Space Mappings with Bounded Distortion YU. G. RESHETNYAK
TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
Space Mappings with Bounded Distortion
TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
VOLUME 73
Space Mappings with Bounded Distortion YU. G. RESHETNYAK
American Mathematical Society
Providence
Rhode Island
10. T. PEIUETH$IK
HPOCTPAHCTBEHHbIE OTOSPA)ICEHHSI
C OI,PAHHLIEHHbIM HCKAWEHHEM «HAYKA >, MOCKBA, 1982
Translated from the Russian by H. H. McFaden Translation edited by Ben Silver 1980 Mathematics Subject Classification (1985 Revision). Primary 30-02, 30C60; Secondary 30C20, 30C35, 30C85, 46E35, 53B20, 31B15, 35J99, 53A30. ABSTRACT. This monograph is an exposition of the main results obtained in recent years by Soviet and foreign mathematicians in the theory of mappings with bounded distortion. The book relates to an active direction in contemporary mathematics. The mathematical apparatus contained in it can be applied to a broad spectrum of problems that go beyond the context of the main topic of investigation. A number of questions in the theory of partial differential equations and the theory of functions with generalized derivatives are expounded
for the first time in the world monograph literature. The book is intended for research workers, graduate students, and university students concerned with questions in analysis and function theory. Bibliography: 183 titles
Library of Congress Cataloging-in-Publication Data Reshetnyak, lUrii Grigor'evich. (Pro stranstvennye otobrazhenifa s ogranichennym iskazheniem. English) Space mappings with bounded distortion/Yu. G. Reshetnyak. p. cm. - (Translations of mathematical monographs; v. 73) Translation of: Prostranstvennye otobrazheniia s ogranichennym iskazheniem. Bibliography: p. 347 ISBN 0-8218-4526-8 (alk. paper) 1. Conformal mapping. 1. Title. II. Series. QA646.R3813 515.9-dc 19
1989
Copyright ©1989 by the American Mathematical Society. All rights reserved. Translation authorized by the All-Union Agency for Authors' Rights, Moscow The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America Information on Copying and Reprinting can be found at the back of this volume. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
89-72
CIP
Contents
Foreword to the English translation
From the Author CHAPTER I. Introduction § 1. Some facts from the theory of functions of a real variable 1.1. Sets in R" 1.2. Classes of functions in R" 1.3. Differentiation of measures on the space R. Lebesgue points of a function and points of density of a subset of R^
1.4. Approximation of integrable functions by smooth functions §2. Functions with generalized derivatives 2.1. Definition of a function with generalized derivatives 2.2. Sobolev imbedding theorems 2.3. Tests for a function to belong to the class WP1.10C(U) 2.4. Transformations of functions with generalized derivatives 2.5. Dependence of the coefficients in the imbedding theorems on the size of the domain 2.6. A theorem on differentiability of WP 10C-functions almost everywhere §3. Mobius transformations 3.1. Motions and similarity transformations of a Euclidean space 3.2. Mobius transformations. Definitions 3.3. Mobius transformations and cross ratios. Construction of Mobius transformations 3.4. Mobius transformations and spheres
xi
xiii 1
I 1
4
6 10 12 12 15 17 21
24 25 28 28 33 36 39
CONTENTS
vi
3.5. The hypersphere bundle and linear representations of Mobius transformations §4. Definition of a mapping with bounded distortion 4.1. Orthogonal invariants of linear mappings of Euclidean spaces. A measure of nonorthogonality for a linear mapping 4.2. Mappings with bounded distortion 4.3. Examples of mappings with bounded distortion §5. Mappings with bounded distortion on Riemannian spaces 5.1. Riemannian metrics in domains in R" 5.2. Mappings with bounded distortion on Riemannian spaces
CHAPTER II. Main facts in the theory of mappings with bounded distortion § 1. Estimates of the moduli of continuity and differentiability almost everywhere of mappings with bounded distortion 1.1. Some auxiliary facts 1.2. An estimate of the modulus of continuity of a mapping with bounded distortion 1.3. Differentiability almost everywhere of mappings with bounded distortion §2. Some facts about continuous mappings on R" 2.1. The degree of a mapping 2.2. The degree of a mapping and exterior differential forms 2.3. Change of variables in a multiple integral §3. Conformal capacity 3.1. The capacity of a capacitor 3.2. Sets of zero capacity 3.3. The concept of a Hausdorff measure. Cartan's lemma 3.4. Capacity and Hausdorff measures 3.5. Estimates of the capacity of certain capacitors §4. The concept of the generalized differential of an exterior form 4.1. General facts about exterior forms 4.2. The concept of generalized differential of an exterior form 4.3. Properties of the generalized differential of an exterior form
44 53
53 61
63 67 67 72
79 79 79 82 83 85 85
90 93 103 103 110 114 118 120
129 129 131
133
CONTENTS
4.4. The homomorphism induced on the algebra of exterior forms by a mapping of the domain 4.5. Weak convergence of sequences of exterior forms §5. Mappings with bounded distortion and elliptic differential equations 5.1. A description of a certain class of functionals of the calculus of variations 5.2. Variational properties of mappings with bounded distortion 5.3. The classes Wp(U/A) and W,(U/A) 5.4. The Dirichlet problem, extremal functions, and generalized solutions of the Euler equation for functionals of the calculus of variations 5.5. The maximum principle for extremals of functionals of the calculus of variations 5.6. Harnack's inequality and its corollaries 5.7. The concept of the flow of a stationary function in a capacitor 5.8. The set of singular points of stationary functions for functionals of the calculus of variations 5.9. Liouville's theorem on conformal mappings in space 5.10. The property of quasi-invariance of conformal capacity §6. Topological properties of mappings with bounded distortion 6.1. Continuous mappings with nonnegative Jacobian 6.2. Satisfaction of condition N for mappings with bounded distortion 6.3. Topological properties of mappings with bounded distortion 6.4. A theorem on removable singularities 6.5. On the method of moduli 6.6. Bi-Lipschitz mappings §7. Local structure of mappings with bounded distortion 7.1. Preliminary remarks 7.2. Some estimates of a solution of an elliptic equation having one singular point 7.3. A measure of the distortion of a small sphere under a mapping with bounded distortion 7.4. Behavior of a mapping with bounded distortion near an arbitrary point of the domain
vii
134 138 141
141
144 149
158 161
163
164 168 171
172 173 173
176 182 187 188 190 194 194
196
200 201
CONTENTS
