue value of u at x. Then for all r > 0 we have 1
(1)
B(x,r)
1u(y) -u(x)Ip dy y - xIp
dt f IDu(y)IPdy . B(x,tr)
0
In particular Morrey's inequality 1f
I u(y) - u(x) I P dy < r '
(2)
B(x,r)
J0
f IDu(y) I P dy
dt
.
B(x,tr)
holds. If moreover p > n, then I u(y) - u(x) I dy < p
(3)
1/p l n r( f I Du(y) I P dyl l \ B(x,r)
p
B(x,r)
Proof. Of course we may assume x = 0 and u(x) = 0, thus we have to prove
f f 1
IyIrpdy,
(4)
<
dt
I
B(0,r)
IDu(y)IPdy
B(0,tr)
0
provided u E Wlo' (RI) and f u(y) I dy -> 0 as r --+ 0. B(0,r)
Step 1. (4) holds for u E C1(Rn) such that u(0) = 0. In this case in fact
Du(ty) y dt
u(y) 0
so that
f
u)IP IyI
f
dy
B(0,r)
B(0,r)
f
P
1
dy
IDu(ty)I dt
0
1 11
dy f B(0,r)
0
IDu(ty)IPdt =
Jdt f IDu(y)Ipdy 0
B(0,tr)
Step 2. We now prove (4) when u E C1(IIPn \ {0}) and
f
B(O,r)
r -> 0. Assume
I u(y) I dy
0 as
3. Cartesian Maps
194
J0
dt
IDu(y)IP dy < +oo
-f B(O,tr)
otherwise there is nothing to prove. By changing the order of integration we get f1f
dzJ IDu(tz)dt = B(0,r)
fdt
f
JDu(y)j1 dy < 00
B(0,tr)
0
0
from which we deduce that for a.e. z E B (0, r) the function u, ,(t) := u(tz), t E (0, 1) is absolutely continuous. In particular there exists
,\(z) := limu(tz) t-.0
f Ju(y)j dy --+ 0 as r --40 we infer that lA(z)J = 0 for a.e. z. Therefore
Since
B(0,r)
we conclude
u(z) = u(z) - 0 = fDu(tz).zdt. 0
The claim then follows as in the Step 1. Step 3. Let now u E Wio' (]R'n) and
f
Jul dy
0 as r --f 0. Again we may
B(0,r)
assume 1
/
f dt I JDu(y)Jpdy < +oo 0
B(0,tr)
First we approximate u by a sequence of maps {uk} for which step 2 holds, as stated in the following Lemma 1. Let u E Wlo'p(IRn), p >p 1, and suppose that
+ Judy->0 B(0,r)
as r --* 0. Then there exists a sequence {uk} c Wjof (Rn) fl C1(Rn \ {0}) such that Uk --+ u in WloP (W) and for any k = 1, 2, .. .
f
lukl dy ---> 0
as r
0
B(0,r) (5)
f JDuk &p dy < c B(0,r)
f
B(0,2r)
JDulp dy
dr > 0.
1.2 Differentiability Properties of W" P Functions
195
We postpone the proof of the lemma and conclude the proof of the proposition. By construction all the Uk satisfy the hypotheses in Step 2, hence for all k = 1, 2,... we have 1
Iuk (y) I p
B(0,r)
dy < fdt
IyI
IDuk Ip dy
.
B(0,tr)
0
Passing to subsequences, we can suppose that uk(y) --f u(y) for a.e. y. Therefore by Fatou's lemma and Lebesgue's dominated convergence theorem, using also
(5), we get (4) and finally (1). Morrey's inequality (2) then trivially follows. Similarly we have
f
f1
rJ dt
lu(y) - u(x) I dy
B(x,r)
+
IDuldy
B(x,tr)
0 1
< rJ dtI 0
f
1/p
IDuIpdy
B(x,tr)
if p > n, we can estimate the last integral by
f
p
0
f B(x,r)
1/p
1/p
IDuIp dy I
J
dt = 1- n/p
f DuIp dy I B(x,r)
f
which yields (3).
Proof of Lemma 1. We construct the sequence Uk by mollifying u so that its "value" at zero does not change. Consider the sequence of balls Bk = B(0, a-k), where for reasons to be seen later we choose a = ' 2, and define A0 = III" \ B2, Ak := Bk \ Bk+2, k = 1, 2, .... The open sets Ak form a locally finite covering of Rn \ {0} so that there exists a decomposition of unity {cpk} associated with {Ak}.
Choose a sequence Ek in such a way that spt ('Pk * Pek) C Bk-1 \ Bk+3
where p is a standard mollifier and pe(x) := E-np(E). Then the function (6)
u := E(u(pk) * PEk k
is well defined and of class C°° in Rn \ {0}.
Fix r > 0 and let k be the largest integer such that a-k > r. By the choice of a, only the values of u for x with I x I < a2-k < a3r enter in the definition of fi(y) with jyj < r, thus
3. Cartesian Maps
196
f
(7)
lalvdy < c
f
lulldy
B(0,2r)
B(O,r)
where c is a constant depending only on n. Similarly we get
f IDaIP dy < c
(8)
f
IDuIP dy
B(0,2r)
B(O,r)
We shall now prove that for every b > 0 we can choose {Ek} in such a way that
Ilu-ally,,, < 38 This together with (7) and (8) clearly concludes the proof of the lemma. We have
a - u = X:((Wk)*PEk-Wk) k
so that E Il (uWk) * PEk - Wk IILP
u IILP
II
k
therefore, choosing Ek in such a way that ll (Wk) * PEk - UcPk Il L, < S12k+1
we get
Ilu - UIILP <s . On the other hand
Dii = ED((ucpk) * PEk) k
Y, (DU - Tnk) * PEk + (u D7'k) * PEk k
Du + E((Ducpk) * PEk - Ducpk) + E((u D(pk) * PEk - u DcPk) k
k
thus, choosing Ek in such a way that II (u Dcpk) * PEk - (u Dc0k) ll LP
< b/2k+1
II (Dupk) *PEk - (Ducpk) IILP < $/2k+1 we conclude
II Du - Du II L < 2S Next theorem is a simple consequence of Proposition 1.
1.2 Differentiability Properties of W1'P Functions
197
Theorem 1. Let u E Wio (R ), with p > 1. Let x be a Lebesgue point for u and let u(x) be the Lebesgue value of u at x. Then we have
f
(9)
Iu(y) - u(x) - Du(x) (y - x) I P dy < VP (Du) (x) Iy - xIP
B(x,r)
If moreover x and Du(x) satisfy the additional condition
VP(Du)(x) -> 0
as r --> 0
then (10)
f
B(x,r)
I u(y) - u(x) - Du(x) (y - x) I P dy ---> 0 Iy - xIP
as r-+0.
In particular, for all x E Ru, we have
Iu(y) - u(x) - Du(x)(y - x) I ly - xI
B(x,r)
dy -+ 0
as r-+0.
provided also Du(x) is the Lebesgue value of Du at x.
Proof. The claims easily follow by applying Proposition 1 to the function y u(y) - Du(x) (y - x). Theorem 1 yields at once
Theorem 2 (Calderon-Zygmund). Let u E WloP(Rn), with p > 1. Let x E .Cu and u(x) be the Lebesgue value of u at x. Then (11)
r
f
lu(y)-u(x)-Du(x)(y-x)IPdy :5 VP (Du) (x)
lim 1
f
B(x,r)
Consequently
(12)
r.o+ rP
Iu(y) - u(x) - Du(x)(y - x)IP dy = 0
B(x,r)
for a.e. x E Rn, Du(x) being the Lebesgue value of Du at x.
If (12) holds one says that u is differentiable in the LP-sense at x. Therefore Theorem 2 says that every function u E W110 f (Rn) is almost everywhere differentiable in the LP sense. Remark 1. Let p* := n-P be the Sobolev exponent of p, 1 < p < n. Sobolev's inequality yields
198
3. Cartesian Maps 1/p'
Cf
Iv(y) - v(x)lp dy
B(x,r) 1/p
1/p
+c
Dv(y)lpdy) B(x,r) 1/p
J
iv(y)-v(x)1pdy)
B(x,r)
+cr
f Dv(y)Ipdy
1/p
1
( B(x,r)
11
J0
dt f IDv(y)lpdyl
f
B(x,tr)
Applying the previous inequality to v(y) := u(y) - u(x) - Du(x)(y - x)
we infer
( f u(y) - u(x) - Du(x) (y - x)
Ip*
dy
B (x, r)
1/p
< cr (
}
IDu(y) - Du(x)Jp dy
B(x,r)
/ 1/p
1
+cr
I dt -} IDu(y) - Du(x)IP dy \(0'0J B(x,tr) J
< 2cr(Vp(Du)(x))1/p Thus, we conclude that every u E WloP(R"), with 1 < p < n, is almost everywhere differentiable in the LP'-sense.
A function u c W 1,p (]Rn ), with 1 < p < n, is not in general almost everywhere differentiable as it is shown for instance by the following example Consider the function cp : R2 _4R fi(x)
log log -y 0
if IxI < otherwise
u belongs to W1"2(]R2). For a denumerable set of dense points {xk} in JR2 we now define
00
u(x)
E 2-kco(x - xk) . k=1
Then it is easily seen that u(x) E W1'2(1R2), and that u is not almost everywhere differentiable in the classical sense since it is unbounded in a neighborhood of 0 every point.
1.2 Differentiability Properties of W1,P Functions
199
x
Denote now for y
u(y) - u(x) - Du(x)(y - x)I
R(y; x) :_
(y - xI
and by
q(x, r) := I R(y ; x) dy B(x,r)
Even if u is not differentiable in the classical sense, we have the following
Theorem 3. Let u E Wio" (R'), and let C be a closed set for which lim q5 (x, r) = 0
T--.o
holds uniformly on compact subsets of C. Then uIC is differentiable in the classical sense, i.e.,
lim R(y; x) = 0
y-.x
,
yEC
moreover the limit is uniform on compact subsets of C.
Proof. Let r
Ix - y1. We have
Ju(y) - u(x) - Du(x)(y - x) < I f u(z) dz -
u(x) - Du(x) (y - x)
B(y,r)
+I
f
u(z) dz - u(y)
-f
B(y,r)
Using (2) of Proposition 1, we deduce I u(z) dz - u(x) - Du(x)(y - x)I
B(y,r)
f
= I + [u(z) - u(x) - Du(x)(z - x)] dzI B(y,r)
< 2n I lu(z) - u(x) - Du(x) (z - x) I dz B(x,2r)
< 2n+lr O(x, 2r) and,, similarly,
I I u(z) dz - u(y) = I B(y,r)
+ (u(z) - u(y) - Du(y)(z - y)) dzI B(y,r)
r 0(y, r) .
3. Cartesian Maps
200
Therefore
R(y ; x) < c(n) {O(x, 2r) + 0(y, r) }
r = Ix - yj
,
.
The claim easily follows from this inequality. In fact, for any e > 0, b > 0 such that 0(y, s) < e holds for s < 8 and Iy - xo1 < 1, we deduce
R(y; x) < 2c(n)e for anyx,ywith Ix - xol
< 2min(l,b).
2l y
Also, as consequence of Theorem 2, we can deduce that the functions in W1oc (RI) possess almost everywhere classical directional derivatives in almost all directions.
Theorem 4. Let u E W1oe (1R'n). Then for almost every x we have (13)
in Lloc
k [u(x +
Iz) - u(x)] --> Du(x) z,
k -> oo
,
z E Rn
(Rn).
Proof. From (12) we deduce that for a.e. x and all R lim
k
I
k-+oo R
lu(x + w) - u(x) - Du(x) wl dw = 0 ;
B(O,R/k)
changing variable, w =fz/k, we then get
km R
lk[u(x + k) - u(x)] - Du(x) zj dz = 0
-}
.
B(O,R)
We shall now prove that functions u E W1oc(WZ), with p > n, are "almost everywhere differentiable". More precisely we have
Theorem 5 (Morrey-Sobolev). Let u E W1o'cP(l$n) with p > n. Then in the equivalence class of u there is a function ii which is locally Holder-continuous with exponent a := 1 - n/p, ii E Coa(Pn) n Wl,,(IISn), in particular A. = Rn, u(x) is the Lebesgue value of u at x, and for x, y E B(0, r), r > 0 we have u(x) - u(y) l <_ e(n, p) Ix -
yIl-n/P
( f jDu(x) jP dx)
1/p .
B(O,r)
Moreover u is almost everywhere differentiable in the classical sense, i.e. (14)
lim y-.x
Iii(y) - u(x) - Du(x)(y - x) 1
=0.
Ix - yl
for a.e. x E Rn, Du(x) being the Lebesgue value of Du at x.
1.2 Differentiability Properties of W" Functions
201
Proof. Fix some Ro > 0, and for x, y E B(0, Ro), set b := x - yI and
S = B(x, b) fl B(y, J)
.
Let v any smooth function in R. For all z E S we have
Iv(x) - v(y)I < v(x) - v(z)I + V(z) - v(y)I hence, integrating over S with respect to z, we deduce v(x) - v(y) I <
f
Iv(x) - v(z) I dz + -}- Iv(z) - v(y) I dz
.
S
S
As in the proof of (3) of Proposition 1, we easily deduce
f lv(x) - v(z)l dz < c(n) f Iv(x) - v(z)l dz S
B(x,a)
1/p
<
c(n, p) S
jDv!p dy I
-F
B(x,b)
Similarly we estimate f-I v(z) - v(y)I dz , and we conclude that S
(15)
Iv(x) - v(y) I < c(n, p) b
f IDvlp dy B(x,2s)
Let now {uk} be a sequence of smooth mappings which converges to u in W1oP(R'`2). Applying (15) to the approximating sequence {uk}, (16)
luk(x) -uk(y)l C c(n,p)s(
t
IDuklpdy)1/p
/
B(x,25)
we get
Iuk(x) -'uk(y)I < c(n,p) Ix - yli_m'/p(
f
IDukIpdx)
1/p
B(0,3Ro)
Thus we deduce, possibly passing to a subsequence, that {uk} converges uniformly on bounded sets (by Ascoli-Arzela theorem, and of course in W' ( 2) ) to a continuous function a (which is in the same class of equivalence of u), which is Holder continuous with exponent 1 - n/p; moreover (15) holds for u. This proves the first part of the theorem. Since (15) holds also for v(y) := i!(y) - u(x) - Du(x)(y - x)
3. Cartesian Maps
202
we get
lv(y) - v(x) l
<
I
c(n, p) 5 L
IDu(y) - Du(x)11 d
s(x,zu)
c(n,p) [V b(Du)(x)]'11Ix - yj Thus
R(y;x) < c(n,p) [V26(Du)(x)]1/P and the claim follows.
1.3 Lusin Type Properties of W" Functions In this subsection we shall prove, as consequence of Theorem 1 in Sec. 3.1.2 the following Lusin type properties for functions in Wlo p (IRn). (i) the set of regular points R,,, := L.,, n ,CD,, n { x I u(x) = A,, (x) } of u E
Wlop(Rn) can be decomposed in a denumerable union of measurable sets {Fk} such that ulF, is a Lipschitz function, (ii) except on arbitrarily small open sets u is of class C1. Let us begin by discussing property (i). Recall
Definition 1. Let A be a subset of IRn. A function u : A --> R is said to be Lipschitz in A if there exists a constant L such that (1)
Ju(x) - u(y) 1
< Lax - y
b x,y E A
.
The infimum of all constants L satisfying (1) is called the Lipschitz constant of u in A and denoted by Lip(u, A). By considering the function u : Rn -p R defined by
fi(x) :=
inf
(u(y) + Ljx - yj)
L := Lip(u, A)
it is not difficult to prove the following extension theorem
Theorem 1 (Kirszbraun). Let A C Rn and let u : A -+ R be a Lipschitz map. Then there exists a Lipschitz function u : Rn -+ R such that is = u in A and Lip(u, FRn) = Lip(u, A).
In fact Kirszbraun theorem holds for Lipschitz maps u : IRn
RN, N _> 1,
see e.g. Federer [226]. We also have
Theorem 2. A function u is Lipschitz in some open set if and only if it has bounded distributional derivatives.
1.3 Lusin Type Properties of W 1'P Functions
203
Proof. Let u be Lipschitz. By mollifying u we find a sequence of smooth maps with equibounded gradients in LO° and converging strongly to u in any Wioc (R'). Conversely if it has bounded distributional derivatives then u belongs to any W1o'(Rn), p > n, thus the Holder-continuous representative a of u satisfies (15) in Sec. 3.1.2, hence u is actually Lipschitz. Next theorem, which contains the basic differentiability result for Lipschitz maps, is an immediate consequence of Theorem 2 and of Morrey-Sobolev theorem in Sec. 3.1.2.
Theorem 3 (Rademacher). Let u : Rn --+ R be a locally Lipschitz function. Then u is almost everywhere differentiable, i.e. for a. e. x lim
u(y) - u(x) - Du(x)(y - x)I
= 0,
y-.x
where Du(x) is the Lebesgue value at x of the distributional gradient of u.
We are now ready to prove Lusin property (i) for functions u E W"I,C (][fin). As the content of (i) is local we shall in fact consider functions u E W""l(iRn) Consider the maximal function of I Du(, M(I Dul ), and for \ > 0 define the closed sets Fa by FA :_ {x E ]E2' I M(IDuD)(x) < Al Then we have
Theorem 4. Let it E W',P(]R'). Then the restriction uIFanRu is a Lipschitz function with Lipschitz constant not greater than c(n)A. Thus for any k = 1, 2,..., one can find a closed set Fk and a Lipschitz function uk : R' -+ JR with Lip(uk) < k such that (i) UIRUnF,, = uk (ii) U Fk D Ru k
(iii)
Du(x) = Duk(x) for a.e. x E Ru n Fk.
Moreover
kp meas(Rn \ Fk) < c(n, p)
(2)
r
J Rn
IDulp dx
and in fact (3)
k2 meas(Rn \ Fk) < c(n,p)
J DuI>k/2
IDuI P dx -+ 0
whenever u E W l'p (Rn ), p > 1.
Proof. Let X, y E Ru n F,\, J:= I x - yl , and let
S = B(x,S)nB(y,b)
as k -+ oc
204
3. Cartesian Maps
For any z E S we have 1u(x) - u(y)1 < 1u(x) - u(z)1+ 1u(z) - u(y)1
thus integrating over S we obtain
f
*(x) - u(y) I <
Ju(x) - u(z) I dz +
f
Ju(y) - u(z) dz .
Using (2) in Sec. 3.1.2 of Proposition 1 in Sec. 3.1.2 we then estimate
f I u(x) - u(z) I dz < c(n) f
I u(x) - u(z) I dz
B(x,(5)
S ri
< c(n) b
J0
dt
f
IDu(y)ldy < c(n) 8M(jDuj)(x)
B(x,ts)
< c(n)AIx-yI. Similarly we estimate f Iu(y) - u(z) I dz. Hence we infer S tu(x) - u(y) 1
< c(n) 2A
x-yI
and the first part of the claim is proved. Renaming Fk the set Fklc(,,,) and using Kirszbraun theorem we extend ulFknR to a Lipschitz function Uk : Rn --, ]R with Lip(Uk) < k. As
R. C {x I M(IDuD)(x) < oo} = UkFk (ii) follows trivially, and (iii) follows from Theorem 2 and Theorem 1 in Sec. 3.1.2.
In fact both functions u and Uk belong to W' and for the difference
Vk := a - uk we have f Ivk(y) - Vk(x) - Dvk(x)(y - x)I
ly -xl
B(x,r)
dy
0
as r-->0,
for every x E Ru, hence for x E Ru fl Fk 1
meas(B(x, r))
I
B(x,r)nFk
Dvk(x)(y-x)d---,
0
as r-->0.
ly - -TI
This implies that DVk(x) = 0, if moreover the density of Fk at x is 1. In fact suppose DVk(X) 0 0 and denote by C the cone
1.3 Lusin Type Properties of W1'P Functions
205
C :_ {y I I Dvk (x) (y - x) I> 2 I Dvk (x) I y- X II I
then
B(Fk fl C, x) := r-O
meas (Fk fl C fl B(x, r)) meas (B(x, r)) /'
1
< 2 lim r-.o meas (B(x, r))
J
FknCnB(x,r)
IDvk (x) (y - x) I l y - -TI
dy
y
0
which contradicts 0(Fk, x) = 1. As almost every point of Fk is of density 1 for Fk, the proof of (iii) is concluded. Finally estimates (2) and (3) easily follow from (6) in Sec. 3.1.1. Remark 1. A similar result can be proved for function u E Wl°1(.2) where .2 is a domain in ]R'. We leave the statement and its proof to the reader. In order to discuss Lusin property (ii), let us recall some simple facts. Let A be a subset in ]R'1 and let u : A -- ]R be a function defined on A.
Definition 2. We say that u : A --f R is differentiable at x E A if there exists a linear map Lx : ]Rn --- R such that
lim R(y ; x) = 0 y-x
(4)
yEA
where we have set for x # y, x, y E A (5)
R(y ; x) :=
I u(y) - u(x) - Lx (y - x) j
Iy-xI
We say that u is differentiable in A if u is differentiable at every point x E A.
Of course the previous definition is meaningless if x is an isolated point of A. We agree upon the fact that on isolated points any function is continuous and every choice of Lx give rise to a differentiable function. Notice that instead L. is unique, if it exists, on points of A of positive density.
Definition 3. Let E be a closed set in I. We say that a function u : E --r 1R belongs to C1(E), or is of class C' in E, if u is differentiable in E, the gradient Du is continuous in E, and R(y , x) -+ 0 as y -- x, uniformly on compact subsets of E.
Notice that if E is also open, for instance E = ]R", the previous definition agrees with the standard one, as it is easily seen from the inequality sup
R(y ; x) < osc B(x,r) Du
yEB(x,r)
Next proposition states that any differentiable function is of class C' except on suitable small sets.
206
3. Cartesian Maps
Proposition 1. Let A be a measurable subset of 1Rn and let u : A --+ R be a differentiable function with measurable differential Du. Then for every e > 0 there exists a closed set FE C A such that u E Cl (FE) and meas (A \ FE) < e. Proof. Fix e > 0. Since Du(x) is measurable, we find by Lusin's theorem a
closed set BE such that Du(x) is continuous in BE and meas (A \ BE) < e/2. Define now f o r k = 1, 2, .. .
ifB(x,1)nA\{x}=0
0 77k (X)
sup{JR(y; x) I
I
y E B(x, 1) n A \ {x}} otherwise
R being defined in (5). Clearly 77k(x) --> 0 as k -* oc for all x E A. Therefore, by Egoroff's theorem we can find a closed set FE C BE such that meas (BE\FE) < e/2 and rik -> 0 as k ---> oc, uniformly on compact subsets of FE, i.e., u E C'(FE); finally
meas (A \ F£) < meas (A \ BE) + meas (BE \ FE) < e . r-I
The relevance of Cl-functions on closed sets of 1R' is due to the following extension theorem due to Whitney that we state without proof.
Theorem 5 (Whitney's theorem). Let E be a closed subset in R and let u : E -* 1R be a function in Cl (E). Then there exists a continuously differentiable function g : R' --> R such that
g=u,
Dg=Du
in E.
Moreover one has the following estimates 11911o,Rn
II Dg l.An
<
CIIuIIOC,E
< c max{11 Du III E, Max Iu(x) - u(y)I } x,yEE Ix - yI
where c is a constant depending only on the dimension n.
Combining Proposition 1 and Whitney's extension theorem we get at once Proposition 2. Let u : A --> 1R be a differentiable function on a measurable subset A of 1R'. Then for every E > 0 there exists a closed set FE C A and a function 9E E C' (1R'2) such that
meas (A \ FE) < e,
u = gE in FE,
Du = Dg5 in FE
.
Combining Kirszbraun's theorem, Rademacher's theorem and Proposition 2 we immediately deduce
1.3 Lusin Type Properties of W 1,P Functions
207
Proposition 3. Let A C R' be a measurable set and let u : A - 1[8 be a Lipschitz function. Then, for every e > 0, there is a closed set FE C A and a C'-function gE : R n - JR such that meas (A \ FE) < e,
u = gE in FE,
Du = DgE in FE
and 11 96 Moo
.'
< C II U Io,A'
II DgE II.,R" <_ c Lip(u, A)
where c is a constant depending only on the dimension n. Combining Theorem 4 and Proposition 3 we derive at once the following theorem of which we give also a different proof in the Cl-context avoiding Lipschitz maps.
Theorem 6. Let u E W" (1Rn). Then for every e > 0 there exists a closed set FE and a function gE E C' (11$n) such that meas (1R
\ FE) < e,
u = gE in FE,
Du = Dg, in FE
.
Proof. Let e > 0. By Lusin theorem one can find a closed set CE such that
u and Du are continuous on CE. Since .FB(x r) R(y ; x) dy -> 0 for x E Ru, compare Theorem 1 in Sec. 3.1.2, by Egoroff theorem one can find a closed set FE C CE n Ru such that meas (Rn \ FE) < e and
f R(y ; x) dy
0
as r-f0
B(x,r)
uniformly on compact subsets of FE, therefore by Theorem 3 in Sec. 3.1.2 u E Cl(FE). Then Whitney extension theorem, Theorem 5, yields the C' extension gE on the whole of 1R' of UI FE .
Although we shall not need it, we conclude this subsection by proving a quantitative form of Theorem 6, that is, by proving that not only u agrees with 9E E C' (]Rn) except on some open set of small measure, but also that gE is not far from u in the W1,P-norm. Theorem 7 (Liu). Let 0 be a domain of Rn and let u E W1"P(Q) with p > 1. Then for every e > 0 there exist a closed set CE C 0 and a function gE E C1 (0) such that
meas(Q\CE) <e,
gE =u in CE,
Dg, =Du in CE
and
I u - 9E IIW1,P(S2) < e
First we consider the case I? = 1Rn, and we prove
3. Cartesian Maps
208
Theorem 8. Let u E W"(R') with p > 1. Then for any A > 0 there
exist a
closed set Ba and a function ga E C1(ll8n) such that
ga = u , Dga = Du on Ba meas (R'\ Ba) < c(n)A-pll u llwi,P(Rn) l
l 9a l
l
l9a I + sup l D9a l < A . i,. := Sup an Rn
Moreover
jga jji,. meas (l$n \ Ba) --> 0
(6)
as A -f oo
in particular (7)
as A -->oo.
0
IIu-9allwl,p
Remark 2. Of course relabeling Ba and ga we can arrange so that meas (lf8n \ BE) < E,
and
Dg, = Du on BE
gE = u on BE
IIu-ge1IWI,p <E.
In order to prove Theorem 8 we define for A > 0
Ha := {x E an l M(lul)(x) +M(IDul)(x) < A} . As in the proof of Theorem 4 we easily deduce
Lemma 1. Let u E Wl°r(1[8n), p > 1. Then the function UIHAnR is Lipschitz continuous. Moreover for all x, y E Ha fl Ru we have lu(x)l < A,
lu()
I Du(x)I < A,
-
uI(y)l
< c(n) A
.
Proof of Theorem 8. From Lemma 1 we have: UIHAnRu is Lipschitz-continuous and
II U II.,HAnR,. +max{ II Du Il. HanR,,,
sup
x,yEHanR,,
Iu(x) - u(y)I } < c(n)A. Ix - yI
If meas (Rn \ Ha) = 0, then the result is an immediate consequence of Proposition 1. Suppose now that meas (Rn \ Ha) > 0. Applying Theorem 6 we find a closed set Ca C Ru such that u E C'(C)) and meas (an \ Ca) < meas (l8n \ Ha). On Ba := Ca f1 Ha we have u E C'(B)) and II U
Max{ 11 Du Il OO B, ,
sup 1U(x) - u(y) I } < c(n)A x,yEB,\
x - yl
.
Thus applying Whitney's extension theorem, we find a function ga of class C1(R) such that
1.3 Lusin Type Properties of W1'P Functions
209
ga = u , Dg ,\ = Du on B. and II 9A II1,00 < c(n)A .
(8)
On the other hand
Rn\HA c {xEW2 I M(u)(x)+M(IDul)(x)>A} C {x E Rn I M(u)(x) > 2} U {x E R
I
M(IDuD)(x) > 2} ,
hence the weak estimate in Proposition 1 in Sec. 3.1.1 yields meas (i[8n \ BA) < 2meas (]Rn \ HA) < c(n)A-P11 u while Proposition 3 in Sec. 3.1.1 (ii) yields 11 9a IL 00meas (Rn \ BA) < c(n)A-Pmeas (Rn \ HA)
as A
0
oo
taking also into account (8). Finally, we have IIu-9AIIW1.P(R) Ilu-9AIIW=,P(R'\Ba)
<_
II U Iiwl.»(R-\B,) + II 9a II W1,P(R
\B,)
The first integral tends to zero by the absolute continuity of integral, while the second integral tends to zero because of II 9A II W1.P(R-\Bx) <
II ga IIi,. meal (Rn \ BA) -> 0 as A --> oo
thus also (7) holds.
Now Theorem 7 follows from Theorem 8 by means of a suitable partition of unity.
Proof of Theorem 7. Let {Ci} be a sequence of non empty compact sets such that
Ci CCi+1
UiC2 = 02 .
,
0. For each i, let cpi be a C°° function on RI such that 0 < cpi < 1, sptcpi CCi+1 and Ci is a subset of {x (pi (x) = 1}. Define Set C_1
0
I
Oo
po ,
"
'J'
Then 4'i E
wi - ,i-1
wi
for i > 1
0
.
and spt 45i CCi+1 -Ci_1, moreover, for x E S2, '2bi(x) ,-4- 0 for at most two values of i, and 00
E 0i (X)
i=0
=1
VxE(2.
210
3. Cartesian Maps
For each i = 0, 1, 2, ... we now define
J u(x)'bi(x) x E .f2
ui(x)
0
By Theorem 8, we find function ge,i E C1(Rn), which we may assume with support in Ci+1 -Ci, such that O
meas 1x
E
I
ge,i(x) r ui (x)} < 2i+1
It is now easily seen that the functions 00
ge =
E
ge,i ,
e>0,
i=0
satisfy the claim.
1.4 Approximate Differential and Lusin Type Properties We shall now prove that Lusin type properties (i) and (ii) of Sec. 3.1.3 are in fact equivalent to a kind of pointwise differentiability called approximate differentiability. We recall, compare Sec. 1.2.1, that for any measurable set A C Rn, the upper
density 0*(A,x) of A at x is defined as lira sup
r-O
meas (B(x, r) fl A) meas (B (x, r))
Similarly the lower density 0. (A, x) of A at x is given by lim-O f
meas (B(x, r) fl A) meas (B(x, r))
The density of A at x is defined whenever 0* (A, x) = 0* (A, x) as such a common value 0(A,x) = 0*(A,x) = 0*(A,x)
By Lebesgue's differentiation theorem or Radon-Nikodym theorem, 0 (A, x) = 1 for a.e. x E A and 0(A, x) = 0 for a.e. x E 1l \ A, compare Ch. 1. Let us introduce the concept of approximate limit and approximate continuity for a measurable function u : A -+ R.
Definition 1. Let A be a measurable subset of R n, and let x E Rn be such that 0* (A, x) > 0. Let u : A --> R be a measurable function. We say that P is the approximate limit of u at x when y tends to x in A, and we write
1.4 Approximate Differential and Lusin Type Properties
211
Q := aplimu(y) yEA
if for all e > 0, the set
AE = {y
E A I Iu(y) - QI > ej
has density 0 at x. We say that u : A -> JR is approximately continuous at x, if
u(x) := aplimu(y)
(1)
.
yEA
From the definition, one easy sees that (i)
The condition 0* (A, x) > 0 is essential. In fact in the case 0* (A, x) = 0 every
real number would be the approximate limit of u at x for y tending to x. (ii)
(iii)
aplimv-A u(y) = 2 if for any open set W C R containing P, the set A \ U-1 (W) has density 0 at x. The existence and the value of the approximate limit do not change if we redefine u on a set N of density zero at x. In particular if 0(A, x) = 1, one may think of u as defined on the whole of R', since 0(R' \ A, x) = 0. Thus we may simply write P := aplimu(y)
y-.x
when 0(A, x) = 1.
Proposition 1. Approximate limits are unique. Proof. Let A be a measurable set, x E A with 0* (A, x) > 0, and let u : A -> R be a measurable function. Suppose that for 2, t' with i# £' we have
0(AE, x) = 0(A' x) = 0 ,
`d e
where
AE = {y E A I Iu(y) - e( > e},
Choosing e =
e
AE
= {y E A I Iu(y) - e'I > e}
8 L, we have
3e = jQ - e'I < Iu(y)-el+Iu(y)-t'I for any y E B(x, r) fl A, hence
B(x, r) fl A C (AE fl B(x, r)) U (AE fl B(x, r)) consequently
0(A, x) < 0(AE, x) + 0(A'E, x) = 0 a contradiction.
,
212
3. Cartesian Maps
The next proposition relates standard limits and approximate limits
Proposition 2. Let A be a measurable subset of RI, and let x E R" with 0*(A,x) > 0. Let u : A -- R be a measurable function. The following two claims are equivalent (i) P := aplimy-A u(y) (ii) There exists a measurable set E C ll
with 0(A \ E, x) = 0 such that
1 = lim u(y)
(2)
UEE
In particular u is approximately continuous at x if and only if there exists a measurable set E with 0(A \ E, x) = 0 such that uIE is continuous at x. Proof. Suppose that (2) holds. Then clearly for any e and r sufficiently small we have
E f1 A. fl B(x, r) _ {y E E I bu(y) - PI > e} fl B(x, r) _ {x} Thus 0(AE, x) = 0(AE fl E, x) = 0.
On the contrary, if aplim beA u(y) = 2 then one can find a decreasing sequence {rk} of radii, such that (3)
Ju(y) - $l > 1/k} fl B(x, rk))
meas ({y E A
meas (B(x, rk)) <_
2k
Define now
E := A \ U 00 (B(x, rk) \ B(x, rk+1)) n {y I Ju(y) - .£l > 1/k} . k=1
Clearly E is measurable and limy-X uIE = 2. We now prove that 0(A\E, x) = 0. We have
0 A \ E = U (B(x, rk) \ B(x, rk+1)) n {y I Hu(y)
- ej > 1/k}
k=1
For any r > 0 denote by k the integer for which rk+1 < r < rk. We have meas (B (x, r) fl (A \ E)) < 00
E meal ((B(x, rk) \ B(x, rk+1)) fl {y I lu(y) - ej > 1/k}) < k=k 00
E meas ((B(x, rk) n {y Ju(y) - 2j > 1/k}) < k=k
0 meal (B(x, rk) E k=k
2k
< meas (B (x, r))0"0 k=k(r)
1
2k
from (3). Since as r --> 0, k = k(r) -* oe we get the result.
1.4 Approximate Differential and Lusin Type Properties
213
Approximate continuity yields also a kind a local characterization of measurability as shown by next proposition and remark.
Proposition 3. If the function u : R' --+ R is measurable then u is approximately continuous at a.e. x E R'. Proof. Suppose that u is measurable. Then for each k E N by Lusin theorem, one can find a closed set Fk with meas (R' \ Fk) < 1/k such that ujF, is continuous.
Denote byEk:={xEFkI O(Fk,x)=1} and byE:=UkEk. Let x E Ek, and e > 0. Since u,F,. is continuous B(x, r) U {y E FkI bu(y) - u(x)j > E} _ 0 for small r. Thus
B(x,r)n{yER'j Iu(y)-u(x)j >e} C B(x,r)\Fk and the claim follows letting r -* 0, since 0(Fk, x) = 1.
Remark 1. One can define densities and approximate limits even for non measurable sets replacing in the definition Lebesgue's measure with Lebesgue's outer measure L'`. Then one can reverse the previous proposition, proving, compare Federer [226]: u is measurable if and only if u is approximately continuous almost everywhere.
We finally observe that the previous proposition is an easy consequence of the Lebesgue differentiation theorem in the case of functions u E Lioc (R'). We have in fact Proposition 4. Let u E Lioc (R'2) If
f
I u(y) - Ql dy -> 0
B(x,r)
as r -> 0 then
aplimu(y) = B y-.x
In particular the approximate limit exists at each Lebesgue point x and coincides with the Lebesgue value of u at x.
Proof. Simply note that for e > 0 meas ({y
I
lu(y) - 2l > e} n B(x, r))
bu(y) - fj dy
.
B(x,r) F-I
214
3. Cartesian Maps
1 The converse of Proposition 4 clearly holds, if moreover u is bounded nea x. However it is false in general, as the following example shows. 1} and let ue(x, y) be Let Q(0,1) be the cube in 1R2 {(x, y) IxI < 1, 1y the function given by uE (x' y)
_f
x-2E if0<x<1, 0
0
.
EE and It is easy to check that 'Ue E L2'(Q(0, 1)) for p < 22 lue(x, y) I dx dy = 4(21 e) r-E -
oc
as r
0
Q(x,r)
while
ue = 0.
aplim (x,y)-.(0,0)
Let A C ]Rn be a measurable set and let x E R' be such that 0* (A, x) > 0. For any measurable function u : A --+ ]R and any t E IR we consider the measurable sets
Et = {xEAI u(x)>t} Et = {x E A I u(x) < t} Definition 2. The approximate upper limit of u at x is defined as the number (eventually -boo or -oc) given by
aplimsup u(y) := inf It 10* (Et +, x) = 0} Y-.x
YEA
and similarly the approximate lower limit is given by
apliminf u(y) := sup{t 10* (ET, x) = 0} YYEA
Of course
apliminf u(y) < aplimsup yes y- u(y) YEA
YEA
and whenever
apliminf u(y) = aplimsupu(y) =eER yes y...z YEA
YEA
the approximate limit exists and we have
aplimu(y) = e YEA
In the same spirit of approximate limits and approximate continuity, we may now introduce the notion of approximate differential. Let A be a measurable subset of IR" and u : A --> R be a measurable function. Suppose that x E R be such that 0* (A, x) > 0.
1.4 Approximate Differential and Lusin Type Properties
215
Definition 3. We say that u is approximately differentiable at x if there exists a linear mapping L : Rn --} IIR such that
aplimsup
(4)
Ju(y)
- ul(y) L(y - x) = 0
YEA
From the definition it is not difficult to show If u is approximately differentiable at x then u is approximately continuous at x. (ii) The approximate differential, denoted by apDu(x) or simply by Du(x) if no confusion may arise, is unique whenever it exists. (iii) If u : A -> R is differentiable in the classical sense at x and 0* (A, x) > 0 (i)
then u is approximately differentiable at x and apDu(x) is the ordinary differential Du(x) of u at x. (iv) u is approximately differentiable at x if and only if there exists a measurable set E with 0*(A\E, x) = 0 such that uIE is differentiable at x in the classical sense. Moreover D(ulE)(x) = apDu(x). (v) If u is approximately differentiable at x and 0 is differentiable at u(x), then 0 o u is approximately differentiable at x and
apDO o u(x) = DO(u(x)) apDu(x) . (vi)
If u : A -> R is approximately differentiable in A, A measurable, then
(vii)
apDu : A -> Ilk is measurable. If u : A --> Ilk is a.e. approximately differentiable in A and v = u a.e. in A then v is almost everywhere approximately differentiable in A and apDv = apDu
a.e. in A. That is, the notion of almost everywhere differentiability does not depend on the representative chosen in the equivalence class, compare Lemma 1 in Sec. 3.1.5. For u : A -> R measurable, A measurable, one can also define for almost every x E A the approximate partial derivatives of u as the numbers apDiu(x) = gi(x) for which
aplim t-.o
x}t1jEA
lu(x + tei) - u(x) - gi (x)tj =
0
itl
Denote by Ai the domain of existence of apDiu(x). Then one can prove
Theorem 1. Each Ai is measurable, the functions apDiu : Ai surable, u is approximately differentiable at a.e. x in liAi with n
apDu(x) (v) = E apDiu(x) vx i=1
whenever v E R1.
R are mea-
216
3. Cartesian Maps
Since in the sequel we shall not use Theorem 1, we omit its proof for which we refer to Federer [226, 3.1.4]. In terms of approximate differentiability Theorem 2 in Sec. 3.1.2 yields now
Theorem 2. Let u E Wlo' (R'). Then any representative u of u is approximately differentiable at each point x E Ru with approximate differential given by the Lebesgue value Du(x) of the distributional gradient Du at x.
Let us point out that the existence of the approximate differential apDu(x)
almost everywhere does not imply that u E W" . In fact apDu(x) does not agree in general with the distributional derivative of u. 2 The function sign x has zero approximate differential at all points x 0 0, 1 : its distributional derivative is twice the Dirac mass at zero. An example of a continuous function with zero approximate differential a.e. is given by Cantor-Vitali function, which is not in W", compare 02 in Sec. 1.1.3.
but u W"
In Ch. 4 we shall see that if u has derivatives Du which are measures of locally finite total variation then u is approximately differentiable and its approximate differential agrees with the absolutely continuous part of the measure Du with respect to Lebesgue measure. We also observe that Theorem 4 in Sec. 3.1.2 extends as Proposition 5. Let u be approximately differentiable at x0 and let fk(z)
u(xo + f) - u(x0) - Du(x0) k
ll k
then fk --> 0 in measure. In particular there exists a sequence ki such that fk;(z) -> 0 for almost every z. Proof. The claim follows at once since from the approximate differentiability meas { z E B(0, R) I l fk (z) I > e } -> 0 as k -b oo
foralle>0. We conclude this subsection proving the equivalence of Lusin type properties in Sec. 3.1.3 with the almost everywhere approximate differentiability. Let A be a measurable set in R' and let u : A - R be a measurable function. Consider the set of points of approximate differentiability of u
AD(u) :_ {x E A I u is approximately differentiable at x} and the set AL (U)
{x E A I aplimsup
v-s UEA
I u(y) - u(x)I Coo}
y-x
Trivially AD (u) C AL (u) and by definition
AD(u) C AL(u) C {x E A 1 0*(A, x) > 0}
1.4 Approximate Differential and Lusin Type Properties
217
Theorem 3 (Federer). The following claims are equivalent (i) meas (A \ AD (u)) = 0. (ii) meas (A \ AL (u)) = 0. (iii) There exist a non decreasing sequence {Cj} of measurable sets and a sequence
{uj} of Lipschitz functions in R such that meas (A \ Uj Cj) = 0, (iv)
u=uj onCj .
There exist a non decreasing sequence {Fj } of closed sets and a sequence {vj} of functions of class C'(lPn) such that
meas (A \ UjFj) = 0,
u = vj on Fj .
Moreover
AD(U) C AL(u) = UjCj
(5)
and in (iii) and (iv) we respectively have
apDu(x) = Duj(x) a.e. x E Cj apDu(x) = Dvj (x) for all x E Fj Proof. As AD(u) C AL(u), clearly (i) implies (ii). Let us prove that (ii) implies (iii). Let x, y E R' and S := Ix - yI. Let j3 = /3(n) denote the absolute constant given by meas (B(x, b) fl B(y, S)) meas (B (x, 8))
Define for x E A and j = 1, 2,... the sets
Ej(x) := {y E A I Iu(y) - u(x)I >! i l y - xI } and set Bj
{x E A
meas (B (x, r) fl Ej (x)) < a meas ( B (x, r)) 4
Vr0
Clearly Bj is measurable. Let us prove that (6)
AL(U) C Uj Bj
Suppose that x 0 Uj Bj. Then one can find a sequence rj, rj -> 0 such that meas (B (x, rj) fl Ej(x)) meas (B(x, rj ))
4
and, since for k < j Ej (x) c Ek (x), meas (B (x, rj) fl Ek(x)) > Q meas (B(x, rj)) -4
`d j > k
218
3. Cartesian Maps
thus 0*(Ek(x), x) > 0 V k, therefore x 0 AL(u). Next we observe that
lu(y) - u(x)I
(7)
2.7 Iy - xj
whenever x, y E Bj, Iy - xI < 1/j. In fact if S = y - x1, from the definition of Bj we get meas ((B (x, S) n Ej (x)) U (B (y, S) n E; (y)))
meas (B(x, S)) =
2meas (B(x, S) n B(y, 5))
so there exists at least one z for which
Ix-zI <S 1u(X) - u(z)I < jIx - zI,
Iy-zj <S u(Y) - u(z)I <- iIy - zI
and by the triangle inequality we get (7). Finally we express Bj as a union of measurable sets Bj,1i ... , Bj ...... with diameters less than 1/j. Note that UIB, k is Lipschitz. We then glue together the Bj,k to form the non decreasing family {C; } of measurable sets and {u1c } is Lipschitz. As u and uj are approximately differentiable almost everywhere and coincide on C3 we conclude, compare the argument in Theorem 4 in Sec. 3.1.3,
that Dud (x) = Du(x) if 0*(Cj, x) > 0. Kirszbraun's theorem yields then the extension uj. Clearly
U;C; = U3Bj J AL(u) Let us prove that (iii) implies (i). By Rademacher's theorem uj is differentiable on a set Nj with meas (Rn \ N5) = 0, hence ujN3 is approximately differentiable at every point in (8)
N3 n {xECj IO(C;,x)=1},
As O(C;, x) = 1 also it turns out to be approximately differentiable in the set in (8). The conclusion then follows as the set in (8) differs from C3 by a zero set.
The same argument with C3 replaced by Fj proves that (iv) implies (i). Finally Proposition 2 in Sec. 3.1.3 yields at once that (iii) implies (iv).
Remark 2. Since every Lipschitz function it has partial derivatives almost everywhere which coincide with ap Du(ei), from Theorem 3 we infer that every almost everywhere approximately differentiable function has approximate partial derivatives almost everywhere. The converse is also true, compare Theorem 1.
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps Let u : R' --* RN, n < N, be a Lipschitz map, and let A be a measurable subset of Rn. The counting function or Banach's indicatrix of it in A is defined for all y E R N by
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps
N(u, A, y) = #{x
(1)
I
219
x E A , u(x) = y}
Then the classical area formula states that N(u, A, y) is 7-c-measurable in y and (2)
f r N(u, A, y) d?-G"(y) J, (x) dx = J J RN A
,
Jv, :=
det (Du)*Du
compare Sec. 2.1.2. In particular, if u is one-to-one, we deduce
7-('(u(A)) =
(3)
JA
Ju(x) dx
and more generally the following change of variables formula holds
ff(u(x))Ju(x)dx = J f (y)N(u, A, y) dR'(y)
(4)
RN
A
for all measurable f : u(A) -+ R, whenever one of the two sides is meaningful. We shall now show that (2) (3) (4) can be extended to any L1 map which is almost everywhere approximately differentiable, in particular to W','-maps, provided u(A) and N(u, A, y) be suitably defined or provided a suitable representative of u is chosen. In doing that the key point is the rectifiability theorem in the form given in Theorem 3 in Sec. 3.1.4 and a relevant role is played by the set AD (u) or Ru, of points where the approximate differentiability gives a differential control. Of course the set s
AD(u) := {x I u is approximately differentiable at x} depends on the pointwise values of u. However we have
Lemma 1. Let u, v : 0 -> R' be two measurable maps which agree almost everywhere in 12. Suppose that, for a given x, u is approximately differentiable at x and v is approximately continuous at x. Then v is approximately differentiable at x, too; moreover apDv(x) = apDu(x).
Proof. In fact for any e > 0 the sets {y I
1u(y) - u(x) - L(y - x) I
Iy-xl
> e} and {y I
Iv(y) - v(x) - L(y - x) I > E}
Iy-xI
with L := Du(x) differ by a null set. Lemma 1 shows also that the notion of almost everywhere approximate differentiability depends on the equivalence class of u and not on the chosen representative u. Introducing the set of approximate continuity of a representative of u Ac (u) :_ {x E .f2 I u is approximately continuous at x}
220
3. Cartesian Maps
we can state Lemma 1 in formula as
AD(u)DAD(v) C A
denotes the symmetric difference.
Definition 1. Let u : IR" --, RN be a map which as almost everywhere approximately differentiable. We define the Banach indicatrix of u by
N(u, A, y) := # {x I x E A n AD (u), u(x) = y}
(5)
Then we have
Theorem 1. Let Q be an open set on IIBn and u be an almost everywhere approximately differentiable map, in particular let u E W1"((1,RN) Then for any measurable subset A of (2fN(u,AY)dfl'(Y), we have
fJ(x)dx =
(6) A
Ju :=
det (Du)*Du ;
RN
more precisely, N(u, A, .) is 7-l'-measurable and (6) holds. Also Ju E L'(A) if and only if N(u, A, ) is 7-C2-summable in RN. Moreover, for any non-negative measurable function f : RN --> IEB or any f : RN --> R such that either
f
N(u, A, ) is 7{n-summable in
R
or
E L1(A) we have (7)
J
f (u(x))Ju(x) dx =
A
f
f (y)N(u, A, y) d7-ln(y)
RN
Proof. It suffices to prove (6), (7) follows in a similar way. Using Theorem 3 in
Sec. 3.1.4, we can cover AD(u) by a non decreasing sequence of disjoint measurable sets {Fk} and find Lipschitz functions Uk E Lip (RI) such that
UkFk D AD(u),
Duk = Du a.e on Fk
11 .k = u on Fk,
From the standard area formula for Lipschitz maps, compare Sec. 2.1.2, we infer
f
Ju (x) dx =
AnAD(u)nFk
f
Juk (x) dx
AnAD(u)nFk I
1 N(uk, A n AD (u) n Fk, y) d7{n = I N(u, A n AD (u) n Fk, y) d7-ln
J
RN
RN
observing that AD (u) n Fk T AD (u), meas (,f2 \ AD (u)) = 0, N(uk, A n AD (u) n
Fk, y) T N(u, A n AD (U), y) and, (6) follows at once, taking into account the definition of N(u, A, y).
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps
221
If we define the rectifiable image of A by u as
u(A)r := u(A n AD (u)) = {y J N(u, A, y) # O} we can write (7) as
ff(u(x))J(x)dx =
(8)
f
f (y)N(u, A, y) d7-C' (y)
u(A)''
A
Notice that u(A)r is rectifiable and just the definition in (5) of N(u, A, y) allows us to identify it.
In the definition of the Banach indicatrix and of the rectifiable image of u we have taken out the singular points, i.e., .(2 \ AD(u). This way ujAD(u) maps null sets into null sets. This is one of the key properties which makes Theorem 1 valid, as clearly appears also in the next proposition. Recall that a map u : 0 C 1[x'1 --> R' has Lusin's property (N) in f2 if 71'1(u(E)) = 0 whenever 7-ln(E) = 0, E C Q.
Proposition 1. Let u : P C
H
--> RN be an almost everywhere approximately
differentiable map. Then (i) UTAD(u) has Lusin property (N), (ii) UTAD(u) maps L"-measurable sets into 7-ln -measurable and countably nrectifiable sets, (iii) if u = v a.e., then v is a.e. approximately differentiable, and for any A C P
N(v, A, y) = N(u, A, y)
for a.e. y E RN ;
in particular u(A)r and v(A)r differ by a null set. Proof. By Theorem 3 in Sec. 3.1.4 we cover AD(U) with measurable sets {Bj} such that uIB3 is Lipschitz. Then (i) and (ii) clearly follow. To prove (iii) denote by S the symmetric difference between {x I x E AnAD(u), u(x) = y} and {x I x E AnAD(v), v(x) = y}. If y E S, then either u(x) = y or v(x) = y and u(x) v(x), hence
S c u(EnAD(u))av(EnAD(V)) where E
{x j u(x) 54 v(x)}. Since meas (E) = 0 we then infer NT (S) = 0.
Let u E Ll(.Q,RN), 0 C 118'. Recall that the Lebesgue set of u, denoted by Lu is the set of points x E P for which there exists A = .Xu(x) E RN such that
I Ju(y) - A dy ---> 0
as r -> 0,
B(x,r)
.u(x) is then called the Lebesgue value of u at x E Lu. As a result of the differentiation theorem for integrals we have seen in the previous subsections
222
3. Cartesian Maps
that u(x) = X, (x) for almost every x E Q, hence av,(x) is a measurable function 0. A Lebesgue representative of u E L' is then just on L,, and meas (Q \ an extension of X, (x) to the whole of 0. We finally set
R,, := G n AD (u) Notice that u(x) = Au(x) for x E R,,,, since u is approximately continuous for every x E AD (u).
Definition 2. Let u E Ll(Q, RN). Any representative of u which maps if null sets in 17 into 7{n-null sets is called a Lusin representative of u. As consequence of Proposition 1 we readily get
Corollary 1. Let u E Ll (.R, IRN) be almost everywhere approximately differentiable. Then we have
(i) Any representative iii of u restricted on Ru has Lusin property (N) in Rv,. In particular a Lusin representative of u in 17 is given simply by
f u(x) yo
if x E Rte,
if
where yo E RN is arbitrary. (ii) Any Lusin representative of u maps measurable sets into fn-measurable and countably n-rectifiable sets.
iii) For any two representatives u and v of u we have
N(u, A, y) = N(v, A, y)
for 7-ln-a. e. y
and the sets u(A)T and v(A)r differ by a 7-n-null set. From Proposition 1 and Theorem 1 we also infer
Corollary 2. The change of variables formula (4) holds for any map u E Ll which is almost everywhere approximately differentiable provided in (4) we take as u a Lusin representative of u, and of course one of the two sides is meaningful.
Remark 1. In many respects it would have been more convenient to define as Lebesgue representative of u the map A,,(x) with domain G and as Lusin representative the restriction of A,,,(x) on 7Z,,,. This way one could simply state that the area formula hold for the Lusin representative of an a.e. approximately differentiable map. Though sometimes confusing, we preferred however to follow the tradition in introducing the Definition 1 and Definition 2. Remark 2. We notice that even if every point in .f1 \ 1? is a Lebesgue point, for instance even if u is continuous, the null set 1 \ 7Z,,, may be mapped by u into a set of positive measure, making (4) not true, while of course (8) holds. This is for instance the case for the Cantor-Vitali map V, compare 2 in Sec. 1.1.3. V
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps
223
is continuous in [0, 1] with zero a.e. differential. It maps [0, 1] \ E, E being the Cantor set, into the dyadic set
D :_ {x = Zk l k = 1, 2, ... , j E N, j E [0, 2k] } and E into [0, 1] \ D. Consequently
N(u, A, y) = 0
for a.e. y
while
#{x l V(x) = y} = 1
.
However as a simple consequence of Theorem 1, or Corollary 2, we have
Theorem 2 (R.ado-Reichelderfer). Let u : Q --> 1RN be a continuous map. Assume that u has Lusin property and is differentiable almost everywhere in .fl. Then (4) holds whenever one of the two sides is meaningful.
Remark 3. Formula (7) does not hold in general for maps u : ( -+ Rn which are in W1,n (Q, R,), if N(u, A, y) is the classical Banach indicatrix in (1). In fact, given a domain S2 in the plane with boundary a simple curve for which the Lebesgue measure of any arc is non zero, one can find a conformal map u of the half disk {z E C I I z I < 1, Im z > 0} onto (2. The map u is continuous and
belongs to W1"z. We extend u to the disk { lzl < 11 by setting f (z) = f (z) if Im z < 0. One then sees that u is continuous and of class W1"2 in the disk { I z I <
1 }. However u has not Lusin property, the segment Im z = 0, -1 < Re z < 1, is mapped into some arc of non-zero measure. But we have
Theorem 3. Let u belong to Wl,p(Q,RN), fl C Rn, with p > n. Then the continuous representative of u still denoted by u has Lusin property (N) and (4) holds for it.
Proof. Let A C .f2 be such that JAI = 0. We need to show that 7-ln(u(A)) = 0. Fix E > 0 and choose an open set V C .Q with I V I < E, V D A. Let V = be a cube subdivision of V and r, the length of an edge of Q. We have 00
u(A) C U u(QU) V=1
and in view of Morrey-Sobolev estimate in Sec. 3.1.2, the essential diameter of the set does not exceed
cri-nlp (f Q.
Consequently
IDulpdx)1/p
3. Cartesian Maps
224
00
00
7ln(u(A)) <
xn(u(Q,))
r,(1-n/p) (j IDulp dx)
1
n/p
Q.
Applying Holder inequality we infer 00
xn(u(A)) < c (E,=1 rv)
1-n/p
00 Ev_1 (f IDulp dx) Q
cE1-n/P(f
civil-n/p(f IDulpdx)n/p < v
n/p
IDulpdx)n/p
n
Since e > 0 is arbitrary, this gives us 7-Ln(u(A)) = 0.
Let us consider now the special case of maps from a domain .(l C Rn into Rn+N of the type
id x u :.fl -., Rn+N
given by
(id x u) (x) :_ (x, u(x))
.
Let u E Ll (Q, RN) be a map from dl C W1 into RN which is almost everywhere approximately differentiable in .fl.
Definition 3. We define the graph of u in ,fl x RN as !'u,s?
:= {(x, u(x))
I
R,
9u, j? = (id >a u) (Ru n 0) = (id x u) (,f2)' .
Notice that again we are taking out the singular points of u, that is we are considering the rectifiable image, which also turns out to be the image of a Lusin representative of u, so that the area formula holds. In particular notice that we have
7-ln(cu,.ft n (A x RN)) = 0
whenever A is a null set in Q. Also 9v,,n can be computed in the 7-ln a.e. sense
using any other representative v of u. In fact, if v = u a.e., then v is a.e. approximately differentiable and Rv and Ru differ by a null set. By the area formula we then conclude that I L.
u,.n'G'v,rz) = 0 .
With respect to the standard notion of graph of a map graph u :_ { (x, u(x)) I X E ,fl }
the graph gu,,{Z of an approximately differentiable map is clearly a subset of graph u, and in general we may have
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps
225
7-ln(graphu \ 9u,n) > 0 From this point of view it would be more reasonable to call 9Ju,n the 1-graph of u, or in Federer's terminology the countably rectifiable graph, to distinguish it from the usual graph. In fact, as we have seen, 9u,n is 7-1n-measurable and countably n-rectifiable and therefore for almost every point of 9u,n the (approximate) tangent plane to cu,n exists and does not contain vertical vectors, compare Sec. 2.2.1 and Sec. 2.2.4. In the sequel we shall in fact use also the terminology 1-graph, but if no confusion may arise we just speak of graphs. The discussion above can be formally stated as
Theorem 4. Let it E L' (0, RN) be an almost everywhere approximately differentiable map in .1? C Rn. Then 9u,n is 7-Ln-measurable and countably nrectifiable,
Hn(gu,n) = f
1+
nV
IMa (Du(x)) 12 dx JaI+i I=n
where {M« (Du(x))} denotes the set of minors of the approximate differential Du(x). In particular Qju,n has finite area, i.e. is n-rectifiable, if and only if all Jacobian minors of it are in L'(.f2). We emphasize that in particular at 7"Ln-a.e. point zo E (xo, yo) in Gu,n there exists an approximate tangent plane Lx which is obtained as limit Hn L Lx,,
,Hn L 17zo,a(9u,n)
as A -f 0, compare Sec. 2.1.4, where
z - zo 7]=o,a(z)
A
For xo E 0 and A > 0, set U'\ (X)
(u(xo + Ax) - u(xo))
Then ua converge in measure to the approximate tangent map x -+ Du(xo)x at xo
u,\(x) --f Du(xo)x
in measure ,
if u is approximately differentiable at xo, compare Proposition 5 in Sec. 3.1.4, and
ua(x) --4 Du(xo)x
in L1
if u is differentiable in the L1 sense, compare Theorem 4 in Sec. 3.1.2. We have
3. Cartesian Maps
226
Theorem 5. Let u E L' ((2, ]E81') be an almost everywhere approximately differentiable map with 7-ln(9u,n) < oo. Then for Hn-a.e. x0 in .(2 or 7-Ln-a.e. (xo, u(xo)) in 9u,fl both the approximate tangent map Du(xo)x and the approximate tangent plane Lxo to 9u,.a at (xo, u(xo)) exist and agree Lxo = {(x, y) E ][8n x ]E81'
Proof.
I
y = Du(xo)x}.
Being u a.e. approximately differentiable, by (iv) of Theorem 3 in
Sec. 3.1.4 we can decompose .(l as ,fl = US?, and find functions uj c C1(Wn ] such that
uj = urn,,
Ln(.(lo) = 0,
)
Dud = apDu a.e. in Q.
Denote by IM(Du(x))I the area density which by assumption is in L1 and for j > 1 let x0 E (23 be such that Duj(xo) = apDu(xo) x0 is a Lebesgue point for I M(Du)
(9)
0*(y,Rn\f2j,xo)=0 where p is the measure
p(A) = f I M(Du) I dx. A
Of course a.e. point x0 in ,fly meets the requirements in Set
77a(x,y)
/x - xo y -u(xo)\ l\
A
J
A
and let f be any smooth function with support in B(0, 1) C IEBn
R'L77A(gu)(f) = f f(x,y)d7{n 77a(c )
An
f
f (x -xo y
u(xo)) A
J
d7-ln
G,.
Jf f
(x -xo u(x) _u(xo))
an
J
Rn
ff B(o,1)
Set now Fj,A (10)
{l;
\
M(Du(x)) I dx
u(xo + A) - uo(xo)) IM(Du(xo + A6)1 d A
J
I xo + Al; E .(lj } and compute the last integral as
1=! ...+ f
B(0,1)
B(0,1)nF,,,,
B(o,1)\F,,,,
....
1.5 Area Formulas, Degree, and Graphs of Non Smooth Maps
227
For A --f 0 we have
M(Du(xo + AC)) -----> M(Du(xo))
XF;,AnB(o,1) -' XB(o,1)
in L'
pointwisely
u(xo + AC) - u(xo) = uj (xo + AC) - uj (xo)
Dud (xo) l; uniformly,
therefore the first integral on the right hand side of (10) converges to
f f ((;, Du(xo))
dx
I
which of course is just 7-l" L Lx0 (f), while the second integral can be estimated by
f
f
IIfII.
I M(Du) I dl;
B(O,a)\,f2j
B(0,1)\FF,A
which tends to zero by the third condition in (9).
11
Consider now a map u : 12 C R" --> 1R? which is approximately differentiable almost everywhere. We can define the degree of u at y with respect to a measurable set A C .0 by
#{x E A n 7Zu I u(x) = y, det Du(x) > 0}
deg (u, A, y)
- #{x E A n Ru I u(x) = y, det Du(x) < 0} N(u, A n {x l det Du(x) > 0}, y) -N(u, A n {x I det Du(x) < 0}, y) i.e. we set
Definition 4. The degree of u at y with respect to a measurable set A C .fl is the integer defined by (11)
deg (u, A, y) _
sign (det Du(x)) xEu-1(y)nAnizu
whenever the sum on the right exists. We then infer
Theorem 6. Let u be an almost everywhere approximately differentiable map, in particular let u E W1"1(Q,R") and let A be a measurable subset of .fl. Then deg (u, A, y) is measurable and deg (u, A, Y) E L' ([8") if and only if det Du(x) E L'(A) and (12)
f det Du(x) dx A
deg (u, A, y) dy Rt
Moreover deg (u, A, -) = deg (v, A, -) a. e. in W1 if u = v a.e. in D.
228
3. Cartesian Maps
Proof. It suffices to apply the area formula (7) to the subsets
A+ := A n {x E 7Zu det Du(x) > 0} A_ := A n {x E Ru I det Du(x) < 0} I
0 Remark 4. We observe that, although for every u E W1,1 with det Du(x) in L1 the degree deg (u, A, y) is well defined, it is not very useful in such a generality. In fact an important property of the degree for smooth maps is its homological invariance, i.e., if u, v E C' (0, fan) and u = v on 8.2, then
deg (u, ,R, y) = deg (v, 0, y)
for a.e. y .
But this property is not anymore valid for W1'1-functions with det Du E L'. For instance, consider the functions u, v : B(0,1) c In -+ Rn x u(x)
x ,
v(x) .1xI
We have v(x) E Wl,p(B(0,1), Rn) for all p < n, u = v on 8B(0,1), but deg (u, B(0, 1), y) = 1,
deg (v, B(0,1), y) = 0
for a.e. Y E B(0, 1). We shall return to this point in Sec. 4.3.
2 Maps with Jacobian Minors in Li Motivated by the results of Sec. 3.1, and in particular of Sec. 3.1.5, we shall be concerned in this section with maps u in L1(f2, R N) which are approximately differentiable almost everywhere in .f2, for instance Sobolev maps, and with Jacobian minors in L1. We shall see that the geometric-differential properties of such maps are described by the integer multiplicity rectifiable current Cu defined as integration of compactly supported n-forms in Q x IN over the graph Cu, s2, compare Sec. 3.2.1, i.e. as the current carried by 9,,,,s2. For smooth maps, u E C' (S?, R N) Stokes theorem yields that the current Gu is boundaryless in ,fl x RN, (1)
BGu L .f2 X IRN = 0.
This is still true for maps u E Wl,n(S? IRN) n := min(n,N). However it turns out that in general (1) does not hold for u E W1,P(.(1,R N), p < n, compare Sec. 3.2.2.
The geometric condition (1) is related to suitable analytical formulas of integration by parts. For instance in the scalar case, N = 1, it is equivalent to the
2.1 The Class A'(fl,IRN), Graphs and Boundaries
229
classical formula of integration by parts for the derivatives, which expresses the coincidence of the approximate differential and the distributional derivatives. This is no more true in the vector valued case, N > 1. For smooth functions several formulas of integration by parts for the minors are known, e.g., the so called Piola identities corresponding to the differential identities
Dj(adj (Du)")' = 0
.
We shall see in Sec. 3.2.3 that condition (1) is equivalent to the stronger integration by parts formulas
fDj{(x,u(x))](adj (Du)a)'dx = 0
V 0 E CC(fl x RN) .
jEa
In Sec. 3.2.4 we shall also discuss in terms of boundaries the notion of distributional determinant and certain classes of maps denoted Ap,9(.f2,Rn) C Wl,n-1(Q,JRn) where p > n - 1, q > n/ (n - 1), whose elements turn out to have graphs with no boundary in $2 x RN. For maps in these classes we shall also prove an isoperimetric inequality for the determinant from which one can easily deduce higher integrability of the determinant. We conclude Sec. 3.2.4 with a brief discussion of a different approach to higher integrability in terms of Hardy spaces. Finally relations between traces and boundaries in 812 x RN will be discussed in Sec. 3.2.5.
2.1 The Class A' (,f2, RN), Graphs and Boundaries Let 0 be a bounded domain in
IR
Definition 1. We denote by Al (,f2, RN) the class of functions in Ll (0, RN) which are approximately differentiable almost everywhere in .(l, with approximate differential Du, and with the property that all minors of the Jacobian matrix Du are in Ll (.fl), i.e., A'(,(l,ISBN) (1)
{u E L1(12,ISBN)
I
u is approximately differentiable a. e. Ma (Du) E L1(Q) V a,,3 with Ial + 10 1 = n}
.
Here we have denoted by Mg (Du) the (3, a)-minor of Du, compare Sec. 2.2.1. We set (2)
IIuIl.al
f (Jul + IM(Du)l) dx
where we recall that M(Du) is the n-vector in A Rn+N given by
230
3. Cartesian Maps
M(Du) = (e1 + Diu) A ... A (e,,, + Dnu) N
N
(e1 + E D1u'ei)n ... A(en +
Dnutei)
=1
o(a, a) Ma (Du) eanea. IaI+IQI=n and
IM(G)l :_ {1 + E 1Mg
(G)12}112.
aIZI=n IRI>o
We remark that A' (0, R N) is not a linear space and II ' Il.,al is not a norm. Also observe that every function W1,°(12, IRN), with p > n, where
n := min(n, N) ,
(3)
belongs to A1(9,1RN). In fact any k-minor of Du, being a linear combination of products of k derivatives of u, belongs to Ln1!(Q).
Let u E A'(0, RI) and let w E Dn(P x RI) be an infinitely differentiable n-form with compact support in .fl x IRN. We can define the pull-back of w onto .(2 by the map id x u : 0 -> 1? x IR'n, (id < u) (x) := (x, u(x)), as in the smooth case, just replacing ordinary derivatives by approximate derivatives. In coordinates, if
w(x, y) _
wap(x, y) dxandyA IaI+IQI=n
then for a.e. x E fl waQ(x, u(x)) dxaAdup(x)
(id pa u)#(w)(x) IaI+Ipl=n
E o,(a, a) w.o(x, u(x)) Ma (Du(x)) dx IaI+Isl=n
_ < w(x, u(x)), M(Du(x)) > dx where < , > denotes the duality between covectors and vectors in IRn x particular we find (4)
I(id xu)#(w)(x)J < Ilwlloo IM(Du(x))J
RN. In
a.e. in 1 .
Therefore to any u E A' (17, IRN) we can associate the n-dimensional current G,,, E Dn, (,fl x IRN) defined for w E Dn (,f2 x IRN) by (5)
G,,(w)
=
f(idu)#w
= Q.f2J((id xu)#w)
Q
= I < w(x, u(x)), M(Du(x)) > dx . 92
2.1 The Class A' (.(2, RN), Graphs and Boundaries
231
From (4) we also see that G.,, has finite mass given by
M(G,,,) =
(6)
As for smooth graphs, and actually for any current in D,,(S2 x IRN), the kcomponent of Gu, 0 < k < min(n, N), is defined by testing Gu with the part of w with k-differentials in y E RN, Gu(k)(w) := Gu(w(k)) and, obviously
Gu(w(k)) =
(7)
f
JQ
<w(k) (x, u(x)), M(k) (Du(x)) > dx
.
The (a, /3)-component of Gu, (G,,) '13, is the distribution, actually the measure, in ,(2 x RN given for 0 E Co ° (.(2 X RN) by
f(x,u(x))M(Du(x))dx.
G"a(O(x, y)) :_ o,(a, d)
(8)
S'2
As M(Gu) < +oc, we can think of Gu as being defined on the space of all n-forms w with Borel bounded coefficients in Rn X RN. In fact if {wk} C Dn(Q x IRN) converges pointwisely tow in ,f2 x RN, and SUPk II Wk LI.,QXRN < oo,
by Lebesgue dominated convergence theorem, we can define Gu (w) as
Gu(w) := lira Gu(wk) k-.oo
compare Sec. 1.1.4.
Let u E A' (.(2, R'), and let yu,,(Z be the graph of u defined in Sec. 3.1.5. From Theorem 4 in Sec. 3.1.5 we known that 9u,,{l is n-rectifiable and
x'L(9u,n) =
f I M(Du(x))I dx = M(Gu) 0
and by Theorem 5 in Sec. 3.1.5
M(Du(x))
9u,n(x,u(x)) = M(Du(x))I is an n-vector which orients 9u {l. Again the area formula, Theorem 4 in Sec. 3.1.5, then yields that
Gu (w) = f < w, j u,n > 0-0 L 9Ju,,f2
,
and
,Hn(Qu ,{l f1 (A x RN)) = 0
therefore we can state
if and only if fn(A) = 0
,
232
3. Cartesian Maps
Proposition 1. Let u E A' (Q, RN). Then G,, is an i.m. rectifiable current with multiplicity one, more precisely Gu = T(9u,n) 17 u,n)
Definition 2. We say that a sequence {uk} C Al(S2,RN) converges weakly in A' u, if and only if Ai to u E Al (.f2, RI), uk
Uk -+ u
strongly in L'
M(Duk) - M(Du) weakly inL' .
(9)
Notice that in Definition 2 we require strong convergence of uk to u in V. This is due to the fact that the Duk are not in general the distributional derivatives of uk. If this is the case, we can as well replace strong convergence of Uk to u by weak convergence. We also notice that the a.e. convergence of Uk to u would do the same. The Al-weak convergence reads as convergence of currents.
Proposition 2. Let uk, u E A' (0, RN) and suppose that uk ` U. Then Guk (w) -> Gu (w) for any n-form w with continuous and bounded coefficients in R'n x RN, i. e.
Gu, - Gu
in Dn (R x R")
Gus - Gu
in D,, (1? x RN) .
in particular Proof. Possibly passing to a subsequence from Uk
A'
u we deduce
M' (Duk(x)) - Ma (Du(x)) weakly in L1 a.e. in 0 uk (x) -> u(x) a.e. in S2 cb(x,uk(x)) - q(x,u(x)) supk IO(x, uk (x)) l
< +oo
for any a, Q with I a I + 101 = n and for any 0 : R" x RN -> R which is continuous and bounded. Thus (i) of Proposition 1 in Sec. 1.2.4 implies
(Gu,)"'(0(x, y)) = a(a, a) f O(x, uk(x))Mg (Duk(x)) dx 0
-- 0'(a, a) f O(x, u(x))Mg (Du(x)) dx =
y))
.n
We then conclude that any subsequence of {uk} has a subsequence {uh,} such that Ga3 (q) -' Gup(0) O
V.
Since the limit is independent of the chosen subsequence, the claim follows at once.
2.2 Examples
233
Let u : Q - ]RN be a smooth map, for instance u e C2 (,f'2, RN ). Then u E Al (Q, RN), and, due to Stokes's theorem G,,, is boundaryless in (2 x ]RN, i.e.,
8G,, L I2 x IRN = 0.
(10)
In fact (id x u)#w is an (n - 1)-form in .fl with C, -coefficients for any w E Dn-1(.fl X RN), hence n
(id >< u)#w = E(-1)i-1fi(X) dx2,
fi E Cc' (n) ,
i=1
therefore
8Gu(w) = G.(dw) = Q Q I (d(id >< u)#w) =
r
Jn
div f dx =
ff.vQd_1 = 0 . an
As we have seen in Ch. 1, and as we shall see later on, property (10) is crucial in a number of situations and especially if we want conserve homological properties of regular maps to their weak limits. But examples in Sec. 3.2.2 below show that, for a generic u E .A1(,f2,1RN), (10) does not hold. In general, if u CA' (Q, RN) nW1"P(.f2, RN), Gu does have boundary in ,f2 x RN whenever p < n, n := min(n, N). We then set
Definition 3. Let .f2 C W be an open domain. The class of Cartesian maps is defined by
carts (0, IRN) := {u E A1(.f2, IRN)
I
8Gu L .fl x IRN = 0}
We have
Proposition 3. Let u be a map in Wl,T(f?, IRN), n := min(n, N). Then u E cart' ((2,RN).
Proof. Since any minor of Du is a linear combination of products of k derivatives of u, 1 < k < n, it belongs to Ln/k, thus u E Al (,f2, IRN). Let {uk} C Cl (Q, ]RN) n Wl,n(Q,IRN) be a sequence of smooth maps converging to u in W"((2, IRN) Then uk - u weakly in A' (.f2, RN), and Proposition 2 yields Guk -s Gu weakly
in the sense of the currents in .f2 x RN. As BGuk - 8Gu in .f2 x RN and BGuk L 12 x ]RN = 0, the claim is proved.
0
Notice that in particular 8Gu L 0 X ]RN = 0, i.e. u E carts (S-2, RN) if u is a Lipschitz map.
2.2 Examples We shall present in this section a few examples of maps in Al (,f2, IRN) whose graphs have holes in (2 x RN and the corresponding currents Gu's have non zero boundaries in f? X ]RN.
234
3. Cartesian Maps
0 Let us begin by showing that there exist maps in Al (Q,1RN) n W',p (Q, RN), p < n, whose graphs have homological holes in .fl x IRS". Let B(0,1) be the unit ball in 1Rn and let u : B(0,1) -> B(0,1) the map x
U := ICI
.
Clearly u E W1"P(B(0,1), Rn) for all p < n, hence all minors of order less or
1
1
1-E
Fig. 3.1. The maps rE (t) and pE (t).
equal to n -1 belong to Lr/(n-1) (Q) C L1(Q), while for the last minor which is of order n we have det Du(x) = 0 for x # 0
since u maps B(0,1) \ {0} into Sn-1 C Rn. Thus u E A'(0,IR'). We now claim
that (1)
aGuL .flxRn = -8oXaI[{yEIRn
yI<_1} or in other words, that CgT z T,a(o,l) has the set {0} x B(0,1) in Rn x 1n as a hole, which lies above the origin of Rn.
In order to prove (1) we approximate u(x)
x in W1"p(B(0,1), IRn),
p < n, by the Lipschitz-continuous maps
ue(x) := re(IxI) where re(t) is the function from [0, 1] to [0, 1] that is linear in (0, e) and in (e, 1) and satisfies rE (0) = 0, rE (e) = 1 - e, re (1) = 1, see Figure 3.1. We observe that an easy computation yields that in the sense of measures
det Due - B11bo where So is the Dirac mass at zero. We shall now prove that in fact (2)
G, -G,, +8o x QB(0,1)
,
2.2 Examples
235
therefore, since 8Gti,E LB(0,1) x W1 = 0, d e, we deduce
a(G,, + 6o x B(0, 1) ]I) L B(0,1) x TR' = 0
(aGu) L B(0,1) x R7 = -go x aQ B(0,1) D which is (1).
To show (2) we reparametrize cuE,B(o,l) as follows. Let yE : B(0,1) -* B(0,1) be the map -YE(z) := pE(IzI) IzI
pE being the map from [0, 1] in [0, 1] which is linear in (0, 1/2) and in (1/2, 1), and satisfies p6(0) = 0, pE(1/2) = e, p(l) = 1, see Figure 3.1. Then defining 0E . B(0,1)
B(0,1) x B(0, 1)
k(z) := (yE(z), uE(7e(z)))
,
,
we have (x, uE(x))
0E(z)
,
x ='YE(z) ,
thus
GuE = 0E# [B(0,1) l On the other hand 0E is a family of equi-Lipschitz maps which converge uniformly to the Lipschitz map 0 given by (0, 2z)
if IzI < 2 I
O(z)
{ ((2IzI -1)11, (0, 2z)
{ (y(z), u(y(z)))
if 2 <
<1
if IzI < 2
if 2 < z < 1
where y(z) _ (2z - 1) IzI . Thus we deduce
GuE -0#B(0, 1)
_ Gu + {0} x B(0,1) j
which proves (2). The previous example shows also that it is not possible to approximate weakly GTw-T by a sequence of smooth graphs Guk, since in this case we would have
aGzf L B(0,1) x R = 0 while as we have seen
aGz L B(0,1) x R' = -Jo x QB(o, 1)
0
.
3. Cartesian Maps
236
z
in W1, P, In a more analytic form, this shows that, while we can approximate for any p < n, by a sequence of smooth functions, no such a sequence {uk} may satisfy in L1(B(0,1)) det Duk --p det Du I
or even
I
det DUk - det Du
in the sense of measures.
0 Similarly to 1 we can discuss the boundary in B(0,1) x RI of graphs of
functions which are homogeneous of degree zero.
Let cp a smooth map from S"i-1 C W into IRI' and let us denote by u the homogeneous extension of cp to B(0,1), i.e.,
u is singular at the origin (except in the case cp = constant), moreover u E Wl"P(B(0,1),IRN) for any p < n, and also u E A1(B(0,1),IRN). Approximating u by
rE(jxj) V(IX
uE(x)
rE being the same function in 1 , setting as in
0, (z) :_ (ryE(z),uE(-yf(z))),
we get
OE#QB(0,1)Y = Guc and
Gue -O#QB(0,1)11 = 0#QB(0,1)\B(0,1/2)l+0#QB(0,1/2)1 where
O(z) :_
(0, 2IzjW(z )) (Y(z), P( IryY(=) I ))
z 2
_
z z <1
z
-y(z) := (21zl - 1) z
I
0#QB(0,1) \ B(0,1/2) j = Gu, aO# Q B(o, l) l L B(O, l) x 1RN = o ao# QB(o,1/2)1 L B(O, 1) X RN = aO# IB(0,1/2) J _
= 0# 8QB(0,1/2) l = bo x W# Sri-1 ]I denoting the current integration over
aIB(0,1) . Therefore we conclude that
QS'-' I
,
S'n-1 oriented so that Q
Sn-1
]I _
2.2 Examples
237
aG,, L B(0, 1) x ]RN = -bo x and that there exists a sequence {uk} of smooth maps Uk such that Guk -k Gu if and only if 5Gu = 0 in B(0,1) xIRN or equivalently if and only if cp#QSn-1 I = 0.
Very roughly we can say that even if the graph Gu of the map u(x) has a hole above zero, one can approximate Gu by graphs Guk of smooth maps Uk if and only if the integration of forms on (p(Sn-1) is identically zero. In the special case cp : Sn-1 -* Sn-1, by the constancy theorem, compare Sec. 4.3.1 and Sec. 5.5.4, we have = deg cp Q
W # Q Sn-1
Sn-1
j
Sn-1 __+ Sn-1, thus
where deg cp is the degree of the map cp :
S'-1
0Gu L B(0,1) x RN = -deg cp bo x Q
j
and we can conclude that there exists a sequence of smooth maps Uk : B(0,1) C Rn --f ln, uk(x) = ;O(x) for x E aB(0,1) such that
Guk -Gu
u
x
O (x
I)
if and only if deg cp = 0.
3 In ]R2 which we identify with the complex plan C we consider ,
W(Z)
We have deg cp = k, thus by
Izlk
zEC, kEZ.
0
aGu L B(0, 1) x IR2 = -k 5o x Q Sl j where
u(x)
(IXx1
In particular we have
M(aGu L B(0,1) x 1R2) = 27r kI
,
that is, one can find in A' (Q, IRN) mappings with non zero boundary in Q x RN of arbitrarily large mass.
Actually there exists maps u in A' (0, RN) with M(aGu L I? x IRN) = oo. Let 0 : iR -> R be a Lipschitz function which is zero outside the interval (-1, 1) and satisfies (0) = 1. For k = 1, 2,... we denote by xk the points in IR2 4
with coordinates Xk = (k,0)
238
3. Cartesian Maps
and we set 1
Ak := min(Ixk - xk-1IH Ixk - xk+1I) =
k(k + 1)
Then we define u : B(0, 2) C R2 --->
0 U(X)
_ lk
k=1
-k
k 1 lx
- Xk l )
An easy computation shows that u c W1"P(B(0, 2),1R2) for all p < 2, det Du E L' (B(0, 2)), i.e., u E A' (B (0, 2), R2). According to the computations in Example 3 one also sees that
aGw L B(0,2) x R2 =
b"k x Q6B(0,
Ak)
k thus
Ak = +00
M(aGv, L B(0, 2) x 1R2) = 2ir
.
k
2.3 Boundaries and Integration by Parts In this section we discuss in a more analytic way the nature of the condition (1)
aGti, L .f2 x R N = 0
which defines the class cart 1(0, RN), for functions u E A' (fl, RN).
If a = (al, ... , aP) is a multi-index in I (p, n), 0 < p < n, we say that the positive integer j belongs to a if j is one of the indexes a1, ... , c ,. If j E a we denote by a - j the multi-index of length p -1, obtained by removing the index j from a. Similarly, if j 0 d, we denote by a + j the multi-index of length p + 1 obtaining reordering naturally the multi-index (al, ... , a,,, j). For any submatrix Ga of an N x n-matrix G, 1 <_ la = Ian a:= min(n, N), we define the matrix of adjoints of G13 by the formulas (2)
(adj G13) := v(i, Q - i) o(j, a - j) det Ga-j
where i E,3 and j E a. Of course, if G is an n x n-matrix and if I a l _ 1)31 = n, we simply write adj G for adj Ga = adj G8. Also we define adj G13 := 1 if Had _ IaI = 1. Notice that we also use the notation det Ga-j = Ma-i (G)
.
The previous definition is set up in such a way that the usual Laplace's formulas for the determinant hold, i.e, for i, h E /3
2.3 Boundaries and Integration by Parts
G (adj Ga)
(3)
jEa
=
Sihdet GO a
=
239
bih Ma (G)
and, for i, j E a (4)
Y G (adj GO)h = Sij MQ (G) hEO
Let u be a function in Al (.fl, IRN), where .R is a domain of R'. Since G,,, has finite mass, we may regard Gu as a linear functional on the space of all n-forms with compact support in IRn X RN, and not only in .(1 x IRN, defined by the same formula
f<w(x,u(x)),M(Dz1(x))>dx,
Gu,(w) :=
n that is, we may regard Gu as an element of D,, (R' X RN). The measure theoretic boundary of Gu is then defined by 8Gu (w) := Gu (dw)
for any (n -1)-form w with compact support in IR' X RN. The product structure in IRn x IRy induces a natural splitting of the exterior differential operator d in IR"" X RN as
d=d,+dy
and 8Gu splits into its components (8Gu)(k), 0 < k < n, defined by (8Gu)(k)(w) := 8Gu(w(k)) ,
i.e., by testing 8Gu on the (n - 1)-forms with exactly k differentials with respect to y. Since for any form w with exactly k differentials in y, obviously d.,w has again k differentials in y while dyw has (k + 1) differentials in y, we can write (5)
(aGu) (k) (w) = G.(k) (dXw) + Gu(k+,) (dyw)
First we would like to discuss the boundary of Gu in .(1 x IRN. If u is smooth we know that (6) 8Gu L .(l X RN = 0 .
Equivalently we can write (6) as: for all (n - 1)-forms w with compact support in (1 X RN and for all k, 0 < k < min (n - 1, N), we have (7)
Gu(k) (d.w) + Gu(k+,) (dyw) = 0 ,
i.e., (6) provides a linking between the k and k+1 components of Gu, which looks like, and in fact is, a formula of integration by parts. We shall now restate (5) in a more explicit and analytic way for functions u E A'(f?, RN) thus interpreting (6) and (7) which, as we have seen in Sec. 3.2.1, do not hold for all u E A' (D' We begin by considering the 0-component (5Gu)(o) of aGu. RN).
240
3. Cartesian Maps
Proposition 1. Let u E A1(.f2,RN) The following claims are equivalent (i) (,9G.)(o) L 0 x RN = 0, d 0 E C°°(.fl x lR") (ii) fn D3[o(x, u(x))l dx = 0 the approximate differential of u is the distributional gradient, iii)
(iv) u E W1,1(f2
]RN)
Proof. An (n - 1)-form with no differentials in the vertical directions can be written as CbiECi,,(Qx RN) .
(x, y) dx2
W(x, y) i=1
Thus dxw
n` dyW
=
L:
[
Lam(-1)n+i-2
i=1 j=1
a0i dxi A dy7 aye
n N E(-1) i=1 j=1
-i'907 dxi A dyi y
57. v(i, i) aye dxi n dyHence (5Gti.)(o) (W) = Gu(o) (dxw) + Gu.(1) (dvw)
i=1
f
a0i (x, u(x)) dx + axi i=1 7=1
120i (x, u(x)) Diuj (x) dx y7 92
n
DiOi(x,u(x))dx i=1
If U E W1'1(0,]RN) and Oi E Cc°(.fl X IRN), then Oi(x,u(x)) E W"1(0), so that (3Gu,)(o)(w) = 0. Denote by the approximate differential of u, and suppose conversely that (aG,,)(o) = 0. For all cp c Coo (S2 X RN) we have (8)
J
N fix; (x, u(x)) dx + E
n
Apply now (8) to
J
j=1 n
Oyj (x, u(x)) vi (x) dx = 0
2.3 Boundaries and Integration by Parts
241
co(x) yj XR(y)
O(x, y)
where cp E C°°(Q), and XR(Y) is a cut off function in ]RN, i.e., X E C0°(IRN)
0<XR<1, XR=1 on B(O,R), in RN\B(O,2R)
XR=O
R on B(0, 2R) \ B(0, R) .
DXR
We get
J DiceujXRoudx+ J cpviXRoudx n
n N
+ E f (P (x) u' (x) ayh (u(x)) v (x) dx = 0 h=1 s2
and, as R -* oc, we deduce
fDicoudx+fcovdx = 0
(9)
n
n
since
f n
O(x) u'j (x)
ayh (U (X))
J
vi (x) dx < R
f
JullvZIdx
R
IviIdx
R
f
lvz I dx --> 0.
R
Finally, (9) implies that vz are the distributional derivatives of u.
Let us consider the other components (BGu)(k), 1 < k < min(n - 1, N) of BGu.
Proposition 2. Let u E A'(f2,RN) and let a,/3 are multi-indices with IaI + LB = n, Ql = k + 1 where 0 < k < min(n - 1, N). Then, for any i E 3 and any function 0 E C'(Rn x RN) with bounded derivatives, the function
x -' > Dj [O(x, u(x))] (adj (Du(x))a)'j jca
3. Cartesian Maps
242
belongs to L1(Q) and we have
BG.,(O(x, y) dxa A dyR-i) _ (10)
(-1) a o (a, a) o (i i3 - i) jE« f Dj { (x, u(x))] (adj (Du(x))") dx Since every (n - 1)-form with k differential in y is a linear combination of dy"-i
, dal + 101 = n, i E Q, and 1/31 = k + 1, we forms of the type O(x, y) dxa A immediately deduce from Proposition 2
Corollary 1. Let u E A1(Q, RN) and let 1 < k < min(n - 1, N). Then (BG,,,)(k) L ,fl x RN = 0 if and only if
r (11)
E Dj [O(x, u(x))] (adJ (Du(x))«)'dx = 0
JP jEa
holds for all a, /3, with ja + 101 = n, 101 = k + 1, for all i E /3 and for all
0EC°0(0xRN). Proof of Proposition 2. Fix a, 0, i, ¢ as in the assumptions and consider the (n - 1)-form c/(x, y) dxa A
w(x, y)
dyp-i
.
We have d.w(x, y)
= 1: o(.7, a) txi (x, y) dxa+j A dyQ-z jE« N
a(h, Q - i) 0yh (x, y) dx' A
dyw(x, y)
dyA-i+h
h=1 thus
G,,(k) (dxw)
_ jE&
(12)
(a
a - ) (7, a)
fn
Oxj (x, u(x)) M"= (Du(x)) dx
Gu(k+1)(dyw) N
_ (-1) al v(a, a)
v(h, /3 - i) h=1
_ (-1)1a1 a(a, a)
f
J
Oyh(x,u(x))M«-Z+h(Du(x)) dx
Q(h, /3 - i) f Cpyh. (x, u(x))Mg-i+h(Du(x)) dx. hE$+i
,S?
2.3 Boundaries and Integration by Parts
243
Now we claim that
a(a+j,a-j)a(j,a) = (-1)Ial a(a,a)a(j,a-j)
(13)
This can be checked using the following strategy to reorder (a + j, a - j): first
shift to the left the index j with a(j, a) permutations, then shift j after the a's with (-1)IaI permutations, then insert j in a - j at the right place with a(j, a - j) permutations; now a and a are in increasing order, so with a(a, a) permutations we complete the reordering of (a + j, a - j). From (12), taking into account (13), we then deduce (14)
(3G.)(k) (w) = G.(k) (dxw) + G..,(k+1) (dyW)
{a(i, / - i) E a(j, a - j) jE&
f 0, (x, u(x)) M'-j
N
+ a(i,13 - i)
i) h=1
r
(Du(x)) dx
0yh (x, u(x)) Ma-Z+h(Du(x)) dx}
J 9JA(x)
a)a(i, /3 - i)
dx Q
where
A(x) :_
OXj (x, u(x)) (adj
jEa
0
N
+ a(i, - i) E a(h, 0 - i)oyh (x, u(x)) Mg-x+h(Du(x)) h=1
Clearly A(x) E LI(fl); we shall now prove that for a.e. x Dj [O(x, u(x))] (adj (Du(x))«)'j
A(x)
(15)
jEa
which clearly proves (10), taking into account (14). From Laplace's formula Mg-Z+h(Du)
_ >Djuh (adj
(Du)_2+h)
a
jEa
EDj uha(j,a - j)a(h,,3-i)MM-j(Du) jE&
we deduce
Djuh(adj (Du)q)
a(h Q - i) a(i,Q - i) Ma-i+h(Du) jEa
.
3. Cartesian Maps
244
thus
N
0yh (x, u(x)) Djuh(x)] (adj (Du(x))")
A(x) = 1:[Oxj (x, u(x)) + jEa
h=1
which yields at once (15).
Remark 1. Taking into account Laplace's formulas, it is easily seen that (11) and therefore the condition (aGu) (k) L ,2 x I[8N = 0
is equivalent, in terms of determinants, to the following family of formulas of integrations by parts: for all i E /3 (16)
> cr(j, a - j) jE&
f
Oxj (x, u(x))
dx
dl
+ o-(i,,3 - i)
J
Oy; (x, u(x)) Ma (Du(x)) dx = 0
.
S?
Remark 2. It is easily see that (11) and (16) hold also for all 0 E C'(.fl X RN) which have bounded derivatives in f2 x RN and spt o c fix RN, with ( cc ,2, compare the proof of Proposition 1. In fact it suffices in (11) (16) to test with 0 of the type cp(x)g(y), as linear combinations of such products are dense. Moreover (11) (16) hold for any O(x, y) = cp(x)g(y) where V(x) is a Lipschitz function, spt 0 cc ,2, and g E C1(RN) with bounded derivatives. Finally we observe that (10) holds for all 0 E C1 (D x RN) with bounded derivatives.
A consequence of (11) and of Remark 2 are the classical identities N
j=1
axj (adj (Du)") = 0
to be understood now in the sense of distributions.
Corollary 2 (Piola identities). Let u E A' (,2, RN) and let a, 0 be multiindices with IaI + 1,31 = n, 131 = k + 1, 1 < k < min(n - 1, N). Suppose that
(BGu)(k) L 0 x RN = 0. Then for all i (=-,3 and all Lipschitz functions cp in .2 with spt W CC .2 we have (17)
f E DjW(x) (adj (Du(x))a)j' dx = 0 . jEa
2.3 Boundaries and Integration by Parts
245
Proof. It suffices to apply (11) with O(x, y) = cp(x), taking into account Re-
mark 2.
However conditions (17) for k = 1, 2, ... , min(n - 1, N) are weaker than (6) i.e., they do not imply in general that Gu is boundaryless in .fl x RN. In fact we have
Proposition 3. Let u E W1,n-1(0,RN) Then (17) holds for all k. Proof. Consider a sequence of smooth functions converging W1°'-`-1 to u. Then all the adjoints of Uk converge in L' to the adjoints of u. Since (17) holds for smooth functions, we also get it for u.
In particular we see that (17) holds for the function in Al u(x) := xl Ixl for which, however, we have 8Gu L B (0, 1) x R''
0.
Remark 3. Let u E W1.P(Q, RN), where p is an integer with 1 < p < n. The argument in the proof of Proposition 3 yields also at once that (8Gu)(k) L .fl X
= 0
for allk
for all k < n - 2, and the only non zero component of the boundary 8Gu in B(0,1) x RN can be
(aGu)(n-1) L (2 X RN
.
In (11) it could sound very natural to test only with functions 0 which
1
depend just on x and on the dependent variables which appear in the submatrix
(Du) , i.e. with 0 of thetype W(x,y) = 70(-T' V1,...,yQk+l)
Of course, as in Corollary 2, we deduce that, if (11) holds, then (18)
f
Dj [W(x, uRl (x),
.
,
u)3k+1 (X))] (adj
dx = 0
.
7Ea
However (18) is again strictly weaker than (11) as it is shown by the following example.
Consider the map cp : S1 C R2 --+ R3, defined by
if 0 < 0 < 2
(cos 40, sin 40, 0) p(B)
if 2
(1,0,B-2) (cos4(0 - 7r), sin 4(8 - 7r),
(1,0,27r-8)
2)
if 7r < 0 < 27r
if 2 <0<27r,
246
3. Cartesian Maps
and its homogeneous extensions of degree zero u : B(0,1) C R2 cp(x/Ixl). As we have seen in F2 in Sec. 3.2.2
aGu L B(0,1) x ]R3 = -6o x cp# JS1
R3, u(x) :=
.
Hence, in our case,
p# Sl J = Q {y E R3
-
{yEE RI (
I
(y1)2 + (y2)2 = 1, y3 = 0} 1
(y1)2+(y2)2=1, y3= 21
aGu L B(0,1) x R3 i4 o
0
.
For any 0 E 1(2,3) set
vo(x) :_ (u 31(x), U12 (x)) The maps v,3(x) are homogeneous of degree zero, and again
L B (0,1) x R2 = -8o x vp# Q S1 But vp# Q Sl is the projection of cp# Q S1 hence vQ# Q S' Jj = 0 and therefore
over the coordinate plane (y$i , y132 ),
aGvp L B(0,1) x l2 = 0 . This implies that
f
D.i
u11(x) ua2 (x))] (adj Dv,3(x)); dx = 0 ,
j=1,2
equivalently
f
1: Dj [cp(x, u11 (x), U12 (x))] (adj (Du(x)) a ) i dx
=0
,
92 j=1'2
for any /3, with 1/31 = 2, i.e. i E /3, and cp E C'°(fl x R2), while, since aGu L B(0,1) x R3 # 0, for any /3 with 1/31 = 2, there is some i E /3 and 0 E Cf (Q x R3) such that
f
Dj [W (x, u1(x), u2(x), u3(x))] (adj j=1,2
(Du(x))g)'dx # 0.
2.4 More on the Jacobian Determinant
247
We shall now discuss an equivalent formulation of the zero boundary condition 8G,,, L 12 x RN = 0 in terms of the commutator of the exterior derivative and pullback. Recall that for any w E Dh(i) and ri E Dk(12), h + k < n - 1 we have d(wAr7) = dwhr7 + (-1)hwnd77
.
This allows us to define the differential dry for any k-form ry with coefficients in L' (Q) by
r
fwAd-y=
(-1)n-k
j
dwAy
dw E
Dn-k-1(12)
n
Dn-k-1(Q) and 77 E Dk(RN), 0 <
Proposition 4. Let u E A' (0, RN), w E k < n. Then
u#drl)) = (-1)n-k9Gu(wnr7)
Proof. In fact [Q]1(wndu#rl)
=
(-1)n-kq.f2l(dwAu#77)
= (-1)n-kGu(dwAr7) = (-1)n-k3Gu(wAr7) +Gu(wndrl) = (-1)n-kaGu(wni]) +Q.f2l(wnu#drl)
From the density of linear combinations of forms of type wnr7 in Dn-1(12 x we readily infer
Corollary 3. Let U E A'(12, RN). Then 8GuL,f2XTR
=0
if and only if
u#d = d u#
.
2.4 More on the Jacobian Determinant In this subsection we discuss in term of boundaries some subclasses of the Sobolev space W1,n-1(Q IR) 12 C Rn, and we collect a number of recent results con-
cerning the Jacobian determinant. For the reader convenience we first state some of the results of the previous
subsection in the special case of maps in W" 1(fl fin) Proposition 1. Let u E W1,n-1(12,PJl) Then we have (i) u satisfies the Piola identities
f
fi(x) (adj Du (x))'. dx = 0
S2 j=1
for any i = 1, ... , n and any
E C,'(12).
3. Cartesian Maps
248
(ii) u c A' (.f1, Rn) if and only if det Du (x) E L' (.f2). (iii) aGu L .f2 x Rn = 0, i. e. u E cart' (fl, R') if and only if det Du E L' (Q) and (1)
f E DjO(x)g'(u(x))(adj Du(x))'dx S? i,j=l
+ J O(x)divg(u(x)) det Du(x) dx = 0 0 for any 0 E Cc', (0) and any g E C' (][8n, Proof. (i) is precisely Proposition 3 in Sec. 3.2.3, and (ii) is obvious. By Remark 3 in Sec. 3.2.3, a(k)Gu L ,f2 x Rn = 0 for k = 1, ... , n - 2. Thus (iii) follows since (1) is equivalent to a(n_1)Gu L,f2 X R' = 0, i.e. f n
J
E[DjV)(x, u(x))] (adj Du(x))' dx = 0 j=1
since linear combinations of the functions O(x)g(y), 0 E Cc' (.f2), g E C' (Rn) are dense in CI (f2 x 118') (in the C' topology).
Let u E A' (.f2 x Rn) n Wl,n-' (0 fin) Considering the n-vector fields or E
L'(Q,1R'), i = 1,2,...,n o (x) := (adj Du(x)) and denoting by Div the divergence operator in the sense of distributions
< Diva, cp >:= -
J
Q Dcp dx ,
cp E C (Q) .
S2
to distinguish it from the div operator in the sense of approximate differentiability, (1) is equivalent to
Div (g'(u)az) = div (gZ(u)a
(2)
since, taking into account the Piola identities and Laplace formulas we have div (g°'(u(x))ci(x))
_ 1: gu,,(u(x))Djuh'(x)a, (x) +9Z(u(x))divo (x) j,h
=
(u(x))82hdet Du(x) = divg(u(x))det Du(x)
A sufficient condition ensuring (2) is given in the next proposition. Let v be a function in W',r and let a be a vectorfield in L" (Q, Rn) where P +'1 - -1 < 1. Since va E L' (,f2, R'), we can consider the distribution Div (va) Then we have
2.4 More on the Jacobian Determinant
249
Proposition 2. Let u E Wl°P(Q) and a E LQ(Q,Wi) where 1 + 1 - I < 1. Suppose that Div or and Div (vv) are locally absolutely continuous with respect to the Lebesgue measure,
Div a = diva (x) dx , div a(x) E L 00(Q) Div (va) = d(x) dx , d(x) E L%O(,(2) .
Then for almost every x E .(2 we have
d(x) = Dv(x) a(x) + v(x) div a(x)
(3)
Actually if Div (va) is merely a Radon measure and
Div (va) = d(x) dx + (Div (vv))(3) then (3) still holds. Proof.
Suppose first that diva = 0. From Lebesgue's differentiation theorem
and Calderbn-Zygmund theorem on the LP* -differentiability of v, p* := n P, of Sec. 3.1.2 we have, for a.e. xo E .(2, limo
(4)
r-
I
,
B(xo,r)
ja(x) - a(xo)lq dx = 0 ,
lim
(5)
ld(x) - d(xo) I dx = 0
r--.0 B(xo,r)
and (6)
lim 1
r-.o r
lv(x) - v(xo) - Dv(xo)(x - xo)ll4 dx = 0
} JJJ
B(xo,r)
Fix such a point xo, choose l E C°O(B1) ,
> 0, f zb(x) dx = 1, and let B,
4'r(x) = r-n4'(x
xo )
Then we have, by the differentiation theorem for measures, compare Ch. 1,
limo = lim
d(x)zbr(x) dx = d(xo) B(xo,r)
On the other hand letting
P(x) := v(xo) + Dv(xo) (x - xo) we have
250
3. Cartesian Maps
_ -
J
Di r(x) v(x) of (x) dx
S2
f
DjV)r(v-P)ojdx-
J B(xo,r)
B(xo,r)
f DjOr (v - P) aj dx +
(7)
DjOr (v - P) aj dx +
Or(x) Djv(xo) of (x) dx
J
')r(x) Djv(xo) (oj(x) - of (xo)) dx
B(xo,r)
B(xo,r)
+
J
B(xo/,,r)
B(xo,r)
J
Dj rPojdx
f ')r(x) Djv(xo) of (xo) dx . B(xo,r)
The last term is of course Djv(xo)oj(xo) = Dv(xo) a(xo). The first term in the last expression is bounded by
r-n-1 sup IDOI f Iv - PI Jul dx B(xo,r) 1/p*
/
f
rT\r( f Iv(x) - P(x)Ip' dx) /
B(xo,r)
1/q
Iolg dx
B(xo,r)
and thus converges to zero as r --+ 0 by (5) and (6). Finally the second term in the last identity in (7) is bounded by
clDv(xo)I f Ia(x)
- o(xo)I dx
B(xo,r)
and therefore it converges to zero as r -* 0. From this it follows that n d(xo) = E Djv(xo)oj(xo)
.
j=1
For the general case, div o # 0, we choose xo so that we have (8)
divo(x) - divo(xo)I dx = 0
lim
r--.0
B(xo,r)
in addition to (4) (5) (6). Now in (7) the additional term
f B(xo,r)
0, (x) P (x) div a (x) dx
2.4 More on the Jacobian Determinant
251
appears. We write it as
f '0r(x)(P(x) - P(xo))divo,(x) dx+ f 'br(x)P(xo)(diva(x) - divo(xo)) dx+ B(xo,r) B(xo,r)
f
V)r(x)P(xo)div o (xo) dx .
B(xo,r)
Since the last term is clearly v(xo)div o (xo), while the first two terms can be estimated by
clDv(xo)I r f Idiva(x)I dx B(xo,r) and
cv(xo)I
}
I divo(x)-diva(xo)Idx
B(xxo,r)
which go to zero taking into account (8), the proof is concluded.
D
1 The distributional determinant. Let u be a smooth map from .(l C Rn into RI. From Laplace's formulas we deduce n 19
i =1
j (u1(adj Du)') 1
axj (adj Du)' j=1
axj (adj Du) = det Du
u1
,
j=1
i.e., that det Du can be expressed as a divergence. Suppose now that u E W11n+1(Q,Rn). From Sobolev's embedding theorem we have u E Ln2 (.fl, ][fin), in fact n2 is the Sobolev exponent of n+1 ; since the
dual exponent of n-? is just n2 and (adj Du) E LT (.2), we then get 11
u1 (adj Du)l E L1(,R) Therefore, for any u c W1, +1 (dl, ][fin), we can define the distributional determi-
nant of Du as the distribution Det Du
Dj (u1(adj Du)')
i.e., as the distribution
E C((2)
f
-EDjcp(x)u1(x)(adjDu(x))' dx 12 j=1
3. Cartesian Maps
252
In general Det Du is not a function and
Det Du # det Du , x one easily verifies that det Du = 0 while Det Du = for instance, if u(x) B1Ibo. Similarly for the map u(x) := (Ixl2 + a2)1/I2 Ixl we have
and
det Du(x) = 1
Det Du = 1 dx + IB(0, a) Iao
.
But we have
Proposition 3. Let U E cart' (0, Rn) fl WI, n+l (,fl, Rn). Then Det Du is a function, and actually
Det Du = det Du .
(9)
Proof Since aG,, L .f2 x Tn = 0 we have, compare Remark 3 in Sec. 3.2.3 and Corollary 1 in Sec. 3.2.3, (10)
J
[
0, (x, u(x)) (adj Du(x))
+Y 0,,e(x,u(x))Djue(x)(adjDu(x))] dx = 0 j,Q
for all 0 E C'(Q x Rn) , sptcp CC .fl x Rn and IDcpI E L°°. Since u E W'1n+l and Djue(adj Du)i E L', we now deduce as in Corollary 2 in Sec. 3.2.3 that (10) holds also for 4(x, y) = cp(x)y' , cp(x) E C' (12), therefore
f
(cp u') (adj Du) dx = 0
E
(.fl)
Q
On the other hand, since det Du E Ll we have
-< Det Du, co > + f cp det Du dx = JDj(cou1)(adjDu)dx n
n
thus (9) holds. We also have
Proposition 4 (Miiller). Let u belong to W'°Ty+l ((1,Rn). If the distributional determinant Det Du is a Radon measure then (11)
Det Du = det Du dx + p
,
µ l dx .
In particular, if Det Du is absolutely continuous with respect to Lebesgue measure, then (12)
Det Du = det Du dx .
2.4 More on the Jacobian Determinant
253
Proof. Apply Proposition 2 with 2
v := ul E W 1' n+l and for j = 1, ... , n,
2
:= (adj Du) E Lnn=1
o,j
.
Since Div v = 0 , we get by Proposition 2 that n
Djv(x)o-j (x) dx + µ = det Du(x) dx + p
Det Du = Div (vo,) =
.
j=1
Finally we observe that a natural question is whether the converse of Proposition 3 holds, i.e., whether, for functions u E Al fl Wl' n+l , (9) is equivalent to 8Gu, L .(2 x Rn = 0. But we do not know the answer. 2 Isoperimetric inequality for the determinant. Next we prove an isoperimetric inequality for the determinant of an almost everywhere approximately differentiable map u : 12 C Rn -+ 1n in which the absence of boundaries play an important role.
Proposition 5. Let u be an almost everywhere approximately differentiable map for which adj Du, det Du E L'(fl, Rn) and for which (8G,,)(n_1) L .f2 x W = 0, i. e.,
n (13)
DjO(x)g''(u(x))(adj Du(x))'dx
J
f(x)divg(u(x))detDu(x)dx = 0
+ n
holds for any 0 E Cc' (.2) and any g c C' (Rn, IRn) . Then for all xo E .2 and a. e. r, 0 < r < ro := dist (xo, 8.f1) we have (14)
f divg(u(x))det Du(x) dx B(xo,r) r
n
J E gi(u) (adj Du)' vj aB(xo,r) i,j=1
for any g E CiC(Rn,lRn). Here v= (vl,...,vn,) := yyoul. Moreover for any O E Cc' (R') and a.e r we have
d7-Cn-l
254
3. Cartesian Maps
(15)
(u(x))det Du (x) dx
f
B(xo,r)
II0I1.(
JadjDuJd7-l--1)
J 8B(xo,r)
We first prove (14). Fix g E Cc',(R'). Using Fubini theorem or the (adj coarea formula we find that for a.e. r E (0, ro), Ei E Ll(8B(xo,r)) and the map Proof.
r ---; h(r) :=
f E 9ti (u) (adj Du)'vj d7-l"''-1(x) 8B(xo,r)
a,3
is in L'(0,ro). Moreover using (13)
rte 2e
Jh(s)ds = T-E
fr,e(1)diV9(U(X))detThL(T)dX .R
where
r,E=
Ix-xpl
1 z-2E(IxI-r)
if Ix-x01 >r+E
0
Thus for e - 0 we get (14) for a.e. r and fixed g. Now (14) follows taking into account that C,, (lR) has a denumerable dense subset. We notice that in the proof one uses just the general slicing argument. We now use (14) to prove (15). By the area formula, compare Sec. 3.1.5, setting dr(y) := d(u, B(xo, r), y), we have
f
divg(u(x))det Du(x) dx = fdivg(y)dr(y)dy
B(xo,r)
R
which, together with (14) says that
fdlV9(Y)dr(Y)dY <
I IgI I00
f
Iadj Dul d7-1n-1
.
8B(xo,r)
1R°
In other words for a.e r, dr is a function of bounded variation in Rn, compare Ch. 4, and IDdrI(WW) <
f
IadjDuId7-ln-1
8B(xo,r)
By Sobolev embedding theorem, taking also into account that dr(y) E Z, we then get
2.4 More on the Jacobian Determinant
255
(I Idr(y)Idy)
IV
1 dy)
( f I dr (y) I
n" 1
< c(n)
f I adj Du I
J 8B(xo,r)
dHn-1
and therefore
f
(16)
<-
f
(u(x))det Du(x) dx =
B(xo r) 10IIoo
f
IR"
Idr(y)I dy < c(n)II0II. (
O(y) dr(y) dy
f
ladj Dul
dHn-1)
8B(xo,r)
Il2n
11
Remark 1. Notice that (14) and consequently (15) hold for any r E (0, ro) if u is a smooth map. Going trough the proof of Proposition 5 it is easily seen that in fact we have (17)
J
0(u(x))det Du(x) dxl
B(xo,r)
f
< c(n) II*I I.
{E[
8B(xo,r)
2
n.1
1/2
2
(adj Du)'vj] }
dHn-1
3
c(n) being the isoperimetric constant in Rn. If u is smooth and 1-to-1, (17) amounts just to the isoperimetric inequality n/(n-1)
c(n)
r)))] since the integral on the right hand-side of (17) is just the ?in-1-measure of I u(B(xo, r)) I
<-
u(8B(xo, r)), compare Sec. 2.1.2. 3 The class Ap,q(f2,Rn). Consider the class of functions u E W1,p(fl,Rn) where p > n - 1, such that adj Du E Lq (.f2, IIBn) for some q > n/ (n - 1)
Ap,q(n, Rn) :_ {u E W1,P(Q, IRn)
I
adj Du E Lq(.f2, I[2n2)}
We claim that
Ap,q (Q, Rn) C cart' (., IRn) if q > p/(p - 1). In fact from (i) of Proposition 1, we have (18)
f E Dj cp (adj Du)' dx = 0 S? j=1
3. Cartesian Maps
256
for any cp E C' (.fl). Since (adj Du)' E L4 (Q), (18) holds also for all cp in W1"4'(Q), where ,- + 1 = 1. On the other hand for any 0 E CC(Q x ]E$n) the map p(x) := O(x,u(x)) belongs to W'1P(S2), thus, since P + 1 < 1 one deduces that u E A' and that
fDj[(xL(x))] (adj Du(x))' dx = 0
.
By (iii) of Proposition 1 we then conclude that u E cart' (Q, Rn). Actually we have
Theorem 1 (Muller-Tang-Yan).
C cart'(,fl,
In order to prove Theorem 1, we need the following lemma
Lemma 1. Let u E W1,n-1(.fl Rn), xo E ,f1 and ro := dist (xo, 8Q). Then for a.e. r E (0,ro) we have gi (u) (adj Du)'vj
(19)
d7-Ln-1
aB(xo,r) 2+x=1
< c(n) ldivgll.(
J aB(xo,r)
adjDuld-W-')=1
where v = (vl, ... , vn) :--Ix-xol' `0 Proof. Suppose first that u is a smooth function. Then we may apply Proposition 5 to u and (19) follows from (14) and (15). For the general case, let {uk} be a sequence of smooth maps which converges in W'"'-' to u. Since 'uk -> it and adj DUk -+ adj Du strongly in L1, we can select a subsequence uhk such that H uhk, - U IIL1 < 4-k,
11 adj Duhk - adj Dull L1 < 4-k.
By Fubini theorem we then infer that for a.e. r 00
J
I E 2kJuhk - ul +
E 2kI adj Duk -adj Du l dx < cc,
hence
uhk -f it,
adj Duhk -+ adjDu strongly in L' (8Br).
Writing (19) for uhk and passing to the limit the result then follows at once. Proof of Theorem 1. If F denotes any n x n-matrix, by (3) in Sec. 3.2.3 we have
F . (adj F)T = det F id
2.4 More on the Jacobian Determinant
hence
257
Idet Fln-1 < Idet (adj F) I < c(n) ladj FIn
(20)
This proves that if u E Ari_1,,,,/ (n_1) then det Du E Land u E A' (S'2, RI`'). We show now that 0Gu L .(l x RN = 0. In view of Proposition 1 (iii) it suffices to prove (1). For any i = 1, ... , n, denote by a' the vector field whose components are given by
ail (x) :_ (adj Du)'
.
We have o' E Lq (f), q = n' 1 and Div o,i = 0. Let g E C' (IRn, IRn). Then v(x)
9(u(x))
belongs to W1'', p = n - 1, and 1
1
1
p+q
Ti
_
2-n n(n-1)+1<1
If Div (gi(u) 01i) E L'
(21)
for any i = 1, ... , n, we may apply Proposition 2 and conclude that Div (gi o u vi) = Ei,j,h 9,,
h
(u(x))Djuh (x) (adj Du)? dx
_ Ei,j,h 9y,, (u(x))bihdet Du(x) dx = divg(u(x))det Du(x) dx which is precisely (1). Therefore it remains to prove (21). Consider a radial mollifier p(x) = OxI) and let pE :_ E-np(E ). Denote by T the distribution
T := Div (gi(u) ui) Fix D0 CC .f2. Clearly the distributions T * pE belongs to L1(Qo) and converges as distributions to T. Thus (21) holds if we show that in fact T * pE converges weakly in L'(,f2o) to T. Let e < 2dist (S ?O, 912) and z E .(l0. Then we have T * p6(z)
= - f (9Z(u(x)))(adj Du(x))'Djp6(x) dx Rn
- f (g2(u(x)))(adj
x
E
zl )vj dx
Rn e
E-(n+l) f l 0
f gi(u(x))(adj Du(x))'vj d1-1n-1}z)'(r) dr 8B(z,r) '"
258
3. Cartesian Maps
Thus by Lemma 1 we get E
IT * PE(z)I <_ c(n)
< c(n) E-(n+l) J r0
J B(z,E)
adi Dul
d7-tn-1 } -1 dr
f
OB(z,r)
0 Ef
< c(n) E-n
f
e-(n+l) f {
{f
adj DuI n=1 d7-1n-1 }r dr
6B(z,r)
adj DuJ7
n-1
dx .
and, assuming without loss of generality that PIB(0,1/2) ? IB(0,1)1-1
we deduce that IT * pE(z)I < c adj DuI n=1 * P21(Z)
in f2o
The claim now follows since the functions I adj Dul n 11 * p2E (z) are equi-absolutely continuous. 4 Higher integrability of the determinant. As a further consequence of the isoperimetric inequality (15) we now prove
Theorem 2 (Muller). Let u E W1,1(ffl, Rn) be such that Ln/(n-1) (,fl Rn2 ) (i) adjDu E (ii) 8(,,_1)G,,, L 1? x RN = 0, i.e., (13) holds (iii) det Du > 0.
Then for any ,!2 CC ,f2 we have det Du log(2 + det Du) E L' (h) and IJ det Du log(2 + det Du) JIL1(n) < c(Q, ,R, jjadj Du
IjLn/(n-1)(,n,yR_2 )) .
Proof. The proof of Theorem 2 relies on the characterization of the class L log L, Proposition 5 in Sec. 3.1.1. It suffices to show that M(f) E L1(.Q), ,f2 CC 1 CC d2 where
f (x) =
det Du(x) x E { 0 xV l
In fact for d:= dist (.fl, 80) it suffices to estimate
f B(x,R)
if(y)I dy
2.4 More on the Jacobian Determinant
259
for all x E (2 and allR< d/2, as forR>d/2 we have
f
Duln/(n-1) dy
I f (y) I dy 5 c(d) J det Du dy < c(d) J Iadj
B(x,R)
11
1?
because of (20).
From (15) we infer that for a.e. r, R < r < 2R (R < d/2) we have n-
J B(x,r)
< c(n)
det Du dy
J 8B(x,r)
Iadj Dul
d7-ln-1
Integrating with respect to r from R to 2R and using (iii)
/
RI
\n
r
J
f(y)dyI
2R
1
<
n
/
B(x,R)
< c(n)
7 dr
/
n
Iadj DuI dy
J B(x,2R)
hence
det Du dyJ
\B(x,R)
R
/
J
(
\ n-1
/
f (y) dy < c(n) I
f
n
Iadj DuI dy)
n-T .
B(x,2R)
B(x,R)
Setting 9(x)
Iadj Du(x) I
=
0
x E .fl x Q,
we can then state
M(f)(x) <_ c(n)
[M(g)(x)]nl(n-1)
+ c(d) II g II n/(nn1),
Integrating on .(l, and taking into account Proposition 2 in Sec. 3.1.1, the result follows at once from Proposition 5 in Sec. 3.1.1. One can show, compare Muller [501], that the result of Theorem 2 is optimal in the sense that the sign condition on det Du cannot be dropped, nor can .fl be replaced by .2. Moreover det Du log(2+det Du) cannot be replaced by -y(det Du) with ry(t)/(tlog(2 + t)) -> oo as t -f oc. An immediate consequence of Theorem 2 is the following
Corollary 1. Let u E W l ,n (,fl, R'). Suppose that det Du > 0. Then
det Du log(2 + det Du) E Ll ,(.fl)
3. Cartesian Maps
260
0 BMO and Hardy space Hl (R n). There is a more general approach to results of the type of Theorem 2 in terms of the Hardy one space. We recall that a locally summable function f : 1R -+ R is said to have bounded mean oscillation if
f If (y) - fz,r I dy <
sup
IIfII.
x,r
00
B(x,r)
where
f f (y) dy =
fx,r
B(z,r)
I B(x, r) I
f f (y) dy B(x,r)
The class of functions of bounded mean oscillation is denoted by BMO and often is referred as John-Nirenberg space. Note that II f II. = 0 if and only if f is constant a.e. One then can show that BMO modulo constants is a Banach space with respect to the norm II f II..
Let g E L'(Rn) and let 0 be any smooth function with support in the unit ball and satisfying
f(x)dx = 1 . Set
fg(,)O(x-
r y)dy rn where 0, (x) = r-no(x/r). Notice that at every Lebesgue point x for g we have 9* (x) := sup r>O
lg(x)l < I9*(x)l hence
119* IIL'(R")
The Hardy space 7{i (Rn) is defined as
7ll(]
)
:= {g E L1(1
)
I
g* E Ll(Rn)}
It turns out that 7{'(R') is a Banach space with norm 119 Ilnl
II 9* IIL1(1n),
which does not depend on the chosen function 0. In fact one can show that f E H' (1R) if and only if the function sup sup{I0r * 9I r>O
I
0 E Cc(B(O, 1)),
11 Do 11,,.
belongs to L1.1 Trivially 1 Notice that in the sequel working with a specific 0 or with all 0 with 11 Do Il is just equivalent.
<1
2.4 More on the Jacobian Determinant
261
xl( R
) c L'(Rn), and the behaviour at infinity together with the summability of f * yield that (1Rn) necessarily has zero average. In particular one easily sees that every f E no function of L' (R') with compact support and non vanishing integral belongs
to 7{1(Rn). On the contrary one can show that every function f c LP(IRn), p E (1, +oo], spt f compact, and f f = 0 belongs to Finally we observe that every bounded sequence {fj } in 7{1(1Rn) has a subsequence which converges in the sense of distributions to a function f E 7{1(8n) 7{1(Rn).
with
in fact passing to a subsequence { fj } converges in the sense of distributions to some f and it is not difficult to see that
f * (x) < lim inf f (x)
d x E 1R"''.
j -OO
A fundamental theorem in the theory of Hardy space 7.11 developed by C. Fefferman and E. Stein asserts
Theorem 3 (Fefferman). The dual space of f'(IRn) is BMO. More precisely, L is a continuous linear functional on HI (Rn) if and only if it can be represented as
L(9) = f .f9 Rn
for some function f in BMO, moreover for any f E BMO and any g E f1(1Rn) we have
ff dx < c(n) 11 f 11* 119Ijx'
(22)
Rn
Theorem 3 plays an important role in many fields, among them harmonic analysis, complex analysis, interpolation theory and partial differential equations. By means of it we shall now deduce a few interesting results in our context.
In order to do that let us first state a simple lemma.
Lemma 2. Let u E W1,P(Rn) and v E LP '(RI), p > 1, p' = p/(p - 1). Then we have
f sup rn+1
( f lu(y) - uz,rl jv(y)I dy) J dx < c(n) J1 Du LP B(z,r)
Proof. Chose s and q so that
max(1, s*) < q < p < s < p*
p*:=p-n, 1
1
1
S*S+n 1
1
1
V L9'(Rn)
3. Cartesian Maps
262
Using Holder and Sobolev-Poincare inequalities we get
f sup 7 +1 ( f Iu(y)
- ux,rl Iv(y)I dy) dx
B(x,r)
IlPri
<
r sup
J r
1 r \( f
1/s
I U(y) - ux,rls dy/1
\ B(x,r)
B(x,r)
IEYn
f
f IDu(y)Igdy)l
1/q
1/s'
f Iv(y)Is dy J/ 1/s'
f
\ B(x,r) Iv(y)IB(x,r) s f [M(IDulq)(x)] ° [M(IvIS')(x)] dx r
IIt^
dx
dy I
dx
yen
)r ( fn M(IvI
1/p
(J
)
dx \ 1/p,
M(I DuI g) 9 dx
Du II L1
I
VIILP'
because of Proposition 2 in Sec. 3.1.1. We can now prove
Proposition 6. Let u E W 1,r (Rn) and let v E Lp (Rn, lRn) be a vector field with zero divergence in the sense of distributions, 1 < p < oo. Then Du v belongs to 1-L'(Rn) and (23) II Du vllrcl(pn) < c11 Du 16 IIVIILP'
Proof. Clearly Du . v E L1(Rn) and
JDu.vdx Rn
Jdiv(uv)dx = 0
= Rn
It remains to estimate the L1-norm of (Du v)*. Using the fact that v has zero divergence we have
f
f
Du(y) . v(y)'Yr(x - y) dy
B(x,r)
Rn
(24)
div (uv) (y) 0, (x - y) dy
=
f
(u(y) - ux,r)V (y) DOr(x - y) dy -
B(x,r)
<
r +1
f I u(y) - ux,r l Iv(y) I dy , B(x,r)
whence (23) follows at once from Lemma 2.
2.4 More on the Jacobian Determinant
263
Late we shall need the following localized version of Proposition 6
Proposition 7. Let u E W1,P(Rn) with sptu CC 1R'' say sptu C B(xo,R), and let v E LP (][fin, Rn) be a vector field with zero divergence in the sense of distributions, 1 < p < oo. then Du - v belongs to ?-1' (R) and II Du v II9-(1(IIY") < C II Du IILP(B(xo,R))II V IILP'(B(xo,2R))
Proof. Choosing a standard cut-off function 71, rl E Cc°(B(xo, 2R)), 71 = 1 on B(xo, R), I D771 < 2/R, we get similarly to (24)
f
(Du - v) * 0r(x) =
u(y) rl2(y) v(y) DOr(x - y) dy
B(x,r)
f
(u - ux,r)172 v . Dor dy +
B(x,r)
f
f (u-ux,r)v772DOrdy-2 f B(x,r)
ux,rD77-v(y)0rdy
B(x,r)
Therefore (25)
ux,r v?72 DO, dy
B(x,r)
f
I(Du. v) * Or(x)I <
sup r +1
u(y) - ux,rl IV (Y) rl2(y) dy
B(x,r)
R"
r
f
1
+R
I ux,r I f I vr71 dy B(x,r)
Using Lemma 2 we infer
Jsup r +1
(26)
f MY) - ux,rl I v(y) y2 (y) I dy dx B(x,r)
R"
< C 11 Du IILP(R")II V712 IILP'(R")
while as in the proof of Lemma 2, we also have
R f sup r l ux,r I IIt^
f
j v771 dy dx
B(x,r)
R II U II LP(Rn) II v77 II
c II Du IILP(Rn) II v77 IILP'(R-)
The claim then follows easily from (25) (26) and the last inequality.
A similar conclusion to that of Proposition 6 can be inferred for the commutator or the Jacobian minor of two functions u E W1,P(R') and v E W1,P' Set
264
3. Cartesian Maps
[u,V]ij = uxi vxj - uxi vxi which of course belongs to L'(R'). Due to its divergence structure, namely
[u,v]jj =
vxj)xi - (uvxi)xj
in the sense of distributions, we have
Proposition 8. Let u E W1,P(IRn), V E W1,P' (R7), 1 < p < oo. Then [u, v]ij belongs 7-l1(Rn) and (27)
II [u, v]ij IIV(Rn) : c II Du IILP(R') II Dv IILP'(an)
Proof. Clearly [u, v]ij E L' (1R )r and
[u, v] ij dx = 0
J
.
Rn
As in the proof of Proposition 6 we now have
r flu, v] 1
f
-
ij dy
(u(y) - ux,r) vyj (y) Or,y^ (x
- y)
B(x,r)
+ f (u(y) - Ux,r) Vy (Y) Or,yi (x - y) B(x,r)
Thus 1
rn
f
[u,
<-
v]ij dy
r +l
f
I u(y) - ux,r I I Dv I dy
B(x,r)
and the claim follows again from Lemma 2.
Let u E
W",n(Rn,ln). As
det Du = Djul v,1 where v1 is the vector field
V1 := (adj Du)3 which is divergence free and V1 E
Ln/(n-1)(Rn)
we can apply Proposition 6 and infer
Corollary 2. Let u E W l,n (Rn, Rn). Then det Du belong to the Hardy space
xl (n)
2.5 Boundaries and 'Daces
265
Actually in a similar way one can get
Corollary 3. Let u E Ap,q (Rn, Pn) where 1/p + 1/q < 1. Then det Du E xl(Rr ). We can localize the previous result recovering this way Corollary 1. If u E W1" (.f2,Rn) and 77 E C°°(Q), 77 < 0 write
77 det Du = Dj (u177) vl - ul D;7? vJ
and observe that
D ul
vl E ll(IIBn)
u1D
vl E LP
1<<
n
Assume now that det Du > 0, then
M(r7detDu)(x)
It follows that M(77det Du) E L' (Rn) and consequently 77det Du log(2+77det Du) is summable, recovering this way Corollary 1.
We conclude this subsection by stating one more consequence of Proposition 6. Since every vector field E in LP (R'), 1 < p < n, with curl E = 0 can be written as
E = Du
for some scalar function u E W1,P(Rn), we readily get Corollary 4. Let E E LP(IEBn, IIBn), B E LPG (1[Bn,1Rn), 1 < p < oc, be two vector fields in IIBU satisfying in the sense of distributions
curl E = 0
divB = 0.
Then E - B E Ri(1[Bn)
2.5 Boundaries and Traces Let Q be a bounded domain in W with smooth boundary and let u be a map in A'(0, IRN) n W1"1(f2, RN). The trace of u on an, ul an is well defined in the sense of Sobolev spaces. Also the current Gu is well defined in 1R X ISBN and, when aGu L ,f2 X RN = 0, the boundary aGu of Gu has support in an x ]RN. A first question which naturally arises is whether the trace of u on an determines aGu , that is, whether we have
aGu = aGu for u, v E carts (,fl, IIBN) satisfying ulan = vlan
The answer to this question is in general negative as shown in M below. In fact the trace in the W1,1 sense just fixes the 0-component of the boundary.
3. Cartesian Maps
266
Proposition 1. Let u, v E A' (f2, RN) n W1'1(.f2RN) Then u = v on aQ if and only if (aG.)(o) =
First we note that (aGu)(o) L f2 x RN = (8GV)(o) L (2 x ]RN = 0 by Proposition 1 in Sec. 3.2.3. Now we observe that any (n - 1)-form with no component in the vertical direction is a sum of forms of type Proof.
W=
dx3
x RN) and, by Proposition 2 in Sec. 3.2.3, compare also the when 0 E proof of Proposition 1 in Sec. 3.2.3, (3Gu)(o)(w)
=
JDj{(x,u(x))]dx
(aGv)(o)(w)
=
JDj[(xv(x))]dx. n
As O(x, u(x)) E W1,1 (Q) and trace O(x, u(x)) = O(x, trace u(x)), we then infer
(8Gu)(o)(w) - (aG,,) (o) (w) =
f{(x,u(x)) - O(x, v(x))l v
dfn-1
= 0.
an
Conversely, if (aGu) (o) =
(o) , then for any 0 E C' (R' X RN) we have
f Dj [o(x, u(x)) - O(x, v(x))l dx = 0 and by approximation, as in the proof of Proposition 1 in Sec. 3.2.3, we deduce, for O(x,y) = co(x)yz, W(x) E C,' (R')
0 = fDj[(z)(ui_vz)]dx = J(x)vj(ui_vt)dflTh_1 .a
an
that is u = v on a.Q. As consequence of Proposition 3 in Sec. 3.2.1 we instead have
Proposition 2. Let (1 be a bounded Lipschitz domain in Rn. Let u, v be maps in W1,n(Q RN), a:= min(n, N). Then u = v on a.2 implies aGv, = 8G,,. Proof. Clearly the claim is local. Fix a point xo E a!2 and choose a neighborhood U of xo for which there exists a bi-Lipschitz map cp : B(0, 1) - + U such that
cp(B+(0,1)) = ,fl nU, and
B+(0,1) = {x E B(0,1) 1 x1 > 0}
2.5 Boundaries and paces
w(F)=aQnu,
267
r=B(0,1)n{xl=0}
Denote by O(x, y) the map O(x, y) := (cp(x), y). As
O#Gvow = G in U x R'
O#Guow = Gu, and
uoW = voce
onF
in the sense of traces in W""n, it is not restrictive to assume that u, v belong to
Wl,n(B+(0,1)) with u = v on F. Consider now the function w defined on B(0,1) as w(x)
(1)
{
v(-x1, ) if x1 < 0
Clearly w belongs to Wl"n(B(0,1),EN). Thus by Proposition 3 in Sec. 3.2.1
aGw L B(0,1) x ]RN = 0. From this equality one easily infers the claim. For that fix a,
aI + 1,31 = n,
k+1, 0
{ 0(x x , x) if xl < 0 .
By Corollary 1 in Sec. 3.2.3 and Remark 2 in Sec. 3.2.3 we have for i E Q
O= f B(0,1)
'Ea
f ... + f
Dj['t/,(x)g(w(x))](adj
B+(0,1)
B-(0,1)
rDj[cp(x)g(u(x))](adj (Du(x))a) dx B+(0,1) .7E&
D j[zb(x)g(v(x))] (adj (Dv(x))«)'dx B+(01) .7E5
_ (-1)1 al o(a, a)o(i,Q - i)(aGu -
Actually the proof of the previous proposition shows also
Proposition 3. Let 0 be a bounded Lipschitz domain in R'. Let u, v be maps
in Wl"p(Q,RN) with 1
3. Cartesian Maps
268
We also have
Proposition 4. Let ,2 be a bounded Lipschitz domain in Rn. Let u, v be maps in An-l,n/(n-1). Then u = v on an in the sense of traces implies WG,, = aG,,. Proof. We proceed as in the proof of Proposition 2. By Binet's formula, see e.g. Lemma 3 in Vol. II Sec. 2.5.2, we have adj D(u o go) = adj Dgo
adj (Du) o g,
which yields at once
w E An-l,n/(n-1) (B(0,1),lf8n) , w being the map in (1). By Theorem 1 in Sec. 3.2.4 we again infer
aGw = 0
on B(0,1) x ll
thus we can conclude as in the proof of Proposition 2.
Let u, v E cart' (.2, RN). We can have u = v on aQ while aGti, # aG,,. Let go : S' C R2 -> R2 be a smooth map and let v(x) := cp( x ) be its homogeneous extension to B(0, 1) C R2. We choose a cut off function 77(jxj),
for 0
3
17/(t)I < 8, i/(t)<0,
and we consider the function u(x), restriction of v(x)77(lxl) to the upper half ball B+(0,1) = B(0,1) n {(x', x2) x2 > 0}. Evidently u(x) E W1'P(B+(0,1),12) for any p < 2; moreover, since det Du = 0 on B+(0, 1/2) and u is smooth outside B+(0,1/2), u belongs to A'(B+(0, 1/2),1[82) fl W1,1(B+(0,1/2),R2). The trace of u(x) on aB+(0,1), in the sense of W",P, is given by I
u(x) :=
,P(1, 0) 770x1)
if x2=0, x'>0
;o(-1, 0) 77(lxl)
if x2 = 0
10
,
x' < 0
if (x')2 + (x2)2 = 1
,
x2 > 0
In particular, if we choose go so that ;p(1, 0) = go(-1, 0) = 0, we have ujaB+(o,1) _ 0. For instance, this is the case if we multiply by 77(jxj) the function (2)
fi(x)
((1)2 + (x2)2, (x1)2 +)(x2)2)
,
(xl,x2) E B+(0, 1)
We shall now show that, though uI&B+(o,l) = 0, the boundary of Gu is different from the boundary of the current carried by the graph of the constant map into 0 E1R2.
2.5 Boundaries and Traces
aGu # 8G0 = aB+(0,1) x 80,
269
80 = Dirac mass at 0 E ]1 2 .
Observe that aGu L B+(0,1) x 1R2 = 0. As in 2 in Sec. 3.2.2, we set
U--(X) = re(IX'N x(Il) 77(1XI) where r, (t) is linear in (0, e) and re(0) = 0, r, (e) = 1 - E, and re(1) = 1. The functions uE(x) are Lipschitz-continuous, and have trace equal zero on aB+(0,1). Hence
aGu. = aGo = aB+(0,1) x bo
(3)
for every E > 0. Reparametrizing cu,,B+(o,1) as 0E(z)
('Ye(z),U ('YE(z)))
where pe(z)
'Ye(z)
z
and p,(t) is linear in (0,1/2), (1/2, 1), and p(O) = 0, p(1/2) = E, p(l) = 1, we get 0e# Q B+(0, 1)
Gue
Since 0E is a family of equi-Lipschitz functions converging uniformly to the Lipschitz function O(z)
zi
(0, 2jzjcp(T'j--,)
(-Y(z),W(
z I
Y(z) I
77 (7(1'x1))
2 < IzI
we deduce that (4)
Gue -0# 1B+(0,1) \ B+ (0, 1/2) JI+ 0# QB+ (0, 1/2) 1
As
0#QB+(0,1) \ B+(0,1/2) 1 = G. we deduce from (4), taking into account (3) 8G0 = aGu + a0# Q B+(0, 1/2)
i.e., denoting by I S'+ the current integration on the positive half circle
aGu = aGo - o#af B+(0,1/2) ]I = aGo - cp# Q Si l if we also take into account the expression of O(z) for jzj = 12 For the map in (2) we therefore get ujaB+(0,1) = 0
3. Cartesian Maps
270
and
aGv, = aGo - So x aQ B((0,1/2),1/2) since obviously
# 8Go
,
QS'I
= aQ B((0,1/2),1/2) l Essentially a similar phenomenon occurs for the mapping W#
w(z) _ (Izl)2k in complex coordinates. The associated current has a non zero boundary in {0} x R2 given by So x 2kQ S1 , while the boundary of the restriction it of w(z) to Imz > 0 is given by
49G,, = Go - k So x j S' D
.
Notice finally that 2ii can be obtained from w(z) = if by a rotation R of 4 degree in the z-plane plus a translation in the target plane 2 4G(xl, x2) = w(Rz) + (0, 1) ,
z = x1 + ix2
The previous considerations show that in terms of graphs we have a large choice in imposing to Gu Dirichlet's type conditions in vector valued problems: we may fix the 0-component of aGu (aGu)(0)
to which sometime we shall refer as to the weak anchorage condition, or (.9G.) (o), (3G.)(1), ... , (0G,) (k0)
0 < ko < n -
or we may fix the entire boundary 8G,,, i.e., (aGu)(o), ... , (aGu)(n-1)
to which we shall refer also as to the strong anchorage condition. Notice that, since
aGuk - 8Gu and even
for i=0,1,...,n-1
(&Guk)(ti) - (&Gu)(z)
whenever Al
u, the corresponding subclasses of the u's in A' (Q, RN ), for which Uk
(OG)(o), ..., (aG)(k)
0
are prescribed, are closed with respect to the convergence in A1.
2.5 Boundaries and Traces
0
271
2
Fig. 3.2. The map v.
2 It is worth remarking that the weak anchorage condition, i.e. prescribing is far from the intuitive idea of prescribing the the trace in the W1,1-sense,
boundary. This is better shown by the following modification of the example in
0 Set
if0
1
77(1xI)
x12
and consider
if 1 < IxI < 2
x E B+(0,2) C R2
v(x) := rl(lxl)u(x)
where u is the map in (2). Clearly V E W1"p(B+(0, 2), R2) n A1(B+(0, 2), ]R2), p < 2, and one easily sees that
v(B+(0, 2)) = B((0,1),1) \ B((0,1/2),1/2) and
v(aB+(0, 2)) = aB((0,1),1) in the sense of traces.
Let (2 be a Lipschitz domain in ][fin. For 7-Ln-1-a.e. x E an the exterior unit normal vector v(x) is well-defined. As it is well-known flux integrals can be transformed into integration of differential forms, or in terms of vectors, to every vector v we can associate the (n - 1)-vector *v orienting the orthogonal (n - 1)-plane to v so that vn * v = e1n... nen defined by
n
n *v=E(-1)i-I vie-.
i=1
if v
vZei i=1
272
3. Cartesian Maps
Defining now the i.m. rectifiable (n - 1)-current Q an carried by the boundary of fl as Q a fl
= T(8fl,1, *v)
Gauss-Green theorem reads as
aQ.?I = QanI. In fact, as every (n - 1)-form w can be written as w dw = div cp dx, we have aT ll(w) =
f
div cp dx =
Sz
J
Wi(x) dxi so that
fco.udfln_1 M
<w, *v>
dH"'-1
= Q8Q]f(w)
W2
Let u be a map of class C2 from a neighbourhood of ,fl into RN. The i.m. rectifiable (n - 1)-current (id < u)# Q an
(5)
is well defined and Stokes theorem says that (6)
a(G,, L (2 x RN) = G,,,,an
In fact for any differential (n - 1)-form in Rn x RN we have Gu L (n x IRN) (dw) _ I fQ D ((id Pa u)# dw) _ V ,fI JI (d(id >a u)# w) = aQ Q ((id > u)#w)
= 8Q ((id x u)#w) = Gan(w) In coordinates, if jal + JQj = n, 1/3 > 1, i E 0 and w = q(x, y) dxandy with -0 E CO" (Rn x RN) we have taking into account Proposition 2 in Sec. 3.2.3 and Corollary 2 in Sec. 3.2.3
f D, (O(x, u(x))] (adj
a - i)
dx
7Ean
a(a, a) a(i, Q - i) f
Dj [0(x, u(x)) (adj
dx
n jEa
f
0(x, u(x)) (adj
aSl ,jE&
Recalling the definition of the adjoint minors in Sec. 3.2.3, we therefore conclude for any w E DT (Q x RN) that
2.5 Boundaries and Zfaces
f<w(x,u(x)),A(Du(x))>dw_1
G,,,,an(w) = 3G,,(w) =
(7)
273
an
where A(Du(x)) denotes the (n - 1)-vector in Rn+N given by
A(Du(x)) _
Aa°(Du(x)) eaAEp
(8)
ICI+IaI=n-1
Aap(Du(x))
a(j, a - j) Ma-j (Du (x)) vj (x)
(-1) I0Io.(a, a) j=1
Of course A(Du(x)) depends only on u and on the tangential derivatives of u on asz,
Gu,an =
A(Du) A(Du)
and
M(Gu,an) =
J IA(Du)Idf"i-1 an Moreover we notice that the same holds, by approximation, under the weaker assumption that u be of class C1 in a neighbourhood of Q. Because of that we can now set
Definition 1. A map u : a,Q -+ R' is said to be in A' (aft, RN) if u belongs to Ll (a,(2, RN) and is 7-Cn-1-a.e. approximately differentiable in an with A(Du) E Ll(a,f2), where, for instance, 7-1n-1-a.e. x E an A(Du(x)) can be computed by extending constantly u in the normal direction to an.
Definition 2. Let u E A' (an, IRN). We define Gan by Gu,an (w) :=
J an
<w(x, u(x)), A(Du) >
dfn-1
Of course Gu,an is a well defined i.m. rectifiable (n - 1) -current in Rn X RN.
To be more explicit, we can assume that in a suitable neighbourhood of a point of a,fl the boundary of an is flat, for instance, 0 = B+(0,1), where
B+(0,1) := {x E B(0,1)
I
x1 > 0}
Let u E A' (a.fl) and let i
:
Rn-1 -i Rn=RXRn-1
denote the immersion Y -+ (0, Y), and
u(x) := u(i(x))
xj < 1
.
3. Cartesian Maps
274
Then u E A' ({Y E Rn-1 JxI < 11,R N), and, extending ii constantly in the normal direction to Y = 0, we easily see that I
in Dn-1(B(0,1) X RN)
Gu,an = i# Gi
(9)
,
i.e., that Gti,,an is the current carried by the graph of the restriction of u on afl. More generally, if is a bi-Lipschitz transformation which preserves the orientation,
W#Q,A = QQL by Gauss-Green formula we have (10)
4 =aQflhI _ Tafl
#Qaa ='# 9Ia]
therefore we have 0# Guow,aa = Gu,a,fl
(11)
where
O(x,y) In fact
Guow,aa (w) = Q as ((id a u 0 co)#q1#w) = Qao]] (W#(id xa u)#w) = Qafl ((id a u)#w) = Gu,an(w)
Let u E cart' (fl,RN) with trace ulan in the sense of W1,1 in Al(a,fl,RN). As we have seen in [1], in general aG,,, 5 Gu,an. However we have
Theorem 1. Let 0 be a Lipschitz bounded domain in Rn and let u belong to Wl°n(,fl,RN), respectively u E A,_l,n/(n_1)(0,ll?n). If the trace of u on afl belongs to Wl,n'(afl,IRN), respectively to An_l,n/(n_1)(a,fl,1Rn), then
cGu = Gu,an Proof. We proceed as in the proof of Proposition 2 defining w as w(x)
I SI
u(xl, x) u(0, x)
if x1 > 0 if x1 < 0
instead of as in (1). By Proposition 3 in Sec. 3.2.1 respectively by Theorem 1 in Sec. 3.2.4, we have
aG, L B (O,1) X IRN = 0.
2.5 Boundaries and Traces
275
For E > 0 denote by pE : R -* R the piecewise linear function with pE (t) = 0
for t < -e, p,(t) = 1 for t > 0, and pE (t) = 1 + t/e for -E
-
t -< 0, let
C (RN) By Corollary 1 in Sec. 3.2.3 and Remark 2 in Sec. 3.2.3 we have for any
cp E
a,8, kaI+IQI=n, i E 0
0=
f I:Dj[ (x) pe(xl)g(w)] (adJ
(Dw)«)' dx
B(0,11) jEa
=
J
J
B-(0,1)
B+(0,1)
EDj[ (x) g(u)] (adJ (Du)«)' dx B+(01) jE&
+s
]x' -E
f w(x) g(u(0, Y)) (adJ (Du(0,
d?in-1 + 0(6)
2I<1
(-1)1'l or(a, a) or(i, Q - 1) aG,,,(W(x) g(y) dxandyA-2)
+
f p(0, T) g(u(0, Y)) (adJ (Du(0, x))«)i d7-1n-1 + 0(6)
.
IYI<1
This concludes the proof as, we recall,
(adj (Du))«)' = o(i, $ - i) o(.7, a - j) Ma=d (Du) .
Though
aGu 0 Gu,aB(0,1) for U E cart'(B(O, 1),RN), even if M(aG,,,) < oo and u smooth, we have a(Gu L B(0, r)) = Gu,aB(o,r) for almost every r E (0, 1). More precisely we have
Theorem 2. Let u E cart' (B(0,1), RN). Then for a.e. r, 0 < r < 1, u is HI-1 a.e. approximately differentiable in aB(0, r), u(x) = u aB(o,T)(x) in the sense of
traces, A(Du) E L'(aB(0,r)), and (12)
Proof.
0(G u L B(0, r)) = Gu,aB(0,r)
Observing that by Fubini theorem u, A(Du) E L'(aBr,Rn) and the
restriction of u on aB, is approximately differentiable with approximate differential equal to the tangential derivative of u on OBr for a.e. r, we see that A(Du)
276
3. Cartesian Maps
depends only on the values of u on BBr; hence (7) holds with (1 replaced by Br. Equality (12) then follows as in the proof of Proposition 5 in Sec. 3.2.4 or by the slicing theory as (7) reads as G,L,aB,. = < Gu, d, r >
where d(x,y) := xl.
3 Cartesian Maps In this section we shall deal with the weak continuity of minors. More precisely, let {uk} be a sequence of maps in A' (S?, RN) such that
uk -> u
(1)
strongly in Ll (fl, RN) weakly in L1 va Ma (apDuk) weakly in L1
apDuk - v
for all multi-indices a and 0 with I a I + 101 = n, 101 > 2, and functions u, v and va in L1(0) 2. We ask whether almost everywhere
v(x) = apDu(x)
and
vq(x) = Ma (apDu(x)) .
The answer to this question is negative already in the scalar case, as the following example shows.
1 Consider the sequence of functions Uk : (0, 1) - R given by
Uk(t) := t -
[
]
2k
[x] denoting the integer part of x. Clearly Uk -+ 0 strongly in L°°(0, 1) while
apDuk = 1
a.e. in (0, 1)
v
weakly in L' (R, R')
.
Recall however that if Duk
,
Duk being the distributional derivative of Uk, then v is the distributional derivative of u. In view of Theorem 2 in Sec. 3.1.4 we therefore conclude that our question has a positive answer provided 2 compare the remark following Definition 2 in Sec. 3.2.1
3.1 Weak Continuity of Minors
277
apDuk(x) = Duk(x)
(2)
in the L1 sense, i.e. {uk} is a sequence in W1'1(r2,IR) By Proposition 1 in Sec. 3.2.3 condition (2) amounts to
aCukLQXR = 0. In the vector valued case the situation is totally similar. Even assuming that the approximate derivatives apDuk agree with the distributional derivatives, Duk, i.e.,
uk --> U
in L' (S?, RI) weakly in Ll Ma (apDuk) -k vq weakly in L'
Duk - Du
it is not true in general that v. = Ma (Du), compare Sec. 3.3.1. Again the key condition is that all Guk be boundaryless in .f2 x RN,
aGuk L ,fl x RN = 0. Actually, we shall see that va = Mg (apDu), if (1) holds and the boundaries of Guk in .f2 x RN are controlled in the sense that sup M(aGuk L ,f2 x RN) < 00 k
This is just a special case of the closure theorem of Federer and Fleming. This way we shall generalize a well known theorem of Reshetnyak concerning the continuity of the Jacobian determinant with respect to the weak convergence in Wl,n+E 6 > 0. For this reason, after presenting a few examples and Reshetnyak's theorems in Sec. 3.3.1, we shall study in Sec. 3.3.2 the space of Cartesian maps carts (0, RN) := {u E A1(.f2, lR')
I
aGu L Q x lRN = o}
for which we prove a closure and compactness theorem, with respect to the weak L'-convergence. The subclasses cartP(d2,RN) of cart 1((2,RN) of maps with Jacobian minors in LP will finally be discussed in Sec. 3.3.3.
3.1 Weak Continuity of Minors Let {uk} be a sequence of maps in A' (,fl, RN) n W1,1(Q, RN) such that (1)
uk - u in L1 Duk -k Du in L1
for some u E A'(Q,RN) n W1,1(Q,RN) Suppose that for all multi-indices a and 3 with I a I + 1 QI = n , 10 I > 2, there exist functions va E L1(fl, R) such that
278
(2)
3. Cartesian Maps
weakly in L'
ve
Ma (Duk)
.
We shall see in 03 and in ® below that in general ve(x) 0 M, (Du(x)) and we shall discuss some circumstances in which equality holds. A more thoroughly discussion is delayed to next section. In many respects it is also reasonable to replace the convergence in (2) by the convergence in the sense of measures (3)
Mg(DUO
as measures
ve
or by the related convergence of graphs {Guk } C D, (0 x RN) (4)
S
Guk
where S is the current in D,,,(.fl x RN) whose components are given by 500(0)
(5)
f(x,u(x))dx
f
SZ' (0)
O(x, u(x)) Djui(x) dx
Sap(s) := a(a, a) f O(x, u(x)) ve(x) dx n
where 0 E CI(f2 x RN) and Ic + IQI = n , L31 > 2, compare (5) in Sec. 3.2.1. Thus, before going into our main question, let us first compare the convergences in (2), (3), and (4). Clearly the weak convergence in L1 implies the convergence in the sense of measures, and the proof of Proposition 2 in Sec. 3.2.1 easily yields that the weak convergence in Ll implies also the convergence of graphs, i.e., convergence in (2)
convergence in (3)
convergence in (4)
If the minors M(Duk) are also equi-integrable, then the convergence as measures
implies, and therefore is equivalent, to the weak convergence in L', compare Theorem 2 in Sec. 1.2.4. The following two examples show that in general the convergence in (3) or in (4) does not imply the convergence in (2), i.e., convergence in (2)
convergence in (3)
11`
convergence in (4)
Finally a few words concerning the comparison between the convergences in (3) and (4). If the sequence {uk} is equibounded in L°°, it is easily seen that the
3.1 Weak Continuity of Minors
279
convergence of graphs implies the convergence in the sense of measures of the minors, compare also Proposition 1 in Sec. 4.2.3, i.e. convergence in (4)
-
equiboundedness in L°° of {uk}
convergence in (3)
But it seems to be not clear whether the convergence in (4) imply the convergence in (3), compare Theorem 3 and 6 in Sec. 4.2.5. 1 Consider the sequence of radial mappings
k =1,2,...
uk:B(0,1)cIRn -+Rn, uk(X) := rk(IxI)
I2II
where rk(t) is the continuous and piecewise linear function
kt if 0
rk(t) :=
if
0
It is easily seen that the sequence {uk} is equibounded in Wl,n(B(0, 1),W ), and 1
II
k Ilw=,n(B(o,1),R-)
<-
c(n)
f tn
l
In
(Irk'(t)I + I rktt) In) dt < c(n)
,
0
n-det
thus, since Uk -> 0 a.e. and Iuk I < 1, Uk - 0 weakly in W1'n(B(0,1), Rn) as k -+ oo. Also one easily computes
Duk(x) -
(rk(xD)'F(H)
where kn
- k)n-lk
Wk(t)
= Wk(IxI)
if 0
k
if k
10
thus {det Duk} is equibounded in L1, fB(o,1) Idet Duk I dx < c(n). Observing now
that
fJ
det Duk dx = 0
B(0,1)
for all k, and writing for any W c- Cc` (B (0, 1)),
f
det Duk dx =
cp(0)
B(0,1)
+
J B(0,1)
J B(0,1)
det Duk dx
[cp(x) - cp(0)] det Duk (x) dx
,
3. Cartesian Maps
280
we easily deduce that in the sense of measures.
det Duk -10
On the other hand, if 8 > 0 and k < 8, we have b
r to-1 ((rk (t)) n-1
f I det Duk I dx
c(n) J
t
I rk (t) I dt
0
B(O,d)
2/k
c(n) f Irk (t) In-1 I Irk (t) I dt =
2c(n)
n
0
This shows that {detDuk} is not equi-integrable in B(0,1), and therefore {detDuk} does not converge weakly in L'. Go where One can also show, compare the examples in Sec. 3.2.2, that Guk Go = [B(0,1) 1 x S0 is the graph of the null map from B(0,1) into lRn. Notice that this example shows also that the map u --> det Du is not sequentially weakly continuous from W1'n(B(0,1),W) into L'(B(0,1)).
A similar example to 0 is the following
0
Let n = 2, Q = (-1, 1) x (-1,1). Consider the sequence of maps Uk from Q
into R2 k-112
uk(x, y) :=
(1 - IYI)k (sinkx,coskx)
We have II uk III
<
k-1/2 1
II2
II Duk
2
1
= 2k f dx f[(i -1
- y)2k + (1 - y)2(k-1)] dy =
0
1
4k [
1 2k+1+ 2k-1] < 8
hence Uk converge weakly in W1,2 to zero. Moreover
det Duk(x, y) = k (1 - Iy1)2k-1 signy Thus one easily sees that in the sense of measures det Duk k ->oo a as
fJIdetDukIdxdY = 2k a -a
f
J -a
dx
f (1 -
0. However for
IyI)2k-l dy
-a
= 2a [1 - (1 - a)2k] --* 2a thus {det Duk} does not converge in L' to zero. Again one could show that Guk Go.
3.1 Weak Continuity of Minors
281
Let us return to our main question. A by now classical result concerning the weak convergence of determinants is the following theorem
Theorem 1 (Reshetnyak). Let ,f2 be a bounded domain of RI and let Uk 0 --+ ][fin be a sequence of continuous maps in Wl,n(Q,lRn). Suppose that the maps uk converge uniformly in (2 to u and that the sequence of Jacobian determinants {detDuk} is equi-integrable in L1((). Then det DUk -s det Du weakly in L'(Q).
Proof. Recall, compare (3) in Sec. 3.2.3, that Laplace's formulas yield
MO(Du) _ > Dju2 (adj Du)a) 7Ea
_
u(i, /3 - i)o(j, a - j) Ma=d 3Et
where IaI = IaI = 2 , 1 < Q < n, and that, in the sense of distribution we have for every u E W1'n, and i = 1, 2, ... , n
Dj ((adj Du)a) = 0
,
compare Proposition 3 in Sec. 3.2.1 and Corollary 2 in Sec. 3.2.3. Thus for all
a,3, al+I0I=2,1
J
Ma (Du) co dx =
-J
u'
n
Deco (adj (Du)')' dx jEa
holds for all cp E C°° (.fl) .
Since {det Duk } is equi-integrable, in order to prove the weak convergence of {det Duk} in Ll it suffices to show that
f co det Dudx
JcodetDudx
for all cp E C°° (t2).
We shall now prove by induction on the order of the minors that (7)
fcoM(Duk)dx -4 fcoM(Du)dx
V cp E Cc-
for all a,,3 with IaI=IaI=f, 1
3. Cartesian Maps
282
fM(Duk)cdx = - J -f
uk (adj (Duk)
,R
Dj cp dx
(ux - 4Z' k) (adj (Du)") Djcpdx.
uti (adj (Duk)') Djcodx + J i,j
«)
i,j
By the inductive assumption the first integral on the right tends to
-
ui (adj (Du)') Dj cp dx i, j
which is equal by (6) to
J
Ma (Du) dx
while the second integral on the right tends to zero. This proves (7) for £ and therefore the theorem. An immediate corollary of Theorem 1 is the sequential weak continuity of the operator u --- det Du from W1°r(0, IR') , p > n, into L' (Q). Actually, going through the proof of Theorem 1 one immediately gets the following
Theorem 2. Let {uk} C Wi,p(Q,llgn), where p > n. Suppose that Uk --" u
weakly in W 1,P
then
det Duk
det Du
weakly in Lp1n
and actually
Mlel (Duk) - M(t) (Du)
weakly in Lp/e
Notice that in fact Theorem 2 is a closure theorem. Given an equibounded sequence {uk} C W""p(Q,IRn) , p > n, passing to subsequences, we may assume that {uk,, } converges uniformly to u by the Sobolev embedding theorem, and
that M(t) (Duk) -s vt weakly in L' and actually in LP/e. By Theorem 1 vi = M(t) (Du). Therefore, if u is also independent of the chosen subsequence, we conclude that the entire sequence converges together with all minors in Ll-weak to the map u and the corresponding minors of Du. Theorem 1 and Theorem 2 are optimal in the sense that, as we have seen in 1 and 2 det is not sequentially weakly continuous from WI,n into L'. However we have
Theorem 3 (Reshetnyak). Let {uk} C
Uk - u
W1,n(0, Rn). Suppose that 1.
weakly in
W1'n
Then
det Duk
det Du
in the sense of measures.
3.1 Weak Continuity of Minors
283
Proof. From Theorem 2 we infer that
adj DUk - adj Du
weakly in L.-
and by Laplace's formula and Piola's identity, compare Sec. 3.2.3, that
fdetDukcodx = -
I: J u1 (adj Duk) i
W dx
j=1 S2
S2
holds for any cp E C' (.fl). As uk Dj cp -+ ul Dj w in Ln (92), we conclude that
fdetDukgodx -*
(8)
fdetDugodx n
n
for any cp E C' (0), and by approximation that (8) hold for any cp E C°(.(2).
Taking into account the result on the higher integrability of the Jacobian determinant in Theorem 2 in Sec. 3.2.4 we can also state
Theorem 4. Let {uk} C Wl"n(.0,Rn). Suppose that detDuk(x) > 0 a.e. and that
Uk - u
weakly in Wl,n .
Then
det Duk - det Du
weakly in Ll
.
The same argument in the proof of Theorem 3 actually yields the following two results, compare 31 in Sec. 3.2.4 and 1 in Sec. 3.2.4.
Proposition 1. Let {uk} be a sequence such that
Uk - u
weakly in W1'P(Q,11$n)
adj Duk - adj Du
weakly in Lq
withp>n-1 andq> P2l. Then det Duk -k det Du in the sense of measures. Proposition 2. Let {uk} C W1"P(,f2, ]R'), p > +1, be a sequence such that u weakly in W1,P. Suppose that uk, u E Wl,n(Q,Rn). Then the distribuUk tional determinants Det DUk converge to the distributional determinant of u in the sense of distributions. We finally observe that the determinant is not weakly lower semicontinuous
in W1,P for p < n, under no extra assumptions, as shown by the next two examples.
3. Cartesian Maps
284
0
LetQ1]-CR ,n>1.ForEQweset X
u(x) :=
XEQ.
x,
IIxII
where II x II := maxl R' is clearly continuous except at zero and vanishes on aQ; for x E Q \ {0} we have au2
bk
__
axk
x2
{
axk
k
11x112 axk
11x11
all x 11
ai
ail x 11
signxk 0
if II X II = Ixkl
if IIxII=1x'I, ilk,
moreover
au2 axk
11x211 + 1
From this one easily deduces that u E W1'P(Q,Rn) for all p with p < n. Since u = 0 on aQ, we may extend u as a Q-periodic function to the whole of RI, and denoting again by u this extension, evidently we have u E Wll, (Rn ][Sn)
for allp
uk(x) := x + k-lu(kx) ,
x E 12
Then, compare Proposition 1 in Sec. 1.2.3, Uk - x in W1,P(.fl, ][2n) for 1 < p < n; furthermore Uk -+ x strongly in LOO (.fl, ]fin). But for almost all x E 12 we have
DUk = id + Du(kx) = D II Y II
y= kz
hence
det Duk = 0
Vk
.
Therefore we have found a sequence {uk} such that Uk Uk
Vp
in W''P (.fl, Wi)
x x
in L°O (Q, Rn)
det Duk = 0
t1 k
but
det Du = det id = 1
.
Passing to the graphs, it is easily seen that
supM(Gu,,) < 00 k
while
M(3Gti,,k) --'1 00 As it will clearly appear in the sequel of this section, the unboundedness of the masses of the boundaries aGUk is responsible for the loss of the sequential weak 0 continuity of the determinant.
3.2 The Class cart 1(J2,lR ): Closure and Compactness
285
4 Let Q = [-1, 1]n c Rn, n > 3 and 1 < p < n-1 and let r := maxi=l,n-1 Ixil For k = 1, 2,..., let Qk :_ [-1,1]n-1 x [1 - 2k, 2k - 1] and ak := k-P/(n-P-1).
Set
r
ak
Vk,i(x) =
r > 2ak
for i = I,-, n - 1, and
r<2ak
0
Vk,n(x) _
x,,(-L- - 2) 2ak < r < 3ak xn r > 3ak
We extend vk to [1 - 2k, 2k - 1]n in such a way that vk,i(x) - x is 2-periodic in the variables x1i...,xn and set uk(x) := Then, compare Proposition 1 in Sec. 1.2.3, {uk} converges to the identity map in LP(Q) and det Duk = 0 a.e. for every k. Actually kvk(x).
Uk - x
det Duk = 0
weakly in W1,P,
since one can easily prove that the W1,P norms of the uk's are equibounded. In fact
IDuklPdx = k f IDvklPdx Q
Qk
and
IDvk(x)I IDvk (x) I
c1
1
r
< C2(r + k) ak
for r > 3ak, for r < 3ak
hence
f I Dvk I P dx < C3 k f tT"-2-P dt +
J
Qk
J (0,
kPak-P-1
< ck
which implies that fQ I Duk IP dx < oo. Passing to graphs, we obviously have aGuk L Q x R' = 0, while one can check
that
M(Guk) -* 00.
3.2 The Class cart' (i7,ll ): Closure and Compactness We shall now discuss in more details the class of Cartesian currents we already introduced in Sec. 3.2.1.
3. Cartesian Maps
286
Definition 1. The class of Cartesian maps is defined by
carts(Q,RV) := {u E A'(.fl,RN)
(1)
BGu,L.fl x RN = O}
I
By Proposition 1 in Sec. 3.2.3, we obviously have
cart' (Q,RN) C
W','(,fl,RN)
In particular every u E cart' (Q, RN) is n/(n - 1) summable by Sobolev embedding theorem, but we cannot expect any extra summability of u as consequence of the summability of Jacobian minors. For instance the map (u(x), u(x)) from W1,1(,f2, R2). Notice a domain of R2 into R2 belongs to cart' (,fl, R2) for any u E also that in general the i.m. rectifiable current Gu associated to u E cart' (.fl, RN)
is not a normal current in RI x RN since a priori we may have in R x RN M(8Gu) = +oo. We have
Theorem 1. Let {uk} be a sequence of maps in cartl(,f1,RN). Suppose that W','(S? RN) and that there exist functions vo E V(Q) for there is a map u E dal+101=n, 1/31>2 such that
Uk - u DUk
(2)
Ma (Duk)
weakly in L' weakly in Ll va weakly in L'
Du
Then va"(x)
(3)
= Mg(Du(x))
.
Before proving Theorem 1, let us state a proposition which explains the geometric meaning of the conclusion in (3).
Proposition 1. Let u E Ll (,fl, RN) be an a.e. approximately differentiable map, and let va E L'(,fl), jai + 101 = n, 1,31 > 1. Define the current S = Su,,, E Dn(Q X RN) by S00 (0) := f q(x, u(x)) dx 0
S"' (0)
o,(a, a) f O(x, u(x)) va(x) dx .a
,
,
Jai + 101= n ,
101
>_ 1
Then the following claims are equivalent (i) va(x) = Ma(Du(x)) for a.e. x E £1 and for all a,,3 with Jai +1,31 = n , 1
(ii) u E A' (.fl, RN) and S = Gu (iii) S is an i.m. rectifiable current in (1 x RN.
1,31 >
3.2 The Class carts (Q, T N): Closure and Compactness
287
Proof. If va(x) = Ma(Du(x)) for a.e. x, then u E Al(f2,lE.N), S= Gu and, by
Proposition 1 in Sec. 3.2.1, S is an i.m. rectifiable current. Suppose now that S is an i.m. rectifiable current, S = -r(S, 0, S). Denote by ii the n-vector given by
1:
Q(a, a)va ea n Ep,
ja!+jBI=n
and let M :_ {x E Q I I MDu(x) I < oo} As 9u, f2 is 71' measurable and countably n-rectifiable from the area formula we deduce S(w)
f < w(x, u(x)), v(x) > dx n
:_
I<w(x,u(x)),
M
IM(Du(x))I >
=
IM(Du(x))I dx
T(cu,0nM, A, )(w)
for any w E D'n(.f2 x1[8 N). Here we have set
((x) :=
v(x)I
'
a(x)
IM(Du(I ))I
Since S is i.m. rectifiable S = T(S, B, S), we have S 7{'-a.e. and,
0
v(x)
I, (x) I thus
Iv(X)I = cu,n(x,u(x)) =
(4)
On the other hfor
f
any
IM(Du(x))I C°(.Q x we have
O(x, u(x)) dx = S(O(x, y) dx) =
M
on M
f O(x, u(x)) A(x) dx .fl
thus M = ,f2 a.e., A = 1 a.e. and (4) yields
v,(x) = Ma (Du(x))
for a.e. x E
Proof of Theorem 1. As in the proof of Proposition 2 in Sec. 3.2.1 we deduce
that
288
3. Cartesian Maps
G,,,k - S
in Dn,(.fl x RN) .
On the other hand, by Proposition 1 in Sec. 3.2.1, all currents Guk are i.m. rectifiable, and
skpM(G,hk) = skpJ IM(Duk)I dx < oo M(BG,,,,,L Q x RN) = 0. Therefore Federer-Fleming's closure theorem yields that S is an i.m. rectifiable current, and the result follows from Proposition 1. As Federer-Fleming's closure theorem holds for sequences of i.m. rectifiable currents Gu,, with equibounded masses satisfying also
supM(BGuk L.fl X RN) < +0 , k
the same proof actually yields
Theorem 2. Let {uk } be a sequence in A' (.f2, RN). Suppose that strongly in L'
uk --+ u and for each a, (3, IcI + 1,31 = n, 1,31 > 1
Ma (Duk(x)) --> va(x)
(5)
weakly in L'
If supM(oGuk L.f2 X RN) < +oo ,
(6)
k
then we have
v"(x) = Ma(Du(x))
?- n-a.e. in Q
.
Remark 1. Actually it is easily seen from the proof that both Theorem 1 and Theorem 2 hold under the possibly weaker convergence
Guk - S
in Dn (.f2 x RN) .
The proof of Theorem 1 and Theorem 2 we have presented strongly relies on Federer-Fleming closure theorem; actually in our situation the proof is much simpler, as by Calderon-Zygmund theorem we already know that the limit current r(9,,,,{Z, 1, ) has as set of integration the rectifiable set yu,, and therefore it suffices only to show that is the orienting n-vector of 9,,,,,{2; see Theorem 4 in Sec. 2.2.7. Therefore let us give a direct
Proof of Theorem 1 and Theorem 2. From the assumptions the currents Guk carried by the graphs of the uk's converge to the n- current in ,f2 x RN given by
3.2 The Class cart' (0, RN): Closure and Compactness
289
f<w(x,u(x)),v(x)>dx
S(w) :=
n where
v(x) :_ E a(a, a) va eanEQ,
vo° := 1.
We now consider the blow up of S in RI x RN with center (xo, u(xo)) to be chosen suitably. Set for A - 0 7la(x,y) : = (
x - xo y - u(xo) A
,
A
and let w E Dn (,(2 x RN). We have
lla#S(w) =
S(77,\ W) An
<w(x Axo, u(x) Au(xo)),v(x)>dx
J
J u(xo + Az) - u(xo)
J<w(z,
A
Rn
If xo is such that
v (xo + Az) -f v (xo) u(xo + Az) - u(xo) A
strongly in Li o as A --> 0
-> Du(xo)z
1
strongly in Lloo as A -40,
(notice that those xo are almost all points of .(2, by Lebesgue differentiation theorem and Calder6n-Zygmund theorem) we can conclude that Sxo,a := 77a#S -s Sxo,oo for a.e. xo E .R where Sxo,oo is the current integration over the tangent plane to u at xo but with "orienting" vector the constant n-vector v(xo), i.e. Sxo,oo = (f{n LTx,Gu,n) L v(xo) or
f Sxo,oo (w) =
J
<w(z, Du(xo)z), v(xo) > dz.
In the case of Theorem 1 in Sec. 3.3.1 we clearly infer (7)
aSxo,00 = 0 for a.e. xo E ,(2;
we shall now prove that (7) holds also under the assumptions of Theorem 2. By lower semicontinuity M((aS L ( x RN) < oo, consequently the set E {z E Rn+N I 9n*(1I aS 11, z) = +oo} has zero hn-measure, and for a.e. xo E ,S?
290
3. Cartesian Maps lim sup II aS (B((xo, u(xo)), p) _ c(xo) < oo. wnpn P-0
Therefore, if for instance w E Dn-1(B(0, r)), we infer aS(77#w) I < A II as II (B(( o, u(xo)), Ar) II w III, hence
aSa(w) = rla#S(dw) = as(rla w) = O(A) as A -> 0
i.e. (7). The claims in Theorem 1 and Theorem 2 follow at once from (7) and the next lemma.
Lemma 1. Let P be an n-plane in Rn+N, v a constant n-vector and L the current defined by
L(w) :=
J
<w, v > d7-ln LP.
If aL = 0, then v = cP where c is a real constant. Proof. Since the claim is rotationally invariant we can assume P =RnX{0}So
that
L(w) = f<w(x,y),v>dx. Assuming IQI > 1, choose w as
w :_ 0(x)yz dxandya-i,
IaI + IQI = n, i E 0-
Then dwIp = o -(i,,3 - i)o(x) dxaAdxo hence
0 = aL(w) = L(&,)) = o,(i, /3 - i) f O(x)va(x) dxn consequently v = 0 a.e. Being Q arbitrary, 1,31 > 1 we conclude that v(x) v00e1A... Aen.
A more precise analysis then shows that we may replace condition (6) by the weaker condition (9) in the next proposition. Denote by Fp the class of all (n - 1)-forms which are linear combinations of forms of the type t(x, y2) dxa A
dyQ-i
where lal+IaI=n,1
3.2 The Class cart' (fl, RN): Closure and Compactness
291
Proposition 2. Let {uk} be a sequence of maps in A'(Q, RN) such that Uk -4 U in L' and weakly in Ll M (Duk (x)) - va (x) (8)
where Ial+1QH =n, 1< 1)3I
(9)
I W E .7=p,
(WI < 1} < 00
k
then
ve(x) = Mg(Du(x))
(10)
7-Ln-a.e. V O, I01 < p.
Proof. As in the proof above we find for a.e. xo E ,fl 3Sxo, (w) = 0 V W E .7=p. Let R : Rn+N _,., Ian+N be the map R(x, y) := (x, y - Du(xo)x). It is readily seen that R#w E Fp if w E 7'p. Hence aR#Sxo,.(w)=R#aSx,,,,,.(W)=0
VwEFp.
This means that a(/
J
<w(z),R#v(xo)>d7-ln) =0 `dwE.Fp.
1R x{0}
As in the proof of Lemma 1 we then find
R#v(xo) = c (eln ... Aen +
(R#v)«eaneQ) IRI>p
equivalently
R#(v(xo) - M(Du(xo))) _
(R#v)aeaAep, Ial>p
va (xo)
= Mg(Du(xo))
for 101 < p.
0 As a corollary we have
Corollary 1. Let {uk } be a sequence of maps in Ll (.fl, RN) such that uk --> u in Ll and Ma (Duk(x)) - vc(x) E L'(,fl) weakly in Ll where IaI + 1131 = n, 1 < IQI
for all j, 0 < j < p - 1 and all k, then ve(x) = Ma(Du(x)) In-a.e. V/3, 1 <101
292
3. Cartesian Maps
Remark 2. Notice that, under assumption (6), we may replace the hypothesis (8) by the possibly weaker condition in Dn(,fl x RN)
G,k - S,,,,,,
(11)
and the same proof yields the result (10). The proof instead fails if we replace (8) by (11) and we insist in requiring only (9) instead of (6). We may consider Al (.fl, RN) as embedded in Ll (fl, IRN) x Ll (.fl, AnRn+N) by the map u --f (u, M(Du)). The previous theorems then read as weak closure theorems in L1(.fl,RN) x Ll(.Q,AnRn+N). In particular we have
Proposition 3. The class of Cartesian maps carts (,fl, RN) := {u E Al (.fl, RN)
I
BG,,L .fl X RN = 0}
is sequentially weakly closed in L' (.fl, Rn) x Ll (,f1,
AnRn+N).
We conclude this subsection stating a compactness theorem. Recall that a well known result that we shall discuss in Vol. II Ch. 1 states that, given a non negative and convex function F(l;) in ]R and a sequence {uk}in L1(,fl,R') with Uk - u weakly in Ll(dl,R'm), then (12)
J F(u) dx _< lim inf k--.-boo
J
F(uk) dx
.
Taking into account the Ll-weak compactness criterion in Theorem 2 in Sec. 1.2.4 we can now easily prove
Proposition 4. Let 0: R+ -r IR+ be an increasing and convex function satisfying (t)
t
, oo
as t -> oo .
For allKEIR+thesets
cart'
{u E cart'
I
II u IIL1(n,IlYN) + f O(IM(Du)I) dx < K}
are sequentially compact with respect to the weak convergence in Ll (Q, RN) X Ll (0 AnRn+N).
Proof. Let {uk} C cart'. Passing to a subsequence we may assume that
Uk - u , Duk - Du weakly in Ll M« (Duk) - va
weakly in Ll
,
by the sequential compactness criterion in Ll-weak Theorem 2 in Sec. 1.2.4.
By the closure theorem va = M (Du), thus Uk -k u weakly in A', and u E carts (,fl, RN). Finally (12) yields at once
3.3 The Classes cartP(Sl, RN), p > 1
I
Iu dx +
12
J
O(I M(Du) l) dx < liminf Li Iukl + k-oo
f
293
¢(I M(Duk) I) dx
n
hence u E cart }.
3.3 The Classes cartP(S2, RN), p > 1 Analogously to Sec. 3.2.1, we define the class AP(.f2,IRN), for all p > 1, as the collection of all mappings from 0 C RI into RN which have approximate differential Du(x) at a.e. x E S2 and which are p-summable together with all minors Mg(Du(x)) , Ial + I,OI = n, (1)
Ap(S1, RN)
:= {u E Al(Q,RN)
I
Jul, IM(Du)I E LP(SQ)}
In AP we introduce the "norm" (2)
= II U II LP(.a,aN) + II I M(Du)I II LP(n)
II U IIAP
and we say that a sequence {uk} in AP (0, RN), converges weakly in AP to u E AP (S2, RN)
uk
u in AP
if and only if
Uk -> u
strongly in L"(S2, RN)
Ma (Duk) -k Ma (Du)
weakly in LP (f2)
for all a, Q with lal + 1,31 = n and 101 > I.
Evidently the map u -> (u, M(Du)) from AP(.f2, ][8N) into LP (S?, RN) X LP(S2, AnRn+N) is injective and II u IIAP is just the norm of (u, M(Du)) , in LP (0, RN) x LP (S?, A,,R '+N). From now on this immersion will be understood.
If p is larger than one, we expect that AP(S2,RN) enjoys better properties than A'(Q,RN). This is in fact true, but only partially. For instance 3 in Sec. 3.3.1 shows that in general AP(S2,RN) is not weakly closed in LP (0, RN) x LP (fl, AnRn+N). Therefore, compare Sec. 3.3.2, we are led to introduce the following subclass of AP (fl, RN).
Definition 1. The class of p-Cartesian maps is defined by AP(SQ, RN) fl cart1(Q, ]RN)
cartP(S2, RN) (3)
{uEAP(SZ,RN)
I
As in Sec. 3.3.2 we have
cartP(Q,RN) C W1,P(S2,
BG,,,LS?XRN=0}
294
3. Cartesian Maps
moreover for u E cartP(Q, RN) the approximate differential agrees with the distributional derivative. Often we shall write instead of I IuIIAP
I Jul lcartp
From the closure theorem of Sec. 3.3.2 we then deduce
Theorem 1 (Closure theorem). The class cartP(0, IRN) is closed under the weak convergence of LP(Q,RN) x LP(Q,A,,,IR"+N). That is, if uk and u belong to caxtP (j?, RN) and
Uk - u (4)
weakly in LP (0, ]RN)
Ma (Duk(x)) -k ve(x) weakly in LP (0),
for all a, / with Ial + IQI = n, then va(x) = Ma(Du(x)) for a.e. X E 1?. The sequential weak compactness of bounded sets in LP, p > 1, together with the previous theorem, readily yields the following
Theorem 2 (Compactness theorem). Let {uk} be a sequence of maps in cartP(S2,IRN), p > 1. Suppose that suP II uk IIAP < OC k
then there exists a subsequence {uk;} Of {uk} and a map u E cartP(ul,IRN) such that in AP uk: - k u
In contrast with the case p = 1, for p > 1 the weak convergence of sequence {uk} in cartP(Q, ]RN) is equivalent to the convergence of the graphs Guk in Dn(.fl X RN) together with the equiboundedness of the norms II uk IIA-
Theorem 3. Let {uk} be a sequence in cartP(Q, RN) , p > 1. Suppose that suP II uk IIAp < 00 k
and that there exists a current T E D,,,(.(2 x RN) such that
Guk -"T
in Dn,(dl x R N)
Then there exists u E cartP(Q, RN) such that
uk-ku
in AP
and T = Gu. In particular {uk} C cartP(Q, IRN) converges weakly in AP to u if and only if Guk- Gu
(5)
suPII uk H AP k
in D,(QXRN)
< 00
3.3 The Classes cartP(Q, RN), p > 1
295
Proof. Since Supk II uk II AP < oo, from any subsequence of {uk} we can extract a subsequence {uk, } such that ukc u and u E cartP(.fl, RN). Thus Guki Gu
and T = Gu. The first part of the claim then follows since the limit u is independent from the chosen subsequence. In fact suppose that for two subsequences & v(2), v(1) 54 v(2); {vk1)} and {v(2)} of {uk} we had vk1) P v(1) and vk2) from T = G,(i) = Gu(2) we infer
f(x,v(2))dx
f O(x, v(1)) dx = for all
E C°°(f? X RN), which contrasts v(1) 0 v(2). P
U. The converse is conseIn particular this shows that (5) implies Uk quence of Banach-Steinhaus theorem in LP, in fact, if Uk AP u, then we have quence supk II uk II AP < oo, and as {uk} C cartP(.f2, ]RN), Gu,.
Gu.
Remark 1. Theorem 3 is false for p = 1. In fact we have seen in 1 in Sec. 3.2.2 and 2 in Sec. 3.2.2 that
+ J udx) < +00 /
sup (M(GUk)
T,
Guk
Sa
do not necessary imply weak convergence of Uk in A'. We shall see in the next section that, even if
Uk - u in L',
sup 11 Uk IIAl < 00, and Guk -k T in D,,,(.(2 x IRN) k
in general T is only partially related to u. Finally let us state explicitly the following simple corollary of Theorem 3
Corollary 1. Let {uk} C cartP(Q, ]RN), p > 1. Suppose that
Uk - u
weakly in LP,
sup / k
I M(Duk) I P dx < 00
.l
then
Mg(DUk) - M« (Du)
weakly in LP
for all a and )3 with Iaj + IaI = n. Remark 2. In the definition of the classes AP (fl, RN) we may require that each component of M(Du), M(k) (Du), or, for a fixed system of coordinates, even each Ma (Du) be summable with exponent depending on the component itself min(n,N)
IWO dx < +oo S2
,
E k=1
r
J S2
IM(k) (Du) IPk dx < +oo
.
296
3. Cartesian Maps
This way we can define the classes
AP(Q,RN)
,
cartP(S2, RN)
and so on with p a multi-index
p
(p0, P1,
, pmin(n,N) )
With the obvious changes, the whole theory evidently extends to these classes. As we shall see later this is not just a formal extension, as those kind of spaces arise quite naturally.
4 Approximability of Cartesian Maps One of our main questions is to characterize weak limits of smooth maps from S2 into RN with equibounded energies. It is therefore natural to introduce the class SW-limAP (C1(Q,RN) nAP(Q,1RN)) ,
(1)
p> 1
consisting of all u E A" ((2, RN) which can be obtained as sequential weak limits in AP of maps u E C'(0, RN) with equibounded AP-norms. Of course, by the closure theorem in cartP(S2,RN)
sw-limAP (Cl (52,RN) n AP (Q, RN)) C cartP(0, RN) since for every map u E C1 (52,RN) n AP(0,RN) we have BGu L f2
= 0.
In particular no function in AP \ carte can be weakly approximated in AP by smooth maps, and the class in (1) agrees with the class of sequential weak limits in carte of maps u E C' (S?' RN) with equibounded carte-norms sw-hmAP (Cl (Q, RN) n cartP(Q, RN))
p>1.
Remark that, while for p > 1 every bounded sequence in AP has a subsequence weakly converging in AP, this is not true for p = 1. Consider now a map u in SW-limAP (sw-limAP (C' (f2, RN) n AP (fl, RN)) )
By definition there are uh,k E Cl (f2, RN) n AP (S?, ]R ) and Uh belonging to SW-lim,AP (C'(S2,RN) n AP (fl, RN)) such that uh
and for fixed h
u
weakly in AP
4 Approximability of Cartesian Maps
297
weakly in AP . As a consequence of Banach-Steinhaus theorem, we obtain Uh,k ---" Uh
11 Uh IIA,, C C
,
II uh,k II AP < c(h)
Nevertheless, even in the case p > 1, it is not clear whether the whole family {II uh,k IIAr} is bounded. Hence it is not clear whether sw-limAp (sw-limAP (C1(Q, RN) n AP(,(l, RN)))
= sw-limAp(Cl(Q,RN) nAP(.Q,RN)), equivalently whether SW-limAP (C1 (Q, RN) n Ap (.(2, RN)) is sequentially weakly closed in cartP(,(1, RN), and we in fact do not know the answer for p > 1. Therefore we set Definition 1. We denote by CartP (0, RN) the smallest subset of AP (f2, RN)
which contains C' (S?, RN) n AP(.(2,RN) and is closed under the sequential weak convergence in LP of the maps and of all their minors.
A more precise definition of CartP(12, RN) is the following. Denote by j the immersion i : AP(Q,JRN)
,
LP(Q,RN) x LP(f2,AnRn+N) ,
j(u) = (u,M(Du))
Let C := j (Cl (.(2, RN) n Ap(f?, RN)) and let C be the smallest subset of LP(Q, RN) x L3' (0, AnRn+N) which contains C and which is (sequentially) closed under the weak convergence in L3 ((, RN) x LP((2,AnRn+N), then by definition
CartP(.(l, RN) := j-1(6) . Since j(cartP(,(l,RN)) is sequentially weakly closed in LP(f2,RN) x LP (fl, AnRn+N) we have C C j (carte (0, RN) ), thus CartP (0, RN) C carte (.(2, RN )
Evidently the compactness theorem is valid in CartP(Q,IRN), i.e. Theorem 2 in
Sec. 3.3.3 for p > 1 and Proposition 4 in Sec. 3.3.2 for p = 1 still hold if we replace carte by Carte both in the hypotheses and in the theses. As it is well-known, the weak sequential closure of a subset of Banach space is a quite complicated object, so it is in principle Cartp(S?,RN)3. It is obtained in general by a transfinite inductive process (up to the first uncountable ordinal) of limits. For its relevance for us, we shall now explain with some details this procedure in Sec. 3.4.1. We have sw-limAp (C' (.(2, RN) n AP (,(2, RN)) C CartP (.(2, RN) C Carte (.(l, RN )
and the following two questions arise naturally 3 Remark that Cl ((2, RN) n AP (.(l, RN) is not convex.
3. Cartesian Maps
298
I.Does
sw-limAP(Cl(.fl,][8N)nAP(Q,RN)) = CartP(Q,RN) hold or not ? II. Does
CartP(0, RN) = cartP(Q, RN) hold or not ? If not in general, in which circumstances the previous equalities hold? Recall that functions in AP \ carte cannot be weakly approximated in AP by smooth maps. Are there reasonable conditions guaranteeing that a given map u in cartP(.f2, RN) or in Carte (0, RN) could be weakly approximated in AP by a sequence of smooth maps? We might also consider the strong closure of C' (.fl, ]ESN) n AP ((, RN) in LP (.fl RN) x LP (( A,Rn+N) denoted strong-cl p (C1(.?, RN) n AP (Q, RN) ) or
CARTP (S?,
The strong convergence of {uk} to u in the sense
f IU - ukIPdx+ J IM(Duk)-M(Du)IPdx -> 0 as k -->oo
J
implies, and in fact for p > 1 is equivalent by Radon-Riesz's theorem 4, to the weak approximability of u in AP and in p-energy, i.e.,
r
J
Uk - u weakly in AP(Q,RN) IM(Duk)IPdx ->
We may then ask III. Does CARTP(0, RN) := strong - CIA, (C' (.Q, RN) n .,4P(Q, RN)) agree with CartP(Q, RN) or with sw-limAP (C1 (.f2, RN) n AP(0, RN)) ?
We do not know the answers to questions I and III, except in the case p = 1. This will be discussed in Sec. 3.4.2 where we also show that question II has in general a negative answer, i.e.,
CartP(.fl,RN) c cartP(.f2,RN)
.
54
4 Recall that this is not true for p = 1, compare 71 in Sec. 1.2.2.
4.1 The Transfinite Inductive Process
299
4.1 The Transfinite Inductive Process To each set we can associate its cardinal number which roughly tells us the number of its elements. To totally well ordered sets we can associate an ordinal
number. Recall that a totally ordered set is a pair (X, <) where < is a partial order (i.e., a reflexive, antisymmetric, and transitive relation on X x X) satisfying the trichotomy condition: for all x, y E X either x < y or y < x. A totally ordered set (X, <) is called well-ordered if every its subset has a smallest element. Every set can be well ordered and this claim is equivalent to the axiom of choice, or
to Hausdorff maximality principle, or to Zorn's lemma. Finally the following principle holds
Principle of transfinite induction. Let (X, <) be a well ordered set and let A C X be such that a E A whenever
{xEX I x
1 < 2 < 3 < 4 < ... or by
2 < 3 < 4 < ... < 1;
the first order has no maximum element, while the second does. Let A and B be totally ordered sets. An order isomorphism from A onto B is a one-to-one function f from A onto B such that x < y in A implies f (x) < f (y) in B. We write A B, to say that there is an order isomorphism between A and B. One easily sees that .: is an equivalence relation: the equivalence classes are called order type. This way every set A has an order type noted ord A. If A is well ordered, its order type is called an ordinal number. A natural order is defined among the ordinals. If a and /3 are ordinals and ord A = a, ord B = /3, one says a < 0 if there is x E B such that A ~ {y E B I y < x}, and we write a < /3 to mean that either a < /3 or a = /3. One easily sees that the order defined this way does not depend on the choice of well ordered sets A and B. With the order defined previously any set of ordinals is totally ordered and even well ordered. Moreover, for any ordinal a > 0, the order of the set P,, := {ordinals /3 /3 < a} 1
is a,
ord Pa = a. Also, for any cardinal number a, there exists an ordinal number a such that
300
3. Cartesian Maps
cardinality of Pa = a . We say that a is countable if Pa is countable. We set
ord ¢ = 0 ord {1, 2, ... , n} = n ord {1, 2, 3, ...} = wo ord {2, 3, ... ,1} = wo + l
ord{n+1,n+2,...;1,2,...,n} = wo+n The first ordinal following these numbers is wo + wo denoted by w0.2
ord { 1, 3, 5, 7.... ; 2,4,6 .... } = wo + wo . The numbers following w .2 are
wo 2+1,wo 2+2,...,wo - 2+n,... Continuing this process we find all numbers of the form
The number following those numbers is denoted by w2. Continuing this process we define numbers of the form wo .no
+wo-i.nl+...+wo
-nk-i +nk
The number wo ° follows these numbers, and the process restarts with wo ° + 1 Wo
till the number wo ° ° . The number following numbers of the form w0 "& is denoted by e and so on. Notice that all these numbers are countable since only a denumerable set of numbers are included in the process. One then proves
Proposition 1. We have (i) There exists a smallest uncountable ordinal number wl, i.e., such that P
1
is uncountable. In particular Pc. is countable for all a E P4J1. (ii) For every countable subset C of P°11 one can find a Q E P(1 such that a < /j
`daEC. such that ao E A and a belongs to A whenever all 3 with ao < /j < a belong to A. Then
(iii) Transfinite induction up to w1. Let A be a subset of P,
A= PL,,.
1
4.1 The Transfinite Inductive Process
301
We notice that the continuum hypothesis is the assertion that w1, or more precisely, P,,, has the caxdinality of R. Finally we remark that every countable ordinal a, i.e., every a E P,,,1 either is the successive of another ordinal, that is there exists ,3 such that
a = Q+1 , and in this case we say that a is a non-limit ordinal, or is a limit ordinal, i.e., for all )3 < a, there exists y with 3 < ry < a. We are now ready to turn back to the sequential weak closure of a set in a Banach space X. Let X be a Banach space and let C be a subset of X. We define
SW-limX C := {x E X 13 {xk} C C, xk - x} The following example shows that in general sw-limx C is not sequentially closed in X.
0 Let H be a separable Hilbert space and let {e,} be an orthonormal basis. Define
wn,k := en + n ek ,
C :_ {wn,k}n.,k=1,2,...
Obviously 11 en +nek11 >-
11nek11-IIen1I = n - 1 .
Therefore the only weakly converging sequences in C are the sequences {wn,k}keN, for n E N, and we have Wn,k -s en as k --k oo. Thus
sw-limX C = C U {en}n=1,2,... But {en}n=1,2,.. C SW-limX C, en - 0, 0
SW-limX C.
Set C(°) := C, and for any countable ordinal number a define C(a) := sw-limX C63)
if a = /3 + 1 is a non limit ordinal, and
C(a) := U C('3) O
if a is a limit ordinal. Notice that
C(") C C(a)
if a<0
C(1) = sw-limX C . Finally, define the class of successive limits up to w1 as
succw-limX C := U C(-) a<W1
302
3. Cartesian Maps
and denote by
SW-C1X C
the sequential weak closure of C, i.e., the smallest sequentially weakly closed set in X containing C.
Proposition 2. We have sw-clX C = succw-limx C . Proof. Let {xk} C succw-limx C with Xk
x. Then for each k E N there exists
ak < w1 such that Xk E C(k) and by Proposition 1 (ii) there exists p < w1 such that ak < Q for all k E N. Therefore {xk} C C(O), hence x E C(0+1) C x. This shows that succw-lima C is sequentially weakly Ua<,,,, C(U) as xk closed. Hence
sw-clx C C succw-limx C . Let us prove the opposite inclusion by means of the principle of transfinite induction up to wl. For simplicity write C for sw-clX C. Obviously C(°) = C C C. If a = /3+1 and Co) C C, then C(a) C C as C is sequentially weakly closed. If a is a limit ordinal and if C(O) C C V,3 < a, then obviously C(a) = C Therefore by Proposition 1 (iii) we conclude
UC«> cC. LY <w1
Remark 1. Suppose that X is the dual of a Banach space. Then we may consider on X also the weak* convergence. Evidently we may repeat the same construction and proof above replacing weak by weak*, in particular we then conclude, with the obvious meaning of the notation
sw*-clx C = succw*-clX C
2 Returning to the example in 1 , C:= {wn,k}n,k=1,2,... we readily see that
C(°) = C
,
C(1) = C U {en}FEN
,
C(2) = C U {en} U {0}
Since C(2) is trivially sequentially weakly closed, we conclude in this case
Sw-ClXC = C(2) = CU{en}U{0}
We emphasize the fact that sw-clx C is the sequentially weak closure of C. The closure of C in the weak topology of C is in general strictly larger than sw-clx C as shown by a classical example of von Neumann, see below. Recall that the weak topology in X is defined as the weakest topology in X for which all
4.2 Weak and Strong Approximation of Minors
303
linear functionals x -- < f , x > f E X' are continuous. A base of neighborhoods of a point xo E X is obtained by taking the sets
V :_ {xEX I II<_-ViEI} where I is a finite family of indexes, f;, E X', and e > 0. We shall never work with the weak topology. 3 von Neumann's example. In the Hilbert space £2 we consider the subset E of vectors e,,,n, 1 < m < n < -boo, the m-th component of e,,,,n is one, the n-th component of e,nn is m, and all other components of emn are zero. Trivially E is weakly sequentially closed as E contains no weakly converging sequence except for the ones which definitively are constant. However 0, which is not in E, is in the weak closure of E. In fact neighborhoods of 0 in the weak topology of P2 have the form cc
V
{xE22
M I
bnZ>xnI <e, i=1,2,...,k E(bn)2<+oc}
I
n=1
n=1
As for all b')) as above we can choose m, n so that
jbM+mbni)I < e it follows that 0 belongs to the weak closure of E.
4.2 Weak and Strong Approximation of Minors We begin by giving a characterization of the strong approximability property by smooth maps. The trivial observation contained in the following proposition allows us to replace C1-maps by Lipschitz-continuous maps.
Proposition 1. Let u c cartP(Q,AN). Suppose that there exists a sequence of Lipschitz-continuous maps uk such that
Uk ` u M(Duk)
-->
strongly in LP(,f2,RN)
M(Du) strongly in LP(f2,A,,
n+N )
Then there exists a sequence of Cl-maps uk which again are such that ilk
M(Dilk)
u
strongly in LP(.f2,IRN)
M(Du) strongly in LP(l2, AnRn+N)
Proof. It suffices to approximate each Uk or each Lipschitz-continuous extension Uk to Rn by smooth maps in W1"n(.f2,1RN) or in W1'n(f2,1RN) .f2 DD S?.
The next theorem contains a key result concerning the approximation property
3. Cartesian Maps
304
Theorem 1. Let u E cartP(.Q,1RN) , p > 1. The the following two claims are equivalent (i) There exists a sequence of locally Lipschitz maps {uk} C cartP(.fl,IRN) such
that Uk -- u
(1)
weakly in AP
and the family {jM(Duk)IP} is equi-integrable in Q
(2)
5
There exists a sequence of locally Lipschitz maps {wk} C cartP(,fl,IRN) such that (3) strongly in LP (S?, RN) Wk --+ u M(Dwk) -> M(Du) strongly in LP(,fl, An1Rn+N)
(ii)
(iii) There exists a sequence of locally Lipschitz maps {wk} C cartP(,fl,1RN) such that
wk} < 1/k
meas {x I u
and
f
J Wk #U
IM(Dwk)IPdx ---> 0
Proof. Trivially (iii) implies (ii) and (ii) implies (i). Let us prove that (i) implies (iii). Let {uk} be a sequence in (i). We have
skpllukIIAP < c1
,
IIuIIAP < c1
for some constant c1; moreover, possibly passing to a subsequence, we can assume
that
Uk -> u
uk -> u
strongly in L1(.f2, IRN)
a.e. in Q .
First, for any S > 0, we construct a Lipschitz map ws in IRn and a closed set Fs such that, if Gs
we have (4)
.f2 \ Fs
,
meas Ga < S
uwb
Uk -+ u uJ
wb
in F6 uniformly in F5 a.e. in Ga , for all j = 1, ... , N .
By Lusin type theorem for functions in W1,P, compare Sec. 3.1.3, we find a closed
set F1,8 and a Lipschitz map fb such that Note that, when p = 1, the condition of equi-integrability of {IM(Duk)1} is already contained in (1).
4.2 Weak and Strong Approximation of Minors meas (.f2 \ F1,&) < 6/2
and
305
u = fs in F1,5 ,
applying Egoroff's theorem we find a closed set F6 c F1,6 such that u = fs on F6 and U k -* u uniformly in F6 .
meas (.fl \ F6) < 5,
Then we finally modify fs on G6 :_ 0 \ F6 by setting, for j = 1, ... , N,
wa = fb + A dist (x, F6) where Ad's are chosen by considering the family Ha of disjoint sets
HA := {x E G6
I
f 5 (x) + A dist (x, F6) = u& (x)}
and choosing A3 so that measHaj = 0. Clearly the map ws we constructed this way is Lipschitz continuous I Dws(x)I < c2(5)
(5)
and satisfies all requirements in (4). Finally, since uk - u II.,F5 -* 0 and I uk(x) - u(x) I > t} tends to zero as k --* oc, passing possibly to meas {x I
a subsequence, we can also assume that (6)
II uk-UII.F6< 2-k meas {x
I
Iuk(x) - u(x) I > 2-k} < 2-k
We now define the sequence of locally Lipschitz maps wk = (wk,
.... wk )
as
(7)
wk := max{min{wb, uk + 2-k} uk - 2-k}
and we shall show that for k > k(8) (8)
meas {x I wk(x)
(9)
J
u(x)} = 0(1)
M(Dwk) - M(Du) I1 dx = o(1)
d2
as S -+ 0+. Evidently this implies (iii). From the expression of wk we have
I uk + 2-k if wb > uk + 2-k wk (x) :=
wb
if
1W6 - u' I < 2-k
7 -k uk - 2-k if w6
in particular, from (6) and ws = u on Fs, we deduce that wk = u on Fs. In order to compute the minors of Dwk let us define, for any multi-index e _ RN as (Sl, S2 ... , err) E {-1, 0, 1}N, the map zk :_ (zkl, ... , zkN)
306
3. Cartesian Maps xkj :=
I uk + 2-k if +1 w6j if Cj = 0 u3k -2-k
if Cj=-1,
and the subsets of G6
Ek := {x E Gs
wk(x) = zk(x)}
I
.
Since the values of wk (x) are just the values of zk' (x) for some C E {-1, 0,1}N, we have
G6=UEk
We now divide G6 into a (good)k part Ak given by
U
Ak
Ek
and a (bad)k part given by
Bk := G6 \ Ak
.
On Ak we have DWk = Duk, hence M(Dwk) = M(Duk). On Bk DWk is mixed in the sense that some its rows are rows of Dw6 and others from Duk. If X E Ek , ,Q' _ that 62
0 {-1,1}N, a,,3 are such that Ial + IaI = n, t3 = (Ql......3n) and is the submulti-index obtained by choosing the Pi's in such a way 0, by Laplace's formula we deduce I Mg(Dwk) I < II Dw6 II1)31,Faa
I Md' (DUk) I k ' I+J0' I=n
Therefore, taking also into account (5), we deduce (10)
J Bk
IM(Dwk)
Pdx < C3 (J) f(M(Duk)dx. Bk
We now prove that
lim meas Bk = 0
(11)
k-.oc
In fact, if x E Bk, then there exists j E {1, ... , N} such that wk (x) = wb (x), i.e., I uk (x) -w(x) I < 2-1. This yields either I uk (x) - ui (x) I > 2-k or I wa (x) -
ui(x)I < Iw'(x) - uk(x)I + Iuk(x) - u&(x)I
Bk C {x E G6
I
2-k +2-k = 2-k+1 Therefore
Iuk(x) - u(x) I > 2-k} U
N
U {x E G6 j=1
I
Iu(x) - wd
(x)J<2-k+1
}
4.2 Weak and Strong Approximation of Minors
307
Since, by (4), 00
n{xEG6 I Iui-w,I<2-k} = {xEGs
I
uj=wb}
k=1
has measure zero, taking into account (6) we get (11). Finally, let us prove (8) and (9). We obviously have (8) as {x I wk(x)
u(x)} C G6
,
while, taking into account (10),
J G6
IM(Dwk)IP
= f IM(Duk)IPdx+ J IM(Dwk)Irdx < Ak
Bk r
<
r IM(Duk)IPdx+c3(5) G6
J Bk
IM(Duk)IPdx
The first integral is o(1) in virtue of the equi-integrability of I M(Duk) I p, while, again by the equi-integrability of IM(Duk)IP and (11), also the second integral is o(1) as 6 --> 0.
O
An immediate consequence of Theorem 1 is the following corollary
Corollary 1. We have Carts (!l, IRN)
= SW-limas (C1 (Q, IRN) n Al (fl, RN)) = strong-clA, (Cl (.(2, RN) n A' ((2, RN) )
= CART' (S?, RN)
Proof. By Theorem 1 the class strong-c1Al (Cl (( RN) n A' (Q, RN) )
is sequentially weakly closed. In fact if uk u weakly in A', and {uk} E strong-c1Al (Cl (,(l, RN) n A' (.fl, RN)), then for each k we can find {uk,h} C Cl (dl, RN) such that
Uk,h --- uk
strongly in Al
Thus we can choose h = h(k) so that uk,h(k)
--' u
weakly in Al .
Applying Theorem 1, we then see that u belongs to the strong closure in A' of smooth maps. Consequently Cart' (.2, RN) C strong-c1A (C1(.Q, RN) n ..4'(Q, 1RN))
and equality holds as the opposite inclusion is trivial.
308
3. Cartesian Maps
Later in this subsection we shall see that in general Cart' (.fl, RN) = CART' (d1, IRN)
#
cart' (t2,
Remark 1. Though for every u E Cart 1(Q, RN) we can find a sequence of Lipschitz maps Uk such that Uk --> u,
M(DUk) --> M(Du) strongly in Ll
it is not clear whether a Liu type theorem holds. In particular it would be of some relevance to decide under which conditions on u we can find for every A > 0 a Lipschitz map ua with Lipschitz constant /less than A such that {xI ua
u}-->0,
1 M(Du,\)I dx-*0
J
as A - oo. For p > 1 we introduce the following definition
Definition 1. Let U E cartr(Q,RN). (i) We say that u has the q-strong approximation property, q < p, if there exists a sequence of smooth maps, of class C' or Lipschitz, {uk} such that
Uk --- u
strongly in L4
and
M(DUk) -3 M(Du) (ii)
strongly in L'
.