§8. Characterization of mappings with bounded distortion by the property of quasicgnformality 8.1. The concept of a mapping which is quasiconformal at a point and in a domain 8.2. Differentiability almost everywhere of quasiconformal T-mappings 8.3. The condition of absolute continuity for a real function of a single variable 8.4. The analytic nature of quasiconformal T-mappings 8.5. Main result 8.6. Homeomorphic quasiconformal mappings §9. Sequences of mappings with bounded distortion 9.1. A theorem on local boundedness of sequences of mappings with bounded distortion 9.2. A theorem on the limit of a sequence of mappings with bounded distortion 9.3. A sufficient condition for relative compactness of a family of mappings with bounded distortion § 10. The set of branch points of a mapping with bounded distortion and locally homeomorphic mappings 10.1. The measure of the set of branch points 10.2. Some lemmas on local homeomorphisms 10.3. The measure of the image of the set of branch points for mappings with bounded distortion 10.4. A local homeomorphism theorem 10.5. A theorem on the radius of injectivity § 11. Extremal properties of mappings with bounded distortion 11.1. The homomorphism generated on the algebra of exterior forms by a mapping with bounded distortion 11.2. Main theorem § 12. Some further results 12.1. Classes of domains in R" 12.2. Stability in the Liouville theorem on conformal mappings of a space and related questions 12.3. Stability of isometric and Lorentz transformations 12.4. Quasiconformal and quasi-isometric deformations. Semigroups of quasiconformal transformations 12.5. Mappings with distortion coefficient close to 1
204
204 205
208 210 213 215 216 216 218
220 221 221
224 227 229 234 240
240 242 246 246 251
260 266 277
CONTENTS
12.6. The general concept of stability classes 12.7. A characterization of quasiconformal mappings as mappings preserving the space Wn
CHAPTER III. Some results in the theory of functions of a real variable and the theory of partial differential equations § 1. Functions with bounded mean oscillation §2. Harnack's inequality for quasilinear elliptic equations 2.1. Preliminary remarks 2.2. Main inequalities 2.3. Consequences of the integral inequalities in §2.2 2.4. Boundedness of generalized solutions of equation (2.3). Harnack's inequality §3. Theorems on semicontinuity and convergence with a functional for functionals of the calculus of variations 3.1. Weak convergence of sequences of functions in measure spaces 3.2. Some lemmas about convex functions 3.3. Theorems about semicontinuity of functionals of the calculus of variations 3.4. Corollaries to Theorems 3.1 and 3.2 3.5. The convex envelope of a function 3.6. A theorem on convergence with a functional 3.7. Corollaries to the theorem on convergence with a functional §4. Some properties of functions with generalized derivatives 4.1. A theorem on differentiability almost everywhere 4.2. Proof of Lemma 1.1 in Chapter II 4.3. An estimate of the modulus of continuity of a monotone function of class W §5. On the degree of a mapping Bibliography
ix
283
287
289 289 295 295 296 300 304 310 310 312 315 319 321
325
329 330 330 335 338 341
347
Foreword to the English Translation This text differs from the Russian original in the following respects. First, Chapter I has been completely reworked. The author has tried to give a more complete exposition of the preliminary facts needed for reading the main text. In particular, complete proofs are presented for all the required properties of Mobius mappings. In Chapter II some errors in the original are corrected, and improvements in the text are introduced where it seemed possible by slight changes to strengthen individual results or to make the presentation more complete and clear. Moreover, § 12 is added to Chapter II. It contains a survey of certain further investigations of questions close to the main topic of the book. Here the author has confined himself mainly to an account of work carried out in Novosibirsk and little known outside the USSR. The bibliography has been enlarged accordingly. Chapter III remains almost without changes.
Xi
From the Author The book before the reader is devoted to an exposition of results of investigations carried out mainly over the last 10-15 years concerning certain questions in the theory of quasiconformal mappings. The principal objects of investigation-mappings with bounded distor-
tion-are a kind of n-space analogue of holomorphic functions. As is known, every holomorphic function is characterized geometrically by the fact that the niapping of a planar domain it implements is conformal. In the n-space case the condition of conformality singles out a very narrow class of mappings. As Liouville showed back in 1850, already in threedimensional Euclidean space there are no conformal mappings besides those which are compositions of finitely many inversions with respect to spheres. Such mappings are called Mobius mappings. They form a finitedimensional Lie group which includes the group of motions of the space R" and is only slightly broader than this group. However, if one weakens the condition of conformality, replacing it by the condition of quasiconformality, then a considerably broader class of mappings emerges.
To give the reader an idea about the subject of the book we present some explanations (the exact definitions are contained in the main text). A mapping of a domain in an n-dimensional space is called a mapping with bounded distortion if it satisfies definite requirements of regularity, preserves the orientation of every small domain, and (the main point) satisfies the following condition. There exists a constant q, 1 < q < oo, such that an infinitesimally small sphere is transformed by the mapping into either a point or an infinitesimally small ellipsoid for which the ratio of the largest semiaxis to the smallest does not exceed the constant q. If, moreover, the mapping is also topological (a homeomorphism), then it is said to be quasiconformal. The regularity requirement mentioned here is that the components of the vector-valued function determining the given mapping must have firstorder generalized derivatives that are locally integrable to the power n. The X111
uiv
FROM THE AUTHOR
condition about preserving the orientation of a small domain is analytically
equivalent to the Jacobian of the mapping being nonnegative. For an arbitrary mapping with bounded distortion there can exist points such that the mapping is not a homeomorphism in any neighborhood of them. These points are called branch points of the mapping. The dimension of the set of branch points does not exceed n - 2. For example, the set of branch points can be a curve in three-dimensional space. For a holomorphic function of a single variable the branch points are simply the zeros of its derivative.
We note that in the two-dimensional case the study of arbitrary mappings with bounded distortion is easily reduced to the consideration of homeomorphic quasiconformal mappings and holomorphic functions of a single variable.
The theory of planar quasiconformal mappings arose at the end of the 1920's in work of Grotzsch and M. A. Lavrent'ev. This theory is now a far-advanced area of the theory of functions of a complex variable which has important applications both in function theory itself and beyond its boundaries, in particular, in applied areas. The concept of a quasiconformal mapping in n-space introduced by Lavrent'ev in 1938 in searching for a suitable tool to construct mathematical models of certain hydrodynamics phenomena. He formulated also a number of problems whose solutions later played an essential role in the development of the theory of quasiconformal mappings in n-space. However, the beginning of intensive investigations in this area dates from 1960. Further, as in the planar case, only homeomorphic quasiconformal mappings were considered at first. The systematic study of general mappings with bounded distortion was begun in 1966. There are two basic methods in the theory of mappings with bounded
distortion. One of them goes back to the classical work of Grotzsch and is based on the use of a certain quantity characterizing a family of curves or surfaces in space and called the modulus of the family. This method depends on inequalities describing the behavior of the modulus of a family of curves or surfaces when the family is transformed by a given mapping, as well as on certain estimates for the moduli. The other method consists in the use of a certain apparatus in the theory of differential equations. As is known, the real and imaginary parts of a holomorphic function of a complex variable are harmonic functions. Analogously, the components of a vector-valued function representing a mapping with bounded distortion are solutions of a certain elliptic partial differential equation. The method consists in the use of this fact (and
FROM THE AUTHOR
xv
certain generalizations of it) and properties of elliptic equations, in particular, the maximum principle for elliptic equations, Harnack's inequality, etc. Some estimates relating to the concept of the capacity of a capacitor are also used. In this monograph the second method is used to study mappings with bounded distortion. All the needed facts about elliptic equations are given. A significant part of the book is devoted to an exposition of material that is auxiliary with respect to the main topic, though it is definitely of independent interest. In particular, a proof is given of the well-known theorem of Moser and Serrin on Harnack's inequality for elliptic equations; theorems are proved on semicontinuity and convergence with a functional for functionals of the calculus of variations; the necessary facts are given about the concept of the degree of a mapping and the metric properties of mappings connected with these facts; etc. In this connection the author hopes that the book will prove to be useful not only for specialists in the theory of mappings in n-space, but also for a broader circle of readers. The investigation of mappings with bounded distortion is based on the concept of the generalized differential of an exterior form [146]. In particular, a detailed study is made of the properties of the generalized differential, and this, in the author's opinion, can be of interest, for example, in connection with certain recent investigations of the topology of Lipschitz manifolds by analytic means [49]. Quasiconformal mappings in n-space have been used in the theory of spaces of functions with generalized derivatives ([177], [178], [46]), as well as in investigations of compact Riemannian spaces of constant negative curvature [106]. The theory of mappings in n-space with bounded distor-
tion is one of the areas in the general metric theory of space mappings that is being intensively developed at present. Among the investigations in this area one can cite work on the theory of quasi-isometric mappings [60], the theory of quasi-Lorentz mappings [51 ], a series of investigations
in the theory of Kleinian groups in space [75), papers on the theory of homeomorphisms of class W,,' ([165], [112]), and other publications. Many interesting questions in the theory of mappings in n-space close
to the topic of the book had to be omitted for lack of space. In choosing the material the author was oriented toward results used in studying the problem of stability in Liouville's theorem on conformal mappings in space.
Yu. G. Reshetnyak
CHAPTER I
Introduction §1. Some facts from the theory of functions of a real variable
1.1. Sets In R". Below, R" denotes the n-dimensional additive Euclidean space of points x = (x1, , x"), lxi is the length of a vector x E R", and (x, y) is the inner product of vectors x and y in R, i.e., for
x= (x1, x,,) and y= (y,, ,y") we have (X.Y) =X1Y1 +...+xnynr X
xi +...+x,4,
(X. Y) =
If a and b are two arbitrary points in R", then [a. b] denotes the segment joining them, i.e., [a, b] is the collection of all points x of the form x = via + µb, where A + p = I and A, p > 0.
Denote by e;, i = 1_., n, the vector in R" with ith coordinate equal to 1 and the other coordinates equal to 0. The vectors e1,... , e" form a basis in R", called the canonical basis.
Let P(x) be some expression containing the variable x, and let A be a set. Then {x E AIP(x)} denotes the collection of all elements x in A such that the expression P(x) is true. We employ the usual symbolism of mathematical logic; in particular, d ("for all"), 3 ("there exists"), ("if ..., then...-), a ("is equivalent to"), & ("and"), V ("or"), and - ("not"). The symbol 0 denotes the empty set. Let X be a given topological space. The closure of an arbitrary set A C X
is denoted by A, CA is the complement of A, CA = X\A, A° = C(CA) is the open kernel or interior of A, and 8A = A\A° = ;in CA is the boundary of A.
A domain in a topological space X is defined to be a connected open subset of X. Let U be an open set in X. We say that A C X lies strictly inside U if A is compact and U D A. I
1. INTRODUCTION
2
LetA cX. ForxEXletXA(x)= 1 ifxEAandXA(x)=0ifxfA. The function XA : X - R so defined is called the indicator function of the set A.
Let X be a metric space and p its metric. For X E X and A C X the distance from x to A is defined by
p(x, A) = yEA inf
p(x. y).
Obviously, (p(x, A) = 0) a (x E A). The function x " p(x, A) is continuous. Further, for x1, x2 E X p(x1, A) - p(x2, A)I <_ p(xi. x2) Let
d(A) =
sup p(x, y). xEA,yEA
The quantity d(A) is called the diameter of A. For arbitrary A, B c X let dist(A, B) =
inf p(x, y).
xEA.yEB
The quantity dist(A, B) is called the distance between the sets A and B.
For ACIlet Uh(A) = {x E AI p(x, A) < h}
and let Uh (A) be the closure of Uh (A). The set Uh (A) is called the hneighborhood of the set A, and Uh(A) is called the closed h-neighborhood of A. Let U be an open subset of a metric space X. Then we let (Jh = {x E XIp(x, CU) > h}.
The set Uh is open. If U # 0 and h is sufficiently small, then Uh # 0. The supremum of the values of h such that Uh is a nonempty set is called the interior radius of the nonempty set U. It is clear that U °h for every h > 0, Uh, Oh, for h 1 < h2, and Uh>o Uh = U. For an arbitrary metric space X with metric p let Bx(a, r) denote the
open ball, Bx(a, r) the closed ball, and Sx(a, r) the sphere about a with radius r > 0, i.e.,
Bx(a,r)= {xEXIp(x,A)
by w"_ 1, w"_ I = 2x"/2/r(n/2 + 1), and Q" = w"_ I In is the volume of
B(0,1) in R.
§ 1. THEORY OF FUNCTIONS OF A REAL VARIABLE
3
Let a = (a, .... , a") E R". Denote by Q(a, r) the n-dimensional open cube
{(x1.... x") E R"JVi = 1, ... , n Ix; - ail < r/2}, ,
and by U(a, r) the closed cube
{(xl , ... , x") E R"Idi = 1.... , n Ix; - ail < r/2}. The point a is called the center of each of these cubes, and the number r is the length of an edge. A dyadic cube in R" is defined to be an n-dimensional rectangle P of the form
P= [LI, k, +1\ J
2r
x [L2, k2+11 2r
2r
J
xx [k",k"+112 L 2r
2r
J
where k1,... , k", and r are integers. The number r here is called the rank of the dyadic cube P. Two different dyadic cubes of the same rank do not have common points. Let P and Q be dyadic cubes of respective ranks r and s, with r > s; if P n Q 54 0, then P C Q. For each r and every x E R" there exists a unique dyadic cube Q of rank r such that x E Q. Denote this cube by 9,(x). Let U be an open subset of R", and x an arbitrary point of U. Take the smallest integer r such that r > 0 and the closure of Or(x) is contained in U. The cube Or (x) corresponding to this value of r is denoted by 9U(x). The set of all cubes 9U(x) is countable, since the set of all dyadic cubes is countable, and the set of cubes 9'u is obviously infinite. If the cubes 9U(xl) and 9'U(x2) are distinct, then they do not intersect. Indeed, let r, be the rank of 9'u (xl) and r2 the rank of OU (x2), and let r, < r2. If
9U(x,) n9U(x2) # 0, then 9U(xl) D 9U(x2), and hence x2 E 9u(x,). We have that 9'U(x2) = 9,2(x2), and since x2 E 9u(x,), it follows that 9U(XI) = 9,,(x2). This contradicts the fact that r2 is by definition the
smallest integer r > 0 such that c U. We number all the cubes 9'u(x) and let Q. be the cube with index v. Then U = u;° Q, U c U for each v, and Q,,, 0 for v, # v2. This partition of U will be called the dyadic subdivision of U. Every compact set A C U is covered by finitely
many cubes of the dyadic subdivision. Indeed, p(x, CA) > 6 = const > 0 for all x E A; hence, for x E A the rank of the cube .v(x) does not exceed some number ro < oo, and thus the number of cubes such that Q n A 34 0 is finite. If A is an arbitrary subset of R", then (AI will denote the outer Lebesgue measure of A; that is, IAA is the infimum of the measures of the open subsets
of U containing A. For every subset A of R" there exists a measurable set H D A such that H = Al. C
4
1. INTRODUCTION
1.2. Classes of functions in R". Let A be a measurable subset of R", and u a measurable real-valued function on A. For p > I let 1/p
IIuIIp.A =
{ fAlu(x)I°dx}
.