We say that u has the p--strong approximation property if there exists a sequence of smooth maps {uk} such that
Uk ---* u strongly in L9 (.R) M(DUk) --> M(Du) strongly in L4(.R) for all q < p. Of course the p--strong approximation property is just the q-strong approximation property for all q < p. We also define
Definition 2. We denote by Carte (Q, R') the subclass of elements u in cartp((1,RN) which have the p -strong approximation property. As a consequence of Theorem 1 we infer
Corollary 2. Cartr (11, RN) is sequentially weakly closed in cartP(S1, RN). In particular Cartr(f1,RN) C Carte (Q,RN) .
4.2 Weak and Strong Approximation of Minors P r o o f.
309
Let {uk} be a sequence of map in cartP(Q, ]LAN) which h a v e the p -strong
approximation property and such that Uk sequence of smooth maps {vk} such that
J
P
u. Fix q < r < p. We can find a
IM(Dvk) - M(Duk)IT dx < 2-k
> u in LT .
Vk
,
In particular vk A' u and the family { JM (Dvk) Je } is equi-integrable. Hence, by Theorem 1, there exists a sequence of smooth maps {wk} such that
f IM(Dwk)-M(Du)IQ -+ 0. Choosing an increasing sequence {qk} converging top, we now find by a diagonal process a sequence {wk } such that
f IM(Dwk) - M(Du) I4 dx
0
Vq
For p > 1 we therefore have CARTP (.f2, RN)
:= strong - clAP (C1(Q, RN) n AP (Q, RN)) C SW - lirAP(C;1(f2,RN) n Ap(.f2, RN)) C
CartP(Q, RN) C Carte (Q, IRN)
It is an open question to decide whether some of the previous inclusions are actually equalities or strict inclusions.
We shall now prove that in general Carte (.f2, RN) c cartP(Q, 0 consequently
Carte (,f2, j N) c carte (.Q, RN )
for p > 1, and actually also for p = 1. Denote by Br = B(0, r) the open ball in 1R2 of radius r centered at the origin. We shall consider maps u : B1 -> ]I82 satisfying the following condition (A) u takes values on a fixed domain f2 := CO \ Uk=1Ck
where C1, ... , Ch are open balls with disjoint closures contained in a ball Co
centered at zero.
310
3. Cartesian Maps
Let u E cart' (B1i R2) satisfy condition (A). Set
Ru :_ { r E (0,1)
j
J aB,
IDul dal' < oo , u(BB,.) C .(l }
and
U(r) := u restricted to aB,. . Clearly 7'((0,1) \ Ru) = 0 and, by Sobolev theorem, u(,.) E C°(BBr., .(2) for all r E Ru. Then we have
Theorem 2. Let u E cart' (B1i R2) satisfy condition (A). Suppose that there exists a sequence of locally Lipschitz maps wk E cart' (B1i R2) such that
strongly in L'
wk --> u
M(Dwk) - M(Du) strongly in L' Then u(r) is homotopic in (1 to a constant map for almost every r E Ru.
Proof. Choose e > 0 such that e < 72 (C2) ,
(12)
i = 1, ... , h
.
where Ci, i = 1, ... , h are the balls in the definition of 12. From det Dwk -> det Du strongly in L' it follows that the sequence of functions det DWk is equiintegrable, hence there exists 5 > 0 such that for all k
fIdetDwk I dx
(13)
<e
if meas E < b
.
E
Also, there exists a subsequence of {wk} such that 00
J
Gwk - ul + IDWk - Dul) dx < oo
k=1 B,
thus we can find R° C Ru such that
7' (Ru \ R°) = 0 (14)
E °°
f
J
(Iwk - ul + IDWk - Dul)dx < oo
for rER°
Consequently, for each r E R° we deduce
wk -> u
strongly in
W1,1(aBr, R2)
4.2 Weak and Strong Approximation of Minors
311
and by the Sobolev embedding theorem
uniformly on 9Br for r c Ro
u
wk
(15)
We shall now show that for r1, r2 E R°, 0 < r1 < r2 < 1 satisfying S jr,
(16)
-
r2l
< Tir j
we have
u(r,) and u(r2) are homotopic in
(17)
Let v > 0 be such that (18)
,(2Q
:= {x E 1R2
1
dist (x, u(aBri) U u(aBr.2)) < al c .(2
.
For ko sufficiently large we deduce from (15) wko (8BT, U
(19)
aBr2) C ,(2Q
.
\ Br1) < S, thus by (13) From (16) it follows ?-ll(B,.2fdetDwko(ri,r2)ldx
<E
(20)
where we have set Wko(ri,r2)
wko l Br2\B,
From (12) and (20) we then see that the set Wko(rl,r2) (Br2 \ Br,) does not cover a n y of the Ci, i = 1, 2, ... , h. Therefore we can find 7o, 'Yi , ... , 'Yh > 0 and y1 E C1, ... , yh E Ch such that h
(21)
,(2
:= B-Y" (0) \ U B7; (yi)
(2
i=1
and
_
(22)
.(2 : wko (ri ,r2) (Br2 \ Bri )
Since Wko(r,,r2) is a homotopy between wko(rl) and wko(r2) we deduce that wko(r,)
and wko(r2) are homotopic in h and, as it can be easily seen, in (2. On the other
hand from (18) and (19) we also see that for i = 1, 2 wko(r,) and u(r;) are homotopic, a homotopy being given by
ht (x) := t wko (x) + (1 - t) u(x)
x E aBri
thus (17) is proved. In a similar way we can prove that u(r) is homotopic to a constant map for
r < 2 5. This together with (17) proves that u(r) is homotopic to a constant map for each r E RU O.
312
3. Cartesian Maps
Let u : B1 --f R2 be a map which is homogeneous of degree zero and, more precisely, is of the type (23) u(x) = cP( I
where cp is a smooth map from S' = tB1 into 1182. Clearly, compare 17 in Sec. 3.2.2, u E W1,P(B2, R2) for all p < 2, det Du = 0 in B1, thus u E AP(Q,1182)
for any p < 2 and aG,, L B, X R2 = -So X Q cp(S1)
.
In particular, if for instance W(S1) is covered twice with opposite orientation, it follows that 8Gu L B1 x 1[82 = 0, hence u E cartP(B,,I182) for p < 2. It follows from Theorem 1 that any map with the following properties (i) (ii) (iii)
u is of the type (23) and u E cart' (BI, 1R2 ) 1[82 \ W(S1) has a finite number of connected components cp is not homotopic to a constant map in 1182 - {X1, ... , xk} where x1, ... , xk are points respectively in each bounded connected component of 1E82 \ cp(S1).
does not belong to Cartr'(B1i R2), p < 2. An explicit example is given for instance
by the map (24)
u(x) = cP(I x I
where cp is defined by
(25)
cp(0) =
(-1 + cos 40, sin 40) ( 1 - cos 40, sin 40) (-1 + cos 47r, - sin 47r)
( 1 - cos40, - sin 40)
for 0 < 0 < 7r/2 for it/2 < 0 < 7r for 7r < 0 < 3/27r for 3/2ir < 0 < 27r
Clearly the image of S1 = aB, through cp in (25) is the boundary of the union of the two unit discs centered at (-1, 0) and (1, 0) of R2, cp(S') is covered twice with opposite orientation so that u E cart1(B,, R2), and one easily sees that cp is not homotopic to a constant map in 1182 \ {(-1, 0), (1, 0)}. We finally observe that, though u in (24) is not in Cart' (B(0, 1),1R2) we can always find by Y2 in Sec. 3.2.2 a sequence of smooth maps {uk} such that Guk
in B(0,1) x 1182.
Gu
4.3 The Join of Cartesian Maps
313
4.3 The Join of Cartesian Maps Given two maps u, v from a bounded domain .f2 of 1R' respectively in RN1 and RN2, the join of u, v, denoted by u 'a v, is defined as the map from ,f2 into RN,
N=N1+N2, u Di v(x) _ (u(x),v(x)) . In general the join of two Cartesian maps is not a Cartesian map as it is shown by the map Tx from B(0,1) C R2 in R2. In this subsection we shall give sufficient I
conditions in order that the join of two Cartesian maps be a Cartesian map. These kind of results may be regarded as a generalization of Proposition 3 in Sec. 3.2.1 which, on account of W1,1'(.f2,1R) = cartp(.f2,]R)
,
may be restated as
Proposition 1. Let u :
,fl
118N
be a map with components v.1,... , uN in RN).
cartp(Q, I[8). Suppose that p > n := min(n, N). Then u E
Showing that u
u x v belongs to some A'' (0, I[8N) aG,,,.v L (2 X ISBN = 0.
Answers to question (i) are easily given in terms of standard formulas on the minors of the sum of two matrices and Holder inequality. More delicate is the question of the vanishing of the boundary on f2 x RN of the join. In fact essentially
the only operative way we known to ensure the vanishing of 9G,,,, L ,(2 x R N is by approximating G,,,,, by currents which are boundaryless in .(2 x ISBN. This was actually the the argument in the proof of Proposition 1. Let us begin by recalling the standard formula for the minors of the sum of two matrices
Lemma 1. Let A and B be two N x n-matrices. Then for all multi-indices a,,3 with lal = 131, 0< 1,31 < n := min(n, N) we have
M« (A + B) _
or(a , a")
11
Ma, (A) Ma (B)
a,+a"=a la'I=1p'I
the s t an d ard b ases i n ][87L an d in N respectively, and denote still by A and B the linear operators from R'll6into ISBN associated to A and B in those bases. Setting I a I = p, we have P r o o f . Deno t e by e1, ... , en and El, ... , EN
3. Cartesian Maps
314
(A+ B)n ... n(A+ B) (e,,,) _ (A + B) (e«1)n ... A(A + B) (e«p )
(1)
E Ma(A+B)EQ QEI (p,N)
and analogously
(An ... AA) (e«) = E Mg (A) co ,3EI(p,N) (2)
(BA...AB)(e«) = E M«(B)Ea /3EI(p,N)
Thus from the k-linearity of the wedge product and (2) we infer
(A+B)n...A(A+B)(e«) = = A(e«1)AB(a«2 )n ... AB(e«,,) + .. . a") (AA ... AA) (e«,) A(BA ... AB) (e« )
_
a") Mme' (A) M« (B) Ea, ct Q'EI(p',N)
Mc (A) M (B) }E0 QEI(p,N)
where p' = Ia'I,p" = Ii3"I. The result then follows comparing with (1). As
n a,. nei 9, =
v
an immediate consequence of Lemma 1 is Lemma 2. Let u, v be maps in Al (f2, IRN,) and Al (Q, RN2 ), respectively. Then
G cIM(Du)IIM(Dv)I where c is a constant depending only on n, N1 and N2.
Taking into account Holder inequality, we can therefore state
Proposition 2. We have (i) Let u E .Ar (.0, lRNi ), r > 1 and let v : Q - RN2 be Lipschitz. Then u x v E
Ar(1,RN), N=N1+N2.
(ii) Let u E Ap(Q,1RN1) and v E Aq(fl,IRN2). If 1/r := 1/p + 1/p < 1 then u x v belongs to A"(Q,RRN).
4.3 The Join of Cartesian Maps
315
Next lemma in conjunction with the closure theorems for Cartesian maps will allow us to give sufficient conditions in order that BG,,,,, L .(l X R V = 0.
Lemma 3. Let v, vk E cartP(f2, IRS'), Vk - v weakly in AP and let w belong to cart4(!2, IRN) where 1/r := 1/p + 1/q < 1. Then
v Xw
Vk Xw
weakly in A'
.
In particular v D4 w E cart' (S?, RN) provided vk Pa W E cart' ((l, IRN)
Proof. Recall that
weakly in L' , if fk - f weakly in LP, g belongs to LQ and 1/r := 1/p + 1/q < 1. Taking into account Lemma 1 and applying the previous remark to fk := (DVk) and fg
fkg
M0,,,,
g := Ma,, (Dw) we then infer
weakly in L'
Ma (D v xw)
Ma (Dvk Xw)
.
This concludes the proof as the second part of the theorem is a trivial consequence of the weak closure of cart'. Proposition 3. Let v : Si --+ RN2 be a Lipschitz map. Then we have (i) u X v E cartr (Q, RN), if u E cart' (.f2, RN1) (ii) U X v E Cart' (Ql, IRN ), if u E Cart' (.(2, RN1)
Proof. The claim (i) follows at once as the current G,,Dav is the Lipschitz image of the boundaryless current Gu by
V : Rn+Nl
Rn+N
(x, y) '-' (x, y, v(x)) GUNV = V#Gu, and taking into account Proposition 2(i). In order to prove (ii) we consider the set
Cl := {u E cart' (Q, RN1)
I
u X v E Cart' (,(l, RN) }
It suffices then to show that C1 is closed under the weak convergence in A', so that Cart' (.(l, IRN1) C C1 .
To show this consider a sequence {uk} in C1 such that Uk -f u in A'. From Lemma 3 we infer uk Da v
u Xv
in A'
and it ca v E cart'(.(l, RN). Since each Uk cc v belongs to Cart'(Q, RI`'), it follows that also it ca v belongs to Cart'(,11, RN), hence it ca v E C1.
We are now ready to state the main result of this subsection.
3. Cartesian Maps
316
Theorem 1. Let p, q be such that 1/r := 1/p + 1/q < 1. (i) If u E carte(f1,l 1) and v E Cart" (fl,RN2) then u r v E cartr(fl,RN). (ii) If u E Carte (f1, R") and v E Cart" (fl, lN2) then u x v E Cart' (fl, RN)
Let moreover fl1 C I[8N1 be a bounded open set
(iii) If u E carte (Q, IRN1) , v E Cart" (.fl, RN2) and 9G,,, L W1 X f21 = 0
(3)
then BGu,,v L R"` X fl1 X RN2 = 0
(4)
Proof. (i) For any w E cart" ((2, RN2) we set
C2 := {v E cartP(f2,RN1)
V D4 W E cart'(f2,R N)}
I
From Proposition 3 we know that
Lip (0, RN1) C C2 . We shall now prove that C2 is sequentially closed with respect to the weak-AP convergence, hence
Carte (fl, RNl) C C2 .
This clearly proves (i). Let us prove that C2 is sequentially weakly closed in carte (f1,RNi) .
Consider a sequence {vk} C C2 weakly converging in AP to some v which of course belongs to cartP(Q,RN1). By Lemma 3 VkLW
VLaw
in A'
and v x w E cartr(fl,RN) (ii) For any w E Cart" (,f1,RN2) we set
C 3 := {v E carte(fl,1N1)
I
v x w E Cart'(Q,R N)}
Again from Proposition 3
Lip (fl, R' 2) C C3 and C3 is sequentially closed in cartP(fl, RN1). In fact if Vk -1 v in AP, we have v E cartP (f1,1f8N1) and vk La w - V N W
in A',
v x w E cart'(Q,
by Lemma 3. On the other hand Vk La w E Cart'(fl, EN), hence v x w c Cart'(fl, RN) by closure theorem.
4.3 The Join of Cartesian Maps
317
As Cartr(Sl,lfBN) is the smallest sequentially closed set which contains the class Lip (Sl, ]18N), we conclude
C 3 D Carte (0, RN, ) which proves our claim. (iii) In order to prove (iii), for any u E cartp(SQ, ]f8 N,) satisfying (3) we consider the class C4 :_ {v c Cartq(.f2,1RN2) u x v satisfies (4)1. I
It suffices then to show that (c) C4 D Cl (7 , RN2 )
(d) C4 is weakly closed in To prove (c) we consider associated to v e Cl(t2,]18N2) the map cart9((,1RN2).
V:
(x, y) E lR x RN,
--+
(x, (y, v(x))) E II8''1 x
RNl
x 1R.N2
.
As in the proof of Proposition 3 we see that GuD< = V#G,,,. For any (n-1)-form w with spt w C R x S21 x 118N2 we now compute
G,,,xv(dw) = V#G,,(dw) = G,,,(V#dw) = G,,(d(V#w)) and
spt V#w = V-l(sptw)- C V-1(lR x Sll x RN2) = Il8n x S21
,
consequently Gu(d(V#w)) = 0, i.e., 0. This proves (c). Observing that condition (4) is weakly closed, the claim (d) follows immediately from Lemma 3.
Remark 1. It is an open question to establish whether u D< v belongs to
cartr((2,R") or not under the weaker assumptions that u E cartP(Sl,RN1), v E cartq(S1, RN2) and 1/r := 1/p + 1/q < 1. We conclude this subsection with a simple but useful remark concerning the continuity of the join of Cartesian maps with respect to the weak convergence in Ll with equibounded cart-norms. Theorem 2. Let {uk } C carte (Sl, ]RN1), {vk } C cart4 (S2, RN2) where 1/r 1/p + 1/q < 1, be sequences such that Uk -s u weakly in
L', Vk - v weakly in L'
and
supI k
uk X Vk II.ar
< oc .
Then
Uk D< vk -' u D< v
weakly in Ar
318
3. Cartesian Maps
and u x v E cart''(.f2,RI"), if at least one of the two sequences {uk} and {vk} is contained in Cart' (Q, RNi) or in Cart4 (,f2, RN2) respectively. Finally, the claim remains true also for r = 1, if moreover we assume that the functions JM(Duk)] IM(Dvk)I are equi-integrable. Proof. It is an immediate consequence of Proposition 2, Theorem 1, and of the
closure theorem in cart, as Uk << vk - u Da v
weakly in Ll
Remark 2. Let u and v be two maps from (2 C Rn into RN. We can consider the map
u+v:Q-) RN
It is not difficult to see that all results in this subsection remain true with unchanged formulations and proofs if we replace > by +. In fact we could have proved all theorems for the sum, and then deduce them for the join as
ux v = (u,0)+(0,v) .
5 Notes 1 For the introductory content of Sec. 3.1.1 the reader is referred for instance to Stein [611, Chapter 1], Stein [612], Stein and Weiss [613], Torchinsky [634] and their bibliographical notes. For the theory of Sobolev spaces the reader may consult e.g. Adams [4] Maz'ja [463] Kufner, A. and John, 0. and Fucik, S. [416] Ziemer [686]. Theorem 2 in Sec. 3.1.1 appeared in Giusti [304]. A better estimate of the exceptional set in terms of Riesz or Bessel capacities can be found in Federer and Ziemer [231] Ziemer [686]. From such an estimate for instance it follows that 7- C`-' (,fl \Lu) = 0 if u E W"(Q), compare Remark 3 in Sec. 4.1.4.
2 The theory of pointwise differentiability of non-smooth functions, as well as the notion of approximate differential, goes back to the beginning of this century in particular to the work of Lebesgue, Khintchine, Denjoy, Rademacher, and Stepanoff among others, compare Saks [570]. Relevant contributions are due to Whitney [673] and Federer [226, chap. 3.1]. Examples of absolutely continuous functions in dimension n < 2 which are not differentiable on a set of positive measure can be found e.g. in Caccioppoli [128] Saks [569] Cesari [141] and Serrin [585]. In particular in Serrin [585] it is shown a function in W' "(B") fl C°(B") which is nowhere differentiable in a classical sense.
Pointwise differentiability properties of Sobolev maps from a modern point of view are discussed for example in Ziemer [686], compare also Stein [611, Chapter 8]. The basic LP-differentiability theorem was proved in Calder6n and Zygmund [131], see also Calder6n and Zygmund [132]. The Holder continuity of W"'-functions, p > n, is a classical well-known result due to Morrey [486] [487], while the a.e. differentiability seems to be proved by Cesari [141]. From these results the a.e. approximate differentiability of Sobolev maps follows easily.
5 Notes
319
3 Given an a.e. approximately differentiable function u, Federer in 1945 proved in Federer [220] [221] that the set of points of approximate differentiability of u can be decomposed as a countable family of measurable sets such that the restriction of u to each member of the family is Lipschitz. More precisely he essentially proved the equivalence of the claims (i) (ii) and (iii) in Theorem 3 in Sec. 3.1.4. In conjunction with Whitney extension theorem this yields that every a.e. approximately differentiable function is of class Ci except on small open sets, Whitney [673]. This way the theory developed in Sec. 3.1.3, apart from the estimates, could be regarded as a consequence of Theorem 3 in Sec. 3.1.4, that is, as a special case of the rectifiability theory of Federer, via Calderon-Zygmund result. However we preferred to present Lusin type properties for Sobolev functions in the context of Sobolev spaces theory as this provides us also nice estimates. 4 The proofs of Lusin type properties in Sec. 3.1.3 are inspired by Liu [440], in particular Theorem 7 in Sec. 3.1.3 and Theorem 8 in Sec. 3.1.3 are taken from Liu [440], in connection with Theorem 4 in Sec. 3.1.3 the reader may also compare Acerbi and Fusco [3]. Related papers are Michael [470] Goffman [312] Chen and Liu [148]. Lusin type properties for Sobolev functions with higher order derivatives are discussed in Zierner [686] and in Liu and Tai [442] [443]. The proof of Rademacher theorem we have given in Theorem 3 in Sec. 3.1.3 can be found in Morrey [490], compare also Simon [592]. For the proof of Whitney Theorem, Theorem 5 in Sec. 3.1.3, we refer e.g. to Whitney [672], Stein [611] Ziemer [686]. Sec. 3.1.4 closely follows Federer [226, chap. 3.1], compare in particular Federer [226, 3.1.8, 3.1.16, 3.1.17].
Some extensions of the theory to rectifiable sets of higher order of differentiability can be found in Anzellotti and Serapioni [50].
5 Since long time and even nowadays the concept of area, as well as representation formulas, area formulas or the change of variables formula, have been of central importance in many questions. A recent historical analysis of the different contributions seems to be missing, though it would surely be of interest. To quote only a few books we refer the interested reader to Saks [570] Rado [545] Rado and Reichelderfer [546] Cesari [142]. Eventually the most reasonable approach to area formulas was formulated by Federer [220], in terms of rectifiable sets. Our presentation in Sec. 3.1.5 follows in fact Federer's ideas. The reader may find interesting consulting also e.g. Besicovitch [84] Besicovitch [85], Goffman and Liu [315] [316], Goffmann and Ziemer [321], Nishiura and Breckenridge [514], Nobeling [521], Liu [441], Maly and Martio [450].
Motivated by Federer's approach, the notion of graph, or 1-graph, of an a.e. approximately differentiable map in Sec. 3.1.5 is modeled after a similar notion and first appeared in Giaquinta, Modica, and Soucek [279], see also Liu [439]. Theorem 2 in Sec. 3.1.5 is taken from Rado and Reichelderfer [546], while we could not trace back Theorem 4 in Sec. 3.1.5. It appears in Marcus and Mizel [458] Rickman [555]. For Remark 3 in Sec. 3.1.5 compare Reshetnyak [554] and also Morrey [490, p. 352] and Besicovitch [85].
6 The classes AP (17, RN) were essentially introduced in Giaquinta, Modica, and Soucek [279], where however it was assumed that their elements were Sobolev functions. The relationship between the old and new definition given here is formulated in Proposition 1 in Sec. 3.2.3. The present definition in terms of a.e. approximately differentiable maps makes clear the independence of the geometric property of rectifiability of graphs from the vanishing of their boundary in 17 x RN. Apart from the considerations of Sec. 3.2.4, the results of Sec. 3.2 are a reworking of similar results appeared in Giaquinta, Modica, and Soucek [280] [279].
320
3. Cartesian Maps
The classes A,,,(0, ') in Sec. 3.2.4 were introduced by Sverak [618] in connection with nonlinear elasticity. Still in connection with nonlinear elasticity, the notion of distributional determinant was introduced by Ball [63]. The div-lemma, Proposition 2 in Sec. 3.2.4 is due to Muller [500]. It is proved in Miller [503] that the singular part of the distributional determinant may concentrate on a set of prescribed Hausdorff dimension a, 0 < a < n, compare 9 in Sec. 4.2.5. One could also introduce a notion of distributional minors, in particular of distributional adjoint, Adj Du, but we have not pursue this point. For further information about this subject we refer to Ball [63] and Dacorogna [166].
The results concerning the isoperimetric inequality and higher integrability of the determinant are taken from Muller [501]. There one can also find examples showing the optimality of that theorem. The class An_l,n/(n-1) has been introduced in Muller, Tang, and Yan 504], where Theorem 1 in Sec. 3.2.4 is proved. Results in 5 in Sec. 3.2.4 are taken from Coifman, Lions, Meyer, and Semmes [156] [155] where general results connected to the theory of compensated compactness, compare Murat [506] [508] [507] Tartar [626] [627], are developed. A direct proof of Proposition 6 in Sec. 3.2.4, which does not use Fefferman-Stein's duality theorem, can be found in Chanillo [146] Chanillo and Li [147], compare also Wente [658]. Finally, for the proof of Fefferman-Stein duality theorem we refer the reader to Fefferman [232] Fefferman and Stein [233], compare also Torchinsky [634]. For a primer on Hardy spaces the reader may consult Semmes [583].
7 The results in Sec. 3.2.5 are partly new, and suggested by previous similar results in Sverak [618] and Muller, Tang, and Yan [504]. We have only slightly touched the problem of characterizing currents which are boundaries of i.m. rectifiable currents in cart' (0, R"). Two problems might deserve further studying. First characterize maps
in cart' such that 8G, = G,,,an, and, secondly, characterize the boundary space of cart' (.(2 RN )
8 After the results of Reshetnyak [550] [551], Theorem 1 in Sec. 3.3.1, Theorem 2 in Sec. 3.3.1 and Theorem 3 in Sec. 3.3.1 on the weak continuity of determinants, a new flowishing interest in the weak convergence of minors was motivated by the work of Ball [63] in nonlinear elasticity and Tartar [626] on the so-called compensated-
compactness methods. Other motivations were provided by the interest in studying variational problems for functionals with integrands depending on minors, compare e.g. Dacorogna [166] [164].
The relevance of the vanishing of the boundaries 8Guk L Q x RN in studying the weak continuity of minors was pointed out in Giaquinta, Modica, and Soucek [280] [279] where the classes of Cartesian maps were introduced. In Giaquinta, Modica, and Soucek [280], see also [279] we also proved the closure theorem for Cartesian maps relying on Federer-Fleming closure theorem. Later Muller [499] pointed out that a simpler proof, which in fact reminds of part of the proof of the rectifiability theorem, compare with Proposition 2 in Sec. 3.3.2 and Theorem 4 in Sec. 2.2.7, could be given directly.
9 The examples in in Sec. 3.3.1 3 in Sec. 3.3.1, due to Ball and Murat, and the example in in Sec. 3.3.1, due to Tartar, are taken from Ball and Murat [71]. The example in ® in Sec. 3.3.1 is taken from Mali [448], compare also Besicovitch [84] Nobeling [521] Goffman [312]. Modifications of this example provide sequences of smooth diffeomorphisms {uk } which converge weakly in W''p (Q) for p < n - 1 to the
identity but
ki
f det Duk dx < J dx Q
Q
.
5 Notes
321
More generally, it is proved in Hajlasz [337] that for each u in Ap,q with n - 1 < p < n
but q < n/ (n - 1) there exists a sequence {u,} c A,,q such that each u has range in a (n -1)-dimensional simplicial set and u converge weakly in Ap,q to u; in particular det Du,, = 0. Proposition 1 in Sec. 3.3.1 and Proposition 2 in Sec. 3.3.1 were proved by Ball [63], see also Ball, Currie, and Oliver [70]. Dacorogna and Murat showed in [170] that Proposition 2 in Sec. 3.3.1 is optimal. Finally we mention that biting type convergence of minors is studied in Zhang [684]. 10 The classes CartP(0, RN) were introduced in Giaquinta, Modica, and Soucek [280], see also [279]. The content of Sec. 3.4.1 is classical, the reader may consult e.g. Banach [73] Natanson [512]. The results of Sec. 3.4.2 are slight improvements of results due to Mali [447]. Finally the content of Sec. 3.4.3 is new, it will be relevant in studying the composition of weak diffeomorphisms in Vol. II Ch. 2.
4. Cartesian Currents in Euclidean Spaces
Let {uk} C C1(0,RN) be a sequence of regular maps from a bounded domain Q into RN satisfying (1)
SUp{II Uk II LP(fl,R") + II k
M(Duk) II LP(.f2,A,R +N)} < 00
for some p > 1. We saw in Ch. 3 that there exists a subsequence, that for the sake of simplicity we call again {uk}, and a map u E cartP(d2,RN) such that
Uk - u (2)
weakly in
Ma(Duk)
W1,P(d?,RN)
M.3, (Du)
weakly in
LP(Q)
for all a,)3 with Ial + 101 = n, Q > 2. Furthermore, we saw that (2) implies the weak convergence in Dn((2 x RN) of the currents Guk to the n-current Gu and also the equiboundedness condition (1), i.e.
Guk - Gu (3)
in Dn (.(2 x RN)
SkP{II Uk IILP(Q,RN) + II M(Duk) IIL P(n,AnR
+N)} < 00
.
Actually, we saw that (2) and (3) are equivalent. All those nice properties are lost in general if we replace the bound (1) by the weaker bound in Ll (4)
SkP{II Uk II L1(.n,RN) + II M(Dnk) II L1(f2,AnRn+N) } < 00 .
The lack of weak compactness of bounded sets in Ll forces us to embed Ll into the space of measures with bounded total variation and to replace, for instance, W1'1 (.(2,RN) with the space of functions with bounded total variation BV(f2, RN). Therefore from (4) we can only infer the existence of a subsequence of {uk}, that we again call {uk}, of a function u E BV(Q, RN) and of measures p , IaI + 101 = n, Q > 2 such that (5)
Since
Uk - u
weakly in
Mg(DUk) - µap
BV(.(2,RN) as measures.
324
4. Cartesian Currents in Euclidean Spaces
M(Guk) = f M(Duk)I dx
,
the n-currents Guk are equibounded in mass; again passing to subsequences, there exists a current T in D,,,(Q X RN) with finite mass such that
Guk -k T
(6)
in Dn (Q x
But in general u, ,u , and T are unrelated in the sense that we cannot recover 4110 from u and T from p and u, as shown by the following examples, compare also Sec. 4.2.5.
0 Bubbling of circles. Consider the sequence of C1-maps Uk from the interval (-1, 1) into Sl C R2 defined by (
(cos kO, sin k8)
0 E [0, 27r/k]
(10)
0E(-1, 1) \
[0, 27r/ k]
.
Clearly,
S uP k
Ju kI+ILk1)dO
Uk -s uo. - (1, 0)
sup2(1+ k)
,
that is Uk -+ (1, 0)
Uk -i (0, 0)
strongly in Ll ((-1,1), IR2) as measures.
But the 1-dimensional currents Guk, which are just integration of 1-forms over the curves 0 -* (0, uk(0)) in R3, clearly converge to the current integration over the graph of the constant map u,, plus integration over the circle S1 which lives above 0 = 0 G,,,k -1 Gz,,, + 8o x S1 I .
In this case, the graph of the limit map u,,,, is only a part of the whole limiting current. Note also that, if we regard the Ilk as maps from S1 into S1, the limit map u,,,, has degree zero, though all the Uk have degree one, while the limit graph T has "degree one". Also the length of the curves (0, uk (B)) tends to 2+2vr which is exactly the mass of T, while the length of the curve (0, u,,,, (B)) is 2.
The previous example shows clearly that the weak limit of Guk is not the graph of the map u,,,,, nor of any other map, and cannot be recovered from the map u,,,,. While in codimension one the vertical part of the "graph" of a BVfunction u is uniquely determined by its pointwise values, compare Sec. 4.1, in
4. Cartesian Currents in Euclidean Spaces
325
the vector valued case we are forced to regard weak limits of Guk as currents in Q x IRN
In Sec. 4.2.1 we shall see that limit currents T of sequences of smooth maps satisfying (4) belong to the class of Cartesian currents defined as
{ T E Dn(.fl x RN)
cart(,R x RN)
I
T is an i.m. rectifiable current
in.(lxIRN, M(T) 0 }
Such a class plays a relevant role in the study of variational problems for vector valued maps as we shall see later. Its properties will be discussed in Sec. 4.2 where, in particular, we prove a closure and compactness theorem. There we shall also prove a structure theorem for Cartesian currents T, which states that each T has the form
T = Guy, + S where GUT is the current associated to the 1-graph of a function uT in BV(,fl,RN) and S is a vertical current, i.e., 7r#S = 0. Rightly the class cart(. x RN) may be regarded as the counterpart of the classes carte (.fl x 1RN ), p > 1, similarly to as BV(Q) is the counterpart of Sobolev
spaces W1'p(Q), p > 1, in the scalar case. However in the vector valued case cart (fl x R') turns out to be the natural substitute even of W'J'(Q, RN), at least for integer values of p < min(n, N), as we shall see in the sequel of this monograph and as the following example already shows. 2 Bubbling of spheres. Let a : S2 C 1R3 -+ R2 denote the stereographic projec-
tion from the south pole PS := (0, 0, -1) into R2. Its inverse c-1 : R' ---t S2 being conformal satisfies Dv-ll2
2
= IM121(Dc-1)I
hence we have 1
2
f Do,-1
12
fM(2)(Da_1) I = ?-l2 (S2) = 47r .
dx R2
R2
Consider now the sequence of maps
Uk(X) := Q-1(kx)
x E IR2,
k = 1, 2,-
Then clearly 1 2
J
IDukI2dx = 47r
`d k
R2
and
uk(x) -> uo. (x) Uk -1 uoo
south pole pointwisely in 1R2 . weakly in W1,2
326
4. Cartesian Currents in Euclidean Spaces
On the other hand, from the isoperimetric inequality for parallelograms we have
IM(2)(Duk)I s 2 IDuk12 , whence we infer sup{ II uk IILI(n,I2) + II M(DUk) IILl(n,1R2) } < 00 ,
P CC 1R2
k
Notice that we have no better control on the two by two minors. Consequently the currents Guk have locally equibounded masses, and it is not difficult to show that
Guk - Gu- + 5o x Q S2
in D2 (1R2 X 1R3) ,
compare Sec. 4.2.5.
Regarding the maps Uk as maps from S2 into S2, uk(x) := Q-1(kv(x)), similarly to 1 , all maps Uk have Brower degree one while u,,, has degree zero. Though all Uk have Dirichlet energy equal to 47r, the Dirichlet energy of u,,o is zero. On the contrary the current Gv,_ + do x Q S2 I has "degree one". We therefore see that maps with equibounded Dirichlet integrals provide examples of the same nature as in M, and their limits are better described in the context of Cartesian currents than in the Sobolev spaces W1'2(1R2,1R3) or W1,2(S2, S2).
Due to the relevance of functions of bounded variations in dealing with Cartesian currents, we shall begin presenting in Sec. 4.1 the theory of BV-space. After a few simple but important functional results, as for instance the approximation theorem, we shall mostly deal with structure property of BV-functions. In particular we shall present De Giorgi's rectifiability theorem for Caccioppoli sets,
i.e., sets whose characteristic function are in By, in Sec. 4.1.3, and, based on it, a structure theorem for BV-functions in Sec. 4.1.4, describing the pointwise behaviour of BV-functions. Sec. 4.1.5 provides a link with Federer's approach to BV-functions in terms of boundaries of subgraphs; while in Sec. 4.2.4 we shall
see that in codimension one BV-functions can be identified to Cartesian currents. Typical pointwise properties are best visualized by means of the classical Cantor-Vitali function that we shall discuss in Sec. 4.2.4. Sec. 4.2 is dedicated to the study of Cartesian currents. There we prove closure, compactness, and structure theorems, and we present a few examples in Sec. 4.2.5. Cartesian currents in codimension one are discussed in Sec. 4.2.4, and, finally, in Sec. 4.2.6 we discuss radial functions and radial currents. In Sec. 4.3, after discussing n-dimensional currents in R and in particular proving the so-called constancy theorem, we shall develop, on the basis of those results, the degree theory for currents, and particularly for Cartesian currents. In Sec. 4.3.2 we shall in fact see that currents and BV-functions provide a natural setting for degree theory. We shall then see that the degree is continuous with respect to the weak convergence of currents. Contrary to weak convergence in
1 Functions of Bounded Variation
327
Sobolev spaces, in particular we shall see that the weak convergence of Cartesian
currents preserves the degree. Further results on degree will be presented in Sec. 4.3.3 and Sec. 4.3.4.
1 Functions of Bounded Variation In this section we shall present some of the main results of the theory of functions of bounded variation in dimension larger than one. The class of such functions, denoted BV (.(2) , is defined as the family of functions u : Q C IEBn -> 1[8 with distributional derivatives which are measures of finite total variation in Q.
We shall see that BV-functions arise naturally as weak limits of smooth functions with equibounded Ll-norms and L1-norms of the gradients. Moreover we shall see that every function u E BV(.(2) can be approximated "strongly" by smooth functions in the sense that for every u E BV (fl) there is a sequence of smooth functions Uk in .f2 such that
u---u
f n
in L' (,f2)
IDuk(x)I dx
f jDul D
where JIDUI D
denotes the total variation of the vector valued measure Du in (2. An immediate consequence of this result is the extension of Sobolev and Poincare type inequalities , as well as of compactness properties, to BV-functions. In particular one sees that bounded sets in BV (.(2) with respect to the natural energy or norm
f Jul dx + n
Jn
I Dul
are relatively compact in BV(1(2). This makes BV(f2) the natural space to work with, instead of W1'1, when dealing with energies with linear growth at infinity as for instance the area functional
f
1 + IDu dx
D
for nonparametric surfaces u :(2 ---* Iii Those as well as other basic properties of B V-functions will be discussed in Sec. 4.1.1. There we shall also prove the coarea formula for BV-functions. An important subclass of BV(f2) is that of characteristic functions XE of sets E for which DXE is a measure in (2. Sets E for which XE E BV.,,(') are called
328
4. Cartesian Currents in Euclidean Spaces
Caccioppoli sets and the total variation of DXE in 0, IDXEI (fl), is called the perimeter of E in ,(2 and denoted by P(E, f?). Caccioppoli sets were introduced in the fifties by Caccioppoli and named after him by De Giorgi, who essentially developed a full theory, which we shall present in Sec. 4.1.2 and Sec. 4.1.3. Their relevance is mainly, though not only, in connection with the problem of defining a notion of generalized (n - 1)-surface and of area suited for the purposes of calculus of variation, i.e., in order to find surfaces of minimal area, or, in other
words, in order to identify limits of sequences of smooth (n - 1)-parametric surfaces in .(2 with equibounded areas.
The idea is to regard oriented (n - 1)-surfaces in RI as boundaries of ndimensional sets E. The naive idea would then be to take as area of aE, or perimeter of E, just the 7-1n-1-measure of OR However one immediately realizes
that one can make the topological boundary aE of E as large as one wants, by just changing E by a set of measure zero. In other words the topological boundary is not the right object in order to identify surfaces of finite area. Caccioppoli sets, and the perimeter, instead do it, as they are defined by duality and therefore depend only on the equivalence class of E. Clearly, identifying the
(n - 1)-surface "boundary of E" together with its "differential" properties is related to the possibility of extending the classical Gauss-Green formula (1)
- J divg dx =
J(g, v) d7-l"-1
b g E C,1(.(2 Rn)
aE
E
to Caccioppoli sets, and more precisely to the question of identifying an (n -1)rectifiable set "M" together with an "inward normal" v to E so that (1) holds. From the definition of Caccioppoli set we have
- f Dig dx =
fYdDixE
d g E C"(0)
.
E
Defining the reduced boundary of E, a-E, in the sense of De Giorgi, as the set of points x at which the Radon-Nikodym derivative of DXE with respect to I DXE I exists
d DXE
d IDxEI
:= n(x,E) and I n(x, E) I = 1
we shall see that a-E is locally an (n - 1)-rectifiable set with tangent plane normal to n(x, E) for 7.1n-1-a.e. x E a-E and DXE = n(x, E) 7.1n-1 L a-E . This is the celebrated De Giorgi's rectifiability theorem for Caccioppoli sets. And now, generalized (n - 1)-surfaces are identified as the reduced boundaries of Caccioppoli sets, and their perimeters as the 7.1n-1-measure of the reduced boundaries. Existence of minimal surface of dimension n - 1 in Rn, under for instance prescribed boundary value, clearly becomes a trivial fact, taking into
1.1 The Space BV (,(2, R)
329
account the semicontinuity of the total variation, i.e., of the perimeter, and the compactness property of the class of Caccioppoli sets. We shall see in Sec. 4.1.3 that the reduced boundary a-E agrees apart from a set of zero (n - 1)-dimensional measure with the so called measure-theoretic boundary of E, B,2E, which consists of the points which are neither of density zero for E (i.e., of rarefaction for E), nor of density one for E (i.e., of rarefaction for the complement of E). Later in Sec. 4.1.4 we shall then show that in fact
8-E = 8µE = a*E
H`1-a.e.
where a*E is the set of jump points , defined again in a measure-theoretic way, for XE with jump equal to one. In Sec. 4.1.4 we turn back to general BV-functions proving a fine structure theorem which states that, accordingly to our simple one-dimensional idea, we can decompose every BV-function in a part which is almost everywhere approximately differentiable, in a jump part which lives on a locally (n - 1)-rectifiable set, and in a Cantor part. Finally, in Sec. 4.1.5 we shall prove that a measurable function is in BV((2) if and only if its subgraph is a set of finite perimeter in ,(2 x JR. This way we shall link De Giorgi's approach with Federer's approach to BV-functions. Such a result will be used in Sec. 4.2.4 when proving that the class of Cartesian currents in codimension one agrees with the class of BV functions . It has to be mentioned that, contemporary to De Giorgi's theory of Caccioppoli sets, related measure-theoretic ideas were being developed by Federer and by Reifenberg in order to deal with general k-dimensional surfaces in JRn. Those ideas eventually led to the theory of integer multiplicity rectifiable currents and varifolds.
In this section we address our attention mainly to structure properties of BV-functions. We shall report on other results, without proof, in the notes of Sec. 4.4.
1.1 The Space BV (,fl, lid) Let 0 be an open subset of lien, n > 1. Definition 1. A function u E Lion (0) is said to be a function of locally bounded variation, u E BVoc(f2), if its distributional derivatives, Diu, i = 1, ... , n, are signed Radon measures.
In other words there exists a vector valued measure Du in M((2,Rn), Du = (Di u, ... , Dnu), such that (1)
J f2
< g, d Du > = -
J (2
u divg dx n
for any vector field g E Q1 (,(2, Rn). Here div g = EDigi, as usual. i=1
330
4. Cartesian Currents in Euclidean Spaces
We say that a function u has bounded variation in .(2, if its total variation IDuj is finite in .fl, IDuI (.(2) < +cc. We shall also write instead of IDuI(f2),
fdIDuJ n
and even with some abuse of notation
f
IDui
n
Definition 2. The space BV (.(2) is defined as the space of all functions in L' (n) with bounded variation.
Obviously every function u E C' (0) belongs to BV0,((1). In fact, integrating by parts, we find r
-J udivgdx = n
/
n
EDiug'dx ,
Jn z=1
Du = dLn L Du(x)
.
Similarly we have W11' (f2) C BV0,:(f). Notice that actually
W1'1(.(2) # BV(Q) as shown by Heaviside function in (-1, 1) or Cantor-Vitali function, compare Sec. 4.2.4.
As we saw in Sec. 1.1.4, by Riesz's theorem, the total variation of Du, IDuI can be computed as the mass of the vector valued measure Du. If U C .(2 is an open set, then, compare (7) in Sec. 1.1.4, (2)
I DuI (U) := sup{
Jn
u div g dx 19 E C,, (U, R ),
11g J
< 1}
.
Therefore we can state equivalently
BVoc(f) = {u E L oc(Q)
I
Mu(Du) < oo V U open U CC ,(2}
BV(dl) _ {u c L'(Q) I Mn(Du) < oo}
.
From (2) one easily infer
Proposition 1 (Semicontinuity). Let {uk} be a sequence of L' functions in .(2 which converges weakly in L' to u, uk u in L1. Then Dul(S2) < 1im of IDukl((2) k-oo
In particular u E BV(l), if all Uk belong to BV(f2) and have equibounded total variations.
1.1 The Space BV (0, R)
331
The next proposition is just a rewriting of general properties of Radon measures, compare Sec. 1.2.1.
Proposition 2. Let uk, k = 1, 2, ... , u be functions in BV (.(2) (i)
If uk - u as distributions, then Duk
.
Du as measures if and only if
sup IDukI(U) < c(u) < oo,
for all U CC 0
k
(ii)
If Duk - Du as measures and limk.,,,, IDuk (0) = IDuI (,fl), the following equivalent three claims hold IDuj as measures, a) IDukI b) IDuk I (B) I DuI (B) for any Borel bounded set B in .f2 such that I DuI (8B) = 0,
c) for any open set A C P we have IDuK(A) < liminfk_,,,,, Dukl(A) and for any compact set F C 12 we have I DuI (F) > lim supk_. I Duk I (F) . In particular it follows from (b) that IDukI (B(x, r)) -+ IDuI (B(x, r)) for all x E .f2 and almost every r for which B(x, r) C P.
It is easy to show that BV (.f2) equipped with the norm IIUIIBV(Sz)
(3)
IIUIIL1(fa)+IDuK(f2)
is a Banach space. However such a structure is of no real use since C' (.f2) fl BV (17)
is not dense in BV (.fl) and BV (.fl) is not separable.
By considering the family F of characteristic functions Xa of the intervals 71 (a,1), a E (0,1), we see that F C BV(0,1) and for a # Q we have IIX
-XOIIBV(rz) = 2+Ia-QI
This of course shows that BV(fl) is not separable. Since the closure of C' (() fl BV (.fl) with respect to the norm (3) is just the Sobolev space W" (() and W',' (D) C BV (.fl), we see that C' (.fl) fl BV is not dense in By. . 34
It is instead more convenient to regard BV(12) as subspace of L', as we did in the semicontinuity theorem, or endowed with the weak*-convergence or the convergence of measures which is induced from the immersion of BV (Q)
into the space of vector valued Radon measures M(Q,R'+') by the mapping u --+ (u, Du).
Quite a number of useful properties of BV-functions are consequences of the following approximation theorem.
Theorem 1 (Approximation by smooth functions). We have (i) Let u E BV°°(Q). Then there exists a sequence {uk} C C°°(Q) such that
in L o°(.fl) Uk -+ u dC L IDukI - IDuI as measures in P
4. Cartesian Currents in Euclidean Spaces
332
(ii) Let u c BV((l). Then there exists a sequence {vk} C C°°(.(2) such that
in L1(Q)
Vk --+ u
-.
and
JDuj(,n)
.
12
Moreover, if .(2 is unbounded, we can choose the vk's to be zero near infinity. Proof.
(i) p,,_ (X) := o
Let p be any symmetric mollifier. For or > 0 and u E L1 ° set p(x/or) and ua := cpa * u, where is := u in .(2a, u := 0 outside ,(2Q
and .(2a
{x E (I dist (x, 8.(2) > v} n B(0,1/a)
.
Then obviously ua -f u in Li (D) as a -+ 0 and it suffices to show that
J(x)IDu(x)Idx f o r any c
-V
E C°°(12), w>0
As
f cp(x)dIDuI = sup{ J udivg 19EC'(,(2), Ig(x)l <W(x)1 we infer at once
f(x)dDu l < 1im ionf
J
w(x) IDuQ(x)I dx
.
Thus we only need to show that (4)
limsup f cp(x) lDu,(x)ldx .< J(x)dDuh
For g E C': (0) with Ig(x) I < W(x) and or < dist (spt g, an) we have
f dx
- J ua(x) divg(x) dv = - J(pa * u) divg dx - J u (pa * div g) dx = - J u div (pa * g) dx .
(5)
For any e > 0, since Pa * I9(x) 1
I pa * 9(x) I
_<
cP(x) + E
if we take a close enough to zero, we can estimate the right hand-side of (5) by
WJ
(W+E) IDuI
1.1 The Space BV (fl, IR)
333
where
WQ :_ {x E .R
I
dist (x, spt cp) < v}
,
getting this way
f(+E)IDuI.
f dx <
Wa
Taking now the supremum in g and letting E - 0 we get (4). (ii) We still use a mollification procedure but stepping down the size of the mollification when going to the boundary. More precisely, let {Qk}kEN be an increasing sequence of open sets for which h o CC Q, CC Qk+i, Uk.(lk = (2 and define Al := ,(21, Ak := 12k \ 72k-2 for k > 2. The open sets Ak form a locally finite covering of ,(2 so that there is a partition of unity {cpk} associated to the Ak's. Choose now a sequence {Ek} in such a way that
SPt (W * Pek) C Qk+1 \ ?7k-3)
(2-1 = 0
where p is a standard symmetric mollifier and pe(x) = e-" p(x/E). Then the function
u := E(Wk) * Pek k
is well defined and belongs to C°°(.0). We shall now prove that for every S > 0 we can choose {Ek } in such a way that
II u - i 4L1(D) < S
(6)
and
f IDii(x)ldx < DuI (Q) + d
.
Sz
This of course implies that one can construct a sequence {vk} for which
Vk -* u
in L1(h2)
and
lim sup k-.oo
f IDvk(x)Idx < IDul(f2) S2
The semicontinuity of the total variation with respect to the L'-convergence then yields IDul(.f2) < liminf I Dvk(x)I dx k-oo
f
and this completes the proof of (ii). Let us prove (6). We have
u - u = 1:((uWk) * Pek - ucpk) k
so that
4. Cartesian Currents in Euclidean Spaces
334
II
- U IILI(.a)
<
II (W) * Pek - ucPk IILI(0) k
Therefore we get the first claim in (6), if we choose {Ek} in such a way that II (u(Pk) * PEk - (Wk) IILI(.fl) < b/2k+1
(7)
On the other hand in the sense of distributions we have
Du = E(Ducok) *Pek +>((uDPk) *Pek -uD(pk) k
k
as >D`Pk = 0. Therefore, choosing Ek in such a way that (7) holds and k
I I (u DcOk) * Pek - u DcPk II LI (f) < 6/2k
tl
we also get II DuIILI(a)
j:II(Duck)*Pek IILI(Q)+8. k
To conclude the proof of (6) we now observe that for any g E C' (Q, Rn), II g Il < 1, we have
J .a
<(Ducpk)*PekI9>dx =
fk(
* Pk) dDu < fk
dlDul
thus
fkdDUI
(Du Wk) *Pek HL' S
= Duk(,2) .
k0
k
L7
Remark 1. Notice that the sequences {uk} and {vk} in Theorem 1 also satisfy dGn L IDiuk l
i = 1, ... , n
IDiul
as measures in (2, and
fIDivkIdx
IDiul(Q)
Therefore
dGn L Duk dGn L Dvk
dGn L IDivkl dGn L IDvkl
Du Du IDiul
i = 1...
IDuI
in the sense of measures in 12, because of Proposition 2.
,n
1.1 The Space BV (.(2, ]R)
335
We shall now discuss a number of interesting and useful results which can be easily obtained by the use of Theorem 1.
A variational characterization of BV-functions. An immediate consequence of the semicontinuity and approximation theorems is the following characterization of BV-functions
Proposition 3. A function u belongs to BV(Q) if and only if there exists a sequence {uk} of smooth functions such that
Uk -1 u in L' (D)
with
sup k
Duk I dx < oo .
J
This claim may be formulated as follows. Consider the total variation functional
V(u,0) := f 12
with domain the functions u E C'(S?) with V (u, ,(2) < oo. As we have seen V (u, ,(l) is lower semicontinuous in C' (S?) n {u V (u, () < oo} with respect to the Ll-weak convergence. Thus, following Lebesgue, we can extend it to Ll I
as the highest lower semicontinuous extension, by setting for any u E Li (.f2) (8)
V(u,.fl)
inf{lim nfJ I Duk(dx k-oc
I
Uk E Cl(.(2),
S-2
uk -+ u weakly in Ll (.(2) }.
Clearly for u E BV(f2) we have IDul(.(2) < V(u,,fl). Proposition 3 actually says
Proposition 4. In L1 (Q) we have (9)
V(u,0)
IDul(Q) 11 +oo
if u E BV(.(?) otherwise.
This way BV (.fl) is characterized as the proper domain of V (u, U),
BV (,f2) = {u E L' (.fl) I V (u, .fl) < oo}
.
3 Sobolev and Poincare inequalities. The approximation theorem is a useful tool to prove in a very short way Sobolev-Poincare type inequalities and embedding theorems for BV-functions. For instance, for each u E BV(R') we can find a sequence {uk} C CoO(Q) such that
336
4. Cartesian Currents in Euclidean Spaces
Uk -+ u in L1
,
J Duk I dx
ID K
Uk --+ u a.e.
Rn
From Sobolev embedding theorem for smooth maps, we then infer II uk IIL'n/("-1)
<_ c(n) II Duk IIL=(R
)
and passing to the limit: For any u E BV (JPJ ) (10)
II U IILn/cn-1)
<_ c(n) DuI(
In particular BV(W1)
--,,
Lnl (n-1) (Rn)
with continuous immersion. Similarly, for a smooth (Lipschitz) domain .fl in IRn we get (11)
c(n,1?) (II U IIL1(n) + I DuI (Q))
II u IILn/(n
The same procedure yields Sobolev-Poincare inequality. Let u E BV(Q). Set
f u(y) dy ,
UX,P
B(x, p) C D .
B(z,P)
Then
f
I u(y) - ux,P I
dy)1-1/n < c(n)
P1-n
DuI (B (x, P))
B(x,p) or
Poincare inequality. Let u E BV(B(x, p)). Then
f
u(y) - ux,Pl dy <_ c(n) P I DuI (B(x, P))
B(X,P)
4 Compactness in By. The approximation theorem in conjunction with Rellich's theorem for Sobolev spaces W1"1 or Wlo'. yields immediately
Proposition 5. Let {uk} C BV3,(.Q). Suppose that sup II uk II BV(U) < K(U) < oc k
1.1 The Space BV (f2,1 F)
337
for any open set U cc (2. Then there exist a function u E Bv°,,(.f2) and a subsequence {uk,, } of {uk} such that uk;
Duk;
in Loc (Q)
u
-->
-
Du as measures in f2
If moreover {uk} C BV(.fl), .f2 is Lipschitz and SUPk I1 Uk Bv(n) < oo, then there exist u E BV(fl) and a subsequence {uki} of {Uk} such that I
u
Uk;
Du
Duke
in L1(f2) as measures in (2
.
In other words the immersion BV (.f2) -+ L1(f2) is compact, provided an is sufficiently smooth. Actually one can easily see that the immersion BV (.(2) LP (n) is continuous for p < n/(n - 1), and compact for p < n/ (n - 1).
5 The coarea formula for BV-functions. We conclude this subsection proving an extension of the classical coarea formula for smooth functions. In order to do that we first anticipate a remark which actually belongs to next subsection.
Let E be a smooth subset of ][fin, say with boundary of class C2 and let g E C' (.f2, Rn). From the classical (12)
J
XE div g dx =
Gaussforfmula we infer
fdivg E
=-J
v g dl-['i-1
BEnQ
where XE is the characteristic function of E and v denotes the inward unit normal to M. From (12) we deduce at once DxE I (,fl) < 9-1n-1(0E n (2)
.
On the other hand, extending v to a smooth, say C', function v with Ivl < 1 in 1R
we find, using (12) with g = 077,
J Mnfl
77 dfn-1 = -
J
XE div (L77) < I DXEI ('a)
for any 77 E Cc°(S2) with 1771 < 1, thus
hn-1(aE n ,fl) < DXEI(0) and we conclude (13)
xn-1(aE n ,f2) = IDxEI (Q)
Recall now the classical coarea formula for smooth function u, say C°° +00
(14)
f
= ffln_1(Qfl{XEQ u(x) = t}) dt I
_C0
.
4. Cartesian Currents in Euclidean Spaces
338
By Sard's theorem for a.e. t the set {x E (2I u(x) = t} is a smooth hypersurface which agrees with the boundary of the upper-level set
Et(u) :_ {x E .(l
I
u(x) > t}
.
therefore by (13) we can write (14) as +00
fDu(x)dx =
(15)
fDXE()I(n)dt.
n
-00
We shall now prove that (15) actually holds with obvious variants for any u E BV(Q). First notice that for any function u E L' (,(2) we have
f
+OC
(16) u(x) = u+(x) - u-(x) :=
XEt(u)(x) dt -
](1_xE)(x))dt.
Let now ( E C,1 (9,1R ). Taking into account that div ( has zero mean value in f2 we infer from (16)
r
J.a
0 u(x) div (x) dx = JdtfxE(U)(x)div(x)dx n
0 00
- J dt
f
(1 - XEt(u)(x)) divC(x) dx
n
-00
+00
f f dt
X_-t(u) (x) div ( (x) dx
n
-00
which for u E BV(d2) can be written as +0"
(17)
fdDu
=
fdtfdDxE(U) -00
Theorem 2 (Fleming-Rishel). Let u E Li ,,(,f2). Then +oo
(18)
DuI(Q) = fDxE(U)I(n)dt. -00
In particular u c BV (fl) if and only if the function
1.1 The Space BV ((l, R) t --+
339
(0)
I
is summable on T1 and for almost every t
IDXEt(u)I(f2) < 00 Proof. Taking the supremum for all S E C' (Q, lRn), I(I < 1 in (17), we get +00
fIDxEuIndt.
I Dul (Q) <
-00
To prove the opposite inequality we apply the standard coarea formula for smooth functions to uk, where Uk ---> u in L' and (19)
JIDukIdx
>
IDuI(Q) ;
.a
thus
+00
f I Duk I dx
(20)
f IDXEt(uk)I(Q)dt -00
d2
Now we observe that by (16) +00
uk(x) - u(x) =
f
(XEt(uk)(x) - XEt(u)(x)) dt
-00
and actually +oo Iuk(x) - u(x)I =
JIXEtuk)(x ) - XEt(u)(x)I dt -00
since sign(uk(x) -u(x)) = sign(XEt(uk)(x) -XE,(u)(x)) for all t. The convergence in L' of Uk to u then yields
in L'(Q)
XEt(uk) -ti XEt(u)
for almost every t. Passing to the limit in (20), using (19) and the semicontinuity of the total variation we then get +00
IDuK(Q)
f
um. IDukI(n) =
DXEt(uk)I(Q)dt
-00
f
+00 ,
-00
which concludes the proof of the theorem.
11
340
4. Cartesian Currents in Euclidean Spaces
An immediate consequence of the coarea formula is that max(u, t) E BV (Q), if u E BV (Q), moreover
ID max(u,t)I(Q) = IDul(Et(u)) Consider in particular the truncation of u E BV(f2) ut(x) :=
t
if u(x) > t
u(x)
if - t < u(x) < t
-t
if u(x) < -t
and set ht (x) := u(x) - ut(x)
.
As ut E BV(Q), we find +00
JDhtl(fl) = f IDXE..(ht)I(Q) = -00
f
DXEt(u)I(Q) ds
jsI>t
and consequently, taking also into account that ht converges strongly in Ll to zero we can state
Proposition 6. We have 0 as t -> oo, i.e., the total variation cannot concentrate near oo. infinity, i.e. for Jul (ii) BV(Q) n L°°(1l) is dense in BV(12) with respect to the norm topology. (i) IDhtI(0)
1.2 Caccioppoli Sets Let E be a measurable set in Ilk' and let ,(l be an open set in r. Definition 1. We say that E has finite perimeter, or locally finite perimeter, in ,(l if its characteristic function XE belongs to BV(Q), respectively to BV°°((l). The perimeter of E in 0 is defined as
P(E,.fl) := IDXEI(Q) Definition 2. A set E with locally finite perimeter in R''1 is called a Caccioppoli set.
Clearly P(E, ) defines a measure, therefore for any Borel set B we can compute P(E, B), i.e. the perimeter of E in B. From the definition of BV-functions it is immediately seen that Caccioppoli sets are exactly the sets for which the following Gauss- Green formula holds
1.2 Caccioppoli Sets
- f XE Di9(x) dx =
(1)
f9dDixE
341
d g E C, (III"), i = 1, 2, ... , n .
From the differentiation theory for measures we infer that
DXE = IDXEI L n(x, E) where
n(x,E) :=
(2)
lira DXE(B(x,p)) P-.O+ IDXEI(B(x,p))
is well defined and n(x, E) I = 1 for IDXEI-a.e. point x. Thus (1) can be written as
(3)
r
-J div g dx =
f
J
9Z (x)ni(x,E)dl DXEI
d9EC'(3[2",Rn)
This justifies the terminology of inward normal for n(x, E). The set of points x where the inward normal n(x, E) exists plays an important role. Thus we set
Definition 3. The reduced boundary (in the sense of De Giorgi) a-E of a Caccioppoli set E is defined as (4)
a-E
{x E R'
I
DXE(B(x, p)) =: n(x, E) P'-'0+ DXEI (B(x, p))
3 lim
and In(x,E)I = 1} Notice that
IDXEI(R'\ a-E) = 0. Consequently, a-E is IDXEI-measurable and (5)
IDXEI = IDXEI L a-E .
For convenience we set n(x, E) = 0 for any x 0 a-E. We already saw that
P(E, .(2) = Hn-1(aE n fl) if E has smooth boundary. It is immediately seen from the classical GaussGreen formula that in this case n(x, E) agrees with the classical inward normal
to 3E, a-E = aE, and IDXEI =
Hn-1 L OE .
However, if E is not smooth this may not be true and we may even have P(E, .(7) < oc but -H--1(aE n .fl) = oo, see [U below. But first, let us restate in terms of Caccioppoli sets the compactness theorem for BV-functions and the semicontinuity result for the total variation.
342
4. Cartesian Currents in Euclidean Spaces
Proposition 1. Let {Ek} be a sequence of Caccioppoli sets such that sup P(Ek, .(2) < c(.(2) k
for any bounded open set f2. Then there exist a Caccioppoli set E and a subsequence {Ek, } of {Ek} such that XEk; --i XE
in Ll c(R''),
DXEk, -k DXE
as measures
and
P(E, .fl) < liminf P(Ek;, 0)
.
0 Let .f2 be the open unit cube in R', n > 2, and let {qk } be a denumeration of the points in Q with rational coordinates. We select a sequence {Pk} of positive real numbers such that cc
00
Epk-1
B(qk, Pk) C .f2,
0Pk<1.
k=1
k=1
For any h the sets h
Ah
U B(qj, pi) z=1
have piecewise CO° boundary and obviously h
P(Ah.a) < xn-1(8B(0,1))
P2 -1 i=1
From Proposition 1 we therefore deduce that the open set A := U°_1B(gti, pi)
is a Caccioppoli set. On the other hand, since A is dense in Q and Cn(A) < 1, we have
G'n(aAn.fl) = 1-.Cn(An.(2) > 0. Hence 7-Cn-'(,9A n ,(2) = +oo.
The following result yields in particular an approximation theorem for bounded Caccioppoli sets.
Theorem 1. Let E be a set of finite perimeter in D. Then there exists a sequence of sets Eh such that aEh n 0 is of class C°O, XE,, -} XE
in L'
and
P(Eh, 0) --p P(E, 0)
.
1.2 Caccioppoli Sets
343
Proof. By the approximation theorem for BV-functions, Theorem 1 in Sec. 4.1.1, we find a sequence of functions uh of class C°O such that
f IUh - XEIdx --+ 0 n
fDlihldx --+ P(E, fl)
Replacing Uh by Ph(uh) where 'Ph E COO (R) is such that
'h(t) = t VtE(1/h,1-1/h)
0<'Ph<1, 0
we may assume that 0 < Uh < 1, and also that uh converge a.e. to XE. From Fatou lemma and the coarea formula we then get 1
if
f lim nfP(Et(uh), fl) dt < lim n 0
= liminf h-oo
J0
P(Et(uh), t2) dt
f JDuh(x)I dx = P(E,fl) 0
where
Et(uh) := {x E fl
I
uh(x) > t}
.
As XE,, -F XE in Ll(fl), by semicontinuity we have for any t E (0, 1)
P(E, fl) < liminf P(Et(uh), fl) h-'oo
and we conclude that
P(E, fl) = liminf P(Et(uh), fl) h-+oo
for a.e. t. On the other hand, by Said lemma, for almost every t E (0, 1), caEt(uh) n Si is of class C°°. Therefore we can find t E (0, 1) such that the sets Et(uh), h = 1, 2.... have boundaries in Si of class COO and
liminf P(Et(uh), f2) = P(E, fl)
.
h--.oo
A suitable subsequence of {Et(uh)} satisfies the claim in the theorem.
Remark 1. In general it is not possible to approximate a Caccioppoli set E by C°O sets contained inside E, nor it is possible from the outside. This is easily seen from the example in 1 Let E be a bounded Caccioppoli set in R. Applying Sobolev inequality (6)
(f
Iulni(n-1) dx) (n-l)in < S(n) f IDul
,
4. Cartesian Currents in Euclidean Spaces
344
valid for all u E BV (R'), to the characteristic function XE of E, we find the following
Isoperimetric inequality : Let E be a bounded Caccioppoli set in R'n. Then (7)
IEI(n-i)/n < I(n) P(E, Rn)
with I(n) = S(n). Actually we can see that (7) implies also (6) with S(n) = I(n). In particular the best Sobolev constant S(n) in (6) and the best isoperimetric constant in (7) agree.
In order to prove that (7) implies (6) consider the set
At(u) := {x
I
lu(x)I > t}
and the function ut obtained from u by truncation at heights t and -t and let In/(n-1) dX) (n-1)/n f (t) := ( f IUt
.
For h > 0 we have
ut+hl s ut1 +hxAt(u) hence
f(t+h) < f(t)+hlAt(u)
i.e.
f'(t) < IAt(u) (n-1)/n . From (7) we then deduce 00
luln/(n-1) dX)(n-1)/n
= f(oo ) - f(0) =
J
f f'(t) dt 0
cc
00
f 0
lAt(u)1(n-1)/n
dt < I(n)
JPAtU, R n ) dt = I(n) f 1Dul
,
0
taking into account the coarea formula. From the previous remark, if we also take into account the approximation
theorems for BV-functions and Caccioppoli sets, and Sard lemma, it is not difficult to show the following. The isoperimetric inequality for smooth sets implies Sobolev inequality for smooth functions, via the coarea formula for smooth functions. By approximation, Sobolev inequalities for smooth functions implies Sobolev inequality for BV-functions, which in turn implies the isoperimetric inequality for bounded Caccioppoli sets, and in particular for smooth sets. Moreover the best constants in both inequalities are the same. As a symmetrization
1.2 Caccioppoli Sets
345
procedure shows for instance in the context of smooth sets that the best constant in the isoperimetric inequality is realized by the unit ball, we then conclude that
S(n) = I (n) = IB(0,
(8)
1)1(n-1)/n 1,9B(0,1)1-' = n-1 wn 1/n
where wn is the volume of the unit ball in 1[8n, wn r(n+2) We conclude this subsection by proving two more isoperimetric inequalities which will be used later.
Proposition 2. Let E be a Caccioppoli set in R' and let Br be any ball of radius r in 1Wn. Then
G2(Br \ E) P(E, Br) < Ci(n) rn-1 rn
Gn(E n Br) rn
(9}
Moreover
min{Gn(Br n E), Gn(Br \ E)}n < c2(n) P(E, Br)
(10)
Proof. As the mean value of XE on Br is given by Gn (E n Br) Gn(Br)
(XE)B,. = and
IXE- - (XE)B, I dx = 2
Gn(E n Br)Gn(Br \ E) Cn(Br)
B,
(9) follows at once applying Poincare inequality of Sec. 4.1.1 to XE on Br. Similarly we have
f XE - (XE)B, I nl(n-1) dx B,.
Ln(Br \ E) ]n/(n-1) Gn(Br n E) G (Br) + L- (Br n E) ] n/(n-1) Gn(Br)
£n(Br \ E)
If £'(B, \ E) > L-(Br n E) then Ln(B,. \ E) > 2n(Br), 1G otherwise we find L- (B, n E) > 2Ln(Br). Thus in both cases IXE - (XE)B, I r`l (" -1)
dx > (2)- min{G'(Br n E), L'(Br \ E)} .
B,.
Applying Sobolev-Poincare inequality of Sec. 4.1.1 to XE in Br we infer at once (10).
4. Cartesian Currents in Euclidean Spaces
346
1.3 De Giorgi's Rectifiability Theorem As we saw in Sec. 4.1.2, the most important object in dealing with Caccioppoli sets E is the reduced boundary
a-E := {x E Rn 13 lim DXE(B(x' p)) =: n(x, E) and In(x, E) I = 1} P
0+
I DXEI (B(x, p))
which carries the measure DXE, (DXEI =IDXEI a-This L.E. subsection is devoted mainly to prove the following important result of De Giorgi
Theorem 1 (De Giorgi). Let E be a Caccioppoli set in RI. Then The reduced boundary a-E is locally (n - 1)-rectifiable and its approximate (n - 1)-plane at x is normal to n(x, E) for xn-l-a.e. x E,9-E. DXE I = n'-1 L ,9-E and DXE = 7in-1 L n(x, E). The following Gauss-Green formula
- J div g dx =
J
(g, n(x, E)) d7-ln'1
V 9 E CC (Rn, R n)
8-E
E
holds.
The key point in the previous theorem is the (n -1)-rectifiability of a-E which will be obtained as consequence of the rectifiability theorem for measures, Theorem 2 in Sec. 2.1.4. We begin by collecting a few remarks. Let E be a measurable set in 1n and let x E R'. The set E n 6B (a;, p) is ,hn-l-measurable for a.e. p and we have Pf
L' (E n B(x, p)) =
(1)
J0
9.1n-1(E n aB(x, t)) dt
.
As p - Gn(E n B(x, p)) is nondecreasing, hence a.e. differentiable, for almost every p we also have dtGn(E n B(x, t))I t=p =
(2)
R'-1(E n aB(x, p))
Proposition 1. Let E a Caccioppoli set in W. Then (i) For all u E C' (Rn) we have UXE E BV (R') and (3)
I D(UXE)I <- f Jul dIDXEI + f IDul dx E
(ii) For all x E Rn and almost every p > 0 (4)
P(E n B(x, p), Rn) < P(E, B(x, p)) + ,Hn-1(E n aB(x, p))
1.3 De Giorgi's Rectifiability Theorem
347
(iii) For all v E R' with I v I = 1, for all x e IPJ1, and for almost every p > 0
IDvXEI(B(x,P)) <
(5)
xn-1(En,9B(x,p))
Here DvXE denotes the distributional derivative of XE in the direction v =
(v1,...,v )
n
Vi DiXE
DvXE i=1
Proof. (i) Since u E Cc' (R-') we can assume that E is bounded, hence XE E BV(Rn). We choose {ck} C CO°(lRn) such that (k > 0, Ck(x) -a XE in L1 and almost everywhere, and Ln L JDCk (x) - JDXE I as measures, compare
Theorem 1 in Sec. 4.1.1. Obviously
f JD(u(k)Jdx <
fuHDckdx+fckIDnIdx.
Thus (i) follows passing to the limit.
(ii) We choose p in such a way that (2) holds and we approximate the characteristic function of B(x, p) by Ck(y) where Ck(y) := cpk(Jx - yl) and
if0p+1/k
1
cpk(t)
-k(t - p - 1/k) if p:5 t < p + 1/k 0
We have
flDkIdx
= k [L-(E n B(x, p + 1/k)) - Gn(E n B(x, p))]
E
Thus applying (i) with u = Ck and passing to the limit we infer
P(E n B(x, p), R') < P(E, B(x, p)) + d L' (E n B(x, t)) I t=p The claim then follows since JDXE 1(8B(x, p)) = 0 for a.e.p. (iii)
Of course it is not restrictive to assume that v is the direction e1. From
J CdDiXE = - JDidX E
C E CO°(II$n)
we deduce
fdDlxEI < f Di((x)I dx
.
E
Using this inequality with = Ck, £k as in the proof of (ii), we infer
348
4. Cartesian Currents in Euclidean Spaces
ID1XE(B(x,P))I < d L' (E n B (x, t)) I t=p
and the claim follows since for almost every p, j DXE I (8B(x, p)) = 0 and (2) holds.
A consequence of Proposition 1 (i) is that the derivatives of the characteristic function of a Caccioppoli set are absolutely continuous with respect to the (n-1)dimensional Hausdorff measure.
Proposition 2. Let E be a Caccioppoli set in Rn. If 7-l'-1(C) = 0 then Hn-1.
IDXEI(C) = 0, i.e., IDXEI <
Proof. It suffices to prove the proposition in the case that C is compact. As ,Hn-1 (C) = 0, for any e > 0 we can find a finite number of balls B(xi, pi) such
that
P
P
C C U B(xi, pi)
> p7-' < E .
and
i=1
i=1
For some 0 E (0, 1) we then have P
C C U B(xi, 6Pi) i=1
We now choose a function cp E CC (B(0,1)) such that cp =_ 1 on B(0, 0), 0 < cp < 1, and I Dcc' ([fin) < c(n), and we set p
w s(x)
=
- xi)) (1
i=1
- w(x
XE(X) Pi
Clearly co (x) __ 0 in a neighborhood of C, thus by Proposition 1 (i) P
IDwEI(Q)
<
IDXEI(f2\ C) + I Dwl (Rn) T
<
IDXEI(12 \ C) +C(n) E .
i
-1
i=1
where [2 denotes a bounded open set containing C. Passing to the limit as E -f 0 we then get IDXEI(f) <- IDXEI(f2\ C) , since cpf -+ XE in L', and this trivially implies I DXE I (C) = 0.
We begin the proof of the rectifiability of the reduced boundary. The next lemma yields rough upper and lower estimates for densities at points x E 8-E. Lemma 1. Let E be a Caccioppoli set in ]Rn and let x E E. Then there exists a positive constant c depending only on the dimension n such that for sufficiently small p, 0 < p < PO (E, x), we have
1.3 De Giorgi's Rectifiability Theorem
Gn(E n B(x, p)) (
6)
pn
>
£n(B(x, p) \ E) > pn
(7 )
and
349
c
c
1 > P(E, B(x, p)) > C. pn- 1
(8)
-
C
-
Proof. To simplify the notation, we may assume that x = 0 E a-E and n(0, E) = (0, 0, ... ,1); also we write Bp for B(0, p). First we observe that for a.e. p small we have
IDXEI(Bp) < 2 I DnXE(Bp)I as consequence of
DXE(Bp)
(0, 0,
... ,1)
I DXE 1(13p)
Using (5), we then infer
P(E, Bp) < 2 ,Hn-1(E n aBp)
(9)
.
In particular we find
P(E, Bp) < c(n)
pn-1
Moreover, since c(n)pn-1 is continuous in p and p --+ P(E, Bp) is nondecreasing the previous inequality holds for any p, p small. This proves the left inequality in (8). Let us prove (6). Set
a(p) := Gn(E n Bp) . From the isoperimetric inequality (10) in Sec. 4.1.2 we infer
a(p)(n-1)1n < c(n) P(E n Bp,Rn) and by (4), (2) and (9) that
a(p)(n-1)/n < c(n) {P(E, Bp) + a'(p)} < 2c(n) a'(p) Integrating this inequality we find (6). Inequality (7) can be subsumed to (6) as 1Rn \ E is a set of locally finite perimeter and DXRt \E = -DXE Finally, from the isoperimetric inequality (10) in Sec. 4.1.2 we find P(E, Bp) > c min{Gn (E n Bp, Gn(Bp \ E)} nn 1 > C pn-1
taking into account (6) and (7).
350
4. Cartesian Currents in Euclidean Spaces
The next lemma contains the main facts for the proof of Theorem 1. In order to state it we find convenient to introduce the following notation. Given a point x E R7L and a set E, we denote by Ex,a the blow-up of E with factor 1/a from the point x,
Er,a := {z E 11
x + A(z - x) E E} .
I
Given an unitary vector field v(x), and we have in mind the case v(x) = n(x, E), x E a-E, we denote by E°,,, the (n - 1)-plane through x which is orthogonal to v(x)
Ex := {z E IRn
I
(z - x, v(x)) = 0}
.
The positive and negative halfspaces generated by v(x) are denoted by
E2
{z E R'
Ex,,,
{z E IR'
(z - x, v(x)) > 0} (z - x, v(x)) < 0}
I
Lemma 2. Let E be a Caccioppoli set and let x E a-E. Then the Caccioppoli sets Ex,p converge in Llor as p --+ 0+ to EX ,(X), v(x) := n(x, E). Moreover I DXEa, I
DXE,,a
-' -'
7-1n-1 L E0,.(x)
v(x) xr -1 L EO,U(x)
in the sense of measures.
Proof. To simplify the notation we may assume x = 0 and v(0) = n(0, E) _ (0, 0,
... ,1). We write Ea for Eo,a, Ea :_ {z E Pn
I
Az E E}
and Br for B (0, r). We divide the proof into four steps Step 1. Convergence of the blow-up sequence. Observe that
Gn(Br n Ea) = A-n Gn(E n Bar) .C"(Br \ EA)
_ A-n L'(Bar \ E)
I DXEa I (Br) _
Al-n
IDXE I (Bar)
From Lemma 1 we therefore deduce
(10)
G"(Br n Ea) > crn C'(Br \ EA) > crn C rn-1 > P(Ea, Br) = I DXEa I (Br) >
crn-1
provided rA is small enough. Using the compactness result in Proposition 1 in Sec. 4.1.2 we can then find a sequence of positive numbers Ak, Ak - 0+, and a Caccioppoli set F such that
1.3 De Giorgi's Rectifiability Theorem (11)
XEAk --> XF
in L o,(R)
and
DXF
DXEak
351
as measures.
In particular, for almost every r we have
DXEak (Br) -, DXF(BT) .
(12)
We also have (13)
DnXEak (Br)
DnXE(BAkr)
IDXEak I (Br)
IDXEI(B),kr)
as k - oo
(0, ... , 0, 1). Therefore from (12)) and (13), and being 0 E 8-E and n(0, E) the semicontinuity of the total variation we infer for a.e. r
DnXF(Br) =
r IDXEak I (Br)
klim.
I DXFI (Br)
and, as the opposite inequality is always true,
DnXF(Br) = IDXFI (Br)
(14)
for a.e. r .
From (12), (13) and (14) we then finally get
IDXEak I - I DXFI as measures.
(15)
Step 2. The blow-up limit is a half-space. From (14) we in particular infer that
DiXF = 0 for i = 1, ... , n - 1 and DnXF > 0
IDXF
I-a.e.
Hence XF is independent of x1 , .. , xn_1 and is a nondecreasing function of xn;
but, since it takes only values 0 and 1, we conclude that F must have the form
F = {x
I
xn > p}
for some Q E R. Step 3. The blow-up limit F is E0,,,, v = (0, 0, ... ,1). It suffices to show that
0 = 0. Suppose 0 > 0. Then for a small ball B centered at 0 we get on account of (10)
lira a_n G'(Ea
0 = C-(F n B) =
A
0+
n B) > c > 0
a contradiction. Similarly, if 3 < 0
0 = Ln(B \ F) = lim A-n Gn(B \ Ea) > c > 0 . A
0+
Therefore p = 0. Step 4. Observing that in step 1, step 2, step 3 we have in fact proved that from any sequence {Xk} converging to zero, we can extract a subsequence {.k,} so that (11) and (15) hold for EAk, , the claim in the lemma follows easily, since
F = Eo does not depend of the chosen subsequence.
El
352
4. Cartesian Currents in Euclidean Spaces
Lemma 2 and a simple scaling argument easily yield the following precise estimates of densities at a point x in the reduced boundary
Theorem 2. Let E be a Caccioppoli set in lRY and let x E
E. Then
9(E, x) = 2 and, setting v(x) := n(x, E), 1
9(E n E (X), x) Moreover
lim
P(E B(x' P Wn-1 P'-1
P-+0
2
=1
Proof of Theorem 1. We consider the measure p := IDXEI, and its blow-up, in the sense of the Definition 5 in Sec. 2.1.4, ux,.\. Clearly lLx,a = IDXE.,a i and Lemma 2 yields
Px,A -'
7..1n-1 L EO'v(x)
v:= n(x, E) .
Therefore a-E is contained in the set {x e Rn
I
El 9(x) E (0, +oo) and an hyperplane Px such that
µx,a -
9(x)7-1'-1
L Px} .
Since IDXEI = IDXEI L a-E and a-E C M, we have
IDXEIL(R' \.M)=0. From Theorem 2 in Sec. 2.1.4 we infer that M, and consequently a-E is an (n - 1)-rectifiable set, moreover
IDXEI = 97'1'-iLM hence, using again that IDXEI = IDXEI L a-E and a-E C M, we conclude that L E. IDXEI = This complete the proof of Theorem 1, since the other claims follow easily. 7-L'-1
0
Besides the topological boundary and the reduced boundary we may also introduce the notion of measure theoretic boundary
Definition 1. The measure theoretic boundary of a set E C R' is defined as the complement of the set of points of density 0 or 1 of E
{x E R' 19* (E, x) > 0 and 9* (E, x) <11
1.3 De Giorgi's Rectifiability Theorem
353
We have
Proposition 3. Let E be a Caccioppoli set in R1. Then
a-Ec 3,,,E and
xn-1(a,,E \ a-E) = 0 .
In order to prove Proposition 3 we first state the following
Lemma 3. Let E be a Caccioppoli set in ]Rn. If for some x E Rn lim
(16)
p_..o
P(E, B(x, p)) =0 pn-1
then either 0(E, x) = 0 or 0(E, x) = 1. Proof. On account of Proposition 2 in Sec. 4.1.2, (16) implies (17)
L- (E n Bp) Ln(Bp \ E) = 0
lira
pn
p-.o
Set now
pn
.
Ln(E n BP) a' (P)
on
From (17) we infer that for any E > 0, 0 < E < 4, there exists 5 > 0 such that for any p E (0, 5)
a(P) (1 - a(p)) < E a(p) E [0, 4e) U (1 - 4E,1] Since a is continuous we infer that either
0 < a (p) < 4e
for all p < 5
or
1-4E < a(p) < 1
forallp<S.
Letting e --p 0+ we conclude that either a(p) --+ 0 or o(p) , 1 as p -* 0+ and this clearly proves the claim.
Proof of Proposition 3. By Lemma 3 we have 8-E C 8E, and 8µE C Ua>oA5
,
Aa
{x 19*n-1(lDXE I, x) > 5}
By Theorem 5 in Sec. 1.1.5 we know that S7-1n-1(A6 \ a-E) <
I DXEI (Aa \ 8-E) = 0 ,
hence In-1(Aa \ a-E) = 0. This clearly yields ln-1(8µE \ a-E) = 0.
4. Cartesian Currents in Euclidean Spaces
354
One can also define
6E:= {x 10 < JE n B(x, p)j < IB(x, p)j by > 0}.
Clearly 8E C 8,AE and it is easily seen that 8E is closed. In fact it turns out that
aE=8E*
where
E*=(EUEl)\Eo Ei
{x I ]p > 0 such that JE n B(x, p)I = IB(x, p)j} Eo := {x 13p > 0 such that IE n B(x, p) I = 0}. In particular we have
a-ECa-EC5E.
Also, if x E IRn \ a-E, we find B(x, p) with B(x, p) n a-E = 0, and by GaussGreen formula,
b g E C,(B(x, p)),
= f 9DcpE = 0 a-E
i.e., cpE is constant in B(x, p). It follows that x 0 aE, that is, aE C a-E. Consequently
aE = a-E. From now on we shall in principle think of a Caccioppoli set E as identified with
E*, so that aE = E. Notice that in principle we can have Rl-1(aE\a-E) > 0, but we have spt I DcpEl = aE. We conclude this subsection by stating without proof the following characterization of Caccioppoli sets
Theorem 3 (Federer). Let E be a measurable set in R. Then E zs a Caccioppoli set if and only if 7-1n-1(aN,E n K) < 00 for any compact set K C lll;n.
1.4 The Structure Theorem for BV Functions In this subsection we shall deal with pointwise properties of functions of bounded variation and we shall prove a structure theorem for them. The way this is done is by recovering information about u from information on its level subsets (1)
Et(u) := {x E 12
I
u(x) > t}
.
1.4 The Structure Theorem for BV Functions
355
Therefore the main tool will be the coarea formula we proved in Sec. 4.1.1 which states that (2)
J
IDul = Jdtf IDxE( -00
1?
I
12
or, taking into account the rectifiability theorem of Sec. 4.1.3, I (3)
+oo
IDuj =
I 7-(n-1(a-Et(u) n £2) dt -00
Given now a function u E BV(f2) we define
(4)
u_ (x)
apliminf u(y) = sup{t
u+(x)
aplimsupu(y) = inf{t
I
v_X
yx
Ju
{x E 0 I u_ (x) < u+(x) }
I.
{x E .f2
I
I u+(x) I
j
O' (Et (u), x) = 1}
O'(Et(u), x) = 0}
= oc} U {x E 0 I lu_ (x) I = oo}
.
Notice that u_(x) and u+(x) are Borel functions in .f2. This is easily seen by reminding that for any measurable set E the upper and lower density functions B*n(E, x) and 0 (E, x) are Borel functions (compare Sec. 1.1.5), and by observing
that, for instance a point x satisfies u_ (x) > t if and only if Bn ({z
j
u(z) < t - 1/i}, x) = 0
for every positive integer i.
Theorem 1. Let u E BV(.f2). Then (i) Ju is 7.I1 -measurable and countably (n - 1)-rectifiable, (ii) IDuI L (B \ Ju) = 0 for any 7-1n-1-measurable set B with 7-1n-1(B) < oo, (iii) We have
I Dul L Ju = (U+ (X) - u_ (x))7-1n-1 L Ju
.
Ju agrees with the set of points where IDuI has positive density
(iv) 71n-1-a.e.
with respect to 7..1n-1,
Ju = {x I
On-1(IDul,x) > 0}
7-cn-l-a.e.
.
and in fact
0n-1(IDuI,x) = u+(x) -u_(x)
7-(n-1-a.e.
in Ju.
Proof. As all claims are local we can assume that u E BV(I[$n). (i) From the coarea formula we known
356
4. Cartesian Currents in Euclidean Spaces +oo
IDuI(Rn) =
ftDxE()I(r) dt < 00. -00
We can therefore select a denumerable and dense set Q in ]l such that for each t E Q Et (u) has finite perimeter, hence, by the results in Sec. 4.1.3, 19-Et(u) is (n - 1)-rectifiable and xn-1(BµEt(u) \ 8 Et(u)) = 0 . From the definition of u_ (x) and u+ (x) we immediately see that
B*n(Et(u), x) > 0 if t < u+ (x)
B*(Et(u),x) < 1 if t > u- (x) Thus for each t we have (5)
Jt := {x E Rn
and
I
u-(x) < t < u+(x)} C a,,,Et(u)
Ju = UJt C tEQ
UajEt(u). tEQ
Consequently,
U a-Et(u)) = 0
.
tEQ
by Proposition 3 in Sec. 4.1.3. This shows that Ju, is 9-1n-1-measurable and countably (n - 1)-rectifiable. (ii) Let B be a Borel set. We observe that the sets (B \ Ju) n 8-Et(u) are disjoint for different t. In fact if x E 8-Et(u) n a-E8(u) and t s, t, s e Q, then B(Et(u),x) = B(ES(u),x) = 1/2, hence u_(x) < u+(x), i.e. x E Ju. From
this we infer in particular that
,Hn-1((B \ Ju)
n a-Et(u)) = 0
for a.e. t ,
if 7-1n-1(B) < oo. Finally, I DuI (B \ Ju) = 0 for all Borel sets with xn-1(B) < 00 follows from the coarea formula +oo
IDuI (B \ Ju) =
ffl''((B\J)naE(u))dt. -00
(iii)
Using the coarea formula we get for every Borel set B +oo
IDuI (Ju n B) =
ffl_1(8_Et() n Jun B) dt -00
f f +OC
=
dt
-00
B
Xa-E,(u)nJu drln-1 = f(u+(x) B
- U_ (X)) d7-1n-1
1.4 The Structure Theorem for BV Functions
357
(iv) From (iii) and the countably rectifiability of Jv, we see that
0'-'(IDuj,x) > u+(x) -u_(x)
7-l'-'-a.e. in JtL
hence
Ju C {x 19''-1(IDuI, x) > 0} . On the other hand for any Borel set B C Ju with 71r-1 (B) < oc, by (iii) we have (6)
lim sup
I Dul(B(x, P)) < limsup
P_0+ 7{
DuI (B (x, P))
(B(x, p)) - p-.o+ 7-'(B fl B(x, p)) ",I
= u+(x) - u_(x)
for 7-l"i-1-a.e. x E B, consequently for 7{"-1-a.e. x E Ju recalling that Ju is or-finite with respect to 7-1ri-'. This concludes the proof.
Remark 1. Let u be a BV-function with integer values. From Theorem 1 (iii) we infer that Ju is (n -1)-rectifiable, as u+ (x) - u_ (x) > 1. On the other hand for any t Et (u) = E., (u)
for some s
t, hence
8-Et(u) C Ju and by the coarea formula IDuI is absolutely continuous with respect to 7-l'-' L Ju. Consequently
IDul = IDuI L Ju
.
Let u E BV (Q). By Lebesgue's decomposition theorem for measures, compare Theorem 1 in Sec. 1.1.4, we can decompose the Radon measure Du into its absolutely continuous part (Du)' with respect to Lebesgue measure and its singular part (Du)-'
Du = (Du)' + (Du)'
(7) and
(Du)8
C
,
(Du)' = C"° L F(x),
F E L1(Sl, R") .
We now define the jump part (Du)U) and the Cantor part (Du) (C) of Du by (8)
(Du)(') := DuLJu = (Du)SLJu
and (9)
(Duff) := (Du)s L (D \ Ju)
,
so that
Du = (Du)' + (Du)(j) + (Duff) . Notice that (Du)a, (Du)(U), and (Du)(C) are mutually orthogonal. Moreover note that the measure (Du)(U) lives on the countably (n - 1)-rectifiable set Ju, while (Du)(C) lives on a Cantor type set C, that is,
358
4. Cartesian Currents in Euclidean Spaces
(Du)(') = (Du) (C) L C
if
W2(C) = 0
,
and
(Du)(C) (A) = 0
7-ln^1(A) < oo
if
.
We shall now discuss fine properties of the measures (Du)a and (Du)(i). Let us begin with (Du)( ). First, we fix some terminology. Let u E L' (If87). We say that a point x E Rn is a regular point for u if the approximate limit of u at x exists and is finite. Non regular points are said to be singular. Among the singular points we now identify the jump points as follows
Definition 1. We say that a point x E Rn is a jump point of u with parameters (a, /3, v), if there exist a unit vector v and real numbers a and /3 such that a > /3 and
aplim u(y) = a ,/--.x
aplim u(y) 7, -.x yEEa
HEEi
where we set
E Ex
{yERn {yEW
I
I
(y-x,v)>0} (y-x,v)<0}
and
Ex o,, := {y E R
I
(y - x, v) = 0}
.
It is easy to see that the parameters a, /3, v are uniquely defined at each jump point. Suppose in fact that at the point x there are two different sets of parameters (a, b, i) and (a, /3, v). If rl = v, then a = a and b = 0, as consequence of the uniqueness of the approximate limit, compare Proposition 1 in Sec. 3.1.4. Suppose now that rl and v are independent. Then the four sets
E nEX,,,
,
EX,,, nEX,17
are non empty cones which have non zero density at x. From this it follows at
once a = a = b = 0, a contradiction. If finally rl = -v, then a = /3 and b = a, again a contradiction since a < b and a < /3. Let x be a jump point for u with parameters (a, )3, v). We shall refer to v as to the jump direction of u at x and to a - /3 > 0 as to the jump value, or, simply, to the jump of u at x. Note that the jump value is always finite and positive. Let us consider the special, but important case of characteristic functions of measurable sets. We can then characterize jump points in terms of densities
Proposition 1. Let E be a measurable set in W2. Then x E 1R7 is a jump point for xE with parameters (1, 0, v) if and only if (10)
or equivalently
Bn(EnEx-, x) = 0
1.4 The Structure Theorem for BV Functions (11)
Bn(EnE,,,,x) = 0
359
0 ((R'n\E)nE ,,,x) = 0
or
(12)
on (EnE,,x) = 2
on(E,x) =
1
2
Proof. First, observe that (10), (11) and (12) are equivalent also to (13)
Bn(E n EE ,,x) = 2
on ((R' \ E) n Ex x) = 2
Now, since XE takes only values 0 and 1, the set E agrees with the set
AE :_ {x E IRn
I
I XE (x) - lI < E}
for any E, 0 < e < 1. Therefore the first condition in (13) is equivalent to on (A, n EE ,,, x)
1
=
2
for any e, 0 < e < 1. This in turn is equivalent to aplim XE (y) = 1
,
7/EEm,
compare Sec. 3.1.4.
Similarly we see that the second condition in (13) is equivalent to
aplim XE(y) = 0
,
YE
and this clearly concludes the proof. Suppose now that E is a Caccioppoli set in R. From Theorem 2 in Sec. 4.1.3 we know that (12) with v = n(x, E) holds for all point x in the reduced boundary 3-E of E. Thus we immediately infer
Corollary 1. Let E be a Caccioppoli set. Then every x E 8-E is a jump point for XE with parameters (1, 0, n(x, E)).
Actually, we have even more. Denote by 8*E the set of jump points of XE with jump value one,
8*E :_ {x E IRn
I
x is a jump point of XE with jump value 1}
and recall that 8E denotes the measure theoretic boundary of E, compare Definition 1 in Sec. 4.1.3. Then taking into account Proposition 3 in Sec. 4.1.3 we readily deduce
360
4. Cartesian Currents in Euclidean Spaces
Proposition 2. Let E be a Caccioppoli set in R. Then
8-EC a*EC 8µE and
xn_1(a*E \ a-E) = 0
,
xn-1(8µE \ a.E) = 0
.
Of course every jump point of u is in J,,. The next theorem gives a complete description of the set of jump points of a BV-function.
Theorem 2. Let U E BV (,fl) . Then (i)
For fn-1-a.e. point x E Ju and all t with u_(x) < t < u+ (x), x is a jump
point for XEt (u) with parameters independent oft and equal to (1, 0, n(x, Ju)). (ii) We have
Du L Ju = n(x, Ju) DuJ L Ju = (u+(x) - u-(x)) n(x, Ju) (iii)
7-Cn-1-a.e. point x in Ju (u+ (x), u- (x), n(x, J.)).
Hn-1 L Ju
is a jump point of u with parameters
Proof. (i) We shall prove that (i) holds for all points x in Ju \ N1,
N1 = U (BµEt(u) \ 8 Et(u)) tEQ
where Q C R is the set defined ha the proof of Theorem 1. First we claim that (a)
If fortswe have B'(Et(u),x) = en(Es(u),x) =
2
and x is a jump point for XEt(u) with parameters (1, 0, v), then x is also a jump point for XE., (u) with the same parameters. In fact, we have O'(Et(u) \ E8(u), x) = 0, t < s, as E3(u) C Et(u). Setting now v = n(x, Et(u)) we deduce
O'(ES(u) n Ey,,,x) = Bn(Et(u) nEx,,,x) -Bn(Et(u) \ES(u),x) 1 2-0 = 21
taking into account that 0' (Et (u) nEE ) = 1/2. Thus the claim (a) follows from Proposition 1. We now observe that x E BµEt(u) n 8,E., (u)
whenever x E Ju and u_ (x) < t < s < u+ (x), while
1.4 The Structure Theorem for BV Functions
361
X E a-Et(u,) n a-E, (u)
if x E J. \ N1 and t, s E Q, u_ (x) < t < s < u+(x). In this last case, in particular, from Theorem 2 in Sec. 4.1.3 and Proposition 1, x is a jump point for both XE2(v,) and XE,,(v,),
O' (Et(u),x) = 0''(E3(u),x)
=I
.
2
and n(x, E. (u)) = n(x, Et (u))
by the claim (a). Since Q is dense in (u_ (x), u+ (x)) we also have
O'(Er(u),x) = 1 for all r E (u_(x),u+(x)). Therefore, again from the claim (a) we deduce that x is a jump point for XE, (u) for any r E (u_ (x), u+(x)) with jump direction n given by
n = n(x,E5(u))
d s E Q.
In particular n(x, Ju) defined as n(x,Ju) = n
is orthogonal to the approximate tangent plane TanxJu to Ju at x. (ii) Using the coarea formula in its vectorial formulation, (17) in Sec. 4.1.1,
r
JB
dDu = fdt( fdDxE(u)) -OC
B
one derives
+r0
Du(Ju n B) =
f
dt J Xa-Et(u)nJ n(x, Et(u)) dIDXE,(u) I J co -B
JB
n(x, Ju) dIDuI L Ju =
J(u+(x)_u_(x))m(x,Ju)dTh-1
B
taking into account (iii) of Theorem 1. (iii)
(14)
First let us show that
Hn-1(Ju n Iu) = 0
By Fubini-Tonelli Theorem and (5) we infer
.
4. Cartesian Currents in Euclidean Spaces
362
cc
f(u+(x) - u_ (x))
d7-l'-1 =fu n {x : u_ (x)
J.
0
0ffl-1(a_E(u))
00
dt < IDu
< f xn-'(8 Et(u)) dt 0
0
Consequently u+(x) - u- (x) is ln'-1-a.e. finite in Ju. This yields (13) as x E Jun Iu if and only if u+ (x) - u_ (x) = +oo. Fix now a point x E Ju \ Iu for which TanxJu and n(x, Ju) = n(x, Et (u)) for a.e. t with u_(x) < t < u+(x) are well defined. Notice that this happens 7-lam'-1-a.e. in Ju. Define J. :_ {y E W1
I
(y - x,n(x, Ju)) > 0}
.
For almost all e > 0 with
u_(x) < u+(x) - E < u+(x) we then infer
en(Eu+(x)+e(u), x) = 0 from the definition of u+ (x) and Bn(Eu+(x)-E(u) n Jy ,x)
-2
since n(x, Ju) = n(x, a-Eu+(x)-e(u)). Therefore, for AE :_ {y
I
U(y) - u+(x)I < e} ,
we find
9n(AE n JJ ,x) = Bn(Eu+(x)-E(u) n J , x) - Bn(Eu+(x)+e n JJ ,x) This yields at once
aplimu(y) = u+(x) y-x yEJ.
compare the proof of Proposition 1. Similarly one proves
aplim u(y) = u_(x) . x
yERn\Jx V
Corollary 2. Let u E BV(Q). We have
1
2
1.4 The Structure Theorem for BV Functions
363
(i) For IDul -a.e. x in Ju, equivalently for fn-1-a.e. x E Ju,,
n(x, Ju) =
d Du (x) = d IDul (x) dI (Du) U) I
(ii) For IDul-a. e. x E .(2 \ Ju
IdDu
n(x, 8 Eu+(x)(u)) = d Dul (x) Proof. The first equality in (i) follows at once from Theorem 2 (ii), the second is trivial because of the definition of (Du)(j).
Let us prove (ii). First we observe that if x E (2 \ Ju n 8-Et(u), then u_ (x) = u+(x) and 8(Et (u), x) = 1/2; hence u_ (x) = t = u+(x). Being DXE,(u) I = 7-1n-1 L 8 Et(u) using the coarea formula we infer for any cp E C0°(()
J
cp(x)dDU L0\Ju +r0
f
= J dt J p(x)n(x, C7 Et(u))
dxn-1 L L9-Et(U)
\ Ju
- 00
+00
=
fdt f W(x)n(x, a-Ev,+(x) (u)) d7-1n-' L a-Et(u) \ Ju -00
= J co(x)n(x, 8-E,,+(x) (u)) djDuj L .R \ Ju ,
since n(x, a-Et(u)) is a Borel function in x and t. This of course proves (ii). We saw in the proof of the previous theorem that
xn-1(Iu n Ju) = 0 . Now we want to show that in fact 7-1n-1(Iu) = 0
It is immediately seen that Iu = (Ju n Iu) U Iu
Iu
{x
I
u+(x) = -oo} U {x
I
u_ (x) = +oo} .
Therefore it suffices to show that the set
I := {x
I
u_(x) = +oo}
has zero ?in-'-measure. We shall do that by means of the estimate on the approximating measures 'H'-' contained in the next lemma.
364
4. Cartesian Currents in Euclidean Spaces
Lemma 1. Let E be a Caccioppoli set with finite measure and finite perimeter. Denote by E1 the set
El := {x E ]n
0'(E, x) = 1}
I
.
Then there exist two constants cl and c2 depending only on the dimension n such that
Hb-'(El) < c2IDXEI(Rn) where
[cl 1
IDXEI(Rn)11
Proof. For any x E El the map g : R+ -> R given by 9(P)
Gn(E n B(x, p)) Ln(B(x, p))
is continuous in (0, +oo), g(p) --> 1 as p --> 0, and g(p) --> 0 as p -> oc, since Gn(E) < oo. Then we can find for every x E El a radius r_- such that
L' (E n B(x, rx)) Gn(B(x,rx))
_
1
2
and from the isoperimetric inequality (10) in Sec. 4.1.3 rx-1 < c1(n) I DXEI (JRn) =
bn-1
Using Besicovitch covering theorem, Lemma 2 in Sec. 1.1.5, we find a covering
of El by balls B(xi, rr,) such that the balls B(xi, r., /3) are disjoint. Now we have
IDXEI(Pn) i
DXEI(B(xi, 1rxi) ? 31_nc1(n)_1 Ern--1
> c2(n)-1H6n-1(E1) where the last inequality follows since all radii rx, are less than 6.