(1.1)
Assume that U is an open subset of R" and u(x) is a measurable function defined almost everywhere on U. We say that u(x) is locally integrable to the power p > 1 on U (and write u E Lp,1a(U)) if IIUIIp.A < oo for every compact set A C U. By the Borel covering theorem, this is equivalent to the following: every x E U has a neighborhood V such that IIuII p.v < oo. The collection of all measurable functions u(x) such that II uIIp.u < co is denoted by Lp(U). In the case U = R" the quantity IIuIIp.u is denoted by IIuIIp and Lp is written instead of Lp(U). Here we assume known all the facts in the theory of the Lebesgue integral on R" and all the properties of the space Lp(U), in particular, the Holder and Minkowski inequalities, the completeness of the spaces L p (U), and so on.
Let (u,"), m = 1, 2, ... , be an arbitrary sequence of functions in the class 410C(U), where 1 < p < oo. Then we say that a sequence (um), m = 1, 2,... is bounded in L p,1OC(U), or locally bounded in Lp(U), if the sequence (II um IIp.A), m = 1, 2, ... , is bounded for every compact set A C U. A sequence (um ), m = 1, 2, ... , of functions of the class L p,10(U), 1 < p < oo, is said to converge in Lp,10(U) to a function uo E Lp,10(U) if Ilum - uII p.A - 0 as m - co for every compact set A C U. Let X be an arbitrary topological space, and Y an arbitrary vector space.
For an arbitrary function f : X - Y let S* (f) be the set of x E X such that f (x) 0 0. The closure of S'(f) is called the support off and denoted by S(f). A function f is said to be compactly supported if its support is compact. In the special case when X is an open subset of R", endowed with the
topology induced by R", the support of a function f : X - Y is the set So (f) n X. Here the closure is in the topology of R. Assume that the vector space Y is a normed space. In this case C(X, Y)
denotes the collection of all continuous mappings f : X - Y. If no confusion is possible, we write f E C(X), or simply f E C, instead of f E C(X, Y). The collection of all compactly supported functions f E C(X, Y) is denoted by C0(X, Y); the abbreviated notation C0(X) or Co will sometimes be used instead. For an arbitrary function f : X - Y and an arbitrary set A C X let II!IIM(A) = suPIf(x)IY, xEA
(1.2)
§ 1. THEORY OF FUNCTIONS OF A REAL VARIABLE
5
where I l y denotes the norm in Y. If f E C and A is compact, then -
clearly IllIIM(A) < 00. In particular, if f E Co, then IllIIM(A) < oo. Let IllIIC(A) = IIf IIM(A) in the case when f is a continuous mapping.
LEMMA 1.1. Let U be an open subset of R". Then for every. p > 1 the set Co(U) = Co(U, R) of functions is dense in Lp(U).
The proof is a simple exercise in the theory of functions of a real variable, and we omit it (see also [ 148]: the lemma is a direct consequence of the construction of the Lebesgue integral presented there). Let X be a locally compact topological space. A sequence (fm: X -' Y), m = 1, 2, ... , of functions is said to converge locally uniformly to a function fo: X - Y if fm converges to fo uniformly on any compact set A C X as
m - oo. Let U be an open subset of R", and Y a normed vector space. We say that a function f : U - Y belongs to the class Cl (U, Y), where k > 1 is an integer, if f has all partial derivatives of order at most k in U, and these derivatives are continuous on U. The collection of all functions f : U -+ Y of class Ck(U, y) with compact support in U is denoted by Co (U, Y). If f E Ck(U, y) (f E Co (U, Y)) for all integers k > 1, then we say that f E Co (U, Y) (respectively, f E Co (U, Y)). In all this notation the symbols U and Y are omitted whenever no confusion can result. Suppose that U is an open subset of R" and f : U --' R" is a mapping of class C I (U. R"). (According to the conventions announced above, we can simply write f E C I.) Then at each point x E U the linear mapping f I (x) : R" - R" is defined, the derivative off at x. For any vector h E R"
f'(x)h = lim f (x + th) - f (x) _ 1-0
of
X7, (x)hi-
The matrix of the linear mapping fl(x) is the Jacobi matrix of the mapping f at x. A mapping f : U R" is called a diffeomorphic mapping or a diffeomorphism of class Ck (where k > 1 is an integer) if f is in the class Ck (U, R"), f is one-to-one, and the Jacobian det f'(x) of f is
nonzero at each point x E U. In this case the set V = f(U) is open, and the inverse mapping f - I is a diffeomorphism of the same smoothness class Ck. In particular, every diffeomorphism is a topological mapping (homeomorphism). Let X and Y be metric spaces, with px the metric in X and py the metric
in Y. Let f : X -, Y be a given function. A modulus of continuity of f is defined to be any function w: [0, oo) -' [0, oo) satisfying the following conditions:
I. INTRODUCTION
6
A) to is nondecreasing, and w(t) - 0 as t - 0.
B) For anyx,yEX Py[.f (X)- f (Y)]:!:- w[Px(x. Y)l
A function f : X - Y has a modulus of continuity if and only if it is uniformly continuous. If X is compact and f : X -' Y is continuous, then f is uniformly continuous. If a function f : X - Y has a modulus of continuity w(t) = Kt°, where 0 < K < oo and 0 < a 1, then f is said to satisfy a Holder condition with constant K and exponent a. If f satisfies a Holder condition with constant K and exponent a = 1, then f is also said to satisfy a Lipschitz condition with constant K.
Let U be an open subset of R", and Y a normed vector space. The symbol C°-° (U), where 0 < a < 1, denotes the collection of all mappings f : U - Y satisfying a Holder condition with exponent a on every compact
set A C U. A mapping f : U - Y will be said to belong to the class where k > 1 is an integer and 0 < a < 1, Ck.°(U, Y) (or simply f E if f E C"(U) and all the kth-order partial derivatives off belong to 1.3. Differentiation of measures on the space R". Lebesgue points of a function and points of density of a subset of R". Here we give a summary of classical results on differentiation of set functions on R. The symbol B" will denote the collection of all Borel subsets of R. Let A E B". The collection of all sets E E B" contained in A will be denoted by B(A). The union and intersection of any at most countable family of
sets in B(A) belongs to B(A). The difference of any two sets in B(A) belongs to B(A). A measure on a Borel set A C R" is defined to be a function P : B(A) -+ R such that P(UEm) = E#(E) m
in
for any at most countable family (Em), m = 1, 2, ... , of disjoint sets in B(A). In this definition it is not assumed that the set function P is nonnegative.