Theorem 3. Let u E BV(Q). Then Hn-1(II,) = 0. Proof. Without loss of generality we can assume that u belongs to BV (JEBn) and
that u has compact support in R. Because of +00
I Dul (fin) =
J
IDXEi(u) (W1) dt
-00
we can choose a sequence {tk}, tk --> oo, such that I
DXE,k (u)
I (Rn)
0
as k -* oc .
1.4 The Structure Theorem for BV Functions
365
Since for any t {x
I
u_(x) > t} C {x B(Et(u),x) = 1} CC Rn I
we can apply Lemma 1, deducing Hbk-1({x I u-(x) > tk}) < cl IDXEtk(u)I(Rn)
with Sk
`C2IDXEtk(u)I`in)]nll Letting tk -* oo, we then obtain for each 6 > 0
H' ({x
u-(x) _ +oo}) < c1 1 m of IDXEtk(u)I(1n) = 0 k-oo
hence 7-tn-1({x
I u_ (x) _ +00 }) = 0
.
Concerning the pointwise values of a BV-function u, Theorem 2 (iii) says in particular that Hn-1-a.e. point x E Ju is a jump point with
u_ (x) = aplim u(y) ,
u+ (:r) = aplim u(y) ,
v = n(x, Ju)
Ey
-Z-
while Theorem 3 says that 7-1n-1-a.e. point x E f2 \ Ju is a regular point
aplimu(y) = u+(x) = u_ (x) E y-X
These claims can be in fact improved as follows.
Theorem 4. Let u E BV(Q). Set 1* = n/(n - 1) (i) For 7-Ln-1-a.e x E 1? \ Ju we have
u_ (x) = u+(x) =: u(x) E R and
urn f
(15)
Iu(y)-u(x)I" dy = 0.
B(x,p) Set
B+ (x, p)
B(x, o) n Ey ,
BL (x, P) := B(x, p) n E;
Then for 7-1n-1-a.e. x E J. and v = n(x, Ju), we have (16)
plu
0+
1
Iu(y) - u+(x)114 dy = 0
By (fx,p)
(17)
lim
po+
-I JJ
B (x,p)
I u(y) - u- (x) I1* dy = 0
4. Cartesian Currents in Euclidean Spaces
366
In order to prove Theorem 4 we need the following variant of SobolevPoincare inequality
Proposition 3 (Sobolev-Poincare inequality). Let u E BV(B(xo, p)). Set 1* := n/(n - 1) and ap
_
cn({x E B(xo, p) u(x) = 0}) £n(B(xo, P)) I
Then we have
f
c(n)
f lu(x)ll* dx)l/l* <
(18)
1
1 - (1 - ap)1/n Pn-1
B(xo,p)
I Dul
.
B(xo,p)
Proof. From Sobolev-Poincare inequality in 0 in Sec. 4.1.1 lull*
(
<
dx )1/l*
f
(
J
I u(x) - ux0,Pll* dx )1/l* + luxo,P I
B(xo,p)
B(xo,P)
<
f
P(ni
IDul + uxo,pl
B(xo,p)
The result then follows since by Holder inequality Iuxo,PI
<-
1
f
1B( o,P)I
I u(x) I dx
{xEB(xo,P) I u(x)00}
< (1-LYp)l/n(
f lull*dx)l/l* B(xo,P)
We also need
Lemma 2. Let u E BV(Q). Define Nu := {x E .fQ
1
0*n'-l(lDul,x) > u+(x) - u_ (x)} .
Then'H'-l(Nu) = 0. Proof. We split N,,, as Nu = N1UN2 where Nl := Nun(.Q\J,,,) and N2 := NunJu, and we prove that N1and N2 have zero (n - 1)-dimensional Hausdorff measure. By Theorem 1 clearly 7{n-l(N2) = 0. To prove that 7-ln-l(N1) = 0 it suffices to show that for k = 1, 2, ... we have 7-ln-1(Nk) = 0 where Nk
{x E 0 \ Ju
1
9*n-l(IDuI,x) > k}
.
1.4 The Structure Theorem for BV Functions
367
From Theorem 5 in Sec. 1.1.5 we know that
?-1n-1(Nk) < k jDuj(Nk)
(19)
hence Hn-1(Nk) < cc. Since Nk C .f2 \ J, from Theorem 1 (ii) we then infer IDul(Nk) = 0, and therefore')-L'-1(Nk) = 0 again by (19). Proof of Theorem 4. The proofs of (15) (16) (17) are similar. We shall therefore prove (16) and leave the proof of (15) and (17) to the reader. Consider first the case in which u E BV(f2) fl L°°(f?). Then we have
u+(x) = aplim u(y) E a_z
(20)
for ?in-1-a.e. x E Jz, by Theorem 2 (iii). As u E LO° (,f2), it is then easy to show that (20) is equivalent to (16), compare 1 in Sec. 3.1.4. In the case in which u Loo (.f2), we consider of truncated of u, {uh}, defined by
uh(x) = max(min(u(x), h), -h)
.
Of course
uhf (x) = max(min(ut (x), h), -h) . hence Juh C Ju
.
For each h = 1, 2.... denote by Eh C Ju, the set on which (16) fails for uh. As uh E L°° (.fl) we have
,ln-1(L'h) = 0
.
We also consider the set
Nh := {x E .fl
I
0'-1(ID(u - uh)I, x) > (u - uh)+(x) - (u - uh)-(x)}
for which we have ,{n-1(Nh)
=0
by Lemma 2. Then we set
E = U Eh h
N = U Nh h
and
I = Iu = {x
lu+(x)I = oo or ju- (x)I = oo} . Of course E C Ju, xn-1(E) _ ?-1n-1(N) = 0, while from Theorem 3 xn-1(I) _ I
0. Thus it suffices to prove (16) for all x in ,fl \ (E U N U I). Fix one such an x. We have
4. Cartesian Currents in Euclidean Spaces
368
f
(
Iu(y) - uh(y)I1` B (x,p)
By (x,P)
+(
(21)
dy)1/1.
<(
-u+(x)I"dy)1/1"
Iu(y)
f Iuh(y) - uh+(x)I' dx) 1/1* + Iuh+(x) - u+(x)I B. (x,p)
As x
E, for each h we have lim
(22)
As x
'-0
}
uh(y) - uh+(x)I'F dx = 0 .
I, max(Iu+(x)I, Iu-(x) E R; hence, if we choose h > max(Iu+(x)I, Iu_(x)I)
,
we have uh+(x) = u+(x) ,
(23)
uh-(x) = U-(X)
Therefore (24)
lim sup ( P-+0
fJ
Iu(y) - u+(x)I1* dx)
1/1'
B (x,p)
u(y) -uh(y) I1 dy)1/1`
21imsuP( p-.0
B(x,P)
We now observe that the set {y E ,fl u(y)I < h} has density 1 at x for the chosen h, and u - uh is zero on that set. We then apply the Sobolev inequality in Proposition 3 and find I
lim-sup ( f I u(y)
- uh(y)I1* dy
)1/1.
B (x, p)
(25)
ID(u - uh) I B(x, p)) pn-1 P-0
< c(n) limsup
= c(n)O*n-1(ID(u-uh)I,x)
as x
< (n) [u+(x) - uh+(x) - U-(X) + uh- (x)] = 0 N. The claim then follows at once from (21)-(25).
Remark 2. If u E Ll (.f2), we know that the convolutions u * cp, (x) converge to u(x) at every Lebesgue point x of u when o -+ 0+. From Theorem 4 we therefore see that convolutions with a symmetric kernel cp converge pointwisely to U+ (X) + u_ (x) 2
7.1n-1-a.e. in J, provided u E BV(Q).
1.4 The Structure Theorem for BV Functions
369
Remark 3. Let u E W1'1(12). Of course then u E BV(S2) and (Du)(j) = 0, i.e., 7-Ci'-1(Ju) = 0. Therefore from Theorem 4 we deduce that for l'-1-a.e. x in d2
f
lu(y) -
u(x)l"("-1) dy
asr--0,
--; 0
B(x,r)
in particular
H--1 (f? \ Gu) = 0
.
Next theorem states that BV-functions are almost everywhere approximately differentiable and, indeed, the approximate differential is the density of the absolutely continuous part (Du)a of Du.
Theorem 5. Any function u in BV (,f2) is approximately differentiable at almost every point x c .(2 and
(Du)' = C' L apDu
.
In particular we have BV (t2) C A1(.(2, ]I8)
(26)
Actually a stronger statement holds, which in fact implies Theorem 5.
Theorem 6. Let u E BV (,(2) and let x be a Lebesgue point for u, x E L, u(x) the Lebesgue value of u at x. Suppose moreover that the Radon-Nikodym
derivative aDn (x) exists at x, and x is a Lebesgue point for Then
Ju(y) - u(x) - L(y - x)j
lim
P-0
Iy - xI
x E Gam
dy = 0
B(x,p)
where L is the Lebesgue value of aDn (x) at x. In particular
apDu(x) =
(x)
.
The proof of Theorem 5 proceeds along the same lines of the proof of Calde rSn-Zygmund LP-differentiability theorem in W1,r, Theorem 2 in Sec. 3.1.2. The key step is the following proposition which replaces Proposition 1 in Sec. 3.1.2.
Proposition 4. Let u E BVoc(W ), let x E Lu and let u(x) be the Lebesgue value of u at x. Then for all r > 0 we have 1
Iu(y) - u(x) dy <
(27)
B(x,r)
ly-x1
(B (x,
tr) dt IB(x,tr)l
Dul 0
370
4. Cartesian Currents in Euclidean Spaces
Proof. Of course we may assume that the right hand side of (27) is finite.
From Proposition 1 in Sec. 3.1.2 step (i) and (ii) we know that (27) holds for u E BVoc, (1R') n C'(1Rn \ {x}). The same construction in Lemma 1 in Sec. 3.1.2
yields now a sequence {uk} C C1(R' \ {x}) such that
in L' and a.e. Uk --> u Ln L I Duk (x) I - I DuI as measures and forr --* 0
f Iuk(y) -u(x)I dy - 0 B(x,r)
I Duk I dy < c I DuI (B(x, 2r)) .
Applying (27) to Uk, passing to the limit, and taking into account Fatou's lemma and Lebesgue's dominated convergence theorem we then get
B(x,p)
IUM - u(x)I dy <_ liminf k-oo Iy - xI
< lim of k-oo
1
dt
Iuk(y) - uk(x)I dy IV - xI
B(x,p)
IDuk(y)I dy = fo 1
fo
B(x,tp)
I DuI (B(x, tp)
L (B(,tp))
dt
0 Proof of Theorem 6. Denote by F(x) the Radon-Nikodym derivative of Du with respect to L'. The assumptions yield x E LF and
F(y) - F(x) I dy ---p 0 ,
p-' I(Du)SI(B(x, p)) --> 0 .
Thus applying (27) to the function v(y) := u(y) - u(x) - F(x) (y - x)
we get fIu(y) - u(x) - F(x)(y - x) I
dy
ly - xI
B(x,p)
1 IDu - F(x) dGn(y)I (B (X, tp)) o
G
sup
(
o
f
Cn(B(x, tp))
B(x,r)
I F(y) - F(x)I dy +
dt I (Du)sI (B (-T, r))
Ln(B(x, r))
which concludes the proof, as the right hand-side tends to zero as p -, 0+.
1.5 Subgraphs of BV Functions
371
In conclusion we can state
Theorem 7 (Structure theorem). Let u E BV(S7). Then The set Ju is 7-1n-1 measurable and countably (n - 1)-rectifiable and agrees 7-1n-1-a.e. with the set of jump points of u. (ii) Hn-1-a.e. point x in (2 \ Ju is a Lebesgue point for u. (iii) u is almost everywhere approximately differentiable in 0. (iv) The gradient measure Du splits as (i)
Du = apDudx + (u+(x) - u- (x)) n(x, Ju) dhn-1 L Ju + (Du)(C) Moreover
(Du)(
= (Du)(C) LC
if
7-Cn(C) = 0
and
(Du) (C) (A) = 0
if
Hn-1(A) < oc
.
1.5 Subgraphs of BV Functions The aim of this subsection is to show that properties of BV-functions can be read equivalently in terms of their subgraphs. As usual, .f2 will denote a bounded domain of R', n > 2, 7r :.f2 x Il8 -> (2 the orthogonal projection of .f2 x III in .f2, and we denote by (xi, ... , xn, y) _ (x, y),
x E (, the coordinates in (2 x R. For any measurable function u : 0 -> R the subgraph of u is defined as the measurable subset of .f2 x I[2 given by
y) E Q X J I y< u(x)}
(1)
.
When no confusion may arise we simplify the notation writing Su
for
Sg.,,a
-
Correspondingly, we shall denote by Xsu the characteristic function of SGu,12
in .f2 x R, and, for the sake of simplicity, we write DiXs instead of aa Xs i = 1,.. . , n, and Dn+i Xsu instead of ay Xsu. Finally we introduce the R'+'valued distribution in P defined by (2)
u(Du) := (Diu, ... , D, u, -L')
.
Our first theorem characterizes BV-functions in terms of their subgraphs.
Theorem 1. Let u E L1(.f2). Then u belongs to BV (.f2) if and only if the subgraph of u, Su, has finite perimeter in .f2 x IR,
P(Su, .f2 x lR) < 00
.
372
4. Cartesian Currents in Euclidean Spaces
Moreover we have (3)
7r#DiXsv. = pi(Du)
(4)
7r#IDiXs,,I
(5)
7r# IDXs, I = I u(Du) I
i = 1 ... , n + 1
i=1,...,n+1
= Ii'i(Du)I
Proof. Suppose first that P(S,, Q x R) < oo. In this case the measures DiXsu can be extended as linear functionals acting on continuous and bounded functions in Q x R by means of Lebesgue theorem. In particular setting
IyI
1
PN(y)
N+1-IyI N
0
we get for anycpEC0°(Q)andi=1,...,n
f V (x) DiXsu = -limn f2 x 1
PN (y) Xs,, (x, y) d-- dy
J
.f2 x li
(6)
u(x)
- N-oo lm fD(x)(
f
PN (y) dy) dx
.
- 00
,fl
As z
JPN(Y)dY < z + N + 1
-n
ifz>-N-1,
z
f
---' z
PN (y) dy - N - 1
as N --r 00
,
-00
by definition of PN (y), and as u c L' (Q), we then get (7)
f
- fn
u(x) dx
.f2xR
letting N --Y oo in (6). Equivalently we can write (8)
7r#DiXs,, = Diu
n
and consequently (9)
IDiul(A) <
for any open set A and i = 1, ... , n. This proves that u E BV (f?) and that (3) holds for i = 1, ... , n.
1.5 Subgraphs of BV Functions
373
On the other hand we have
f p(x, y) Dn+iXsu. = - f Wv(x, y) xs (x, y) dx dy = - / p(x, u(x)) dx nxa
.f2 x ]R
12
in particular (10)
Dn+i Xsu
< 0,
while, proceeding as previously, we infer
f p(x) Dn+iXsu = - f cp(x) dx
(11)
.
nx1R
This concludes the proof of (3) and also gives 7r# jDn+iXsu l (A) = L'1(A)
(12)
taking into account (10), and Iµ(Du) (A) < DxsJ (A x III)
(13)
for all open sets A C fl, taking into account (9). Suppose now that u E BV (f2). By the approximation theorem, Theorem 1 in Sec. 4.1.1 (ii) we choose a sequence of smooth maps uk such that uk --f u in
L' and (14)
f IDiukjdx --> IDzul(Q),
fDukdx
n
n
-' IDuI(Q)
as k --> oc, and we observe that for any cp E C, ° (fl x R) we have
I
p (x, y) Dixsuk = -
f2xa
f
pxi (x, y) Xsu,, (x, y) dx dy
QXR
-
f ,fl
Uk (x)
dx
f cpxi(x, y) dy -o° 'UUk (x)
- f [dxi f w(x, y)dy - w(x, uk(x))Diuk(x)] dx 0
-00
=
f(xuk(x))Diuk(x)dx
f
o(x, y) Dn+lxsuk
f2x1R
=-
f 12x11
py (x, y) Xsu, dy dx = - f cp(x, uk (x)) dx Q
374
4. Cartesian Currents in Euclidean Spaces
Taking the supremum for all cp E C,00 (A x IE8), Icpl < 1, A c .2, A open, we then get
IDZXsuk I(A x R) < Diukl(A)
IDn+lxsuk I(A x R) < L'(A), and similarly
IDxsuk I(A x R) < L'L
1 + IDukI2(A) .
Letting k -; oo we then conclude Dzxsu l (A x IIg)
(15)
<_
D2ul (A)
Dn+1Xsu1(A x II2) < Ln(A)
DXs, I(A x Il) < a (A) , taking into account (14) and the lower semicontinuity of the total variation with respect to the Ll-convergence. This proves that Su has finite perimeter in .f2 x R. Also (15), in conjunction with (9), (10), (12) and (13), clearly concludes the proof of (4) and (5).
Remark 1. It is worthwhile noticing that the assumption u c Ll (,f2) is not necessary: the measurability of u would suffice. The summability of u follows from the isoperimetric inequality. Suppose in fact that P(Su, .2 x R) < oc, then
f l u(x)I dx = Ln+1(S1ul n {(x, y) n < c(n) P(Slul n {y >
I
y > 0} 0})1/(n-1)
On the other hand, arguing as in Proposition 1 in Sec. 4.1.3,
P(Siui n {y > 0}) < P(Siui,.f2 x R) + IQI and the summability of u is proved if one shows that
P(SJuJ,1? x R) < P(SS, ( x Ill;)
.
This actually follows from a slicing formula for the perimeter of a set or more generally for the total variation of a BV-function, compare Miranda [475].
Let u be a function in BV(Q). By the previous theorem then, and actually equivalently, the subgraph Su has finite perimeter in ,fl x R. By De Giorgi's rectifiability theorem in Sec. 4.1.3, we have (16)
DXsu = n(z, Su) 7C L 8 Su
,
z = (x, y)
Now we would like to write (16) in terms of u and Du, by recovering the set a-S,, and the inward normal n(z, a-S,,) to the subgraph Su in terms of u and Du for which we have proved a structure theorem in Sec. 4.1.4.
1.5 Subgraphs of BV Functions
375
Recall that
u+(x) := aplimsupu(y)
u_ (x) := apliminfu(y)
y-.x
y-.x
and that the jump set of u is defined by
Ju := {x E R'1
I
u-(x) < u+(x)}
.
We have seen in Theorem 1 in Sec. 4.1.4 that Ju is a Borel and countably (n-1)-
rectifiable set so we can single out the part of the "graph" of u related to the jumps of u by setting DXS) := DXsu L Ju x R = n(z, Su) 7-C L (8-Su n Ju, x R)
(17)
and the approximately continuous part of the "graph" of u by DXSant)
(18)
DXsu - DXSj) = DXsv. L ((2 \ Ju) x ]R .
Also set Du D(c)
{(x,y)E.(2xIIF {(x, y) E .(2 x R
u_(x)
Obviously VU(0) C Ju x
Concerning the approximately continuous part of the "graph" of u we have
Theorem 2. For any cp E C°° (Q x R) we have (19)
DiXScont)
(W(x, y)) = f 0 (x, u+(x)) Diu
i = 1, ... , n
and Dn+1XSCOnt)
(20)
(x, u+ (x)) dx
n Concerning the "jump part" of the "graph" of u we have
Theorem 3. We have (i) 7{nL(a-SunJuxR) = 7-C' LDU(,°) (ii) For fn-a.e. (x, s) E DU(°) the inward normal to Su exists, x is a jump point
for u and (21)
n((x, s), Su) = (n (x, Ju), 0)
4. Cartesian Currents in Euclidean Spaces
376
(iii) Consequently, for any cp E CC°(Q x l18) we have u+(x)
(22)
D2Xs (cp(x, y)) =
fni(xJ)( f cp(x, s) ds) dhn-1(x)
J
u_(x)
for i = 1, ... , n and (23)
Dn+1Xs) (cp(x, Y))
=0
In order to prove Theorem 2 we first state
Lemma 1. We have
B,SU C Du where a ,,S,, denotes the measure theoretic boundary of S. Proof. Suppose that (x, t) Du. Then either t > u+ (x) or t < u_ (x). In the first case we observe that for s with t > s > u+(x) we have on (E., (u), x) = 0 by the definition of u+(x). On the other hand if (y, 7-) E Su fl B((x,t), p) then y E ET (u) , l y - x j < p and It - 7-1 < p. Thus for 0 < p < t - s we have
Su f1 B((x, t), p) C (E,(u) fl B(x, p)) x [t - p, t + p] Hence
en+l (Su,
(x, t)) = 0
,
,
consequently (x, t) 0 a,,,Su. In the case t < u- (x) we similarly prove that en+1(Rn+1 \ Su, (x, t)) = 01
i.e., that again (x, t)
i9m&-
Proof of Theorem 2. Since
a-S' C 3,'Su C Du by Lemma 1, the function cp(x, y) and cp(x, u+(x)) agree I DXs,. L ((!2 \ Ju) x R) a.e.. Therefore DZX(cont) (cc)
DiXs,. L (Du n (f? \ Ju) x R)(cc(x, y)) DiXs,. L (((2 \ Ju) X IR)(cp(x,u+(x))) 7r#DiXsu L ((2 \ J-) Mx' U+ (x)))
fcp(x,u+(x))DiuL(S?\Ju)
,
taking into account (3) of Theorem 1. Similarly one proves (20).
In order to prove Theorem 3 we first state
1.5 Subgraphs of BV Functions
377
Lemma 2. Let x E Ju and t E (u_ (x), u+ (x)) be such that x is a jump point for XEt(u) with parameters (1, 0, n). Then (x, t) is a jump point for Xs with parameters (1, 0, (n, 0)). Proof. Recall that z E RN is a jump point for XA, A C RN, with parameters
(1, 0, v), v E RN, if and only if ON
(A n EZ,,,, z) = 0
and
ON ((RN
\ A) n Ez ,,, z) = 0
,
compare Proposition 1 in Sec. 4.1.4.
Now let x E Ju and let s be such that u_ (x) < s < t. If (y, T) E Su n E ly t) (n o) n B((x, t), p) then y E E T (u) n E n and I t -TI < p. Thus for 0 < p <
t - s we have Su n
n B((x, t), p) C (E,, (u) n Ex-'. n B(x, p)) x [t - p, t + p)
Consequently
Bn+1(Su n E(i,t),(n,o) (XI t)) = 0
Similarly using s with t < s < u+ (x) one shows that Bn+1((Rn+1 \ Su) n
(x, t)) = 0
which concludes the proof.
Proof of Theorem 3. With the notation in the proof of Theorem 1 in Sec. 4.1.4 set
All
U (,9,, E, (u) \ a- E, (u)) rEQ
and recall that 7-1n-1(N1) = 0, so that for N2 := Ni x R we have
7- n(N2) = 0 Let now x E Ju \ N1. By Theorem 2 in Sec. 4.1.4 (i), for every t with u_ (x) < t < U+ (x), x is a jump point for XEt(u), hence, by Lemma 2, (x,t) is a jump point for Xs with parameters (1, 0, (n(x, Ju), 0)). Thus Duo) C 8*Su n (Ju x R) u N2
(24)
On the other hand for the set N3
(Du \ Duo)) n (Ju x R) _ {(x, y)
I
y = u_(x) < u+(x)} u {(x, y)
I
u_(x)
we have (25)
Nn (N3) = 0
by the area formula, since N3 projects in Ju and for any x in such a projection there are at most two real yl, y2 such that (x, yl) and (x, y2) belong to N3.
4. Cartesian Currents in Euclidean Spaces
378
Using now (24), Lemma 1, and the inclusion relations among reduced boundary jump points, and measure theoretic boundary, we get
a-S, nJuxR C a,,,SunJuxR c D,,,nJuxR c D(°) U N3 c a*Su n Ju
x R U N2 U N3
hence
?1 L(a-SunJuxIII) = H' LD,(°) = x"'L(DunJux]IF) This proves (i) and (ii). (iii) follows easily from the previous two steps taking into account the (17) and the fact that
7-l' L (Ju x R) = 7-l'-1 L Ju x L' which follows from the rectifiability of J.
o
As a corollary of Theorem 2 and Theorem 3 we can now state
Theorem 4. Let u E BV (,fl) . Then (i) for I u(Du) I-a.e. x in .(2 \ Ju we have
dp(Du)
(x) = n((x,u+(x)),Su)
d ItL(Du) I
(ii) for I u(Du) I -a.e. x in Ju
d p(Du) = (n(x, Ju ), 0) d y(Du) (x)
Finally, let us note that from Radon-Nikodym theorem we know that (Du) (coast) = (Du)' dx + (Du) (C) and
(Du) (C) _ (Du) (coat) L Z where
Z = {x E f2
d(Du) (coast) I
dGn
(x) = +oo}
.
Then we infer
(Duff) _ (Du) (cont) L W where
W={xEO1 ddGl=+oo}xEQI
dGn=O} dlµl
and therefore
(Du)(C) = Du L ((Q \ Ju) n {x E Q I n.+1 ((x, u+(x)),Su) = 0}) by Theorem 4. In conclusion we can then state
2 Cartesian Currents in Euclidean Spaces
379
Theorem 5. Let u E BV(Q). Then
(Du)' = Du L {x E (2 u+(x) = u- (x), nn+1((x, u+ (x)), Su) > 0} = Du L {x E 0 u+(x) = u- (x), nn+i ((x, u+ (x)), Sv,) = 0}
(Du) (C)
(Du)
=
DU LIx E 0 I u- (x)
2 Cartesian Currents in Euclidean Spaces This section in dedicated to the study of Cartesian currents. After a few preliminaries in Sec. 4.2.1, we introduce in Sec. 4.2.2 the class of Cartesian currents cart (S2 x ]RN) :_
{ T E Dn(.f2 X RN) I T is an i.m. rectifiable current in ,f2 x IRN, M(T) < oo, T < oo, aT L 0 x RN = 0,
Too >0, it#T=Q,f2l } and the class of graphs graph(Q x IRN) :_
{
T E Dn(Q X RN) I T is an i.m. rectifiable current in .f2 x RN, M(T) < oc, 11 T II1 < oo, M(8T) < oo, T00 > 0, 7r#T = Q .f2 j }
and we prove closure and compactness properties for those classes, with respect to the weak convergence in cart, i.e., with respect to the weak convergence of currents with equibounded masses and L1-norms.
In Sec. 4.2.5 we shall see that not every T E cart(f2 x RN) can be weakly approximated in cart by graphs Gu, of smooth maps uk. For this reason we also introduce the sequential weak closure of the graphs of smooth maps in
cart(. x RN) Cart(.f2 x IRN) := sw*-cl{Gu
I
u E C1(,f2,IRN) n Al(.f2,]RN)} .
Of course we have
Cart(.f2 X IRN) C cart(f2 x RN) C graph(,f2 X ]RN) . In Sec. 4.2.3 we prove a structure theorem for graphs, showing that for every T E graph(,f2 x 1RN), in particular for every Cartesian current T, we have
T = GuT +S where GUT is the current associated to the 1-graph of a BV-map UT and S is a vertical current, i.e., 7r#S = 0. We shall also discuss relations between the components of T and the minors of (DuT)a
380
4. Cartesian Currents in Euclidean Spaces
In Sec. 4.2.4 we shall consider Cartesian currents of codimension one and we prove that they can be identified with BV-functions in (. In particular we then deduce that Cartesian currents of codimension one are just weak limits of smooth graphs. We conclude Sec. 4.2.4 by brief discussion of the Cantor and jump part decomposition of Cartesian currents. After presenting a few examples of Cartesian currents in Sec. 4.2.5, and in particular discussing with further details the examples in the introduction to this chapter, in Sec. 4.2.6 we shall discuss radial currents, proving in particular that they are weakly approximable by smooth graphs.
2.1 Limit Currents of Smooth Graphs Let {uk} C C'(2, RN) be a sequence of smooth maps from a bounded domain 0 in Rn into RN satisfying (1)
sup (II Uk II L1(S2,IRN) + II M(Duk) IIL1(S2,A, k
+N)) < 00 .
Passing to a subsequence, we may suppose that (2)
uk -k u
weakly in L' (,2,
Ma (Duk) - µ as measures
for lal + 101 = n, 101 > 1, where µa are Radon measures of bounded total variation in .2. Also, since M(Duk) IIL1(,(2,A,
M(Gu.k)
1
+N)
and therefore
sup M(GUk) < oc k
we can suppose that (3)
Guk - T
where T is an n-dimensional current in Dn (.2 x RN). One of our main questions is to characterize such currents T, i.e., all currents T which are limits of regular graphs in the sense (1) (2) (3). Unfortunately, in this general context, this is beyond our possibilities, and it will remain an open question. Therefore, we shall instead try to describe the class of such limits tentatively by successive approximations, by detecting their properties. The lower semicontinuity of the mass with respect to the weak convergence
M(T) < lim inf M(G,,,k ) k-.oo
yields evidently (4)
M(T) < oc
.
As by Stokes's theorem aG,,,k L fl X RN = 0, we also have
2.1 Limit Currents of Smooth Graphs
381
BTL,2XRN = 0.
(5)
Since {Gu,k } is a sequence of i.m. rectifiable currents with equibounded masses and no boundary in .2 x RN, Federer-Fleming's closure theorem yields that
T is an i.m. rectifiable current, T E R,,,(.(2 x RN) .
Consider now the 0-component of T, or equivalently the 00-component in
3xRN
T°D(go(x, y)) := T(co(x, y) dx) _: T(°)(go(x, y) dx) b cp E
x RN) .
Since GOO
0k =
fxnk(xdx
we see that G0k is a positive measure in f2 x RN, thus passing to the limit as k -> oo, also T00 is a positive measure,
T00 > 0.
(6)
Actually the previous remarks can be stated evidently in the following slightly more general form.
Proposition 1. Let {Tk} be a sequence of i.m. rectifiable currents satisfying
supM(Tk)0 Vk. k
Suppose that
Tk
T
in D,,, (.f2 x RN) .
Then T is an i. m. rectifiable current with finite mass and no boundary in f2 x RN, moreover T00 > 0.
Of course under the mere assumptions of Proposition 1 the sequence of currents Guk may just disappear at infinity in the y direction. What prevents this to happen is the equiboundedness of the Ll-norms of Uk. For this reason we now introduce the Ll-norm of a current in ,(2 x RN. Let T be a current in D,(37 x RN) whose 60-component T00 is a Radon measure with bounded variation in .f2 x RN, i.e., (7)
II T II0
:= sup{T(W(x, y) dx)
I
W E C°°((2 x RN), ICI < 1} < oc
We define the L'-norm of T by (8)
II T II1
:= sup{T(W(x, y)IyI dx)
I
W E C2(D X RN), ICI <_ 1}
.
Notice that, being T0° a measure with bounded variation in ,f2 xRN, T°° extends, by Lebesgue's dominated convergence theorem, to all continuous functions with,
4. Cartesian Currents in Euclidean Spaces
382
compact support in .f2 x IRN, and even to all bounded and continuous or bounded and Borel functions in .fl x IRN. Also, taking into account that IToo I
(f (x, y)) =
sup
T" (f (x, y) co(x, y) )
WEC'O (fl XIRN)
wl<1
for all f > 0, we easily deduce
cp Borel in ,fl x RN, I(pI < 1} _ cc Borel in f x 1RN, 0 < cp < 1} sup{IT00I (;o(x, y) Iyl) sup{T00(cp(x, y) Iyl)
ITII1 (9)
I
I
.
Moreover, if II T II, < oo, then we can extend T00 to all Borel functions cp(x, y) in 0 x RN satisfying I cp(x, y)I < c(1 + lyl), compare the end of Sec. 1.1.4 and we easily deduce (10)
II T Ill =
ITool(lyl)
f
=
IyI dlTool
OXRN
Finally, we have
Proposition 2. Let Tk, T E Dn, (.f2 x R'). Suppose that sup(IITk110+11Tk111)
(11)
Tk-sT inD,,,(Qx1RN).
k
Then, obviously IITIIo<_1imnf11TkIlo,
11 Till
_ liminf
In particular II T II1 < oo. Proof. From (11) we deduce that T00 is a measure with bounded variation and
T°°(p(x,y)IyI) = k Tk°(cc(x,y)IyI) for all cp E C°(.f2 x RN)1, thus for Icp < 1 T(cp(x, y)Iyl dx) < lim of 11 Tk ll1 k-oc
For T = Gu where u E Al (.f2, RN) we trivially have 11
GU Ill
sup{ f so(x, u) lul dx
I
c E CO(S?x RN), Iccl <_ 1}
S2
fhuhdx. S?
1 Notice that the previous equality is not true in general for cp E C°(dl x RN).
2.1 Limit Currents of Smooth Graphs
383
Thus Proposition 2, applied to the sequence {G,,,,k } in (3), yields
IITII1 < oo.
(12)
Let T be such that II T IIo < oc, and let 7r denote the orthogonal projection 7r
:
l x R N ---+ f2
,f
7r(x, y) = x
,
.
In these circumstances the push forward of T to the n-dimensional current 7r#T in Dn(Q) is well defined. In fact, as 7r is linear and I ITIIo < oo, T extends to all n-forms with continuous and bounded coefficients in Q x ISBN, in particular on forms of the type V(x) dx, cp E C°(.fl). By the dominated convergence theorem, (13)
7r#T (W(x) dx) := R T(co(x) XR(y) dx)
where for R > 0, XR is a cutoff function in the vertical directions, XR E C o(I18N) and,
0 SXR<1, XR=1inBR(0),
XR = 0 in RN \ B2R(O)
,
IDXRI<2/R.
However 7r# : Dn,(Q x ISBN) -->'D (Q) is not continuous, compare Sec. 2.2.3. Turning back to our initial sequence G,,,, we readily see that 7<#Guk (W(x) dx) = G00
W(x) dx = J f2J (W(x) dx)
i.e. 7r#Guk = Q fl I for all k. We now claim that, because Guk sup II Guk II1
T and,
:= Sup II uk IILI < 00
k
k
we have 7r# T = Q .f2 1.
In fact, more generally, we have
Proposition 3. Let {Tk} be a sequence of currents in Dn(,f2 x ISBN) such that Tk - T in D,,,(,R x ISBN)
and
sup(JI Tk 110 + II Tk II1) < oo k
Then
7r#Tk - 7r#T
in Dn(Q) .
In particular 7r#T = ,f21 if 7r#Tk = Q f2l for all k. Proof. As previously, if XR(y) is a cut off function and cp E C°(Q) we have 17r#Tk (cp(x) dx) - 7r#T(co(x) dx)I
< ITk°(co(x)XR(y)) -TO°(cc(x)XR(y))I + ITk°(co(x)(1 - XR(Y)) dx) - T00(cc(x)(1 - XR(Y)) dx) I
4. Cartesian Currents in Euclidean Spaces
384
The last term can be estimated by
thus, for R sufficiently large, it is small uniformly in k, while the first term on Oc for R fixed. From that the claims the right hand-side tends to zero as k follows at once.
In conclusion we can state
Proposition 4. Let {uk} C C' (,f2, RN) be a sequence for which (1) holds and G,,,,k - T in Dn(.(l,RN). Then T is an i.m. rectifiable current with finite mass, finite norms I I T IIo and I I T IIi, without boundary in QXRN, and satisfying Too >
-
0 and 7r#T = Q Q]J .
The previous considerations can be easily extended to sequences of maps {uk} in A' (12, RN) for which (1) holds, G,,,,k -s T in Dn(Q, RN) and moreover
sup M(aGUk L fl x RN) < oo .
(14)
k
In this case we have
Proposition 5. Let {uk} C Al(.f2,RN) satisfy (1). Suppose that supM(aG,,k L.f2 X RN) < Oc,
Guk - T
in Dn(.f2
k
Then the current T is an i.m. rectifiable current with finite mass and finite boundary mass; moreover II T IIo and II T II1 are finite, and T00 > 0, 7r#T = 101.
Of course (14) is essential in order to get in general the rectifiability of T, compare the examples in Sec. 2.2.4.
Finally we observe that in the previous claims we could replace the norm II T IIl by (15)
II T Ilh
:=
y)h(I yI ))
I
'p E C°°, ICI <_ 1}
where h is a non decreasing function h : [0, oc) -* R, h(t) -> oo as t -+ oo, for instance h(t) = 0, Q > 0, without altering the conclusions (as well as the proofs). However we shall not insist on this point.
2.2 The Classes cart(Q x RN) and graph(, X RN): Closure and Compactness Theorems Motivated by the considerations in Sec. 4.2.1 we now introduce the class of Cartesian currents as follows
2.2 The Classes cart (S2 x RN) and graph(Q x ][BN)
385
Definition 1. The class of Cartesian currents is defined as
cart(. x RN) :_ { T E D,,(Q x RN) I T is an i.m. rectifiable current in 12 x ISBN, M(T) < oo, J T 111 < oo,
8T L .f2 x RN = 0, Too > 0, 7r#T = Q ,f2
}
For future purposes we also introduce a larger class, the element of which are allowed to have non zero boundary in .f2 x RN.
Definition 2. The class of graphs is defined as graph((2 x RN)
T E Dn(f? x ISBN) I T is an i.m. rectifiable current
in .fl x IIBN, M(T) < oo, 11 T I1 < oo,
M(8T)0, 7r#T=If ,fll }.
I
Fig. 4.1. (a) A Cartesian current in (0, 1) x R, (b) A graph in (0, 1) x R.
Of course elements of graph(.fl x RN) are in general not approximable by graphs of smooth maps, and cart(, x RN) C graph(.fl x ISBN) 4
.
We could also introduce intermediate classes, as for instance the class of T in graph(, x RN) for which we have (8T) (o) L 12 x IEBN = ... = (aT)(k) L 17 X RN = 0
up to a fixed k < n - 1. Particularly important is the case in which (8T)(0) L.Q x I[8N = 0
as this condition expresses the standard formula of integration by parts for derivatives. But for the moment we do not dwell on that. An immediate consequence of Proposition 1 in Sec. 4.2.1, Proposition 2 in Sec. 4.2.1 and (3) in Sec. 4.2.1 is the following theorem which makes useful Definition 1
386
4. Cartesian Currents in Euclidean Spaces
Theorem 1. Let {Tk} be a sequence of currents in cart(Q x RN). Suppose that sup{IITk III +M(Tk)} < 00,
Tk
T in Dn(.f2 x RN).
k
Then T belongs to cart(f2 x RN).
Actually, taking into account Federer-Fleming closure theorem, we readily see that the same theorem holds for sequences of graphs provided the masses of their boundaries in (2 x RN are equibounded, that is
Theorem 2. Let {Tk} be a sequence of currents in graph(Q x RN). Suppose that sup M(8Tk L,f2 x ISBN) < 00
sup{II Tk III +M(Tk)} < oo ,
k
k
and
Tk - T
in D,(Qx
Then T E graph(, x RN).
Remark 1. Let T belong to graph(,R x RN). Since M(T) < oo, T naturally extends, by means of Lebesgue's dominated convergence, to all n-forms w = >wap dxandyO with bounded and continuous or even bounded and Borel coefficients in Si x IRN, not necessarily with compact supports in P X RN. Notice however that, if
Tk - T
in D,,(PxlRN),
then Tk(w) does not converge in general to T(w) if w has not compact support in .(2 x RN, even if we assume that supk{M(Tk) + II Tk III} < oo. Remark 2. From Remark 1 it follows in particular that every Cartesian current,
and in fact every graph T in P x RN may be regarded as an i.m. rectifiable current in R X I[BN. In this case however we have no information on OT in
R'n x RN. In particular Cartesian currents in (2 x RN are not necessarely normal currents in Rn x RN.
Finally we would like to observe that in fact Theorem 1 and Theorem 2 are closure theorems. In order to see this, we introduce the subclass of all ndimensional currents in 0 x RN
C :_ {TED,,,(QxRN)
I
M(T)+IITIII
and we set
IITIIC := IITIII+M(T) . Evidently C is a linear space and II IIc is a norm on C. For any n-form
w = E wao (x) y) dxa Adya IcI+IAI=n
2.2 The Classes cart(.f2 x RN) and graph(, x RN)
387
in £2 x RN we set
+
(woo(x,Y) 1 + Iyl
IwIts := sup
czxRN
Iwas(x,y)I) IAI>1
sup E Iwap(x,y)
IIWII. :=
and we denote by Co (12 x RN) the space of continuous functions in 12 x RN with
zero limits as points (x, y) go to infinity or to 80 x RN. We then consider the subclass of all n-forms in £2 x RN woo
B :_ {w Also B with I I-
1+ I
IyI
,
w,,,3 E Co(f2 x RN), Ial + 1,31 = n , 1,81 > 1}
.
s is a normed linear space.
Proposition 1. We have (i) B is a Banach space with the norm II
IIg. Moreover B is separable and
-
D"`(.(2 X RN) is dense in B
(ii) C is a Banach space with the norm II IIc. (iii) C is the dual space of B. (iv) A sequence {Tk} converges weakly* in C if and only if -
sup(II Tk III + M(Tk)) < oo,
Tk
T
in D,,,(0 X RN)
k
In virtue of Proposition 1, Theorem 1 now obviously reads
Theorem 3 (Closure theorem). The class cart(, x RN), and for any K > 0 the class
graph(S2xRN)n{T I M((8T)L0xRN)
,
which are contained in C, are closed in C with respect to the weak* convergence in C.
Definition 3. From now on we shall say that {Tk} c cart(, x 3R') converges weakly in cart
if
in cart (j? X RN)
Tk - T Tk
T
weakly* in C
or, equivalently, if
Tk - T in D,,,(.f2 x RN)
with
sup(II Tk II1 + M(Tk)) < oo k
Moreover we set for T E cart(12 x RN) IITIIcart
11 TIC = ITIII+M(T)
.
388
4. Cartesian Currents in Euclidean Spaces
Proof of Proposition 1. Since Co(.f2 x IRN) is a separable Banach space with respect to the sup norm and C°(.f2 x RN) is dense in Co(Q x RN), (i) follows immediately. The claim (ii) is trivial. Let B' be the dual of B and let T E B'. Then for any w E B we have IT(w)I
(1)
<_
II T IIB, II w IIB
IIB being the norm of B'. For w E Dn (,f2 x IRN) we trivially deduce
IT(w)I < IITIIBIIIWII. Applying (1) to w = f (x, y) I yI dx with f c- C° (92 x 1RN) and If (x, y) I < 1, we get
thus M(T) < +oo and M(T) < II T
hence we also have I I T I
I <_
IITIIB, sup
<_
IT(f(x,y)Iy!dx)
IIB,.
If(x,y)IVI
(x,y) 11 T 1 B ,1
< IITIIB,
1 + Y1
We conclude
IITIII = IITIII+M(T) < 2IITIIB-
13' CC,
Conversely suppose that T E C and let w E B. We split w as w = w1 + w2 where w1
WOO
=
)+
dx-AdyQ I+IPI=n IAI>_1
woo(x,y)IYI
``'2 =
1+IyI
From M(T) < +cc, we get IT(wl)I <_ M(T) IIw1 IIB
,
while from II T I I1 < oc we get T(w2)
IITIII sup
(x,y)
1+I I)I < IITIII IIwiIIB
Therefore IT(w) I
< (M(T)+IITIII) IIwJIB
,
that is, T extends to a linear and continuous functional on B. This proves (iii). Let us finally prove (iv). Suppose that Tk _. T in Df(Q x ]RN)
and
sup 11 Tk llc < cc k
By Proposition 1 in Sec. 4.2.1 and Proposition 2 in Sec. 4.2.1 we get II T III < oo, hence, using the density of D' (fl x ]RN) in B, we deduce
2.2 The Classes cart(.2 x RN) and graph(,2 x IRN)
389
dwEB,
Tk(w) -* T(w)
T weakly* i.e. {Tk} converges weakly* to T in C. Conversely, suppose that Tk in C. Clearly Tk (w) -+ T (w) for all w E Dn (S? X RN), moreover, Banach-Steinhaus theorem yields sup 11Tklls, < 00 k hence
sup IITkllc < 00k
O
Remark 3. We observe that C = 13' is not metrizable, while, being 13 separable, the weak* topology on bounded sets of C is metrizable, thus defined by the weak* convergence.
It is an open question to decide whether cart(,2 x RN) is the smallest sequentially weakly* closed set in C which contains the graphs of regular maps, i.e., with the terminology of Sec. 3.4 whether cart(.2 x IRN) = sw*-cl (f {Gu
U E Cl (,fl, RN) n Al (,2, RN)}) .
I
We therefore set
Definition 4. The smallest sequentially weakly* closed set in C, equivalently, the smallest sequentially weakly closed set in cart(.2 x RN), which contains the graphs of regular maps with bounded C-norms is denoted by Cart(.2 x IRN). Of course
Cart(,f2 X RN) C cart(.fl x IRN)
and Cart(fl x R') is closed with respect to the weak convergence in cart. The previous question amounts then to ask whether the inclusion above is strict or not.
The class Cart(Q x RN) is in principle obtained by transfinite induction up to the first uncountable ordinal, i.e., in C with the weak* convergence
Cart(Q x IRN) = sw*-cl({Gu
I
N)})
.
U E Cl(.fl,IRN) n Ai(.f2,IRN)})
,
u E Cl (2, IRN) n A' (,2
We may again ask, compare Sec. 3.4, whether Cart(,f2 X RN) = sw*-lim({Gu
I
but we do not know the answer. In Sec. 4.2.5 we shall see that cart(.fl x RN) sw*-lim({Gu
I
u E C'(0,1RN) nA'(12,RN)}) ,
Finally it would be natural try to characterize, compare Sec. 4.1, the currents T E cart(Q x RN) for which there is a sequence of smooth maps uk : f --p RN such that
390
4. Cartesian Currents in Euclidean Spaces
Guk - T
and
M(G,,,k) -> M(T).
Such a question is connected with the problem of relaxation of the area integral for non parametric graphs which will be discussed in Vol. II Ch. 6. There we shall see some partial answers. We also remark that cart(Q x TRN) and Cart(.f2 x RN) are neither linear subspaces nor convex subsets of C or of D,,, (,(1 x RN ). This is trivial as a linear combination of i.m. rectifiable currents is not in general an i.m. rectifiable current. One could think of defining the sum of two Cartesian currents at least in Cart(.f2 x RN) by a transfinite approximation process, i.e., defining for instance
T ® S = weak limit of Guk+vk in cart if Guk
c
T
Gvk c- T
and
,
but in general sup II Guk+vk IIc = +oo. k
In fact, for the equiboundedness of the masses of Guk+vk, it is essentially necessary that the L'-norms of Duk and Dvk be equibounded. This simple observation has an important consequence for variational problems with integrand depending on minors of Du. In fact, it roughly means that if one insists on working in a linear space, then one is forced to work essentially with energies which control the Ln-norm of the gradient of the map u.
For future purposes, it is convenient to introduce a more general class of Cartesian currents. Set n = min(n, N) and denote now by p a multi-index p = (pa, p1 i ... , pn) where pi E 1R, pi > 1, i = 0,1, ... , n. For T E D,, (U) we set IITIILPo := sup{T(I yI PO (x, y) dx) 10 E D(U), sup 101 < 11; U
for a n-form w in U, w = II4IIMPk :_
aI
pk/2
f1supf Sl
pk
waa(x, y) dxandyp, we set
Y
IwaR(x,y)I2}
1/pk
dx}
,
aI+IAI=n
being the dual exponent of pk, p' = p', and IIWIIMP
km oax,-
IIwHIMPk.
Finally, for T E Dn(U) we define IIT II MPk :=
Sup{
Ta13(w.0) I w E D'(U), IIwIIMPk < 1} I
f3 =k
2.3 The Structure Theorem
391
and IITIIMP
k
oax
IITIIcartp := IITIILpo + IITIIMp
ITIIMpk,
and we consider the Banach space CP = {T E Dn(U) I IITIIcartp < DO I
Observe that CP is the dual space of the separable Banach space { w I I w I Mp < oo } and that the weak* convergence in CP is equivalent to the weak convergence of currents with equibounded carte norms. Notice that if u E C' (.f2, IRN), T = [ G,, j and Mk (Du) denote all minors of order k, then IITIILpo = IIuIIL1o(2,RN),
IITIIMp
k
f
1/Pk
IMk(Du)Ipk dx}
and in general IITIIMpk is the total variation of the vector valued measure (Tap) IHI+1a1=' if pk = 1, while for pk > 1 IITIIMpk is the LPk-norm of the Al=k vector valued Radon-Nykodym derivative of with respect to J31=k
Lebesgue's n-dimensional measure.
Definition 5. For p = (po, ... , p,,,), pi E IR, pi > 1, we set cartP(,f2 x RN) := CP n cart(.f2 x RN)
and define Carte (.2 x RN) as the smallest set in CP containing the currents in CP integration over smooth graphs and sequentially closed with respect to the weak* convergence in CP.
Of course carte (12 x R') is sequentially weak* closed,
CartP(Q x RN) C cartP((2 x RN) C cart(.? x RN). We shall return to this class in Vol. II Ch. 4 and Vol. II Ch. 5 in connection with the study of harmonic maps with values in S2.
2.3 The Structure Theorem In Sec. 4.2.2 we introduced the class of Cartesian currents. Our goal in this subsection is to prove a structure theorem which shows that every T E cart(Q x RN) is in a suitable sense integration over graphs of maps with vertical parts. For the sake of clarity we shall split such a structure theorem into several statements.
Let T be a current in graph(, x RN). Since II T II1 + M(T) < +oo we may define an RN-valued measure in (2
4. Cartesian Currents in Euclidean Spaces
392
u(T) = (u'(T),...,uN(T)) by
T(co(x) y2 dx) =
(1)
yj)
V W E CO (j?),
j = 1, ... , N, and the measures Ma (T) in Q, Ja1 + (8 = n, 1,31> 1, by Ma (T) (po(x)) = o-(a, a) T (po(x) dx'AdyA)
(2)
= o(a, a) Taa (c2(x))
for all cp E CO (Q). Since T is an i.m. rectifiable current T = T(M, 0,T) where M is an n-rectifiable set in Rn+N 0 an H'-measurable and integer-valued function,
and f the n-vector orienting M, we may also assume that the tangent space exists for all z E .M, and that 9(z) exists and is larger or equal to 1 for any
zEM.
We denote now by M+ the set of points z in M for which the tangent space to M at z does not contain vertical vectors. Proposition 1 in Sec. 2.2.1 yields
M+ = {z E JVt
(3)
I
T00(z) > 01 .
Our first theorem describes some properties of T restricted to M+ and in particular shows that the Young measure T00 is a function.
Theorem 1. Let T E graph(.( x RN), T = T(M, 0, T). Then the measures uj(T), j = 1, ... , N, are absolutely continuous with respect to Lebesgue measure Gn, i.e.,
u& (T) = uT(x) dx
.
If we set (uT(x),...,uT(x)) E L1(Q,RN)
UT(X)
,
then UT (X) is a. e. approximately differentiable, UT E A' (0, RN), and
TLM+ = GUT Actually we shall prove
Theorem 2. Let T E graph(f2 x RN), T = T(M, 0,T ). Then we have (i) C'(fl \ 7r(M.+)) = 0. There exists a set f2 C ir(M+) such that Ln(,f2 \
1,) = 0 and for every x E Q1 there exists a unique ii(x) E RN such that (x, fi(x)) E M+ and
T (x,y)Eir-1(x)nM+
0(x,y) = 1
2.3 The Structure Theorem (ii)
393
The measures uj (T), j = 1, ... , N, are absolutely continuous with respect to the Lebesgue measure L', i.e., u3 (T) = uT(x)dx
where (uT(x),...,uT(x)) E Ll(Q,RN)
uT(x) = Moreover,
uT(x) = u(x)
a.e. in .f2
(iii) H (M+nAxRN) = 0, whenever 7-C (A) = 0. In particular 0(z) = 1 7-1? -a. e. in M+, the map illn, has the Lusin property (N) in ,f21 and,
M+ = {(x, y) E 52 x RN I x E ir(M+), y = u(x)}
7-l"' a. e.,
(iv) UT is a.e. approximately differentiable and M+ _ !9uT,n 7{", a.e., 9,,,,.r2 being the 1-graph of UT. ( v ) UT E Al (.f2, RN) and T L Nl+ = GuT
.
Proof. First observe that II T II0
= sup{T(cp(x, y) dx)
= sup { =
(4) II T 111
=
f f
B
E Co (,f2 x RN), ICI < 1}
I
f cp(x, y) B(x, Y) T 00(x, y) d7-Cn L M }
y) T
60
(x, y) d7-l"' M
y)Iyl dx)
I
E C°(Q x RN), ICI <_ 1}
IyI9(x,y)To0(x,y)d7-l'1L.M+ .
Since II T I10 < oc, the function cp(x) 0(x, y) T 00 (x, y) is summable with respect to 7{n L M+ for any cp E C° (,f2). Thus, since by Proposition 1 in Sec. 2.2.1 (iii)
the area formula, compare Sec. 2.1.5, yields (5)
f
o (x) B(x, y)
to" (x, y) dfl L Nl+ =
f p(x) N(x) dx ir(M+)
where N(x) is the Lebesgue summable function
N(x) Comparing (5) with
yE7r-1(x)fM+
B(x, y)
4. Cartesian Currents in Euclidean Spaces
394
J2
co(x) dx =
J ir(M+)
po(x) N(x) dx
,
which follows at once from n#T = Q 0 , we readily deduce that G71 (Q\7r(M+)) =
0 and N(x) = 1 7-p -a.e. in (l. Since O(x, y) > 1 for all (x, y) E M and N(x) = 1 a.e. in 7r(M+), we also deduce the existence of a uniquely Nn -a.e. defined function u(x) _ (u1(x),... , uN(x)) such that (x, u(x)) E M+. This proves (i). 60
< oo, also the functions W(x) yi O(x, y) T (x, y) j = 1, ... N Since 11 T are R' Nf+ L summable for all cp E C° (S?). Applying again the area formula we
therefore infer that the function y19(x, y)
4T(x)
(6)
yEir-' (x)nM+
is f'-summable in 12 and for all cp E C°(Q)
f
f(x)Yio(x,Y)°°(x,Y)dnLM+ =
u' (T)
(x) UT (x) dx .
ir(M+)
This yields that the measures u3 (T) are absolutely continuous with respect to Lebesgue measure and in conjunction with (6) we get u3T(x) = u(x) a.e. in 17, UT E L1(Q,RN). This proves (ii). For any measurable subset A of 17 we have
M(T) > 7-l,(M) > 7in(m+ n A x R N) oo (x,
1
=
2 00 (x,
y) d7i' L (M+ n A x RN)
y)
Since M(T) < oo, the area formula yields that the function
K(x)
_
1
VE7r-'(x)nM+ T oo(x, y)
1
00(x, UT (x))
is summable in 1 and
7-l"`(M+ n A x IAN) = fK(x)dx. A
Thus 7l" (M+ n A x RI) = 0 if 70(A) = 0. In conjunction with step (i) we infer that
0(z) = 1
(7)
7f a.e. in M+
and (iii) is proved.
Since M is rectifiable, taking into account (i), one can decompose M+
as M+ =
with 7- '(Mj) < oo, fl' (Mo) = 0 and for each j =
2.3 The Structure Theorem
395
1,2,..., Mj = {(x,y) I y = u(x), x E ir(.Mj)}, 1r(Mj) C -121i Mj C N; where N; is a C1-submanifold with Tang, (z) = T (z) V z E M2, and u is the function in Step (i). Since for every z = (x, ii (x)) E M j we have Do
Tanv (z)O" = T (z) > 0
(8)
the tangent plane to N; at z does not contain vertical vectors. Therefore there exist a neighborhood U of y and a map v : U --> RN, v E C1(U) such that N; n (U X RN) is the (smooth) graph of v, Nj n (U x IRN)
(id >a v) (U)
and by Proposition 1 in Sec. 2.2.1,
TanN, (x, v(x)) =
(9)
M(Dv(x)) M(Dv(x))
Since Mj C .N,, u and v agree on 7r (M;) n U. This way we can decompose Uj_1ir(Mj) n.f21 as union of measurable pieces in which u coincides with a C1-function. This implies that u is approximately
differentiable at a.e. x E (7, and therefore at a.e. x E (, by Theorem 3 in Sec. 3.1.4. Moreover, taking into account the Lusin property (N) of u, we have 9Cu,n, = {(x, y)
I
x E (,, y = u(x)} ?C a.e.
Therefore UT is a.e. approximately differentiable, too, DtT = Du a.e. and,
uT,sz = u,ftl = M+
(10)
1-l' a.e.
This proves (iv).
Finally we also have for a.e. X E ir(M) n U apDu(x) = Dv(x). Taking into account (8) (9) we get (11)
T (x, fi(x))
_ M(apDu(x)) M(apDu(x))
a.e. X E .fl.
and collecting (7), (10) and (11) we then conclude that T L- M+ := T(M+, 9, T) = T (9UT,SZ J)
M(DUT)
I M(DUT) I) = GuT F-I
Next theorem yields a formula of representation for the zero component Too of T.
Theorem 3. Let T E graph(f? x IRN), T = T(M, 9, T), and let UT E Al be the map associated to T in Theorem 1. Then we have
4. Cartesian Currents in Euclidean Spaces
396
(2)
For all continuous functions O(x, y) in ,f2 x RN satisfying I q5(x, y) I < c (1+ IyI) we have Too(O(x,
y)) = J(xUT(x))dx.
In particular
IITII1= JkLTdX (ii)
The function UT E BV (.f2, RN) if and only if the L1 -norm of 9T, defined as II aT II1
:=
E CC°(Q,Rn), ICI < 1} ,
I
is finite. In this case we have
IIDiu'T-M (T)II < 1aTII1 and
< M(T) + II aT IIi In particular UT E BV(Q, RN) and Diu' = M? (T) if T E cart(Q x RN). (iii) Finally for any integer k the truncations at levels k of each component of II DUT II
UT
UT k := max(min(u', k), -k)
belong to BV (Q), and II Diu' k II
< M(T) + c(k) M(ST)
In particular UT E BV (Q,RN) if UT is bounded.
Proof. Note that by (4) co(x, y) O(x, y) T 00(x, y) is 1-ln L M+-summable for all 0 E C° satisfying I O(x, Y) I < c(1 + Iyl). Moreover we have T00
T oo (O(x, y) dx) = f O(x, y) 9(x, y)
d7-ln L M+
= (T L M+) (O(x, y) dx) = J 0(x, UT (x)) dx a
by (iv) of Theorem 2. This proves (i). Let us prove (ii). Consider the (n - 1)-forms on Q x where co E C' (,f2) and DpI E L°°. We have
w := y' c (x) dx2
dw = (-1)2-1 Dice y' dx + cp(x) dy' Adxi . Since the coefficient of (dw)(°) grows linearly in y and the coefficient of (dw)(1) is bounded, we can compute T(dw), by approximation, as limit of T(drl), the y's
2.3 The Structure Theorem
397
being smooth (n - 1)-forms in ,f2 x RN with compact support and 17 = rl(°). We then deduce
T(dw) _
(-1)i-1 { fD(x)dx+M(T)((x))}. From this the conclusion follows at once. To prove (iii) it suffices to choose as w the form
w := max(min(yi, k), -k) cp(x)dxz and proceed as in the proof of (ii). 1 It is easily seen that Theorem 3 (vi) is optimal in the sense that in general UT does not belong to BV(f2, RN) if T is merely in graph(,f2 x IRN). Consider for instance the function u : B(0, 1) C I[ R2 -> JR given by
u(x) = k2 on Brk \ Brk+1 ,
rk := k-2 ,
k = 1, 2, .. .
one easily sees that u E L'(B(0,1)), Gu E graph(B(0, 1) x R), but u BV(B(O, 1)).
We denote by M9 (T) = (Mg (T)) a dx + (Ma (T)) s
and, when UT E BV (D, RN) , by
DUT = (DUT)a dx + (DUT)s the Lebesgue decompositions of the measures MM (T) and DUT with respect to the Lebesgue measure in 12. Clearly
(DUT)a(x), Mg(T)a(x) E L1(Q)
Theorem 4. Let T E graph(,f2 x JN), T = T(M, 0,T'). Then we have (i) For any a,Q with ia1+ 101 = n, 1,31 > 1, (12)
Mg(T)a(x) = Mg(apDuT(x)),
Consequently, for any (13)
for a.e. x E .f2
.
E C°(12 x RN) we have
(Tca L M+)(O(x, y)) = a(a, a)
f(x,uT(x))Mg(T)a(x)dx. .a
398
4. Cartesian Currents in Euclidean Spaces
(ii) If (7r#IIT" II)S = 0, then T"Q 11 i_ (.M \ M+) = 0, consequently, in this case,
T" (0(x, y)) = Q(a, a) f O(x, UT (X)) Mg (aPDuT(x)) dx
.
In particular, if (7r# II T")3 II)S = 0 for all a, a with I a I + 10 I = n, then
II T II (M \ M+) = 0 ,
i.e. M = M+ 1f -a.e.,
and this implies that T = G,,T.
Proof. Taking into account that the total variation of the measure T"3 L M+ is finite, we have (14)
(T "a L M+)(cp(x) dx) = a(a, a)
f(x) Mg (apDuT(x)) dx
.
.a
Let N be a Borel set such that ?in(f2 \ N) = 0 and fn(M+ \ (N x RN)) = 0, which exists because of Theorem 2. Observing that -7r# (To"' L (M \ M+)) is concentrated over fl \ N, the identities 7r#T"Q = r#(T "Q L M+) + -7r# (T" 3 L M \ M+) (15)
7r#IIT"III =
#(IIT" JILM+)+7r#(IITIIIILM\M+)
yield the Lebesgue decompositions of the measures -7r#T"Q and 7r# II T"p I
From the definition of Mg (T) and the uniqueness of Lebesgue's decomposition we therefore deduce that
7r#(T"RLM+) = o(a,a)Mg(T)a(x)dx or, equivalently, from (14) (16)
Ma (apDuT(x)) = Mg(T)a(x)
for a.e. x E (2
.
This proves (i). In order to prove (ii) it suffices to observe that from the second identity in (15) we deduce (ir# I
T"Q
II)S = 7r#(
IT-" II LM\M+),
therefore (7r# (I T",6 II )S = 0, which yields at once TO L M \ M+ = 0.
11
Next theorem concerns the convergence of uT,k and Ma (Tk) when the Cartesian current Tk converge in cart to T.
2.3 The Structure Theorem
399
Theorem 5. Let {Tk} C graph(fl x RN) be such that sup(M((8Tk) L f2 x RN) + II (8Tk) L f2 x R N IIi) < oo; k
in particular, let {Tk} C cart(f2 x IRN). Suppose that
Tk - Tin D,, (f2
and
sup(II Tk II 1 +
M(Tk)) < 00
k
then
(17)uTk -+ UT strongly in L'
and
DuTk
DUT as measures in f2
Moreover, for all a, Q with IaI + IQI = n and 10 > 1
as measures in it
Mg (T)
Ma (Tk)
(18)
provided the uTk 's are equibounded in L°° (f2, RN).
Proof. Since
II uTk II L'+IIDuTkII 5 IITk1II+M(Tk), the sequence {uTk} is equibounded in BV(f2,RN). Thus there exists a subsequence uTk which converges strongly in Ll(f2,RN) to some v E BV(f7,IRN).
For0ECO°(f2xRN) TkO(O(x, y)) = Tkh (cb(x, y) dx) --+ T"(4'(x, y))
on the other hand TkO0(O(x, y))
f(x,u)dx
= f o(x, uTkh (x)) dx -+ n
n
thus
f(x,v(x))dx n
= fc5(x,uT(x))dx. n
Since the last equality holds for all 0 E C°°(f2 x RN), we conclude that v = UT,
and, as the limit is independent of the subsequence, that the entire sequence {uTk} converges and (17) holds.
If {uk} is equibounded in L°°, choosing test functions O(x, y) of the type (X, Y) = cp(x)XR(y), cp E C°°(Q) and XR a cut off function in RN, for large R we readily get (18).
We can in fact prove a slightly more general result on the convergence of minors at least for sequences of maps in cartl(f2,RN). In order to do that we need the notion of ACP weak convergence we introduced in Sec. 1.2.7.
400
4. Cartesian Currents in Euclidean Spaces
Proposition 1. Let {uk} C cartl(Q,1RN) be such that (19)
Uk --> u strongly in L'(Q,RN), and sup II Uk Ilcartl < 00.
Suppose that (20)
O_R := limsupM(GUk L .(l x (8N \ B(0, R))) -- * k-oc
0
as R -> oo. Then u E Al ((2,RN), (Du)a =apDu and for each a, p, l al + A = IQI>-1 n, Ma (Duk) A2.. MI(apDu) Proof. First let us assume that moreover
Gv,k - T
(21)
weakly in cart((2 x RN). Then we claim that (22)
Ma (Duk) = Ma (GUk) - Mg (T)
as measures.
To prove this for R > 0 we consider the function
z'R(y) := max(min(1, R + 1 - IyI), 0)
(23)
which belongs to CO(RN) and satisfies "bkI < 1, and any function cp
E
C°((2), I W I< 1. We have
lim sup IGua(W) -Tap((p)I <- limsup[IG"(co R) k-oc
k-oo
+ IGup(cP(1 - GR))I +
2 UR
because from (20) and (21) we have
M(TL.(lx(RN\B(0,R)) < QR. Letting R -> oc, we obtain (22). By Theorem 5 (17) we have u = UT and by Theorem 4 (i) we have (24)
MM (T)°'
= Ma ((DUT)a) = Ma (apDu)
.
Fix a, 0; for every subsequence {uk, }, uki u, we can extract a further subsequence {uk. } such that (21) holds. Applying the above reasoning we deduce from (22) and (24)
,a = M(T) a = M« (apDu)
and this proves the proposition.
2.3 The Structure Theorem
401
Remark 1. Proposition 1 extends Theorem 2 in Sec. 3.3.2, showing that equiintegrability of minors near infinity is sufficient to ensure convergence. We do not know whether assumption (20) is really necessary. In fact we are not able to prove or disprove the implication (21)x(22) without assuming (20). This is related to the following question. Let U, Uk E cart' (fl, RN), suppose that uk -> u in L' (,fl, RN)
and
sup 11 uk Ilcart1 < 00 k
We have two kinds of convergence (a) (b)
as currents Guk -s G.,, as measures M(DUk) -1 M(Du)
Without assuming equi-integrability of the sequence {M(DUk)} we are not able to prove or disprove complete relations between (a) and (b). Remark 2. We collect here a few observations which aim to illustrate the meaning of the previous theorems.
(i) Theorem 2, Theorem 4 and Theorem 5 hold for graphs T. In particular they hold for all Cartesian currents. Notice however that we have not used the full information
8T L ,fl x RN = 0.
(25)
(ii) Examples in in Sec. 4.0 and 2 in Sec. 4.0 of the introduction to this chapter clearly show that a Cartesian current T, even though in sw*lim(C'(Q,RN) n A'(Q,RN)), is not determined by its UT. There may be infinitely many different T's to which it is associated the same map UT in Theorem 2 and Theorem 4, compare also Sec. 4.2.5 and next chapter.
(iii) The structure theorem yields a complete description of currents T in cart(,fl x RN), T = r(M, 0, T), when restricted to M+. Examples in Sec. 4.2.5 show that in general we may have (i) (ii)
?-Ln (M \ M+) > 0
On M \ M+ the density 0 can be any positive integer.
In fact the situation can be even more complicated. Let B2 be the unit ball in R2, let yo, yl be the point in R2, yo := (0,1), yl :_ (0, 1/2), and let Go be the current in B2 X R2 integration over the graph of the zero constant map. As in T in Sec. 4.0 and 02 in Sec. 4.0 of the introduction of this chapter it is easily seen that the currents R S
Go+Q{0}j x 8QB(yo,1) Go +2Q{0} x 8QB(yl, 1/2)]
are approximable in cart(B2 x R2) by smooth graphs, therefore they are Cartesian currents, and obviously different, but UR = us, and for all a,)3
402
4. Cartesian Currents in Euclidean Spaces
r#R00 = 7r#S«p and even
ir# RaR 11 = ?r#
Saa
Also
7r# R`0 = 0 ,
7r# R°° 11 = 2 8o
0.
This shows that no information on 7r#T can allow to recover T on M \ M+, and that Theorem 4 (ii) is optimal. The vertical part T L M \ M+ of a Cartesian current is essentially uncontrollable in terms of UT.
(iv) In terms of functions the previous remarks say the following. Consider a sequence {uk} of smooth maps such that
sup(J iUkidx+ J IM(Dnk)I) < +00 and suppose for simplicity that the Uk's are equibounded in LO°. Passing to a subsequence, we may assume that there exists a Cartesian current T in Q x RN such that Uk ` UUT
strongly in Ll (,f2, RN )
o- (a, a) M« (Duk) xn
M(Duk)I
L
uw,n
as measures in f2 x RN
TaQ
o (a, a) Ma (D'ak) -i o (a, a) Ma (T)
as measures in 0
M"(T) = M«(DuT(X))dx + Ma (T)3
In general M(T) s # 0; even if Mg(T) ' = 0 for all a and Q, T`-' need not be equal to
a(a,a)Ma(DuT(x)) 'H' LNl+
.
I M(DuT (x))
Moreover, while all formulas of integration by parts are available for uk and Ma (Duk) as well as for T (compare Sec. 3.2.2), they do not hold for UT and Mg(DUT(x)) Remark 3. For future purposes we observe that the following claim is an immediate consequence of the area formula: Let T E V n (T Y X RN) be an i. m. rectifiable n-current satisfying
M(T) < oo
,
Too > 0
,
7r-#T = I01
for some open set (2 in R'. Then spt T C (2 x RN.
2.4 Cartesian Currents in Codimension One
403
Finally we explicitly state the following easy consequence of the previous results
Proposition 2. Let T E graph(,f2 x IRN), T = T(M, 0,T) - Then we have (i) There is u E cart'(.(? X RN) such that T = Gu if and only if and
(7r#11T11)s=0 "ii)
aTL.f2 x1RN=0.
There is u E cartr(.fl, IRN), p > 1, such that T = Gu if and only if
f(T(z))1dLM < oo
.
and aT L .f? x RN = 0.
2.4 Cartesian Currents in Codimension One In this subsection we consider more specifically Cartesian currents in codimension 1, i.e., elements of cart(fl x IR). We shall prove that they are boundaries of subgraphs of BV-functions, and indeed they can be identified as BV functions in Q. The theory of Sec. 4.1 then applies and provides us with explicit structure formulas for those currents.
In doing that the simple relevant remark is that, as it is well known, flux integrals can be transformed into integrations of differential forms, or in terms of vectors, to every unit vector v E Rn we can associate the (n - 1)-vector *v orienting the orthogonal (n - 1)-plane to v, so that
vA*v = e1A...Aen, and defined in coordinates by n
(1)
n
*v=
(-1)i-1 vi
ei,
if v =
v2 ei
Given a measurable set E C ]Rn, we easily see that the distributional derivative DiXE of the characteristic function XE of E are related to the components of the current a Q E as follows Proposition 1. For any cp c C°O (IRn)) we have (2)
0Q E (W (x) (@) = (-1)Z
J
W(x) d DiXE,
i = 1, ... , n
Consequently E is a Caccioppoli set if and only if
Mu (a! E J) < oo for any open bounded subset U of R. In this case we also have (3)
aQ E I (w) = -
J
<w, *n(x, E) > dIDXE l
.
404
4. Cartesian Currents in Euclidean Spaces
cpxa (x) dx1A... ndxn and
Proof. In fact, if w = W(x) dxi, then dw = .QE"
(x) dxi)
= (-1)Z-1
JE
DiW(x) dx
= (-1)x f W (x) d DiXE If
n
n
wi (x) dx' ,
w=
dw = (-1)2-1 wi
(x) dxl n ... ndxn
i=1
i=1
and E is a Caccioppoli set we find
aQEJ(w) = f wi(x)(-1)in2(x,E)dIDXEI
= - f <w, *n(x, E) > djDXEI We note that the sign minus in (3) is due to the fact that n(x, E) is the inward normal to 8-E. We can now restate De Giorgi's theorem in Sec. 4.1.3 as
Theorem 1. Let E C iR' be a measurable set for which +oo
for any open bounded set U C R'. Then aQ E is a locally i.m. rectifiable current with density 1, and
afEj(w) _ - f <w,*n(x,E)>dRn-1La-E. In particular BQE] (x) = - * n(x, E), and aQE] = -r(8-E,1, - * n(x, E)). Let now u : !2 C II -> R be a measurable function. We denote by SGu the (n + 1)-current in W2 x R defined as integration over the subgraph of u
SGu := QS9u,n For the sake of simplicity we also write Su
for Sgu,.n
.
Applying Proposition 1 to SGu we then get aSGu(W(x, y) dxiAdy) = (-1)z .f W(x, y) dDi,Xsg,.,n (4)
aSGu(w(x, y) dx) = (-1)n+1 f W(x, y) dDn+1Xsgu,r: for all co E C°° (Q x R) and i = 1, ... , n.
2.4 Cartesian Currents in Codimension One
405
Fig. 4.2. Subgraph of a BV function.
Theorem 2. Let u E L1(Q). Then u E BV(f2) if and only if M(8SGU) < 00. Moreover the current T defined by
T := (-1)n BSG,, L ,(2 x
IR
is a Cartesian current, T E cart(Q x R), and
T (w) = -
J
<w, *n(x, Su) > d?-l' L C9-Su .
Finally, 7r#IT°°I = Gn
,
rr#TxoI = IDiul ,
7r#IIT II
= Iµ(Du)I
where p(Du) is the measure in 0 (Dlu, ... , Dnu, -L'))- in particular as M(T)=IITII(f2 xR)=rr#IITII(f2), we have M(T) = sup{T(w) w = rr#(r7), rl E Dn(Q), IrII < 1} (5) I
Proof. By Proposition 1 M(aSGu L ,f2 x 1[8) = P(Su, f2 x R). Therefore all claims,
except for T E cart(.(2 x ]IP), follow at once from Theorem 1 in Sec. 4.1.5. We now observe that
T0'(w(x,y)) =
f(x,n(x))dx n
by (4), thus Too > 0, II T II1 < oc and rr#T = QQj. Obviously we have in 0 x III
aT = (-l)naSGu = 0. In order to conclude that T E cart(.(2 x R) it remains to show that T is an i.m. rectifiable current. This is exactly the content of De Giorgi's theorem as restated in Theorem 2 above.
We emphasize the fact that (5) says that the mass of T can be computed in terms of the projections of its components. This is typical of codimension one,
406
4. Cartesian Currents in Euclidean Spaces
and in fact not true in codimension larger that 1, compare Remark 2 in Sec. 4.2.3 (ii).
We shall now prove that actually every Cartesian current, T E cart(.f2 x IE8), can be written as
T = (-1)nBSG,,,LQxR
for some u in BV(Q). Therefore cart(.f2 x l) is characterized as the class of currents which are boundaries of subgraphs of BV-functions. From Theorem 4 in Sec. 4.2.3 we know that every T E cart(.f2 x R) can be written as T = GuT + ST where uT E BV (,R) is given by
UT =
dir# (T L y dx)
dJn
We have
Theorem 3. Let T E cart(.f2 x Ti). Then T = (-1)naSGUT know that Proof. From Theorem 3 in Sec. 4.2.3 we (i)f(xuT(x))dx, T(O(x, y) dx) =
(-1) n 5SGu.T (O(x, y) dx) = in O(x, UT(x)) dx
.
The claim then follows at once from Proposition 2 below.
Proposition 2. Let T E E),,(1? x R) be a boundaryless current, aT L (1 x R = 0. Then T = 0 provided T(cp(x, y) dx) = 0 for all cp E C,° (Q x R). Proof. It suffices to show that T(w) = 0 for all form of the type
w = cp(x, y) dxiAdy ,
cp E C,-(f2 x IR)
,
i = 1, ... , n .
Set
O(x, y) :=
J
W(x, s) ds
,
-00 and
e := cb(x, y) dxx
for a fixed i, i = 1, ... , n. Clearly dl; = a(x, y) dx+ (-1)1-1 w. Being 0 bounded in .f2 x IR, we then conclude
T(w) = (-1)n-1
-T(a(x,y)dx)] = 0
since aT L (1 x JR = 0 and T(a dx) = 0 by assumption.
2.4 Cartesian Currents in Codimension One
407
It is worthwhile noticing the homological character of the argument in the proof of Proposition 2. On account of the structure theorem for BV-functions, compare Sec. 4.1.4 and Sec. 4.1.5, we can now give an explicit representation formula for currents in cart (S? x Ii).
Let T be in cart(Q x li), T = T(M, 1, T). Denote by M+ the set of points z E M at which the tangent plane Tan,z.M is not vertical, or equivalently the projection map 7r restricted to TanzM has maximal rank n. From Theorem 3 we have
T = (-1)"` BSG,,, for some u E BV(.Q), actually for u = UT, and from Theorem 2 0'-1(7r#IITII,x)
0"-1(IDuI,x)
= 0'i-1(IIL(Du)I,x) =
hence the jump set of u can be written, because of Theorem 1 in Sec. 4.1.4, as JJ,
= {x E .fl {x E s2
I
I
u-(x) < u+(x)} 0n-1(ir# II T II, x) > 0}
.
We now set T(a)
:= T L M+
T(i)
T L (J,,, x R)
:= TL(M\(M+U(Ju xIR)))
T(C)
Obviously, T decomposes as
T = T(a) +T(i) +T(C) and the three measures II T(a) T(A II and II T(c) II are mutually singular. Taking into account Theorem 2 in Sec. 4.2.2 and the structure theorem for BV-functions, equivalently for subgraphs of BV-functions, we can then easily infer
Theorem 4. Let T E cart (Q,li). Then there exists a unique function u E BV(Q), given by u = UT such that, decomposing T as
T = T(a) +T(j) +T(C)
,
we have
T(-) = GUT. and
T0) (cb(x, y) dx) = T (C) (c(x, y) dx) = 0
V 0 E Cc°(Q x IR)
Moreover, for any n-form w = O(x, y) dxindy, i = 1, ... , n, we have
.
4. Cartesian Currents in Euclidean Spaces
408
T(a)
(-1)n-Z
(W) _
J O(x, u+(x)) (Du)' dx u+(X)
T(1)(W)
f q5(x, s) ds) n2(x, Ju) d7-l'-1 L Ju U- (x)
T(C)(w)
O(x, u+ (x)) (Diu) (C)
An immediate consequence of Theorem 4 is
M(T) = f V-(Du)a dx+ f [I(Du)(C)I+I(Du)U)l]
(6)
n
S?
Finally we notice that as consequence of the approximation theorem for BVfunctions in Sec. 4.1.1 we infer at once
Proposition 3. Let T E cart(Q x R). Then there exists a sequence of smooth maps Uk : Q --> li8 such that
Guk - T
and
M(Guk) -> M(T)
,
in particular
Cart(Q x R) = cart(Q x IR)
.
Cantor- Vitali functions. Let V be the Cantor-Vitali function in Sec. 1.1.3. It is easily seen that (i) (ii)
V E BV(0,1), since it is non-decreasing, and V(0) = 0, V(1) = 1. On each interval in [0, 1] \ Ek we have V (x) = fk (x) = constant. In particular, V (x) is differentiable on each x E [0, 1] \ C, and (approximately) differentiable almost everywhere in [0, 1] since C1 = 0, and
apV'(x) = 0 (iii)
for a.e. x E [0,1]
.
Since V is continuous, V' has no jump part, and therefore
V' = VIM and V' is concentrated in C, V' - C = V'. (iv) Obviously V([0,1]) = [0, 1], V maps [0, 1] \ C into the denumerable dyadic
set
D := {y E I[8
I
y = Zk , j E [0, 2k], j, k E N}
.
Therefore V maps C onto [0, 1] \ D. In particular V has not Lusin property (N).
2.4 Cartesian Currents in Codimension One
409
From the structure theorem we infer that -BSGV is an i.m. rectifiable current 7(M,1, C) and, setting
M+ := {(x, y)
(C., Sy), ex > 0}
I
we have
-BSGv L M+ = Gv = GV L ([0,1] \ E) x R , whence ¶
M+
lj,k 1 X Syk,,
ykj = j/2k
j,k
In particular
M+ _ {(x, y) E [0,1] x [0,1]
I
= (1, 0) } = {(x, y) I y = 0}
and
M = {(x, y)
I
Sy = 0} U {(x, y) I G = 01
.
Notice the difference between the 1-graph of V and the graph of V in the sense of continuous functions. In terms of the points
+ bk+1 ' V(bk,j +
Ak,j
(bk,j
Bk,j
(bk,j + 5k (1 - a), V (bk,j + 1 jk))
2
Sk))
k = 0,1, ... , j = 1,... , 2k, we can write the 1-dimensional current graph of V as 00
>
Gv
2k
I[
k=0 j=1
where denotes the oriented line segment from A to B. Its boundary o0
2k
8Gv = 1: E(QBk,j]1 - Ak,j k=0 j=1
is a 0-dimensional current, in fact a flat chain. In principle it can act on Lipschitz functions, indeed for Lipschitz functions cp(x, y) we have 00
Ic7Gv(cp)I
2k
< Lipcpj:Edist(Bk,j,Ak,j) = k=0j=1
k=0 j=1
00
<
2kbk
c
k=0
c26 < 00
4. Cartesian Currents in Euclidean Spaces
410
but not on Borel functions. In fact we have M(aGv) = +oo, corresponding to the intuitive idea that the graph of V has infinitely many holes. In order to prove that M(aGv) = +oo we can argue by contradiction. Assume M(aGv) < oo, then aGv L B is well defined for any bounded Borel set and obviously we have
M(aGv) > M(8Gv L B)
.
By taking as B the set 2k}
U{< Ak,3, Bk,j > I k
we then infer
M(aGv L B) = 2(2k+1 _ 1)
which for k --> oo, yields M(aGv) = oo: a contradiction. For future reference we would like to present also a direct proof. Let 77(x, y)
cp(x)O(y) be a test 0-form with cp and
Lipschitz functions, so that dry
cp'-tp dx + cpzb' dy and
aGv(rl) = Gv(drl) = J['(x)(V(x)) +w(x)O'(V(x))V'(x)] dx cp'(x)V)(V(x)) dx 0
as V' = 0 on the 1-graph of V. We then have 1
M(aGv) = sup
{ f cP'(x)0(V (x)) dx
1}
101 :5
ICI
0
We now choose cp and 0 suitably. First we choose cp and 0 as piecewise linear functions satisfying
VW1-b))=cp(5) 0(0) = &(1/4) = 0(3/4) = 0(1) = 0
,
b(1/2) = 1
Then p'=-1and7P = 1 onandcp'=1,
=0on.
Thus f cp'(x)zb(V(x))
dx = -1
.
By induction and using the self-similarity o"f. V we now ow set c01 := cP and
,
4'1 = Y
.
2.5 Examples of Cartesian Currents (b-2x)
411
for x > b Ok (4y) , lk+1 = 0 for y > 1/4 . Ok+i (y) By construction the functions Wk(x)Ok(y) have disjoint supports, hence, setting PPk+1(x) = Pk
O(x, y)
,
Ok+1 = 0
Pi (x)i (y) + ... Pk (x)k (y)
we have 101 < 1 and
aGv(cb) = -k, from which we infer at once M(aGv) = +oo. Notice that in fact we have proved that the mass of 8Gv is infinite in every neighbourhood of the origin. Taking
into account the self-similarity of V we actually see that the mass of 4Gv is infinite in every neighbourhood of (x, V (x)) for every x c C. _ Consider now the "inverse function of V" denoted by V : (0, 1) C R -> R, defined for instance by
xlR = aSGvL(0,1) xlib where
gv := {(x, y) E (0,1) x (0,1)
1
x < V(y)} .
Then V is a non-decreasing function which belongs to BV((0, 1)), and is continuous in [0, 1] \ D, consequently
aSG9 L (0,1) x 1R = 7(M,1,) It is readily seen that
M+ _ {(x, y) E M 16y > 0} = .M \ .M+ and
?(M+) = (0,1) \ D while
r(.M \ .M+) = D where ? : R x I[8 -> l[2 denotes the orthogonal projection into the second factor.
Also, every point in D is a jump point for V. Consequently V has no Cantor part, but only a jump part which is concentrate on an everywhere dense set D, which is countably 0-rectifiable but not 0-rectifiable, i.e. finite. Finally, of course V is almost everywhere approximately differentiable with V' = 0 a.e.
2.5 Examples of Cartesian Currents In this subsection we would like to discuss a few specific examples of Cartesian
currents. We shall see that in contrast with the case of codimension one, in general the vertical part of a Cartesian current may be quite complicated.
412
4. Cartesian Currents in Euclidean Spaces
An easy way of producing a Cartesian current with a vertical part is the following. Consider any function u in A' (Si, RN), ( C Ifs", and let S be an n-dimensional i.m. rectifiable current in R' +N such that
7r#S = 0 ,
aGu + aS = 0 in (2 x RN
and
Then obviously the current
.
T := Gu+S
belongs to cart(Q x RN). However what makes Cartesian currents interesting is the fact that they arise as weak limits of smooth graphs: this will be our main point in the following examples. Let us begin by reconsidering the examples in the introduction to this chapter.
1 Bubbling off of circles. Let .(l := (-1, 1), n = 1 and N = 2. We consider the sequence {uk} of Lipschitz maps Uk :.(l -+ R2 given by uk(t)
r (coss)kt, sin kt) t E (0, 27r/k) otherwise.
Clearly Uk -k uo := (1, 0) in the sense of measures in (-1, 1) and the length of the graph of Uk is given by
iJi+u'(t) M(Guk)
2 dt
-1
which converges to 2+27r as k tends to infinity. As remarked in the introduction to this chapter it is easy to convince ourselves that
Guk - G(l,o) + 8o x Q Sl
(1)
where Sl denotes the current integration over the circle Sl := {y
(y1)2
+
(y2)2 = 1} in R2 with the usual clockcounterwise orientation. Moreover we have M(Guk) --* M(T) = 2 + 27r. In particular T E Cart(.(2 X R2).
Now we would like to supply the proof of (1). We change the "time coordinate" in order to describe the curves x -+ (x, uk (x)) in R x IR2. Denote by yk : [-1, 1] -+ [-1, 1] the function which is linear on each of the intervals [-1,0], [0, 1/2],[1/2,1] and takes the values yk(-1) = -1, yk(0) = 0, 'yk(1/2) 27r/k, yk(1) = 1, and consider the curves
Sk(t) := (yk(t), uk(yk(t)) It is easily seen that the curves Sk are equi-Lipschitz and converge uniformly to the curve
8(t) :_
(t,1,0)
if -1
(0, cos 47rt, sin 47rt)
if 0 < t < 1/2 if l/2 < t < 1
(2t - 1, 1, 0)
,
2.5 Examples of Cartesian Currents
413
Fig. 4.3. Bubbling off of circles: density of the vertical part equals 1 in (a) and 3 in (b). moreover
Guk = 8k# Q (-1, 1) From this it immediately follows that
G.., = Sk# Q (-1, 1) 1 - S#
G(i,o) + So X Q Si J
.
Similarly, considering the current in cart(.f2 x R2) given by
T := G(l,o) +pSo x QSi ,
pEZ it is not difficult to construct a sequence of Lipschitz maps uk such that Guk and conclude that T E Cart(f? x 1R2). It suffices to consider the sequence uk
(t)
(coss)kpt, sin kpt)
T,
27r/k)
t G (0,
/k)
and proceed as above. 2 More generally, we can choose a sequence {y2} of smooth closed curves in 1[82,
yti
: Si --+ ]I82, and a sequence of points {xi} C (-1, 1) and consider the
1-dimensional current T in (-1, 1) x JR2 given by .
T := Gu +
00
Syi x yz# Q Si
where u is a smooth map from (-1,1) into Il82. If jC M(ryti# Q Si ) < oc, then it is easily seen that T E cart (f?, JR2). Actually, essentially in a similar way as in the previous example, it is not difficult to construct a sequence of Lipschitz maps Uk such that
414
4. Cartesian Currents in Euclidean Spaces
Guk - T,
M(Guk) - M(T),
showing therefore that T E Cart(,fl x R2).
Fig. 4.4. Stereographic projection.
0 Bubbling off of spheres. Consider the stereographic projection a ]18n+1 -,
Sn C ggn from the south pole P. It maps a point (y, z) E Sn C Rn x :
R, Iy12 +z2 = 1, to y/(1 + z) E I[8n, while its inverse o-1 : Rn ^- Rn x {0} --+ S"
maps x E 1n to (2)
1- x2
2
9-1(x)
)
\ 1 + Ixl2 x' 1 + IXI2
A simple computation shows that for i, j = 1, ... , n (3)
(Dj or-' (x)
I
Di or-1 (X))R-+1 = (1 +
4
Ixl2)2
In particular
Dia-1(x)I = IDS v-1(x)I and (Dio,-1(x)
Dj O'_1 (X))R.+l = 0 for i
i.e., c-1 is conformal. Consequently DO-1 2
and
=n
ID1u-1 2
4n
= (1 + x12)2
j
2.5 Examples of Cartesian Currents
415
I M(n) (Dv-1(x) I2 = ID1Q-1A ... hDna-1 I2
=
= (1 IDQ-112 )n
ID1Q-112 ID20--112... DnQ-112
Hence we have
(4)
nni2
f
I Do,-1I n dx =
Rn
f
(DO.-1)
I M(n)
I dx =
xn
(Sn)
Rn
as consequence of the area formula, taking into account that a-1 is injective. Finally, a simple checking of the orientation yields that (-1)n1Sn¶
a#1QRn =
where Sn I denotes the current integration over Sn with the usual orientation in which the normal to Sn points outward. Later, compare Proposition 1 in Vol. II Sec. 4.1.1, we shall see that for any fixed e > 0 we can modify o,-1 outside a large ball B(0, S), S = Se, to a map u actually uE,
u
:
R' -> Sn c Rn+1
in such a way that (5)
_- J Rn
1DuIn dx < un/2
IDQ-11n dx + Rn
u = PS outside B(0, 2b),
(-1)n Q Sn
u# J Rn
Consider now the sequence of smooth maps {uk} from B(0,1) into S' given by
Uk(X) := u(kx) . Using the conformal invariance of f IDuIn dx we infer
f
I Duk (x) I n dx =
Rn
f
I Du(x) In dx
RT
and, taking into account the rough isoperimetric inequality IM(7)(Duk)I
-< cn(1+IDukIn),
7=1,...,n
that the currents Guk have equibounded masses. Moreover one has (6)
Guy. - Gu0 + (-1)n 5o x ii Sn
in Dn(1[8n x
Jn+l)
where uo(x) - (0, ... , 0, -1). The convergence in (6) can be proved by just changing parametrization in the representation of Gu,k as in Sec. 3.2.3. Let in fact yk : B(0,1) --> B(0,1) be
4. Cartesian Currents in Euclidean Spaces
416
P5
Fig. 4.5. Bubbling off of spheres.
where fk(t) : (0, 1) -* (0, 1) is linear in (0,1/2) the map ryk(z) := and in (1/2, 1) and takes values 1(0) = 0, 1(1) = 1, and 1(1/2) = 1/k. Set vk
:
B(0,1) ----f B(0,1) x
lR
n+1
Vk(Z) :_ (-Yk(z),uk(yk(z)))
It is easily seen that the maps Vk are equi-Lipschitz and converge uniformly to the map v : B(0,1) --> B(0,1) x Rn+1 given by
0 < IzI < 2
(0,Q-1(2z)) (21zl - 1) ( z
,
Ps)
2 < IzI < 1
Moreover
Guk = Vk# Q B(0,1) l - v# Q B(0,1) and
_ (-1)n ao x Q Sn v# Q B(0,1) \ B(0,1/2) ]I = Gu0 . v# Q B(0,1/2)
This proves that the current
Gu0 +(-1)nbo x QSnI which trivially belongs to cart(B(0,1) xRn+1) actually belongs to Cart(B(0,1) x Rn+1)
Finally, notice that for any k = 1, 2, .. .
2.5 Examples of Cartesian Currents
I
1
417
IDuklndx < 7f(S")+E =
B(o,1)
Therefore, by a diagonal process, compare Proposition 2 in Vol. II Sec. 4.1.1, we can in fact find a sequence of smooth maps uk such that Guk
GV,o + (-1)n So x Q Sn
and also satisfy
fIDuklndx -- M((-1)na0 X QSn
nn/2 B(0,1)
M(Guk) - M(Guo + (-1)nbo x U Attaching 141 Attachinga acylinder. cylinder.InInthe theprevious previousexample examplewe wehave have seen seen that that vertical spheres can be attached to a graph of a constant map as result of the weak convergence of smooth graphs. We shall now see how a vertical cylinder can be attached and in fact produced by weak convergence. Let .f2 be a bounded domain in R'+P. Choose any function uo E A' (,f2, Rn+l ) and a p-dimensional i.m. rectifiable current L in Q. Clearly the current
T := GV0 + L x Q Sn belongs to cart (,fl x ]E8n+1) provided
3Gu0 = -3L x Q S"
in P x Rn+1
We now consider the simple case in which (2 = B(0,1) x [0, $]P, B(0, 1) being
the unit ball in RT, uo - (0, 0, ... , 0, -1) E II81+1, and L is the p-dimensional current integration over the p-dimensional cube {0} x [0, e]' C W x W. Denote the coordinates in W x R as (x, z), x E W, z E R and consider the sequence of smooth functions Uk
vk(x,z)
Uk(X)
Clearly the graphs of Vk have equibounded masses, moreover
GVk - G,0+Q(0,E)P X QSn
.
This simply follows by rescaling the x-variable as in the previous example and by considering z as a parameter. We therefore see that Gu0 + Q (0, $)p x Q Sn belongs to Cart(S2, xl[8n+1)
4. Cartesian Currents in Euclidean Spaces
418
Notice that for any k = 1, 2, ..., n2/2
f
IDvkIndxdz = £p_
f DukJndx
B(o,l)x[O,2]P
M(Q(0,q)
X Sn1)+e
by construction, hence by a diagonal process, we could in fact find vk so that
GVk - Q(O,e)pI X QSn and
n ;T/2 X [O,t]P 11
5 The procedure of the previous two examples can be modified to produce by weak convergence of graphs general surfaces as vertical parts. Let {uk } and u0 have the same meaning as previously, so that
Guo + JO x Q S'
Guk
.
Choose any Lipschitz map 0 : Sn -* IRn+1 and set
Vk(X) := ql(uk(x)) We have
(id Da 0)# Guk and (id D4 0)# (Gu,, + bo x Q Sn ) = Go0u0 + bo x 0# Q Sn
This way we produce as vertical part the n-dimensional i.m. rectifiable current
OAS' F-
For example, if we split IRn+l as RV x R, and set O(y, z) :_ (y, co(I yDz)
with
W(t) :_
0
J0
2t-1 1/2
then 0# S' I is just the current integration over standard n-dimensional torus Tn, and conclude that Gu0 +bo x Tn E Cart(B(0,1) x ]R+1) 6 Concentrations at infinity. Consider the family of maps uE from the unit ball of R2 in IR3 defined by
uE (r, cp) := 1 + (cos cp, sin W, log r) e
2.5 Examples of Cartesian Currents
419
in the annulus B(0, 1) \ B(0, E), (r, cp) being the polar coordinates in 1E82, and by
2)+
0
2 uE(x) = v_1(E
Ixl
0
log S -
1-
4S2
in the ball B(0, S), or being the stereographic projection.
0 It is easily seen that each uE is conformal and in particular 2
JB
IDuE12 dx = f2(S2) + O(E).
Consequently, by the isoperimetric inequality, we have sup
0<E<1
Of course uE
11 Gv.E
cart < 00-
uo, uo - 0, weakly in W1"2, and GuE - Gu0, in particular
M(2) (Du,) - 0
but I M(2)(Du) - H2(S2)So.
7 Vertical parts are produced in the weak convergence procedure not only as the result of concentrations, but also as needed to close graphs. Consider for
420
4. Cartesian Currents in Euclidean Spaces
instance the map x/jxj from B(0,1) C lRn into
R,
n > 1. In 1 in Sec. 3.2.2,
we proved that there is a sequence of smooth maps Uk from B(0,1) into R' with equibounded L°O-norms and equibounded masses of their graphs such that Gu,
Gx/Ixl +So x QB(0,1)
Hence the current
T := Gx/I xl + S0 x Q B(0,1) l belongs to Cart (B(0, 1) x R'). While of course this is the only way to close the graph of x/jxj by a vertical part with support in {0} x R', it is worthwhile noticing that there are other ways of closing the graph of x/jxj. For that consider any 1-dimensional i.m. rectifiable current L in B(0, 1) such that aL = SO in B(0, 1). Then the current
T := Gx/Ixl + L X Q
Sn-i ¶
has no boundary in B(0,1) x RI and 7r#(L x Q S"'_1 ) = 0 in B(0, 1). Therefore T E cart(B(0, 1) x l[8n). Later, compare Vol. II Sec. 4.2.6, we shall see that actually T belongs also to Cart(B(0,1) x lR'). Of course we can extend the previous considerations to the other examples in Sec. 3.2.2. Consider for instance a smooth map cp : Sn-1 -+ Sn-1 and let u(x) = cp(x/jxj). From 02 in Sec. 3.2.2, it follows that
aGu = -deg cp So x Q
Sn-1
and that
Gu+degc 8o x B(0,1)1 belongs to Cart(B(0,1) x lien). For any 1-dimensional i.m. rectifiable current L in B(0, 1) such that
a(LxQS'i-11)=degVSoxISn-1
7r#(LxQS'
(e.g. for a current integration on a line going from the boundary of B(0,1) to the center {0} with multiplicity equal to deg cp), we have G, + L x S1-11 E cart(B(0,1) x W2); and in fact one can prove that G,, + L x may be QSn-1
weakly approximated by the graphs of smooth maps, compare Vol. II Sec. 4.2.6. 8 We would like to present two -examples of currents which in some sense provide an extension of the notion of vector valued BV-functions. We shall return later on this point. First we consider a map u0 : B(0,1) --+ RN which takes constant values on
the upper and lower half balls, i.e., u° (x)
a
if x E B_ (0, 1)
b
ifxE B+(0,1)
2.5 Examples of Cartesian Currents
a,b E RN, B+(0, 1)
{x E B(0,1)
I
421
xnc0}. Of course uo E A1(B(0,1),RN)
and Gu,0
= Q B+(0, 1) ]1 x bb + Q B_ (0,1) l x Sa
Hence
BG,,,o = LX5b-Lx5a = Lx(8b-ba)
inB(0,1)xRN
where
L := aQ B+(0,1) L B(0,1) Choose now a smooth curve y : [0, 1] -> RN joining b to a and let r = y#Q (0, 1) ]]. Then L x F is an n-dimensional i.m. rectifiable current and
L x al" =
a(L x r)
L x [5a - ab]
in B(0,1) X RN. Thus the current
Gu3+(-1)n-1LxF has zero boundary and belongs to cart(Q x RN) with 12 = B(0,1). Actually it is easily seen that Gu0 + (-1)n-1L x F belongs to Cart(Q X RN); in fact the graphs of the functions a
if xn < 0
y(kxn)
if O < xn <
b
ifxn>
1
weakly converge to Gu0 + (-l)'-'L x F, and also that
M(Guk) -> M(Guo +
(-1)n-1L
x F)
We now consider the case in which the function uo takes three constant values. For the sake of simplicity we consider the case in which n = 2. Denote by (p, 0) the polar coordinates in R2 and set D1 := {(x, y) E B(0,1)
I
{(x, y) E B(0,1) {(x, y) E B(0, 1)
D2 D3
B E (0, 21r/3)} 9 E (27r/3, 47r/3)1 9 E (47r/3, 27r)}
and
E1
E2
{(x, y) E B(0,1) {(x, y) E B(0,1)
E3 :_ {(x, y) E B(0,1)
9 = 4ir/3} 9 = 0} 0 = 2ir/3}
422
4. Cartesian Currents in Euclidean Spaces
We shall denote by Li the currents integration over the segments Ei oriented in the outward direction and we consider the map a1
if xE l
a2
if xEd2
a3
if XEd3
{
where al, a2i a3 E IAN. Clearly uo E A1(B(0, 1), IRN) and
8Guo = (L2 - L3) X Jai + (L3 - L1) X Jag + (Li - L2) X Jai L1 X
Ja2) + L2 X (Jai -
J,,,
)
Choose now three curves yi, 72, rya in RN, yl from a2 to a3, 72 from a3 to a1,
y3 from al to a2 and let ri = yi#Q (0,1) . Then
a(Ll x f) = 8L1 x Tl - L1 x aP1
= -Joxfl-Llx(Ja3-Jag) and similarly
a(L2 x T2) = -Jo x T2 - L2 X (Ja,, - 6a3) a(L3 X T3) = -Jo x T3 - L3 X (6a2 - Ja1) Setting r := Tl + T2 + I'3 we therefore have 8I' = 0 and 3
8(Gv,o+ELixri) = -Joxr. i=1
Choose now a two-dimensional surface A in RN so that 8z = T; then
8(Joxa) = JoxaA = Jox1'. In conclusion we then infer that 3
Li x j'i + Jo x A
T := Gv,o + i=1
is a Cartesian current, T E cart (B(0,1), RN). Such a current provides an example in which different vertical parts appear: jump parts Li x Ti over the 1-dimensional sets Ei, and a completely vertical part 4 of dimension two over the origin. Tough slightly length, one can prove the existence of a sequence of smooth maps uk such that 3
Guk - T:=Gu,+Lixfi+Joxa i=1
and also
M(Guk) -) M(T)
so that, in particular, T E Cart(B(0,1), RN), but we leave it to the reader.
2.5 Examples of Cartesian Currents
423
9 A Cartesian current with Cantor mass on minors. We conclude this section by showing a Cartesian current
T = GUT +ST in Q x 1R2, Q := (0, 1) x (0, 1) C R2, which is the weak limit of smooth graphs such that (i) UT E W' P(Q) for all p < 2 and det DUT = 0 a.e. in Q (ii)
0, spt ST(2) C 7r-1(C x C) where C is the Cantor set in Sec. 1.1.3, and actually ST(o) = ST(1) = 0, ST(2)
7r#STC2> = Det DuT elne2 = V' ®V' elne2 where Det DUT is the distributional determinant of UT and V is the CantorVitali function. For that we shall construct a map u : Q --+ R2 which belongs to W l,p n for all p < 2, s := log 2/ log a-1 such that IDul I IDu2 1 = 0 a.e. but
Co,s (Q)
DetDu = V' ®V' , i.e. the derivatives of u have no masses, but the distributional determinant has a Cantor-type mass. The current graph of u, G, is very similar to the graph of Cantor-Vitali function Gv: the role played by V' in C is here played by Det Du in C x C. The graph of u has infinitely many holes, and in fact
M(aGu) = +oo
.
The current Gu can be "completed" to a boundaryless current, and in fact to a Cartesian current T, so that UT = u, by an approximation argument and this give rise to our example. To accomplish such a program we need several steps where we use notation and results of Sec. 1.1.3. We begin with
Lemma 1. Let k > 0, 81 := as - 1 E (0, +oo). For j = 1, ... , 2k consider the open intervals
Jk,j := bk j+bk(-bl,1+51) Then {Jk, j } form a disjoint covering of Cantor set C and .Lk
(7)
dist ([0,1] \ U Jk,j, C) > bl . dk
Proof. The Jk, j obviously cover C as Jk, j D Jk, j . Set ek,j
:= bk,j -
818k ,
dk,j := bk,j +6k(1+81)
424
4. Cartesian Currents in Euclidean Spaces
so that Jk,j = (ck,j,dk,j). To show that the cover Jk,j is disjoint it suffices to show that
for j = 1, ...
dk,j C Ck,j+1
(8)
2k
-1
For k = 0 there is nothing to prove. Assume (8) holds for k, from Ck+l,j = SCk,j
Ck+l,j+2k = 1 - b + SCk, j
dk+1,j = Sdk,j
dk+1,j+2k = 1 - b + bdk,j
we infer dk+1,j <- Ck+1,j+1
if j 542k
For j = 2k we infer from the definition of bk, j bk,2k
<
dk,2k
= bk,2k + 8k(1 + 81) < 1 + 818k
1 - 8k
hence
dk+1,2k = 8dk,2k < 8(1 + 818k) < 8(1 + 81) = 82b = 2
while, as Ck,1 = -8k81, Ck+1,2k+1
= 1-b+8Ck,1 =1-b-bk+181
> 1-8(1+81)=1-b2b
2
Finally, since
dist (R \ Jk,j, Jkj) = bl 8k and the Jk,j are disjoint we get (7). As next step we shall define an auxiliary function f : Q = [0, 112 -> ]R which interpolate between Cantor-Vitali function V and the identity. We set
1-28
1-8
2
and for (xl, x2) E [0, 1] x [8, 1] we set (1 - 82(1 - x2))-1x1
f(°) (xl,x2) :=
2
if x1 E [0, (1 - 82(1 - x2))]
if xl E
2
[2(1-82(1 -x2)), 2(1+82(1 -x2))]
1-(1-82(1-x2))-1(1- x1) ifxl E f (O) (xl, x2) is piecewise linear and f
2 on the segments [(8, 8), (2, 1)] and
[(1 - 8, 8), (2,1)]; also f (°) (-, 1) = V° := id
[2(1+82(1-x2)),1]
,
.f (°) (', b) = Vi
2.5 Examples of Cartesian Currents
425
Vk, k > 0, denoting Cantor-Vitali approximate functions. Inductively we now define
f(k)
:
[0, 1] X [5k+1 5k] -: R
by
f(k+1)(XI, x2)
2 f (k) (s-1x1, J-1x2)
if x1 E [0, b]
1
if x1 E [S,1- b]
2
f (k) (1
2+
- S-1(1 - xl), J-1x2)
if x1E[1-5,1]
2
Taking into account that f (1) (., 8) = f (°)
f(k+1)(, bk)
Vl we infer
= f(k)(.,ak) = Vk ,
and we set f(k) f :=
1V
on [0, 1] X [6k+1 bk] , k = 0, 1, .. . on [0,1] x {0} .
We also introduce the following approximation of f fk (xl, x2)
foon [0, x2 E [0
Vk (x1 ) 2)
JI]
1]
r
Then
Lemma 2. We have (i) f, fk are Holder-continuous with exponent s and locally Lipschitz in Q, i22) .fk(0, ') = 0, f (1,') _ fk(1, -) = 1, f (', 0) _ f(',1) = fk(',1) = id, f (0, V, fk (', 0) = Vk,
iii) For (x1, x2) E Jk,j X [0, sk] f (xl, x2) = V (bk,j) + 2-k.f (a-k(xl - bk,j), b-kx2), 2k
'v) For (x1, x2) E ([0,1] \ Ek) x [0, 8k], Ek := UJk,j, we have j=1
f (x1, x2) = V (xl), and D f (xl, x2) = 0
if moreover x2 E (0, bk). As usual D f denotes the approximate gradient of f(v) Set 8o := 1/2 - J. Then
f (xl, x2) = V (XI)
if
x2 < 55 ldist (xl, C)
426
4. Cartesian Currents in Euclidean Spaces
Proof. Assertion (ii) follows at once. (iii) Inductively we shall prove the following assertion (P,,,,): If 8m+1 < x2 <
8m, k < m, j = 1, ... , 2k, and 0 < xl - bk,j < 8k, then f (x1,x2) = V (bk,j)
+2-k f(5-k(xl
- bk,j),S-kx2)
(Po) is trivial; suppose (Pm_l) holds, and assume moreover that j < Writing
xl = bk,j + SkiFl
we have
V (bk,j) = 2 V(bk-lj)
bk,j = 8-1bk-1,j and
1f(8-1x1 8-1x2)
f(xl x2) =
=
1
1 f (bk-1,j + 2
(Sk-lyl 8-1x2)
[V(bk-l,j) +2-(k-1)f (x1, S-kx2)] 2-k f(S_k(xl - bk,j), 8-kx2)
V (bk,j) +
l
1
2k-1
Similarly one proceeds in the case j > 2k
(iv) Let 0 < x2 < 8k, x1 E [0,1] \ Ek, Ek = U Jk,j. As j=1
k-1 2[0, 1] = Ek U U U Im,j m=0 j=1
we find m < k - 1 and i < 2m such that
xl E Im,i = bm.i + 8m(8, 1 - 8) Therefore we can write
xl = bm,i + 8m2l
with
8
If m = 0, we then have xl = 2l, k > 1, and
f(xl
x2)
= f(xl,x2) =
2
= V(x1)
If m > 0, by (iii) and the self-similarity of V, we get
2k-1
2.5 Examples of Cartesian Currents
f (xl,
427
= V(bm,i) + 2-m f(xl, 8-mx2)
x2)
V (bm,i) + 2--V(V) V(bm,i) +
2-mV(8-m(xl
(v) By (ii) the assertion is trivial if xl E C. If x1
- bm,i)) = V (xl) C, we find k and j such that
xl E Ik,j = bk,.j + 8k(8,1- 8) Assume first k = 0, i.e., xl E (8,1 - 8). Then
dist(x1,C)=min(Ixl-8I,Ixl-1+8l)=8o-1x1-2I
.
Therefore f (x1, x2) = a = V (xl) if x2 < J. Instead, if 8 < x2 < 80 1dist (x1, C)
=1-801x1-21,we have < 80(1 - x2) < Ixl - 11 2 -
2
b2
(1 - x2)
82 :=
1 - 28
1-8
hence again f (xl, x2) = z = V (xl) by the definition of f (0). Assume now k > 0. Then
x2 < 80 ldist (x', C) < 8k Setting
xl := 8-k(xl
- bk,.7)
,
x := 8-kx2
we then infer
i52 < 80 1dist (xl, C)
,
xl E (8,1 - 8)
Therefore by (iii) and using the case k = 0, we get
f(xl,x2) =V(bk,a)+2-kf(x1,X
)=V(bk,a)+2-kV(xl) = V(xl)
It suffices to show that there is a positive constant c such that for all pairs (xl x2), (yl y2) E Q we have (i)
(i)1
I f (yl, y2) - f (yl, X2)1 < c ly2 - x21 s
(i) 2
I f (yl, x2) - f (xl,
X2)1
!5 c l yl - x118
We may assume y2 < x2, and x2 > 0 (the claim being trivial for x2 = 0), x2 E (8k+1, 8k] for some k. Let k = 0. Then either y2 > 82, and (i) l follows from the Lipschitz continuity
off on [0,1] x [J2, 1], or y2 < 82; but then Ix2 - y2l > 8 - 82 and (i)1 follows as
0
If k > 0 and yl
k
U Jk,j, (i)1 follows from (iv). If instead yl E Jk,j we let
9=1
y := 8-k(yl - bk,,j). Then by (iii) and the case k = 0 we infer
428
4. Cartesian Currents in Euclidean Spaces
If(y1,y2)-f(yl,x2)I < 2-kIf(y1,b-ky2)-f(y1,8-kx2)I < 2-kcla-k (y2 _ x2) IS < c I y2 _ x2ls
as28s=1. This proves (i)l. We shall now prove (i)2 by induction on k assuming x2 E (8k+1 8k]. For k = 0 we use the Lipschitz continuity of f on [0,1] x [6, 1]. Suppose now k > 0, i.e., x2 E (8k+2 dk+1] If xl, yl E (8,1 - 8), then (i)2 follows from the definition of f (k+1). Assume xl (8,1 - 8) and say xl E [0, 8]. If yl E [0, 8], then by the definition of f (k+l) and the inductive assumption for x2 E (8k+1, 8k1 we get I f (yl, x2) - f (x1, x2) I = 2 If
<
1c I8-1(x1
(b-lyl 8-1x2) _ f(b-1x1 8-1x2) I
- yl)IS = c x1 - ylis
If yl E [1- J, 1], then (i)2 holds as I xl - yl I > 1- 28 > 0 and 0 < f < 1. Finally, if yl E (8, 1 - 8), then f (yl, x2) = 1 = f (8,x2) and
If(yl,x2)-f(xl,x2)I C clx1 -8IS < clxl - ylls We now abbreviate dist to d and define the map u = (ul,u2) : Q -- ]182 by ul(x1,x2) u2(xl, x2)
f(xl,ap ld(x2,C)) ld(xl, f (x2, 80 C)) = ul (x2, xl)
With such a definition we can state
Theorem 1. We have (i) uEWhpnC°,s(Q,1R2) forallp<2 (ii) IDul I IDu2I = 0 a.e. (iii) The distributional determinant of u is a non-negative measure supported on
the Cantor-type set C x C, more precisely
Det Du = V' ®V' ; moreover there are positive constants cl and c2 such that
a1H2sLCxC < Det Du < C2H 2sLCxC. Proof. (i) As d(., C) is a Lipschitz function we infer at once that the map u is Holder-continuous with exponent s and locally Lipschitz in Q, and uxi(xl,x2) = f11i(x1 ,8 ld(x2,c))
ux2 (xl, x2) = fy2 (xl, 8 ld(x2, q) j-1 a'2 d(x2, C)
.
2.5 Examples of Cartesian Currents
429
As IDd(-, C) I < 1 a.e. we only need to show that the function g(x)
1d(x2,
I Dyf (x1, SO
C'))
belongs to LP(Q) for all p < 2. For y2 E [Sk+1, Sk] we have by Lemma 2
Df (yl, y2) = 0 whereas
2k
if
Ek
y1
U [bk,j, bk,j + Sk] j=1
IDf(y)I < c(2J) -k = CS-k(1-s) <
c(y2)-(1-s)
if y2 E Ek. Thus for x2 with So ld(x2, C) E [Sk+1, Sk] we infer 1
C)-(1-s)p < c(d(x2, C))-(1-s)(P-1)
f I g(xl, x2)I p dxl < c2k8kd(x2, 0
Taking into account that the set {(xl, x2)
I
d(x2, C) = 0 or So} has measure
zero, we then conclude 1f
f I9IP dx < cJ (d(x2, Q
C))-(1-s)(p-1) dx2
o
In order to show that the last integral is finite we consider the distribution function
SP(A) := xl({x2 E (0, 1)
(d(x2,C))-(1-s)(p-1) I
> A}
We claim that, as consequence of Lemma 1,
,P(A) <
c,\-1/(P-1)
,
and this of course concludes the proof of (i). In fact if Jk,j
bk,j + Sk(-Sl, 1 + S1) ,
Sl = 1
is the covering of C in Lemma 2, and 2k
Ek
U Jk,j
j=1 we have
d((0,1) \ Ek, C) > cbk
.
2J 2S
CS
430
4. Cartesian Currents in Euclidean Spaces
Since
ill (Ek) < C 2kbk =
bk(l-s)
C
we then deduce (P(Cb-k(1-s)(p-1)) <
which yields V(A) <
Cj(l-s)k
cA-l/(p-1)
(ii) Consider the closed set
M :_ {(x1, x2)
1
d(xl, C) = d(x2, C)}
we claim that (ii) 1 7-12 (M) = 0
(ii)2 for a.e. (x1, x2) E Q either ul(xl, x2) = V(x1) or Du2(xl, x2) = 0 (ii)3 for a.e. x E Q either Dul (x) = 0 or Due (x) = 0. For that we consider the map
h(xl, x2) := d(xl, C) - d(x2, C)
.
As C is closed with H1 (C) = 0 we have jhxi I = Ihx2 1 = 1 a.e., hence DhI = a.e. Setting
h1((
Mk
))
the coarea formula yields
2,H2(Mk) =
f
J M,;
1/k
DhI dl2 = f 7-l1
(h-1(t)) dt
.
-1/k
As t -> V (h-1(t)) is summable, we then infer f2(Mk) --+ 0, which proves (ii)l. Let us consider (2l, 22) E Q \ M. Then d(21, C) d(22, C), say d(xl, C) > d(22, C), and Lemma 2 (v) yields
u1(x) = f(x1,b01d(x2,C)) = V(xl) for x close to 2 _ (21i 22). Thus Dul = 0 a.e. in a neighbourhood of 2. Similarly, we find Due = 0 a.e. if d(2l, C) < d(22, C). Therefore the set of points where (ii) 2 or (ii) 3 fail has density zero in Q \ M. As H2 (M) = 0, this proves (ii)2 and (ii)3 and in turn (ii). (iii) Recall that the distributional determinant is defined by
Det Du(W) =
f(_u1u22i Q
Let
+ u'u2i cpxz) dx
V WE CC. (Q)
2.5 Examples of Cartesian Currents
Ik,j
Ik,j x [0, 1]
bk,j +
Ik,j
,
431
We observe that the Ik,j are disjoint and
,H2(Q\ \
00
2k
U Ulk,j/ = 0. k=0 j=1
Consider first Io,l = (b,1- b) x [0, 1]. From V = 1 on (5,1- S) and (ii) 2 we infer
ul (xl, x2)u2j (xl, x2) = V (xl)u2j (xl, x2)
a.e.
thus
_J
(-ulu2z(Pxi + ulu2icx2) dx =
Io,l
2JIo,i
1
(-u2x2wxl +uxiWx2) dx
2(P 2 (u2 xix2 - uxlx2) dx + 2 J u Pxz
2
xi=1-b xi=b
dx2
0
Ia 1
Since S and 1-S belong to C we have u2 (b, x2) = V(x2) and u2 (1-b, x2) = V (x2)
.
Thus we compute the last integral as 1 V(x2)Wx2(xl,x2)Ix1l-=(5
2
dx2
=
7 V(x2)7'xlx2(x1,x2)dx
1
2
0
f10,1
V(x1)V(x2)(pxix2 dx
.
Io,1
Since the Cantor-Vitali function V is constant on Ik,j we obtain similarly f (-uluxzCPxl + uluxlWx2) dx = Ik,j
J
V (x1)V
dx
Ik,j
and letting W(xl, x2) = O(xl)X(x2) DetDu(cp)
= f V(xl)V(x2)0'(xl)X'(x2)dx Q
/1
11
= J V*' dxl J VX' dx2 = V'(0)V'(X) = V' 0
0
This proves the first part of (iii). To prove the second part we consider the Radon measures
432
4. Cartesian Currents in Euclidean Spaces
µ := V'®V',
v
7i2sLCxC
and an arbitrary rectangle R :_ [a1, a1] x [a1, a2] C Q
Let {Fm} be a covering of R n (C x C). Choose a cube F,,,, parallel to the coordinate axes with side £m = 2 diam Fm, Fm D Fm. By the Holder continuity of V we have
µ(F,,,,) <,u(Fm) <2m <22s(diamF,)2s
.
Since A L (Q \ (C X C)) = 0 we infer
µ(R) = µ(R n (C x C)) < >2 µ(F,,,,) < 22s
(diam Fm)2s ,
m.
hence
µ(R) < c7t2s(R n (C x c)) . To prove the converse bound we take k E N and choose integers mi, 7ni, i = 1, 2 in such a way that bk,mi < ai < bk,mi+1
bk,m,; -1 < ai < bk,nx;
As V (bk,j) = 2 and V' > 0 we have 7hi - mi < 2k (V (ai) - V (ai)) + 2 . Then the collection of cubes
F :_ {Jk,7i X 432
I
m1 < il < ml, m2 < i2 < in2}
covers Rn (C x C) and each cube in F has diameter V,"2- bk. Thus, being 251 = 1,
(diamF)2'
< cb2sk[2k(V(a1)-V(al))+2][2k(V(a2)-V(a2))+2]
F(=.F
< c(V (al) - V (al))(V(a2) - V (a2)) + 0(2 -k) < cp(R) +O(2-k) Letting k -> oc it follows
1-12s (R n (C x C)) < cu(R)
.
Being V continuous, a particular consequence is that v(H) = p(H) = 0 for any straight line H parallel to the coordinate axes. As every open set U C Q can be written as countable disjoint union of open rectangles and parts of lines both parallel to the coordinate axes, we then infer
f2s(U n (C x C)) < cp(U) and this concludes the proof of the second part of (iii).
2.5 Examples of Cartesian Currents
433
Next we shall construct a sequence of Lipschitz maps which approximate the map u in Theorem 1 in terms of the functions fk introduced before Lemma 2. Notice that if y2 >
fk(yl,y2) = .f(yl,y2)
bk
We set for (xl, x2) E [0,1]2
uk(xl,x2) :_.fk(xl,So ld(x2,C)), uk(xl,x2)
(9)
uk(x2,xl)
The maps Uk are Lipschitz continuous in Q and Uk = u if d(x2, C) > tS06k
d(xl, C) > 8o8k
and
Proposition 1. The sequence of Lipschitz maps Uk in (9) satisfies (i) SkPII Uk IIWI,P(Q,R2) < oc and Uk -> u strongly in W l'P (Q, R2) for all p
< 2, u
being the map in Theorem 1 (ii) sup k
f
Idet Duk I < oo.
Q
Therefore, passing to a subsequence we have Guk - T = Gw + S, moreover S(o) = S(1) = 0, T(2) = S(2) 54 0, and lr#S(2) = Det Du e1 Ae2 .
spt S C (C x C) x R2,
Proof. Recall that Jkj := bk,j + 8k (-81i 1 + 81) form a disjoint open cover
2k
Ek = U Jk,j j=1
of C, and l2(Ek) < 2k8k(1 + 281) = O(6(1-s)k)
---*
0,
d((0,1) \ Ek, C) > CBk.
Set
Uk :_ {(x1, x2) E Q I d(xl, C) < 8oSk or d(x2, C) < 8o5k}
We then have ll2(Uk) -> 0, U1 D U2 D
.
, and uk = u on Q \ Uk, in fact
Uk = Uk+1 = ... = u
on Q \ Uk .
As in the proof of Theorem 1 (i) one proves that {uk} is equibounded in Wl'P(Q, R2) for every p < 2 and since H2({uk ; u}) --* 0 one easily concludes the proof of (i) by a simple use of Holder's inequality. Vk or u'(x1i ) = Vk. In the first case we On Uk we have either get for instance
434
4. Cartesian Currents in Euclidean Spaces
detDuk = Vk(x')(fk(x2,60ld(x1,C))X2 and integrating with respect to x2
IdetD2
= V(x l )
as fk(1, ) = 1 and fk(0, ) = 0. Therefore, using also that
I
det Duk dx < 1
[0,1] x [o,Sk]
we conclude that sup k
J
det Duk I dx < oo .
Q
The second part of the claim now follows easily. In fact (i) yields at once S(0) = S(1) = 0. From det Du = 0 we infer T(2) = S(2), from spt det Dul C Ek x Ek, for f x k, we infer spt S C Ek X Ek d k, i.e., spt S C (C X C) X R2. Finally from 1
J det Dukgo -+
r#S(2) (W)
0
we infer 7r# S(2) = Det Due, Ae2 .
Next proposition concludes the proof of our claims
Proposition 2. Let u be the map in Theorem 1. Then
M(aG, L Q) = +00
.
Proof. We divide the proof in three steps. Step 1. First we explicitly compute the boundary current aG,,,. As Gz, ha finite mass, we can test WGu on forms with bounded and Lipschitz coefficients, so that
M(aGu) = sup{G.(dw) I w E Lip(Q,A'(R2 X R2),
11
w 1), < 1}
On forms of the type w := cp(x)g(y)dy', where g E Lip (R2 ) n L°°(3R2) and cp E C°O([0,1]2), we have, compare Proposition 2 /in Sec. 3.2.3,
aGu(w) =
J Q
g(ul,u2)(cpx,u52 -cpx2u',)dx+ J cp(x)gy2(ul,u2)detDudx
From (ii)3 in the proof of Theorem 1
Q
.
2.5 Examples of Cartesian Currents
det Du(x) = 0
435
a.e.,
while, from (ii)2 and recalling that u1(xl,x2) _
we get that
u2(x2 x1)
Dul(x1,x2) 54 0 implies
ul(xl,x2) = u2(x2,x1) = V(x2)
.
We then conclude aGu(w) =
Jg(u1,V(x2))(xi2 -tpx2uXi)dx
.
Q
Step 2. We now exhibit a form w of the type w = W(x)g(y) dyl with small support for which aGu(w) 0. Recall that by construction (10)
u1(xl,x2) =
1/2 1/4
on (8,1 - 8) x (0, 8)
on (8(8,1- 8)) x (8(1- 8),1)) .
Denote by L the segment joining the points ab
bi
8(1- 2 ))
(8, 8(1 -
2
))
and set c(8)
cp(x)
:=
min(8(1 - 28),82/2)
:= max(0,1 -
dist (x, L)
Clearly cp E Lip (Q), II cp 11 < 1, spt cp C (82,1 - 8) x (8(1 - 8), 8) and, for each cp(xl, x2) is constant in x1, i.e., x1 E (5(1 W(x1,x2) = p(x2)
with p(t) := max(0,
It
-
J/2)1
Hence for any (x', x2) E (8(1 - 8), 8) x (5(1 - 8), 8)
t E IR .
we have
Oxi(xl,x2) = 0 1
Ox2 (xl, x2) =
i
if x2>8(1-8/2)
(b)
if x2 < 8(1- 8/2)
C(J)
On the other hand the interval (8(1 - 8), 8) is one of the intervals which occurs in the construction of Cantor-Vitali function, and
4. Cartesian Currents in Euclidean Spaces
436
{x
I
V(x) E (V(8(1- S/2)), V(8))}
has positive measure. In particular V(8(1 - 8/2)) < V(8), and for any positive function g : R --> R with support in (V(8(1 - 6/2), V(8)), for convenience with g < 1, we then have
fg(V(t))dt>
(12)
0.
R
For w = cp(x)g(y2) dy1, with cp and g defined as above we then compute, taking into account (10), (11) and (12),
8G,, (w) = f g(V
(px2ux1) dx1 dx2
Q
dx2 J
C(8)5(1-5/2)
g(V(x2)uxi dx1
5(1-5)
5
4c(8)
f g(V(x2)) dx2 > 0 , 5(1-5/2)
since for almost all x2 E (8(1 5
f J
u
dx1 =161(8,x2) -ul(8(1-8),x2) _
5(1-5)
Step 3. Let cp, g, w be as in Step 2. Consider the self-similarity transformations
21 :=Q(1-8)+8x1
x2=8x2
,
,
U=0,1
and set po (x1, x2)
co(x1, x2)
9o (1J) = 9(1J18)
wQ
cPQ (x1, x2)9Q (y) dy1
From 1
1
f(x1,x2) = Q 2 + 1f(x1,x2) we deduce
Df (xl, x2)
28
Df (x1, x2)
Using 8C = C n [0, b] and 0 < x2 < 8, we see that d(;_-2, C)
thus
= d(:t2, 8C) = d(8x2, 8C) = 8d(x2, C)
2.5 Examples of Cartesian Currents uxl (xl,
x2) =
(x1, 50 ld(x2, C)) 28fx1 (x1, 50 ld(x2,
25 fx1 (xl,
q)
437
5-150 ld(x2, C))
= 25u11 (x1, x2)
and similarly U
1
2
l' x2) = 25U12 (xl, x2) Therefore in both cases o = 0, 1, taking into account V(22) = 8V(22), we get
aGu(wv) =
fg(v(2))(co,xi U 2 - (O ,x2
ux1) 01211122
= 23Gu(w) .
Q
Moreover, notice that the supports of w, wo, and wl are disjoint. Iterating the previous construction, i.e., setting cpo,l = co, 9o = g and (x1
co
k+1,j (x
Wkj 1 b
2
1 ,
x)
x21
if j= 1, ... , 2k
aJ x
1 _ (1
'Pk,j-2'
-
x
5)
z
'5
5
if j = 2k +1,...,2 k+1
z
9k+1(Y2):= 59k( 2 )
Wk,j (x, y)
dy'
we then obtain that the forms Wk,j have disjoint supports for all k and j and moreover
d j = 1, ... , 2k
aG,(wk,j) = 2 kaG,,,(w) Finally, defining for all k = 0, 1, .. . k
wk
2k
Y, W kJj k=0 j=1
we have lwJI < 1 and k
2k
aG,,,(wk) = E 2-kaG,,,(wk,
k+1)aGu(w)
k=0 j=1
i.e. aGu(wk)
oo ask
oc.
El
10 cart # sw* - lim({Gu I u E Cl (Q, lN) n Al (,fl, RN)}). Let B2 be the unit ball in R2, and let T E cart(B2 x R2) denote the Cartesian current
T := Go + ST
438
4. Cartesian Currents in Euclidean Spaces
where Go is the current carried by the graph of the null map u - 0 and
ST := Q9B(0,r) x 3B(yo,1) where 0 < r < 1, yo E R2, IyoI > 2. We claim that if Uk : B2 --+ I[82 are smooth maps such that Guk weakly in D2(B2 x R2) T then
sup M(Guk) > 4 dist (aB(yo,1), (0,0)2).
(13)
k
Let us postpone discussing this claim and let us first show how we can then construct a Cartesian current T for which there is no smooth weak approximation, i.e. a current T such that
T E cart(. x RN)
T O sw* - lim({Gu
I
UE
C1 (S?, RN) n Al (,f2, RN)}).
Consider in fact the current T := Go+ST where ,(2 and ST this time is defined by
B(0, s), Go is as previously
00
Ti := Q aB(pi, ri) x aB(yi, 1) fl
Ti
ST i=1
where pi := (23-i, 0), ri := 2-i yi (2i+1 0). It is readily seen that T belongs to cart(. x R2). Actually, one easily sees that'T belongs to Cart(f2 x R2). Set Ui := B(pi, 21-i). Clearly Ui n Uj 0 and
TLUU xR2 =GoLUU xR2+Ti Suppose now that T be weakly approximable, i.e., assume that there exists a sequence of smooth maps Uk : 0 -> R2 such that Guk -k T, then, in particular, we would have
sup M(Guk) < oo. k
On the other hand by the claim above (2i+1 _ 1)
sup M(Guk L Ui x 1R2) k
2
and summing on i
sup M(Guk) _ +00, k
a contradiction. Returning to our initial claim, one easily convinces himself that it holds by considering the simpler example which follows. In R x 1R2 consider the Cartesian current T:= Go + Q aB((0, d),1) 1
2.6 Radial Currents
439
Clearly, for any sequence of smooth uk for which Gu,, -k T, spt Gu,r must intersect (-E, s) times a neighborhood of 8B((0,d),1), therefore
fDukdx d Consequently M(Gu,,) is or the order of d for large d.
2.6 Radial Currents A map u : B(0,1) C Rn --> ]R' is said to be radial if it has the form
u(x) = U(1x1)
1 XI
where U : (0, 1) --> ll is a nonnegative function. In this case of course
U(Ix) = u(x)I The aim of this subsection is to discuss graphs of radial maps and their weak limits in cart(B(0,1) x RI). We begin by considering radial maps for which U E C°([0,1]) f1 C'(0,1), U > 0. Our first proposition provides formulas for computing the area and the boundary of their graphs. As usual we denote by wn the volume of the unit ball in W' so that xn-1(Sn-1) = nW n Proposition 1. Let u be a smooth radial map, possibly singular at zero, i.e., u(x) = U(1 x1) x
IxI
where U E C°([0,1]) fl Cl(0,1), U > 0. Then we have (1)
f
lu(x)l dx = nwn ]Pn_1U(P)dP
B(°,1)
0 1
(2)
M(Gu) = nwn f (p2 +UZ(P))
(Ul (P))2)' dp
(1 +
0
(3)
8Gu L- B(0,1) x Rn = _Sox a B(0, U(0))
Proof. Equality (1) is trivial. To prove (2) we note that
Diui =
Psii+(U'-PU ) IZI2
x
,
P=1xl
440
4. Cartesian Currents in Euclidean Spaces
so that the Jacobian matrix Du is symmetric with eigenvalues given by u(P) p ,..., u(P) p ,U'(P) Hence we can rotate coordinates by R in such a way that
RDuRT = diag(U(P)/P,...,U(P)/P,U'(P)) and consequently
M(Du(x)) = M(RDuRT)
_
(e, + uPp) e1)^ ... n(en-1 + uPP)
En-1)^(en
+u'(P) )
where ei := Rei, Ei = Rezi i = 1, ... , n and p = x1. Since the vectors
ei+u(P)ei, P
i=1,...,n-1,
and ee"n+U '(P)en
are mutually orthogonal, we then infer )n-1 I M(Du)12 = (1 + U2(P) (1 + (U'(A))2) 2
from which (2) follows taking into account the area formula. Equality (3) follows as in 1 in Sec. 3.2.2, 2 in Sec. 3.2.2 or in 0 in Sec. 4.2.5 by changing the parametrization of the graph of u. 1 Notice that the condition M(Gu) < no does not imply that the length of the graph of U is finite. For example, if U(p) = p2 sin p_a
it is easily seen that fo (1 + (U')2)112 dp = no for 0 > 2 while
M(Gu) = ncvn ](p2+u2)(1+(uI)2)1/2dp < no
for Q < n+ 1
.
0
Of course M(Gu) < no implies that the graph of U over (b, 1) has finite length for any d > 0. Moreover, if U(0) > 0, we have for some positive 8 p2 + U2 (P) > b2 +
U2 2
(0)
,
thus, in this case, M(Gu) < oo if and only if the graph of U over (0, 1) has finite length.
a
2.6 Radial Currents
441
In order to characterize the graphs of smooth radial functions we introduce the Lipschitz map h : R x ]l8 x ]18'' -, Rn x R given by h(p, r, z) :_ (pz, rz)
(4)
.
Since h is linear in z, for fixed p and r, the tangent map h*
(p,r B)
at (p, r, 0)
maps any vector of the type (0, 0, v), v E ]R' into h*,(p,r,e) ((0)0)v)) = (pv, rv) hence
lh'*,(p,r,B)((0, 0, v))l = 7+ r2 lvl
(5)
and (6)
h*,(p,r,e)((0, 0,
v)) I
h*,(p,r,e)((0, 0, w)) ,
if v and w are orthogonal in 118". Moreover any vector of the type e _ (1;1i 1;2i 0), 1, S2 E ]l8, is mapped by h*, (p,r,O) CCto
h*,(p,r,e) ((S1, 2, 0))
= (0,60)
hence
h'*,(p,r,e)((b1,6,0))I = 161 101
(7)
and (8)
h*,(p,r,e) ((S1, e2, 0)) i h*,(p,r,e) ((0, 0, v)) ,
if t:1, '2 E R, and v is orthogonal to 0. In terms of the map h we have
Proposition 2. Let
a(x) = u(ix() x,
x E B(0, 1)
be a radial map with U E C°([0,1]) n C1(0,1), U > 0. Then (9)
G.
h#(Gu x Q Sn-1
on D((R' x R') \ 10, 0})
Moreover (10)
h#(60 x (0,U(0)) x [Sri-1f) = s° x QB(0,U(0))
Proof. Denote by eu = eu(p,U(p)) the tangent vector to the graph of U in 1182 eu
(1,u'(p)) 1 + (u' (p)) 2
and choose a basis ee...... ee,_, for the tangent plane to S'i-1 C JR' at 0 = (01, ... , 0,) so that the standard orientation of Sn-1 is given by eB = eel n ... nee, _1, i.e., so that
4. Cartesian Currents in Euclidean Spaces
442
OAeel A ...nee,-1 = e1A ... /den
el,... , en being as usual the standard basis in R . Finally we think of 1[82 and Rn as of the factors of (IR X R) x R'. The vectors
h*,(p,u(p),e) (eu) ,
i = 1,...,n - 1
h*,(p,u(p),e) (eo )
generates the tangent plane to Gv, at (x, u(x)), x = p8, and by (6) and (8) are mutually orthogonal. Also, from (5)) and (7) we infer _1
jh*,(p,u(p),e)(eunee)I = (P2 +u2(P))2 Let now w be an n-form in DI((IIBn X Rn) \ {0, 0}). We have, taking into account (2), Gu(w) = J <w,GU>d7-LnL.cu,B(o,l) 1 +U2)2
f(i + U12)1/2 (P2
f < w, S'e-1
0 1
f(i + U'2)1/2 o
h*,(p,u(p),e) (euAee) h*,(p,u(p),e) (eune9) I
f J
>
d?{n-1(g) dp
dHn-1(8)
<w, h*,(p,u(p),e) (euAee) >
S-1 1
= f(i +U'2)1/2
J
d?-Ln-1(8) = Gu X Q Sn-1
I (h#w)
which proves (9). Equality (10) follows easily since for t E (0,U(0)), 8 E
Sn-1
we have h(0, t, 0) = (0, t8) E W x I[Bn.
Motivated by the above we now set
Definition 1. An n-dimensional current T in IRn x Rn is said to be radial if there exists a 1-dimensional current L E Dl (I[8 x I[8) with support in + x R+ C II8 x IR, I[8+ :_ [0, +cc), such that
T = h#(L x Sn-1))
on D'((JR' x ]Rn) \ {0, 0}) .
By definition the support of a radial current T is contained in the image of the map h in (4)
(11) sptT C Imh={(x,y) EI[82n
I x=Dory=0or
x IwI
Consider now the map g : JR' x Rn \ {0, 0} -> ((
by
=
y
}_: E.
IYI
xN+)\{0,0})xRngiven
2.6 Radial Currents
g(x,y) = (Ix1' M,
443
X+Y 1XI +
IyI) .
Clearly g E C°((Rn x ]Rn) \ {0, 0}) and
< 2+
Dg(x, y) I
on Rn x Rn \ {o, o} .
Ixl + m
When restricted to E \ {0, 0} the map g turns out to be the inverse of the map h with domain ((IR+ x III+) \ {0, 0}) x Sn-1. Thus for any differential k-form w in Dk(((R+ X I[8+) \ {0, 0}) x Sn-1) we have
h#g#w = W.
(12)
Finally notice that, if (x, y) E . \ {O, 0} and x # 0, then g(x, y) agrees with the map gl(x,y) given by gl (x, y) = (1xWy1, I ) l
hence
w = h#gw = h#g1w
(13)
for any w E Dk((0,1) X IIB+ X
Sn-1)
Given a radial current T, next proposition allows us to recover L from T. Proposition 3. Let T = h#(L x I Sn-1 ) E Dn((II$71 x ]EBn) \ {0, 0}) be a radial current with spt L C l[8+ x R+. Then (14)
QSn-1I
Lx
= g#(T)
on Dn(((1+ x K+) \ {0,0}) x Sn-1)
In particular L(q5) = n-1 wn 1 g#T (OAwsn-1)
(15)
for all 0 E Dl ((IIB+ x IIB+) \ {0, 0}), ws,.-1 being the volume form in
Sn-1
Proof. Let w E Dn((R \ {0, 0}) x Sn-1). Then g#w E Dn(1R2n \ {0, 0}) and by +
(12)
g#T (w) = T(g#w) = L x Q
Sn-1
I
(h#g#w) = L x Q
Sn-1
(w)
This yield (14) at once. Applying (14) to w = OAwsn-1 we infer g#T (OAwsn-1) =
Sn-1 Q
(ws,
n wn L(O)
which yields (15).
In general we cannot say very much on the "restriction" of a radial current at zero. A characterization of Cartesian currents, which are radial, is instead given in the next theorem.
444
4. Cartesian Currents in Euclidean Spaces
Theorem 1. Let T E cart(B(O, 1) x R') be a radial current,
T = h#(L x QS'-1 ])
on D7L((B(0,1) x Rn) \ {0, 0})
Then we have
(i) The BV function UT associated to T is a radial function, x UT(x)=ZT(IxI) ICI.
(ii) M(T) = nWn
f
+r2)n21
dIi L II
(p2
Rp x 1Rr
(iii) Moreover, UT E BVo°(0,1), UT (0) := aplimp-0+ UT(p) exists finite and
L=-BSGuT+8o x Consequently, T = h# (-BSGuT x Q Sn-1 ) + bo x Q B (O, UT (0)) . (iv) If T = Gu, u E cart' (B(0,1), RN), then UT (0) = 0.
Proof. We split the proof into several steps. Step 1. Let Q be a rotation in R', Q E SO(n). Taking into account the form of h and that spt T C E we easily see that in R21 \ {0, 0}
(Q x Q)#T = (Q D4 Q)#h#(L x
= h#(L x
QSn-'1)
QSn-1
U) = h#(L x
Q#ISn-11)
= T,
that is, T is Q D4 Q-invariant. Let UT be the BV-function associated to T =T(M,V,T'). We know that (x,uT(x)) E M for a.e. x, hence UT (X) = InT (x) I x
as M C E. Finally, by the rotational invariance of T,
f
T(IyI(v(x)dx) =
f = f IuT(Q-'X) I o(x) dx
holds for any cp E Therefore
{0}), and actually for any 0 E C°°(B(0, 1)). UT(X) = UT(IxI)
ICI
for some UT > 0. This proves (i). Step 2. L is locally i.m. rectifiable in IR+ \ {0, 0}. In fact, being T of finite
mass and i.m. rectifiable, and being g locally Lipschitz on R2n \ {0, 0}, it
2.6 Radial Currents
445
follows from Proposition 3 that L L (R2 \ B(0, e)) x Q Sn-1 , and consequently LL (IR2 \ B(0, e)) is i.m. rectifiable for any e > 0. Step 3. We now prove (ii). Denote by (eLAeq)(p r,B) the tangent n-vector to L x I Sn-1 ]I in R2 X S'n-1 at (p, r, 0), p + r 0. Since T = h# (L x Q S ' ' ) , the n-vector h*,(p,r,B) (eLAes)
generates the tangent plane to T = h#(L x Q Sn-1 JJ) at (p9, rO), compare the proof of Proposition 1. By (5), (7), (6), (8) we have
h*,(p,r,6)(eLnee)I = (p2+r2)2 eLIIeeI = (P2+
r2)!
Therefore
T(w) = L x TSn-1(h#(w)) = fdIILx f(p2 + r2)
{
J
QSn-1 II
<w,T'>Ih(p,r,e)dx"-1(0)}dHI LII
taking the supremum for w E Dn(IR2n \ {0, 0}), IwI < 1, we then find
M(T) = M(TL (B(0,1) \ {0}) x R') < nwn f (p2 +r2)' dMI L II To prove the opposite inequality we use the characterization in Proposition 3
Lx
T(g#w)
.
We choose w E Dn((I[g+ \ {0, 0}) x Sn-1) with Iw(p, r, 0) I < (p2 + r2)
L x Q Sn-1 (w) =
.
Then
fdMTII
= f <w,eLnee>Ig(X,Y)(IHI2+yI2)2dMITII <_ f dIITII. Taking the supremum in w we then infer nwn Step .4.
f(P2+r2)IdIILII < M(T) .
Set
LE := LL((E,1)xR),
E>0.
We shall prove that LE E cart((,-, 1) x R) for every E > 0. From step 2 we know that LE is i.m. rectifiable, M(LE) < oc. As a consequence of Proposition 3 we also see that 9LE = 0 in (E, 1) x R. Consider now the 1-form 0 := cp(p, r) dp in D' ((0,1) x JR). By (13)
446
4. Cartesian Currents in Euclidean Spaces
0nwsn-= = h#g As
n
and
90(o) = P(Ixl, yl)dlxl
II dxi
9(wsn i=1
we deduce
9 #(AAw sn- =)
y) = `P(xl, Ixln-1 dx
,
hence (16)
QSn-1(OAwsn-1)
nwnL(O) = L x
= T(`P(I " IyI) dx)
.
This implies that
L(O) = and, choosing
0
if cp > 0
= W(p) dp
L(O) = T( Ixln--I1) =
Jo
AP (P) dp
i.e. 7r#L = Q (e,1) , 7r denoting the projection of (e, 1) x IEF into (e, 1). As finally (16) implies obviously that the Ll-norm of L is finite in (e, 1) x IR, we conclude that LE E cart ((e,1) x R).
Step 5. From step 4 and the results in Sec. 4.2.4 it follows that L = -aSGu for some U E BVo, ((0, 1), IR). We now prove that
U=UT
in (0,1).
This follows from the equalities below which are valid for any cp E C' (0, 1) 1
fU(p)(P)dP =
r dP) = n-1 wn 1 7'(Ixl In-I1 Iy dx)
0
-1 -1 nwn
f
P(IxI) IuT(x)I Ixln-1 dx
B(0,1) 1
UT(xj)
lll1 dx = fUT(P)(P)dP.
B(0,1)
0
Step 6. Finally let us prove that
L = bo x Q (0,U+(0)) l - BSGu Consider the map c
:
Il8+
-+ FR+ given by
in
2.6 Radial Currents
(p, r) -' (P,(P+r)n)
447
.
It has range {(P,S) E JR+
S > pn}
I
and inverse 3 given by f3(P, S)
(P,
S1/n - P)
Clearly ,3 is Lipschitz continuous away from (0, 0). Consider now the map
V:
(0, 1)
II8
,
V (P)
(P + U(p)), .
We claim that V E BV(0,1). In fact V E BVo,(0,1) as U E BVo,(0,1). Moreover, for any 1-form w (p, r) we have Ia#w(P, r) l < n (P + r)n-1 jw(a(P, r)) j
,
hence 11
M(a-SGv) < n f (p + r)n-1 d1-l1 L a-SGu < +oo 0
by step 2.
From the trace theory, the current
b0 x [ (0, V+(0)) - aSGv has boundary -S(0,0) in [0, 1) x IIS and
V+(0) = aplimV(p) = aplim(p+U(p))72 = U+(0) P-.0
Therefore the current
5o x Q (0, U+(0)) - aSGu = /3#(bo x Q (0, V+(0)) l - aSGv) has no boundary in R2 \ {0, 0}. It follows that
Z := L-5o x Q(0,U+(0))}[+aSGu has no boundary in R2 \ {0, 0} and by step 5 has support in {0} x R. By the constancy theorem, compare Sec. 4.3.1, we then infer that
L=c8Ox
IR
+ + bo x Q (0, U+(0)) 1 - 8SGL1
in R+ \ {o, o}
for some c E JR. Since L has finite mass outside B((0, 0), E), we finally infer that c = 0, which concludes the proof of (iii). Finally, under the assumptions of (iv), we have
Q{0}xB(0,UT(0))J[ = G,,,L{0}xJR72 = 0 hence UT (0) = 0 .
448
4. Cartesian Currents in Euclidean Spaces
An immediate consequence of the definition of radial currents and of Proposition 3 is the following
Theorem 2 (Closure theorem). Let {Tk} be a sequence of radial currents in cart(B(O, 1) x R), suppose that Tk converge weakly in cart(B(0,1) x Il8n) to T. Then T is a radial Cartesian current.
Proof. In fact from
T k = h#(Lk x
TSn-1
l
on
Dn((Rn
x R n) \ {0, 0})
and Proposition 3 we infer that Lk -s L as currents in R2 \ {O, 0}. Then, it follows at once that
on Dn((R' x Rn) \ {0, 0})
T = h#(L x Q Sn-1
We conclude this subsection by proving the following approximation theorem for radial currents.
Theorem 3. Let T E cart (B(0,1) x R) be a radial current
T = h#(Lx
on
Dn (Rn X Rn \ {0, 0})
.
Then there exists a sequence of smooth radial functions uk (x) = Uk (IX l) ' with Uk E C1([0,1]), Uk(0) = 0 such that
Guk - T
and
M(Guk) -+ M(T)
Proof. From Theorem 1 we know that L = -BSGuT + So X Q (01 uT (0))
UT E BVo,(0, 1) and ZIT (p) = uT (p), limp-o ZIT (p) = UT (0) for pin (0,1) \
E with JEl = 0. Therefore we can find a sequence Ek 1 0 such that UT E BV((Ek/2,1)), ZIT (Ek)
= U1 (Ek) ,
uT (Ek) - UT +(O)
and
(17)
11 T 11 ((B(0, Ek) \ {0}) x Rn)
nwn
f
(p2 + r2)'i
dll BSGuT 11
-> 0 ,
(O,Ek)x]R
and, by Proposition 3 in Sec. 4.2.4, a sequence of functions Uj E C1((Ek/2i1)) such that
2.6 Radial Currents
-aSGu3 - -aSGuT
449
in (Ek/2i 1) x R
II aSGu, II ((Ek/2, 1) x R) --> II aSGuT II (Ek/2, 1) x R)
Since U (Ek) = UT (Ek) we can also assume, compare Note 4 in Sec. 4.4,
as j/-- 00
ujp(Ek) - UT (Ek)
(18) (19)
!
-->
co dII aSGuu II
w dII aSGuT II
J
(Ek,1)
as j --> o0
(Ek,1)
for any W E C°(R2).
Using (18) and then (18) (19) we see that we can choose a subsequence of Ek (still denoted by Ek) and a subsequence {jk} such that, setting I Uk(P)
P ? Ek
ujk(P) ujk(Ek) Ek
P O
and
xEB(0,1),
Uk(IxI)
Uk(X)
X71
we have
II Guk ((B(0,1) \ B(0, Ek)) x Rn) -'
(20)
nwn f (p2 +r2)
dII aSGuT II
(0,1)
and (21)
Uk(Ek) --+ UT (0) By a direct computation we then see that II Guk II (B(0, Ek) X 1Rn) Ek
nwn
f(P2+U2(P))(l+Uk(ek))1/2dP k
0
Ek
nwn (1 + Uk (Ek) E2k
pn-1 dp
) 0
converge to
wn (UT (0))n = f'(B(O,UT (O)) The result then follows taking into account (20), since by Theorem 1
M(T) = nwn f (p2 +r2) 2 dII aSGuT Il +rcn(B(O,UT(0)) (0,1)xJR
and by construction trivially Guk - T.
11
450
4. Cartesian Currents in Euclidean Spaces
3 Degree Theory In this section we begin by discussing in Sec. 4.3.1 n-dimensional currents in R. A key result we shall present is the so-called constancy theorem which states that any n-dimensional current T in an open and connected set of Rn satisfying aT L ,fl = 0 is a real, and in fact an integer, in case T is also i.m. rectifiable, multiple of the current integration over 0, i.e., T=cQ,fl]1
.
Such a result may be regarded as the formulation in terms of currents of the well-known fact that a distribution with zero derivatives is constant. We shall then see that n-dimensional normal currents in ]Rn, that is, currents
T such that
M(T) + M(3T) < 00
may be identified to BV-functions, and that n-dimensional normal currents in Rn+N cannot concentrate, i.e. have masses, on closed sets of dimension smaller than n. We finally conclude Sec. 4.3.1 discussing some relations between rectifiability of T and of 8T. The result of Sec. 4.3.1 are then used in Sec. 4.3.2 where we shall briefly develop a degree theory for currents. In fact we shall see that, in some sense, the natural context in which degree theory lives is that of currents, and of Cartesian currents, compare also Ch. 5. Let T E cart(.fl x An), .2 C ]Rn, be a Cartesian current and let A C 0 be a Borel set. Denote by 7r ,fl x Rn -+ Rn the linear projection onto the second factor, and consider the current projection of T L 7r-1(A), (T L 7r-1(A)) and the open set :
FT,A
Rn\sptaF#(TL7r(A))
.
For any y E FT,A consider also its connected component C. in PT,A. Of course (T L 7r-1(A)) has no boundary in C,,, thus by the constancy theorem
#(TL7r-1(A)) = m lCy
,
mEZ.
Clearly, if T = G,,,, u being a smooth map from 0 into ]Rn, then m is the classical degree of u at y with respect to A
m = deg (u, A, y) . Such an approach will be pursued in Sec. 4.3.3 where we shall see in which sense the image of an integral current splits into its connected components. More generally in Sec. 4.3.2 we shall see that the notion of degree is build
into the area formula and the projection mechanism. For any normal current T of dimension n in Rk X Rn, k not necessarily equal n, a degree mapping of T with respect to the projection F, counting with orientation how many times y E Rn is covered is well-defined as an integer-valued BV function. It enjoys all
3.1 n-Dimensional Currents and BV Functions
451
properties of the classical degree and agrees with it if T = Gw, k = n and u is smooth. Moreover, while as we have seen the degree is not continuous with respect to the weak convergence in Sobolev spaces, one instead sees that the degree mapping is continuous with respect to the weak convergence of normal currents provided their supports are equibounded. We shall then see that those results when read for Cartesian currents lead to state that the weak convergence of Cartesian currents preserves the degree. We shall conclude Sec. 4.3.2 by showing how one can easily recover the classical notion of degree for merely continuous maps from the notion of degree mapping.
3.1 n-Dimensional Currents and BV Functions It is usual to regard functions, say smooth functions from an open set Q of ]R' into R, as distributions in D'(.0) by identifying u to the distribution
f(x)u(x)dx,
A.(cp) :=
cp E
Such an identification turns out to be convenient in many instances; however it depends on the choice of the basis in R as it is clearly shown by the formula for changing variables in integrals. From this point of view it is better to work in the dual context of n-forms and identify functions u as n-dimensional currents acting on n-forms with compact support in 0. More precisely we identify u to the n-dimensional current in .f2, Tu := Q I[$n L u, given by (1)
f<w(x)eiA...Aen>u(x)dx
Tu(w) :=
or if w(x) = f (x) dx1A... ndxn T,, (w) :=
Q
Rn L u(w) =
r
J
f (x) u(x) dx .
Since for any change of variables ¢ : IEBn -* Rn we have 0# (w) (x) = f (cb(x) )
det DO(x)dx'A ... ndxn, the current T,, is well defined and independent from changes of coordinates which preserve the orientation. Of course, if we fix a basis , we have a one-to-one correspondence between distributions and n-currents given by
Tu(w) = µu(<w(x), ejA ... Aen >) From now on this correspondence will be understood.
Next we note that the boundary of Tu, 8T, is clearly related to the distributional derivatives of u. In fact, if w is an (n - 1)-form written as n
w(x) _ Y,(-1)i 1 fi i=1
d, -Ti
fi c Cc°(Q)
452
4. Cartesian Currents in Euclidean Spaces
we then have
LU(dw) =
QRB
f>2Dif(x)u(x)dx x=1
n
(2)
n
_ -El1Du= (fi) _ - < Diu, fi > i=1
i=1
A basic theorem in the theory of distributions states that any distribution µ on an open set f2 with zero distributional derivatives is constant, i.e., has the form cp r cJ cpdx, cER on each connected component of Q. In the dual setting of n-dimensional currents this amounts to the following simple but important theorem.
Theorem 1 (Constancy theorem). Let T E D,,(!7) be an n-dimensional current in an open set ( of Rn, and let U be an open set in Q. Suppose that
aTLU=0 or equivalently spt OT C 0 \ U. Then for each connected component Ui of U there exists a constant ci such that
T = >ci QUi]j
in U
.
i
In other words, for any f E C°O(U) we have
T (f (x) dx) = f f (x) c(x) dx where c(x) is paecewise constant, c(x) = ci on U.
Proof. We fix an open set W CC U, choose a < dist (W, 0U) and a mollifier cp, and we let cpQ := v-ncp(x/a). For any form w = f (x)dx in Dn(W) the form w, (x) := f * co, dx belongs to D' (U), hence we can define the current To in D- (W) by Ta(w) := T(wo).
Since wo --> w uniformly together with all derivatives as a -+ 0, we have (3)
TQ - T
in Dn(W)
.
Also, for every a we can find a constant c depending on a such that 11f*`pvlloo,U
c
ff1 dx, U
3.1 n-Dimensional Currents and BV Functions
453
hence
TT(w)I
(4)
cQ
J
IwI dx
.
U
Consequently TQ extends to a linear continuous functional on forms with LP(W) coefficients for any p > 1. By Riesz theorem, compare Ch. 1, then there exists a Radon measure y, on W and a unit n-vector v, (x) which is j -measurable such that
T,(w) =
f<w,v>d.
By (4) p, is absolutely continuous with respect to Lebesgue measure, thus we can write
T,(W) =
(5)
ff.9d ?In
On the other hand by assumption we have for any w E Dn-1 (U) with spt w C W
TT(dw) = 8T(wo) = 0. Therefore, taking w := fdx-j so that dw = (-1)j-1Djf dx and using (5), we find
Djf . Ba dxn = 0 j = 1, ... , n J for f E C,"°(U) with spt f C W. This evidently implies that 0, =constant depending on o on each component of W. The result then follows from (5) by letting o 1 0 and W T U if we take into account (3). By inspecting the previous proof one realizes that in fact we have the following more general result
Theorem 2. Let T E Dn(f2) be an n-dimensional current in a bounded open set .(2 C Rn. Suppose that
MQ(aT) < +oo
.
Then there exists a function u E BV0,(Q) such that
T(f (x) dx) = Jf(x)u(x)dx
V f E C,° (0) .
n Moreover (6)
IIaTII(A) = IDuI(A)
for any open set A C 0. Proof. We proceed as in the proof of Theorem 1 until we get (5). We now deduce from our assumptions, using I I f * oQ 11 oo < 11 f I I oo,
ff(x)O(x)dx l
cM(T)II f II,,.,U
454
4. Cartesian Currents in Euclidean Spaces
and, for i = 1, ... , n,
fDfxoxdx I
< cM(aT) II f J .'U
where c is independent from o, f E C°O (U), spt f c W, o < dist (spt f, aU). From the compactness theorem for BVOO-functions, we infer the existence of a sequence {crk}, 0k -40, such that Bok
0
in Lo 0(U)
where 0 E BV0O(U). From (3) we then get
T(f(x) dx) = ff(x)0(x)dx Choosing then w = (-1)i-lfidxti, fi (7)
`d f E C°O(0)
.
we finally infer
E
aT(w) = fdivf(x)9(x)dx
from which equality (6) follows at once.
An immediate corollary of Theorem 2 is the following characterization of normal n-dimensional currents in ,fl
Corollary I. An n-dimensional current in .f2 C Rn, T E Dn(fl) satisfies
Mn(T) + Mn(3T) < 00 if and only if it is representable as
T(f(x)dx) = ff(x)u(x)dx for some u E BV(.f2). In this case we also have
Mn(T) =
fu(x) I dx, Sl
Ms(aT) _
f
IDul
n
A consequence of Corollary 1 is that an n-dimensional current in (2 C RI with Mn(T) + MQ(3T) < no cannot concentrate on closed sets of dimension less than n. More generally we have
Theorem 3. Let T E Dn(fl) be a current with Mn (T) + Mn (aT) < oo, where .f2 is an open set in Rn+N Denote by Ira, a E I(n, n + N), the orthogonal projection of p.n+N into the n-plane coordinate determined by a given by xn+N\ _ (xa', ... , xa ), and let E be any closed subset of ,(2 with 7r(xl
H'(7ra(E)) = 0 Then TLE=0.
daEI(n,n+N)
.
3.1 n-Dimensional Currents and BV Functions
455
Proof. It suffices to prove the theorem in the case in which E is a compact set. First we show that
MU(TLE) < cMu(TL(IRn+N\E))
(8)
for any open set U with E C U C f2. To prove (8) we consider for any form w E D' (U), w = lal_n wadxa the currents in Dn, (]a1 x ... x IRan) given by
Ira#(T Lwa(x))(T) = T(wai)
T E D'(IRal X ... X Ran)
Obviously
M(7ra#(TLwa)) < Mu(T) suplwaI 0
(9)
.
As for any 77 E Vn-1(0) we have
a(T Lwa)(77) = T(wa dij) = T(d(war7)) - T(daA77) we also infer (10)
M(a7ra#(TLwa)) < M(a(T Lwa(X))) < M(5T) sup IwkI +M(T) sup dwal < oo .
From (9) and (10), taking into account Theorem 2, we deduce the existence of functions Ba E BV (IRal x ... x Ran) such that
Ira#(TLwa)(T) = fO(y)dy for every T E Dn(IRa1 x
... x IfBan). Hence 7ra#(TLwa)L7ra(E) = 0,
being xn (7ra (E)) = 0. Since now
M(7ra#(T L wa)) < M( 7ra# (T L wa) L (R'\ Ira (E))) = M(7ra# (T L wa) L Ira 17ra (E))) M((TLwa) L (iR +N \7ra17ra(E))) M(T L (R71+N \ 7r, 17ra (E))) sup wa
M(T L (R'+N \ E)) suplwaI a
and
T(w) = 1:7ra#(TLwa)(dxalA ...Adxan) a
we get (8) at once.
4. Cartesian Currents in Euclidean Spaces
456
Consider now a non increasing sequence of open sets Uk, Uk+l C Uk, such that lkUk = E. Then we have lim Muk (T L koo
(Rn+N
\ E)) = 0,
consequently T L E = 0 if we take into account (8). Let us discuss now in which way the previous results may be improved under rectifiability assumptions. The following claim readily follows from Theorem 1.
Theorem 4 (Constancy theorem). Let T be an i.m. rectifiable n-current in an open set (2 of Rn. Suppose that ,fl is connected and that
8T L .(2 = 0. Then
T = cQQ] where c is an integer.
Clearly, any i.m. rectifiable n-current T = r(M, 0, T) in I2n must have the form
T(f (x) dx) =
ff(x)u(x)dx
where u(x) is an integer-valued summable function on M. In fact T = e(x) e1A ... Aen with e(x) = ±1. Therefore it suffices to take u(x) := e(x)8(x). Next theorem allows us to draw an interesting information on n-dimensional currents in ][8n under the assumption that 8T is i.m. rectifiable.
Theorem 5. Let T be an n-dimensional current in an open connected subset f? of IIBW, T E D, (D). Suppose that 8T = -r (T, 0, e) in 0 is i.m. rectifiable. There there exists a constant c E Jl such that T - cIQ( is an i. m. rectifiable current. More precisely we have
T(f (x) dx) = cJ f (x) dx + n
J n
f (x) v(x) dx
where v E BV(Q), and for almost every x, v(x) is an integer. Proof. From our assumptions and Theorem 2 we deduce that there exists u E BV0,((2) such that T (f (x) dx) =
Jf(x)u(x)dx
d f E C°° (Q) .
3.1 n-Dimensional Currents and BV Functions
457
Moreover (11)
- < Diu, f > = fu(x)Df(x)dx = 5T((-1)x-1 f (x) dxZ)
If = E l;i(x)ei, we denote by n
n(x)
(-1)2-ll;iei i=1
the unit normal vector to the plane defined by the n-vector 1;, oriented in such a way that n(x)A (x) = e1A... new, .
Then we have aT((-1)i-1
(12)
f (x) dxi) = =
f(_i)' f (x) 2(x) 0(x) d7-l'-1 L T ff(x)n(x)O(x)dN'T.
From (11) and (12) we then infer
Diu = -ni(x) B(x)
dHn-1 LT
.
In particular the distributional derivatives of u have only jump parts with integral jump values. The conclusion then follows from the result that we state as Proposition 1 below.
Proposition 1. Let u E BV(.?), where .fl is an open and connected subset of Rh. Suppose that Du has only jump parts, Du = (Du)(i), and that the jump value is integer-valued. Then there exists a constant c E R such that
u(x) = c + v(x) where v(x) is an integer-valued function in BV(fl).
Proof. Clearly it suffices to prove that
u+(y) - u+(x) E z for almost every x, y E B(xo, R). In this case in fact we easily infer that
u+(y) - u+ (x) E Z
for a.e. x, y E .f2
by covering fl with balls, and that for some xo E 1
u+(y) - u+(xo) E Z if we take into account Fubini's theorem.
for a.e. y E .fl
,
458
4. Cartesian Currents in Euclidean Spaces
First we extend u as a function of the variables x E Rn, z E R-, t E IL8 by setting u(x, z, t) := u(z) ;
we may think of u as defined on R'. Then clearly u E B V (R2'+1) Du = (0, Du, 0), u+(x, z, t) = u+(z), u- (x, z, t) = u- (z) and, by the assumption, Du has only a jump part Du = (Du)el with integral jump value at N2n-a.e.
(x,z,t)EJu,=R'xJv,xRRS. We now consider the map O(x, y, t) := (x, x + t(y - x), t) and the function v := u o 0. Clearly v E BV(lR2n+1) and Dv has only jump part Dv = (Dv)(U) concentrated along 0-1(Ju) with integral jump values at l2n-a.e. (x, y, t) in
ill = 4-1(Ju). Applying Theorem 6 in Sec. 4.4 in Note 4 in Sec. 4.4 with i = 2n+1, so that t = 0} we then infer that for 7-12n-a.e. (x, y) E IT the function
17 = { (x, y, t)
I
t _% vx,y (t) := v(x, y, t) belongs to BV(R). Moreover its gradient has only a jump part, i.e.,
(Dvx,,)(a) = (Dvx,y)(c)
= 0,
and finally
Dv,,,y(t) = E kjStj (t)
k3 E Z
since
{t (x, y, t) E Jv} vyy(t) = u+(x y,t) = u+(x+t(y-x)) vy,y (t) = u- (x, y, t) = u- (x + t(y - x)) . The claim then follows since for 7.12'-a.e. x, y we have
u+(y) - u+(x) = vx y(1) - v5 y(0)
kj bt)
EZ
Remark 1. We notice that the proof of Theorem 5 simplifies considerably if we can prove that there exists an i.m. rectifiable current E in 0 such that aE = aT. In fact we then have a(T - E) = 0 and by the constancy theorem
T-E=cjQf. This is the case for instance if Si is star-shaped with respect to a point xo, as for E we could take the cone over aT from {xo}.
3.1 n-Dimensional Currents and BV Functions
459
We conclude this subsection proving a special case of the boundary rectifiabality theorem we stated in Ch. 1, as consequence of De Giorgi's rectifiability theorem Theorem 1 in Sec. 4.2.4, and a decomposition theorem for n-currents in
Theorem 6. Let T be an i.m. rectifiable n-current in £2 C ]Rn. If MQ(aT) < oo, then aT is i.m. rectifiable in (2. Proof. Since T is i.m. rectifiable T has the form
ff(x)u(x)dx
T(f (x) dx) =
where u E BV(u2) and u(x) E Z for a.e. x E (2. From Remark 1 in Sec. 4.1.4 and Theorem 2 in Sec. 4.1.4 we infer
Du = ('u+(x) - u-(x)) n(x, Ju) 7-( 1 L Ju . Using (7) we can then write
OT = r(Ju, u+ - u_,
J,,,)) F-I
Theorem 7. Let T be an i.m. rectifiable n-current in .f2 C 1RI such that MQ(3T) < oo. Then one can find a decreasing sequence of Caccioppoli sets Uj, j E Z, such that T
00
00
+00
j=1
j=0
-00
and
+00
IlaTil = T = C'' L u with u E BV (,fl, Z). For any
T j E 7G, set
Ui :_ {xE.f2 I u(x)>t}.
Being u integer-valued, we can easily check that 00
Do
j=1
j=o
u(x) _ E XU, (x) - E Xn\U-, (x) from which the decomposition formula for T follows. On the other hand
Et(u):={xE.fl I u(x)>t}=Uj
for
j
4. Cartesian Currents in Euclidean Spaces
460
and by the Fleming-Rishel formula, Theorem 2 in Sec. 4.1.1 we conclude that the sets U, are Caccioppoli sets and, +°o
+°o
IlaT11(B)=IDuI(B)= f IDxE,(u)I(B)dt=E11aQU;I11(B) -00
-00
for any Borel set B in Rn. The decomposition formula for aT follows now from the decomposition of T.
3.2 Degree Mapping and Degree of Cartesian Currents Let Q be an open set in Rn and let u :.fl --> IRn be an almost everywhere approximately differentiable map with Jacobian determinant det Du in L' (Q). Here Du(x), as usual, denotes the approximate gradient of u at x. In Sec. 3.1.5 we have seen that the area formula leads naturally to the notion of degree of u. In fact defining
deg (u, 0, y) :_
(1)
sign (det Du(x)) zEu-' (y)fAD (uu)
the area formula says that
deg (u, .fl, .) is a well defined function in L' (lPJ), and, as L'-function, depends only on the equivalence class of u. (ii) deg (u, Q, y) takes integer values for hn-a.e. y (i)
(iii)
for any cp E C°(IRn) n L°°(Rn)
f (y) deg (u, S?y) dy = f(u(x))detDu(x)dx.
(2)
Of course the sum in (1) is extended only to points x where u is approximately differentiable.
Suppose now that u E A' (Q,W2). Then (2) can be written in a more geometric way in terms of Gu. Consider in fact the current graph of u, Gu, defined on n-forms w E Dn(I? x Rn) by
Gu(w) :=
f<w,M(Du)>dx. S?
Then for any cp E C°(R') we have (3)
f(u(x))detDu(x)dx
= Gu(cp(y) dy'A ... Adyn) .
n Denoting as usual by ? the orthogonal projection on the second factor of Rn x Rn,
r(x, y) = y, from (2) and (3) we then infer
3.2 Degree Mapping and Degree of Cartesian Currents
461
Proposition 1. Let u E A' (,f2, Rn). Then
fdetDu(x)I dx
(4)
n 5i# G. ('o (y) dy) = f p(y) deg (u, Q, y) dy
(5)
Proposition 1 says in particular that deg (u, ,fl, y) is the Radon-Nykodim derivative of the measure p :_ :R# GU L dy'A... ndyn with respect to Lebesgue measure Gn. Therefore counting points over a given y with their sign, that is the degree, is an information totally build into the mechanism of projecting the graph of u on the target space ll$n, i.e., into the current :R#Gu. In view of Proposition 1 one can easily extend the notion of degree to any Cartesian current in cart(Q x IRn), and actually to any current T E Dn(Rk+n), where k is not necessarily equal to n, provided r#T exists. We shall consider the case of n-dimensional i.m. rectifiable currents T =
T(M, 9,T) in Rk x R n. Since M(T) < oc, T extends to forms with bounded and continuous coefficients on the whole of Rk x I. In particular the current F#T, #T (w) = T (7r#w), is well defined. Denoting now, as usual, by (e1, ... , ek) and (El, ... , En) the canonical bases in IRk and in IRn, the tangent n-vector T (X, y)
of T at (x, y) can be written as E T as (x, y) ea AE, 3.
T (x, y)
aH+1i31=n
Clearly, compare Proposition 1 in Sec. 2.2.1, =Too
(x, y)
#,(x,y)T (x, y)
,
I JM (x, y) I = IT
oo (x, y) l
hence
#T(W(y)dy'A...Adyn)
= f <0(x,y)d7-LnL,M =
(6)
JW(Y)O(x,y)T06(x,y)dHn(X' y) L M
We now set
Definition 1. The degree mapping, or simply the degree of T at y with respect to F is defined by (7)
deg (T,
y) :=
E 0(x, y) sign (T xE*-'(y)nM
whenever it exists as a real number.
00
(x, y))
4. Cartesian Currents in Euclidean Spaces
462
By definition deg (T, R, y) is an integer whenever it exists. From (6) and the area formula we immediately infer
Proposition 2. Let T = r(M, 0,T) be an n-dimensional i.m. rectifiable current x RI. Then in 1 (i) deg (T, ?, y) E L' (RI) dy) = f W(y) deg (T, F, y) dy
(ii) F#T
In particular #T is an n-dimensional i.m. rectifiable current in Rn
F#T = T(M, A, 6) where
Nt := {y
I
deg (T, i, y)
0}
)t (y)
,
Ideg (T, Fr, y) I
and
(y) := sign (deg (T, , y)) r1A ... nen From (5) and (6) we obviously have
deg (u, Q, y) = deg (Gu, 5r, y)
whenever u E A' (Q, Rn). Later we shall be mostly interested in the degree of Cartesian current. To simplify notations and to be consistent with the classical notations we set
Definition 2. Let T E cart(. x RI), S2 C RI, and let A be a Borel set, A c Q. The degree of T over A at y E Rn is defined by
deg(T,A,y) := Of course if T = G, we have
deg(G.,A,y) =
deg(u,A,y).
Before discussing properties of the degree, and in particular before proving that the degree defined above enjoys all properties of the classical degree, and moreover is continuous with respect to the weak convergence of currents, let us consider a few examples.
0 First we consider the map uo (x) := x/ x( which belongs to Al (B(0,1), Rn), B(0,1) being the unit ball in R'. We have detDuo(x) = 0
for a.e. x
hence
deg (uo, B(0,1), y) = deg (Gu0 F, y) = 0
In contrast, if we consider the Cartesian current
for a.e. y
.
3.2 Degree Mapping and Degree of Cartesian Currents
463
T := Gv,o + So x Q B(0,1)
we have #T = Q B(0,1) 1 hence if Iyj < 1
deg (T, , y) = deg (T, Q, y) = jl 10 ifjyj >1. Finally, for the Cartesian current
T := G.0+LxQS'n-1JJ,
L being a one-dimensional i.m. rectifiable current in B(0,1) such that aL L B(0,1) = do, we have r#T = 0. Hence deg (T, F, y) = 0
for a.e. y E IE8"
2 Consider now the Cantor-Vitali function V : [0, 1] -> [0, 1], see Sec. 4.2.4. Being V continuous and nondecreasing with V (O) = 0 and V(1) = 1, its classical degree at every point y E (0, 1) is 1. In contrast to that, in the sense of
Definition 1 and Definition 2 we have
deg (V, (0, 1), y) = 0
for a.e. Y E (0,1)
,
since V'(x) = 0 for almost every x E (0, 1). However, considering its subgraph, SGv, we have (
deg (-aSGv, i, y)
1
jl 0
for y E (0, 1) otherwise.
In other words, the notion of degree introduced here distinguishes between the graph of V (x) (in the "almost everywhere differentiability" sense) and the boundary of the subgraph of V (x) as Gv and -aSGv are different. This corresponds to the fact that one can find sequences of smooth maps Uk : (0, 1) --> (0, 1) such that
G,,,, - aSGv
,
while there is no sequence of smooth maps vk such that
Gvk - Gv ,
(8)
the fact that 8Gv # 0 in (0, 1) x ]R being an obstruction to the approximation in (8).
In order to state the basic properties of the degree let us define the set
TT :_ {y E R' We have
I
y 0 spt (8r#T)} .
464
4. Cartesian Currents in Euclidean Spaces
Theorem 1. Let T = r(M, 0, 1;) be an n-dimensional i.m. rectifiable current in IRk x Wz. Then we have (i) deg (T, F, ) is an integer-valued function in L' (Rn), which vanishes outside
the support of#T. (ii) deg (T, r, y) is constant on the connected components of FT.
(iii) M(8F#T) < oo if and only if deg (T, F, ) E BV (Rn). In this case 8--7r#T is an i.m. rectifiable (n - 1)-current and the gradient of deg (T, r, ) has only jump part with integral jump values. (iv) (Excision). If {Ai} is a numerable family of pairwise disjoint Hn-measurable sets in IRk+n and A:= U Ai, then i
for a. e. y e
deg (T L. A, r , y) = > deg (T L Ai, F, y)
n
i (v)
(Homological invariance).
Let T, S be two i.m. rectifiable n-currents in
k+n. Then
if and only if
deg (T, r, y) = deg (S, F, y)
for a. e. y E W1
.
(Homotopic invariance). Let Tt be a 1-parameter family of i.m. rectifiable n-currents, t E [0, 1]. Suppose that the map t --> F#Tt (w) is continuous for any w E Dn(Rn). Then deg (Tt, F, -) is independent from t (vii) (Leray product formula). Consider any Lipschitz continuous map co IRn -* R and the map ¢(x, y) = (x, co(y)) in ]Rk x IRn. For k E Z and (vi)
F.k := {y E IRn
I
deg (T, 9r, y) = k}
we have k deg (co, Ek, y)
deg (q#T, Fr, y) =
for a. e. y E IRn
kEZ
(viii)
(Continuity). Suppose that Th, h = 1, 2, ..., and T are i.m. rectifiable ncurrents in IRk x W2 such that
Fr#Th - r#T . Then
deg (Th, r, y) -+ deg (T, F, y) in the sense of measures. In particular if deg (Th, F, y) = constant = A with respect to h on some open set U, then deg (T, F, y) = A on U.
3.2 Degree Mapping and Degree of Cartesian Currents
465
Proof. The area formula and the constancy theorem yield at once (i) and (ii) respectively. The first part of (iii) follows easily by just computing air#T from (6)). The rest of the claim then follows as in the proof of the Theorem 6 in Sec. 4.3.1. The excision formula in (iv) is an immediate consequence of the trivial equality 00
(T L A) _
r# (T L Ai) i=1
taking into account that deg (T, F, y) E L1(Rn). To prove (v) we observe that T - S is i.m. rectifiable and aF# (T - S) = 0, hence by the constancy theorem
c[R n
cEZ.
From M(T - S) < M(T) + M(S) < oc, we infer c = 0, hence F#T = F#S, from which the claim in (v) follows at once. The claim in (vi) follows easily from the continuity in t of F#Tt taking into account the fact that the degree mapping is integer-valued. To prove (vii) we notice that O#F# (f (y) dy) = f ('P(y)) det Dco(y) dy hence
f f(y) deg (O#T, ?, y) dy = F#O# (T) (f (y) dy) = T (O#Fr#(f (y) dy)) =
_ >h
ff
det
f
f (co(y)) det Dcp(y) deg (T, Fr, y) dy
dy = >h f f(y) deg (co, Eh, y) dy hEZ
hEZ Eh
Finally (viii) is trivial.
Remark 1. We explicitly remark that in the case in which a5r#T has finite mass, Theorem 1 (iii), the i.m. rectifiable current afr#T and the BV-function deg (T, F, y) are related by
air#T(w) =
if w =
f
div f (y) deg (T, F, y) dy = -
1(_1)i-1fi(y) dyi
f
fi(y)Dideg (T, F, y) dy
compare (7) in Sec. 4.3.1 and Theorem 6 in
Sec. 4.3.1.
As consequence of Theorem 1 (ii) and (iii), on each connected component of FT := ]EFn \ spt the degree of T, deg (T, F, y), can be identified with an integer constant. If more-
over a7r#T has finite mass, such an identification holds in the BV sense, that is, 7-1n-1-almost everywhere, in particular, at every point in which the degree is approximately continuous. That identification being understood, it is then useful to state a pointwise version of Theorem 1.
466
4. Cartesian Currents in Euclidean Spaces
Fig. 4.7. The degree is locally constant on the complement of 8u(fl).
Theorem 2. Let T be an i.m. rectifiable n-current in IRk+n (i)
(Excision). If {Ai} is a numerable family of disjoint Borel sets in JRn, and
yEnI1FTLAi
-n(n1 -TLA2f
TTLU;Ai,
then
deg (T L U°_1 Ai, F, y)
00
deg (T L Ai, r, y)
i=1
(ii) (Homological invariance). If T, S are i. in. rectifiable n-currents in ]Rk+n and
aF#T = 8F#S then Is = 1T and deg (T, F, y) = deg (S, F, y)
VYE
(iii) (Homotopic invariance). Let Tt, t E [0, 1], be a family of i.m. rectifiable n-currents in IRk+n. Suppose that t --> r#Tt is a continuous map from [0, 1] into Dn(IRk+n) and that yo is an interior point of
n fTti t1] Then deg (T, , yo) is independent of t. (iv) (Leray product formula). Let cp : ]Rn -> Rn be a Lipschitz map and let 0
(x, co(x)). Denote by Ci the connected components of 1T and by yi any point in Ci. Then for any y which is not in the closure of O(IRn \ JT) we have
deg(c#T,?,y) = i
3.2 Degree Mapping and Degree of Cartesian Currents (v)
467
(Continuity). If Th, h = 1, 2, ..., and T are i.m. rectifiable n-currents in Rk+n such that
#Th - F#T , then
deg (Th, F, y) - deg (T, 7, y) for any y which is an interior point of
h FTh Proof. (i) From Theorem 1 we know that for a.e. y E FT L An fl TT L A; i
deg (T L Ai, 5r, y)
deg (T L A, 7r, y)
The claim then follows since all degrees in the previous formula are constant on the connected components containing y.
(ii) We obviously have FT = Ts. For a small ball B around y E FT we have
deg (T, r, y) Q B = 7r#T L B = #S L B = deg (S, F, y) Q B if we take into account the constancy theorem. This proves the claim. (iii) Considering again a small ball B around yo we have
deg (Tt, F, y) Q B
=
#TtLB,
from which we see that deg (Tt, i, y) is continuous. The claim then follows since the degree is an integer.
(iv) By the assumption we can find a ball B centered at y and contained in the complement of the closure of O(R \ FT). We can also assume that B is contained in the connected component of y E Tf#T. Thus O-1(B) C FT and B C Rn \ 0(8Ci). We have for f E C'°(B) deg (O#T, F, y)
ff()dy
f
=
f (y) deg (O#T, F, y) dy
= Eh f f (y) deg (0, 1h, y) dy h
where
Zh = {y
I
deg (T, r, y) = h} .
Since
fT n L'h = U{Ci we then find
I
deg (T, Ci, y) = h}
4. Cartesian Currents in Euclidean Spaces
468
Yh f f (y) deg (0, 11h, y) dy = Eh
f
f (y) deg (0, Zh n 1'T, y) dy
_ >deg (T, Ci, y) f f (y) deg (0, Ci, y) dy i
and the claim follows by choosing as y in deg (0, Ci, y) a fixed point yi E Ci. Since (v) is trivial, this concludes the proof.
Let us return to the situation we considered at the beginning of this subsection, i.e., let us consider an almost everywhere approximately differentiable map u : .(2 C Rn ---* R' with Jacobian determinant in L'. We can then consider the n-dimensional current in R defined by / f p(u(x))det Du(x) dx = J W(y)deg (u, Q, y) dy nu,Q(P(y)dy)
and it is easily seen that the properties stated in Theorem 1 and Theorem 2 remain true with the following replacements
T -u,
deg (T, fr, y) ---> deg (u, .R, y),
*#T -i 17u,n.
In particular we explicit mention (i)
Homological invariance If alfu,n = 81f,,,f then deg (u, ,fl, y) = deg (v, .fl, y)
(ii) We have (9)
MT., g (w) =
fdivg(Y)deg(u, .R, y) dy Rn
if w = Ei
.,(-1)Z-lgi(y)dy2, g E Cc(1Rn,]Rn).
Of course if u E A'(Q,1Rn), we have 17u,,n = fr#Gu Let u E A'(Q,R"1). In dealing with the degree of u a certain precaution is necessary, as the previous general results holds away from fraGu. In particular if it E Al (fl,Rn) we may infer that (10)
deg (u, S?, y) = deg (v, 0, y)
if and only if (11)
fr#aGu =
equality of the traces on an in general being not sufficient, compare Remark 4 in Sec. 3.1.5 and 3 below.
3.2 Degree Mapping and Degree of Cartesian Currents
469
Notice that the condition (11) is fulfilled if BGv, = aG,,. This is the case, for instance, if u, v E cart' (.fl, 1R?),
,fl DD .fl,
u = v on ,f2 \ l .
The results of Sec. 3.2.5 now allow us to read (11) in terms of equality of traces. For instance from Proposition 4 in Sec. 3.2.5, we infer at once
Proposition 3. Let 12 be a bounded Lipschitz domain in RI and let u, v E Ap,q(f2, R') with p > n - 1, q > n/(n - 1). Suppose that u = v on a(2 in the sense of traces. Then deg (u, ,fl, y) = deg(v, .fl, y).
3 Of course in the case of the maps x and x/jxj from B(0,1) into B(0,1) the failure of equality (10) is due to the fact that Gx,1 does not belong to cart'(,fl,R''). Example 1 in Sec. 3.2.5 where y is chosen strictly decreasing shows the existence of a one to one map u E cart'(B+(0,1),82), B+(0,1) C R2, 1
with on
deg (u, B+(0,1), y) = 1
u(B+(0,1))
,
for which
u(x) = (0, 0)
on aB+(0,1)
in the sense of W1''(B+(0,1), IR2), Vp < 2 .
Finally, it is worthwhile mentioning that (9) can be written as a boundary integral provided u is sufficiently smooth. In fact we have
Proposition 4. Let ,f2 be a bounded Lipschitz domain in R' and let u be a map in Ap,q(.fl, Rn), p > n - 1, q > n/n - 1. Suppose that also the trace of u on a(2 belongs to Ap,q (a,fl, Rn ), then
f divg(y)deg (u, .f2, y) dy = 1R
J an 2,7=1
g(u(x)) (adj Du (x))' vj (x)
d7-ln-'
for all g E CC (R', Rn), where v is the exterior unit normal to an. Proof. In fact from (9) and Theorem 1 in Sec. 3.2.5, we get
J
divg(y)deg (u, Q, y) dy = aflu,n (w) = afr#Gu(w)
Rn
= #aGu(w) _ ft#Gu,an(w) Consider now a Cartesian current T E cart(.f2 x lR' ), ,f2 C R'. In analogy with Definition 2 we write for any Borel set A C .2 FT,A
rTLAxl.n = A\spt #a(TLA
470
4. Cartesian Currents in Euclidean Spaces
We could then rewrite Theorem 1 and Theorem 2 for Cartesian currents. We leave it to the reader; instead we shall explicitly state two propositions which will be used later on.
Proposition 5. Let 1? CC ,f2 be two bounded domains in Rn and let T, Tk E cart(, x W) be such that Tk T and sup,, M(Tk) < oo. Suppose that
(Tk-Th)L7r-1(.(2\f2) = 0
Vh,k.
Then for all y E TT D we have
deg (Tk, f2, y) = deg (T, ,fl, y)
Proof. For all k we have
TkL.7r-1(.R\,f2) = TL7r-1(.f2\,(2) hence
8(Tk L 7r-1(.(2)) = -8(Tk L 7r-1 (, \ D))
-8(T L 7r-1(f \ ,(l)) = 8(T L 7r-1(,f2))
.
The result then follows from the homological invariance of the degree.
Proposition 6. Let u, uk E cart7' (0, WL ), p > 1, and let A be a Borel set in fl. Suppose that and sup 11 uk 11Cart, < oo Gu. uk k
Then
deg (Guk, A, y) --> deg (Gu, A, y)
provided y is an interior point of Fcu,A fl
k
FGuk,A
Proof. In fact if w = 0(y) dy is an n-form with compact support in the connected component of y in FGu,A n Fcuk,A with w = 1, we find k
deg(Guk,A,y) = i#(Guk L7r-1(A))(O(y)dy) f(uk(x))detDuk(x)dx A thus
k i m deg (Guk, A, y) =
f 0(u(x)) det Du(x) dx = deg (Gu, A, y) A
.
3.3 The Degree of Continuous Maps
471
Again a certain care is necessary. Next example may serve to classify the situation. 4
Consider the sequence of Lipschitz maps Uk : (0, 1) ---r (0, 1) given by r
uk (x)
kx if0<x<1/k
Sl
1
ifl/k<x<1.
Clearly
b y E (0,1) . The currents Guk of course belong to cart((0,1) x I[8) and converge weakly in cart((0, 1) x R) to the Cartesian current in cart((0,1) x R) Guo where uo(x) = 1 deg (uk, (0, 1), y) = 1
for all x E (0, 1), for which deg (Guo, (0, 1), y) = 0
`d y E 118 \ {1} .
However, if we regard the Guk's as i.m. rectifiable currents in I[8 x R, we see at once that Guk - T := Jo x Q (0,1) + Q (0,1)1 x bl and
deg (T, F, y) = deg (T, [0, 1), y) = 1
b' y E (0,1) .
3.3 The Degree of Continuous Maps In this subsection first we show how one can recover the classical notion of degree for merely continuous maps, and secondly that the degrees in the sense of currents and of continuous maps agree for currents T with spt T C { (x, u(x)) x E f2}, u continuous.
Let ,f2 be an open bounded set in 1R' and let u r be a continuous map. We extend u outside of £2 to a continuous map that we still denote by u. Chosen a standard mollifier cp and a, 0 < a < 1, we let cp0(x) := Q-n(p(x/a) and
uQ = u * co,
.
Obviously uo(x) converge uniformly to u in U. Consider now the continuous map H : (0,1) x .f2 -> I1 x RI given by
H(v, x) := (x, u,(x)) For any r and a, 0 < r < a < 1, H is of class CO° in [r, a] x ,f2 and in 1R'
x I18n
we have
=H#(8[(T,a)Jj x QQ]j-I(T,a) x Qa21) H#(Q(T,a) x af2 = Gu., -G, -H#([(T,a)I x 9QQ]j)
aH#(Q(T,or)I x jf2
= H# (J, x IQ1)
H# (J, x QQ1)
472
4. Cartesian Currents in Euclidean Spaces
Consequently the n-current
T := G,,,, -GuT
-H#(Q(T,a)I x aQQ])
has no boundary and finite mass. Hence
deg (T, :, y) = 0
for a.e. y
.
Choose now y E W \ u(a.Q) and let U be a neighborhood of u(a.(l) such that dist(U, y) > 0. Then for o < co = co(y) we have uQ (z) E U
`d z E aS?
and spt (8Q 0 f) C B.C. Therefore, since H ([-r, v] x 8f1) C U
sptH#(Q(-r,o)
CU
for 0 < Tr < v < co. This implies that
x 9 S2fl,F,y) = 0 .
deg (H#(Q(T,o
Consequently deg
F, y) = deg (Gur , ?, y)
for0<7-
11 v - u I an < E/2 and the homotopy
H(t,x) :_ (x, tu, + (1 - t)v) o, close to 0, the argument given above shows that for any point y in Rn \ UE
UE := {y E Rn
(1)
I
dist (y, u(8.f1)) < E}
we have
deg (v, Q, y) = deg (u,, ,fl, y) for all o, close to zero.
Of course this allows us to define the degree of u at any point y E Rn \ 8,f1 as
deg,(u, Q, y) := deg (Gu,,, Fr, y) for o- sufficiently close to 0 and state that degc(u, (1, y) does not depend on the approximating sequence u, we chose. Moreover from the previous argument we see that deg . (u, Q, y) = deg, (v, ,fl, y)
if u, v E C°(Q) with u = v on 8f1. We then have
3.3 The Degree of Continuous Maps
473
Proposition 1. Let T be a current of finite mass in (,R x RI). Suppose that there is a function u E C°(f2, ]R) such that spt aT C {(x, u(x)), x E a.(2} or, in particular, that T E cart (S-2 x Rn) and spt T C {(x, u(x)), x c .f2}. Then deg (T, fl, y) = deg,, (u, f2, y) for ally E I[8n \ u(aQ).
Proof. We shall prove that for any positive e
deg (T, Q, y) = deg,(u,12, y) V y E UE where U is the set defined in (1). From this the conclusion follows at once. In order to prove (2) we consider a map v E C1(Q,Rn) such that (2)
IIv - uII.n < e/2. As we have seen
deg,(u, O, y) = deg (v, Q, y) = deg (G,,, ,f2, y)
(3)
for all y E Rn \ l.[E. Next, we consider the homotopy Ot : ,f2 x ll.' --> (2 x ]Rn given by Ot(x,y) :_ (x,ty + (1 - t)v(x)) .
It is easily seen that {Ot#T t E [0, 1]} yields a family of Cartesian currents and in fact a continuous family of currents with 0o#T = Oo# o 7r#T = O°# Q (2]] = Gv and 01#T = T. Moreover I
spt aot#T C (an x Rn) f1 {(x, y)
I
x E aQ, y = tu(x) + (1 - t)v(x)} ,
in particular
spt #aot#T C (tu + (1 - t)v)(aQ) c UE
V t E [0,1]
By Theorem 2 in Sec. 4.3.2 (iii) we then infer deg (T, (2, y) = deg (Gz,12, y)
d y E l1 ] \ uE
,
which together with (3) yields (2).
Remark 1. Of course from Proposition 1 we infer that if T E cart(.f2 x Rn) and T = Gu, for some u E C°(72, RI), then
deg (G.., fl, y) = deg.(u, 0, y)
R'
Remark 2. For the Cantor-Vitali function V (x) we then recover the well-known relation deg,, (V, (0, 1), y) = 1 V Y E (0,1) and compare 2 in Sec. 4.3.2, deg,, (V, (0,1), y) = deg (-aSGv, (0, 1), y)deg (Gv, (0, 1), y)
.
474
4. Cartesian Currents in Euclidean Spaces
3.4 h-Connected Components and the Degree Let T be an integral current in Rk+n, i.e. T E Rn(]Rk+n), M(T)+M(8T) < oo, so that (by the boundary rectifiability theorem) also OT E Rn-1(Rk+n), aT T(M, 0, aT ). We have seen that deg (T, fr, y) is an integer-valued BV (Rn)function with ID deg (T, fr, y) I (][8n \ fr(.M)) = 0. More precisely, setting
Ek :_ {yEJR'
I
deg(T,W,y)=k},
we have seen that Ek has finite perimeter and
F#TLEk = k{EkJ
.
We shall now see that we can decompose RI \ 1r(M) into "connected" components on which the degree is constant, paralleling the classical theory. This is achieved by modifying the classical topological notions of closed sets and connected components into corresponding homological ones. A byproduct of those kind of results may be an alternative definition of the degree, but we shall not pursue any further this point. Definition 1. Let F C ][fin be an (n - 1) -rectifiable set and let A C lRn be a set
of finite perimeter in R, P(A) < oo.
We say that F is homologically closed, in short h-closed, if there exists a current T E Rn(Rn), with OT E Rn(Rn), OT = T(M, 0, ), 0 > 1 on M, such that F = M. (ii) We say that A is homologically disconnected if there exists B C A with (i)
P(B) < oc, 0 < IBI < oc, IA \ BI > 0 such that a-B C a-A,
and
consequently 8-(A\ B) C a-A, in the xn-1-sense. Otherwise we say that A is h-connected.
Equivalently we can state
Proposition 1. F C Wn is h-closed if and only if there exists u such that
F = Ju
E BV (,f2, Z)
xn-1 a.e.
where Ju is the jump set of u.
Proof. From Corollary 1 in Sec. 4.3.1 we know that T is an n-dimensional i.m. rectifiable current with finite boundary mass if and only if
T = £n L. u for some u E BV (,fl) with integer values. Theorem 1 in Sec. 4.3.1 then yields
8T = r(Ju, u+ - u-, *n(-, Ju))
3.4 h-Connected Components and the Degree
475
Proposition 2. Let A C R' be a set of finite perimeter, P(A) < oo. Then the following claims are equivalent
A is h-disconnected There is an essentially non-constant function u in BV (.(l, Z), ess infA u < ess SUPA u, such that
(a)
(ii)
IDuI((A)1) = 0 (iii)
,
(A)1 := {x
I
O(A,x) = 1}
There is B C A, with O < B < oo and I A \ B I > 0 such that
P(A) = P(B) + P(A \ B)
.
In particular, if u E BV(.f2, Z), A is h-connected and IDul((A),) = 0, then u is constant in A.
Proof. ForBCAwesetB1 :=B, B2:=A\B. We have
a-Bi = a-A = and
(a-Bi n a-A) u (a-Bi \ a-A)
i =1, 1,2
(a-Ana-B1) u (a-A \ a-B2)
xn-1((a-Ana-B1) u (a-A\ a-B2)) = 0
as in the Hn-l-sense points in a-A cannot belong both to a-B1 and a-B2. Moreover
a-B1 \ a-A = a-B2 \ a-A
.
This shows that (i) is equivalent to (iii). Let B C A satisfy all conditions in Definition 1 (ii). Then u:= XB satisfies
Ju = a-B c a-A, i.e. IDul((A)1) = 0 and u is not constant in A, thus (i) implies (ii).
Let u E BV (R', Z) be non-constant in A. Then there is k E Z, k
0 such
that
B := {x E A I u(x) = k} satisfies I B I > 0, I A \ BI > 0, and I B I < oo as u E BV. Then a- B C Ju and
IDuK((A)1) = 0 imply that J,,, C R' \ (A)1 i.e., a-B C a-A. We can now show that, similarly to the topological situation, every set A of finite perimeter can be decomposed into h-connected components.
Theorem 1. Let A C R' be a set of finite perimeter, P(A) < oo. Then there exists a (possibly finite) sequence {Ak} of disjoint h-connected sets such that
A=UAk (1)
P(A) = EP(Ak) k
and, consequently, (2)
a-Ak c a-A
d k.
4. Cartesian Currents in Euclidean Spaces
476
The proof of Theorem 1 relies on the following two lemmas.
Lemma 1. Let A C W1, 0 < Al I< oc, P(A) < oc. Then there exists e > 0 such that for each (possible finite) disjoint decomposition A = k Ak satisfying (1) we have IAko > e for some ko.
Proof. Choose c > 0 so that
Al I> cnP(A)en where Cl ,is the isoperimetric constant
cn := sup IAkIn1 P(A)
If I Ak I < E for all k, we have
IAkI = lAklnnl lAk1
< cfP(Ak)en
and by (1)
JAI < c,, EP(Ak)en = cn P(A) en , k
a contradiction.
Lemma 2. Let A C Rn, 0 < A < oo, P(A) < oc. Then there exists a subset B of A with IBI > 0 which is h-connected. Proof.
Let e be the constant in Lemma 1. Assume that each B C A with
CBI > 0, P(B) < oc, a-B C a-A is h-disconnected. The collection of all such B's can and will be divided into four families r1, T2, 7-3 and T4 1. B E 7-1 if CBI > E and there is a decomposition B = B1 U B2 satisfying (1) such that IBiI ? E, IB2I >_ E 2. B C T2 if B rl, J BI > E, and there is a decomposition B = B1 U B2
satisfying (1) such that IBuI < e and IB2l < e
and IBl>e 3. BE 4. BE'r4iff0<JBI <e. Clearly for each B E T3 there exists a decomposition such that (3)
B = B1 U B2 ,
P(B) = P(B1) + P(B2)
,
I B1
<e,
IB21>0.
First we decompose A by only using decompositions of the type Tl. After a finite number of steps we arrive at a decomposition satisfying (1) no
A= U Bz
no E N, no > 1
i=1
such that every BZ E T2 U T3. Let us call this decomposition Ro. For each ordinal
a, a > 1, we shall define inductively the decomposition Ra:
3.4 h-Connected Components and the Degree
477
no
A = (UBa)) U (U B)
(4)
i=1
Q
satisfying (1) and such that
Bia) E T2 U r3 Vi and Bp) E T4 V /3 < a ,
(5)
as follows.
For a non limit a: By Lemma 1 some B(o -1) E T3, i.e., there exists a decomposition Bio -1) = B1 U B2 satisfying (3). Then we set B2p) := B1 B(a) :=
B(c) := Bia-1)
i3-,(3a-')
for i 0 io
for 0 < a - 1
B2
For a limit a: We set B(a) :_ n B2Q)
(6)
TO
V i = 1,...,no
/3
B(a)
(7)
da
B(,3+1)
a
By construction we see that (6) is in fact a intersection of a countable decreasing family of sets, so that we have IBia) I > e, because of the inductive hypotheses
JBza)l > e V 0 < a. Note also that no Bzp) is of type T1. This process can be realized for each countable a, but this way we obtain a non-countable family of disjoint sets with JBp ) I > 0, which is impossible since JAI < oc. O Proof of Theorem 1. Suppose A be disconnected. Then there exists B1 C A such
that 0 < B11 < oc, P(B1) < oo, JA \ Bil > 0 and P(A) = P(B1) + P(A \ B1). Set Al := A \ B1, and if Al is h-disconnected we continue this process (possibly transfinitely). After a countable number of steps we however exhaust A, because A is 'H'-o--finite, compare Ch. 1. This way we obtain a decomposition of A
A = UBk, 0
and
P(A) _
P(Bk) k
Using now Lemma 2 we simply decompose each Bk as
Bk = U Ajk)
P(Bk) _
This yields the conclusion since Ek a-A for all k.
A h-connected. P(A) which implies that
0
4. Cartesian Currents in Euclidean Spaces
478
Finally, next theorem provides a decomposition of RI \ F into h-connected components, if F is h-closed.
Theorem 2. Let F be an ?1n-l-rectifiable set which is h-closed in the sense of Definition 1. Then the set G :_ ][fin \ F can be decomposed as a countable disjoint union G := UkGk of h-connected sets Gk, called h-connected components of G, with a-Gk C F ?-1n-1-a.e.. Such a decomposition is unique, up to permutations. Moreover for any A C G with a-A C F, we have A D Gk xn-'-a.e. if AnGkJ > 0.
Proof. Let u E BV ([8n, Z) be the function in Proposition 1, i.e., such that
F = J,. Set
Uk :_ {xE1PJ \Ju We have
I
kEZ.
u+(x)=k}
G=UUk, P(Uk)
kEZ
consequently a-Uk C F. Now we decompose each Uk with UkI > 0, into hconnected components using Theorem 1
Uk = U G,k) We have
'd k .
a-UkCJu=F.
This proves the first part of the theorem.
If JA n GkI > 0, a-A C F, we set B1 := A n Gk, B2 := Gk \ A. Then Gk = B1 U B2 is a decomposition of Gk with
a-B1 c a-Gk U (a-A n Gk) = a-Gk
,
since a-A c F, F n Gk = 0. Similarly we have a-B2 C a- Gk. The h?{n-1-a.e.. This proves
connectivity of Gk then implies IB21 = 0, i.e., A D Gk the last part of the theorem and the uniqueness.
Theorem 2 can now be easily applied to degree theory. Let T E Rn(Rk+n) be a current with M(T) + M(OT) < oo. Write
aT = T(.M, O, T) E R,,-,(R k+n) set
and let
F=
U Gk k
,
4 Notes
479
be the decomposition of IR \ F into h-connected components. The current #T belongs to Rn (1R') and can be written as fr#T = ,C'' L u for some u E BV (P" , Z),
and J., C F H'-1 a.e., i.e., &5i#T = -r (F, , ) . Consequently IDuI(Gk) = 0 for each k, and u is constant on Gk. In other words
deg (T, , ) is constant on each Gk.
Remark 1. Notice that the rectifiable h-closed set F can be dense say in B(0,1) so that spt F#aT D B(0, 1). Nevertheless I FI = 0, i.e., I B(0, 1) \ UGkI = 0. Therefore by the above we still have a degree information on B(0,1), providing T and aT are both i.m. rectifiable.