Let p: B(A) - R be a measure on a Borel subset A of R". Then a certain nonnegative measure IPI called the variation of p is defined. For an arbitrary E E B(A)
IPI(E) = sup {IP(E )I +Ip(E\E')I}. E'E'8(E)
For every E E B(A) we have that IP(E)I < IPI(E).
§ 1. THEORY OF FUNCTIONS OF A REAL VARIABLE
7
Let p be a measure on an open set U in R. We say that p is differentiable at a point x E U if the limit lim p[B(x, r)] _ .p(x)
r-.O IB(x, r)I exists and is finite. The number '?,u (x) is called the density of p at x. A measure p on an open subset U of R" is said to be singular if there exists a Borel set A c U with Lebesgue measure zero such that Iµi (U \ A) = 0.
THEOREM 1.1. Every measure p on an open subset U of R" is differen-
tiable almost everywhere on U. The function x'- gp(x) is integrable, and for every EEB(U)
p(E) = fE 'p(x) dx + a(E), where a is a singular measure on R". (In particular, a can be identically equal to zero).
See, for example, [149] or [48] for a proof. Let us now define the concept of a Lebesgue point of a function. It is expedient first to introduce a certain more general concept. Let 91 be a Hausdorff topological space whose elements are measurable real-valued functions on the ball B(0, 1) c R such that for any I E R the function identically equal to 1 is in 91.
Let U be an open subset of R", and let f : U - R be a measurable function. Take an arbitrary point a E U, and let 0 < h < p(a, 8U). Then the number Fh (X) = f (a + hX) is defined for almost all vectors X E T4(0, 1). We thereby obtain a family of measurable functions Fh defined on the ball B(0, 1). A number I is called the limit off as x -i a in the sense of convergence in 9t if there is an ho > 0 such that Fh E 91 for 0 < h < ho, and the functions Fh, as elements of 9t, converge to the function identically equal to the constant I as h 0. In this case we write I = limx.a(9t) f (x). In the case when f (a) = limx.a(91) f (x), we say that f is continuous at a in the sense of convergence in 9t. We give examples.
Let M be a vector space of bounded real-valued functions F : B(0,1) R, and let the topology on M be determined by the norm
IIFII = sup IF(x)I. 0
(Functions F1, F2 E M such that F1 (x) = F2(x) Vx 96 0 are identified.) It is not hard to see that the limit in the sense of convergence in M coincides with the usual concept of limit.
1. INTRODUCTION
8
Let Wi be the collection of all measurable functions on B(0,1). For F, GE W? let
p(F, G) = 40.1)
I F(x) - G(x)I dx. 1 + IF(x) - G(x)I
The function p(F, G) is a metric on At. (Functions F1, F2 E WI coinciding almost everywhere on B(O,1) are regarded as the same element of Wi.) Convergence in the sense of this metric is none other than convergence in
measure. For an arbitrary measurable function f : U - R the concept of limit in the sense of convergence in Wt coincides with the concept of approximative limit (see the definition in [ 149]; the proof that these concepts are equivalent is left to the reader).
Assume that f is a function of class Lp,1 (U), where 1 < p < oo. Then for a E U and h < p(a, 8U) the function Fh : X p-. f (a + hX) belongs to the class Lp[19(0,1)]. A limit of f at a in the sense of convergence in L p[ff(0,1)] is called an L p-limit off at a. If f is continuous at a point x in the sense of convergence in L p [f(0,1)], then one also says that a is a Lebesgue L p-point off (a Lebesgue point off in the case p = 1). A point a is a Lebesgue Lp-point of a function f if and only if lim h-O
j
o1)
f(a+hX)-f(a)pdX=0.
Performing the change of variable a + hX = x in the integral, we get that a is a Lebesgue Lp-point of f if and only if 1 IB(a,h)I
J
If (x) -f(a)I p dx - 0 ca.h)
as h --+ 0. Textbooks on the theory of functions of a real variable usually define a Lebesgue Lp-point to be one for which the last relation holds.
THEOREM 1.2 (see [183] and [48]). Let U be an open subset of R. Then for every function f E L p,1a(U) (p > 1) almost all points x E U are Lebesgue L p points of f. In the theory of the Lebesgue integral all functions are regarded as defined to within values at points forming a set of measure zero. In some cases the ambiguity arising from this can lead to certain difficulties. This ambiguity can be removed by introducing the concept of a natural value of a function. Let f be a function in L 1,1 (U). A number L E R is called the
natural value of f at a point x E U if L = lim1.x(Lp)f (t). By Theorem 1.2, if f is a function in L p,1a(U), then f (x) is the natural value of f at x for almost all x E U.
§ I. THEORY OF FUNCTIONS OF A REAL VARIABLE
9
Let A be an arbitrary subset of R. A point x in A is called a point of density of A if
IA nli(x,r)IlIB(x,r)I - 1 as r - 0. (Here it is not assumed that A is measurable.) THEOREM 1.3. For every set A c R" the points x E A which are not points of density of A form a set of measure zero.
PROOF. This is a consequence of Theorem 1.2. Assume first that A is measurable. Then the indicator function XA of A belongs to LI,I(R"). Let E0 be the set of points x E R" which are not Lebesgue points of XA. By Theorem 1.2, IEol = 0. Let X E A \ E0 be arbitrary. Then XA(x) = 1, and as B(x. r)I fa(x.r)
I X4(t) - 11 dt -' 0,
which implies that
_
I
I B(x, r) I
-eecx.r) X.+(t) dt =
IAIi(x.r)I ( )I
1
as r - 0, i.e., x is a point of density of A. This proves the theorem for the case when A is measurable.
Let A be an arbitrary subset of R" such that IAI < oo. Then there exists a measurable set H D A such that IHI = IAI. For every point x E A and any r > 0 we have that I H n ff(x, r) I = IA n R(x, r)I. Indeed, assume on the contrary that IH n W(x, r)I > I A n B(x, r)I for some r. We construct a measurable set H' D A n B(x, r) such that IH'I = IA n ff(x, r)I. It can obviously be assumed that H' c B(x, r). Then the set H" = H' u (H \ B(x, r)) is measurable, A c H", and
I< Al
IH'I+IH\B(x,r)I < IHnB(x,r)I +IH\B(x,r)I = IHI = IAI.
We thus get a contradiction, and hence
IAnff(x,r)I =IHnB(x,r)I. Let E0 c H be the set of all points in H which are not points of density of H. Since H is measurable, Eo is a set of measure zero by what has been proved. Let x E A \ E0. Then x is a point of density of H, i.e.,
IHnB(x.r)I IB(x, r) I
0
as r -' 0. By what has been proved, IH nB(x, r) I = IA nl(x. r) 1; therefore we see that x is a point of density of A, and the theorem is proved also for the case when IAI < oo.