4 Notes 1 Functions of bounded variation in one variable. The theory of functions of bounded variation in one variable is classical and plays an important role in many fields as for instance in the theory of Fourier series, in the theory of rectifiability of curves, and mainly in the theory of differentiation. It developed with important contributions of Lebesgue, Vitali, Lusin, Denjoy, among others, and in view of a theorem, essentially, due to Jordan, which states that a function has bounded variation if and only if it is the difference of the two non-decreasing functions, it can be reduced to the study of monotone functions. For instance the classical Lebesgue's theorem about the almost everywhere differentiability of monotone functions implies then the almost everywhere differentiability of functions of bounded variation. Notice that this is typical of dimension one. In fact in dimension larger than one functions of bounded variations, indeed Sobolev functions, though approximately differentiable almost everywhere, may be not differentiable in the classical sense in any point. For the one-dimensional theory of BV-functions we refer the interested reader for example to Riesz and Sz.-Nagy [556] Saks [570], see also Rudin [566] Federer [226]. Here
we recall some of its special features, with respect to the n-dimensional theory, with also the aim of showing that the classical definition agrees with the one of Sec. 4.1.1, when specialized to n = 1. To each function f : (a, b) --+ R, not necessarily continuous, classically one associates its total variation function defined for x E (a, b),by N Tf(x)
sup E 1f(xj) - f(xj-1I j-1
where the supremum is taken over all N and over all choices of {xj } such that a < xo < ... < xN-1 < x. Clearly, Tf (x) is a non-decreasing function, whence Vb(f) :=
lim Tf(x)
exists. If it is finite one says that f is of bounded variation and one calls Vb(f) the total variation of f.
4. Cartesian Currents in Euclidean Spaces
480
The previous definition has the disadvantage that, if we modify f just at one point, then both T f (x) and V ,,b (f) change. Hence it is convenient to find "normalized representatives" of functions of bounded variations. This can be done as follows. First one observes that for x < y we have If (Y) - f(x)I G Tf(y) - Tf(x)
In particular If (xi)} is a Cauchy sequence whenever {T f (x, )} is a Cauchy sequence. Since Tf is a monotone function, and monotone functions have right- and left-limits at all points, and have at most countably many discontinuities, we conclude that the same is true for f. Therefore we can define
g(x) := f(x) - c
c := lim f (t), t
a+
Clearly, g(x) is a left-continuous function, V(g) < V(f) and we can conclude: For every function f of bounded variation, f (x-) exists at every point x of (a, b], f (x+) exists at every point x of (a, b), the set of points at which f is discontinuous, jump points, is at most countable and there is a unique c and a unique function g of bounded variation, which is left-continuous, satisfies
lim g = 0
x--.a+
so that
f(x) = c+g(x)
at all points of continuity of f . Using the previous claim we normalize every f of bounded variation to c + g(x), and we denote by NBV(a, b) the class of normalized functions of bounded variation. It is easily seen that Tf (x) E NBV (a, b)
f E NBV (a, b)
whenever
Theorem 1. We have (i)
If p is a Radon measure on (a, b) and if we set f (x)
(1)
J
dµ
x E (a, b)
(a,x)
(ii)
then f E NBV(a, b) Conversely, to every f E NBV(a, b) there corresponds a unique Radon measure p such that (1) holds; moreover for this p we have
Tf(x) = f d1jul (a,x)
(iii) Finally, if (1) holds, then f is continuous precisely at those points x at which
p{x} = 0 Proof. (i) If xk T x, clearly limk.oo p((a, xk)) = p(a, x), so f (xk) -+ f (X). If Xk I a, then p((a, Xk)) -+ 0, so f (xk) --+ 0. While for a < xo < ... < xN = x N
N
E I f (xi) - f (xi- 1) 1 = 1: Ip([xi-1,xi))1 i=1
i=1
<-
IpI(a,x)
4 Notes
481
whence (2)
Tf(x)
<_
IpI(a,x)
(ii) We have f = u - v where
u :=
2 (Tf +f)
2 (Tf - f)
v
are two nondecreasing functions which belongs to NBV(a, b). We then define two measures AU and µ as follows. To each x E (a, b) we associate the set E,, (in dependence of u, respectively of v) defined by
{+
if u is continuous at x
{u(x)}
[u(x),u(x
)]
if u(x
) >u(x)
and to a set E C (a, b) we associate the set U E. Then we define xEE
ju(E) = G1( U Ey) sEE
Set now
A :_ blu - AV and denote by A the measure pTf associated in the same way to Tf. It is not difficult to show that µ is a Radon measure,
µ([a, 0)) = f (0) - f(a) and
Tf(/3) - Tf(a) Hence we get I,u(E) I
< A(E)
i.e.
Ipl(a,x) < A((a,x)) < Tf(x) which together with (2) concludes the proof of (ii), and of the theorem, since (iii) follows at once.
11
The previous theorem shows of course that functions of bounded variations in the sense of Sec. 4.1.1 just agree with functions of bounded variations in the classical sense.
Denoting by Fc the measure associated to the function f E NBV, and using Lebesgue decomposition theorem, we also get
µ(E) = µ13l (E) +
JE
dal (t) dt
Hence, if we set x E (a, b) h(x) := a ((a,x)) we conclude that f y is a.e. (approximately) differentiable with apD fs = 0.
f is a.e. (approximately) differentiable with apDf =
d 1L .
Finally, fg = 0 if and only if f is absolutely continuous.
4. Cartesian Currents in Euclidean Spaces
482
2 Functions of bounded variation in multidimensional domains were first considered by Cesari [140] in the classical context. Later, after the work of De Giorgi on Caccioppoli
sets De Giorgi [176] [177], the modern development of the theory of BV-functions started with the work of Krickeberg [415] and mainly with the work of Miranda [474] [475] [476], and Vol'pert [651]. But, contributions can also be found in the work of many other authors, some of which will be mentioned later. Starting from the seventies several systematic presentations of the theory of BVfunctions and Caccioppoli sets have appeared in books. We mention in particular Anzellotti, Giaquinta, Massari, Modica, and Pepe [49] Giusti [308] Massari and Miranda [461] Maz'ja [463] Vol'pert and Hudjaev [652] Ziemer [686].
Our presentation here focused mostly on structure type theorems. We would like to give now a short account of further relevant results. 3 Traces of BV-functions. The theory of traces of BV-functions was essentially stated in Miranda [478], and it is presented for example in Giusti [308] or in Vol'pert [651]. Extension can be found in Burago and Maz'ja [116] Meyers and Ziemer [467] Anzellotti and Giaquinta [45] Soucek [605]. Here we shall briefly report on the results of Anzellotti and Giaquinta [45] which however rely on Miranda [478]. We denote by d2 an open set in R' such that P(Q, R-) < oo ,
(3)
7l" -1(an \.9-f2) = 0
2
Using the coarea formula, as in the proof of Theorem 4 in Sec. 4.1.4 one can then show that the Borel functions
sup{t E R I O(Et (f), x) = 0} f+ (x) := inf{t E III O(Et(f),x) = 1} f_ (x)
I
F(x)
2 (f- (x) +.f+(x))
are well defined for 9-l"-1-a.e x E 912, and
f-(x) = f+(x) = F(x)
7-{r-1-a.e. in a,f2
whenever f E BV(Q).
Theorem 2. Let f E BV(l2) fl L°°(,?) where .f2 is an open set in R" satisfying (3). Then (i) For 7-1r-1-a.e. x c a.2
lim 1 F(x) = f- (x) = f+ (x) = p-_o+ Qfl B(x, p) and
pllo+ I,fl f1 B(x, p)
f
J
f (y) dy
f?nB(x,p)
If() - F(x) l ,."` 1 dy = 0
s2nB(x,p)
2 Actually the conditionf"-1(al? \ a- d2) = 0 is not necessary; the results of Theorem 2, Theorem 3 and Theorem 4 below remain true of course with a( replaced by
a-n.
4 Notes (ii)
483
Denote by f the extension off to zero outside Si. Then for any Borel set B C R" we have
f
JB
r
Df =
J Bnf2
Df +
f
FvdHn-1 J Bn8f2
or equivalently
f 9DJ =
f9Dif+fgFvd'
i=1...,n
where v is the inward normal to 3Q. Finally,
f IDfI = f IDfI+J
dN"-1
JFJ
and the Gauss-Green formula
f Df = - f Fvdfl7
1
holds.
In order to extend the previous theorem to unbounded functions one needs a trace estimate, for which some regularity of 8,(7 is essentialFor x E af? we set q ( x, , (2)
:
=
dHn-1
f XA
lm so up l an f DXA ll
I
A c (2 n B, ( x ) AI > 0 , P(A , (2) < oo } ,
,
,
J
5
JJJ
and
Q(Q)
sup q(x)
zESQ
We observe that for open sets with C' boundary one has Q = 1, while for Lipschitz domains one essentially has Q = 1 _+L2 where L is the Lipschitz constant of 85?. Finally, Q < oo for open sets with outward pointing cusps, while the assumption (3) is not sufficient for Q < oo.
Theorem 3. Let 5? satisfy (3) and Q(f2) < oc. Then for any e > 0 and any f E BV(S2) we have (4)
f IFIdHn-1 <_ (Q(f2)+E) f IDfI +c(Q,E) f IfI dx n
an
where c(12, e) does not depend on f . Moreover, dHn-1
IFI
(5)
fn IDfI +c(Q) f
IfI dx
n
n
holds for a domain 0 satisfying (3) and for all f E BV(f2) if and only if Q(,(2) < oo, and in this case
Q(f2) = inf{L
1
(5) holds}
4. Cartesian Currents in Euclidean Spaces
484
Rightfully one calls the function F(x) on 8f2,
F(x) = f+(x) = .f-(x) the trace off on 852, denoted also yf, and one proves
Theorem 4. Let 52 satisfy (3) and Q(f2) < oo and let f E BV(52). Then (i) (ii)
(Local characterization of the trace). All conclusions of Theorem 2 hold. (Continuity of the trace operator). Let { fk } be a sequence in BV (52) such that
fk -+ f in L1(9) and
fDfkI
fIDfI
Q
.a
then yfk -* -yf in L'(852).
We observe that the trace operator y is not continuous with respect to the weak convergence in By. For instance the functions
fk(x) =
k(x + 1) 1
-1 < x < -1 + 1/k -1 + 1/k < x < 1 - 1/k
k(x-1) 1-1/k<x<1
have trace zero on 8(-1, 1) and converge weekly in BV to the function identically 1 which has trace on 8(-1, 1) the constant function 1. Finally we mention that one can show that the trace operator is onto L' (Q), as consequence of the classical Gagliardo's theorem on the trace of H"1-functions Gagliardo [256].
4 Slicing BV functions. Among the various characterizations of functions in BV 3 one which is of some relevance, also from the historical point of view 4 is the following:
A function u belongs to BV(F) if and only if for almost all lines parallel to the axes its restriction is a function of bounded variation (in one varcable) compare Serrin [585] and Hughs [389]. More precisely, we can state this fact by the following theorem which follows from the approximation Theorem 1 in Sec. 4.1.1. Denote by Hi the coordinate (n - 1)-plane orthogonal to ei in 1Rn, i = 1,... , n. For
u E L1(Rn) and y E Hi, i = 1,... , n, denote by ui,, (t) the function
ui,,(t) := u(y+tei)
.
By Fubini theorem ui,y is well defined for 7-1n-1-a.e. y E 17i.
Theorem 5. The function u belongs to BV (R"`) if and only if for all i = 1,... , n and y E 17i the functions ui,y(t) belong to BV(R). Moreover we have
Diul(R') = f ( f Dui'VI) d7-Cn-1(y) IIi
R
It is well-known that the previous theorem can be improved under certain circumstances. For instance Calkin [133] showed that the functions ui,v are absolutely continuous for any i = 1,.. . , n and 'Hr-1-a.e. y if and only if u E W1,1(Wn) and VoI'pert [651] proved that the ui,y have only jump parts if u = EAiXEi, Ei being sets of finite perimeter. In this direction the following general theorem is due to Ambrosio [32]
3 Results in this number could be state for functions in BV(Q), 0 C R' open. Only for the sake of simplicity we preferred to state them for 0 = R. 4 This was in fact the original definition of Cesari.
4 Notes
485
Theorem 6. Decompose the measure Du as Du = apDu dx + (Du) (j) + (Du) (C) For any Borel set B define Bi,y
{tER I y+teiEB}
17i,Bi = {y E 71
I
Bi,y 0 0}
Then we have
JB
apDu dx =
d7-C`-1(y)
J
Ii,Bi
(Du)(j)(B)
J
apDui,, (t) dt
Bi,.d
f (Dui,y)(')(Bi,,)dx"-1(y)
(Du)(C)(B)
f (Dui,,)(C)(Bi,1/)dxn-1(y) ,Bi
Moreover for almost every y E I7i
apDui,y (t) = apDu(y + tei)
for a.e. t E IIt
the jump set of ui,y is given by
Jui y= {t E I
I
y+tei E Ju}
and
u(t) = u (Vi + tei)
uy y (t) = u+(yi + tei)
a
5 Vector valued BV functions. We make only a few remarks. Denote by BV (,R, II.") the space of maps from I? C r into ISSN N > 2, whose components are functions in BV (D). For u = (u1, ... , u N ) E BV (R", IISN) we define the jump set of u as N
J. = U Jui . i=1
From the theory of Sec. 4.1.4 one can easily deduce that Ju is a countably (n - 1)rectifiable set and the matrix-valued measure Du split as
Du = (Du) (a) + (Du)(j) + (Du)(C) (Du) (j) = Du L Ju and
IDuI = I(Du)°I +I(Du)(j)I+I(Du)(C)I Also one sees that
(Du)' = G" L apDu and
IDuI (B \ Ju) = 0 for any Borel set with l"-1 (B) Goo. Finally, one can show that
486
4. Cartesian Currents in Euclidean Spaces
(u+(x) - u' (x)) n(x, Ju)
d7-C"-1
or in other words that the matrix (Du) (3) is of rank one
(Du) (j) = (u+ (x) - u-(x)) ®n(x, Ju)
d7-Ci-1
.
An interesting result of Alberti [7] proves that also (Du)(') is of rank one, that is d (Du) (C) d I (Du) (C) I
(x)
is a rank-one matrix j(Du)(C)I-a.e. In other words, the Cantor part (Du)(C) has a similar geometric structure of the jump part (Du)(i).
6 Compositions and products. Let f : RN -+ JR be a continuously differentiable function and let u E BV(Q,RN), (2 C IR". In Vol'pert [651] Vol'pert and Hudjaev [652] it is proved that the function v := f(u)
is in BV(f2) and
on Q\ Ju
(6)
Dv = D f (u) Du
(7)
Dv = [f (u+(x) - f (u-(x))n(x, Ju) d9-Cr-1
on Ju .
More recently in Ambrosio and Dal Maso [37] a general chain rule in the case f : RN -* Rk is a Lipschitz-continuous function is proved. The main difficulty in doing that consists in extending equality in (6), in fact it may happen that the function f is nowhere differentiable in the range of u. To state the result of Ambrosio and Dal Maso [37], we observe that IDul-a.e. point x in S7 \ J. is a point of approximate continuity for u and of existence of the RadonNikodym derivative of Du with respect to JDul. For such points x we denote by TT the tangent space Ty :_ {y E ]I8N
I
y = u(x) + < d lDul (x), z> for some z E JN}
Then we have Theorem 7. Let u E BV (.(2, IRN) and let f : RN -,. II8k be a Lipschitz continuous function. Then v := f (u) belongs to BV (.fl,1Rk) and (Dv)(i) = [f (u+ (x) - f (u- (x)] ® n(x, Ju) df"-1
Moreover, JDul-a.e. on .(2 \ Ju f T. is differentiable at u(x) and
Dv = Df, s(u(x))Du
on f?\Ju .
In particular for N = 1 we find, similarly to the case of continuously differentiable f , that f is differentiable at u(x) for JDul-a.e. x in f2\Ju and (6) holds. Notice also that Theorem 7 extends the classical chain rule for Sobolev functions W1'1(.(2), essentially due to La Vallee Poussin, compare Serrin and Varberg [587], to a general chain rule for functions in W1'1 (( RN) compare Marcus and Mizel [458]. Let us return to the case of continuously differentiable functions f and let us look at the composition f (u), u E BV(17,IRN). We can write formulas (6) and (7) in a more
4 Notes
487
suggestive form introducing for each point x (apart for a set of zero (n-1)-dimensional measure) the averaged composition if
o u(x) = J f (tu+ (x) + (1- t)u- (x)) dt . T.0
Then it is immediately seen that (6) and (7) can be equivalently written as N
Dif (u) =
(8)
fuk o u(x) Diuk k=1
where the equality has to be understood in the sense of measures. In fact one can show that f (u) E BV (,fl, II8k ) and (8) holds provided the functions f k o u(x) be locally summable with respect to the measures Diuk. An immediate consequence of this, compare Vol'pert [651] Vol'pert and Hudjaev [652] is: Let u, v E BV(Q). Suppose that
u.:= id o u(x) is locally summable with respect to Dv and v := id o v(x) is locally summable with respect to Du then uv E BV (,fl) and
D(uv) = u Dv + v Du . As a special case let us consider the case in which u E BV (,fl) n L°° (17) and v = XE
where E is a set of finite perimeter: we have uXE E BV and
D(uXE) = u+(x) 2 u-(x) n(x, E) d7-tn-1 L 8-E + (XE1 +
2
XE,,,.) Du
where
E,={xEE I 6(E,x)=1},
E1/2={xEE I O(E,x)=2}.
7 The space SBV(Q). Certainly the most complicated part of the gradient of a BVfunction is its Cantor part. In many situations as for instance in problems with free discontinuities or in mathematical problems connected with the study of the image segmentation, see e.g. De Giorgi and Ambrosio [184] Ambrosio [33], one would like to deal with BV-functions which are free of a Cantor part in the gradient.
The space SBV(,fl) is defined as the space of functions u E BV(f2) such that (Duff) = 0, or in other words as the subclass of BV(Q) of functions u E BV(Q) such that Du = apDu(x) dGC(x) + (u+(x) - u-(x)) n(x, Ju) L JJ . What makes useful this class is the following compactness theorem, conjectured in De Giorgi and Ambrosio [184] and proved in Ambrosio [32], compare also Ambrosio [36] and Alberti and Mantegazza [9] for different proofs. d7-1n-1
Theorem 8. Let {uk} c SBV(.fl) n L°°(Q) be a sequence such that
< 00
sup if
for some p > 1. Then there exists a subsequence {uki } converging in L1(,fl) to u E SBV(Q) and
f .a
lim inf
i-00
Jn
japDuk; lP dx
7tn-1(Ju) < liminf7-Cn-1(Juki) i-+oo
488
4. Cartesian Currents in Euclidean Spaces
As we know, Cantor or jump parts cannot be produced by weak convergence in W1'1. Theorem 8 says in particular that Cantor parts cannot be produced by jumps of controlled length. For the reader's convenience we give a proof of Theorem 8 emphasizing its geometric aspects. First let us prove the following characterization of subgraphs of SBV functions with jump part of finite size.
Proposition 1. Let u E BV(fl, II8). Then u E SBV(fl, II8) with 7-l"-1(Ju) < oo if and only if M((&Gu)(o)) < +oo. Moreover M((aGU)(o)) = 2R`(Ju). Proof. From Theorem 3 in Sec. 4.2.4 we now that the current associated to the "graph" of a BV function, T. :_ (-1)naSGu, decomposes as Tu = Gu +T(C) +T(i)
(9)
where G, is the graph current of u, and T(C) and T(i) are respectively the Cantor and
the jump part of T. In fact T(C) (w) = T(j) (w) = 0, if w = f (x, y) dx
(10) and
T(C) (w)
(11)
= ff(X,U+(X))dDUc) n u+(x)
T(j)(w)
(12)
\
f ( f .f (x, y) dy) ni (x, Ju) dHr-1 l_ Ju
if w = f (x, y)dxindy.
Let u E SBV(fl,IIR). Clearly (10) and (11) yield T(C) = 0. Consequently using (9) and (12) we infer M((,9Gu)(o)) = M((aT('))(o)) = 27nrz-1(JJ).
Conversely, (9) yields that M((aT(i) +aT(i))(o)) = M((8Gu)(o)). Assuming that M((aGu)(o)) < oc, the supremum of (aT(i) + aT(j))(o) on forms of the type w = O(x)V)(y)dxindy, 0 E C,' (D), i E C,' (R) is finite and we have (13)
sup S T(C)(O(x)O'(y)dxindy) +T(j)(O(x)'tl/(y)dxindy)} Iml,lwl« .=l,n l
Taking into account that I DuCI and (12), we finally infer from (13) (14)
sup
'Hn-1
M((aC,)(o))
L J. are mutually singular, (10), (11), and
f O'(u+(x))IdIDuCI = M((aT(°))(o)) <- M((OGu)(o)) < oo.
I+GI<_ln
Now, denoting by a the positive measure on 118 given by a (B) = I Ducl (u+1(B)) for all Borel sets B C R, inequality (14) amounts to sup I0I<_1
f'd
4 Notes
489
which is clearly impossible unless v = 0. Therefore I DuCI = 0, hence u E SBV(Q,IR); and, going back to (9) Tu = Gu +T(') which concludes the proof since trivially
M((8T('))(o)) = 2f"i-1(Ju). Proof of Theorem 8. Passing to subsequences, we can assume that uk -> u in L1, Duk
Du as measures and that Du' -r v in L' (S2, R), v E L'(0, R). Also the
currents Guk converge to a current S in Si x R whose components are S(f(x,y)dx)
=
S(f (x, y)dxiAdy)
ff(x,u(x))dx,
f f (x,
u(x))v''(x)
dx.
Moreover, by Proposition 1, supk M((BGuk)(o)) < oo, and this is precisely the condition
(9) in Sec. 3.3.2 with p = 1. Proposition 2 in Sec. 3.3.2 then yields that Dv = Du', consequently S = Gu and therefore M((8Gu)(o)) < lim of M((OGuk)(o)) < oo. which yields u E SBV (S2, I[8) on account of Proposition 1. This concludes the proof as the rest of the claim is trivial.
We notice that the previous argument extends verbatim to show
Theorem 9. Consider the class X of Cartesian currents T = GuT + ST in S2 x RN such that I(Du)YR(Q \ Ju) = 0, i.e. ST°1) = 0. Suppose that {Tk} be a sequence in X such that (i) supk M(Tk) < oo,
(ii) M(Duk) are equi-integrable in S2, supksup1,,I <18Tk(0 (x,y)dxcAdy") < oo ba,,3 I a I + IQI = n, Ia > 0, or more generally (9) in Sec. 3.3.2. Then passing to a subsequence we have Tic - T = Gup. + ST E X, i.e., ST(i) =0 and
GUTk - GuT,
STuk (1) - ST(1).
Of course in the vector valued case it would be more interesting to find conditions ensuring that in the limit jump parts of the minors do not produce Cantor parts of the minors, but this seems to be a quite complicated question. 8 The approximation theorem, Theorem 1 in Sec. 4.1.1, was proved in Anzellotti and Giaquinta [45], see also Krickeberg [415], and first appeared in Anzellotti, Giaquinta, Massari, Modica, and Pepe [49]. Related Lusin type theorems were proved earlier in Michael [470] Goffman [312], see also Chen and Liu [148]. The coarea formula was proved by Fleming and Rishel [241], and by Federer [224] in the context of Lipschitz maps.
The theory of Caccioppoli sets initiated by Caccioppoli [129] is essentially due to De Giorgi [176] [177] [179], compare also De Giorgi, Colombini, and Piccinini [186].
4. Cartesian Currents in Euclidean Spaces
490
Originally, sets of finite perimeter in R' were defined as sets which can be approximated by polyhedral sets. More precisely, the perimeter of a set was defined as
P(E, lr) := inf{lhm f H'-1(8Eh)
Eh polyhedral set with
(Eh\E)U(E\Eh)-*0}. More generally, one could define the perimeter of any set, not necessarily measurable, as
P(E,R') := inf{liminf7-C`-1(8Eh) I(Eh \ E) U (E\ Eh)l -+0} then show that E is measurable, if P(E, IRN) < oo, and, in this case, the perimeter I
coincides with the perimeter defined in Sec. 4.1.2. De Giorgi's rectifiability theorem then amounts to the following. Take as representative of a set of finite perimeter E the equivalent set O(E,x) = 1} {x E I
and set
8E
{x 16(E,x) = 2} .
Then 8E is an (n-1)-rectifiable set which 71"-1 almost everywhere agrees with 8-E. A related approach was developed by Federer [220] [221] [223]. In conjunction with
De Giorgi's results this eventually led to regard sets of finite perimeter, and more generally BV-functions, as n-dimensional currents in R" and to recover a structure theorem for BV-function as a structure theorem for n-dimensional currents in R", compare Federer [226, 4.5]. In particular for Theorem 3 in Sec. 4.1.3 see Federer [226, 4.5.11].
The theory of Caccioppoli sets and of BV-functions was then developed by many contributors including Krickeberg [415] Fleming [238] and especially Miranda [474] [475] and Vol'pert [651].
The results of Sec. 4.1.4 go back to Federer [226, 4.5], see also Vol'pert [651] and Goffman [313]. The results of Sec. 4.1.5 are taken from Miranda [474], and principally from Federer [226, 4.5.9], see also Dal Maso [171]. Actually in Federer [226, 4.5.9(5)] it 8_Sg) is also proved that 7{" (D,,, \
=0.
Our presentation in Sec. 4.1 is a remaking of previous presentations that can be found in Anzellotti, Giaquinta, Massari, Modica, and Pepe [49] Giusti [308] Vol'pert and
Hudjaev [652] Ziemer [686]. In doing that we have taken advantage also of Alberti's thesis [5], respectively for the proof of Theorem 5 in Sec. 4.1.4 and Theorem 6 in Sec. 4.1.4 of Sec. 4.1.4 and of a set of unpublished notes by Ambrosio [34].
9 The notion of Cartesian current first appeared in Giaquinta, Modica, and Soucek [280]. Basic developments of the theory as structure, closure and compactness theorems, including also examples, appeared in Giaquinta, Modica, and Soucek [280], [279] and in the papers Giaquinta, Modica, and Soucek [282], [292] [286] which followed. Sec. 4.2,
and in particular Sec. 4.2.1, Sec. 4.2.2 and Sec. 4.2.3, provides a reworking and a substantially improved presentation of those basic results. The characterization of Cartesian currents in codimension one in Sec. 4.2.4 first appeared in Giaquinta, Modica, and Soucek [292]. Radial currents were discussed in Giaquinta, Modica, and Soucek [280], the approximation theorem is new, however a special version of it was first proved in Marcellini [455]. The construction of Cantor-Vitali type map in 9 in Sec. 4.2.5 due to Muller [503] is related to the one in Ponomarev [534] where it is shown a homeomorphism in W1'P (p < n) which maps a set of measure zero to a set of non zero measure. We mention that
recently in Mali and Martio [450] it has been exhibited a map u E W1,' satisfying
4 Notes
491
det Du. = 0 a.e. (hence Det Du = 0) which also maps null sets into sets of positive measure (compare Morrey [490] Reshetnyak [554]). In our presentation we have followed Muller [503], where in particular Theorem 1 in Sec. 4.2.5 of 09 in Sec. 4.2.5 is proved. The example in 10 in Sec. 4.2.5 was suggested by D. Mucci. Independently it also was pointed out in Coventry [162] where the following result is also proved: If G, + S1 and
Gu+S2 are two Cartesian currents in f2 x RN and G,.+Si E Cart (Q X RN), then also Gu + S2 belongs to Cart(f x RN). Actually in Mucci [495] a map u E cartl(f7, R2), Si C R2, is shown such that for any sequence of smooth maps Uk : (1 -. R2 with Guk - G. one has M(G,Lk) = +oo. The sequel of this monograph will show, we hope, the relevance of the notion of Cartesian currents; we believe it should be useful also in other contexts. In this respect we would like to mention the interesting paper Anzellotti, Serapioni, and Tamanini [51] where it is shown in which way Cartesian currents and the set of ideas which brought to such a notion are connected with an approach to generalized curvature.
10 The literature on the classical notion of degree is quite broad. We just mention Alexandroff and Hopf [10], Radd and Reichelderfer [546], Nagumo [509], De Rham [189], Heinz [366], Cronin [163], Fucik, Necas, Soucek, and Soucek [255], Nirenberg [513], and
the recent papers of Brezis and Nirenberg [112] [113]. A more detailed bibliography of the topic can be found in the above listed papers and books.
The approach in Sec. 4.3.2 is simply based on the area formula, and the definition of degree mapping is modelled after De Rham [189] Federer [226, 4.1.14, 4.1.15, 4.1.16]. Actually one could also define the degree map with respect to any Lipschitz map f : lick x Rte' -. R ' deg (T, f, y)
Similarly to deg (T, r, f ), compare also Almgren, Browder, and Lieb [24]. The properties of the degree in Sec. 4.3.2 were proved in Giaquinta, Modica, and Soucek [279]. Relationships between traces and degree are consequences of Sec. 3.2.5. They are partly new and weaken the assumptions of similar results in Sverak [618] Muller, Tang, and Yan [504].
The results of Sec. 4.3.1, Theorem 1 in Sec. 4.3.1, Theorem 2 in Sec. 4.3.1, Theorem 3 in Sec. 4.3.1 and Theorem 4 in Sec. 4.3.1 are taken from Federer [226], see also Simon [592]; Theorem 5 in Sec. 4.3.1 and Proposition 1 in Sec. 4.3.1 seem to be new, compare Giaquinta, Modica, and Soucek [292]. The results of Sec. 4.3.4, in particular Definition 1 in Sec. 4.3.4 (ii) and Theorem 1 in Sec. 4.3.4, are in some sense a special case of a general theorem of Federer [226, 4.2.25] concerning indecomposable currents. In Federer's terminology our h-connected sets A regarded as currents Q A are called indecomposable. An i.m. rectifiable current T E 7Zk(R") with M(OT) < oc is called indecomposable if and only if there exists no non zero R E 7Zk(F") with M(8R) < oo such that
T-R#0 and N(T)=N(R)+N(T-R). The following theorem holds
Theorem 10 (Decomposition theorem). Let T E 74(W''), M(OT) < no. Then there exists a sequence of indecomposable currents Ti E 74(R"), M(Ti) < oo, such that 0
T=ET, i=1
N(Ti)=EN(Ti). i=1
5. Cartesian Currents in Riemannian Manifolds
In this chapter we deal with non smooth maps and currents between Riemannian manifolds. In Sec. 5.5.1 we discuss Sobolev mappings between manifolds and we focus in particular on topological properties of these maps and on the closely related question of the density of smooth maps. We shall see that a generic Sobolev map u c W 1,P (X, Y) between Riemannian manifolds induces a homotopy map
if p > n while for p < n they have only a well defined [p - 1] homotopy type. In Sec. 5.5.3 and Sec. 5.5.4 we shall see that Cartesian currents and Cartesian maps, which are introduced in Sec. 5.5.2, naturally carry induced homology and cohomology maps. They agree with the classical ones in the case of Cartesian
currents associated to smooth maps and depend continuously upon the weak convergence of currents.
In doing that a preliminary question is the description of the homology of manifolds in terms of currents. This is a quite classical topic which started in De Rham [189] Whitney [674] and it is one of the major acheivement of FedererFleming's [230] and Federer's [228] [226] theory of currents. We begin in Sec. 5.1 by proving a number of additional results on currents, in particular the deformation theorem we have already stated in Ch. 2 and used previously, we discuss the mollifying procedure for currents and introduce the notion of flat chains. In Sec. 5.2 we prove the classical Hodge-Kodaira-Morrey decomposition the-
orems for forms in a compact submanifold X of ]EPn separately in the cases 8X = 0 and 8X 54 0. As a consequence we infer the non degeneracy of Poincare and Poincare-Lefschetz duality in real cohomology. Sec. 5.3 is devoted to discuss the real homology of compact oriented manifolds in terms of currents. As for the cohomology we discuss separately absolute and relative homology focusing on links among Poincare-Lefschetz duality in cohomology, Poincare-Lefschetz duality isomorphism between homology and cohomology, de Rham theorems, and the intersection index. Similar questions in the case of integral homology are illustrated in Sec. 5.4. In particular there we prove weak closure of integral homology classes defined in terms of integer multiplicity currents. A particular interesting feature of this approach is that one can easily represent cosets by minimizers of variational integrals, in particular in any homology class one finds a cycle of least mass. Moreover, if X and Y are smooth compact
494
5. Cartesian Currents in Riemannian Manifolds
oriented manifolds possibly with boundary and B is a closed Lipschitz retract in Y, we recognize a Cartesian current T E cart (X x Y) with spt 8T C 6X x B as a relative homology cycle,
n:= dim X,
[T]ree E H7 (X x Y, BX x B, Z),
and starting from [T]ree one can easily produce the induced homology and cohomology maps between the two long sequences in homology for (X, 8X) and (Y, B) using the Poincare-Lefschetz duality.
1 More About Currents In this section we collect a few facts about currents which are relevant or useful for the following. In Sec. 5.1.1 we prove the deformation theorem. In Sec. 5.1.2 we deal with the mollifying procedure for currents and in Sec. 5.1.3 we deal with flat chains.
1.1 The Deformation Theorem In this subsection we prove Federer-Fleming deformation theorem which we already stated in Sec. 2.2.6. We shall see its relevance in studying the homology of manifolds in the sequel of this section. For the sake of completeness we shall also prove the strong approximation theorem, Theorem 4 in Sec. 2.2.6. According to the notation of Sec. 2.2.6 we deal here with n-dimensional currents in Rn+N. First we fix some notation. We decompose Rn+N into cubes with edges parallel to the axes and of unitary length. More precisely we denote by Ln+N the family of cubes ,Cn+N := {Z E [0, 1]n+N
so that
I
Z E TI.n+N}
Rn+N=U{F I FELn+N},
and for any j, 0 < j < n + N, we denote by L; the collection of all j-faces of the cubes in Ln+N
L; := IF I F is a j-face of Q E Ln+N}. We instead denote by L2 the j-skeleton of the subdivision
L3:= U{F I FE} and, as we also need the translate of L;, we also set
L, (a) :=a+ Lj,
a E 118'+N .
Finally, for any a E I(j, n+N) we denote by ]R the coordinate j-plane generated by e,, A ... Ae,,, and let ir,, : Rn+^' --+ R be the orthogonal projection onto R,.
1.1 The Deformation Theorem
495
Lemma 1. We have (i) For 0 < j < n + N Le (a)
=
U (z + ira 1(a) )
U
aEI(j,n+N) zEZ-+N
(ii) Ln(a) n LN-1 = 0 if a c
(0,1)n+N.
Proof. The claim (i) is trivial since each j-face F E Lj (a) is parallel to an aplane Ian = j, hence F C ira 17r&(a). To prove (ii) we assume by contradiction
that x E Ln(a) n LN_1. As X E LN_1(a), at least n + N - (N - 1) = n + 1 coordinates of x are integers, hence at least n + 1 coordinates of x - a are not integer. As x E Ln(a), the same argument yields that at least n + N - n = N coordinates of x - a are integers: a contradiction. As already stated, the geometric idea in the proof the deformation theorem is that of projecting or retracting n-currents into the n-skeleton Ln of the standard subdivision, and doing that by a suitably chosen center of projection. Next lemma contains the construction of a suitable class of retraction maps. Let q := (1/2, ... , 1/2) denote the center of the cube [0,1]n+N. By reasoning similarly to Lemma 1 (ii) we see that dist (Ln_1(a), L,v) >
1
if a E B(q, 1/4).
4
For any p, 0 < p < 1/4 we denote by Lj (a, p) := {x E 118,+N
I
dist (x, Lj (a)) < p}
the tubular neighbourhood of Lj of radius p.
Lemma 2. For any a E B(q, 1/4) there is a locally Lipschitz map 0 : I[B,+N \ LN_1(a) : Rn+N \ LN-1(a) such that (i) 0([o, 1]n+N \ LN-1(a)) = [0, 1]n+N n Ln (ii) 0([0,1],+N n Ln) = id[o,11-+NnL,a (iii) 4/1(z + x) = 7/J(x) + z V x E Rn+N d z E Zn+N (iv) I DO(x)1 < c/p for a.e. x E 118,+N \ LN_1(a, p), 0 < p < 1/4.
Proof. We split the proof into three steps. Step 1. Let F be a j-face in Lj, j > n + 1. Denote by aF the orthogonal projection of a into F and let OF : F \ {aF} -+ 8F be the retraction map which takes x E F \ {aF} to the point y E 8F such that for some A E (0, 1] (1)
Of course
x = (1 - A)aF + Ay. FjaF = idIaF. Also for the segment daF we have
496
5. Cartesian Currents in Riemannian Manifolds
Fig. 5.1. The map b.
aaF C
(2)
n
1r
17ra(a) C
aG7(j,n+N)
n
7r,
acl(n+1,n+N)
7r
(a),
since aF is the orthogonal projection of a into F. Therefore Lemma 1 (i) yields
that aaF lies in LN_1(a). This shows that bF maps F \ LN-1(a) into 8F\ LN-1(a) (by (1)),
OF : F \ LN-1(a) ---, F \ LN-1(a) is onto, OF is the identity on 8F and is locally Lipschitz. Step 2. Gluing the V'F for each j-face in [0, 11n+N, we define a map (j)
:
[0, 1]n+N n Lj \
LN-1(a) --' [0, 11n+N n Lj-1 \ LN-1(a),
(?)(x) = 4bF(x) if x E F, which is onto and locally Lipschitz. The composition map 00 0(n+1) 0 /,(n+2) o ... p pn+N) 1]nY+N
makes sense, maps [0, 1]n+N \ LN_1(aww) onto [0, n Ln \ LN_1(a) which is equal to [0, 1] 1+N n Ln by Lemma 1 (ii), and is the identity on [0, 1]n+N n L. Furthermore, if x, x + z E [0,1]n+N, z E Z, then (3)
')o(x+z) =
In fact if x, x + z E [0,1]n+N, either x and x + z belong to Ln where identity, or x and x + z belong to the interior of two j-faces F1, F2, Oo is the j >_ n + 1 which are parallel and F2 = z 4 Fl. Hence a1.., = z + and (3) follows from (1). By (2) we can extend by periodicity the map 00 to a map
V; : R" ' \ LN_1(a)
Rn+N
\ LN- 1(a)
1.1 The Deformation Theorem
497
which satisfies the claims (i), (ii), (iii). Step 3. Finally, let us prove (iv). We shall prove slightly more: for all x E [0, 1]n \ LN_1(a) we have c
lD' (x)l < dist (x, LN_1(a))
(4)
where c = c(n, N) and DV) (x) I := urn sup
x-x
I(x') - fi(x) lx'_xI
By induction on N, if N = 1, (4) is trivial as y = '(x) is given by
y=a+A(x-a) whereA= ly-al lx - al /,(n+l)
Assume (4) holds in case N - 1 replaces N and let so that 0 _ If x E [0, 1]n\LN_1(a) and y N = 1, we immediately infer (5)
D(x)
Y
... o l,(n+N-1)
0,+N (X), as
in'
the case
al
n+NlyIx - al
If y E Ln(a) then 7;(y) = y and (4) is trivial. Assume therefore that this is not the case and let F be the (n + N - 1)-face of Ln+N_1 such that y E F. We can assume that F C x {0}. Let 7rF be the orthogonal projection of Rn+N into 1[Pn+N-1 x {0} and let aF E int F be the projection of a on F. It is easily seen that (jn+N-1 x {0}) LN-2(aF) = LN-1(a) n ]EBn+N-1
is the singular set of OJRn+N-1 x{o}, therefore by the inductive assumption (6)
c
D'b(y)
dist (y, LN-2(aF))
By similarity
ly - al Ix - al
_
dist (y, LN_-2(aF))
dist (x,7r_1LN-2(aF)) '
while
7rFlLN-2(aF) C LN-1(a), hence
dist (x,7rF'LN-2(aF)) ? dist (x, LN-1(a)). We then infer dist (y, LN-2(aF))
lx - al
dist (x, LN-1 (a))
498
5. Cartesian Currents in Riemannian Manifolds
and
ly-al p(y)ca
(7)
z - aJ
dist (x, LN -I (a))'
on account of (6). Finally (7) and (5) yield the conclusion.
Theorem 1 (Deformation theorem, unscaled version). Let T be a
nor-
mal current, T E Nn(I$n+N). Then (i)
We can decompose T as
T=P+aR+S
where P is a polyhedral chain made of faces of the n-skeleton Ln of the classical subdivision of Rn+N
P=Ni E
Moreover
sptPCLn, spt9PCLn_1 spt R c u{Q spt S C U{Q
I
I
Q E Ln+N, Q n spt T 54 0} Q E Gn+N, Q n spt aT 0}
and
M(P) < cM(T), M(S) < cM(aT)
M(3P) < cM(aT) M(R) < cM(T),
where c = c(n, N). Moreover
(ii) If T is i.m. rectifiable, we can choose P, R i.m. rectifiable and Q E Z, and, if also aT is i.m. rectifiable, we can choose S i.m. rectifiable, too. (iii) If T is a Lipschitz chain, then P, S, R are Lipschitz chains too. (iv) If aT is a Lipschitz chain, then S is a Lipschitz chain, (v) If aT is supported in Ln_1 then S = 0.
Proof. We would like to project the current T into the n-skeleton Ln according to Lemma 2. But in order to do that efficiently we must choose a in such a way that not much of the mass of T is concentrated close to LN_1(a). Step 1. Let T be a normal current. We claim that there is a constant cl = c(n, N) and a point a E B(q, 1/4) such that for any p, 0 < p < 1/4 M(T L LN-1(a, p)) (8)
cipn+lM(T)
M(OT L LN-1(a, p)) <_ c1p1+1M(8T)
For any c E I(n + 1, n + N) we consider the (n + 1)-face of [0, 1]n+N F", R,, n [0, 1]n+N and let q, be its center. In Fa n B(qa, 1/4) we define a good set
G, by g E Ga if
1.1 The Deformation Theorem
U
(9) M (T L
\
7r 1(B(9 + z, P))
<_ QPn+1M(T)
499
'd p E (0 1/4)
zEZn+Nn]R,
/3 being a constant to be chosen later. We now claim that the bad set B« F« n [0, 1]n+N \ G. is small, actually xn+1(B«)
which of course is small if 8 is large. In fact for any b E B« there is a Pb E (0, 1/4) such that
MI TL
U
7r.1(B(9+z,Pb))) ?
/Pb+1M(T);
zEZn+Nnt,
by Besicovitch covering theorem there is a pairwise disjoint and denumerable sub-collection such that B« C U B(bk, 5Pk), k
but then, setting b = bk, Pb = Pk in (9) we get /3 >k p'+1M(T) < M(T) i.e. Pk+1 < k
consequently
xn+1 (B«) <
and
,Cn+N(7ra 1(B«)
n B(q,1/4)) <
cs (n, N) Q
Finally summing on a E I (n + 1, n + N) we get ,,,`n+N(l
nB(q,1/4)J < C1(n,N) «
consequently n«ira 1(G«) n B(q,1/4) is most of B(q, 1/4) for ,Q large.
Repeating the same argument for aT, and denoting by Ga, Ba the good and bad sets for OT we also find that for 0 sufficiently large the set n i« 1(Ga) n B(q, 1/4)
is most of B(q,1/4). Therefore we can find a in the good set a E n7ra 1(G«)n n7r- 1(Ga) n B(q,1/4), for which (9) holds. Since LN-1(a, p)
= aU U i« '(B(a. + z, p)) zEZn+NnR,
a« = ir« (a)
500
5. Cartesian Currents in Riemannian Manifolds
the claim is proved. Step 2. We now project T into the La-skeleton by means of the projection map in Lemma 2 associated to a point a E B(q, 1/4) for which (8) of step 1 hold. For
0
so that by (8) M(TP) < cpn+1M(T)
(3T)p := aT L LN-1(a, p)
M((aT)P) <- cpn+1M(3T).
Then by Lemma 2 (iv)
pn+1M(T) < cpM(T)
M(b#(Tp - T12)) < (10)
p
M('#(aT)p - #(8T)p/2)) <
1
p n-1
pn+1M(T) < cpM(aT).
Similarly, if h(t, x) :_ (1 - t)x + ta(x), by the homotopy formula we get
M(h#(Q(0,1)l X (Tp - T12))
(11)
Finally, from the slicing theory we find p* E (p/2, p) such that
M(?i#) < pn-1 pn+1M(Tp - Tp12) < cpM(T) (12) M(h# (Q (0,1)
x )) < cpM(aT).
Notice that if T is i.m. rectifiable we can choose p* so that also is i.m. rectifiable.
Now we select p, = 2-"p p* E (2-"-lp 2-"p) in such a way that (12) holds and we let T" := TP.. From (10), (11), (12) it easily follows that the sequence of currents b# (T - T"), h# (Q (0,1)1 x (T - T"))
#(9T-(3T)"), are Cauchy sequences with respect to the mass convergence, and moreover
M()+M(,b#) --- 0. Therefore there are currents P1, S1 E D7 ('+N) and R1 E Dn+1(R'1+N) such
that M(Pi - 0# (T - T")) (13)
M(R1 - h#(Q (0,1)
-4 0, x (T - T"))) -'
0,
M(S1 - h#(( (0,1) JJ x a(T -T"))) - 0, in particular
1.1 The Deformation Theorem (14)
501
M(P1) < cM(T), M(Ri) < cM(T), M(Si) < cM(aT),
and by semicontinuity
M(aP1) < cM(0T).
(15)
Furthermore, by the homotopy formula we have
ah#(1(0,1) x (T - TT)) - h#(Q (0,1)1 x a(T - TT)) and, as 8T, = (aT) - compare Sec. 2.2.5, we infer
T-P1=aR1+S1.
(16)
Observe that we also have
(17)
P1, R1 are i.m. rectifiable, if T is i.m. rectifiable S1 is i.m. rectifiable, if T is i.m. rectifiable P1, R1 are locally Lipschitz chains, if T is a Lipschitz chain S1 is a Lipschitz chain, if aT is a Lipschitz chain.
Finally, since 0 retracts Jn+N \ LN_1(a) into L,,, and
is the identity on L,,,
we have
spt P1 C L spt R1 C U{Q spt S1 C U{Q
(18)
I
I
Q EC-.+N, n spt T O 0} Q E L.+N, Q n spt aT
0}
and
S1=0
(19)
In fact from L71 n LN_1(a,
a(T -
if spt aT C Ln,.
0 we infer (3T) = 0, hence
aT - - aT
consequently
R1 = l m h# (1 (0,1) 1 x a(T - Tv)) = h# (1 (0,1) ]J x aT) = 0, V
0C
being ',b the identity in L. Step 3. Assume now that spt 8T c Ln_1 (for instance T is boundaryless). In this case (17), (19) yield T = P1 + aR1i
spt P1 C Ln
hence spt (aT - aR1) = 0. We can then apply the constancy theorem, Theorem 4 in Sec. 2.2.3 and conclude that P1 is a polyhedral chain
502
5. Cartesian Currents in Riemannian Manifolds P 1 = 1:/3, [ F i
/3 E 1
and easily conclude the proof of the theorem. In the general case we know that P1 E N,,,(IRT+N) and is supported in the n-skeleton L,. By the representation formula for normal currents, Theorem 2 in Sec. 4.3.1, we can represent P1 over any face F E L as
P1 L F(w) =
J
<w(x), F(x)>OF(x) dhn(x)
where P(x) orients F and OF E Bvo,(R ). Moreover M(P1 L F) = JOFId1Th,
M(a(P1 L F) = f I DBF I ;
0
F
F
furthermore, if p E R,
M(P1 L F -I3 F f) =
f IOF -01 dxn 0
F
M(a(P1LF-131Ff))= f ID(XF(BF-Q))I As in the proof of Poincare lemma we select OF in such a way that min(Gn(OF
L' (OF < 1/2)) > ZGn(F)
(notice that OF can be chosen integer if OF is integer-valued, i.e., if P1 is i.m. rectifiable) and get IOFI <2
f
IoFIdxnL F
f I9F--OFIdfnLF
M(P1 L F -OF QF]) S CM(a(P1 L F)) M(&(P1 L F -OFT F 1)) <- cM(a(P1 L F))
and P1 L aF = 0, since P1 is normal. Summing over the faces F E Ln we set
P= E QFQF]I. FEG
1.1 The Deformation Theorem
503
We have
M(P - P&5 cM(aP1), M(a(P - Pl)) < cM(aP1) 3 F < 2M(Pl), M(P) < 1
,
R
therefore, if S = S1 - P1 - P, we get
M(S) < M(Si) +M(Pi - P) < cM(OT) M(aP) < cM(aPi) < cM(T) and for T = P + 8R1 + S, one easily checks that all claims in the theorem hold true. The scaled version of the deformation theorem, Theorem 1 in Sec. 2.2.6 now follows easily by first changing scale x -+ a-lx, then applying Theorem 1 above, and then changing scale back by x -> ex. This way we actually get
Theorem 2 (Deformation theorem, rescaled version). Let T be a normal current, T E Nn(Rn+N). Then we can decompose T as
T=P+aR+S where P is a polyhedral chain made of faces of the n-skeleton Ln,E of the subdivision of R'+N in cubes of side E
P=>/9QFti
OiER.
Moreover
spt P C Ln,E, spt aP C Ln_i,E spt R C U{Q QELn+N,e, Q f1 spt T# 01} spt S C U{Q QELn+N,E, Q f1 spt aT # O} I
I
and
M(P) < cM(T), M(OP) < cM(aT) M(R) < ceM(T), M(S) < cEM(aT) where c = c(n, N). Moreover If T is i.m. rectifiable, we can choose P, R i.m. rectifiable and ,Q E 7L, and, if also aT is i.m. rectifiable, we can choose S i.m. rectifiable too (ii) If T is a Lipschitz chain, then P, S, R are Lipschitz chains too (iii) If 8T is a Lipschitz chain, then S is a Lipschitz chain (iv) If 8T is supported in Ln_1,E then S = 0. (i)
504
5. Cartesian Currents in Riemannian Manifolds
We conclude this subsection with the proof of the strong approximation theorem Theorem 4 in Sec. 2.2.6. For that we state without proof, and we refer e.g. to Federer [226, 3.1.23], the following
Proposition 1. Let M be an n-dimensional submanifold of class k, k > 1, of Rn+N Z E M, 0 < t < 1. Denote by P the n-plane through z which is tangent to M. Then for every sufficiently small positive number r there exists a diffeomorphism f : Rn+N _, Rn+N of class k such that the Lipschitz constants of
f and f -1 are less than t-1, f is the identity outside B(z, r), and B(z, tr) nM =
B(z,tr) n f-1(P) Proof of Theorem 4 in Sec. 2.2.6. First we also assume that aT is polyhedral. We know that T = r(M, 0, T) where M is n-rectifiable and 0 integer-valued; also M is contained in a countable union UMi of C1 submanifolds of Rn+N Furthermore, as we have seen in Ch. 2, at almost every x E M the density of
M and UMi is 1 and there is a single Mi such that M and Mi coincide at x except for a set of density 0. By a covering argument we can now find open balls Bi C Rn+N \ spt aT and C1 submanifolds Ni of Bi such that UNi coincides with M except for a set of II T II-measure small. Covering each Ni with small balls
centered at Ni of sufficiently small radii in such a way that the conclusion of Proposition 1 holds (with t very close to 1) we obtain a diffeomorphism f of Rn+N which flattens most of M into n-planes and for which f#T differs from a polyhedral chain P1 by a small quantity. However the error f#T - P1, although small in mass, may have huge boundary. Now we use the deformation theorem to decompose the error
f#T-Pi =P2+aR+S. As f leaves spt aT fixed, S is actually zero, and R has small mass if the grid size is small; also, as f#T - P1 has small mass, so does P2 and hence so does the remaining term aR. If we set P := P1 + P2 we then see that f#T - P is equal to aR hence has small mass and no boundary. In the general case in which aT is not polyhedral, first we approximate aT as previously
f1#aT = Pi + aR1 with M(R1) and M(aR1) small. Now f1#T - R1 has a polyhedral boundary and can be approximated by an integral polyhedral chain P2
f2# (fl#T - R1) = P2 + aR2 with M(aR2) small. Therefore (f2 o f1)#T = P2 + f2#Ri + aR2
and the error term f2#Ri + aR2 and its boundary have small mass as desired.
1.2 Mollifying Currents
505
1.2 Mollifying Currents Already in Sec. 4.3.1 we stated a representation formula for n-dimensional currents in IR and in doing that we used mollification of currents. Here we discuss these topics for general k-dimensional currents in Rn.
Smoothing kernels. In order to do this we need a standard smoothing kernel p E CC°(B(0,1)), B(0, 1) being the unit ball in Rn with f pdx = 1, and we set pE := e-np(x/e), 0 < E < 1. Since we would like to work on open sets U our smoothing process ought to take finer and finer averages when approaching the boundary of U. For that we find convenient (it is in fact necessary for dealing with general currents) to use a regularized distance function from the boundary of U, that is a function d(x) E C°°(U) such that 0 < d(x) < dist (x, aU),
x E U,
11 Dd 11w < 1
and to introduce the following map Hz : U -+ R defined for all z E B(0,1) HH(x):= x + d(x) z
and the associate homotopy map Hz : [0, 1] x U --+ R',
Hz (t, x)
Htz (x).
It is easily seen that Hz maps [0, 1] x U into U. Actually we have
Proposition 1. For z E B(0, 1) Hz : U --+ U is an orientation preserving diffeomorphism. In particular Hz and its inverse (Hz)-1 are proper maps.
Proof. Hz is injective. In fact, if HZ(x) = HZ(y), then x - y = z(d(x) - d(y)). Hence either x = y and the claim is true or Ix-yl < ld(x) -d(y)I, a contradiction as 11 Dd j,, < 1.
Hz is surjective. Let V E U. Set xN, := y - µz, p E IR, and F(jt) := d(xN). Being U an open set F is defined in a segment (a, b) containing 0; being d > 0
on U F(µ) ---> 0 as y - a+, b- and finally F(0) > 0, then there is µ E (0, b) such that µ = F(µ). In particular xµ = y - i z = y - zd(xT). Finally Hz is non singular since
DH, =Id+z®Dd and
detHZ=1+>Didzi>1-IzIIIDd11.>0. We explicitly note that from the definition of Hz we have Idist (HZ (x), aU) - dist (x, aU) 1 < IzIdist (x, aU).
Consequently for V C U we have
506
5. Cartesian Currents in Riemannian Manifolds
dist (H,,, (V), 0U) > (1 - Iz1)dist (V, aU)
(1)
and
dist (Hz 1(V), W) >
(2)
1
+ zl
dist (V, W).
Mollification of forms and currents. Let w be a k-form with coefficients in L ',(U), w E L ',(U, AkR,) We define the e- mollified w, of w by
wE(x) = fpE(_z)Hz#(w)(x)df(z) d(x)_n f
(3)
pE
(d(x)) (H11
U
' (w)) (x) den (y)
the second equality following obviously by changing variables in the integral. In the next proposition we state a few simple remarks.
Proposition 2. Let w be in L', (U, AkRn). Then (i) For 0 < e < 1 wE is a well defined form in L o,(U, AkRn) n £k(U). (ii) If w E L1(U, AklR ), then WE -+ w in L1 (U, AkRn). (iii) If w E C°(U, Ak1Rn), then WE -+ w uniformly on compact sets of U and II WE 11.'U C c1l w IIW,U
(iv) If w E Dc(U), then wE -> w in Dk(U) and WE has compact support in U, actually
dist (spt wE, 0U) > 1 1 dist (spt w, 49U). e
Let T E Vk(U), we define the e-mollified TE of T by duality TE(w) := T(wE)
TE(w)
(f
pE(-z)H#w(x)d,Cn(z))
=T Proposition 3. Let T E Dk(U). Then
=
f
pE(-z)Hz#(T)(w)d'Cn(z)
(i) For 0 < e < 1, TE is a well defined current in Dk(U). (ii) TE - T in Dk(U) (iii) M(TE) < cM(T) (iv) If T has compact support, then TE has compact support in U, actually
dist (spt TE, 0U) > (1 - e)dist (spt T, aU). Proof. (i) follows as Hz is proper for any z E B(0,1) and 0 < e < 1. (ii) and (iii) follow from Proposition 2 (iv) and (iii), respectively. To prove (iv), we observe
that, for jzj < e spt Hz#T C HH (spt T) C {x I dist (x, aU) > (1 - e)dist (spt T, W) j =: VE by (1). Consequently Hz#(T)(w) = 0 V w E Dk(U), spt w C U \ VE. Multiplying
by p,(-z) and integrating with respect to z, we get TE(w) = 0.
1.2 Mollifying Currents
507
As we have already remarked the map HZ : [0, 1] x U -+ U given by Hz (t, x) =
x + tzd(x) is an homotopy between Ho = id and H. = x + zd(x). Consequently the homotopy formula for forms, compare (15) in Sec. 2.2.3, yields
H#w - w = d[HZ#w](o,l) + [HZ#(dw)](o,l) for w E L'(U, fk]Rn) f1 Sk(U). Multiplying by p,,(-z) and integrating we infer wE - w = dKE(w) - KE(dw)
(4)
where
KE(w) := JPE(_z) [HZ#(w)](0,1) dC'(z)
is the smoothing homotopy operator. Also notice that for z E B(0,1) HZ [0,1] x U -> U is proper, consequently KE maps Dk(U) into Dk-1(U). Also, the homotopy formula for currents, Sec. 2.2.5, yields
HZ#T - T = aH# ([0,1] x T) - H# ([0,1] x aT) in U. Multiplying by p,,(-z) and integrating with respect to z we then infer (5)
TE-T=aHE(T)-HE(AT)
inU
where HE : Dk(U) -> Dk+l(U) is the smoothing homotopy operator given for all W E Dk+l (U) by (6)
HE(T)(w) =
Jp(_z)HG0,11 x T) (w) dL'(z)
Notice that HE(T) has compact support in U if T has compact support in U. Notice also that we have
M(HE(T)) < ceM(T),
(7)
since M(H#(Q0, 1 x T)) < cIzIM(T). From (5) and (7) we then infer ITE(w)-T(w)I (8)
< ce{M(T)IIdwII+M(aT)IIwII} ce sup {M(T), M(aT)} (II dw II + II w Il)
for w E Dk(U). An immediate consequence of the above is the following
Proposition 4. We have (i) If w is a closed form in U, then wE and w are cohomologous, wE -w = dKE. If moreover w has compact support, wE and w are cohomologous with compact support.
508
5. Cartesian Currents in Riemannian Manifolds
(ii) If T E Dk(U) is a cycle, OT = 0, then T and TE are homologous, TE - T 8HE (T). Moreover, if T has compact support then T. and T are homologo with compact support,
spt TE U spt HE(T) C {x I dist (x, 8U) > (1 - e)dist (spt T, aU)}.
Mollifying forms is by definition a weighted mean of pullback of forr of a suitable chosen family of orientation preserving diffeomorphisms of i H,(x) = x + zd(x), x E U, z E B(0,1). As it is clear from the previous consi, erations, the properties stated in Proposition 2 and the hornotopy formula again hold true if we replace the Hz's by another family of orientation preservii diffeomorphisms Hz : U -> U smoothly varying in z in the sense that (9)
sup I Hz (x) - H. (x) I + sup I VHz (x) - VH'w (x) -> 0 XEU
xEU
as z -> w. For instance, if we choose Hz = (Hz)-1 the inverse diffeomorphis, of H, (9) is clearly satisfied. So if we define for w E S"(U) n L1 (U,AkRn) (10)
wE(x) := f PE(-z)(HH 1)#w(x) dL'(z),
then the claims of Proposition 2, (4) and the first part of Proposition 4 hold tri replacing wE with wE. In particular if w is closed then wE and w are cohomologou
and are cohomologous with compact support if w has compact support. It is worthwhile noticing that WE and WE are in general different, but WE
wE in case U = R', cp(x) =const and p is symmetric. This type of mollifie forms appears when we compute the mollified of a current T = Rn L w, w L1(U,An-k]Rn). In this case we have
Proposition 5. Let w E En-k (U) . Then (IRn L w)E = ]Rn L WE.
Proof. Let 17 E Dk(U). Then (11)
(IR"''LW)E(17)
= (RnLW)(r1E) =
f
wn1
f PE(-z)Hf1(x))
fPE(_z)(fwAHi) dL'(z). Since now Hz is an orientation preserving diffeomorphism of U, we have
fwAHi7 = f(H;1)#wA17.
(12) U
U
Hence, using (11) and (12) and changing again the order of integration, the clairr follows at once. I
1.2 Mollifying Currents
509
We now show that the E-mollified T. are currents that can be simply represented as L'AeT,E where 1;T,E is C°° k-covector field. To see that first we notice
that
H#w(x) = Ak(DH1(x))(w(x + zd(x))) = Ak(id + z ® Dd(x))(w(x + zd(x))) hence, changing variables z ---> y = x + zd(x) in (3), we also have wE(x) =
(13)
f
d(x)-npE
X-Y
(-) R.,y(w(y)) dLn(y)
k where for x, y E U Rx,y denotes the linear operator on covectors Rx,y AkRn given by
Rx,y(a)
Ak
(id + d(x) ® Dd) (00)
Notice also that the support of the map cpE : x -+ pE I d(x I is strictly contained in U since (14)
\\\
spt ce C
{x c U
/
d(y) < d(x) < d(y) 1 CC U.
1+E -
- 1-E
Therefore we can find a vector field eT,E : U ---> AkRn such that and in fact defined by (15) _
<&T,(y), >
/
('')
TI d(x)-npR-,y(a))1
for y E U and a E Ak]RTh. It is also `seen at once that CT,E E C' (U, AklPn). Corresponding to CT,, we have the (n - k)-form wT,e E £n-k(U) = Coo (U, A,,_kRn) given by (16)
WT,E(x) :=
where Dk(C) :=1;J dx'A... ndxn is Hodge operator on l[8n, compare Sec. 5.2.2. We can now state Proposition 6. Let T be a current in Dk (U), TE be the E-mollified of T, 1;T,E E CO°(U,Ak1Rn), wT,E E C°D(U,A,,,_kJRJ) be the fields defined respectively in (15)
and (16). Then (17)
Te = Lnn&,e = R' L WT,,
Moreover (18)
and TE,
M(Te) = f 11 T,e II dLn = f 11 wT,e 11 dLn, T,E, and WT,E have compact support if T has compact support.
5. Cartesian Currents in Riemannian Manifolds
510
Proof. Let W E Dk(U). We have T(WE)
Te(w)
=
TT
\
f
d(x)-'APE (-)
R,(w(y))
\
fTX(d(xynPE()Rx,v(w(Y))) den (y) _:
J
< &,E(y), w (y) > dL' (y)
which proves the first equality. To prove the second equality we observe, compare AkRn we have Sec. 5.2.2, that for all E Ak11Pn and g E
elA ... nen > = ' hence, integrating, wT,eAw
<wT,Enw,elA...nen>dL =J
U
Finally (18) is trivial, and the last claim follows from the definitions and (14). 71
Combining Proposition 6 with (5) and Proposition 5 we easily infer
Proposition 7. Let T E Vk(U) be a current with finite mass and c9T = 0. Then T can be decomposed as
T=18nLf+8S
where S E Nk+l(U) and f E En-k(U) n Ll(U, An-kRn).
A representation formula for normal currents. We also have Theorem 1. Let T E Nk+1 (U) and d > 0. Then there exist two forms f : U --> An-k-1Rn with BV coefficients such that
An-kRn, g : U --+ (19)
T=RnLf+a(R'Lg)
in U.
Moreover
M(Rn L f) + M(a(lRn L g)) < (1 + 8)M(T) + 8M(UT), and f and g have compact support if T has compact support in U.
Proof. We first prove that one can find S E Nk+l(U) and f : U --> with BV coefficients such that (20) and
T=RnLf+CS
An-kRn
1.2 Mollifying Currents
511
M(]E8'L f) + M(0S) < (1 + S)M(T) + SM(aT). (21) To prove this, fix a regularized distance d, a mollified kernel p, and let 0 < e < 1/2 to be fixed later. For i = 0,1, ... define inductively ez := Ei+1, To := T and Ti+1 := HEM (aTT)
(22)
where HF; is the smoothing homotopy operator defined in (6). Then according to (22) Ti+1 = Ti - (Ti)e, - aH6 (Ti), (23) and, according to (22) and (7) M(Ti+1) <
CE''+1M(aTi) E(k-1)(i-1)M(aTi).
M(aTi+1) < M(aTT - (0T)6) < Therefore
M(aTi+1) < E(k-1)(i-1)M(aT) 00
1: M(Ti+1) < cM(aT)
E
ei(k-1)Ei+1 < 2cEM(0T).
i=0
In particular Ti -+ 0 in the mass norm. Consequently the currents 00
00
i=o
i=0
HH(Ti)
belong respectively to Dk(U) and Dk+l(U), have finite mass and 00
M(T) < (1 + Ek)
M(TT) < (1 + Ek) [M(T) + 2cEM(8T)]. i=o
Moreover, since (Ti),, = Rn L fi, fi = Sn-k(U), by Proposition 7, we have T = I[8n L f, f= E°°o fi and f E L1(U, An-kW ). Adding (23) for i = 0,1, .. . we then get
T=l Lf+aS
which also gives f E BV(U, An-kRn) as 8(R' L f) = 8T. Moreover
M(aS) = M(T - T) =MIT - TeO \\
- i=1 (Ti)e)
< EkM(T) + 2c(1 + Ek)M(aT). Choosing E so that 2ce(1 + Ek) < S and 1 + 2ek < 1 + S (20) and (21) follow. The claim now follows from (20) and (21) by a downward induction on k. It is trivial for k = n. Suppose it is true for k + 1 then
S=1I2nLg+9(R'Lh) with g E ll
Lf
+a(]ISnLg).
BV(U,An-k-lRn) and h E
BV(U,An-k-2Rn). Consequently T
512
5. Cartesian Currents in Riemannian Manifolds
1.3 Flat Chains In this section we discuss the notion of flat chains. Roughly we may think of flat chain, respectively integral flat chains, as of boundaries of currents of finite mass, respectively of i.m. rectifiable currents. Therefore in general their components are not measures but derivatives of measures. This is a substantial generalization of currents of finite mass we have considered so far. It originated in the work of Whitney and proved its effectiveness in the work of Federer. Their relevance, especially in connection with homology theory, is due to the following properties
The boundary of a flat chain is again a flat chain Flat chains can be pushed-forward by Lipschitz maps. The boundary operator commute with push-forward, (iv) A flat chain with support on a submanifold X is actually a current on X.
(i) (ii) (iii)
This makes flat chains the natural objects to develop a homology theory in Lipschitz category. Such a theory that we shall discuss in the next sections is isomorphic to the classical singular theory but it is preferable for problems involving integration. In some sense flat chains are the most general currents which carry topological
properties of smooth maps. In the Cartesian situation and in codimension 1 a typical flat chain is the "graph" of an L1-function, i.e., the trace of a BV function. Probably it could be of some interest studying boundaries of Cartesian currents.
Flat rectifiable chains. Let U C R' be an open set and let K C U be a compact set. Following Federer we denote by R,,K the class of i.m. rectifiable currents with support in K Rm,K := {T E Rm(U)
I
sptT C K}
and by Rm,cpt(U) the class of all such compactly supported currents Rm,cpt := U{Rm,K(U) I K C U, K compact}.
Definition 1. The class of integral flat chains supported in K and the class of integral flat chains in U are respectively defined by
..m,K(U) :_ {R + 8S I R E Rm,K(U), S E 7Zm+1,K(U)} and
Fm,cpt(U) := U{Fm,K(U) I K C U, K compact}
Definition 2. The integral flat norm of an integral flat chain is defined by
.JK(T) = inf{M(R) + M(S) I T = R+8S, R E Rm,K(U), S E 7Z,,,,+1,K(U)}.
1.3 Flat Chains
513
Fig. 5.2. The integral flat distance between T1 and T2: T2 - Ti = R + aS.
Clearly FK (T) = +oo if spt T ¢ K, 9K (T1 + T2) :5 2K (T1) + *FK (T2),
and .3K(T1 - T2) defines a distance, the flat distance, in F,,,,,K.
Proposition 1. We have (i) .F,,,,K(U) is a complete metric space with respect to the flat metric (ii) The set of integral currents with support in K
Tm,K :_ IT E 7Zm,K(U) I aT E Rm-1,K(U)}
is.PK-dense in .F,,,,,K(U). Consequently .F,,K(U) can be regarded as the FK-completion of the space of integral currents with compact supports in K. Proof. (i) Let {Tk} be a Cauchy sequence for.FK. Passing to a subsequence we
can assume that
00
.FK(Tki -Tk;-1) < 00 i=1
and consequently that
Tki - Tki_1 = Ri + aSi,
Ri E 7Zm,K(U), Si E 7Zm+1,K(U)
00
E(M(Ri) + M(SS)) < oc. i=1
Ri, >°_-1 Si converge in mass. Then 00
Oc
Ri E R,,K(U),
> Si E ?Zm,+1,K(U),
i=1
i=1
514
5. Cartesian Currents in Riemannian Manifolds
and
00
00
T:=Tk,+ERi+al E As
I
\ i-a
i-a 00
FK(T-Tk)<
(M(RS)+M(S,)) --> 0 j=1+i
the conclusion follows at once. (ii) Let T= R+aS, R E Rm,K(U), S E 7Zm+1,K(U), and e > 0. By the slicing theory we know that M (,9(R L B (0, r)) <00
for a.e. r > 0, and, by the boundary rectifiability theorem that a(R L B(0, r)) is i.m. rectifiable. As M(R - R L B(0, r)) -+ 0 as r -+ oo, we conclude that there is R1 E 1,,K(U) with M(R - R1) < e. Since we can do the same for S, the proof is completed.
Flat chains. We now introduce the notion of (real) flat chain. This is done in a more involved way, as we want to be able to approximate flat chains by normal currents, and the naive decomposition T = R + aS with normal currents R and S would not allow this, compare below. Set M,n,,K(U)
Mm,cpt(U)
{T E Dm(U) M(T) < oo, sptT C K} U{Mm,K(U) I K C U, K compact} I
and
N,,,,,K(U) := {T E Mm,K(U) I M(8T) < oo} Nm,cpt(U) := U{Nm,K(U) I K C U, K compact}. Recall the notation
N(T) := M(T) + M(aT). Definition 3. The flat seminorm of an 7n-form co E D'm'(U) relatively to K is defined by
FK(w) := max{sup II,o(x) II, sup II xEK
II}.
xEK
The flat norm of T E Dm(U) relatively to K is defined by FK(T) := sup{T(cp)
I
cp E Dm(U), FK(cp) < 1}.
Then the space of flat chains is defined as the FK closure of normal currents. More precisely
1.3 Flat Chains
515
Definition 4. T belongs to F,,,,,K(U) if there exists a sequence of normal currents Tj E N,,K(U) such that FK(T - Tj) --> 0. Then we set Fm,cpt(U) := U{Fm,,K(U) I K C U, K compact}.
Often we shall simplify F,,cpt(U) in Fm(U). We have
Proposition 2. Let U be an open set in Rn and K a compact set, K C U. (i) (ii) (iii) (iv)
If T E E),,(U) and FK(T) < oo, then sptT C K If sptT C K, then FK(T) < M(T). If T E Dm(U), then FK(0T) < FK(T). If T E Dm(U) and -y E C°°(U), then FK,KnsPtry(TL'y) < sup(Iy(x)I + IDy(x)I)FK(T). xEK
(v)
We have
FK(T) = min{M(R) + M(S)
I
T = R + aS, R E Mm,K(U), S E Mm+l,K(U)}.
Proof. (i) If x E spt T\K, we find B (x, r) C U\K, cp E Dm (B (x, r)) with spt cc C
B(x,r/2) and T(W) > 0. Then FK(Acp) = 0 and T(AV) -> oc a contradiction with FK(T) < oc. (ii) In fact FK(cp) < 1 for cp E Dm(U) implies II co(x) II < 1 for all x E K. From sptT C K we then infer T(V) < M(T) supK II cP II < M(T). (iii) is trivial and for (iv) it suffices to notice that FK(cp) < supK I YI+supK I dyI
if FK(cp) < 1-
(v) The inequality < is trivial as whenever T = R + aS
T(W) =
R(cp) + S(dcp) < sup II V II M(R) + sup II Dcp II M(S) K
K
< FK(cp)(M(R) +M(S)). To prove the opposite inequality we consider the vector space P := D'n(U) x Dm+l (U) with the seminorm v(cP,
II,su II K
),
the linear map Q : Dm(U) -+ P, Q(cp) := (co, dcp) and the image PO of Q
Po:={(co, ) EP I z/b=dcp} Clearly Q-1 is an isomorphism between Po and Dt(U) and the linear functional To := T o Q-1 : PO -> R satisfies on Po
516
5. Cartesian Currents in Riemannian Manifolds ITO(W,V))I =
FK(cp)FK(co) = v(co,,)FK(T),
O = dcp.
By Hahn-Banach theorem we can extend To to a functional L : P --f 112 satisfying (1) L(cc,'b) =v(co,V))FK(T) d (co,'b) E P,
L(cp,dcp) =To(co,dcp) =T(cO).
We now define the currents R and S by R(co) := L(cp, 0),
S(qP)
L(0, 0),
cp E Dm'(U) zb E Dm.+l(U).
Clearly T (cp) = L(cp, dcp) = R(cp) + S(dcp) = (R + OS) (go)
and
R(co) +S() = L(co,') <_ v(cp,O)FK(T) < FK(T) if supl1 cp K
< 1 and sup 1l V,1I < 1. K
Proposition 3. Let U and K be respectively open and compact in JRn with U D
K. (i) If T E Fm,K(U), m > 0, then aT E Fm_l,K(U). (ii) If T E Fm,K(U), y E C°°(U), then T L__ry E F,,Knsptry(U).
Proof. For any E > 0 there exists Tl E N,,,,,K(U) such that FK(T - Ti) < E, sptT C K. Proposition 2 (ii) yields then FK (8T - aTl) < E. Since for S := 3T1
we have sptS C K, M(3S) = 0 and M(S) = M(8T1) < N(T1) < oo, i.e. S E Nm_l,K(U). (i) follows. Proposition 2 (iv) yields instead
FKnsptry(TLy-T1L'Y) <_ (sup lyl+sup ld'YI)FK(T-Tl) K
K
which clearly implies (ii).
Weak and flat convergence. For normal currents supported in local Lipschitz neighbourhood retracts the flat convergence amount just to weak convergence. In fact we have
Theorem 1. Let U be an open set of IR't and K C U be a compact locally Lipschitz neighbourhood retract. Let {T,} E Nm,K(U) be such that sups N(TT) <
+oc. Then FK(T-T3)->0 ifTj - T. Proof. As IT(cp)1 < FK(T)FK(cp) we infer that Tj T provided FK(Tj -T) 0. The other implication is proved by Theorem 2 below.
Theorem 2. Let U be an open set of 1[2'1 and K C U be a compact locally Lipschitz neighbourhood retract. Let Ti, T E Nm,K (U) be normal currents. If Ti -s T with equibounded masses, supi N(Ti) = A < +oo. Then one can find sequences {Ri} C Nm+i,K(U), Si E Nm,K(U) such that M(Ri) + M(S2)
0,
Ti - T = Si + Mi.
1.3 Flat Chains
517
Proof. It suffices to consider the case T = 0. Moreover it is also obvious that it suffices to prove the claim for a subsequence of {Ti}. Divide IR' into congruent
cubes of sufficiently small size o- = UK in such a way that the cubes which touch K are contained in U and let us denote by LK the CW-complex of such cubes. For i = 1, 2,..., we project Ti into the m-skeleton L,,K of LK using the deformation theorem and write
Ti=Pi+BQi+Pti where the Pi are polyhedral chains on G,,,,,K, Qi E N,,,,+l,K(U), sptQi CC U,
Pi E Nk(U) and M(Pi) < cM(Ti), M(Qi)
one can and do assume that each Pi is build with the same faces
Pi = E/3ih[{Fh
Qjh E R.
h
Moreover V h Nihl < M(Pi) < A; passing again to subsequences, 13th ` Oh, Pi -+ P := E /h Q Fh in the mass norm, therefore for a subsequence of Ti's that we still denote by Ti we have
Ti=P+(Pi-P)+P2+8Q; and M(Pi - P + Pi') < coA, M(Qi) < cvA; consequently,
Ti - Tj = (Pi-Pj)+(Pi -Pi')+8(Qi-Qj) with M(P2 - Pj + P, - Pj) < 4coA, M(Qi - Q j) < 2ccA. We can now repeat the same construction for a = UK, UK /2, ... , UK /2n, ... and, by a diagonal procedure, we then select a subsequence of {Ti} that we name again {Ti}, and sequences of currents {Fi} C N, (U) and normal currents Gi E N,,,.+1 (U) such
that
TT-Ti+1=Fi+8Gi with
M(Fi) < ccA2-i, spt Fi C {x l dist (x, K) < QK f 22 } M(Gi) < coA2-i, spt Gi C {x I dist (x, K) < UK',, n2''}. Setting
0
0 Si = 7r#
(1: j=i
Fj),
Ri
l
=7r# (1: Gj /
j=i
where 7r : U -> K is the retraction onto K we infer
M(Si) + M(Ri) < c1UAA1: j=i
2_j
< c2oAA2-i.
518
5. Cartesian Currents in Riemannian Manifolds
Therefore we conclude that TZ = 7r#Ti = 7r#
1:(Tj+2 - Tj)
S, + aR,
j=1
which proves the claim. Remark 1. Theorem 1 may be regarded as the analogous of the classical fact that L1 convergence and distributional convergence are equivalent for equibounded sequences of W1,1 functions.
Density properties of flat chains. We state Proposition 4. Let U be open, K compact, K C U (i)
We have
Fm,K(U)
{R + 8S I R E Fm,,K(U), S E F,,,,+1,K (U) with
M(R), M(S) < oo} (ii) For any T E F,,K (U) there exists S E F,,,,+l,K (U) such that
FK(T) = M(T - (9S) + M(S). (iii) If T E F,,K(U) and M(T) < +oo then there exists a sequence of normal
currents Tj E Nm,,K(U) such that M(TJ - T) -> 0. Proof. (i) Let T E Fm,K(U). Clearly we can find a sequence {Tk} C N,,,,,K with FK (T - Tk) < 2-k so that 00
EFK(Tk-Tk-1) <00, k=1
and we can write Tk - Tk_1 = Rk + aSk with CO
1:(M(Rk) + M(Sk)) < oo.
(2)
k=2
As M(Tk) < oo, we have Sk E Nm+l,K(U), hence Sk E Fm+1,K(U). By Proposition 3 (i) aSk E Fm,K(U), consequently also Rk belong to Fm,K(U). By Proposition 2 (ii)
FK(R-Sk) =FK(ERk) :5 M(Rk) 00 1
similarly
i+1
i+1
i+1
1.3 Flat Chains
519
Being Ei Rk E F,,,,,K(U), Ei Sk E Fm+1,K(U) we conclude that R E F,,,.,K(U) and S E Fm,+l,K (U), and M(R), M(S) < oo, because of (2). On the other hand, if R E F,.,,,,K(U), S E Fm+l,K(U) then as E F,,K(U) by Proposition 3 (i), consequently R+aS E Fm,K(U). (ii) Proposition 2 (v) yields R E Mm,K(U), S E Mm+1,K(U) such that
T = R + as,
FK(T) = M(R) + M(S),
and by the above we get Rl E F,,K(U) and Si E Fm+i,K(U) such that
T = Rl + as,,
M(R1) + M(S1) < 00.
Of course R + as = Rl + as,, and M(a(S - Sl)) = M(R - Rl) < oo, hence S - S1 E N,n,+l,K(U), consequently S E F,,,,+1,K(U) as Si E Fm+l,K(U). (iii) Clearly we have Fk (T - Q) < e for some Q E N,,K (U), and by (ii) there is S E F,,,+l,K(U) such that
FK(T - Q) = M(T - Q - 0S) + M(S). From
M(Q+aS) <M(T)+M(T-Q-DS)
More directly one can prove it essentially as in the proof of Proposition 1 (ii) omitting the use of the boundary rectifiability theorem as we are simply interested in approximation with normal currents. (ii) Flat chains are distributions of order -1. Therefore we notice that claims like T L (2 are meaningless for T E Fm,K (U), and we resort to cut off with Cl maps, T L 'y.
(iii) Clearly for every neighbourhood V of spt T in U we can choose 'y E C°° (V) with -y
=-
1 on spt T, hence T = T L -y, and, if T E Fm,K (U), then T E
Fm,Knclosurev. However it can happen that that T 0 F,,sptK (iv) However if T E F,,cpt(U), spt T C K, K being a compact Lipschitz neighbourhood retract, then T E F,,K(U), see Proposition 7 below. In particular if T E Fm,,pt (R'), then T E F,,C, C = co(spt T) being the convex hull of support of T.
520
5. Cartesian Currents in Riemannian Manifolds
(v) While the Cartesian product of a flat chain times a normal current is a flat chain, the Cartesian product of two flat chains need not be flat. (vi) Not every current with finite mass is a flat chain.
Push-forward of flat chains. We have already defined the image current of a locally normal current under Lipschitz maps which are proper on the support of T, compare Proposition 3 in Sec. 2.2.3, as a byproduct of the homotopy formula. We shall now extend the push forward to flat chains. Let f : U c Rn --' V C RN be a Lipschitz map, U, V open sets in Rn and respectively and let K be a compact subset of U. Assuming T E Nm,K (U), T = R+aS, R E Mm,K(U), S E M,m,+l,K(U) such that FK(T) = M(R)+M(S) we infer that R E N,m,,K(U),
f#T = f#R+Bf#S
S E Nm+1,K(U),
and
M(f#T) < c(Lip f)mM(T),
M(f#S) < c(Lip f )m'-1M(S)
consequently
F f(K)(f#T) < cmax{(Lip
(3)
(Lip f)m+1}FK(T).
Thus the map f# : N,n,K(U) Nm,f(K)(V), can be uniquely extended to a linear continuous map still denoted by f# f# : Fm,K (U) - Fm,f(K) (V ) still satisfying the flat boundedness condition (3). Concerning homotopies for flat chains, let us consider f, g E C°° (U, V), U be
open in ]R'' V be open in ]R" and let K Cc U. Set C = {y E RN
I
y = (1 - t) f (x) + tg(x), x E K, t E [0, 1]}.
If II f - g II .,K is sufficiently small, then C is compact and contained in V, consequently for T E Nm,,K(U) we have h#(Q0,1 l x T) E N,n+1,c(V), h#(Q0,1 D x aT) E N,,,,C(V) and the formula
g#T- f#T=ah#(Q0,1 xT)+h#(10,11 x9T) together with
< constI g - f I1.,KM(T) M(h#(Q0,1 x aT)) constjj g - f 11.,KM(8T) M(h#(Q 0,1]I x T))
compare (16) in Sec. 2.2.3, consequently we get (4)
Fc(f#T - 9#T)
cMM 9 - f MMc,KN(T)
provided C CC V as for example if 11 9 - f IIo°,K is small enough.
1.3 Flat Chains
521
Proposition 5. Let T E Fm,,K(U). The push-forward of T, f#T E F,,,,, f(K)(V) is characterized by the following property:
Let fj : U -> V be Lipschitz maps converging to f uniformly on K with equibounded gradients. Then
FH(f#hT - f#T) -->
(5)
0,
h --> 00
0
for any compact set H such that f (K) C H C H CC V. Proof.
Let T E Fm,K(U). Fix e > 0 one can choose T1 E Nm,K such that
FK(T - T1) < E. Since for h large enough the compact set Ch = {(1 - t) f (x) + tg(x), t E [0, 1], x E K} 0
is strictly contained in H, we infer by (4) FH(fh#Tl - f#Ti) < E for h large enough. Consequently
FH(fh#T - f#T) < FH(ffh#(T -T1)) +FH(fh#T1 - f#T1) + +FH(f#(T -T1)) < (2c+ 1)e. The convergence then follows from the uniqueness of flat limits.
We also emphasize that for f# we again have whenever T E F,,,,,K(U)
sptf#T c f(sptT),
(6)
9(f#T)) = f#(8T).
It is convenient to define also the space of locally flat currents
Definition 5. We say that T E
equivalently T is a locally flat current
if T L y E Fm,cpt(U) for every -y E Cc°.
Then we can also define
f#T E
whenever T E F111 (U) and fIsptT is proper
in such a way that (.f#T)(w) _ [f# (T L'Y)](W) for all w E Dm (V) and y E Cc' (U) with spt T fl f (spt w) Cint {x Again one verifies that (6) holds. From Proposition 4 (iii) we get
I
y(x) = 1}.
Proposition 6. If T E Fm,cpt(U), M(T) < oo, a(x) is continuous in U, f U -> j N is locally Lipschitz, lD f (x) I < o ,(x) a.e., and f is proper on sptT, then 11 .f#T I I :.f# (11 T 11 L 01);
if also g is locally Lipschitz with IDgK (x) < or(x) a.e.,
{(1-t)f(x)+tg(x)
I
xEsptT, t E [0, 1]} C V
and h is the affine homotopy from f to g, then M(h#(Q(0,1)1 x T) < 11 T
I(Ig-.flo-",').
522
5. Cartesian Currents in Riemannian Manifolds
The flatness theorem. The following important support theorem, compare Theorem 1 in Sec. 5.3.1, holds for flat chains.
Proposition 7. Let T E Fm,K (U) and let f, g : U -+ V be Lipschitz maps, U be open in 1Rn, V be open in RN. If f = g on spt T, then f#T = g#T. Proof. First suppose that T E N,,K(U). Consider the map 0, : Rn -+ ][$n,
e > 0, given by if Oe(y) =
<6
if IyI?e.
0E is Lipschitz, D(ae 1 < 110, (y) - y1 < e. Denote by UU the open set UU :_ {x E U
I
I f (x) - 9(x) I< e}.
Clearly spt T fl u C Ue and hE (x) :_ AYE (g (x) - f (x)) vanishes on Ue. Therefore he is proper on sptT and h,#, (w) is null in UU for all w E Dk(IRn), consequently
hE#T = 0. On the other hand I f (x) - g(x) - he(x)I > e, DhEI < IDfI + ID9I, the homotopy formula then yields (f - 9)#T(w) - hE#T(w)I <_ ceN(T)(II w II + II dw II),
and letting e --+ 0 the conclusion follows at once if T E N,K (U). For the general
case, fix e > 0 and let Tl E N,,K(U) be such that FK(T - T1) < E. Being Tl normal, f#Tl = g#Tl and consequently FK (f#T - g#T) < E. The claim follows as s is arbitrary.
Theorem 3 (Federer's flatness theorem). Let C C K C U, C, K be compact and let r : K --+ C be a retraction. Then if T E F,,,,,K(U) and sptT C C then T E F,,,,c(U). Proof. In fact one can extend the retraction to a Lipschitz map r : U --+ C
which coincides with the identity map on C. Thus T = r#T E F,,,,,r(K) (U) = Fm,c(U)
A representation formula for flat chains. We then illustrate the mollification procedure for flat chains and extending the representation formula we have proved for normal currents to flat chains. With the notation of Sec. 5.1.2, if T E N,,,,,K(U), then for any z E 1Rn, IzI < e, H.(sptT) C Ke :_ {x E U I dist (x,K) < e}, hence, using (3), (4) we have FKE (H.#T) < CFK(T) FKE(H#(If 0,11 x T) < cFK(T)
FKe (Hz#T - Hw#T) < cIz - wIN(T)
FKE(H#(Q0,1 xT)-H#(Q0,1}j x T))
1.3 Flat Chains
523
for z, w E B(0, e) C R'. Also the mollified TE's have support in KE, therefore from (7) in Sec. 5.1.2, (8) in Sec. 5.1.2
FK (HE(T) < M(HE(T)) < ceM(T) -+ 0 FKE(T - TE) < cEM(T) , 0.
(7)
By density we then have for flat chains T E Fm,K(U) the following estimates TE E Fm KJU), HE(T) E Fm+1,KE(U), FKe(TE) + FKE(HE(T))
(8)
as E-40. Let U C R'' be an open set. We denote by .flm the space of compactly supported L'-fields with values rn-vectors .flm :_ { E L1(U, Am]R')
I
spt 6 Cc U}.
In tim x flm+i we introduce the norm
f(I+IIDdL' (6,17)E .Qm X .Qm+i, U
and we show that every T E Fm,cpt (U) can be represented by an element of .flm x ,flm+i. More precisely
Theorem 4. We have (i)
If
q) E ,flm x ,flm+i, 5 > 0, and KS := {x E ]R'
I
dist (x, spt U spt 77) < S} C U,
then the current
T:= L'A6 + a(L'A?7) is a flat chain, T E Fm,K6 (U) and FKb (T) < fU(II 6114-1177 11)
(ii) If T E Fm,K(U), b > 0 and
Ks:={xE]RT
I
dist(x,K)<S}CU,
then there exists (6, 77) E tIm x .flm+1 with spt 6 U spt 77 C Kd such that
T :=L'A6+a(.C A7) and ff(IIe 11 + II17II)dC- < FK(T)+8.
524
5. Cartesian Currents in Riemannian Manifolds
Proof. (i) The claim is trivial if C and 77 are smooth, as
f (IIPE*C-6II+IIPE*7j-?7II)dL' --> 0 ase-,0 U
it follows in general. (ii) The proof is close to that of Theorem 1 in Sec. 5.1.2. Set To := T, Ko := K and by induction
Kk := {x
I
dist (x, Kk-1) < 2-k-18} C U.
Given Tk-1 for each k choose Rk E Nm,Kk_1(U) and Sk E N.,,,,+1,Kk_1(U) such
that FKk-1 (Tk-1 - Rk - -ySk) < 2-k-38 M(Rk) + M(Sk) < FKk-1 (Tk-1) +
2-k-28
and Ek such that
ek 5 2-k-16
6kN(Rk) < 2-k-38.
Define Ck E C° (U, A,R) by (Rk)E.
Notice that spt Ck C Kk, and set
Tk := Tk-1 - G'A k - 3Sk E Fm,Kk (U)
< M(Rk) and
We have fU II Sk II dJ
C FKk (Tk-1 - Rk - c 7 S k ) + FKk (Rk - J A k)
FKk (Tk)
< 2-k-38 + EkN(Rk) < 2-k-28 consequently 1
[
I Sk
J
dG" + M(Sk)] < >[M(Rk) + M(Sk)} k=1
U
[F1_1 E k=1
)
+2-k-2j] < FK(T) +8/2.
This implies that 00
00
k E Qm, k=1
S := E Sk E Fm+1,K,/2 (U) k=1
1.3 Flat Chains
525
and
JiE11 dIn+M(S)
If m = n, we actually have Sk = 0 for all k, S = 0, hence the proof is completed. If m < n, by a downward induction the proof is completed as in Theorem 1 in Sec. 5.1.2.
As trivial consequences of Theorem 4 we state
Corollary 1. If T E Fn,cPt (U), U C IRn, then there exists a Lebesgue summable field l; E L' (U, A,,,IRn) with compact support such that
T=,CnLf,
f EL'(U,IRn).
Corollary 2. If T E Fn,cpt (U), U C W', and ,CL(spt T) = 0, then T = 0. Actually for flat chains we have the following theorem which extends Theorem 3 in Sec. 4.3.1
Theorem 5 (Federer's support theorem). Let T E F,,,,,cpt(U), U C W, 0 < m < n. Denote by 7r«, a E I(m, n) the orthogonal projection of 1Rn into the , xn) = (x«1 ... , x«m m-plane coordinate determined by a given by Suppose that there exists a coordinate system (x', ... , x') in R' such that 7r(x',...
Gn(7r«(sptT)) =0
for all a. Then T = 0.
Proof. Let we D' (U). We have w« dx« = > w, A7r, (dx« 1 n ...
w= «I=m
Adx«,n )
laI=m
and
T (w)
T (w« A7ra (dx«1 A ... Adx«"`) )
_ «I=m
E 7r«# (T Lw«)(dx«1A... Adx«m). «I=m
Since T L w« is a flat chain 7r«# (T L w«) is an m-dimensional flat chain in the m-dimensional space 7r«1Rn; moreover spt 7r«# (T L w«) C 7r« (spt T).
Corollary 2 then yields 7r,,# (T Lw«) = 0, i.e., T(w) = 0.
We state here an approximation theorem a slightly more general form of which will be proved in Vol. II Sec. 1.4.3
5. Cartesian Currents in Riemannian Manifolds
526
Theorem 6. Let K and C be compact sets and U be an open set such that K CintC C C C U C R. If T E F,,,,,K(U), e> 0, then there exists a real polyhedral chain with support in C such that
Fc(T - P) < e and IM(P) - M(T)I < e. Flat Cochains. We conclude this section introducing the notion of flat cochain and proving a representation theorem for flat cochains
Definition 6. A linear functional a : Fm,,Pt (U) - R which is continuous in the sense that there exists a positive constant c for which
a(T) < cFK(T)
(9)
for all compact set K C U and all T E F,n,,K(U) is called an m-dimensional flat cochain, and the class of m-dimensional cochains in U is denoted by Fm(U).
The flat norm of a cochain a is defined by (10)
Fa := inf{c E R I
(9) holds V K CC U, V T E F,,K(U)}.
The coboundary da E F-+1(U) of a E Fm (U) is defined by
da(S) := a(8S)
(11)
V S E F,,,,+1(U).
Notice that
F(da) < F(a)
(12)
in fact if S E F,,2,K(U)
da(S) = a(8S) < F(a)FK(8S) < F(a)FK(S) by Proposition 2 (iii). Flat cochains can be represented by couples of forms with L°° coefficients
Theorem 7. Let a E Fm (U) be a flat cochain. The there exist a m-form cp and a (m + 1) -form 0 on U with bounded and measurable coefficients in U such that a(LnA6) = I < 6, W > W"
y6E,elm
U
(13)
a(a(Lnhrl)) =
fdLn
V 77 E ,(1m+1
U
and (14)
f( + <77,
>) dLn = O V 77 E CC(U, Am+iR'),
U
the m -field diiv77 being defined by
JU
dLn := - f <77,dw>dLn V w E D' (U). U
Also F (a) = max ( supUW1 111 , supu 1141 I I) .
2 Differential Forms and Cohomology
527
Proof. To the cochain a corresponds a linear functional ,Q on ,f2m x ,(2,,,,+1 given by f3(e, i1) = a(LThAe + a(Lnn?,)) (C, 77) E .(2,,.,, x ,f2,,.,,+1 and
F(a) := sup {O(e
77)
I
IU
As the dual of L1 is L°° we get at once L°°-forms cp and z/' for which (13) holds. For 77 E
Cwe then have
f
<77,
f
> = a(3(L'A )) =a(-Lnndivr?) _ -J dLn U
U
since Ln ndiv 7j (w)
= fdLn
=-
f
<77, dw > dLn
_ -LnAr)(dw) _ -a(Lnnrl)(w) Finally we remark that in view of the representation formulas for flat chains and flat cochains one also introduce L2-cochains and L4-chains by requiring forms to have coefficients in L9 and fields of rn-vectors to be in L4. If q is the dual of p then those are in duality. But we shall not pursue this development.
2 Differential Forms and Cohomology The goal of this section is mainly to prove Hodge's and Hodge-Morrey theorems, i.e., of representing cohomology classes of a compact Riemannian manifold by
harmonic forms. In Sec. 5.2.5 we look at compact manifolds without boundary where there is a unique harmonic form in each cohomology class. Then in Sec. 5.2.6 we consider manifolds with boundary. In this case we represent cohomology classes of forms tangential at boundary by a unique harmonic form (tangential to the boundary), and show that absolute cohomology classes contain a unique harmonic form normal at the boundary. We shall prove that as consequence of the Hodge-Kodaira-Morrey decomposition theorems. This approach heavily relies on variational methods and strongly uses the regularity theory for elliptic systems.
In the first three subsections we supply some preliminaries. In particular in Sec. 5.2.2 we introduce Hodge's * operator and discuss its algebra, and in Sec. 5.2.3 we briefly introduce Sobolev classes of p-forms. Then in Sec. 5.2.4 we discuss harmonic forms, and in Sec. 5.2.7 we prove Weitzenbock formula expressing the Laplace-Beltrami operator on forms in terms of covariant derivatives and of the curvature tensor. Finally in Sec. 5.2.8 we discuss the Poincare and Poincare Lefschetz dualities in cohomology for oriented compact manifolds.
528
5. Cartesian Currents in Riemannian Manifolds
2.1 Forms on Manifolds Let X be a smooth manifold of dimension m. For y E X we denote by TyX the space of tangent vectors to X at y; the space of covectors is denoted by A'TTX, while ApTTX and APTX denote the spaces of p-vectors and p-covectors
to X at y generated by the exterior algebra construction in Ch. 2. We set also A°TyX = A°TyX = R, and A1TX = TyX. (y) E ApTTX. In the domain of a A p-vector field 6 in X is a map y coordinate chart U C R1, l; may be represented as a p-vector field U in U eu(x) _
Sa(x)ea
aEI (p,m)
and, if two coordinate chart cp : U --> X,
b
: V --+ X overlap in X, then the
vector fields 1=U and 1[[=v on U and V are linked by
cp(x)) = Ap(D4'-1 ° cc) eu(x)
(1)
Similarly, a p-form w on X is a map y --+ w(y) E APTX. In a coordinate chart cp : U --> X, U C lRm, w can be represented as a p-form wu on U wa (x) dxa,
wU (x) = aE1(p,m)
and the transition formula relating its representations on two overlapping charts U and V is given by wU = (0-1 o cp)*wv
(2)
on u n cp-1 o o(V)
i.e. by (3)
wu = AP(D'tl)-1 ° co)
wv('%-1
° cp)
on U n cp-1 o O(V).
Smoothness of p-forms and p-vector fields is usually expressed in local coordinates. Introducing the so-called p-tangent bundle we can regard p-vector fields essentially as maps. For that, one defines another manifold ApTX as follows. Let {U, c u}, cp
: U C R' - X, be a system of charts; for any two overlapping
charts U,V denote by cpvU : U n cp-1(co(U) n z4'(V)) - V n -1(cp(U) n z/'(V)) the change of coordinates from U to V. Consider the new charts {U x ApRm} together with the change of coordinates given by
(xv, v) =
(cpvu(xu), ApDcovu(xu)eu)
Then {U x ApRm} together with gvU defines a new smooth manifold ApTX called the p-tangent bundle to X, which also carries a natural projection map
7r: ApTX -- X
2.1 Forms on Manifolds
529
on the first factor of API[8m. A map s : X --> ApTX, which is compatible with 7r, i.e., such that
iros(x)=x
VxEX
is called a section of the p-tangent bundle. Notice that locally APTX is like X x APR' and sections are roughly obtained by gluing together local graphs, x --+ (x,e(x)). Finally, on account of (1) p-vector fields can be identified to sections of ApTX. Similarly we can define the p-cotangent bundle APTX and p-forms can be identified with sections of the p-cotangent bundles. We shall denote the space of smooth p-forms on X by £P (X) and the space of p-forms with compact support in X by DP (X), or respectively by C°° (X, ApTX )
and C°° (X, APTX) if we want to stress their character of sections. The last notation is particular convenient to denote for instance p-forms in a Riemannian manifold with L2-coefficients, as we can denote them by L2(X,APTX). Let w be a p-form on X, p > 0, and let {wU} be a collection of forms on IR' which represent locally w. Since by (2)
dwU = d(( -1 ° p)#wv) = (,/,-1 o P)#dwv,
we readily see that the collection of (p + I)-forms on RI {dwU} represents a (p + 1)-form on X, denoted dw. The operator d defined this way is the exterior differential operator
d = dx : £P(X) -+ £P+1(X)
and it is easy to see that (i) If f : X --+ IIi, d agrees with the ordinary differential of maps (ii) d is linear
(iii) d2=dod=0
(iv) d(wA77) = dwA77 + (-1)PwAd77 for w E £p(X), rl E £9(X).
Also, if Y is another smooth manifold and
X -> Y is a smooth map, then we
have
(v) dxc# _ O#dy. A celebrated theorem by Whitney states that every smooth manifold X of dimension m may be regarded as an embedded submanifold of some Rn where m < n < 2m + 1. In other words there exists a smooth one-to-one map i : X -> II2n which is a homeomorphism into (i.e., i : X ---+ i(X), i(X) equipped with the relative topology induced by ]Rn, is open) and for which the linear tangent map i#y : TyX ---> Tz(y)I[8n ^_ R',
yEX
is of maximal rank m. Consequently, by the implicit function theorem, i(X) is diffeomorphic to X, and, since i is an homeomorphism into, i(X) is a submanifold of Rn, i.e, for any z E i(X) the are two open neighbourhoods U and V respectively of z and 0, and a diffeomorphism 0 : U --+ V such that
5. Cartesian Currents in Riemannian Manifolds
530
O(Uni(X))=Vn{xER n
I
=xn=0};
x-+1=
in particular the inverse of ¢ gives a local chart w : W C lam -> U n i(X)
(4)
by W(x) := 0-1(x, 0), W := {x E Rm I (x, 0) E V}. Now we discuss relationships between the tangent and cotangent spaces of i(X) and of Rn. Clearly the tangent space TZi(X), z E i(X) is the subspace of TZRn ^ Wn generated by the columns of the Jacobian matrix of cp in (4) at 0, hence of dimension m; as i#yTX C Ti(y)i(X), the image of i#y : T,X -+ Rn is exactly Ti(y)i(X), and similarly i#y(ApTvX) = ApTi(y)X = Ap H
(5)
The dual map iy : ApRn -+ APTX is surjective, hence iy# is an isomorphism between APRn/ ker if and ApT,X i
(6)
: Apr/ ker i# -+ ApTX,
and we say that the p-covectors in keriy are null to i(X) at i(y). Extensions of (5) and (6) to vector fields and forms depend upon relations between smooth maps on i(X) and on Rn. The following proposition is central in this respect
Proposition 1. Let X be a compact submanifold of R n. Then X is a smooth neighbourhood retract, i.e., there is a neighbourhood U of X in R and a smooth map 7r : U --+ X which is a right inverse of i : X '--> lRn, that is, it o i = idx. U is often referred as a tubular neighbourhood of X and 7r as the projection map.
Using the projection map 7r, every smooth function f : i(X) -+ R extends to a smooth function defined in the tubular neighbourhood U of i(X) by g(x) := f (7r(x)). Consequently p-vector fields on i(X) extend smoothly to p-vector fields
on a tubular neighbourhood U of i(X) and the tangent map i# turns out to be an isomorphism between smooth p-vectors fields to X and smooth p-vector fields 1= to U for which e(x) E X E i(X). Similarly the pull-back map i# : EP(U) -+ EP(X) is surjective, and Ep(U)/Nullx(U) ^' EP (X) where
keri# _ {w E E'(U) I i#w = 0} = {w E Ep(U)
I
0V
E ApTi(y)i(y)}.
denotes the null forms to i(X) in EP(U). Finally, the exterior differentiation operators dx on X and d on U commute, i.e.,
2.2 Hodge Operator
i#d = dXi# on £P(U)
(7)
531
,r#dx = dir# on £P(X)
and
dx = i#d7r# on £P(X). In particular the first equality in (7) implies that dw is a null form to X in U whenever w is a null form to X in U. On account of the above we can identify X and i(X) and understood the injection map i : X -* Rn. This way tangent vectors to X are seen as a subspace (8)
of Rn,
ApTyX c APTi(y)i(X) C ApR",
while p-forms in X are seen as p-forms in a tubular neighbourhood U of X modulo null forms, i.e., w E £P(X) if w E £p(U) and <w(y),l;> = 0 V E ApTX. Also, the exterior differential operator is nothing but the ordinary exterior differential operator in Rn, because of (7), (8). Finally we can identify the projection
mapir:U -+X with themapP:=ioir:U--+i1
;
P#w=lr#oi#w=0 = 1 on X we
whenever w is a null form to X, and for any C E CO° (U) with have i#(w (P#w) = i#(w) i#7r#i#w = 0,
-
-
that is, w - P#w is a null form to X for any w E £P(U), or, in other words, .7r#w yields a canonical representative of w modulo null forms.
2.2 Hodge Operator In this subsection we shall collect a few more fact on exterior algebra. Let V be a vector space of dimension n, and let V* be its dual
The operators DP and DP. Given an n-covector Q(n) E AnV and an integer p, 0 < p < n, we may consider the linear map Dp : APV -> A'-PV defined for C E ApV and i E An_pV by := <,fl(n), 77hl;>.
(1)
If el, ... , en is a basis in V and ell ... , en is the dual basis in V* we have (2)
S?(n)
= Ae'A ... hen,
a := <17(n), elh... hen >,
hence
= A<elA... hen, ephea> =
r0 Sl Av(a,a)
consequently
a E I(p, n). a)ea, Dp(ea) Using (2), (3) and the linearity of Dp we easily infer (3)
if/3#a if3 =a
5. Cartesian Currents in Riemannian Manifolds
532
Proposition 1. We have (i) Dp is an isomorphism between ApV and A(n-P)V (ii) DP(C) is simple if and only if C E APV is simple (iii) Dp(C)A = f2 (n) (-1)P(n-p) <e,Dn-p(77)> V C E ApV, 77 E An-pV. (iv) =
Proof. (i), (ii) and (iv) are trivial. To prove (iii) it suffices to observe that for a,,3 E 1(p, n) we have Dp (e«) AeO = Ao, (a, a) e' AeQ = ),e 1 A ... Aen 8Q = < ea, ea > bQ .
Similarly, given an n-vector (2(n) E AnV and an integer p, 0 < p < n, we may consider the linear map DP : APV --f An_pV given for 0 E APV and 0 E An-pV by
= .
(4)
We have (5)
,(2(n) = 7,ce1A ... Aen,
A:= <e1A ... /den, Q(n) >
and DP (e&) = 7lv(a, a) ea,
(6)
a E 1(p, n).
Also
Proposition 2. We have (i) DP is an isomorphism between APV and An_pV (ii) DP(q5) is simple if and only if 0 E APV is simple (iii) vADP(O) = .(2(n) `d v E ApV, 0 E APV (iv)
(-1)P(n_P) V
=
0 E APV,
E
An-pV.
1 Geometric interpretation. First observe that, assuming (2(n) # 0, (1) yields at once
=0
if
Ar7=0.
In particular, if e is simple, e = v1 A ... AvP and 77 is an (n - p)-vector of the type 77 = v2Aa with a E An_p_1V, then
= 0.
(7)
Write now Dp (e) with 6 simple as Dp (1;) = f 1 A ... A f n-P, then (7) reads
f2(v3) = 0
V i = 1,...,n - p, and j = 1,...,p.
Therefore, if we denote by f : V - JRn-p the map f that ker f is the linear space associated to (8)
ker f= {v E V I vAe = 0}.
(f 1, ... , f P) we conclude
2.2 Hodge Operator
533
Conversely, let f : V -> RP be a non singular linear map, i.e., 0 := f In ... n fP # 0. Choose coordinates e1,. .. , e,,, in V in such a way that e1, e,,- p generate
ker f , f = (f',.. . , f P), then < O, v > = 0 for any v of the form v = einw, i = 1, ... , n - p, w E A,,_P_1V. Proposition 2 (iii) then yields d w E An,_P_1V, i = 1, ... , n - p,
einwADP(o) = 0
i = 1, ... , n - p, that is, DP (0) is tangent to ker f .
hence etADP (0) = 0
Interior multiplications. The homeomorphisms DP and DP are special cases of the so-called interior multiplications between vectors and covectors.
The interior multiplication of p-vectors by q-covectors, 0 < q < p < n, is defined as the bilinear form L : APV x AqV
Ap_gV given for t; E APV and
0EAgVby
<eLO,w>_<e,OAw>
(9)
dwEAP-qV
In terms of a basis e1, . . . , e, of V it is easily seen that ea L eR =
(10)
Often one refers to
L
f0 v(Q, a -
if,l0Za if,3c a.
as to the slice of l; by 0. This is motivated by the
following
Proposition 3. Let t; E APV be a non-zero tangent vector to a p-subspace E of V, let :_ (01, . '0q ) be a linear map from V to ]Rq, p > q, and let 0 := Oln ... A0 q. Then C L 0 is non-zero if the restriction cE of to E has maximal rank; moreover, C L 0 is simple and tangent to ker E . Proof.
Without loss of generality we may assume
= e1 n ... nep. If
_
IRI=q 0)3e,3 then e1n... neP L
t h e r e f o r e C L O # 0 iff 0,3 = <,e> 54 0 for some,3 D {1, ... , p}. If we represent
this implies that the q _?< p-submatrix consisting of the is non singular. Therefore OE has maximal rank q. _ first p columns of In order to prove the second part of the claim we observe that ker OE has dimension p - q. Choose now a basis of V in such a way that e1, . . . , e, generate E and e1, ... , ep_q generate E n ker 0 = ker 4z. In these coordinates 0 writes as
0 by a q x n-matrix
0 = E I R I =q 0$ eR, 0,3 = 0 if,3 Z {1, ... , p - q} and, for j = 1,...,p-q, we have ej A( L 0) _
o-(,3,,3')00ej neR' = 0,
as each factor in the sum is zero since either j E 0' or 0,3 = 0. Consequently
5. Cartesian Currents in Riemannian Manifolds
534
E f1 ker 0 C {v
I
vn(1; L o) = 0} _: T L O,
which implies that dimTT L 0 > p - q. On the other hand dimTT L 0 < p - q, hence dim Tg L 0 = p - q. We then conclude that 6 L 0 is simple and
.flkero={v
I
vA(l;L0) =0}
0
L 0 is tangent to ker O_P.
which shows that
The interior multiplication of p-covectors by q-vectors, 0 < q < p < n is defined as the bilinear map J : AqV x APV -p AP-qV given for t; E AqV and
¢EA"Vby
Vr,EAp-qV.
(11)
In coordinates it is easily seen that for a E I (q, n), /3 E 1(p, n) e, J e,3 =
ifa¢13
0
v(a,/ - a)e'3-a if a C Q.
In terms of interior multiplications it is readily seen that D''(0) = S2(n) L 0;
DP(e) = 6J Q(n),
(12)
one also easily checks that 6-1 D8(i)
(13)
Dr+s(OAO) = D?(0) L4.
Hodge operator. We assume now that V is a linear space endowed with a scalar product ( ). Then, as we have seen in Ch. 2 a scalar product is also I
induced on each ApV, ApV, 0 < p < n. If e1i... , en is a basis in V, gij := (ei I ej),
and G denotes the n x n-matrix (gij), the scalar product on ApV and APV is defined respectively by (e. I ea) = MS (G) =: gaa and
(ea I eQ) = M (G-1) =: g"' By classical inversion formulas, compare (5) in Vol. II Sec. 2.3.1 and (6) in Vol. II Sec. 2.3.1, we have (14)
9QO := Ma
(G-1)
or
o, (a, a)
goo
where we recall goo = det G. Also the scalar product induces isomorphisms ,lap
: ApV
ApV,
OP: APV -4 APV
which are one the inverse of the other and are defined, compare (13) in Sec. 2.2.1, by
2.2 Hodge Operator
.- (S 171) A, V,
<0P (0), V) >
535
(0 10) AP V,
i.e., in coordinates (15)
gape"
/P(ea) _ Ia1=Ia1
P(ea) _
(16)
9"'e,3-
Finally, if e1,. .. , en is an orthogonal basis, then Iein ... Ae,,I = 1 and every n-vector has the form Ae1 A ... nen, A E R. Therefore in An V there are exactly
two n-vectors of length one, and choosing one of the two amounts to fix the orientation of V.
Fix now an n-vector on V of length one, denote it by .fl(n) and its dual by Q(n). If DP and DP are the operators associated to .fl(n) and Q(n) by (1) and (5) the Hodge * operators *
are defined for
: APV -> 11n-PV,
*
APV - An"-PV
E APV and 0 E JPV by
(17)
*0 = Qn-P(DP(0))
*C=
If e1, ... , en is an orthogonal basis and sign(e) := <e1A... Ae, Q(n) > = ±1
denotes the sign of the basis e1, ... , en with respect to fl(n), we get (18)
* ea = sign(e)o-(a, d)ea,
*ea = sign(e)o-(a, a)ea
so that (19)
* *ea = a(a, a)a(a, a)ea,
which yields
** =
* * ea = o, (a, a)o-(a, a)ea
(-1)P(n-p)
since v(a, a)Q(a, a) _ (-1)7'(n-P). In terms of a generic basis v1, ... , vn of V we have v1A... Avn = sign (v)(detG)1/2,f1(n) where G = (g2j), Therefore
<eln ...nen, fl(n) > = and
*va =
9ij _ (Vi I VA-
si nv
det G
sign(v) detGC(Q,Q)9a,ovp
(20)
*va - sldet GOr (a, a)0n-P(va) = sldet Go-(a, a)9apve
5. Cartesian Currents in Riemannian Manifolds
536
or by (14) (-1)P(n-P)sign (v)
*va =
It is easily seen in orthogonal coordinates that VC, 77 E A ,V
177)S?(n)
V0, 0 E APV
(21)
VCEA,V, 0EAPV
5>
deeAPV, 0EAn-PV
and, as .(2(n) = *eo, .(2(n) = *e°, (22)
VC, y E AV
* (CA * 77) = *(71A * C) = (C 171)
Integration on submanifolds of R'. Let X C R,-+,k be an oriented submanifold of dimension n. The metric and the orientation of Rn+k induce a Riemannian
metric g on X, a metric (g2j(y)) on TyX and a unit tangent n-vector field T, T(y) E AnTyX, which now plays the role of ,(2(n) above. The corresponding Hodge operator * yields a * operator * : APTyX ---p An-PTX which lifts to a * isomorphism between forms * : SP(X) --+ en-P(X). Denote by S2(n) the dual of Q(n) = T. As
f
J
r
w=J
x
T>d7-ln
x
we see that S2(n) is the volume form on X, i.e.
= r dHn = 7-( (X)
(23)
and, since by the second equality in (21)
(a(y) I,3(y)) = (a(y) I i3(y)) = , we get (24)
f (a I ) d7 In =
x
J x
QA *,3.
2.3 Sobolev Spaces of Forms Let X be an n-dimensional compact submanifold of u : X ---f R which are 7-ln L X-measurable and with
J Iulsd7-InLX
s>1
I[Yn+N. The class of functions
2.3 Sobolev Spaces of Forms
537
is denoted by LS (X) . LS (X) is clearly a Banach space, and for s = 2, L2(X) is an Hilbert space with the obvious scalar product. If (Vi, (pi), Vi : V -> X is
a system of charts for X, we define the class Wl,'(X), s > 1 as the class of u E LS(X) such that u o Wi E Wl,s(V), and we set II u ll wl,s (X) := 11 U 111s(X) + E f I D(u O Wi) IS dx.
(1)
Vi
It is readily seen that the definition of Wl,s(X) is independent of the system of charts, while the norm does depend. However different system of charts give rise to equivalent norms on W""s(X). An intrinsic norm on Wl,s(X) can be defined in terms of approximate tangential derivatives DX to X : one easily verifies that IIUIILS(X)+1
1
1/S
f IDXulsdHnLX)
is a norm equivalent to (1). Properties of the standard Sobolev spaces on open sets of Euclidean spaces are immediately transferred to analogous properties of W1"2(X), we in particular state (i) W"2(X) is a Hilbert space with the scalar product /
(u I V)Wi,2(X) := (u I v)L2(X) +
f(DXU I DXV)TX dfn LX.
LS(X) is compact (ii) The inclusion Wl,s(X) (iii) C°°(X) is dense in W""s(X).
A differential p-form w on X is said to be measurable if <w(y),v(y)> is 7-ln-measurable for any smooth vector field v, equivalently if for any local chart cp2 : Vi -> X, cp#w is L'-measurable, i.e., all coefficients of cp#w are measurable.
Definition 1. Let s > 1. The space of measurable p -forms on X such that
f IwlsdfnLX APT,,X = <7r#w,7r#77>APRn+N
a scalar product on L22 (X) is given by (w,71)L;
f <w(y), 77(y) >APT,,X /Ln =
V w,7) E A°TyX,
/
<7r#W, rr#17 >npRn+N dH-.
538
5. Cartesian Currents in Riemannian Manifolds
Definition 2. The class of p -forms w in L4(X) such that cp#w have coefficients in W1'1 (V,1[2) for every coordinate chart (V,cp) are denoted by WI,"(X). (X) can be defined by choosing a coordinate system of charts A norm on (Vi, fpi) and setting II W II W;'s(X)
II W II LP(X)
+
#wI WP's(Vi).
Though the definition of WP' (X) is independent of the coordinate system, the norm does depend on it, but one easily verifies that different coordinate systems produce equivalent norms. Again properties of WP"°s(X) are inferred from properties of standard Sobolev spaces; we state
(i) W l ,' (X) is a Hilbert space.
L'(X) is compact. (ii) The immersion WP ,3 (X) (iii) Smooth forms are dense inWP(X). Finally we remark that similarly one can introduce the spaces Wp'3(X) and more generally WW ,r (X ), but we shall not dwell any further on this point.
2.4 Harmonic Forms Let X be a smooth compact n-dimensional submanifold of 1[in+k possibly with boundary. We assume that X is a CO° submanifold even if the claims on this subsection hold for Cl,l submanifolds with trivial changes. Let d be the differential operator on forms.
d : C°°(X,APTX) -+ C°°(X,AP+ITX). Definition 1. The codifferential operator
b : C' (X, APTX) ---> C' (X, AP-'TX) is the (formal) adjoint of d, i.e., is the operator defined for all a E CO° (X, APTX)
and p E C,- (X \ 9X, AP-'TX) by (Sa I )3)L2(X,AP-1TX) :=-(a I d,3)L2(X,AP-1TX).
Clearly b2 = 0, and S extends to all
WP,' forms.
Proposition 1. 8 : C°°(X,APTX) -+ C°°(X,AP-1TX) is a differential operator. If x1, ... , x'n are local coordinates and G = (gij) the metric tensor of X, then for w = (1)
aI =P wa
(6w). =
dxa we have Sw = E ,,I=P(Sw). dx-I where
E E E g7ao,(i,
Ial=p-1 i¢a 1QI=p
(a+2)wR)-
2.4 Harmonic Forms
539
Proof. Let 7) E C°° (X \ 8X, AP-'TX) be a (p - 1)-form with support in coordinate chart. We have
-(w,dr1)APTTx = - = gQ7wa(di1)7 IQI=I7I=P
( IQI=I'YI=P
1
\ a+i=ry
I$I=P Ial=P-1 i0«
Integration by parts on X then yields
-(4' dr7)L2 = - f (w, drl)f dx
-
a)Di(f9p(«+i)«)
0(2,
Igdx,
Ii3I=p i s
«1=p=1
Ial=pY_iOa o-(i, a)Dii.e. (1).
consequently g'Y«(Sw)ry =
Proposition 2. If X is oriented and * is the Hodge operator on X then S = (_1)n(p+1) * d * Proof. For a E COD (X, APTX) and /3 E C°° (X \ 8X, AP-1TX) we have
d/3A * a = d(/3A * a) = d(/3A * a)
-
(-1)P-1,3Ad
*a
(-1)p-1(-1)[n-(n-P+1)](n-p+l)/3n * *d * a
= d(/3A * a) - (-1)n(p+1)OA * *d * a. Therefore, by Stokes theorem and (24) in Sec. 5.2.2, we get
(Sa1/3) _ -(aId3)=-J d/3A*a (-1)n(p+l) f /3A * *d * a =
(-1)n.(p+l)(*d
* a 1)3).
Definition 2. The Laplace-Beltrami operator on Coo (X, APTX) is the operator
a=dS+Sd : COO(X,APTX) a form w is called harmonic if Zw = 0. The following claims follow at once
C°°(X,APTX);
540
5. Cartesian Currents in Riemannian Manifolds
Proposition 3. We have (i) A is formally self-adjoint, i.e., (La 1,3) _ (a Ai3) for a, 3 E Cc-(X \ 3X, APTX). (ii) If w E CC°(X \8X,A2TX) then Aw = 0 ifdw = bw = 0, in fact (w dw) _ J8wJ2
+ dwI2 (iii) A is well defined onW2'2(X,APTX). Definition 3. The Dirichlet integral associated to a p-form is D(w) := f(dwI2 + Ibw12) d7-ln
x 1 The Laplace-Beltrami operator on functions. First we compute b on 1-forms. In a chart U, if cp E C° ° (U) and w = > wi dxi we have
f f -
(6w I 0) =
(w, dcc) dy = -
f
-gUwi 28x
U
Dj(V,-gg2'wi)Wf dx
U
where (gij) is the metric tensor on X, gig Sw =
Dj
(gij)-1, and g := det (gij). Hence (-,fg-g2'wi)
and for any function f E C°°(X) A f = (bd + d8) f = bdf =
D; Voij -Lf
axi
Also in local coordinates, i.e., for f E C' (X) with compact support in a coordinate chart we have
D(f) = f Idf12dW2 = f X
8f of dx. /g2j 8xi axi
Notice that, by the maximum principle, any harmonic function on a compact and boundaryless manifold must be constant.
0 The Laplace-Beltrami operator on p-forms in IRn. Given w = E wadxa 1c1=p
we have dw
and, from (1)
=
E wa>z o-(2, a) dxa+i lal=p i a
2.4 Harmonic Forms
Sw = (-1)n(P+1) * d * w =
(2)
Q(i )w(j+i i
1
I01=p-1
541
J dxO.
i j3
We now compute the Laplace-Beltrami operator. Of course
d5w = E(dSw)a dxa, a
and, according to the above
(dbw)a =
/
o(i, ) (dw)a,i =
T Q(i, a)o,(j, Q)wa+j,ji
= E E o(i, a - i)o(j, a - i)wa-i+j,ij ; iE« jva-i now we split the sum in j as
>
joa-i
_ j=i
j$a
to get
o(i, a - i)o(i, a - i)wa,ii
(dbw)a = iEa
o(i, a - i)a(j, a - i)wa+j-i,ij.
+
(3)
iEa
j0a Also 8dw = E(Sdw)a dxa, and (Sdw)a =
o(j, a)(dw)a+j>j =
E a(j, a)o(i, )3)w0,ij j 0a (j+i=a+j
i 0a joaiEa+j
Similarly we split the sum in i as the sum of terms in which i E a and terms in which i = j to get (Sdw)« _
(4)
o(j, a)o(j, a)wa,jj j4za
o(j, a)o(i, a+ j - i)wa+j-i,ij
+ iEa
Summing (3) and (4) we then get (zlw)a = (dbw + Sdw)a = Lwa
+
[o (i, a - i)o(j, a - i) + a(j, a)o(i, a + j - i)]wa+j-i,ij iEa
ja
542
5. Cartesian Currents in Riemannian Manifolds
As fori<j (5)
u(i, a - i) = Q(i, a - i + j)
o-(j, a, - i) = -o-(j, a),
while for i > j (6)
o(i,a-i)+j)
o(j, a-i)=o(j,a),
we see that the term in square brackets in (5) vanish, therefore we conclude
L1w = E (Qwa) dxa
(7)
lal=p
i.e., a p-form in the flat manifold R n is harmonic if and only if its coefficients are harmonic functions. Obviously 1dwj2
+
lawl2
=
IDwal2 + L(w) lal=p
L(w) being a non-zero quadratic form of the derivatives of wa. Since the Euler equations of the two integrals Do(w, U) :=
fI&2 + 82) dx and
Dwa U
2
,
lal=p
as we have seen agree with Awfa = 0, we infer that (8)
Do(w, U) =
JU
IDwa12 dx
V WE W01,2 (U)
Moreover L(w) is a null Lagrangian; in particular fu L(w) dx depends only on
the values of w (wa) at the boundary 8U of U and vanishes if w E C' (U), compare e.g. Giaquinta and Hildebrandt [275]. Of course one can compute bw, L,w, I dw 12 + I dw 2 explicitly in terms of the metric tensor and of its derivative, but, since we do not need it in full generality, we postpone this topic to Sec. 5.2.7 and here we confine ourselves to the following remark. From (1) one sees that (dw).y has the form (9)
(8w).y(x) = api(G)wQ,i + bp(G, DG)wo
where aQi(G) and bp(G, DG) are smooth functions which pointwise depend re-
spectively on the metric tensor and on the metric tensor and its derivatives. Indeed api(G(x)) are Lipschitz functions and bp(G(x), DG(x)) are L°°-functions on X if the metric tensor G is Lipschitz. Moreover we notice that, in case G is the Euclidean metric,
2.5 Hodge and Hodge-Kodaira-Morrey Theorems
543
(8w)7 = E E o (i,,Q)DjwQ+i IQI=p-1 i0Q
consequently b Q (Id , 0) = 0. On account of the previous considerations it is then easy seen that in local coordinates the integrand of the Dirichlet integral has the form (10)
(IdwI2+ 5wI2)/dx = A«Qw«,iwQ,.7 +Bapiwa,iw,+C«QWaWQ
A" (G(x)), BaRi(x) = BaQi(G(x), DG(x)), CCQ(x) _ Cap(G(x),DG(x)) and
where
Aaa(Id)
BaQi(Id, 0) _ C«Q(Id, 0) = 0.
2.5 Representing Cohomology Classes by Harmonic Forms: Hodge and Hodge-Kodaira-Morrey Theorems We are now ready to prove the celebrated theorem of Hodge which states that every cohomology class of a compact smooth manifold without boundary contains exactly one harmonic form. More generally we shall prove a decomposition theorem for forms, Hodge-Kodaira-Morrey theorem, from which Hodge result follows. The proof, based on variational methods and more precisely on Dirichlet principle, shows the entire theory of linear elliptic partial differential equations at work.
In this subsection X is a compact, possibly non orientable, boundaryless Riemannian manifold of dimension n that is thought without loss in generality as isometrically embedded in some RN. We assume that X is a CO° manifold even if one can work on C""l manifolds with trivial changes. One of the key ingredients of the variational proof we are going to present is the following simple but important lemma
Lemma 1 (Gaffney). There is a constant C such that for each point y E X there exist a ball B(0, p) C ]R' and a chart (U,,, cp), cp : B(0, p) -+ Uy C X, cp(0) = y, such that
D(w) ?
2
f
I2
Diwa2 dx - C11 w L2(U,)
B(O,p) 2'a
whenever cp#w =
wa dxa has support in U,,.
Proof. Fix a point y E X and choose coordinates so that the metric tensor at y is the identity, gij(0) = bij. Then using the representation (10) in Sec. 5.2.4 for D(w) we infer D(w)
>
f A'Q(0)w«,iwQ,j dx - J[A(0) - f IBaQi(x)I w«,iIIwQldx - J
ICaQIIWal
dx IwQldx.
5. Cartesian Currents in Riemannian Manifolds
544
As
B(0, p)) = L(O,P)
if A,a(0)wa,tiwg,j dx =
IDwl2 dx, i,a
by (8) in Sec. 5.2.4 the result follows at once choosing p sufficiently small so that A'3 (0)1 be small. El A%3Q
Choosing a finite cover of X by charts for which Lemma 1 holds we readily infer that (i) D(w) +cll w lli2(x) is an equivalent norm on W1"2(X, APTX) for a suitable
constant c depending on the covering, and in fact there is a positive constant cl and a non- negative constant c2 such that (1)
2
2
D(w) >- cl 11 w IIw 1,2 - 0211 w IIL2
and, being ll w II L2(x) continuous with respect to the weak convergence in W1,2 by Rellich theorem, (ii)
D(w) is lower semicontinuous with respect to the weak convergence in W1"2 (X, APTX).
Definition 1. We say that a p-form w E W1,2 (X, JIPTX) is harmonic if dw = bw = 0. The linear space of harmonic p forms is denoted by HP.
Theorem 1. HP is finite dimensional. Proof. Otherwise we could find a sequence {wk} C HP which is orthonormal in L2. As ll wk IIL2 = 1 and D(wk) = 0 a subsequence will converge strongly in L2, which is impossible since wk is orthonormal in L2.
Theorem 2. There exists a positive constant c such that (2)
D(w) > cl w I12 2(x,APTx)
for any w E W1,2 (X, APTX) which is orthogonal to HP. Proof. Otherwise we find a sequence Wk E W1,2 (X, APTX) n HPl such that 1) (LOO < kll wk liL2.
If 77k := wk/I1 wk IIL2, we have 11 77k IIL2 = 1, D(77k) < 1/k. Passing to a subse71 weakly in W1'2, Ink - rl strongly in L2, hence 1171 IIL2 = 1, by quence 77k
semicontinuity 71 E HP' and D(77) = 0, i.e., 77 E HP fl HP' or in other words 71 = 0: a contradiction.
2.5 Hodge and Hodge-Kodaira-Morrey Theorems
545
Given a p-form f E L2 (X, APTX) we are now interested in finding a weak solution of the equation
aw = f, that is a p-form w E W1,2 (X, APTX) such that (3)
(
drl)L2 + (8w l'rl)L2 = (f 7)LZ
V 77E W1,2(X, APTX).
We have
Theorem 3. Equation (3) is solvable if and only if f belongs to the orthogonal in L2 (X, APTX) of HP, denoted by HP. Moreover there exists a unique solution wo in W1"2(X,APTX) n HP1 and the linear transformation from f to wo is a bounded linear transformation from L2 (X, APTX) into W I,2 (X, APTX). Proof. Clearly if w satisfies (3) then f is orthogonal to HP. Conversely assume
f E L2 (X, APTX) n HP1 and consider the integral
F(w) := D(w) - 2(f I U;) L2 defined on W1"2(X,APTX) n HP1. Trivially F is lower semicontinuous with W1,2, W1,2 respect to the weak convergence in n HP1 is weakly closed and, by Theorem 2,
F(W)>CIIwIIL2-21If
IL2IIWIIL2? 2IIWIIL2
- III IL2.
Therefore, taking also into account (1) we immediately infer that there exists a minimizer wo E W1"2 (X, APTX) n HP. From the minimality of wo we now infer (4)
0 = dtF(wo + t() _ (dwo I d() + (Swo 15()
- (f I
for all C E Wl°2(X, APTX) n HP' and hence for any E W1"2 (X, APTX) since any such C can be written uniquely in the form ( = h+(o where Co is orthogonal to HP and h E HP. From (4) we also get D(wo) < 11f IIL21IwoIIL2
which together with (1) and (2) proves the boundedness of the map f -+ wo. Finally, if w1 is another solution of (3) we have for (:= wo - wl I<12 + Ij(I2 = 0 i.e. wo - wl E HP, and if w1 also belongs to HP1, wo - wl = 0.
Equation (3) is the Euler equation of the integral D(w) - 2(f I w)L2
therefore on account of the representation (10) in Sec. 5.2.4, we see that in local
coordinates the coefficients of any solution of (3) are in fact solutions of an elliptic system. The regularity theory for elliptic systems then applies, and in particular we can infer
546
5. Cartesian Currents in Riemannian Manifolds
Theorem 4. Let X be of class C1'1 and let wo solves (3). We have (i) If f E L2 (X, APTX), then wo E W2'2(X, APTX).
If f E LP(X,APTX), then wo E W2'P(X,APTX), p > 1. If f E C°'N'(X, APTX), then wo E C2,4 (X, APTX).
Higher regularity follows, if we assume higher regularity of X and f. For instance, if X and f are of class C°° or analytic, then wo is C°° or analytic. (ii) In particular we see that w is weakly harmonic ifw is smooth and harmonic.
For each p set now
Imd = {da Im5 = {8a H = {a E
a E W1'2(X, AP-'TX) } a E Wi'2(X, AP+iTX)} Wi'2(X,AP-iTX) da = 8a = 0}. I
Then we have
Theorem 5 (Hodge-Kodaira-Morrey). The linear subspaces Im d, Im 6, H of L2 (X, APTX) are mutually orthogonal and closed. Moreover
L2 (X, APTX) = H ®Im 8 ®Imd, i.e., any w E L2(X, APTX) decomposes uniquely as
w=h+8a+d,Q
(5)
with a E Wi'2(X, AP+iTX), /3 E W12 (X, AP-'TX) and h E H, and (6)
IIhIIL2+ aIIw1.2+II/0IIw1,2
and in fact, if X is of class CO°, h E C°°, and for all k (7)
II a IIW1,k + IIQ IIW1,k
Proof. Let w E L2(X, APTX). Denote by h its projection on H which is finite dimensional. Then w - h E H -L and by Theorem 3 and Theorem 4 there is wo E W1"2 (X, APTX) such that
. wo=w-h. Writing a := dwo, /3 = 8w°, (5) and (6) follow at once. Let us prove now that the three subspaces Im 5, Imd and H are mutually orthogonal and closed, from which one easily infer that the decomposition (5) is unique. Let w E H, for a E Wi'2(X,AP-iTX) we have
(dalw)=-(a,bw)=0 and for 0 E W1'2 (X, AP+iTX)
2.5 Hodge and Hodge-Kodaira-Morrey Theorems
547
(5aIw)=-(aIdw)=0; while we also have
(da 15/3) = -(d2a /3) = 0 provided a belongs to W2'2. If a belongs to W1"2 we choose a sequence ak E W1,2
such that ak -> a strongly in W1,2 and we get (da 15p)
lim (dak I SO) = 0.
k
00
Finally let us prove that Imb is closed. Let wk := b77k, 17k E W1,2 (X, AP+'TX), and assume that wk -> w in L2. By (5) we have wk = hk + Sak + dQk,
the uniqueness of the decomposition yields Sak = Wk, moreover 11 ak I I W 1.2 <- C11 wk I
I L2 < cont.
Therefore we infer passing to a subsequence that
ak -k a weakly in W"2,
ak -> a strongly in L2,
and a E W"2. Consequently ba = w, which proves that Im S is closed. Similarly one proves that Im d is closed.
Before stating Hodge theorem, let us recall the definition of the De Rham cohomology groups . A smooth p-form a is called closed if da = 0, exact if there
exists 77 with d77 = a. Because of d o d = 0, exact forms are always closed. Two closed forms a,,3 are called cohomologous if a - /j is exact. This property determines an equivalence relation on the space of closed p-forms and the set of equivalence classes is a vector space over R, called the p-th de Rham cohomology group and denoted by HHR(X)
Theorem 6 (Hodge). In every cohomology class in HHR(X) there exists a unique harmonic p-form. Proof. Let wo be a closed differential form representing a cohomology class. All forms homologous to wo have the form
w=wo+d3; as wo E Im b1, we can write by the decomposition theorem wo = ho + d/30
where ho is harmonic, therefore every w cohomologous to wo has the form
w=ho+d(,3+,do).
548
5. Cartesian Currents in Riemannian Manifolds
It is worth noticing again that if w = h + d/3 is the decomposition of a closed form w, h being harmonic, we also have, assuming X of class C°° II 0Ilwk+l,r < Ck11 W Ilwk.r
J1 Q I1Ck+1,µ < CkI W 11C-,µ.
Later we shall be interested in closed forms in the product manifold f2 x X, where Q is an open set of an oriented Riemannian manifold. The product structure in ,(2 x X induces a canonical splitting of the exterior differential operator
din,(2xXas
d=d,+dy
with respect to local coordinates (x, y), x E ,(2 and y E X and the splitting of DP (f2 x X) as direct sum min(p,n)
Dp(,(2xX)_
®
Dp,k(,RxX)
k=max(O,p-n)
where Dp,k (,R x X) denotes the class of p-forms in the product ( x X with exactly k-differentials in y. A simple consequence of the decomposition theorem is the following result which will be used later
Proposition 1. Let w be a compactly supported p-form in 0 x X which has exactly f-differentials in y, w E Dp,e(.(2 x X), and is dy-closed, i.e., dyw = 0. Then w can written as s
w = 1 cps(x)no" (y) + dyrl (x, y) s=1
where i is a compactly supported (p - 1)-form in .(2 x X with at most £ - 1
[as], Qi harmonic, form a basis of
differentials in y, cps E VP-e, and the cohomology group HHR(X).
Proof. Choose an open covering {Ui} of U so that on each Ui there exists a
basis of simple tangent (n - P)-vector fields eay,i. Denote by i the dual form of ea,i and by {iii} a decomposition of unity associated to the covering {Ui}. On each Ui i jw can be seen as ,.Ijiw
77anwa
where w E CO° (Ui, V (X) ). Thinking of x as a parameter, Theorem 5 applied to yields s
(x) _
cpa's (x)os + dy?7a s=1
with cpa's E C:°(UU) and 77i E C°O(Uj,DP-1(X)). The result then follows summing on i and a.
2.6 Relative Cohomology: Hodge-Morrey Decomposition
549
2.6 Relative Cohomology: Hodge-Morrey Decomposition Let X be a smooth compact, possibly non orientable, Riemannian manifold of dimension n with smooth boundary. We think that X is isometrically embedded in an Euclidean space R' +N and we also assume that X is a C°° manifold with a C°° boundary aX even if our presentation extends to C1,1 manifolds with Lipschitz boundary with only trivial changes. For x E aX, let v(x) be the inward normal vector to aX at X, i.e., the unit vector v(x) E TTX orthogonal to TxX at x and pointing inside X. We infer the existence of special charts for X near each point x0 E 9X. For p > 0 set B+(0, p) := {x = ( x , , . .. , x.) I xI < p, x,, > o} F (O, P) :_ {x E R'-' (x', 0) E B(0, p), x' :_ (x1, ... I
I
Each point x0 E aX has then a chart cp : B+(0, po) --f X such that cp(0) = xo, : T(0, po) --> aX is a local chart for aX, and, for x' E F(0, po), Dgo(x', 0)e,,, = v(co(x', 0)) is the inward unit normal vector field to aX. We refer to these charts as to admissible charts. .
Tangential and normal traces of a form. Let us define the tangential and normal parts to an of a form. Denote by t the restriction map
r : C°° (X, APTX) --+ C°° (aX, APTX ) which associate to each w E C°° (X, APTX) its trace rw E C°O (aX, APTX), rw(x) = w(x), x E aX. As aX is a submanifold of X, the Riemannian structure on X yields a canonical decomposition of TxX in two orthogonal components TxX = Tx5X EB l v(x).
Such a decomposition then extends to p-vectors, and we can decompose every p -vector l; E APTXX in a unique way as
6 = i + 62nv,
61 E APTxax, S2 E AP-1TxaX.
Thus, give a form w E C°° (X, APTX) we can define the tangential and the normal parts of the trace of w, respectively tw and nw, as the functions in C°° (aX, APTX) defined by (1)
_
for
= 61 + 6 Av(x). Obviously we have rw = t w + n w and
(2)
IrWI2 =
ItwI2
+ InwI2
Also, if i : aX --> X is the immersion and it : X fl UEo -* aX where U60 is a neighborhood of aX is a smooth right inverse of i, then
550
5. Cartesian Currents in Riemannian Manifolds
tw(x) _ 7#ti#w(x).
(3)
In local coordinates (x', ... , X1, t) around xo E 8X we can write w E Dr (X) as w _ E1a1=P wao (x, t) dxa + E1a1=p-1 w,,,6 dxaAdt, then
t w(x, t) _ E wao (x, 0) dxa,
nw (x, t) =
wao (x, 0) dx' Adt. 1a1=P-1
IaI=P
We recall now that the restriction at the boundary f -> flan of smooth functions f defined on a Lipschitz domain .f2 extends to a continuous linear
operator from W',8(0) to LS (8Q) for s > 1. Based on that it is then easy to see using local coordinates that the trace operator r : C°° (X, APTX ) C°° (aX, APTX) extends to a continuous linear operator, the trace operator
r : Wl,'(X) -> L'(aX)
for anys>1. From (2) we then infer that the tangential and normal operators extend to linear continuous operators
t,n : WW'S(X) -* LP(X)
for all s > 1. In an oriented manifold, the associated Hodge operator * commutes tangential and normal operators. We have
Proposition 1. Let X be an oriented manifold, let * be the Hodge operator on X, and let S be the formal adjoint of d. Then we have (i) Forw E DP(X) *nw = t (*w),
*t w = n (*w),
t (dw) = d(t w),
n (bw) = 8(n w).
(ii) For wEDP (X),rlED4(X),q
which is also orientation preserving. In such a system, if w = wao dxaAdt as above, one easily computes
wao (x, t) dxa +
o(c , a)wao(x, 0) dxandt = n (*w)
*t W = Ia1=P
*nw =
Q(c , a)
O (x, 0) dxa = t (*w).
lal=P-1
Moreover, from (3) we infer
t (dw) = lr#i#dw = d(7r#i#w) = d(tw).
2.6 Relative Cohomology: Hodge-Morrey Decomposition
551
Equality nSw = Snw then follows as 6 = (-1)P("-P) * d*. Finally, again from (3) we infer t (WA * 77) = 7r*i# (wn * 77) = 7r#i#wAlr#i# * 77 = t wAt (*77) = t wA * n77.
Recall that if X is an oriented manifold, then one fixes an orientation on OX by choosing aX (x) as the unit (n - 1)-vector field orienting TXaX in such a way
that (x) = X(x) We then set for y E C°°(0X, A'-'TX) v(x)A
'Y
ax
f
for X E aX.
> dH"'
<'Y,
ax
Proposition 2. Suppose that X is oriented and let * be the associated Hodge operator. We have (i) If w E W,1,'11(X ), then fx dw = fax i#w (ii) If W E Wp,2(X) and 77 E W112 (X), then (
= fax t w.
I 77)Lp}1(X) + (w I S7I)LP(X) _ f twA * n?]. ax
(iii) If W E WP2'2 (X) and C E W 2 (X ), then (
d()L2P}1 (X) + (6w 16C)Lp_1(x)
= -(nw I OLP(y) + f {t(A *n &-,) + t Swn * n (}. ax
Proof. Since x"' LX-a.eT#W(x) =rW(x)IAPT ox and i#W(x) =tWIAPT,aX, (i) follows from a similar claim for smooth w, on account of the continuity of the trace and of the tangential part operators. To prove (ii) we observe that wA * 77 belongs to Wn' 11 (X), and d(wA * 77) = dwA * 77 + (-1)Pwnd(*77) = dwn * 77 + wA * 877.
Integrating we then get on account of (i) (dw 177)Lz + (w I S77)L2 =
d(wA * 77) = ax
f ax
J x
dwA * 77 + J wA * 677
x i# (WA * 77) = f i#wni# ex
* 77.
552
5. Cartesian Currents in Riemannian Manifolds
On the other hand i#w = t w I A,TaX, hence i#wAi# (*7)) = t wAt (*77) I APTX = twA * bn 77I ApTaX .
Finally (iv) follows at once integrating by parts, (zlw I () = (ddw I ()+(8dw I 11
We now observe that Proposition 2 extends to non orientable manifolds. In fact as any non orientable manifold X is nevertheless orientable in the small, the scalar function
(x) := ,
x E 8X
is well defined if either a E C°° (X, APTX) or 0 E Coo (X, AP+'TX) are supported in a coordinate patch U, and is independent on the local orientation on U. Therefore < a, /3 > is a well defined function in CO° (8X) even if X is non orientable.
Proposition 3. We have Wp,2 then t (dw) = d(tw), n (bw) = b(nw). (i) If W E then (ii) If w E WP(X) and 77 E
f
(dw I fl)LP+,(X)+(w I Si)L2(X) =
dHn-1
ax
(iii) If W E Wp'2(X) and r) E WP+2(X), then (dw I b?7)L2P+, (X)
=0
provided either t w = 0 or n7? = 0. (iv) If w c Wp'2(X) and 77 E Wp"2(X), then (dw I dr))LP+,(X) + (8w I drl)Lp_,(X)
_ -(nw I
77)Lp(x) +
J ax
{ + } d7-l'i-1.
Proof. (i) follows from (i) of Proposition 1 as the claim is local. (ii) and (iv) of Proposition 2 proves respectively (ii) and (iv) if we assume moreover that rl is supported in a local chart. The general cases of (ii) and (iv) then follows using a partition of unity. Finally if tw = 0 and 77j ri in W1"2, the r)j being smooth, we have by (ii)
(dwI drlj)_(dwI arli)+(wI
6277j)=
J d?i'-1=0. ax
(iii) then follows letting j tend to infinity. Similarly one proceeds if nr) = 0 by approximating w.
2.6 Relative Cohomology: Hodge-Morrey Decomposition
553
The Hodge-Morrey decomposition theorem. A key step toward a decomposition theorem similarly to the boundaryless case is a Gaffney type lemma at boundary points. We have
Lemma 1 (Gaffney). For each xo E aX there are constants c, p > 0 and an admissible chart cp : B+(0, p) --p U C X, (0) = xo, such that
f
D(w, U) >
(4)
E JDiwa12 dx - cII w IIL2(U)
B+(O,p)
i,a
whenever cp#W = E wa dxa is compactly supported in B(0, p) and we have either
tw = 0 or nw = 0 on 6X. Proof. Proceeding as in the proof of Gaffney lemma in the interior, we can bound from below the Dirichlet integral D(w, U) by the flat Dirichlet's integral Do(cc#w, B+(0 p)) getting f D(w, U) > lDiwa12 dx - c1l w 11L2(U) p)) 2
J
B+(O,p) i'a
The coercivity inequality (4) then follows at once from the following lemma.
Lemma 2. We have
Do(w, B+(0 p)) = f
IDiwal2 dx,
i'a B+(o,p)
if w = E wa dx' is compactly supported in B (0, p) and moreover either t w = 0
ornw=0 onFp. Proof. From Proposition 2 (iv) we have (5)
Do (w, B+(0, p)) + f (dW I w) = B+(O,p)
r
J
{f wA * n dw + t bwn * n w}.
rv
On the other hand in Rn the Laplacian on forms is just the Laplacian on coefficients, i.e.,
LXW = E / wa dxa, a
consequently, integrating by parts and using Gauss-Green formula
f B+(o,p)
we infer
Digi dx =
f aB+(o,p)
E(-1)x_1gi
i
dxi
V g E C'(B+(O, p)),
5. Cartesian Currents in Riemannian Manifolds
554
(6)
(aw j w) dx
r B+(O,p)
IDw12
I
+
WaWa,n dxlA...
(-1)n-1
B+(O,p)
/ndxn-1.
1p
From (5) and (6) we then get (7)
Do(w,
B+(0, P)) -
IDw12 dx
J
B+ (O,p)
= f{twA *n dw+tbwn *nw+ (-1)n r, wawa n dx1A...
Adxn-1}.
a
rp
We now claim that the right hand side of (7) vanishes if either tw = 0 or nw = 0
on I''. In fact, if tw = 0, then, in I'p we have w = nw =
wa dxa, consequently a3n
t 8w
EE
o (i, a - i)Wa i
dxa-'=(-1)P-1 I
a-i3Jn a3n i
Hence t bwn * nW =
a3n
` 1` wQwa,nu(,a, ) dxa-nAdxQ.
(-1)P-1
3 n, we have dxc
(_i)P-1
twA * nw
wan dxa-n.
a3n
Adxa 54 0 if and only if a = 0, therefore
wawa na(a, a)o(a - n, a) dx1A... a3n
(-1)P-l+n-P
ndxn-1
wawa,n dx1A ... a3n
=
(-1)n-1
E WaWa,n dX1 A ...
ndxn-1
a3n which proves the claim in this case.
Ifnw=0, then in f wehavew=tw=
wadxa, a on
n6w =
E 1: a i, a)wa,i dxa+2 = (-1)P a+i3n aln
Wa,n dxa+n
aon
and
*ndw = (-1)P
Wa,nQ(U + n, a - n) aRfn
Therefore
d?-n.
ndxn-1
2.6 Relative Cohomology: Hodge-Morrey Decomposition
555
t wA * n dw = (-1)P E E wpwa,no-(a + n, a - n) dxandx«-n. )33(n
a
n, we have dxQAdxa-n = 0 unless a
hence
twn * ndw
_ (-1)P E wawa na(a,
a)(-1)n-p-10'(a,
a - n) dx1A... Adxn-1
aon
_
(-1)n-1
EWawa,n
dxlA...
Adxn-1
a5Jn
which concludes the proof.
Gaffney's lemmas in the interior and at the boundary now yield at once
Theorem 1. Let X be a compact Riemannian manifold with boundary. There exist constants cl > 0, c2 > 0 such that D(w) > cl II w IIwp.2(X) - c2II W II2P(X)
for every w E WP,2(X) satisfying one of the boundary conditions
nw = 0 on 8X,
tw = 0 on 8X.
Set now
WPT(X) W12
(X)
HP (X) HPN (X)
{wEWP'2(X) I tw=0} :_ {w E Wp'2(X) I nw = 0} {WEWP"T(X) I dw = 0, dw=0} {W E WP, N (X)
I dw = 0, Sw = 0}.
Then, as in the boundaryless case, one proves
Theorem 2. HT and HN are finite dimensional vector spaces. Theorem 3. There exists a positive constant c such that D(w) > CII W II L2(X) for any w E WP'N(X) which is orthogonal in L2 to HN and for any w E WP'T(X) which is orthogonal in L2 to H. Given a form f E L2 (X), we can now seek a weak solution of the problem
J '6w = f in X (8)
nw=0 onBX
that is we seek a form w E WP,'N (X) such that
5. Cartesian Currents in R.iemannian Manifolds
556
(
(9)
dC)L2 + (SW I SC)L2 = (f I ()
or a weak solution of
V
C E WP,'N(X)
J aw= f inX
(10)
tw=0
on &X
equivalently w E WP 'T (X) such that (11)
(dw I d()L2 + (SW SC)L2 = (f C)LZ
V C E WP'T(X).
As in the boundaryless case, an easy consequence of Theorem 1 and Theorem 3 is the following
Theorem 4 (Hodge-Morrey). We have (i) Problem (8), equivalently equation (9), is solvable if and only if f is orthogonal in LP to HN . Moreover there is a unique solution wo E WP'N (X) n HN
of equation (9), and the linear transformation from f to w° is a bounded linear operator from LP (X) n HN into WP'N(X) n HN . (ii) Problem (10), equivalently equation (11), is solvable if and only if f is orthog-
onal in LP to H. Moreover there is a unique solution w° E WPT(X) n HT of equation (11), and the linear transformation from f to w° is a bounded linear operator from LP(X) n HT into WP'T(X) n HT . Problems (8) and (10) are easily seen to be elliptic boundary value problems, where one has zero Dirichlet data for suitable coefficients of w and zero Neumann
data for the others. Elliptic regularity theory then allows to complement the existence theorem in Theorem 4 with the following Regularity Theorem
Theorem 5. Let w in WP'N (X) or in W' 2 (X) be respectively a solution of (9) or of (11). Then we have (i) w E WP2'2(X) if f E LP(X). (ii) w E WP's(X) n CO, ` (X, APTX), a = 1 - P, if f E L; (X), s > n. (iii) w E C2,- (X, APTX) if f E C°Ia(X, APTX). (iv) Higher regularity of w follows from higher regularity of f, for instance w E C°° or w is analytic if f E C°° or is analytic. (v) In particular harmonic normal forms, i.e., w E HN, or harmonic tangential forms, w E HT, are C°° or analytic up to the boundary if X and aX are of class C°° or analytic. Define now
ImdN := {da ImSN := {Sa then we can state - ---------
a E WP'21,N(X)} ImdT aE
W1,2
N(X)} ImST
{da I a E WP'21,T(X)} {Sa I a E
---------------------------
- - - - ----_
2.6 Relative Cohomology: Hodge-Morrey Decomposition
557
Theorem 6 (Hodge-Morrey decomposition). We have (i) The spaces HN, Im dN and IM SN are mutually orthogonal and closed in
LP(X), and one has the decomposition
L2r(X) = HN ®ImdN ®ImSN, that is, every f E L2 (X) decomposes uniquely as
f =h+da+Sf
(12)
where h E HN, a E Wp 1,N(X), /3 E WP+1,N(X) and IIhIJW,,2+IIaIIw1.2+II13 IIWI,2 <_ cII f IILZ'
(ii)
The spaces HT, Im dT and Im ST are mutually orthogonal and closed in L2r (X), moreover
L22(X) = HT ®ImdT ®ImST, i.e., every f E LP2 (X) uniquely decomposes as
f =h+da+SQ
(13)
where h E HT, a E W l T(X), 3 E
and
IIhIIW1,2 +IIa1W1.2+IIQIIW1,2
Moreover in both cases h E C°° (X, APTX) and also a and Q are of class Cup to the boundary if f is of class C°° up to the boundary. Finally, SQ = 0 if
df =0. Proof. The proof follows the same path of the analogous proof of the decomposition theorem in the boundaryless case: one has only to take into account the formulas of integration by parts in Proposition 6 in Sec. 5.1.3. For the reader's convenience, we only prove (12). Let h be the projection (in L22) of f into HN and let wo E Wp'N (X) n HN be the unique solution of (14)
(dw I d()L2 + (SW I SS)L2 = (f I ()L2
V
E Wp'N(X )
which exists by Theorem 4. By the regularity theorem wo actually belongs to Wp'2, therefore (iv) of Proposition 3 yields (15)
(SW +J{
(dw I dC)L2
+
rr (QW I S)L2
+ } dl-C-1
ax
for all C E WP'2(X). From (14), (15) we infer by taking C E C°°(X, APTX) that
5. Cartesian Currents in Riemannian Manifolds
558
Awo=f - h i.e.,
f = h + da + 60 if a := dwo, /3 := Swo E WP+1 (X) . As n a = n Swo = Sn wo = 0, we see that a E Wp'21,N(X). From (14), (15) we also infer where a E WW_21(X ),
J ax
< t C, n dwo > dx" = 0
V C E Wp,'N (X ),
and, being t( arbitrary on X, that n dwo = 0, hence no = it dwo = 0, i.e. 1,2
E WP+1,N(X)-
Remark 1. It is worth remarking that, since Hodge operator transforms pharmonic forms into (n - p)-harmonic forms and
*nw=t(*w)
*tw=n(*w),
actually on oriented manifolds Theorem 4 (ii) follows from Theorem 4 (i) and Theorem 6 (ii) follows from Theorem 6 (i). Remark 2. One might consider other boundary value problems for the LaplaceBeltrami operator and consequently infer other decomposition theorems. However we shall not pursue this topic any further. For k = 0, ... , n, define
Zk(X, aX) :_ {w E Co(X, AkTX) I i#w = 0, dw = 0} to be the vector space of relatively closed k-forms, i.e. the space of closed k-forms which vanish on aQ, and
Bk(X,OX) :_ {w = da I a E CO(X,.AkTX), i#a = 0, or w = 0 if k = 0}. the vector space of relatively exact k-forms. The relative de Rham cohomology groups are then defined by
HHR(X, aX) := Zk(X, aX)1Bk(X, aX). An immediate consequence of Theorem 6 is
Corollary 1. We have (i) Each cohomology class [w] E HHR(X) contains exactly one harmonic form h
with n h = 0. In particular HHR (X) is finite dimensional. (ii) Each relative cohomology class [w] E HHR(X, 9X) contains exactly one harmonic form h with t h = 0. In particular HHR (X, OX) is finite dimensional.
2.7 Weitzenbock Formula
559
2.7 Weitzenbock Formula In this section we derive a more intrinsic expression for the Laplace Beltrami operator on a Riemannian manifold X in terms of the covariant derivatives and of the curvature of X. Denote by r(TX) the smooth vector fields on X and if p : E -* X is a fiber bundle on X, denote by r(E) the space of sections over E. A connection V on E is an operator
denoted by Vxt; for X E V(TX) and
E r(E) which satisfies the following
properties
(i) V X t; is tensorial in X i.e.
Vx+Y = OX6+VYe,
OfXe = fvXe
for x, y E r(TX ), 6 E r(E), f E C°° (X ). (ii) Vx6 is linear in iii) Vxl; satisfies the following chains rule
fVxe
Vx(fe) =
(1)
for X E r(TX), f E C°°(X), 6 E r(E). Notice that (i) implies that Vxl; at a point x E M depends only on the tangent field X at point x and moreover that V can be also seen as an operator
V : r(E) -* r(E) ® r(A'TX). Moreover, (iii) implies that Vxe(x) depends only on t= near x. Given a connection V on a fiber bundle E one defines a connection V * on the dual bundle E* by the formula
w> _
(2)
for e E r(E), w E r(E*), X E r(TX) and also on all exterior algebra of E. In fact sections
E r(11pE) can be represented locally as
6=
S°(x)eai
(x)A...
Ae%(x)
aEI(p,n)
where a c Coo (X) and eat, ... , eaP are smooth sections in E. Consequently, by setting (3)
Vx f (x) :_ ,
(4)
Vx(t 1A... Abp)
P
i=1
f E C°°(X)
S1^ ... Si-1^OX6ti^Ci+1A ... Al p
560
5. Cartesian Currents in Riemannian Manifolds
E N(E) we define an operator
for
v: N(APE) ®T(TX) --> T(ApE)
which is easily seen to be a connection. Starting with V* on N(E*) a similar construction produces a connection on APE*
APE*
P
Vx(w'A... AW) :_ Ew1A... Awz-1AVxw'Awx+'A... nwP,
(5)
i=1
Proposition 1. Let 6 E N(APE), o E T(AgE), w E N(APE), 77 E N(APE). We have (6) (7)
OxeAo + eAV x o Vx(wAr7) =Vxwnr7+WAVxr7
(8)
<w, Vxe> + = , X >.
Vx
Also, if p < q
Vx(e-J 77) _ CJ Vx?7+Vxl;J 17.
(9)
Proof. Let e1i... , e, be a local basis of sections near xo and fl, ... , fn be the dual basis in N(E*) so that < fi (x), ej (x) > = Jij
near x0.
If a E I (p, n), 0 E I (q, n), from (4) we infer Vx (e,,,Aep) = Vxe,,,Ae,3 + e,AVxeo hence (6) follows. Analogously (7) follows from (5). To prove (8) notice that
V*fa(ea) = f«(Vxeo) = 0 if a
,3
while for a = Q P
Oxfa(eo)+f, (Vxe«)
{Oxfck i(eai +fai(Vxe«i)} i=1
,X>=0. Let us prove (9). Using (6) and (8), for o E T(Aq_pE)
V_ 77)(o,) = -eJ 77(V xo) + < d<77, one >, X > = -77(OxoAe) +
X>
= -77(Vx(o-ne))+,X>+
_ C-J V*77(o) + Vxe-J 77(C)
2.7 Weitzenbock Formula
561
A basic result on Riemannian geometry is that there is only one connection V on TX, the Levi-Civita connection , which satisfies the following additional requirements (1) V is metric, i.e.
(VxYIZ)+(YIVxZ) _ VX,Y,ZET(TX) (10)
(2) V is torsion free, i. e. for X, Y E T (T, X) (11)
T(x, y) =VxY-VyX - [X, Y] - 0
being [X, Y] the Lie bracket of X and Y. In this case, d
[X, Y] := X3
aXj
ei - Yj axi e2
where X := Xiei, Y := Yiei.
One can compute V in terms of the metric on T (TX ), and one has
(VxxlZ) = 2 {x(X Z) - Z(XJX) + x(XIZ) (12)
-(XI[X,Z])+(ZI[X,Y])+(XI[Z,X])}.
In local coordinates, denoting by Gij the metric tensor of X and by e1, ... , en a local basis for I'(TX), we have n
(13)
>rkj ek
Deyej
k=1
where {T kj} are the Christoffel symbols of the connection in the given basis; (12) is then equivalent to (14)
T = 2Gk1{Gil,j + Gj(,i -
Gij,1}.
The torsion free condition (11) implies
T =rk
di,j,k.
Proposition 2. Let X be a Riemannian manifold. Denote by V the Levi-Civita connection in X and also its extension to the algebra of p-vector fields and forms defined on an open set U c X and let dx...... dxn be the dual frame of p-forms. Then dX AVej W in U. (15) dW If moreover ej are orthonormal at xo E U then at xo (16)
dW = >ejJ Ve, w.
j
5. Cartesian Currents in Riemannian Manifolds
562
Proof. The idea is to prove the invariance of both sides in (15) and (16) with respect to changes of coordinates and then proves (15) and (16) in a suitable frame.
Let us prove (15). First we note the the expression
E dxj AVejw
j is independent of the choice of local frame In fact if fl,... , f., is another frame field, we have f j = A ek for some A E COO (U, ]R). For the corresponding dual frame f 1, ... , f' we then have f j = Bkdxk with B := A-1. Consequently
f3AOfjw =
j
8kdxkAVe=w = dxiAV,,w.
w= j,k,i
i,k
i
Since d is also independent on the frame field, it suffices to check (15) for a special frame field. We select around x0, normal coordinates for which we have Gij(xo) = 5ij, Gij,l (xo) = 0 V i, j,1. Hence for the coordinate frame fields ei,... , en, we have
Veiej(xo) = f (xo)ek(xo) = 0 from (13) (14). Thus, if w = codxa, a E I(p, n), f E C°O(X), we obtain at xo
EdxiAvej(cpdxa) _
dxa = d(c'dxa).
dxjA 9
j
To prove (16) we observe that the right hand side is independent of the chosen of
the normal frame at x0. In fact if fl, ... , fn is another orthonormal frame field, f j = A3 ek, A3 being an orthogonal matrix, A A3 = 5ik. Then we have
eiJ V.
Ve;w =
fi-J V f,w =
j
j,k,i
i
Again we chose normal coordinates at xo. The coordinate orthogonal at x0 and V ej = 0. Consequently
ejj Vet (pdxa) =
(17)
j
en) frame is
a e J dxa.
8xj
On the other hand from (1) in Sec. 5.2.4, taking into account that Gij(xo) _ Gti(xo) = V, Gij,a(xo) = 0 we have at x0 1:(5(Wdxa))7 dxl =
6(Wdxa)
(18)
_
a (i,'y) y+i=a
&p
(xi
aacpxe
dxa = iEa
o(i, a) dxa-i
2.7 Weitzenbock Formula
563
and the claim follows comparing (17) (18) since
ifi0a
0
ej i dxa = iS
o-(i, a)
dxa-2
if i E a. 71
In order to express the Laplace-Beltrami operator in terms of covariant derivatives, we introduce the second covariant derivative defined for w E F(APX) and X, Y E TX by
OXYw :_ VX(VyW) -Vvxyw and the curvature tensor associated with a connection (19)
(20)
R(X, Y) :_ VxVy -VyVX : APTX -4 APTX. is tensorial in X, Y, i.e. that OXY is linear in X
It is easy to check that and Y and for all xo E Y
oXY(W)(xo)
depends only on the values X (xo ), Y(xo) of the vector fields X and Y at xo. In fact, for f, g E C°°, X, Y smooth vector fields,
VfX(V9yw) = fVyw+fgVX(VYw) and
VVfx(9Y)W = V fX+f90xY = fVyw + fgVvxyw so
v2fX,YW = f(x)g(x)VXYW. We need also the following algebraic lemma.
(21)
Lemma 1. Let e1,. .. , en E Al V, dx', ... , dx" the dual basis in A'V, w E APV. Then (22)
> {eiJ dxjAw +dxjA(eiJ w)} = 0
(23)
eiJ dxinw + dx1A(eiJ w) = w
ij
Proof. Assume first i
V i = 1, ... , n.
j if j E a, if j a, i 0 a 0 -i if j a, i E a. o, (j, a) o, (i, a + j - i) dxa+j
10
eiJ (dxjndxa) =
ifi a,
10 dx3A(eiJdxa)
ifiEa, jEa
0
v(i, a - i)o,(j, a - i)
dxa+j-z
if j
a, i E a.
5. Cartesian Currents in Riemannian Manifolds
564
a + j - i) = -o (i, a - i)o-(j, a - i), compare (5) in Sec. 5.2.4, (6) As v(j, in Sec. 5.2.4, we see that eiJ (dx3Adxa) + dx3A(eii dxa) = 0
`d a E I(p, n), consequently (22) is proved. To prove (23) it suffices to compute
ifiEa,
(0
eii dx"Adxa = {
dxa
if i
dxineiJ dxa =
a,
r dx' ifiEa
Sl
0
if i
,
c
,
thus
eiJ dx2Aw + dxiheiJ w = w.
Theorem 1 (Wietzenbock formula). Let X be a Riemannian manifold and let V be the Levi-Civita connection on forms. If el, ... , en is a local frame on X which is orthonormal at xo and dxl,... , dxn the dual frame on T*X, then
Qw = Deieiw + dxzA(ejl R(ei, ej)w)
(24)
Proof. By definition, Aw is intrinsic to the Riemannian manifold and depends on the Riemannian structure on X. Also we check that both terms in the right hand side of (24) are intrinsic. In fact, if fl, . . . , f, is another orthonormal frame field, then fj = A ek, A (xo) being orthogonal. Thus (21) implies Vfifiw = 1: AkAt Vekehw =
Dehehw. h
j,h,k
i
Also if fj is the dual basis of fj, fj = Bkdxk, B
R(fi, fj) _
A-1
AkA; R(ek, eh) h,k
fj i R(fi, fj) i
hkj,s
AkehJ R(ek, eh)
AsAi Aj esJ R(ek, eh) k,h
Therefore
ftin(fjJ R(fi, f3)) _
BsAkdxsn(ehJ R(ek, eh)) h,k,s,i
dxkn(ehJ R(ek, eh)). h,k
Consequently it suffices to prove (24) for a special frame. Choose normal coordinates around xo, and dual coordinate frames e1, ... , en, and dxl,... , dxn. At xo we have Veiej(xo) = 0, therefore Vei(dxj)(xo) = 0, Deieyw(xo) = VeiVeiw(xo) and R(ei,ej) =Ve;,Vej - VejVei
2.8 Poincare and Poincare-Lefschetz Dualities in Cohomology
565
Consequently, using Proposition 2 at xo (25)
dxin0ei(ejJ Vejw)
dSw =
dxin(ejJ DeyVejw)
i,j
i,j
and also using Lemma 1
Sdw = EejJ Vej(dx'AVeiw) i, j
E ejJ (dx'AVejVeiw)
- 1: dxin(ejJ Dej Veiw) i=j
i,j n
(26)
+ejJ (dxinvejVei.w)+dx2A(eiJ DeiVeiw)} i=1
E dx2A(ej-i Vej Ve, w) +
Dei Vei
i, j
Adding (25) and (26) we get the claim.
11
Remark 1. If p = 0, i.e. if w = f E C°°(X) then R(ei,ej)f = fR(ei,ej)1 = 0 because of the tensorial property of R, consequently for functions
If =Eoeieyf. 2.8 Poincare and Poincare-Lefschetz Dualities in Cohomology Compact manifolds without boundary. Let X be a compact manifold without boundary. Recalling Hodge's theorem Theorem 6 in Sec. 5.2.5 we first notice that Hodge operator * induces naturally an isomorphism between the homology classes HHR (X) and HHR' (X ), in fact since
*w=*,Aw we see that *w is harmonic if w is harmonic, hence the map * : w ---* *w defines an isomorphism
* : HaR(X) --> HHRP(X). However such an isomorphism depends not only on the orientation of X, but on the metric. An algebraic isomorphism independent from the metric between HHR (X) and the dual of H(X) is obtained as follows. Given two closed forms w E W1'2 (X, APTX) and 77 E W"2 (X, An-PTX) it is readily seen that
<w,77> :=
xJ
wnr7
does not change if we replace w and 77 by cohomologous forms, hence < , > defines a bilinear map over HHR (X) x H(X)
5. Cartesian Currents in Riemannian Manifolds
566
Poinc<, >: HaR(X) x HHRP(X) --> J R. We have
Theorem 1 (Poincare duality). The bilinear form Poinc<, > is non degenerate, that is for every non zero cohomology class [w] there is a non zero class [77] such that Poinc< [w], [77] > 54 0,
consequently HHR(X) is isomorphic to the dual (HaRP(X))* of HdRP(X).
Proof. For each non zero cohomology class that we may assume represented by an harmonic form w we see that *w represents a cohomology class in H.R P (X ) for which
J
wA*w=(w,w) #0.
The second part of the claim follows at once from Lemma 1 below as de Rham's cohomology classes are finite dimensional vector spaces.
Lemma 1. Let E, F be Hilbert spaces and let a(.,) : E x F -+ R be a bilinear map satisfying (i) Ia(u,v)I 0 such that a(uo, v) > co I v I V v E F
(iii) a(u,v)=0VvEF implies u = 0 Then F ^ E*. Proof. Consider the linear map T : F -> E* defined by TT(u) := a(u, v). By (i) E- < k v IIF; also, by (ii)
T is continuous and T
IIVIIF <
IITV IIE
11uo
It follows that T has closed range, is invertible, and T-1 is continuous. It remains
to show that range T = E*. Assume not, then there is 0 E E*, 0 0 0 such that coo E E we then (OITv)E- = 0 V v E F. Representing 0 as 0(u) = (ulcpo), find 0 = (OIT,,)E` = Tv('po) = a('po,v)
(iii) then implies coo = 0, i.e.,
b v E F,
= 0: a contradiction.
Notice that in particular Theorem 1 yields
Proposition 1. Let w be a closed form in X. If fX wnrl = 0 for all rl with d77 = 0, then w is exact.
2.8 Poincare and Poincare-Lefschetz Dualities in Cohomology
567
Compact manifolds with boundary. Given a smooth compact Riemannian manifold X with boundary, we may define, as we have seen, three sets of cohomology groups depending on k = 0, ... , n: The de Rham cohomology groups k HaR(X) _ closed forms of degree k in X
exact forms of degree k in X '
no boundary condition is imposed; for w E Zk(X), we denote by [w]x the cohomology class of w,
The de Rham cohomology groups of the boundary aX k HaR(aX) _ closed forms of degree k in aX
exact forms of degree k in aX
for w E Zk(8X), we denote by [w]ax the cohomology class of w in HHR(aX),
The relative cohomology group of relative closed forms modulo relative exact forms i.e. HkdR (X
ax)
closed forms of degree k in X which are null on OX exact forms of degree k in X which are null on OX
that we have already defined in Sec. 5.2.6. The relative cohomology class of w E Zk (X, aX) in HHR (X, aX) is denoted by [w] rel.
The inclusion map i : OX -f X clearly induces a map i# : HHR(X) HHR(OX ). As forms in Zk (X, OX) and in Bk (X, DX) are just special closed and exact forms, it is easily seen that an inclusion map j* : HHR (X, OX) -# HR(X) is well defined by 7#([w]rel) = [w]x. Finally, one can define a coboundary operator b : HHR(aX) -, Hk+1(aX) by b([w]eX) _ [dw]rel
where ID is an extension of w in X. In fact, if w extends w to X, we have i#(di.-5) = d(i#w) = dw = 0
on aX,
i.e., dW E Zk+1(X, aX ); moreover, if w is another extension of w, then
and i#(w-w)=0, i.e., [dD]rel is independent of the choice of the extension w of w. Also, if w E Bk (aX ), w = da, a E Dk+l (aX ), and a E Dk+1(X) is an extension of a, we have dw = d(w - da)
and i#(w - da) = w - w = 0.
5. Cartesian Currents in Riemannian Manifolds
568
Recalling now that a sequence of vector spaces
+ Vi --f Vi+l - .. .
Vi-1
is said to be exact if for all i the kernel of fi is equal to the image of its predecessor
fi-1
ker fi = Im fi- 1,
we can state
Theorem 2. The long sequence
... ?#
HaR(X)
2#' HHR(aX) a ' HHR1 (X, aX)
'- Hk+1(X)
is exact.
Proof. For the reader's convenience we give the simple proof. (i) First we show that the composition of two successive maps gives zero. Let [w]re1 E HaR(X, aX). We have 2#(7#([w]re1))
= i#(1w)x) = [i#W]X = 0
as i#w = 0. Let [w]x E Hk(X). We have b(i#[w]x) = 6([i#w]ax) _ [di#w]re1 = [di#w]rel = [2#dw]re1 = 0.
Let [w]ax E Hk(aX). We have j#(8([w]ax)) = j#([dw]rel) = [dw]X = 0 as dw is exact in X (though not relatively exact). (ii) Suppose that i# ([w] x) = 0, we need to construct c:i so that j# ([W]re1) = [LO]X.
From the assumption [i#w]ax = 0, i.e., i#w is exact in OX, i#w = dcp in OX. We extend cp to cp in X, define W = w - cp, and verify ill (Lo
- dip) = 0,
j#([w]re1] = [W -
x = [w].
(iii) Suppose that [w]ax E Hk(DX) and 8([w]ax) = 0. Extend w tow in X, i#iv = w; since [dZ]re] = a([w]ax) = 0, dw is relatively exact, i.e., div = dcp with i#cp = 0. If we define = w - cp we have
dw=di3-dcp=0 i#[W]x = [i#w]ax =
[i#cp]ax = [w]ax.
(iv) Finally, let [w]rel E HHR(X,9X) and j#[W]rel = [w]x = 0, W being relatively
closed, du) = 0 and i#w = 0. Then w is exact in X, w = dep, and [i#coiax E Hk-1(aX), as di#cc = i#w = 0, 8(i#cP]ax) = [dcP]rei = [w]rel
2.8 Poincare and Poincare-Lefschetz Dualities in Cohomology
569
Returning to harmonic forms, we recall Remark 1 in Sec. 5.2.6. i.e., that Hodge * operator yields an isomorphism *:xN-'HTk
we easily infer that HaR(X) and HaR k(X, aX) are isomorphic. However the duality given by
* : HdR(X) - HHRk(X,aX) depends on the metric (and on the orientation) of X. On an oriented manifold, an algebraic isomorphism between HdkR(X) and the dual of HHRk(X,aX) is instead given by the map (w, 71) - JwAm
x It is in fact easily seen that fx (w + d17)A(C + do,) = f x wA if w E Zk (X), ii E Dk-1(X), e E Zn'-k (X, aX), and o, E Bn-k-1(X, aX). Therefore such a map induces a bilinear pairing Poinc< [w], [c] > : HaR(X) X HaR k (X, aX) -+ R
between the k-cohomology and the relative (n - k)-cohomology. We have
Theorem 3 (Poincare-Lefschetz duality). The pairing
Poinc<[],[]>:HHR(X) xHaRk(X,aX) _+R given by Poinc< [w], [l;] > :=
J x
wA
- HaR(X,oX)
'-
is non degenerate. Moreover the diagram i#
HHR(X)
'+
x Poinc b
HHRk(X,aX)
HHR'(OX)
x PoincT
'#
x Poinc i#
HHRk(X)
...
HHRk(aX)
b ...
is commutative. PoincT denotes the transposed duality, PoincT< 1771 r'!1' [w] > Poinc< [W], 1771 ]rel >
The commutativity of Al
xF1 A2
B1
xF2 B2
means F2(f(a,),b2) = F1(a,,g(b2)), Val E A1, Vb2 E B2.
570
5. Cartesian Currents in Riemannian Manifolds
Proof. If [w] is a non trivial cohomology class we can represent it by a non-zero harmonic form h with n h = 0. The form *h is harmonic and t (*h) _ *n h = 0,
consequently *h represents a relative cohomology class [*h] E H ' (X,BX), and we have
r
f
Poinc<[w],[*h]>=J hA*h=J Ih12#0. x
x
Similarly one proves that, if [S]rel ; 0, we can find [w] in HHR(X) such that Poinc< [w], 0. The commutativity of the diagram is trivial.
3 Currents and Real Homology of Compact Manifolds Let X be an oriented compact smooth submanifold of dimension n in Rn+N without boundary. We introduce in Sec. 5.3.2 for k := 0, 1.... the k-homology group Hk (X, R) of a X setting Zk(X) Bk(X)
{TENk(U) I sptTCX, 8T=0} {8S I S E Nk+l(U), spt S C X}
and
Hk(X,R) := Zk(X)/Bk(X) where U is a tubular neighbourhood of X, i.e., compare Sec. 5.3.1, as normal cycles in X modulo boundaries of normal currents on X. Two sets of pairings are naturally associated to homology and cohomology. One is the set of de Rham pairings between currents and forms which factors to homology and cohomology
<,>:Hk(X,lI)xHaR(X) defined by < [T], [w] > := T (w). The others are the pairings (w,71) E Dk(X) x
Dn-k(X) _
fwA7]
x which, by Stokes theorem, induce Poincare pairings in cohomology
Poinc<, >: HHR(X) X HHRk(X) -> R given by Poinc < [w], [ri] > := f wnr7. Classical theorems, respectively de Rham's
and Poincare's duality theorems, state in fact that de Rham and Poincare pairings are non degenerate; consequently there are isomorphisms, Poincare duality isomorphisms P : HHRk(X) -' Hk(X,
3 Currents and Real Homology of Compact Manifolds
571
between HHR k (X) and Hk (X, R). In Sec. 5.3.2 we show that the Poincare' duality
isomorphism is actually induced by the map again denoted by P, P : Z,-k(X) -a Zk (X) given by w -+ fX Aw. Moreover, by regularizing a k-cycle T E Zk (X) Dn+N-k(U), and integrating such a we construct an (n + N - k)-form w E form on the fibers of the projection map -7r, we construct a so-called Poincare
dual of T, PT E Dn-k(X). The cohomology class of PT turns out to depend only on the homology class of T, therefore T --# PT yields an explicit way to describe surjectivity of P. Non degeneracy of Poincare duality, which we already proved in Sec. 5.2.8 as consequence of Hodge decomposition theorem, then yields injectivity of the Poincare' duality isomorphism and also yields non-degeneracy of de Rham pairing, as < [T], [w] > = Poinc < [w], [PT] > =
fAPT,
x compare Sec. 5.3.2. We then deduce the weak closure of the real homology classes, hence in particular the existence of mass minimizing real cycles in each homology class.
A completely parallel theory holds in the case of compact manifolds with boundary. One has to introduce the relative homology groups, compare (1) in Sec. 5.3.3. The Poincare-Lefschetz duality isomorphism this time consists of three sets of maps
P# :HdRk(X,BX) -> Hk(X,lR) P' : HHR -k (X) -> Hk (X, 8X, IR)
P
:Hank-1(8X)
- Hk (BX, IR)
which actually give a morphism between the long exact sequences in cohomology and homology, compare (9) in Sec. 5.3.3. The duality forms-currents _ T(w) induces three sets of de Rham pairings at the homological level
<,>a:Hk(X,IR)xH,R(X) ---+
IR
< , >' : Hk (X, 8X, IR) x HdR(X, 8X) -> 1R < , > : Kk (BX, R) x HdR (8X) --+ 1R
which are related each other by the commutative diagram in de Rham's theorem, Theorem 5 in Sec. 5.3.3. As consequence we deduce the closure of absolute and relative homology classes and consequently the existence of mass minimizing cycles in each homology class. Still another set of pairings is relevant in homology: these are the so-called intersection indices of cycles of complementary dimension.
In Sec. 5.3.4 we shall define Kronecker's index for two'arbitrary currents S E Nk(X) and T E N,,-k(X) for which spt 8S f1 spt T = spt 8T f1 spt S = 0 and see that it induces a pairing Hk(X,1R) X Hn_k(X,8X,1R) -> R.
572
5. Cartesian Currents in Riemannian Manifolds
It turns out that
ix(S,T) = J Ps APT,
x therefore non degeneracy of Poincare duality is equivalent to non-degeneracy of the intersection index.
3.1 Currents on Manifolds Let X be a smooth manifold of dimension m. As already mentioned, by Whitney embedding theorem we may and do assume that X is a smooth (CO°) subman-
ifold of some l1P', X C R'. Moreover we can assume that X has a tubular neighbourhood U D X, U C Rn, and there is a C°° retraction map 7r : U -> X, rr(U)=X,ir(y)=yVyEX.
Definition 1. A k-dimensional current T E Dk(U) is said to be a current in X if T(w) = 0 for any null form w E Dk(U) to X. The class of k-dimensional currents in X is denoted by Dk(X). From the characterization of null forms to X, compare the end of Sec. 5.2.1, we easily infer
Proposition 1. Let T E Vk(U). Then T E Dk(X) if and only if for some E C°°(U) with ( = 1 on X we have T = 7r# (T we have
T=7r#(TLc)
Actually, if T E Dk(X),
b'(EC°°(U), (=1 on X.
Proof. In fact for any w E Dk(U) the form w - (7r#w E Dk(U) is a null form
to X, hence T(w) - T((ir#w) = 0. Conversely if q is a null form to X, then 7r#w = 0, hence T(77) = a# (T L () (y) = T ((7r#77)=0.
Remark 1. Proposition 1 extends to the case of Ck manifolds, k > 1, provided T is a vector valued distribution of order -k. In this case one in fact can assume that the retraction map is Ck hence 7r#(TLC) is well defined. In particular Proposition 1 holds true for flat chains on C' manifolds. Trivially sptT C X if T E Dk (X) . In fact, if w is supported outside X and
w is a null form to X, we have T(w) = 0. But in general it is not true that spt T C X implies T E Dk(X). Consider for instance the 1-dimensional current in R2 integration of 1-forms along the real axis el against the normal vector e2 L(w) :=
J
<w, e2 > dxl.
Clearly spt L C l[8xi, while for example on the null form W(x1) dx2 we have T (,p(xl)dx2) =
f(x1)dx1.
Notice that M(T) < oo, while M(8T) = +oo, compare 10 in Sec. 2.2.4. However we have
3.1 Currents on Manifolds
573
Theorem 1 (Flatness theorem). Let X be a closed submanifold of Pn. If T is a flat chain in U, T E Fk(U), and sptT C X, then T E Dk(X). Proof. Theorem 1 is a consequence of Proposition 7 in Sec. 5.1.3. In fact the X and the identity map id : U -+ U trivially agree on retraction map 7r : U
X, hence on spt T. Proposition 7 in Sec. 5.1.3 then yields T = 'r# (T L () V ( E C°°(U), C = 1 on X, and the conclusion follows from Proposition 1. By Theorem 1 normal currents on a submanifold X on Rn may be identified to currents in the ambient space IRn with support in X. We shall now see that a constancy theorem holds also for currents on submanifolds. Let X be an m-dimensional C°° submanifold of RTh with locally finite area. Suppose that X is oriented by a smooth rn-vector field X. Then the current
Q X := T(X,1,..) is a locally m-rectifiable current in R'.
Theorem 2 (Constancy theorem). Let .fl be an open set in ][MTh such that .flflX is connected and letT E D,(X), m = dim X, be such that sptaTfl,fl = 0. Then there exists a constant c E IR such that T= c Q X
in D'(Q f1 U)
U being a tubular neighbourhood of X. Moreover c is an integer if T is i.m. rectifiable.
X be the retraction map, and let ( E C°° (U), = 1 Proof. As usual let -7r : U ,(l2 C D on X. We cover .fl fl X by a system of coordinates Vi : V -> Di f1 X
so that cj#U = X; the maps cpi have inverses, and more precisely there are smooth maps z/ii : U fl it-' (.Qi fl X) -> V which are onto and such that W o Oi = ir,
Oi o wi = id,.
Therefore the currents Wi# (T L () belong to D,,,, (V) and, as spt aT fl .fl = 0, we
have fti# (T L () = 0 on Vi. It follows from the constancy theorem in the flat case, Theorem 1 in Sec. 4.3.1 that
'bi#(TL() =G;QV on V, hence
T = it#(T LO) =Wi#4Ji#(T L() = ciEX
on ,fli fl U,
since cpi#U = X. As the .(li's overlap on X, we infer that the constant ci are all equal to a c independent of the chart T = c QX ]J
on Dm(.fli fl u),
and the conclusion follows by means of a decomposition of unity associated to the covering {.Qi}.
5. Cartesian Currents in Riemannian Manifolds
574
Remark 2. Theorem 1 holds also for flat chains on Cl manifolds, compare Remark 1. More generally we have
Theorem 3 (Constancy theorem). Let E be an r-dimensional oriented, connected and embedded submanifold of U of class C', where U is an open set in ]Rn, and let T E F;°` (U). If (spt T) \Z is closed relative to U and spt 8T C U\E, then there exists a real number p such that
spt(T-pjZ]) CU\E. Moreover p is an integer if T is locally integer rectifiable.
If f : X --+ Y is a Lipschitz map between C' manifolds, every T E Nk(X) has a Lipschitz image f#T E Nk(Y). Also if fo, f, : X - Y are two maps and h : [0,1] x X - Y is an homotopy map between fo and fl, h(0, x) = fo(x), h(1, x) = f, (x), then the homotopy formula holds (1)
fl#T- fo#T=Bh#(Q0,11 xT)+h#(Q0,1J x8T)
for T E Nk (X) . Also a corresponding homotopy formula for forms holds (2)
fl#w - fo#w = d[h#(w)](o,i) + [h#(dw)] (o,l),
[i7] (o,l) being the integration along the fibers of the components of rl which are not
tangential to X. Also the standard estimates for homotopies hold true, compare Sec. 2.2.3.
3.2 Manifolds Without Boundary: Poincare and de Rham Dualities By Poincare duality isomorphism one usually refers to the duality between the k-homology group and the (n - k)-de Rham cohomology group of an oriented compact smooth manifold without boundary. In this subsection we shall describe such an isomorphism in terms of currents and forms.
Poincare duality isomorphism on open sets. Let U be an open set in R". For 0 < k < n consider the vector spaces of currents
Zk(U, R) = {T E Dk(U) 18T = 0, M(T) < +oo} Bk(U,R) = {8S I S E Dk+l(U), M(S) <+oc} and set Hk(U, R) = Zk(U, R)lBk(U, R) Analogously we define de Rham's cohomology groups in U by HdR(U, R) := Zk(U, R)/Bk(U,
3.2 Poincare and de Rham Dualities
575
where
Zk(U,R) _ {w E L1(U,AkRn) n£k(U) I dw = 0} Bk(U,1R) _ {dr) 177 E L' (U, AkRn) n £k-1(U)}
Then we have
Theorem 1.
HaRk(U,R) and Hk(U,R) are isomorphic.
To prove this we consider the map P : Dk(U) I M(T) < +oo} given by
L1(U,An-kRn)
p(W) :_ (-1)k(n-k)Rn LW = As 0(1Rn L W) _
(-1)k-1Wn
J
n £n-k(U) , {T E
Au.
L dw the map P factorizes to a map still denoted by
P P : H a R k (U, R) --+ H k (U,1 [ 8) .
We shall prove that P is an isomorphism, indeed the isomorphism of Theorem 1. Of course this amount to prove
Proposition 1. We have (2)
Let T be a cycle of finite mass, aT = 0. Then there is a (n - k) form PT E An-kR2) n £n-k(U) such that for some S E Dk+l(U), M(S) < +oo L' (U, we have
T= 1 APT+aS.
(1)
UU
(ii) Let w E L'(U, An-kRn) n £n-k (U) be such that 1R L W = aS for some S E Dk+1(U), M(S) < +oo. Then w = dr7, 77 E L1 (U, An-k-lRn) n £n-k-1(U).
Proof. We prove both claims by regularizing T. (i) is in fact already proved in (-1)k(n-k)f. Proposition 7 in Sec. 5.1.2, by taking PT :_ To prove (ii), we choose a convolution kernel p and a regularized distance
d. For0<E
Wi L wE = (Rn L W)E = (aS)E = as'.
On the other hand by Proposition 6 in Sec. 5.1.2 (3)
SE = IEBn L 17e
i 6, E L'(U,
An-k-'Rn) n £n-k-1(U)
From (2) and (3) we conclude that wE = (_l)n-kdr)E. As w6 and w are also cohomologous to a form L1(U,
An-k-lRn) n £n-k-1(U) the claim follows.
576
5. Cartesian Currents in Riemannian Manifolds
Remark 1. We observe that the same reasoning of Proposition 1 implies the following:
(i) if the current T in (i) of Proposition 1 has compact support in U, then the decomposition (1) holds with spt PT and spt S arbitrarily close to spt T; (ii) if the form w and the current S in (ii) of Proposition 1 have compact support in U, then one finds a potential for w with support arbitrarily to spt w U spt S.
(iii) In particular we infer that de Rham's (n - k)-cohomology group of smooth forms with compact support is isomorphic to the k-homology group of currents with compact supports {T E Dk(U) I OT = 0, sptT CC U}/{ 8S I S E Dk+1(U)}.
From now on we shall refer to the map P : Hd k (U, R) - Hk (U, ]R) as to the Poincare duality isomorphism and to (n - k)-forms PT satisfying (1) as to Poincare's dual forms of T. Poincare's dual forms PT, T E Zk(U) are constructed mollifying T, compare Sec. 5.1.2, in particular there are not unique. However they preserve some of the properties of T, e.g., PT E L' (U, do-kin) if
M(T) < oo, sptPT CC U if sptT CC U. Also (ii) of Proposition 1 states that the cohomology class of any Poincare dual form PT E Zn-k (U) of a given cycle T E Zk(U) is uniquely defined by the homology class of T. Let X be a compact oriented and boundaryless submanifold in U of dimension k. A form w E Zn-k(U) with compact support in U such that Q X (7l) =
f7lAw U
for all closed forms ?] E Zk (U) is classically referred as to a Poincare's dual form of X. In principle therefore our notion of Poincare dual form is slightly stronger,
as we require equality (1). Actually this is not the case, in fact one can show that every classical Poincare form is actually a Poincare form in our sense. This amounts to show that for every (n - k)-form w for which (4)
R' Lw(a) = fwAa = 0
V a E Dk(U), da = 0
U
(in particular w is closed) we can find a current S E Dk+1 (U) such that 1R L w = 8S. This is equivalent to say that the linear functional S' : Bk+1(U) --+ Ifs (5)
S'(,(3) :=
J
was
if 3 =
doe
extends as a linear continuous functional on Dk+1(U). We have in fact
Proposition 2. Let U be a smooth bounded open set in Rn and let w be a form LP(U,An-kJn) n £n-k(U), p > 1, such that in
3.2 Poincare and de Rham Dualities
fwAa=0
577
b a E Dk(U) with da = 0,
U
in particular dw = 0. Then there exists S E Dk+1 (U) such that 11 n L W = 3S. Proof. First assume p > 1, and let w = >2 wa (x)dxa. We solve the problem
dQa = wa O6
.Qa E Wo'p(U).
By the elliptic regularity we find a unique .fla which actually belongs to Coo (U) n W2,P(U). Consequently the form (1 := >2 Qa dxa solves
d8ffl + Sdfl = w,
compare (7) in Sec. 5.2.4, 8 being the adjoint of d with respect to zero values on W. Since dw = 0, we infer SdD E ker d n ker 5 and by the assumption, taking into account Proposition 2 in Sec. 5.2.4, Sdfl = w - dSS? E ker J', hence 8d.fl = 0. Therefore we find w = dij, rl E 6,_k-1(U) n W1,P(U, An-k-lRn), and consequently ][8n LW = IRn L drl = (-1)n-ka(Rn L 77).
If p = 1, we can still solve (6), compare Stampacchia [609], and find a weak solution which belongs to C°°(U) n W1,9 (U) for all q < n/ (n - 1). The proof is then concluded as previously.
Integration along the fibers and the lift of currents. Let X be a C2 compact, oriented n-dimensional manifold without boundary in IEPn+N and denote by X : X -* 11nRn+N and by N : X -> ANIRn+N respectively the unit tangent n-vector field to X and a unit normal vector field to X so that
X/vV has the same orientation of Rn+N. Recall now that X is a tubular neighbourhood retract, and more precisely we can find an Eo > 0 such that each point in
U = {x E Rn+N
I
dist (x, X) < Eo }
has a unique point of least distance ir(x) in X, and that -7r : U -+ X is a retraction. Moreover, we have a system of charts cpa : fla -> Va C X of X, and on each chart a system of N independent vectors via), ... , VNa) in such that via)n...nvN) =N on K,; consequently the maps
Ua x B(0, Eo) C Ua x RN -+ 7r- 1(V,) given by N
(7)
Y a (x, t) _ 0a (x) + E tiv(a) (x) i=1
578
5. Cartesian Currents in Riemannian Manifolds
are diffeomorphisms. Of course, taking into account the special form of 0a and denoting by El, ... , EN the standard basis of RN, we see that ANDca (x, t) (Eln ... AEN) = v1A ... AvN (co(x)) = N(W(x))
for x E .(la, t E B(0, Eo). Also note that 1 ° ga (x), A(x)t),
1(x, b) =
where A(x) E MN x N and det A > 0, or in other words 7r : U --> X is an oriented fiber bundle over X. To any (k+N)-form w E Dk+N(U) we now associate a k-form 7r*w in Dk(X) by "integrating w along the fibers of the fibration it : U --> X". We define 7rw by (8)
<7r*w, v > :=
r
J 7r-'(X)
<w(z), vAN(x) > dH'(z) V v E AkTXX.
Clearly 7r* defines a map from Dk+N (U) into Dk (X). In the special case in which
X = .(l is an open set of R'", U = .(2 x RN introducing coordinates el, . .(2 and El, ... , EN in R", we have
X = e1A... hen
N = E1A... AEN,
and any form w E Dk+N l/ f7 x RN) has the form
E
w«Q(x, y) dxandyQ.
aI+j8I=k+N
Then 7r (x, y) = x and (9)
7c*(w)(x) := >
(fWa(X,S)dS)dXa.
Notice that the formula for a product of currents gives (10)
for T E Dk(fl), W E Dk+N(Q X RN)
T x RN(w) = T(7r*(w))
Proposition 3. We have
11
(i) 7<* d = d7r* . (ii) 7r*(7r#77Aw) = 77A7r*w, d w E Dk+N(U), V 77 E DP(X), p + k < n. (iii) For 77 E D,-k (X) and w E Dk+N (U)
f x
7)^7r*w =
f 7#77nw, U
i.e., 7r# (IRn+N L w) = (-1) N(n-k) Q X L -7r*w.
. .
, e,,, in
3.2 Poincare and de Rham Dualities
579
If Px is a Poincare dual form of X in U then 7r.Px = 1.
(iv)
Proof. The first two claims are local, consequently it suffices to prove them for forms 77 and w respectively with compact support in a chart V and in 7r-1(V), we omit here the index a, compare (7). Also we first prove the claims in the case that V =: S2 is an open set of Rn so that -7r-1(.f2) = S2 x RN. In this case we can write 7r(x, y) = x and
(fWa(X)d) dxa.
7r.w := F, Ial=k
Now each component of w takes the form = a(x, y) dxandyO. Therefore
d = ax, (x, y)dxindxandya + v(i, 0) (-1)ka, (x, y)dxandy,3+a hence
0
(i,
7r.dt; =
Q)(_1)k
( l
\
faye (x, y) dy) dxa = 0
(fax(x)d) dxiAdxa
1/31
2.=1
and we conclude n
7r.dw = E T, (
f
axi (x, y) dy) dxindxa = d7r.w.
i=1 dal=k
Writing 7r#77 = E ,7.y (x) dxy the second claim follows easily as 17I=n
7r* (7r# ('y) Aw) (x)
E U 77-y (x)aao (x, y) dy I dx'yAdxa
171=h
Ial=k 77(x)A7r.w(x).
The general case follows from the flat case because (11)
7r.(O#w) = V#(-7r*w),
and cp being the diffeomorphisms in (7) (index omitted). In fact co#[d7r.w] = dco#7r*w = (by (11)) d7r.O#w
= (by (i) in the flat case) 7r.do#w
= 7r*0#(dw) = (again by (11)) cp#(7r.d(.1)
580
5. Cartesian Currents in R.iemannian Manifolds
which proves that in any coordinate chart d7r* - 7rd = 0. Similarly one proves (9).
Let us prove (11). We have ca#w(x, t) = Ak+NDC,(x, t)w(q5(x, t)), therefore for v E AkRn, < 0#W, vAe1A ... ^EN > = <W (O(x, t) ), Ak+NDO(x, t) (vAe1 A ... /BEN) >.
But on the account of the particular form of 0 AkDc(x, t) (v) = AkDcp(x)(v)
ANDO(x, t)Eln... AEN = I ANDO(x, t) I N(((x, t))
... Err) are orthogonal, hence
AkDO(v), ANDO(ElA
< O#W, vAe1 A ... AEN > = < w (q5(x, t) ), AkD p(x) (v) NV (cp(x, t)) >,
consequently <7r* (0#w), v >
r
_ / <w(o(x,t)),AkDV(x)v/
J
<w(z), AkDW(x)vAN(V(x)) > d7-lN (z)
7-1(rv(x))
_ <7r*w(cp(x)),AkDco(x)v> = . Proof of (iii). We use the general coarea formula
f gIVNfI dH =f dnn f g(z)dxN(z) x
x
7r-1(x)
where f : U --+ X and VN f (x) = projection of V f onto the level line of through x. In this case we have for r) E D,-k(X)
f
71A7r*(W) =
f
f (#Aw) = x
x
x
dH f d7-lr'<7r#7]Aw, ex/vV> 7r-1(x)
= f <7r#7)nw, e1A... /den+N>IVN7rI dz = f 7r#7?Aw U
U
since ON7rI = 1. This proves the claim together with
QXj L7r*(w) = (_I)("-k)k
f
r)n7r*(w)
x
_
(-1)(n-k)k f#()Aw
_
(-1)(n-k)k7r#(IRn+N L W).
=
U
(-1)(n-k)[k+N-k]
¶
Ull(WA7r#(7/))
f
3.2 Poincare and de Rham Dualities
581
Finally (iv) follows at once as for 77 E Dn(X) by (iii) and (1) we have
f
fi7ir*(Px) = f 7r#(r1)APx = x U X
771
hence 7r* (Px) = 1.
Fig. 5.3. The lift of a current.
Remark 2. Let cp :.Q -+ L be a diffeomorphism of two open sets of 1R
and let
0: (1xRN -,A x R N be of the type h : 1? x RN __ RN smooth and ah (x, t) > 0.
(12) 0(x, t) = (cp(x), h(x, t)),
Then it is easily seen that cp#7r* simply by changing variables. In particular we obtain that if 7r : E --* X is an oriented fiber bundle, then the integration along the fibers is well defined and gives a map 7r*
: Sk+N(E) -' Ek(X)
since the transition functions of E are maps of the type (12).
One then defines a lift operator 7r* : Dk(X) -> Dk+N(U) on currents by duality *(S)(W) := S(7r*w).
In case X = 1l x {0} C IRn+N and 7r : Rn+N __,, R x {0} is the orthogonal projection on the first factor, 7r(x, y) = x, then
7r*S=SX RN for SEDk(lR X{0}).
582
5. Cartesian Currents in Riemannian Manifolds
Proposition 4. We have (i)1
a7r*s = it*as in U,
(ii) (iii) If M(S) < +oo, then M(7r*S) = vol(BN(0,1))Eo M(S). (iv) If spt S C X, and S is i.m. rectifiable, then 7r*S is i.m. rectifiable in U. Proof. (i) and (ii) follows from (i) and (iii) of Proposition 3.
(iii) For ( E AkTTX,
f <w(x, y), S(x) > dfN(y)
1<7r*w(x), (>I =
7r-1(x)
Taking the sup, we get
7r*w(x)
w(x) vol(BN(0,1))EO hence the conclu
sion.
(iv) We claim that if S = 7-(M, 0, S), M C X then 7r*S = rr(.7r-1(.M), 0 o
7r, F-1(M) ). In fact if w E Dk+N (U) we have
7r*(S)(w) = 7'(M,0, )(7r*w) = =
f<*w(x),(x)>9(x)dflc(x) = M
ff
<w(x, y), S(x)NV(x)>0(x) d7{N(y) d-Hk(x) _
M 7r-1(x)
f
<w(x, y), S(x)NV(x)>0(x) d?jN+k(x, y)
it 1(M)
and the claim follows as SAN(x) is a unit vector orienting 7r-I(M) at (x, y) E 7r-1(M).
Poincare duality isomorphism on compact manifolds without boundary. Let us discuss now Poincare duality between (n - k)-cohomology and khomology groups of oriented, compact and boundaryless submanifolds of dimension n in Rn+N and more precisely prove the following
Theorem 2. Let X be a smooth compact oriented and boundaryless smooth submanifold of dimension n in Rn+N Then the map P : w E D,-k(X) -> P(w) :=
f
Au) E Dk(X)
x induces an isomorphism between the (n - k) de Rham cohomology group of X and the k-homology group of currents Hk (X, R) := Zk (X)/Bk (X) where
Zk(X) = {T E Nk(X) jaT = 0}, More precisely
Bk(X) = {aS j S E Nk+1(X)}.
3.2 Poincare and de Rham Dualities
583
(i) for any cycle T E Zk (X) one can find a closed (n - k) -form PT E Zn-k (X )
such that
T=
(13)
r APT + aS J
X
for some S E Nk+1(X), (ii) if for some w E Zn-k(X) we have fx Aw = as, S E Nk+l(X), then w is
exact in X.
We refer to P as to the Poincare duality isomorphism in X and to any form PT satisfying (13) as a Poincare dual form for T. Proof. Let T E Zk (X). Then, if WT is a Poincare dual of T in U, we have
T(a)=faAwT+aE, EENk+l(U), sptECCU. U
Consequently, taking into account (iii) of Proposition 3, we get
T(a) = 7r#(T)(a) = J '7r#(a)AwT+a7r#E(a) U
r aA-7r*WT + 197r#E(a), J X which proves (i). (ii) From QX ] LW =
(-1)k(n-k) fx
. AW = as, S E Nk+l(U), spt S C X we get
inU by Proposition 4. We then infer from Proposition 1 that 7r#w = d/3, /3 E en-k-1(U) n L1(U,An-k-1Rn+N) which in turns implies that w is exact on X.
Remark S. It is worthwhile noticing that yet another proof of the non degeneracy of the Poincare duality in cohomology on a compact manifold X can be recovered from Proposition 2 using the integration along the fibers. In fact, assume that
fwAa = 0
Va E Zn-k(X).
x Then for /3 E Zn+N-k(U) we have f 7r#c.wA/3 = U
f wAir*/3 = 0 Jx
Then -7r#w is exact in U by Proposition 2 and consequently w is exact in X.
584
5. Cartesian Currents in Riemannian Manifolds
Remark 4. Classically a Poincare dual of T is a closed (n - k)-form in Zn-k(X) such that on Zk(X). T = J . Aa
x
If PT is a Poincare dual of T we also have
T=
J x
APT + 8S,
S E Nk+l (X).
therefore
J?1A(a - PT) = 0
Vrl E Zk(X).
x Consequently a - PT = dQ, and
T=J Aa+BS x
J x
Ad/3=J .Aa+8S1i x
which proves that the classical notion of Poincare dual and the one we have used in this section agree.
de Rham theorem. As we have seen in Sec. 5.2.8, Hodge decomposition theorem yields an isomorphism between HHR(X) and Horn (HaR k (X), IR),
HdR(X) -> Hom (HdR k(X ), l).