1. INTRODUCTION
10
Suppose that IAI = oo, and let Am = A n B(0, m). Then IAmI < IB(0, m)I < oo. Let Em be the collection of points x E Am which are not points of density of Am, and let E0 = U;° Em. By what was proved, IEmI = 0 for each m, so IEo1 = 0. Let X E A \ E0, and take an m such that I xl < m. Then X E Am, and since x 0 Em c E0, it follows that x is a point of density of Am. We have that >
I A n B(x, r)I > I Am n B(x, r)I
B(x, r)I
-
I B(x. r) I
as r -+ 0. Hence, I A n B(x, r)III B(x, r) I -- 1
as r -- 0, i.e., x is a point of density of A. The theorem is proved. 1.4. Approximation of integrable functions by smooth functions.
Here
we describe a certain technique for smoothing functions which, in particular, is used in an essential way in studying functions with generalized derivatives. A function w: R" - R is called an averaging kernel if it is bounded and integrable on R", has support in B(0, 1), and satisfies
Jw(x)dx= 1.
(1.3)
An averaging kernel co is called a Sobolev averaging kernel if w is nonnegative and belongs to Co (R"). We get an example of a Sobolev averaging
kernel by first defining the function (p: R -' R to be p(t) = el/° for t < 0 and lo(t) = 0 for t > 0, and then setting w(x) = yf(IxI2 - 4 ), where y is chosen so that (1.3) holds. Let U be a nonempty open subset of R", and let Oh = {x E UJ p(x, CU)
> h } for h > 0. Take an h0 > 0 such that UU is nonempty. Let w be an arbitrary averaging kernel, and let f be a function in L1,1oc(U). For 0 < h < h0 we have that Uh 0 0. For an arbitrary x E Uh let (wh * f)(x)
flyl
= h"
f f(z)w(x) dz.
(1.4)
This defines a certain function fh = wh * f on the set Oh. It is called the mean function for the function f. The number h is called the averaging parameter. If co is a Sobolev averaging kernel, we call fh the Sobolev mean
function for f . In the case when U = R" it is obvious that Oh = R", and the mean function fh is defined everywhere in R. In the general case the domain
§ I. THEORY OF FUNCTIONS OF A REAL VARIABLE
11
0h of fh is a proper subset of U. For every compact set A there is an ho = ho(A) > 0 such that A C Uh for 0 < h < ho. Let x be a Lebesgue point of a function f E L1.10, (U). Then ash -p 0 the functions Fh (y) = f (x + hy) of y converge in LI [B(0. 1)] to the function identically equal to f (x). Using the first expression in (1.4) for (coh *f )(x), we conclude from this and the boundedness of w that
((Oh * f)(x) - f
_ yI<1
f (x)co(y) dy = f (x)
as h -' 0. Consequently, we get that if f is bounded and locally integrable on U, then for every averaging kernel w li o(wh * f)(x) = f (x)
at every Lebesgue point of f, and hence for almost all x E U. If the averaging kernel co is a Sobolev averaging kernel, then the function wh * f belongs to the class C°° ((Jh ), as can be seen directly from the second
expression for (wh * f)(x). THEOREM 1.4. Let co be an averaging kernel, and let U be an open subset of R^. Then II (wh * f) - f II p.A O for every function f E L,,1(U) and for
every compact set A. If f E C(U), then the functions cob * f are continuous, and (cob * h)(x) -, f (x) locally uniformly in U as h 0.
See, for example, [ 161 ] for a proof of the theorem.
LEMMA 1.2. Let U be a nonempty open set in R". Then for any h > 0 it is possible to define a function Xh : R" R of class C°°(R") such that
0<Xh < 1 forallXER", S(Xh)C (Jh' andXh(x)= 1 for xE 02h PROOF. Let V = fl(3/2)h, and let co be a Sobolev averaging kernel. Let
z = h/4 and
' f / \ for z E V and Xv(z) = 0 for z f V and since
Xh(x) = (wt * Xv)(x) =
I
Xv(z)w 1 z i X J dz.
R
Since Xv(z) = 1
w((Z - X) /T) = 0 if I Z - X1 < T, the last integral here is equal to 1
Tn
w
(Z
-X T
dz.
fVnB(x,,r)
Assume that x is such that Xh(x) # 0. Then V n B(x, r) 9& 0. Take an arbitrary y E V n B(x, r). We have that Ix - yI < 8 and p(y, CU) > 3h/2, from which we conclude that p(x, CU) > 3h/2 - -r = 5h/4. This implies that if x E S(Xh), then p(x, CU) > 5h/4 > h, and hence S(Xh) c Uh.
I. INTRODUCTION
12
Let X E U2h. For every z E B(x, z) we have that p(z, CU) > p(x, CU) Ix - zI > 2h - h/4 > 3h/2; thus B(x, T) C V, and Xh(x) = I for this x. It is obvious from the definition of Xh that Xh E C°° and 0 < Xh(x) < I for all x. This proves the lemma. The next statement follows from Theorem 1.4 and Lemma 1.1. COROLLARY. Let U be an open subset of RR. Then for every function f E Lo,10C(U) there exists a family of functions fh : U R (h > 0) such that fh E C°° for each h, and fh fin Lo,10C(U) as h -4 0.
Indeed, let fh(x) = Xh(x)(wh * f)(x) for x E Uh, and let fh(x) = 0 for x 0 Uh. The functions fh form the required family. §2. Functions with generalized derivatives 2.1. Definition of a function with generalized derivatives. In the exposition to follow, essential use is made of diverse properties of functions with generalized derivatives. Here we present the needed definitions and recall some of the simplest properties of functions with generalized derivatives. The concept of a generalized derivative was introduced by Sobolev [162].
The reader can find the necessary proofs and subsequent results in the theory of functions with generalized derivatives in, for example, [1], 118], [48], [167], and elsewhere.
Let U c R" be an arbitrary open set, and let u: U -+ R be a function of class L 1,1ce (U). Assume that a differentiation operator 912 = 8 +"'+°^ / 8 x j- ' 8 x ° is given, where a = (a1, ,
a = (a 1, ... ,
with integer coordinates will be called a multi-
index. The number 1 = a1 + - - - + an is denoted by jal. A function v E L1,1a(U) is called the generalized derivative 91lu of u if there exists a sequence um : U -, R of functions of class C'(U) such that ur -i u and.'°u,,, - v in L1,10c(U) as m - oo. LEMMA 2.1. Let U be an open subset of R", and u : U - R a function of class L1.1oc(U). Assume that the function v E L1.1oc(U) is the generalized
derivative.g°u of u. Let co be an arbitrary Sobolev averaging kernel. Then for every h the equality
o°(wh * u)(x) = ((Oh * v)(x) holds on the set Ch. PROOF. Consider a sequence (un : U - R), m = 1, 2, ... , of C°°functions such that u,,, - It and .0°ur - v in L1,1a(U). We conclude
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
from the equality fum(x + hy)w(y) dy
(wh * um)(x) =
that for x E Uh
f°um(x+hy)w(y)dy
0°(0h * um)(x) = yi<_1
-f
v(x + hy)w(y) dy = ((Oh * v)(x).
(2.1)
yi<1 __
Using the second representation in (1.4) for the function wh * that for x E Uh 1
0°((0h * um)(x) = T. hn
fu um(z)-XW (Z
fu u(z)-X w (z
h
x) dz =
we find
h x/1 dz
°(wh * u)(x)
(2.2)
asm --goo. By (2.1) and (2.2),
(wh * v)(') =.°((Oh * u)(x). and this proves the lemma. COROLLARY. The generalized derivative-'°u of a function u E L1,1a(U), if it exists, is unique to within values on a set of measure zero.
Indeed, suppose that v1 =.0°u and v2 = 00u. Then ((oh * v1)(x) _ .
(wh * u)(x) and (wh * v2)(x) =
u)(x),
i.e., (wh *v1)(x) _ (wh *v2)(x) for all x E U. We have that (wh *v1)(x) v1(x) and (wh * v2)(x) V2 (x) as h - 0 for almost all x e U. This leads us to conclude that v1 (x) = v2(x) for almost all x E U, which is what was to be proved.
Let 1 > 0 be an integer, and let p
1. We say that a function of
class L1,1a(U) belongs to the class Wp10C(U) if u has in U all Ith-order generalized derivatives, and each is in L p,1a(U). A function u will be said to belong to the class W ,1,C(U) if u E W110C(U) and for every compact
set ACUand any awith lal=I 112°uIIc..4 = ess sup I9°u(x)I < oo. xEA
Let (um : U -+ R), m = 1, 2, ... , be an arbitrary sequence of functions in Wv 10C(U), where 1 5 p < oo. We say that the sequence (um) converges
14
1. INTRODUCTION
in Wp10C(U) to a function u E Wp10C(U) if the functions um converge to
u in L1.1a(U) as m -- oo, and 9aum -,9au in Lp.10C(U) as m -' oo for every a with dal = 1. A sequence (um), m = 1, 2, ... , is said to be bounded
in Wp10C(U), where I < p < oo, if it is bounded in Lp,1a(U) and the sequence (9 aum), m = 1, 2, ... , is bounded in L p,1(U) for any a with lad=1. If u(x) has in U all generalized derivatives of an order I > 1, then it also has in U all generalized derivatives of any order k < 1.
LEMMA 2.2. A function u E L1,10(U) (U an open subset of R") is in W1 10C(U) if and only if for every compact set A C U and any a with lal =1 the family of functions 9a(coh * u) 0 < h < dist(A, CA), converges in LI (A) as h 0. F o r e v e r y f u n c t i o n u E W p 1 (U), w h e r e 1 < p < oo, t h e r e e x i s t s a s e -
q u e n c e (um ), m = 1 , 2, ... , o f C°° f u n c t i on s c o n v e r g i n g to u(x) i n W 1 ( U).
PROOF. The necessity of the condition follows from Lemma 2.1 and Theorem 1.4. We prove the sufficiency. Let U E L 1.Ice(U) be such that for
every compact set A C U the family of functions Da(coh * u) converges in L I (A) to some function VA E L 1(A) as h -, 0. It is required to prove
that u(x) then has a generalized derivative 9au on U. Let Xh be the function whose existence is ensured by Lemma 1.1, and define uh(x) = Xh(x)(cvh *u)(x) for x E Uh, and uh(x) = 0 for x f Uh. Then uh E C°°(U). Let U = UT Q be the canonical subdivision of the open set U. We choose
an arbitrary cube Q and let h < I dist(Q, CU). Then Q, c Ugh, and By assumption, for every a with hence uh(x) = ((Oh * u)(x) for all x E jal = I there exists a function v,,,,: QL -i R such that 9a(coh *u)(x) -, v,,,a
in LI(Q) as h - 0. Let va(x) = v,,,,, (x) for x E Q,,, v = 1, 2, .... Every compact set A C U can be covered by finitely many cubes Q, and this implies that 9°uh = 9a(wh * u) - va in LI(A) as h 0. Let hm = 1/m and um = uhm. We have that um -' u and 9aum - va in LI,1a(U) as m - oo. This implies that va = 9°u. Since a such that IaI =1 was taken arbitrarily, this proves that u E Wi 10c(U).
Assume now that u E Wp1oc(U), where 1 < p < oo. Let Xh (h > 0) be the function whose existence is ensured by Lemma 1.1, and define uh(x) = Xh(x)(c)h * u)(x) for x E Uh, and uh(x) = 0 for x E U. Then uh E COO (U), and the function uh is defined for any h > 0. We let A be an arbitrary compact set and find an ho such that A c U2ha. If x E Ugh with
h < ho, then Xh(x) = 1, and hence uh(x) _ (coh * u)(x). It follows from this and Lemma 2.1 that 9auh(x) = 9'°(c)h * u)(X) = 0)h * 2au(x)
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
Is
for X E Ugh. On the basis of Theorem 1.4,
II0h*0°14 -.
UUIP.A ---' 0 ash - 0.
The desired sequence is obtained by setting h. = 1/m and um = uhm. This proves the lemma. If the function u belongs to C'(U), where l > 1, then u(x) also belongs to Wp 10C(U), as follows directly from the definition above. Let u(x) be an function in W1 10C(U), where U is an open subset of R". Then the symbol Vu(x) will denote the vector
Ca (x).... , a (x)) where the derivatives are understood as generalized. The vector Vu(x) is defined almost everywhere in U. Let U be an open subset of R", and u : U - R a function in the class W1 For an arbitrary measurable set A c R" P10C(U).
II UII L,(A) =
II.
'UII P.A.
(2.3)
IQI=1
II UII r.p.A = II UII W,(A) = IIUII I.A + I AI'-1/P+'""II uII Li(A)'
(2.4)
The factor in front of II uII LI(A) was chosen so that both terms on the right side of (2.4) behave the same way under similarity transformations of the space. We say that u is in the class W.' (U) if II uII i.p.u < oo. The set WP (U) of functions is a Banach space with the norm u - II UII1.p.A (Formally, the
elements of WI(U) are classes of functions differing on a set of measure zero.)
2.2. Sobolev imbedding theorems. The assertions known as the Sobolev imbedding theorems are among the main results in the theory of the spaces WD (U). We formulate these theorems only for the case l = 1, the only case used in what follows. A domain U in R" is said to be starlike with respect to a ball B(a, r) c U
if [x, y] c U for any points y E B(a, r) and x E U. We say that a domain U in R" is a domain of class S if it is bounded and is a union of finitely many domains which are starlike with respect to a ball. Obviously, a ball and a cube in R" are domains starlike with respect to a ball, and in general any convex open subset of R" is starlike with respect to a ball. A continuous linear functional L : WP (U) -a R, where p >_ 1, is said to be projective if L(p) = 1 for the function q, - 1. We have the following theorem.