On the other hand Poincare duality isomorphism gives also an isomorphism between HaR k (X) and the k-homology group Hk (X, R) of X
P : HHRk(X) _ Hk(X, Therefore we can state
Theorem 3 (de Rham). HHR(X) and Hom (Hk(X, IR), IR) are isomorphic. Actually we shall now show that the induced non degenerate pairing between homology and cohomology HaR (X) x Hk (X, IR) -> R is the integration of closed forms over cycles. In fact the bilinear map
: Zk(X) X Zk(X) -+ IR,
:= T(w)
clearly depends only on the homology class of T and the cohomology class of w, therefore it induces a bilinear map, de Rham's duality <[T], [w] > : Hk(X,IR) x HaR(X) -> IR
and we have
3.2 Poincare and de Rham Dualities
< [T], [w] > = T(w) =
(14)
J X
585
WAPT = Poinc< [w], [PT] >.
where PT is a Poincare dual form of T. In particular (14) clearly shows that de Rham's duality is non degenerate i. e. (i) If w E Z"(X) and T(w) = 0 for all T E Zk(X), then w is exact. (ii) If T E Zk(X) and T (w) = 0 for all w E Zk (X ), then T = 8S
for S E
Nk+l(U), sptS C X. De Rham's theorem can be equivalently stated in terms of periods. Let 'Yi...... s E Zk(X) be cycles such that their homology classes ['y],... , [ys] form a basis of Hk (X, IR). If w E Zk (X ), the numbers 'Yi (w)
are said to be the periods of w with respect to the basis [yi]. Theorem 3 can be equivalently stated as
Theorem 4 (de Rham). Let {yi}, i = 1, ... , s be a basis of k-cycles of X. Then (i) If w is closed and all its periods of w are zero, then w is exact. (ii) Given cu,... , as E R. Then there exists a closed form w E Zk (X) with given
periods al, ... , as, i. e., yi (w) = ai, i = 1, ... , s.
In particular homology classes are identified by the periods of a basis of harmonic forms. This yields in particular the closure of real homology classes.
Proposition 5. Bk (X, IR) is weakly closed. The following corollary is then an immediate consequence of Proposition 5 and of the lower semicontinuity of the mass
Corollary 1. In each class -y E Hk (X, JR) there is a mass minimizing current, that is T E Zk (X, III), [T] = -y such that
M(T) = inf{M(S)
[S] = y}.
I
In particular
M(y) := inf{M(T)
I
[T] = y}
defines a norm on Hk(X,IR), called the sphere norm, consequently
M(y)>0 for all yEHk(X,R), y and the coset map Zk (X, IR) -+ Hk (X, R) is continuous.
0
586
5. Cartesian Currents in Riemannian Manifolds
Normal currents and singular real homology. Another path to get the closure of homology classes which also yields a representation of homology classes by Lipschitz chains can be followed by using the deformation theorem of Sec. 5.1.1. Let X be a compact smooth submanifold without boundary of dimension n
in IRn+N. As it is well known X is a neighborhood C' retract, i.e. there is an open set U X and a C' map 7r: U --> X which retracts U onto X and actually one can think of U as a tubular neibourhood, {x E IRn+N I dist (x, X) < Eo }
(15)
for some positive Co and of 7r as homotopic to the identity in U. Fix now E < EQ/Vn and denote by L ,,x the collection of all standard cubes then Q C U because of (15), of size E which meet X. Of course if Q E and one can modify 7r by composing 7r with a projection onto £n,x to another retraction map again denoted by 7r, 7r : L ,,x --> X which is homotopic to the identity in G,,,,X. Given an integer k, 0 < k < n, we denote by Lk,x the collection of all k-faces of cubes in L,,,x, and by Lk,x the k-skeleton of Ln,x
Lk,x := U{F I F E
If F E Lk,x we orient F in the natural way, i.e., if F is parallel to the plane associated to eal A ... Ae, k, Ic = k, then f := ea, A ... Aeak . Finally we denote by Lx the CW-complex of cubes in Ln,,x. Simplicial homology. Given an integer k, 0 < k < n, the simplicial homology of Lx is defined as follows. A chain on Lk,x is simply a polyhedral chain
P := E Oij Fi j,
Oi E
R
F2 E Lk,X .
Then we set Zk,simpl(Lx, IR)
{P I P is a polyhedral chain in Lk,x, aP = 0}
Bk,simpl (LX, IR)
{P = 8Q I Q is a polyhedral chain on Lk+i,X }
and the k-simplicial homology of Lx is defined by Hk,simpl(LX,IR) := Zk,simpl(LX)/Bk,simpl(Lx).
Clearly Hk,simpl(LX, IR) is a vector space over R of finite dimension, since the collection Lk,x is finite.
Homology groups of Lipschitz chains. Similarly to the simplicial homology or to the real homology, by considering Lipschitz chains in X, i.e., Lipschitz images of polyhedral chains, we can define the homology groups Hk,Lip(X,IR). We have
Theorem 5. The spaces Hk(X,IR), Hk,Lip(X,IR) and Hk,simpl(Lx,IR) are isomorphic. In particular Hk (X, IR) has finite dimension.
3.2 Poincare and de Rham Dualities
587
Proof. Let 7r : U ---> X be the retraction map. If P is a cycle in the k-skeleton Lx then 7r# (P) is a Lipschitz cycle on X, and, as any Lipschitz cycle is also a normal cycle, we have the following maps
Hk,simpl(LX, R) -i Hk,Lip(X, R) -s Hk(X, R) Let T E Zk (X, R). By the deformation theorem we can write T = P + OR where P is a k-cycle on Lx and R E Nk+l(U). We then have T = 7r#T = it#P + 87r#R,
i.e. T and i* o 7r*P are homologous, consequently i* o 7r. is onto. Suppose now that L is a Lipschitz k-cycle in X which is a boundary regarded as a current in U, L = OR, R E Nk+l (U), spt R C X. Deforming R into Lx we get
R=Q+8R1+S, with Q E Lk+1,x and S is a Lipschitz chain in U. Therefore L = OR = 87r# (Q + S) is a Lipschitz boundary which clearly proves injectivity of i*. Finally let us prove that ir, is injective. Let L E Lk,X such that 7r#L is a boundary in X. Then being 7r homotopic to the identity in L,,,x, we have
L = lr#L + 8S,
S E Nk+1(U), spt S C 4,.,x
hence Lisa boundary in L ,,,X, L = OR, R E Nk+l (U), spt R C C,,,x. Deforming R into Lk+l,X we get
R=Q+OR1+S, with Q E Lk+l,x and S = 0, being OR = L E Lk,x. Therefore L = OR = ar# (Q) is a simplicial boundary, which clearly proves injectivity of 7r*.
Remark 5. If A is a local Lipschitz retract, one can do the same constructions and define simplicial homology of the skeleton L,,,x, and Lipschitz homology. However it is easy to find examples in which no retraction homotopic to the identity exists. The previous proof applies however partially and we get Theorem 6. The spaces Hk(A, R), Hk,Lip(A, ]R) are isomorphic. Moreover the retraction map 7r yields a surjective linear map ir* : Hk,simpl (LA, III) -> Hk (A, JR) Again Hk (A, IR) has finite dimension.
As we have seen, compare Proposition 5, one of the advantages in dealing with homology groups defined in terms of normal currents is that the cosets are closed with respect to the weak convergence of currents, and this allows of course the use of variational methods. Another proof of Proposition 5 can be done using the following rough isoperimetric inequality, which works also for compact neighbourhood Lipschitz retracts.
588
5. Cartesian Currents in Riemannian Manifolds
Proposition 6. Let A be a compact neighborhood Lipschitz retract. Every current T E Zk(A,1R) which is the boundary of a normal (k + 1) -current in U,
T = aS1 with S1 E Nk+l(U), is in fact a boundary in A, i.e., there exists S E Nk+1 (U) with spt S C A and T = as, and moreover
M(S) < r(k,n)M(T). Proof. Since simplicial boundaries on LA form a finite dimensional space, we
can easily construct a linear map
T : Bk,simpl(LA, R) -> Nk(U)
which is continuous with respect to the mass norm. For that, choose a basis Pi, ... , Pnr E LA and for each element Pi select a simplicial (k + 1)-chain T(Pi) in Lk+1,A so that af(PP) = Pi, then extend T by linearity. For any P in Bk,simpl (LA, R) we of course have
aP(P) = P,
M(F(P)) < ri (k, n)M(P).
Applying the deformation theorem to T = 8S1 we find R E Nk+l (U), spt R CC U, such that
T = P + aR = a(F(P) + R) and
M(F(P) + R) < M(F(P)) + M(R) < riM(P) + c(k, n)EM(T) < c(k, n)(E + ri)M(T). It is now readily seen that S :='r#(r(P) + R) has the desired properties. Remark 6. Notice that in fact we have proved the existence of a current S in Nk+l (U), spt S CC U such that
T = aS,
M(S) < r(k,n)M(T)
independently from the fact that A is a Lipschitz retract. We have then used the last property to retract S to A and find S :_ 7r#S with
T = as,
M(S) < r(k, n)M(T),
sptS C A.
We can now state
Theorem 7. Bk(A, R) is weakly closed.
Proof. Let Ti E Bk(A, R), Ti T. Of course T, = 8Ri, spt Ri C A. Proposition 6 then yields Si E Nk+1(U), spt Si C A such that Ti = asi and M(Si) < r(k, n)M(Ti) < 7- (k, n) supM(Ti) < oo. i
Passing to a subsequence we get Si -s S, spt S C A, T = as.
3.3 Poincare-Lefschetz and de Rham Dualities
589
The following corollary is an immediate consequence of Theorem 7 and of the lower semicontinuity of the mass
Corollary 2. In each class y E Hk(A,IR) there is a mass minimizing current, that is T E Zk (A, ]R), [T] = y such that
M(T) = inf{M(S)
[S] = y}.
I
In particular
M(y) := inf{M(T)
I
[T] = y}
defines a norm on Hk (A, ]R), called the sphere norm, consequently
M(y)>0
for all yEHk(A,1R), y#0
and the coset map Zk (A, I[8) --+ Hk (A, I[R) is continuous.
3.3 Manifolds with Boundary: Poincare-Lefschetz and de Rham Dualities The Poincare-Lefschetz duality isomorphism. Let X be a compact, oriented smooth n-dimensional submanifold of Rn+N with boundary 8X. We define relative cycles in X as
Zk(X, 0X) := IT E Fk,X(r+N)
I
spt 8T C 8X or aT = 0 if k = 0},
relative boundaries as
Bk(X,3X) :_ {R+8S I R E Fk,ax(1Rn+N) S E Fk+l,X(Rn+N)}, and the real relative homology groups as (1)
Hk (X, 0X, II8) := Zk (X, OX) lBk (X, 0X),
k = 0, ... , n.
Since X and 8X are compact and Lipschitz neighbourhood retracts in Rn+N, Federer's flatness theorem yields Fk,ax(Rn+N) = Fk,X(Rn+N) n IT I sptT C 8X};
also observe that Fk X (R'+N) = Fk,X (U),
Fk,ax (Rn+N) = Fk,aX (V )
U and V being neighbourhoods respectively of X and aX which retract onto X and aX. A few words are in order to explain why we are working with flat chains while when dealing with the homology of boundaryless manifold we worked with normal currents. First we should remark that in the boundaryless case, though cosets defined in terms of flat chains are larger than cosets constructed
590
5. Cartesian Currents in Riemannian Manifolds
with normal currents, the homology groups defined in terms of flat chains or of normal currents are the same. This is not difficult to be seen on account of the density of normal currents among flat chains and the closure properties of homology classes. However, while i.m. rectifiable cycles are obviously normal currents, relative cycles, which in general do have boundaries, are not in general
normal as in principle we have no control on the masses of their boundaries: this is the reason why we are forced to work with the more general flat relative cycles. In the next section we shall illustrate relative homology in an even more general setting. Paralleling the similar situation in cohomology, we therefore may define three homology groups depending on k = 0, 1,... on X. The homology group Hk(X,]R)
flat cycles in X flat boundaries in X' if T E Zk(X) is a flat cycle, we denote by [T] x its homology class in Hk(X,R). The homology group Hk (OX, R)
flat cycles supported on aX flat boundaries in ax
Hk(aX,R)
for T E Zk(aX), we denote by [T] ax its homology class in Hk(aX,
lR
The relative homology group Hk (X, 9X, ]R)
Hk (X, aX, R) :=
flat relative cycles flat relative boundaries'
for T E Zk (X, DX) we denote by [T] 1el its relative class in Hk (X, aX, R). The inclusion map i : aX -> X induces a map
i# : Hk(DX,R)
Hk(X,1R).
As cycles are special relative cycles and boundaries are special relative boundaries an inclusion map .7# : Hk (X, RIB) - Hk (X, ,9X, IR)
is well defined by
#([T]x) = [T]rei.
Finally, the boundary operator a : Fk,x(R"1') - Fk-l,x(Rn+1`') induces naturally a map a : Hk (X, aX, R) -4 Hk_ l (aX, ]R)
defined by
a([T]rei) = [DT]ax.
Then it is not difficult to prove
3.3 Poincare-Lefschetz and de Rham Dualities
591
Theorem 1. The long sequence in homology
#Hk(X,R),#Hk(X,aX,R)
... a Hk-,(aX,R)->Hk-,(X,R)-+ # j#
is exact.
It turns out that the long sequence in homology and the long sequence in cohomology are in duality, such a duality being given by Poincare-Lefschetz duality. In the sequel of this section we shall discuss this topic which will lead us to a de Rham theorem for manifolds with boundary. For k = 0, 1, . . . , n we first consider the pairing
<, >: given by
<w, r7> :=
(2)
En-k(X) ,
Ek-l(aX) X
r
J ax
wnr7.
As we saw in Sec. 5.3.2 (2) induces a map P:
Zn-k(aX)
, Zk-1(aX)
given by
ax which factorizes to the Poincare duality isomorphism
P : HHRk(aX) -> Hk-1(aX,
IR
compare Theorem 2 in Sec. 5.3.2. Of course there are other pairings on X
<, >: Ek(X) X En-k(X) --> ][8 again given by
<w,7)> =
(3)
xJ
wAr7.
But, as by Stokes theorem (4)
JX
do-n77 + (-1)k f o-Adrl =
x
f
J ax
oni7
a E Ek-1(X), rl E En-k(X)
we see that there is no way of inducing a map between the absolute cohomology and homology of X unless aX = 0. However, if 77 E Zn-k (X, aX) the k-dimensional current
592
5. Cartesian Currents in Riemannian Manifolds PO77(w)
(5)
(-1)k(n-k) ((X
fwA77 =
IL
w E Ek (X )
L 7? (W),
x is a cycle in X, Par? E Zk(X) by (4). Also the homology class [PO77]x of P077 depends only on [77]rel. In fact, if 77 = da with i#a = 0, then by (4) Pbr7(w)
= J wAda = (-1)k+1
f dwna = (-1)k+1apta(w).
In conclusion we can say that the map Pt : Z,-k (X, aX) -+ Zk (X) induces a map again denoted by Pd
Pt : Hdn k(X,aX) , Hk(X,ltd).
(6)
Finally, in a similar way, one proves that for 77 E Z,-k(X) the current Pb :_ (-1)k(n-k) QX] L7)
(7)
is a relative cycle, P577 E Zk(X, aX), and moreover the map Pb : Zn-k(X) --> Zk (X, aX) induces a map Pb : HHR(X) , Hk (X, aX, ltd).
(8)
We have
Theorem 2 (Poincare-Lefschetz isomorphisms). The maps P, Pt, Pb are isomorphisms, and the diagram S
H,Rk(X,8X)
HdRk(X)
1 Pt
I Pb
(9)
Hk (X, R) i#
-i
Hk (X, aX, IR)
£
Hdn k (aX)
-> .. .
IP
a
Hk-1(aX, R)
#
.. .
is commutative.
We shall refer to the maps Pt, Pb, P as to the Poincare Lefschetz duality isomorphisms. It is quite simple to prove the commutativity of the diagram, though quite long. We already known, compare Sec. 5.3.2 that P is an isomorphism, Poincare duality isomorphism. We shall therefore prove only that Pt and Pb are isomorphisms. This will be done in the next propositions below. Let X be a compact, oriented, smooth n-dimensional manifold with smooth boundary 8X. To be precise we think of X as a C2 manifold with a C2 boundary.
Observing that aX has codimension 1 in X, and X is oriented, the tubular neighbourhood theorem on aX yields
Theorem 3 (Collar theorem). There is an eo > 0 such that the normal geodesic flow to TxaX at x E 0X, g(t, x) : [0, co] x aX --> X is a difeomorphism onto a neighbourhood of aX in x with g(0, x) = x for all x E aX.
3.3 Poincare-Lefschetz and de Rham Dualities
593
Fig. 5.4. The collar map.
A consequence of the collar theorem is then
Theorem 4. We have HHR(X, aX) = H pt(X \ aX) where
HPt(X\8X)
{wEC-(X\aX,AkT(X\aX)) I dw=0} {w=da I aEC0(X\9X,AIT(XQ,aX))}'
In fact
Proposition 1. We have (i) Each relative cohomology class contains a form 6 E Zk(X,aX) which is compactly supported in x \ ax. (ii) Each homology class of X contains a cycle T E Zk(X) of finite mass such
that spt T n ax = 0.
Let Eo and g be as in the collar theorem, Theorem 3. Define r(t) [-Eo, 0] -> [-eo, 0] to be a non-decreasing function such that r(-eo) = -6o, r(-Eo/2) = r(0) = 0, r'(-eo) = 1. Of course (1 - s)t + sr(t) yields a homotopy Proof.
with the identity which is fixed at -Eo and 0. Consider then the mapping O(x)
x g(r(t), y)
if x E X_Eo
if g(t, y) = x E X \ X_E0
which maps X \ X_E0 onto aX and satisfies 0 o i = i on aX. Then h(s, x) := (1 - s)x + so(x) yields a homotopy of ¢ with the identity which keeps the boundary fixed. Consequently, if ri E Zk(X, aX), the form w = 0#rl vanish on X \ X_E0/2i and by the homotopy formula w - 77 = d[h#il](o,l)
5. Cartesian Currents in Riemannian Manifolds
594
This concludes the proof of the first claim as i# [h#r)] (o,l) = 0 being h(s, x) = x
d x E aX and V s E [0, 1]. Analogously, if T E Zk(X), we infer from (i) of Proposition 4 in Sec. 5.1.3 that T is homologous to a relative cycle T' E Zk(X) of finite mass and consequently r#T' E Zk(X) has finite mass. Moreover r#T' is homologous to T' and spt r# TI f1 aX = 0. Actually without loss of generality we can and do assume that, in case aX # 0, X is actually a compact piece of a larger manifold Y and the geodesic flow on X normal to aX extends to both sides of 8X in Y. Denoting such a flow by g(t, X), the collar theorem Theorem 3 then yields, possibly for a smaller EO, a diffeomorphism g(t, x) : [-EO, Eo] x aX - * Y
onto a neighbourhood of aX in Y such that g(0, x) = x for x E aX and g(t, x) E X if t > 0. The equation g(t, x) = y means that x is the point of least geodesic distance in aX from y, Itj is the geodesic distance of y from 9X, and t > 0 if
yEX. For 0
EU we now set
(10)
X_E
{x E X I x 0 g([0, e] x 8X}
X+e
g([-E, Eo] x ax) U X
and X0 = X. Of course any XE, E E [-EO, EO], is a compact oriented manifold with boundary. Given any smooth increasing function r(t) : [-60, Eo] -p [-60, 60]
with r(-Eo) = -so, r(eo) = eo, r'(-eo) = 1, r(-EO/2) = 0, and r(0) = Eo/2, it is convenient to introduce the map e : X+EO -r X+ep
defined by e(x)
fx
if x E X_E0
g(r(t), y)
if g(t, y) = x E X+E0 \ X_60.
It is easily seen that e yields an isomorphism which is the identity on X_EO, maps X \ X_E0/2 onto X+,012 \ X, &X_.012 onto aX and aX onto aX+eo/2 We can also assume that e is homotopic to the identity with a homotopy which keeps the boundary values fixed. The inverse map of e denoted by r r : X+E0 --4 X+EO
which maps the collar X+,012 \ X onto X \ X_,012 and 09X+EO/2 onto OX, is then also homotopic to the identity with a homotopy which leaves aXE0 fixed. Finally, we shall also use the distance function from X. Setting Ue := {x E R"`+N we have
I
dist (x, X) < E}
3.3 Poincare-Lefschetz and de Rham Dualities
595
Proposition 2. There exists eo > 0 such that any point x E U 0 has a unique point 7r(x) of least distance on X. Moreover the map 7r : U6p -- X is Lipschitz in UE and smooth in U6 \ X. Finally, for x E X \ aX, 7r-1(x) is just a flat ball of center x and radius 60 lying in the normal space to T, X . For every 6 in (-60, 60) we consider the enlarged manifold X6 and U6
{x E Rn+rr
dist (x, X6p/2) < 6}.
I
Choosing el so that eo = 461 and taking 61 smaller if necessary a least distance projection map it : UU1 --> X261
is well defined: it maps each x E UU1 to the least distance point in X2,1.
Proposition 3. P is injective, i.e., if w E Zn-k (X, aX) and (11)
S E Fk+l,x (U6y ), then w is relatively exact, that is there exists a E
Dn-k-1(X
)
with i#a = 0 such that w = da. Proof. On account of (i) of Proposition 1, we can and do assume that w = 0 on X \ X_,1. Moreover on account of Proposition 4 in Sec. 5.1.3 we also assume that S has finite mass. Consequently, as in the boundaryless case, we infer Q7r-1(X)J L7r#w = 7r*(f X Lw) = a7r*S
in U,1,
spt,3 C 7r-1(X_61/2), compare Proposition 1 in Sec. 5.3.2 and Remark 1 in Sec. 5.3.2. In particular w is exact on X and zero near aX. hence 7r#w is exact, 7r#w =
Proposition 4. PO is surjective, i.e., if T E Zk(X) there exist a relative closed form w E Zn-k(X, aX) and a current S E Fk+1,x(U61) such that
Proof. First notice that one can suppose that M(T) < oo and spt T C X_61
on account of Proposition 1. We then regularize T in U61 with a parameter 6 << 61/2. This way we have
T=f n6+aS' UE1
where l; E Dn+N-k (U,1 ), spt C 7-1(X_61/2), S' E Fk+l,opt 7r-1(X_61/2), M(S') < oo. Projecting onto X we get (12)
T = 7r#T = 7r#(
f
Ue1
-
a7r#S'
spt S' C
596
5. Cartesian Currents in R.iemannian Manifolds
with it#S' E Fk+1,x(UE,); as for 77 E Sk(X) we have fx 7r#17AC = fx 7 A7r , the conclusion follows since (12) yields also dC = 0, hence ir.dC = 0, and
e = 0 near
1(aX), hence 7r.C = 0 near aX.
Proposition 5. Pb is injective, i.e., if w E Z,-k(X) and
QX]ILw=R+aS
(13)
with R E Fk,ax (UE, ), S E Fk+l,x (UE, ), then w is exact, that is there is a E Bn-k-1(X) such that w = da.
Proof. On account of Proposition 4 in Sec. 5.1.3 we can and do assume that M(R) + M(S) < oo. Applying the extension map e we get Q Xe, I L r#w = R'+ aS'
(14)
where R' = e#R E Fk,aX,012 (U), S' = e#(S) E Fk,XE012 (U), M(R') + M(S') < oo. Since rIx is homotopic to the identity and dw = 0, the homotopy formula for forms yields that w - r#w is exact, hence it suffices to prove that r#w!x is exact too. For that notice that (14) yields
1 XE,12 Lr#w = aS'
in U, fl 7r-1XEO/2.
We then proceed as in Proposition 3 and conclude that 7r#r#w is exact in Ue, n 7r-1Xeo/2 This implies that r#w is exact on X.
Proposition 6. Pb is surjective, i.e., if T E Zk(X, aX) then there are w E Z,-k (X), R E Fk,ax (U,,), S E Fk,x (UU,) such that Proof. We may and do assume that M(T) < oo, compare Proposition 4 in parameter, we infer that Sec. 5.1.3. Regularizing T with a
T= I A +aS
(15)
U.,
Dn+N-k(Uel) S E Nk+1(Ue, ), spt S CUE, fl 7r-1(Xep/2). Therefore, with E projecting on X,.
T = 7r#T =
J
r# () AC + a7r# S
U.,
=
f
xEo/2
. n7r. + a7r#s =
f
. A7r.C + R' + aS'
x
where R' := fxEO/2\x Air.C, S' := -7r#S. As X is a deformation retract of X0012i
denoting by p :X,.12- X the retraction, we infer
3.3 Poincare-Lefschetz and de Rham Dualities
T = p#T =
f
597
. A1r*C +p#R' + ap#S'
X
with R' E Nk+l,ax(UE1), p#S' E Nk+l,x(UE1). The claim now follows observing
that being aT = 0 in X \ 8X, (15) implies that d6 = 0 on 7r-1(X \ aX) which in turns implies dint = 0 on X \ aX and by continuity dir 1; = 0 on X. Remark 1. We emphasize the fact that surjectivity of P4 amounts to the existence for a given absolute cycle T E Zk (X) of a relative Poincare dual form PTE Zn-k(X,aX) such that Pp[PT]rei = [T], i.e.
J T(w) = ox wfPT Vw E Zk(X ) 1
T-
i#
It is reasonable to find for T E Zk (X) a Poincare dual with null value at aX since one can deform with an homotopy T into T' E Zk(X) in such a way that spt T' n aX = 0. Also the surjectivity of Pb means the existence for a given relative cycle T E Zk (X, aX) of a Poincare dual form PT E Zn-k (X) such that PL[PT] _ [T]rel, i.e.
r
T(w)=J wAPT+R+aS x
`dwEZk(X)
where R E Fk,ax (UE1 ), spt R C aX, S E Fk+l (UE1)
Remark 2. The proof of Theorem 2 can follow a different path. First one proves
that P and P are isomorphisms, as previously, then that the diagram (9) is commutative, and finally one proves the bijectivity of Pb using the so-called five-lemma
Proposition 7 (Five Lemma). Suppose we have the following commutative diagram (16)
Al
a+
1h Bl
A2
a2
IA B2
A3
a3
B3
a4
I f4
j f3 _+
A4
G
B4
A5 Lf5
_+
B5
consisting of vector spaces Ai, Bi, i = 1, ... , 5 and of linear maps ai, bi, i = 1, ... , 4 and fi, i = Im bi = Ker bi+i then
1, ... , 5. If the rows are exact, i.e. Im ai = Ker ai+i,
(i) if f2, f4 are surjective and f5 is injective, then f3 is surjective, (ii) if f2, f4 are injective and fl is surjective, then f3 is injective (iii) if fl, f2, f4, fs are isomorphisms, then f3 is an isomorphism.
598
5. Cartesian Currents in Riemannian Manifolds
de Rham duality on manifolds with boundary. As consequence of the non-degeneracy of Poincare-Lefschetz pairings, compare Theorem 3 in Sec. 5.2.8 and taking into account Poincare-Lefschetz isomorphisms, we get
Theorem 5 (de Rham). The pairings <[T]x, [cp]x>9 : Hk(X,R) x HakR(X) --.lI8 < [T]rel, [co]rer >b : Hk (X, aX, R) x Hag.(X, 9X) --> R
induced by the pairings : Ek(X) X Ek(X) ---> R, := T(cp) are non degenerate. In particular
HHR(X,aX) =Hk(X,aX,R)*.
Hk(X) =Hk(X,R)*, Moreover, if we also set
< [T]ax, [W] ax > := T (cp) for [T]ax E Hk (OX, R) and [cp] ax E HaR (5X ),
the following diagram is commutative
...
Fa--
HaR(aX)
HHR(X)
x<, > ...
't-
x<, >a
- Hk (aX, R) * Hk (X, R)
...
HHR(X,aX)
x<, >b '*+
Hk (X, (9X, R)
--a+...
Let Y 1 , .. ,.ys be cycles in Zk(X) such that [Yi]x, [Y2]x, - -, [rys]x form a basis of Hk (X, R). Given a form w E Zk (X), the periods of w with respect to -
yl...... s are the numbers yi(W),
i=1,...,s.
Also if 81i ... , 8, be relative cycles in Zk(X, aX) such that [8i]rel,
- - - ,
[8r]rel form
a basis for Hk (X, aX, R), and w E Zk (X, OX), the relative period of w with respect to 61,. .. , Jr are the numbers 8i (w),
i = 1, ... , r.
Then it is easily seen that equivalently de Rham theorem can be stated as follows
Theorem 6. Let [yi], i = 1, ... , s be cycles in X such that [yj]x forms a basis in homology. Then (i) If w E Zk(X) and all the periods of w with respect to yi are zero, then w is exact. (ii) Given numbers ai E R, i = 1, ... , s. Then there is a form w E Zk(X) with
given periods ai, i.e. 7i(w) = ai, i = I,-, s.
3.4 Intersection of Currents and Kronecker Index
599
Also, if {8i}, i = 1, . . . , s are relative cycles in Zk(X, OX) such that [8i]rei, i = 1, ... , s form a basis for Hk (X, aX, IR), then (iii) If w E Zk(X, aX) and all the periods of w are zero, 8i(w) = 0,
i = 1'. .. , s,
then w is relatively exact, i.e. w = da, a E Ek(X) and i#w = 0. (iv) Given numbers ai E R, i = 1, ... , s. Then there is a form w E Zk (X, aX ) with periods ai, i. e. 8i(w) = aj, i = 1, ... , s. Also, the dual statements holds true, in particular
Theorem 7. We have
(i) T EBk(X) ifT(w)=0VWEZk(X) (ii) T E Bk(X, aX) if T(w) = 0 V W E Zk(X, aX) Of course for T E Zk (X) we have T (w) = 0 b w E Zk (X) iff T (wi) = 0 d wi E
Zk(X), i = 1, ... , s, [wi]x being a basis of HHR(X), and for T E Zk(X, OX) T(w) = 0 V w E Zk(X, aX) if T(wi) = 0 V wi E Zk(X, aX), i = 1, ... , r,
[wi]re1
being a basis of HHR(X, OX). A consequence of Theorem 7 is then
Theorem 8. Bk (X) and Bk (X, aX) are closed with respect to the weak convergence of currents respectively in Zk(X), Zk(X, ax). Consequently we can state Theorem 9. Homology classes can be represented by mass minimizing currents. That is, given y E Hk (X, R) and 8 E Hk (X, aX, IEF) there are currents T E Zk (X) and R E Zk (X, ax) such that
M(T) = inf{M(S), S E Zk(X), [S]x = y} M(R) = inf{M(S), S E Zk(X, aX), [S]rel = y}.
3.4 Intersection of Currents and Kronecker Index The aim of this subsection is to discuss some extensions to currents of the classical notion of intersection of oriented surfaces in generic positions and to introduce the intersection index of homology classes.
Intersection of currents in R'. Let U be an open set of Rn and let S E Dk(U) and T E Dh(U), where h + k > n. Denote by f : Rn x I[8n --> Rn the map f (x, y) := x - y, and let 9: U --> U x U be the diagonal map
g(x) := (x, x). Suppose that the slice, compare Sec. 2.2.5, < S x T, f , 0 > exists and defines a current in Nh+k_n(U X U). Then its support lies in {(x, y) I x = y} = Img,
5. Cartesian Currents in Riemannian Manifolds
600
consequently, by Federer flatness theorem, see Sec. 5.3.1, there exists a unique current, denoted s n T and called the intersection of S and T , in Nh+k-n(U) such that (-1)(n-k)h<S x T, f, 0>. g# (s nT) _ (1) Actually it suffices that <S x T, f, 0> be a flat chain, compare Sec. 5.1.3 below. In the next proposition we collect a few simple properties of the intersection.
Proposition 1. We have (i) If s n T exists, then spt (S n T) C spt s n spt T. S=(-1)(n-k)(n-h)S (ii) S n T exists iff T n S exists, and T n n T. (iii) Assume that S n T, and either S n aT or aS n T exist. Then
a(S n T) = S n aT + (-1)h-k 3S n T.
(2)
Proof. (i) follows as
spt <S x T, f, 0> C spt S x spt T) n {x = y} = g(spt S n spt T). (ii) Using the map r(x, y) = (y, x), which reverse the role of the factors, we infer
for = -f, r#(T x S) = (-1)hkS x T, consequently (-1)hk<S x T, f, 0> = = r# = (-1)nr# = (-1)n.
It follows s n T = (-1)'T n s where r = -hk + n + (n - h)k + (n - k)h = (n - h)(n - k) (mod 2). (iii) From 8(S x T) = 8S x T + (-1)kS x 8T, S E Dk(U), we infer that
8<S x T, f, 0> _ (-1)n = = (-1)n<3S x T, f, 0> + (-1)n-k<S x 8T, f, 0> provided < S x T, f, 0 > and either < 8S x T, f, 0 > or < S x 8T, f, 0 > exist. Therefore 8(S n T) = (-1)a8S n T + (-1)T S n aT where
Q=n+(n-h)h+(n-k-1)h=n-h (mod 2) -r=n-k+(n-h)h+(n-k)(h-1) =0 (mod 2). Intersection theory of currents is in general quite complicated as the intersection of currents may fail to exist. In fact while the existence of the slice < S x T, f , z > is granted for a.e. z by the Lebesgue differentiation theorem, it may fail to exist at a single point, e.g. z = 0. The following examples show some of-the difficulties. -------------
3.4 Intersection of Currents and Kronecker Index
601
0 It is clear by the definition that the intersection defines a bilinear map on Nk(U) x Nh(U) whenever it exists. It is also worth noticing that in general the density of intersection current is the product of the densities of the factors. For example let W E LT'(Rn), 0 E Lq(Pn) be two functions. Consider the ndimensional currents S :_ Rn L W and T = lR L' &. Then the current S x T E Den (RI x JRn) is given by
S x T(w) = f (P (x)'b(y)w(x,y),
w E D2n(Rn x llgn).
R'' xR-
If f (x, y) = x - y, Op (t) = XB, (t)dt' n ... dtn, for
w = E waQ(x, y) dxandyQ I&I+IQI=n
we have 1
wnpn
dx ¢ W(x)o(y)
S x T(f#(Qp)nw)
a(Y, Y)w7ti(x,y) dy h'I
Bp(x)
li
Since
O('Y,7)w77(x,y) - b(x)g#(w)(x)
O(y)
I+
in Lq(PJ) we infer that, assuming 1/p + 1/q < 1, <S x T, f, 0> exists,
<S x T,f,0>(w) = lim
- S x T(f#(S?p)nw) =
f
hence
s n T(w) = f O(x) (x)w(x)
w E Dn(Rn).
Simple examples of course show that s n T may fail to exist.
0 Let Q = Q (0, 1) x (0, 1) 1 E 12 (R2) be the integral current integration over the square in JR2 and let 7r(x, y) = x be the projection on the first factor. Then it is easy to compute 8x1(0,1)1 =
18:,: x Q
0
(0,1)
if0<x<1 if x = 0,1 otherwise
which shows that at specific values the slice need not be rectifiable. Notice that
such a phenomena would not appear if we project into a plane which is not perpendicular to any of the edges of Q.
0
5. Cartesian Currents in Riemannian Manifolds
602
However, by slightly moving one of the two currents, we can and do ensure the existence of the intersection. In fact denoting by Ta, (x) = x + a, x, a E 1I ' we have
Proposition 2. Let S E Nk(U) be a normal current compactly supported in U, and let T E Nh(U), h + k > n. Then for a.e. a E ll close to zero (i) S n Ta,#T exists in Nh+k-n (U), (ii) a(S n Ta,#T) = S n Ta#oT + (-l) k (as) n Ta#T, (iii) if moreover S E Ik (U), T E Th (U) then s n Ta#T E Zh+k-n (U) . Proof.
From the slicing theory <S x T, f,z> exists in Nh+.k_n(U) for a.e.
z E Rn close to zero. On the other hand,
<S x Ta#T, f, 0> = < S X Ta#, f o (id x Ta), 0> = id x Ta#< S x T, f, -a> whenever <S x T, f, -a> exists. This proves (i). (ii) then follows from (iii) of Proposition 1 and (iii) follows similarly to (i) on account of <S x T, f, z> E Zh+k-n(U) for a.e. z E R. Also the intersection is well defined, essentially if we are able to prove continuity of < S x T, f , z > in z. We have in fact
Proposition 3. Let S E Nk(U) and T E £n-h(U). Then S n Rn L T exists and S n (R' L T) = S L T
in U.
Proof. Let w E Dk+h-n(U X U). Consider the k-form O(x, y) = r(y)Aw(x, y) E Dk(U X U). As f #(dtlA... Adtn) = (-1)I o (ry, y) dx"Adyry we have
T(y)Af#(dtlA...Adtn)Aw(x,y) _
'Y)dxryAO(x,y)ndyry, 171<_n
where b=(n-j)+j(n-h)+(n-j)(h+k-n), j Writing 0 = E 0«p dx«AdyQ, we get as p --+ 0
f
B(x,p)
0
if )3
o'('Y,'7)0«y(x, x)
if j3=ry
ry
by continuity. Consequently the averages
I T(y)Af#(dtlA... Adtn)Aw(x, y) B(z,p)
converges, as p goes to zero, to (3)
(x x) dx«Adx1" _ (-1)(h+k)ng#(0)(x) 171
3.4 Intersection of Currents and Kronecker Index since 8 + I ry I
603
I a I = 8 + j (k - j) _ (h + k) n mod 2. Since now S X Rn LT(f#(QP)nw)/wnpn
=
(-1)(n-h)kS w Pn x 1[8n(TAf#(dt'A...ndtn)nw)
I
_ (-1)(n-h)kS(
T(y)Af#(dt'A...Adtn)Aw)
B(x,p)
from (3) we infer that < S X I[2n L T, f, 0 >(w) exists and agrees with (
1)(n-h)kS(g#0) = (_1)(n-k)hS(Tng#w). In other words
g#(sn1
LT) =g#(SLT).
As immediate corollary we notice
Corollary 1. Assume o- E
£n-k
£n-h
(U), T E
(R'
(U), h + k > n. Then
R'
in U.
Of course one is also interested in intersecting currents when both factors are not integration against form, the case of intersection of rectifiable currents being relevant. We shall not go into these topics and confine ourselves to mention the following result taken from Federer [226] which applies for instance to the intersections of smooth surfaces in general position. We have
Theorem 1. Let S E Tk (Rn), T E Zh ([$n) be two integral currents, h + k > n. Assume that spt S and spt T are contained respectively in two smooth submanifolds X and Y of dimension respectively k and h so that
S = r(M,0,X),
T =T(N,cp,y)
where M C X, N C Y. Assume furthermore that (i) 0 is bounded on M, go is bounded on Al. (ii) For each x E X n y, TTX n Ti,)) has dimension h + k - n. (iii) Either k = 0 or spt aS n spt T has a neighborhood E C
Rn such that
E n sptaS is covered by a finite family r of (k - 1) -dimensional smooth submanifolds G of X with E n clos G C G
TXGnTxyhas dimensionh+k-n-1dxEGny. (iv)
Either h = 0 or spt S n spt aT has a neighborhood F C 1R? such that F n spt aT is covered by a finite family A of h - 1-dimensional smooth submanifolds D of y with F n clos D C D and
TXDnTXX has dimension h+k-n-1V X EDnX.
604
5. Cartesian Currents in Riemannian Manifolds
Then the (h + k - n) -vector ((x) given by
9#,.(((x)) = X(x)ny(x) L (df'A... Adfn)(x,x) is non zero and simple, whenever x E spt T n spt S \ (spt aT U spt aS). Moreover Hh+k-n (spt T n spt S \ (spt aT U spt aS)) = 0 and
SnT =T(.tit We omit the proof of Theorem 1 and we refer to Federer [226]. We just make a few remarks on the role of transversality of X and Y. If X (x), y(x) are the two
unit tangent vectors to X and y at x, then X(x)ny(x) generates T,(X) ®T.(y) in the product R' X I. As Im g = ker f , we have TTX n Txy has dimension h + k - n iff
f : Tx X ® T. y has a kernel of dimension h + k - n iff
f,TTX®TTY has maximal rank n.
From Proposition 3 in Sec. 5.2.2, the last claim is equivalent to say that
X(x)AX(X) L (df'A... ndfn) 0 0. Intersection in real homology. In contrast to the many difficulties in defining the intersection of currents, homologically the intersection of cycles is always well defined. Let X be a compact, oriented, n-dimensional submanifold of IIBn+N possibly with boundary. We first consider the intersection of relative and absolute homology classes. Let [S] rel E Hk (X, aX, ]IP) and [T] E Hh (X, Ii), h + k > n, and let P. be a Poincare dual form of T, PT E Zn-h(X, aX). It is easy to see that
the absolute homology of the (h + k - n)-cycle
S('nPT) depends only on the homology classes [S]rel of S and the absolute homology class [T] of T. Then we define the intersection map
nX : Hk(X,aX,R) x Hh(X,]R) -> Hh+k-n(X,I) by (4)
[S]rei nX [Ti :=
Similarly one defines the intersection map
(-1)(n-k)(n-h)[S(
APT)]
3.4 Intersection of Currents and Kronecker Index
605
nx : Hk(X,R) x Hh(X, R) --i Hh+k-n(X,IR) (-1)(n-k)(n-h) for [S] E Hk(X,R), [T] E Hh(X,R), by [S] nx [T] := h+k > n. Finally denoting by PT E Zn-h(X) a Poincare dual form of a relative cycle T E Zh(X, aX), the formula
(-1)(n-k)(n-h)
[Sire, nX [T]rel :_
[S(' APT)]rel
defines a third set of intersection maps nX : Hk (X, aX, ]R)
X
Hh(X, aX, R) -4 Hh+k-n (X, aX, R).
From the definitions it is easily seen that intersection of cycles in homology is Poincare dual of the wedge product in cohomology. For instance for the map nX one has the following commutative diagram Hdn k(X)
X
Hdn h(X,aX) A
Pb
I Pd
I Hk (X, aX, IR)
x
Hh (X, lid)
-n
Hen-h-k(X, l aX)
t
Pd
Hh+k-n (X, l)
where [w]A[77] := [wArf].
Kronecker index in 1EBn. Finally, let us introduce the Kronecker index S E Nk (Rn) and T E Nn-k (Rn) be two normal currents of complementary dimensions. Assume that either k = 0 or spt S n spt aT = 0 and either h = 0 or spt T n spt aS = 0, so that spt (a(S x T)) does not meet the diagonal of J2n, i.e., 0
f (spt a(S x T)).
Then the current (-1)n-kf#(S x T) belongs to Nn,(Rn) and 0 does not belong
to spta((-1)n-kf#(S x T)). The constancy theorem then yields a constant k(S, T) E ][8 such that in a neighborhood of 0
(-1)n-k.f#(S x T) = k(S,T)QRnj The number k(S, T) is called the Kronecker index of S and T, and it is actually an integer if S and T are i.m. rectifiable. (5)
For any cp E C°(JR) and p sufficiently small, applying (5) to cp(t)xBP dt' n ...
ndtn =: p(t)Qp(t), we get
(_1)n-k IBpI
S x T L f #(.Q p )(cp o f) = k(S T)
I B(°,p)
-(t) dt.
606
5. Cartesian Currents in Riemannian Manifolds
Fig. 5.5. SnT =5a-bb, k(S,T) =0. Taking cp = 1, we then conclude in the case that S n T exists that
(S nT)(1) = g# (S nT)(1) =
(-1)(n-k)2+(n-k)<S
x T, f, 0>(1) = k(S,T).
A special interesting case is that of two oriented submanifolds S and T in generic position. By Theorem 1, in this case S n T exists and S n T = >2 njbp. where the ni are defined by S(PZ)nT(Pi) = nzejA ... nen. Therefore we conclude
k(S,T) _ 1: n2, i.e., Kronecker index of S and T is the sum with sign of the intersections. Another interesting case is that in which Dn-k(Rn) S = (-1)k(n-k)Rn La, uE T E Dk(Rn). T = (-1)k(n-k)Rn LT In this case, as we have seen, S n T exists and, by Proposition 4 in Sec. 5.3.2, k(S, T) = S n T(1) = I[gn L (Tnv) (1) = frAa. lin
Kronecker index on manifolds. Of course one can extend the notion of Kronecker index to currents on oriented manifolds. More precisely let S E Nk (X), T E Nn-k(X) where X is a n-dimensional, compact oriented submanifold of lRn+N Assume
k = 0 or spt a8 n spt T = O and k =nor spt aT n spt S = 0. Recall that the lift 7r* (T) on a tubular neighborhood U of X of bounded crosssection has finite mass, and that
3.4 Intersection of Currents and Kronecker Index
607
spt a7r*T n spt S = 0
since spt a7r*T lives on it-1(spt aT) or outside U. Then we can set
Definition 1. The Kronecker index of S and T in X is defined by
kx(S,T) := k(S,lr*T), i.e. as the number such that
(_1)n-kf#(S X lr*T) = kx(S,T)QRn+1v
(6)
near the origin.
From (6) it is easy seen that (i) (ii) (iii)
kx(S,T) is integer-valued if S and T are i.m. rectifiable kx (aR, T) = kx (S, aR) = 0. kx (S, T) = 0 if spt s n spt T = 0.
In particular, if S E Zk (X, aX) is a relative normal cycle and T E Zn-k (X) is an absolute normal cycle which do not meet aX , k(S, T) is well defined and depends only on the relative homology class of S and on the homology class of T, thus it actually defines a map
ix(S,T) : Hk(X,aX,R) x Hn_k(X,]R) -i ][P which is the intersection index of S and T T. As we have essentially seen, if s n lr*T exists, then
s n 7r*T(1) = kx(S,T) = ix(S,T).
In particular if S = the boundary of X, and T = T, then
(-1)k(n-k)
(7)
Q X I LPs, i.e. PP is Poincare dual of S null to (-1)k(n-k)
Q X ] L PT, i.e. PT is a Poincare dual of
1 PAP
x Therefore we can state that the intersection index is Poincare-Lefschetz dual of Poincare Lefschetz duality on forms. Of course one has also an intersection index on the compact oriented manifold without boundary aX iax : Hk (aX, R) x H,,,-k (aX, R) -> R defined by iax (S, T) := Kax (S, T). This yields another way to state non degeneracy of Poincare duality of forms.
608
5. Cartesian Currents in Riemannian Manifolds
Theorem 2. Let X be a compact oriented manifold. The intersection index
ix : Hk(X,8X,R) x Hn_k(X,R) ->
II8
is non degenerate. Moreover, denoting by iX (T, S) := ix (S, T), the following diagram Z#
Hk (X, aX, R) -
Hk(X,R)
Hn_k (X, 8X, IE8)
<.7#
Hn-k (X, Ilk)
s .. .
x iax
X ix
X iX a
Hk-1(3X, R)
4---
Hn-k (8X, lR)
z#
a
is commutative.
3.5 Relative Homology and Cohomology Groups We conclude this section reporting partially and briefly on Federer and Fleming approach to relative homology groups in the setting of local Lipschitz neighbourhood retracts. Let A and B be local Lipschitz neighbourhood retracts in R. Suppose B C A and B relatively closed in A. Denote by 7r : U -> A and p : V B the retractions of neighbourhoods U of A and V of B, and assume, without loss in generality,
that V C U and moreover that (1 - t)x + tp(x) E V whenever x E V and t E [0, 1]. For k = 0, ... , n, we define relative cycles by
Zk (A, B, R) := IT E Fk,cpt (U) I spt T C A, spt 8T C B, or k = 0}, relative boundaries by
Bk(A,B,R) ._ {R + 8S I R E Fk,cpt (V), spt R C B, SE Fk+l,cpt (U), spt S C A}, and the real relative homology groups by Hk (A, B, R) = Zk (A, B, R) /Bk (A, B, R).
We point out that all currents involved have compact supports and that the retractions 7r and p induce continuous maps 7r# : Fk,cpt (U)
Fk,cpt (A),
p# : Fk,cpt (V) - Fk,cpt (B)
The relative homology groups defined above satisfy the Eilenberg-Steenrod axioms for a homology theory in the local Lipschitz category. We recall that the local Lipschitz category consists of couples (A, B) of local Lipschitz neighbourhood retracts, with A D B, B relatively closed in A, and , morphisms f between couples
3.5 Relative Homology and Cohomology Groups
609
(A', B') are Lipschitz maps from A to A' satisfying f (B) C B'. It is not difficult to prove using Federer's flatness theorem that any such morphism induces a map f : (A, B)
f# : Hk (A, B, R) -+ Hk (A', B', R)
compare Federer [226, 4.4.1]. Also if C C B C A are local Lipschitz retracts in W1, the boundary operator acts as
a: Zk(A,B) -* Zk-1(B, C), 8: Bk(A,B) --> Bk-1(B,C), thus induces a map again denoted by 8
8 : Hk(A, B) -* Hk_1(B, C). Here we understood the dependence on R, Hk (A, B) := Hk (A, B, R).
Theorem 1. We have (i) If f : (A, B) -> (A, B) is the identity map, then f* : Hk (A, B) --> Hk (A, B)
is the identity map. (ii) For f : (A, B) -> (A', B') and g : (A', B') --+ (A", B") we have
(g o f)* = g« o f*.
(iii) IfCCBCA, C'CB'CA', f:(A,B)-+(A',B'), fB:(B,C)->(B',C') and k > 0, then
fiB*a = of*. (iv)
If i : (B, C) -* (A, C) and j : (A, C) --+ (A, B) denote the inclusion maps, then the sequence of homomorphism
a, Hk (B, C) (v)
Hk (A, C) '-', Hk (A, B)
Hk-1(B, C) A .. .
is exact. I f h : ([0,1] x A, [0,1] x B) --> (A', B') and h* : (A, B)
(A', B'),
h* (x) = h(t, x), 0 < t < 1, x E A,
then ho* = h1*. (vi)
If i (A, B) --. (A', B') is an inclusion map such that B = A f1 B' and clos (A'\B')f1(clos (A'\A) = 0, or equivalently B C A C A', B' = BU(A'\A) :
and clos (A \ B) n clos (A' \ A) = 0, then i* is an isomorphism. 'vii) If x E R', then Ho({x}) ^ Z and Hk({x}) _ {0} for k > 0.
Suppose now that A and consequently B are compact. Then one can construct a CW-complex LA surrounding A and contained in U which has a sub complex LB surrounding B and contained in V by selecting cubes not far from A of a standard subdivision of IR in cubes with a suitable small mesh size. As in the case B = 0 one proves
610
5. Cartesian Currents in Riemannian Manifolds
Theorem 2. Let A be a compact. Then Hk (A, B, IR) is isomorphic to the relative homology of Lipschitz chains of (A, B). Moreover Hk,,impi (LA, LB, R) can be mapped linearly onto Hk (A, B, R). In particular the vector space Hk (A, B, R) has finite dimension. In this more general context, following the same path of Sec. 5.3.2 one derives an isoperimetric inequality
Theorem 3. Let A D B be compact neighbourhood Lipschitz retracts. There exists a constant r > 0 such that, if T E Zk (A, B) is a relative homology cycle of finite mass, M(T) < oo, with T = X+aY with X E Fk,,pt(V), Y E Fk+1,cpt(U), then T = Bk(A, B), i.e.,
T=R+aS,
with R E Fk,,pt(V), spt R C B, S E Fk,C t(U), spt S C A, and
M(R) + M(S) < TM(T); in particular S E Nk+l (U)
As already remarked it is useful to work with homology group defined as equivalence classes of flat chains as no control on the mass of the boundaries is required. We remark however that as far as homology groups are concerned this is quite irrelevant: replacing flat chains by normal currents leads in fact to isomorphic relative homology groups. This follows from the following proposition, compare Sec. 5.1.3,
Proposition 1. If T' E Zk (A, B), then there is T E Zk (A, B) n Nk (U) such
that T-T'EBk(A,B). As in the boundaryless case, a consequence of the isoperimetric inequality is the weak closure of relative homology classes. In fact we have
Theorem 4. Let {Tz} C Zk(A, B) be a sequence which is equibounded in mass and weakly converging to T. Then T L (A \ B) E Fk,,pt(U) and T L (A \ B) belongs to Zk (A, B). Moreover if the Ti E Bk (A, B) are relative boundaries, then T L (A \ B) is a relative boundary. Proof. First notice that by semicontinuity T has finite mass, hence T L (A \ B) is well defined. In order to prove that T L (A \ B) is a flat chain we set g(x) _ dist (x, B), x E U, and for r > 0
H(r) := {x E U I g(x) > r}.
Being spt aT C B = U \ H(0), from the slicing theory we have for a.e. r,
0
= a(T L H(r)) - aT L H(r) = a(T L H(r)) and
3.5 Relative Homology and Cohomology Groups
1/i
1/rj
f
611
M(a(T L H(r))) dr =
0
J0
M() dr < cM(T).
Consequently we can find a sequence of numbers {rj }, M(a(T L H(rj)) < oc, i.e., we have
rj - 0, such that
T L H(rj) E Nk(U). Also
M(T L H(rj) - T L (A \ B)) = lim M(T L (H(rj) - H(0))) 00 jlim 3-00 hence F(T LH(rj)-T L (A\B)) - 0, which proves that T L (A\B) E Fk,,pt(U) As trivially a(T L (A \ B)) C B, the first part of the claim is proved too. If the Ti's are relative boundaries, the isoperimetric inequality yields
Ti = Ri + 8Si
where Ri E Fk,cpt(V), sptRi C B, Si E Nk+1(U), sptSi C A, and
M(Rj) + M(Si) + M(aSi) < cM(Ti). Passing to the limit on a suitable subsequence, we then infer T = R + aS with R E Dk M, spt R C B, S E Nk+l (U), spt S C A; therefore
T L (A \ B) = R - T L B + aS. Being S normal in U with spt S C A, S is a flat chain, hence aS E Fk (U); by the first part of our claim R-T L B E Fk,cpt(U). Finally, since spt (R - T L B) C B, we conclude that T L (A \ B) E Bk(A, B).
We notice explicitly that we cannot expect in general that T belongs to Fk,Cpt(U). As corollary of Theorem 4 we can now state
Theorem 5. We have (i)
In each relative homology class y c Hk(A, B, ]R) there is a homological minimizer, that is a T E Zk(A, B, IR), [T] rel = y such that
M(T) = inf{M(R) I R E Zk (A, B, IR), [R]r.1 = y} (ii) In particular the function
M(y) := inf{M(R) I R E Zk(A, B, ]R), [R]rel = 71 defines a norm on Hk (A, B,1R) .
612
5. Cartesian Currents in Riemannian Manifolds
Proof. First we observe that in every relative homology class there is a current with finite mass, hence A := inf{M(R) I R E Zk(A, B, R), [R]rel = 'Y} < oo.
The claim in (i) then follows at once from the semicontinuity of the mass and from Theorem 5. The claim in (ii) is trivial.
Actually the previous results extend to the quite more general situation in which one does not assume A compact, but with a number of changes and technical difficulties. For instance homology groups are vector spaces but in general of infinite dimension, in fact they are inductive limits of Banach spaces endowed with a flat norm. Also, in dealing with such a general situation one cannot avoid some compact control on the location of the supports of involved currents. In some sense the homology vector space can be exhausted by a sequence of finite dimensional vector spaces. In fact one has that the vector subspace of Hk (A, B, R) consisting of those homology classes which meet Fk,K(R), K fixed compact, has finite dimension. To keep our extension simple we shall not discuss this general approach and we refer the interested readers to Whitney [674], Federer [226] [228].
Relative de Rham cohomology. Let A, B local Lipschitz neighborhood retracts, B C A, B closed relatively to A, and let it : U -> A the retraction. One then defines the de Rham cohomology groups HHR(A, B)
H4R(A)
Zk(U, B) Hk (A, 0)
where
Zk(U,B):={wEZk(U)1w=0 near B} and for w, 77 E Zk(U) w - 77 if there exist an open set V, A C V C U and a form
a E Zk-1(V,B) such that w - 77 = da in V. One has of course the immersion map j# : HHR(A, B) -> HHR(A), the restriction map i# : HaR(A) -* HHR(B) which restrict forms in HHR(A) to their values in B, and a coboundary operator 8 : HHR(B) - HaR 1(A, B) defined as follows. Choose an open set U' such that B CC U' CC U n V and extend w1u, to w E Dk(U). Of course dw = 0 near B, hence dw E Zk+1 (A, B); one defines b([w]B) :_ [dW]rel
noticing that the relative homology class of w depends only on the cohomology of w.
It is easy to prove that in case A is a compact oriented submanifold X and B = 8X, that the above definitions agree with the ones in Sec. 5.2.6, compare Proposition 1 in Sec. 5.3.3. Moreover a rewriting of Theorem 2 in Sec. 5.2.8 yields
3.5 Relative Homology and Cohomology Groups
613
Theorem 6. The long sequence
? HaR(A) i ' Hax(B) a' HaR1(A,B) - HaR'(A) is exact.
de Rham duality in the Lipschitz category. In the attempt to generalize de Rham duality to the Lipschitz category, Whitney and then Federer introduced the notion of flat cochains which play the role of non smooth differential forms. In order to give a simple presentation we assume A compact, and refer to Federer [228] for general results and proofs. Set Fk(A)
Fk,A(U),
we would have to write instead Fk(A) := U{Fk,K(U) I K C A, K compact} for general local Lipschitz neighbourhood retract A.
Definition 1. We define (i) flat cochains as linear continuous functionals on Fk (A)
Fk(A) :_ {a : Fk(A) -+ R I a linear and continuous} (ii) relative cochains, cocycles and coboundaries as
Fk(A, B)
Zk(A,B)
{a E Fk(A) {a E Fk(A)
Bk(A,B){aEFk(A)
a = 0 on Fk(B)} a = 0 on Bk(A,B)} a = 0 onZk(A,B)}
(iii) cohomology groups as
Hk(A, B) = Zk(A, B)/Bk(A, B). We recall, see Sec. 5.1.3, that Fk (A) endowed with the flat norm is a Banach space. One sees immediately that Zk (A, B) is closed in Fk (A), and, using the isoperimetric inequality, that also Bk (A, B) is closed in Fk (A). It follows that inf{Fk (S) I S E 'y},
'y E Hk (A, B, ]R)
defines a norm on Hk (A, B, I[8) with respect to which the coset map p : Zk (A, B, IEP)
-' Hk (A, B, R)
is continuous, being Hk (A, B, R) finite dimensional. It is worth to notice that instead relative boundaries with equibounded masses are in general not compact
5. Cartesian Currents in Riemannian Manifolds
614
with respect to the flat norm. For instance take A :_ R2 and B :_ R2 fl {x I x2 = 0}. Then the sequence of flat relative boundaries
T i :=iQ(Pl,P2)Jj+iQ(P2,P3)1 +iQ(P3,P4)I
i = 1,2,...,
where
P1 := (0, 0), P2 := (0, i-2), P3 := (i-1, i-2), and P4 :_ (i-1, 0),
converge to the current boner which is not flat. Clearly every relative cocycle a E Zk(A, B) induces by restriction a linear map a : Hk (A, B, R) -* R. Consequently there is a natural morphism Hk (A, B) -* Horn (Hk (A, B, Ilk), R)
which is obviously injective. On the other hand if 2 E Hom (Hk (A, B, R), R), 8 o p defines a linear continuous map from Zk (A, B) into R which extends by Hahn-Banach theorem to a linear continuous map still denoted 2
t: Fk(A) -* R. Therefore we conclude
Proposition 2. Hk (A, B) f-- Horn (Hk (A, B, L), R) Notice that every smooth form w defines by iv(T) := T(w)
a flat cochain iv E Fk(A). The main question becomes now that of realizing Hk(A, B). A key point in this respect is the following
Theorem 7. We have
Bk(A, B) _ {d1 10 E F-1(A, B) On the other hand, a regularization procedure, compare Federer [228], then allows to prove
Proposition 3. We have Hk(A, B) = HdkR(A, B)
Consequently Theorem 7 yields the following extension of the de Rham theorem
Theorem 8. We have
4 Integral Homology (i)
615
For every v E Hom (Hk (A, B, IR), R) there exists a closed differential form w E Ek (U) which vanish in a neighbourhood of B such that v([T])
(ii)
There exists neighbourhood U' C U and V' C V such that, if w E Ek(U), dw = 0, w = 0 in a neighbourhood of B, and d T E Zk (A, B)
T (w) = 0
then w is exact in U', i.e. w = dcp,
WE Ek-1(U') and spt cp C U'\ V'.
In particular the duality between forms and currents yields to a non degenerate bilinear form
<,>:Hk(A,B,lR)xHHR(A,B) -> R. Finally the diagram
...
*-
HHR(B)
{-
HHR(A)
x<, >
x<, > Hk (B
F-
Hk (A, IR)
Hk(A, B)
4 .. .
x<, >
4
Hk (A B R)
is commutative.
4 Integral Homology In this section we discuss the integral homology of a compact, oriented manifold in terms of currents.
4.1 Integral Homology Groups In this subsection we discuss the integral homology of a compact, oriented manifold in terms of currents, in particular we shall prove that homology classes are weakly closed. Actually we shall see that the cosets of homology classes are in fact connected components of integral cycles.
Integral absolute homology groups. Let A be a compact neighborhood retract of IEBn, let -7r
: U --- A be the retraction. We can and do assume that U= {x E ][8'Z
I
dist (x, A) < eA}
for some CA > 0. By decomposing IR' into cubes of size eA/2\, and denoting by G the finite family of those cubes which cover A, we can see that
5. Cartesian Currents in Riemannian Manifolds
616
U{Q I QE.C} cc U. From now on we denote by LA the CW cubical complex over cubes in L and by Lk,A the k-skeleton of LA, k = 0, ... , n. Similarly to Sec. 5.3.2 an integral chain P on Lk,A is simply a polyhedral chain with integral coefficients
/9 EZ, FiELk,A. Setting Zk,simpl(LA, Z) Bk,simpl (LA, Z)
{P P integral polyhedral chain on Lk,A, aP = 0} {P = aQ I Q integral polyhedral chain on Lk,A}
we define the simplicial integral k-homology of LA by Hk,simpl(LA, Z) := Zk,simpl (LA, Z)/Bk,simpl (LA, Z)
which is of course a finitely generated Z-module. We then set for k = 0, ...
,n
Zk(A,Z) :_ {T E Zk(U), 8T = 0, sptT c Al Bk (A, 7L) :_ {aS, S E Zk+1(U), spt S C Al
Then the integral k-homology group of A is defined by
Hk(A,Z) := Zk(A,Z)/Bk(A,Z). Proceeding as for Theorem 6 in Sec. 5.3.2, taking into account that the deformation theorem produces i.m. rectifiable decomposition of i.m. rectifiable currents, we get
Theorem 1. The module Hk(A,Z) is isomorphic to the k-homology of Lipschitz chains in A with integral coefficients. Moreover the retraction map 7 induces an epimorphism 7r,, : Hk,simpl (LA, Z) --y Hk (A, Z). In particular Hk (A, Z) is finitely
generated.
Of course, if A is a C' manifold then Hk(A, Z) and Hk,simpl(LA, Z) are isomorphic, compare Theorem 5 in Sec. 5.3.2. Again a relevant advantage in representing the singular homology in terms
of homology groups of currents is that the cosets in Zk (A, Z) turn out to be closed with respect to the weak convergence of currents. Actually in the case of integral homology we are dealing now, we shall prove that in fact the cosets are connected components of Zk (A, Z). The key step in doing that is an isoperimetric inequality which is more stringent than in the real case. 1 The following example shows that one cannot expect isoperimetric inequality for general currents on manifolds, a size condition is necessary.
4.1 Integral Homology Groups
617
Let S2 = {x E R31 IxI = 1} be the two sphere and let E > 0 be small enough. Let
M =Stn {x E JR3 I x3 >-I+ E} and C = S2 n {x E R3 I x3 = -1 + 2e}. Clearly C is a 1-cycle in M and bounds the surface S := S2 n {x E ][83 I x > -1 + 2e} in M, C = aS, but M(S) 47r,
while M(C) ^-, O(/). If one consider
M=S2nxER3Ix<1-E { } then the cycle C does not bound any surface lying in M.
Theorem 2. For k = 0, ... , n there is a constant e = E(k, n, EA) such that if T E Zk(A,Z) with M(T) < e then T E Bk(A,Z); more precisely T = aR, R E Zk+l(U), sptR C A and M(R) < cM(T) 1+1/k where c = c(k, n).
Proof. Using the deformation theorem one can deform T over Lk,A and write T = P + 8R' where R' E Zk+1(U), spt R' cc U, P = > /i Q Fi , Fi E Lk,A, Qi E 7 G and M(P) < cM(T), M(R') < co-AM(T). As /3 i > 1, M(P) a Therefore choosing e < oA/c, we infer that if M(T) < e then P = 0. Consequently, setting A.
R = 7r# (R'), we infer
T=7r#T=aR, REZk+l(U), sptRcA and
M(R) < c1M(R') <
cl(cM(T))l+11k
The following theorem yields the key decomposition argument, which is relevant also in the context of flat chains, compare Theorem 2 in Sec. 5.1.3.
Proposition 1. Under the assumption of Theorem 2 in Sec. 5.1.3, if moreover all Ti and T are i.m. rectifiable, we can find integral currents Ri E Zk+1(A) and Si E Zk (A, Z) such that
Ti-T=Si+MRi.
M(Ri)+M(Si) -> 0,
Proof. We proceed as in the proof Theorem 2 in Sec. 5.1.3 observing that all currents Pi, Q. , Fi, Gi which are involved in the proof of Theorem 2 in Sec. 5.1.3 this time are i.m. rectifiable and the convergences k
EFj j=1
are in the mass norm.
00
j=1
k
00
E Gj
EGi
j=1
j=1
618
5. Cartesian Currents in Riemannian Manifolds
From Theorem 2 and Proposition 1 we readily infer
Theorem 3. Let Ti, T E Zk (A, Z) be integral k-cycles, Ti -+ T. Then Ti - T = 8R1i Ri E Zk+1(A) for large i. Moreover M(Ri) -- 0. Proof. In fact using Proposition 1 we can write
Ti-T=Si+8R2 Si E Zk(A), R' E Zk+1(A) and M(R'i) +M(Si) -+ 0. The isoperimetric inequality of Theorem 2 implies that Si = OZi, Ei E Zk+l(A), M(Ei) < cM(SS)l+1/k for large i. Thus
Ti-T=a(R'+E'i)
and the claim follows. The following corollary collects some important consequences of Theorem 3.
Corollary 1. We have (i) Bk (A, 7Z) is weakly closed. (ii) Zk (A, Z) \ Bk (A, Z) is weakly closed. (iii) In each integral homology class y there is a mass minimizing i.m. rectifiable cycle, that is, there is T E Zk (A, Z), [T] = y, such that
M(T) := inf{M(S) (iv)
I
[S] = y}.
Given c > 0 there are at most a finite number of integral homology classes y such that inf{M(S) [S] = y} < c. I
(v) For a non trivial integral homology class
inf{M(S)
I
[S] = y} > 0.
Proof. (i) and (ii) follows trivially from Theorem 3. (iii) trivially follows from (i). To prove (iv) assume on the contrary that one has a sequence {Ti} of cycles of equibounded masses. Passing to a subsequence Ti - T and by Theorem 3, all
the Ti but a finite number belong to the same homology class, a contradiction. (v) is just the isoperimetric inequality, Theorem 2.
Relative integral homology. A parallel theory for integral relative homology can be developed. Let A, B be two local Lipschitz neighbourhood retracts in Rn, B C A, B relatively closed in A. F o r k = 0, ... , n, we define the integral relative cycles by
Zk (A, B, Z) :_ IT E
. k (IR')
I spt T C A, spt aT C B, or k = 0},
the integral relative boundaries by
Bk (A, B, Z) := IT + aS I T E .Fk (Rn), spt T C B, S E Fk+l (Rm), spt S C Al,
4.1 Integral Homology Groups
619
and the integral relative homology by
Hk(A,B,Z) = Zk(A,B,Z)/Bk(A,B,Z) One shows that the Z-modules Hk (A, B, Z) satisfy the Eilenberg-Steenrod axioms for an homology theory with integral coefficients in the Lipschitz category, compare Federer [226, 4.4.1], and, being A compact that we have as consequence of the deformation theorem Theorem 4. Hk (A, B, Z) is isomorphic to the k-homology of integral Lipschitz chains in (A, B). Moreover one has an epimorphism of the k-simplicial homology Hk (LA, LB, Z) of (LA, LB) onto Hk (A, B, Z). In particular Hk (A, B, 7G) is finitely generated. From the deformation theorem we also infer the following isoperimetric inequality, compare Federer [226, 4.4.2]
Theorem 5. Let T E Rk(Rn) fl Zk(A, B) fl Bk(U,V). There exists an integral current S such that spt S C A, spt (T - 8S) C B, and M(S)k/(k+l) + M(T - 8S) < cM(T), c being an absolute constant.
Proceeding as in the real case and taking into account Federer-Fleming closure theorem one infers Theorem 6. Suppose that Ti E Rk (U) fl Zk (A, B, Z) are relative cycles converging weakly to T. Then T belongs to Rk (A \ B) and T L (A \ B) E Zk (A, B, Z). If moreover the Ti's are relative boundaries, then T is also a relative boundary, T E Bk(A,B,7G) and consequently, compare Federer [226, 4.4.2]
Theorem 7. We have (i) In each relative homology class -y c Hk(A, B, Z) there is a homological mass minimizer, i. e. a T E Zk (A, B, Z) with [T]rel,z _ 'y and
M(T) = inf{M(R) I R E Zk (A, B, Z), [R]rel = 7} (ii) There are finitely many integral relative homology classes which meet the set
{T E Zk(A,B,Z), M(T) < c}. Real and Integral homology. Let us discuss briefly relationships between integral and real homology in terms of cycles. Let A, B be compact neighbourhood retracts of R'. According to Theorem 1 the integral homology groups are Abelian groups of finite type. The general presentation theorem for such groups says that Hk (A, B, Z) splits as a direct sum of groups with a single generator; more precisely, one can choose exactly q elements
5. Cartesian Currents in Riemannian Manifolds
620
yl...... q E Hk (A, B, Z), where q is the rank of Hk (A, B, Z), independent over Z such that q
Hk(A,B,Z) _
(1)
Z'Y, ®Tk,
where Tk is the torsion subgroup of Hk(A, B, Z), i.e.,
Tk:={7EHk(A,B,Z) 1 2pEZ, py=0}. In other words each y E Hk (A, B, Z) decomposes uniquely as
y = 1: ni'yi + w,
ni E Z, w E Tk.
i
It is also a very well known fact of homological algebra that the integral homology groups fix the homology with any group coefficient. In fact the universal
coefficient theorem, compare e.g. Spanier [607, p. 222 and p.243] express the homology with coefficients in G as
Hk(A,B;G) =Hk(A,B,Z) ®G®Tor(Hk_l(A,B,7G),G),
(2)
where the Tor groups are the "torsion" subgroups. In case G = IR (2) reduces to Hk (A, B, R) = Hk (A, B, Z) ®R,
(3)
which in terms of the generators of Hk (A, B, Z) means
Theorem 8. Let q = rank Hk (A, B, Z) and let S1,.. . , Sq be q integral relative cycles in Zk (A, B, Z) such that [Si]rel,z,
,
[Sq]rel,z
are independent over Z and q
Hk (A, B, Z)
Z[Si]rel,z + Tk (A, B).
Then [Sl]re1,II2,
,
[Sq]rel,R
yield a basis for Hk (A, B, R). In particular dim Hk (A, B, R) = rank Hk (A, B, Z)
A simple consequence is
Theorem 9. Let dim Hk (A, B, lid) = q. Then we can always choose q integral cycles Ti in Zk (A, B, Z) such that
4.1 Integral Homology Groups (i)
621
[Tl]a,...,[Tq]z are independent over Z and generate the free part of the
homology group Hk (A, B, Z). (ii) [T1]lk,... , [Tq]R forma basis for Hk (A, B, ]R).
For the sake of completeness, we give a direct proof of Theorem 8 in terms of currents without any mention of the algebraic theory in the case of the absolute homology. Modulo additional technicalities, the general case of relative homology can be treated in the same way, compare Federer [228, 5.7]. From now on we assume B = 0. It is enough to prove Theorem 8 for the simplicial homology Hk,simpl(LA,Z). For that we need the following lemma
Lemma 1. Let N > 1 and E C QN be a subset of QN, and let r E IRNbe such that (r I a) = 0 d a E E. Then for any e > 0 one can find q E cQN such that
Iq - rI < Eand(gI o,)=0Vo'EE. Proof. We denote by S1, ... , Sk a basis for the Q span of r1, k
. . .
, rN and we write
1 Q, i = I,-, N. sj E IIB, aij E
ri = E aij sj, j=1
Then EN1 riai = 0 is equivalent to EN aij o i = 0 j = 1, ... , k. Choosing now T = (Ti, ... , Tk) in such a way that N
k
EIax7I ITj -Sjj <E i=1 j=1
Ej=1 aijTj has the required properties.
we see that q E QN given by qj w`
0
Notice that ee can choose qj = ri if ri happen to be rational. Next proposition contains one of the key points in the proof of Theorem 8.
Proposition 2. Let T E Zk (LA, Z) fl Bk (LA, ]R), i. e. T is an integral polyhedral cycle, which is a boundary in the real sense, T = 8R, R E Pk+l (LA, R). Then, for each E > 0, one can find a polyhedral chain R' with rational coefficients such
that T = 8R' and M(R - R') < E.
Proof. Denote by {Fi}, i = 1, . . . , I, and {Gj }, j = 1, ... , J respectively the k-faces and the (k + 1) faces of LA with some orientation. We must have I
BQGj j =
E Z. i=1
Moreover, if we write
T=>n4QFjI,
nje7G,
R=I:rjQGjl, rjER
622
5. Cartesian Currents in Riemannian Manifolds
T = aR means ni = Ejj=1 rj Bji, d i = 1, ... , I. Consequently by Lemma 1, given e > 0, one finds sj E (0 such that
ni = E sj8ji. j-1 Therefore, if R' := Ej sjIGj , we have 8T = R' and
N(R - R') < E N(Q Gj Irj - sj I < ce-
j
Remark 1. We observe that T is a polyhedral chain with rational coefficients if for some p E Z, pT is an integral polyhedral chain. Accordingly we could restate Proposition 2.
Denote by ik : Hk (LA, Z) --+ Hk (LA, R) the group homomorphism which associate to each integer homology class [T]z the corresponding real homology class j*([T]a) = [T]R.
Corollary 2. The kernel of j*, ker j*, is the torsion subgroup of Hk(LA,Z), i.e., for T E Zk(LA,Z) [T]R = 0
if
[T]z is a torsion class.
Proof. If [T]a belong to the torsion subgroup of Hk (LA, Z) then for some p p[T]z = 0. Consequently
0,
[T]1 = pp[T]R = p[pT]n = pj*([pT]z) = 0.
Conversely assume that [T]R = 0, that is T E Zk(A,Z) and T = aR, R E Pk+1(LA, R) By Proposition 2 we find R' E Pk+1(LA, Z) and an integer p # 0 such that
pT = aR' therefore [pT]z = p[T]z = 0.
Proof of Theorem 8. We split the proof into three steps. Step I. [T1]R, ... , [TT]R are independent over R.
Assume 9=1 ri [Ti]R = 0. Since R is a vector space over Q, denoting by $1, ... , sk a basis for the Q-span of r1,... , rq, we write ri = Ekj=1 ai7 E Q, sj E R. Consequently q
aij[Ti]IIP=0 i=1
b j=1,...,k.
4.1 Integral Homology Groups
623
Denote by p the 1.c.m. of the denominators of the aid. The current T k
Ek
qaijTi is an integral cycle and k
aii [Ti]R = 0.
[T ]R = q i=1
By Corollary 1 [T] z belongs to the torsion subgroup of Hk (LA, Z) hence
0 = [T]z = j:(gaij)[TT]z. Consequently aid = 0, V i, j and therefore ri = 0. Step 2. Real homology class of rational polyhedral cycles are linear combinations o f [T1]R, ... , [Tq]R with rational coefficients.
In fact we can write any polyhedral cycle T with rational coefficients as
T= 1 S
p where p E Z and S E Zk (LA, Z) . Since [S]z =
ni [Ti]z + w, ni E Z, w E Tk we
get q
[SIR =
57, ni[Tilpp. i=1
Consequently q
[S]R =
n2
[Ti]R-
i=1 p Step 3. Real homology classes of rational polyhedral cycles are dense in Hk (A, R). In fact for T E Zk (A, l) we find, using Proposition 2, a sequence of rational polyhedral cycles Tj such that M(Tj - T) --> 0. As
II[T]R-[Tj ]jII: M(T-Tj) --+ 0 the proof is concluded.
Combining Theorem 8 and de Rham's theorems, compare Sec. 5.3.2 and Sec. 5.3.3, we also get
Theorem 10. Let X be a n dimensional oriented compact manifold. Assume one has q cycles TI, . . . , Tq E Zk (X, Z) such that the corresponding homology classes are independent over Z and generate the free part of Hk (X, Z). Then one can find q closed k-forms w1, ... , wq E Zk (X) such that
Ti(wj) = -5 .
Analogously, given q' relative cycles Ti, ... , T9, E Zk (X, 8X, Z) such that the corresponding relative homology classes are independent over Z and generate the freeqq,part of Hk (X, 8X, Z), then one can find q' relatively closed k - forms 771, ... , E Zk(X, 8X) such that Ti,
51
5. Cartesian Currents in Riemannian Manifolds
624
Proof. By Theorem 8, [Ti]3t generate the real homology Hk(X,IR) of X. By de Rham's duality HaR(X, Ill) is dual to Hk (X, R) therefore choosing dual cohomology class [w1],..., [Wq]
of [Tl]R, ... , [TT]a yields the right forms. The second part of the claim is identical.
In terms of the forms w1, ... , wq, 771, ... , q ' of Theorem 10 we can state
Corollary 3. We have (i) Let T E Zk(X,IB). Then T is (real) homologous to an integer cycle if
T(wi)EZdj=1,...,q.
(ii) Let T E Zk (X, Z). Then T (wi) = 0 V j = 1, ... , q if and only if [T]z belongs
to the torsion subgroup of Hk (X, Z). (iii) Let T E Zk (X, 3X, Ill). Then T is (real) homologous to an integer cycle if
T(wi)EZbj=l,...,q.
(iv)
Let T E Zk (X, aX, Z). Then T (rp) = 0 V j = 1, . belongs to the torsion subgroup of Hk (X, aX, Z).
.
.
, q if and only if [T]Z,rel
4.2 Intersection in Integral Homology Let X be a n-dimensional oriented compact submanifold of I[Bn+N The intersection maps in real homology that we have defined in Sec. 5.3.4 extends to integral homology. The idea is to define [S] n [T] by taking the homology class of the intersection of S with almost any translation Ta#T of T. We first do that in the case of boundaryless manifolds.
Intersection of cycles on boundaryless manifolds. Assume that aX = 0 and for convenience set for T E Dk (X) and x, a E Rn+Iv T. := Ta,#7r*T.
Proposition 1. Let S E Zk(X, R) and T E Zh(X, IR), h + k > n be two normal cycles. Then we have (i) For a. e. a close to zero S n Ta exists in Zh+k-n (X) and [S n Ta]IlZ = [S]l[t nx [T]R-
(ii) If moreover S and T are integral, then for a. e. a close to zero S n Ta is an integral cycle, SnTa E Zh+k_n(X,Z) and its integral homology class depends only on the integral homology classes [S] E Hk (X, Z) and [T] E Hh(X,ZG). Proof.
(i) Let PT E Z,,,-h(X) be a Poincare dual form of T so that T =
fx APT + 3E, E E Nh+l(X), or, setting w :=
T=QXLw+aE.
(-1)h(n-h)PT
4.2 Intersection in Integral Homology
625
Taking into account (iii) of Proposition 3 in Sec. 5.3.2 7r*(QX I LwT) =I U I L7r#wT
and therefore
7r*T= QUj L7r#wT+a7r*E By slightly translating we then infer Ta, = Q U L 7 a7r#wT +
aEa
in U.
near X.
By Proposition 2 in Sec. 5.3.4 we then infer that S n Ea, S n BEa,, SnTa, exists as normal currents on X for a.e. a close to zero. Consequently being aS n Ea, = 0 we infer
a(SnEa)=SnBE,
and (1)
SnTa = S n I U J1 L 7- a7r#wT + a(S n Ea)
near X.
for a.e. a close to zero. Since on the other hand 7- a7r#wT is cohomologous to
7r#WT with compact support in X \ BX, we obtain from (1) that s n Ta is homologous to
SnQU]IL7r#WT=SL7r#wT=SLWT. (i) now follows if we compute
SLWT =
(_1)('n-k)(n-h) S(-APT)
(-1)(n-k)(n-l,)
(ii) We already known, compare (iii) of Proposition 2 in Sec. 5.3.4, that S n Ta
is an integral current for a.e. a close to zero. We now prove that the integral homology class of s n Ta is independent of a for a.e. a close to zero. In fact, being To, and T homotopic,
Ta = To + BE
near X,
E E -Th+N+1(U). Therefore for JbI close to zero, Ta+b = Tb + BE'
near X,
E' E Zh+N+1(U) Proposition 2 in Sec. 5.3.4 then yields that for a.e. b the currents SnTa+b, S n Tb, S n Eb exist as integral currents in X, and B(S n Eb) _ S n BEb. Consequently
SnTa+b-SnTb=a(SnE') inX for each a and a.e. b close to zero. In particular the integral homology of s n Ta is independent of a for a.e. a.
On the other hand, if T = aR, R E Zh+1(X) or S = aQ, QE Zk+1(X), Proposition 2 in Sec. 5.3.4 again yields that for a.e. a close to zero S n (BRa) = B(S n Ra),
being aTa = BS = 0.
(8Q) n Ta = (-1)k+1 8(Q n Ta)
5. Cartesian Currents in Riemannian Manifolds
626
The claim (ii) of Proposition 1 clearly allows us to define the intersection map in integral homology
nx : Hk(X,7L) X Hh(X,Z) -4 Hh+k-n(X,Z) setting
[S]z nx [T]z := [S n Ta]z
for a.e. a,
while (i) of Proposition 1 show that such a map is compatible with the real version of the intersection map defined in Sec. 5.3.4.
Intersection of cycles on manifolds with boundary. More tricky is the
definition of the intersection of cycles for manifolds with boundary. Let us start with the intersection of a relative with an absolute homology class. Let X be a compact oriented manifold of dimension n in ]Rn+N possibly with boundary. Let X1, X2 be two smooth domains in X and let Al C X1 and A2 C OX2 be compact. Since we deal with the relative homology, we suppose even if this is not important for the sequel that Al and A2 are compact Lipschitz neighborhood retracts. Proposition 2. Let S E Zk(XI,A1), T E Zh(X2,A2), h+k > n be two relative normal cycles, and assume that (2)
XlnA2=X2nA1=0,
sptTnX1n8X2=0.
Then for a.e. a close to zero (i) S n Ta exists as an absolute normal cycle, S n To, E Zh+k-n (X1).
(ii) T has a Poincare dual PT E Zn-h(X2) such that PT = 0 on 8X2 n X1 and S n Ta = (-1)(n-k)(n-k) S(. APT) + 8E,
on X1,
Z E Nh+k-n+1(X1) (iii) If S and T are i.m. rectifiable, then S n Ta E Zh+k_n(X1, Z) and its integral homology class depends only on the homology classes [S] z,rel E Hk (Xl, A1i Z) and [T]z,rei E Hh (X2, A2, Z)
Proof. (i) follows from Proposition 2 in Sec. 5.3.4 taking into account that for
a close to zero spt s n spt 8Ta is still empty being X1 n A2 = 0.
(ii) Since spt T n X1 n 8X2 = 0 by (2) then clearly one constructs a Poincare (-1)h(n-h)PT, and for dual form PT with spt PT = 0 on 8X2 n X1. Setting w = 8 := dist (X1, A2),
Xd := {x I dist (x, X1) < 6121,
w extends as a smooth form on X8, w = 0 on X1 \ X2,
T=QX111Lw+R+aZ onX where R E Nh(X2), spt R n Xs = 0, Z E Nh+1(X2). In particular
4.2 Intersection in Integral Homology
627
T=QX11Lw+6E onX5. Arguing now as in the proof of (ii) of Proposition 1 we then conclude that for a.e. a close to zero (3)
SnTa,=SnI7r-1(X1)]Lwa+SnEa,=SLWa+SnEa
(4)
S n aE = a(S n Ea),
where we have denoted by wa the form wa On the other hand wa - w = d,13, 0 E D'-'-1(X2), spt 0 C X2 \ X1. We then have as L Q = 0 and S L (wa - w) = (-1)ka(S L (3) on X1
(5)
Putting together (3) (4) (5) we conclude
onX2 as required. (iii) If now S and T are i.m. rectifiable, then SnTa is an i.m. rectifiable current for a close to zero by Proposition 2 in Sec. 5.3.4. Being Ta and To clearly homotopic we have
Ta:=To+R+aE where E E Nh+N+1(7C-1(X2)), R E Nh+N(7r-1(X2)), sptRn7r-1(X5) = 0, X5 as in step (ii). Consequently for b close to zero Ta+b = Tb+Tb#R+Tb#E near X2, -rb#E E Nh+N+1(7r-1(X2)), Tb#R E Nh+N(7r-1(X2)) and spt75#R n Xl = 0. On the other hand S n Tb# E, S n aTb# E , as n Tb# E, S n Ta+b, S n Tb exist for a.e. b as integral currents,
SnTa+b=SnTb+SnaTb#E onX1
(6)
and
SnaTb#E = a(S n Tb#E) + as n rb#E = a(S n Tb#E)
(7)
being spt rb#E n spt as = 0 for sufficiently small a and b. Putting together (6) and (7) we then conclude that for a.e. a and b S nTa and S n Tb are homologous. To conclude the proof it suffices to note that that for a.e. a close to zero clearly (8)
aR n T a = (-1)k+la(R n Ta),
S n 5Ra = a(S n Ra).
The claim (iii) of Proposition 2 shows that by setting [S]re, n [T] := [S n Ta]
for a.e. a close to zero, [S]rd E Hk(X, aX, Z), and [T] E Hk(X, Z) defines actually the intersection map
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5. Cartesian Currents in Riemannian Manifolds
nx : Hk(X,8X,Z) x Hh(X,Z) - Hh+k-n(X,Z) between the relative and absolute homology classes. Also (ii) of Proposition 2 shows that such a definition is compatible with the intersection map in real homology, compare Sec. 5.3.4.
In order to deal with the intersection of two relative cycles, we need the following variant of Proposition 2. As before we suppose that X1, X2 are two smooth domains in a compact oriented manifold X of dimension n that we think to be a submanifold in Rn+N. Let Al C Xl and A2 C 8X2 be compact Lipschitz neighborhood retracts. Proposition 3. Let S E Zk(X1, Al), T E Zh (X2, A2), h+k > n, be two relative normal cycles, and assume that (9)
X1nA2=0,
sptTnXlnaX2=0.
Then for a.e. a close to zero (i) S n To, exists as a normal current and S n Ta, E Zh+k-n (XI, Al)
(ii) T has a Poincare dual PT E Zn-h(X2) such that PT = 0 on aX2 n Xl and S n Ta, =
(-1)(n-k)(n-k) S(, APT)
+ R + 8E,
in Xl,
R E Nh+k_n(X1), sptR C Al, E E Nh+k-n+1(X1). (iii) If S and T are i.m. rectifiable, then S n Ta0 E Zh+k_n (X1, Al, 7G) and its
integral homology class in Hh+k_n(Xl, Al, Z) depends only on the homology classes [S]z,rei E Hk(Xl,A1iZ) and [T]z,Lel E Hh(X2iA2,Z).
Proof. The proof is quite similar to the proof of Proposition 2. Because of the different hypotheses, we may substitute (4), (5) and (7) respectively by (10)
SnEa=a(SnEa)+(-1)kaSnEa-aE'+R'
R' E Nh+k_n(X1), spt R' C spt S C Al, E' E Nh+k_n+1(Xl), by (11)
S L (Wa - W) = aS L + (-1)ka(S L,8) = R" + aE"
R" E Nh+k-,,,(X1), sptR" C sptS C Al, E" E Nh+k-n+l (XI), and by (12)
SnaTb#E=a(SnTb#E)+aSnTb#E=R'"+aE"'
R"' E Nh+k-n(X1), sptR"' C sptS C Al, E"' E Nh+k-n+1(X1). In order to define the intersection of two relative cycles of X, we recall the notation of Sec. 5.1.2, particularly the extension map
e:X-*Xao,
eo>0.
which is homotopic to the identity on X. Let [S] E Hk (X, 8X, Z), [T] E Hh (X, 8X, Z) be two relative cycles. Then e#T E Zh (XE0 , aXE0,, Z), and x n
4.2 Intersection in Integral Homology
629
aXEO = 0. We then can apply (iii) of Proposition 3 to S and T with Xl := X, Al := aX, X2 := XEO, A2 := OXEO and infer that the homology class of s n (e#T)a,
in Hh+k_,,, (X, aX, 7L) depends only on the relative homology class of S and T for a.e. a close to zero. This then defines the intersection map between relative cycles, nx : Hk (X, aX, 7L) X Hh, (X, aX, 7L) --b Hh+k-n (X, aX, 7L) by
[S]re, nx [T]r.1
[S n (e#T)a]rel
Moreover (ii) of Proposition 3 shows that this definition is compatible with the intersection of relative cycles in real homology that we have defined in Sec. 5.3.4.
Intersection index in integral homology. Let
be two sys-
tems of integral cycles which are independent and generate respectively the free part of Hk (X, OX, 7L) and of Hn_k (X, 7L). Then the intersection index Iij := ix (Si, Tj) takes values in 7L, compare Sec. 5.3.4, and non degeneracy is equivalent to the non degeneracy of the real bilinear form
riSi,Esj Tj) -> ix (S, T) = E ri sj lij. i,j
Thus, being non degeneracy of Poincare duality and of the intersection index equivalent, we have
Proposition 4. Non degeneracy of Poincare duality is equivalent to
dimHk(X,aX,1R) =dimHn_k(X,R) =: q,
(13)
detIii
0.
We can also state
Proposition 5. Non degeneracy of Poincare-Lefschetz duality is equivalent to one of the following equivalent statements: (i) rank Hk(X,OX,Z) =rankHn_k(X,Z) =: q, detIi.7 o 0. (ii) given S E Zk (X, aX, 7L) ix (S, T) = 0 V T E Zn_k (X, 7L) if [S] z,rei is a torsion class. (iii) given T E Zn-k (X, 7L) ix (S, T) = 0 `d S E Zk (X, aX, 7L) iff [T]z is a torsion class.
In fact in case of (ii) non degeneracy of the intersection index on integral cycles amounts to the non degeneracy of the map
Zp X V -> 7L,
(n, m) ---
nimjIi1,
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5. Cartesian Currents in Riemannian Manifolds
that is again to p = q and det IZj 54 0. Therefore we have [S]R,rel = 0, consequently [S]z,rel is a torsion class. The same argument applies for the third claim (iii).
We emphasize the fact that a good basis for Hk (A, B, R) can be obtained only starting from free generators of Hk (A, B, Z). In fact considering a generic basis of H k (A, B, IR), [T1]x, ... , [Tq]R of integral cycles we cannot be sure in principle that [Ti]z,..., [Tq]z generate the free part of Hk(A, B, Z).
An algebraic view of integral homology. Although integral currents are useful in finding representatives of integral homology classes by means of variational integrals, their use in proving the integral version of the Poincard duality isomorphism between homology and integral cohomology is not completely clear. In describing the integral homology of a given manifold X, one usually starts with a triangulation of a manifold and then define the integral chains of k - cells of that subdivision, a boundary operator, hence the homology groups
Hk(X,Z),
Hk(X,aX,7G).
Then, by duality one defines cochains as morphisms on chains of k - cells, a notion of coboundary, and the integral cohomology Hk (X, Z),
Hk (X, aX, 7G)
of the integral cochains. One then proves
Theorem 1. H" (X, Z) = Homz(Hk (X, Z), Z) ®Ext (Hk_ 1(X, Z), Z) Hk (X, aX, Z) = Homz (Hk (X, OX, Z), Z) ® Ext (Hk_ 1(X, OX, Z), Z) In other words Homz (Hk (X, 7G), 7G) and Homz (Hk (X, aX, 7G), 7G) are the free
parts respectively of Hk(X, Z) and Hk(X, aX, Z). If moreover Hk_1(X, Z) (resp. Hk_1(X, aX, Z)) is free, then Hk (X, Z) (resp. Hk (X, OX, Z)) is free, too. Next point consists in defining a pair of morphisms L : Hk(X, (9X, Z) x L : Hk (X, aX, 7G) X
Hn_h
(X, OX, Z) -+ Hh+k_n (X, Z)
Hn-h
(X, Z) -' Hh+k-n (X, aX, Z)
known as the cap product which mimic the role of the interior product between forms and currents in real homology. Applying such maps with the fundamental cycle Q X E Hn (X, OX) as first factor, one constructs the two PoincareLefschetz morphisms PO :
Hn-h(X, aX, Z)
Hh(X, Z) Pb : Hn-h(X, Z) -i Hh(X, aX, Z)
The celebrated theorems of Poincar6 and Poincard-Lefschetz then state
5 Maps Between Manifolds
631
Theorem 2. PO and Pb are isomorphisms of Z-modules. Finally, combining Theorem 1 and Theorem 2 we get Homz(Hk(X, Z), Z) = Hk,free(X, Z) = Ilnfrek(X ax Z) Homz (Hk (X, aX, Z), Z)
= Hk,free (X, aX, Z) = Hnrek (X, Z)
In other words defining the intersection index on X
ix : H,,-k(X, 3X, Z) x Hk(x, Z) -> Z by (14)
ix(S,T) :_ [S]re1 L (P)-1[T],
we conclude that
Theorem 3. ix is unimodular. It turns out that the intersection index that we have defined in terms of currents agrees with the one defined in (14). From the current theory we already known that the intersection matrix {III} for two given systems of generators {S} of Hk ee (X, OX, Z) and } of Hnrek (X, , Z) have integral non zero determinant, 0 # det I2j E Z. The algebraic integral homology theory adds to this information that det I2j = ±1. In particular for k = 0, ... , n one can choose of the free part of Hk (X, aX, Z) and a basis {T3c -k) } of the free a basis {T('L-k)
part of Hn_k(X,Z) such that ix(,(kT(n-k)) =
a?.9.
5 Maps Between Manifolds In this final section we discuss Sobolev and Cartesian maps and currents between Riemannian manifolds focusing on some topological properties of those maps. In Sec. 5.5.1 we deal with Sobolev maps, reviewing some of Bethuel's results
on density of smooth maps and of White's results on the d-homotopy type of Sobolev maps.
After introducing Cartesian maps and currents between Riemannian manifolds and stating a few simple extensions of results we have already proved in the previous chapters for Cartesian currents in Euclidean spaces, we illustrate in Sec. 5.5.3 and Sec. 5.5.4 the homological content of Cartesian maps and currents. In particular we shall prove that, similarly to the case of smooth maps, every Cartesian current induces sets of maps in cohomology and homology.
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5. Cartesian Currents in Riemannian Manifolds
5.1 Sobolev Classes of Maps Between Riemannian Manifolds Let 0 be a bounded domain in R'. If aQ is regular, say Lipschitz-continuous, and ,f2 » d2, then every map u E W',P(Q, RN) can be extended to a map u : !2 -4 RN in such a way that ii E W1,P(d2, RN), ii = u in d2, and even u E Wo'P (dl, RN), with spt u CC S?. A sequence of mollifiers of u is then easily seen to converge in the norm of W1,P to u. Therefore we deduce
Theorem 1. Suppose that a12 is Lipschitz-continuous. Then every map u E W1,P(dl,RN) is the W',P-limit of a sequence ui E C' (f2, RN) Actually, the following Meyers-Serrin H = W theorem holds independently of the smoothness of ad2 Theorem 2. The class COO(d2,RN) n W1,P(.f2,RN) is dense in W1,P(Q,RN)
This is proved by taking weighted mollifiers according to the distance of points x c d2 from the boundary a1, compare also the proof of (ii) of Theorem 1 in Sec. 4.1.1. Introducing the space
H1,P(Q,RN) := strong closure of Cl(d2,RN) nW1,P(Q ISBN) (1)
in W1,P(d2,RN)
we can state Theorem 2 as (2)
W1,P(Q,RN) = H1,P(0,RN)
Moreover, recalling that convex sets of a Banach space are sequentially weakly closed if and only if they are strongly closed, one easily deduces that H1,P(d2, RN), also agrees with the sequential weak closure of C, (Q' RN) n W1,P(,(l, RN), i.e., with the notation of Sec. 3.4, (sw-cl- and strong-cl standing respectively for sequential weak and strong closure in W1,P), we have (3)
H1'P(dl, RN) = sw-clw,,p (C1(d2, RN) n W1,P(.fl,
= strong-clwi,p (C1(Q, RN) n W',P(f2 RN)) This is what makes the space H1"P(d2,RN), equivalently W1°P(1?,RN), very useful in finding minima of energy functionals. Let now d2 be a bounded domain of an n-dimensional Riemannian manifold X and let us replace the flat space RN by a compact and boundaryless submanifold y of RN of dimension m. We set (4)
W"P(Q, y) := {u E W1"P(d2, RN) I u(x) E y for a.e. x E Q}.
We shall see that in general
5.1 Sobolev Classes of Maps Between R.iemannian Manifolds (5)
W1,P(Q, y)
633
strong-clwi,, (C1(,f2, y) n W1°P(rI,RN)), D
the obstruction to equality being the topology of Y. Of course it is to be expected that
Hs ong(Q, Y) := strong-clw,,, (C' (f2, Y) n W1,P(u RN)) has better properties compared to W1,P(Q, Y). On the other hand this class in principle lacks nice compactness properties with respect to the weak convergence and therefore it is not suitable for finding minima of energy functionals. For this reason it is convenient to introduce also the class (6)
Hweak(Q,Y) := sw-clwl,p (c' ((2,Y) n W1,P(Q,RN)),
i.e., the sequential weak closure of Cl (.f2, y) n W1,P (f2, RN) in W l'P (Q, IRN). In general one finds then , Y D Y)) D H1,P Wl'PQ, ( weak ( )
H1,P
strong
Q Y)
In the sequel of this subsection we shall report on some results about the classes W1,P(fl Y), HWeak(,f2, Y) and H t ong(.f2, y) in principle omitting proofs.
The class W1'P(X7, ym) and density of smooth maps. Let X and y be two oriented Riemannian manifolds of dimension respectively n and m. As usual we shall assume that Y is compact and without boundary, while X may or may not have boundary. Moreover, without loss of generality, we think of Y as of a submanifold of RN, and, if we want to emphasize the dimensions, we write X1
and ym instead of X and Y. For any real number p > 1, but actually we shall only consider the case p > 1, we set (7)
W"p(X, y) :_ {u E W1,P(fl, ]RN) I u(x) E Y for a.e. X E X}.
The nonlinear space W1'P(X,Y) depends in general on the embedding of Y into N. This is not essential in case X U 8X and Y are compact, because different embeddings give rise to homeomorphic spaces W1'P(X, Y). However, since the definition (7) works also for non compact X and Y an intrinsic definition should be more suitable. But we shall not pursue this point. Our first question is whether C°° (X, y)nW1"P(f2, JRN) is dense in W1"P(X, Y). Suppose for a moment that X is a domain of Rn and, for the sake of simplicity,
that X = R'. Given u E W1"P(X, y), the easiest and natural approach to our question is to consider the mollifiers uE
J W,, (x - y) u(y) dy , SoE - E-n P( )
where cp is a standard C°°(IR") kernel. Then the maps uE converge in W 1,P (X, RN) to u. But in general the range of uE is not Y. Assuming however
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5. Cartesian Currents in Riemannian Manifolds
that u6(x), x E X, lies in a normal neighbourhood VE of Y in IRN, which shrinks to y as E tends to zero, we might consider the maps
.7ouE:X-->y where n is the nearest point projection map from VE into Y. In this case 17 o uE is readily seen to converge in W1,P to u. Unfortunately, in general we cannot expect that uE(x), x E Si, lies in a small normal neighbourhood of y; for instance, any mollification with a symmetric
kernel of the map xllxl from B3 into S2 has value zero at zero. The previous argument however does work, with minor changes, if p is larger than the dimension n of X because, by Sobolev-Morrey embedding theorem, we know that every u E W 1,P (X, y) is actually Holder-continuous and the mollifiers uE of u converge uniformly to u. In fact, essentially the same argument works also for p = n, and we have
Theorem 3 (Schoen-Uhlenbeck). Let X' be a Riemannian manifold with smooth boundary and let y' be a compact Riemannian manifold without boundary. Then C'(X, y) is dense in Wl°n(X, y).
Proof. By standard extension theorems we may assume that X is compactly contained in a Riemannian manifold k and that any u E W""n(X, y) is the restriction of a map u E W""n(X, y) to Y. We can also assume that X is isometrically embedded in some IRk, k > n. Let U be a normal neighbourhood of X in Rc and let V be a normal neighbourhood of y in RN. First 