I. INTRODUCTION
16
THEOREM 2.1 (First Sobolev imbedding theorem). Let U be a bounded
domain of class S in R", and let 1 < p < n. Take a number q > 1 with q < n/(n-p) if p < n, andq arbitrary ifp = n. Then WL(U) c Lq(U), and there exists a constant C, = C(n, p, q, U) < oo such that for any function u E W,(U) IIuIIq.U <_ C1IIuII1.p.U.
(2.5)
For every projective continuous linear functional L : Wp (U) - R II u - Lullq.u <- C2IIuIILI(u).
(2.6)
where C2 = C2(n, p, q, L, U) < oo is a constant. Every closed bounded set in WpI (U) is compact in Lq(U).
THEOREM 2.2. Suppose that U is a domain of class S in R", and let
p > n. Then for every function u E Wp (U) it is possible to construct a continuous function u' such that u(x) = u*(x) almost everywhere in U.
For all uEW,(U) esssuplu(x)I < C1IIuIll.p,u
(2.7)
xE U
where C, < oo and C, depends only on n, p, and U. For every projective continuous linear functional L : Wp (U) - R there exists a constant C2 = C2 (p, n, L, U) < oo such that for u E Wp(U) ess sup lu(x) - L(u)I < C2II UIILI(u) xEU
(2.8)
Let U be a domain of class S in R", and let A C U be a measurable set with IA I 0. For u E L I (U) let L4(u)
F AI
u(x) dx. JA
Further, let the function ip E Co (U) be such that fu ip(x) dx = 1. For U E L, (U) define
L,(U) =
fu
u(x)g(x) dx.
Obviously, LA and L, are projective continuous linear functionals on W p (U) for any p. These are the concrete functionals L which will be used in Owhat follows. Let W P (U) denote the closure of Co (U) in W.' (U).
THEOREM 2.3 (Poincare theorem). Let U be a bounded domain in R. Then there exists a constant C = C(p, n, U) < oo such that for every u E O WPIM IIuIIp.U <- CII uII L;(U)
(2.9)
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
17
Proofs of Theorems 2.1, 2.2, and 2.3 can be found, for example, in [96].
We make some remarks about the theorems formulated here. In the case p < n the estimate (2.5) holds also for q = n/(n - p). This precise result was obtained by Il'in [56]. For this value of q a closed bounded subset of Wp (U) can fail to be compact in L I (U).
An arbitrary function of class W (U) can be discontinuous in U. It is more convenient for us to give the corresponding example a little later. The concept of a generalized derivative can be extended in a natural way to the case of vector-valued functions.
Let U be an open subset of R", and let f : U - Rk be a mapping. For every x E U we can write f(x) = (f, (x),..., fk(x)); hence f determines k real-valued functions f1, ... , fk: the components of f. The mapping f will be said to belong to the class L p,10C(U) or to the class WP 1.C(U) if each
of j1, ... , fk belongs to L1(U) or Wp 10C (U), respectively. Let f : U - Rk be a mapping of class Wp10C(U), and let a be an arbitrary multi-index of length n and order dal = 1. We have that f (x) _ (fI (x), ... , fk(x)). Now define
Oaf (x) = (O0f1(x), °f2(x)...... T°fk(x)). For an arbitrary measurable set A C U such that 0 < Al J< oo, let k
If Ili.p.. = E
J III.p.A-
!=1
We consider the case l = 1. Let f : U -. R" be a mapping of class Wv 1.C(U). Then for almost all x E U the linear mapping Lx : R" - R" with matrix the Jacobi matrix of f at x is defined. (Here the values of the generalized derivatives at x appear instead of the ordinary derivatives.) The linear mapping Lx will be denoted henceforth by f'(x). For an arbitrary vector h E R" we have that "
f'(x)h
=
ax (x)hl
at each point x E U where fl(x) is defined. The linear mapping fl(x) will be called the derivative of f at x. 2.3. Tests for a function to belong to the lass WP10C(U). THEOREM 2.4. Let u be a function of class L 1,1,. (U).
Assume that there exists a sequence (um: U - R) of functions in Wp,10C(U), where I < p < oo, converging to u(x) in L1.1, (U) and bounded in Wp10C(U). Then U E WP.1,c(U).
I. INTRODUCTION
I8
REMARK. The condition p 96 1 is essential here.
PROOF. Take an arbitrary. compact set A c U and let q be such that 0 < >l < dist(A, CU). Let V = U, (A). The open set V is contained strictly inside U. In view of the condition in the theorem, the sequence
II
m=1,2,..
axi
,
IIp,V
is bounded for each i = 1, ... , n. Hence, there is a subsequence (umk ), M1 < m2 < ..., such that as m - oo the functions (aumk /ax;) converge weakly in L(V) to some function u E L(V) for each i = 1, ... , n. Let w be an arbitrary Sobolev averaging kernel, and take 0 < h < r. Then the functions wh * umk are all defined on V. For X E A
a (cvh * um)(x) = wh *
ax;
k
aumk
(x) =
ax;
aumk(z)w z x T. v ax; h 1
dz.
The right-hand side of this equality tends to the limit I
f
v
ui(z)cv ( z h
x) dz = (wh * ut)(x)
as k - oo. On the other hand, for x E A al (cob * umk (x)) = ax
at
Fn fv umk (z) ax
/ \ Cz-x w h
J dz,
and as k -+ oo the right-hand side of this equality tends to the limit
f u(z)8 w Cz
hn v
h
xl dz
axi(wh * u).
Consequently, we get that for all x E A a (cob * u)(x) = ((Oh * U)(x) ax;
This implies that the functions (a/ax;)(wh*u)(x) converge in LI (A) to ui as h -+ 0. Since i = 1, ... , n and the compact set A c U are arbitrary, we then get that for every compact set A c U the functions a(wh * u)/ax; converge in LI(A) as h -+ 0. By Lemma 2.2, this implies that u E W11 (U). Suppose that A c U is a compact set, 0 < , < dist(A, CU), V = U,I(A),
and 0 < h < >l. By what has been proved, a(wh * u)/ax; - ui in LI(A) as h -, 0, where the u; are functions in L ,(A). On the other hand,
a (wh*u)-- au
ax;
ax,
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
19
in L 1(A) as h -, 0. This implies that 8u/8x; E L(A), i.e., we get that u E WI R (Here Lp(A) denotes the set of all functions v: A
-
such that IIVIIo.A = sup (v(x)I < 00 xEA
in the case p = oo.) The theorem is proved. Another important characteristic of functions in W11 1a(U) is given by Theorem 2.5 below. First we introduce some notation.
Define a mapping pi of R" into R"-' by pi (x) = (x1.... , x,_ 1, x;+1, x") for x = (x1, ... , x") (pi amounts to crossing out the ith coordinate ,
ofapoint xER"). Let Qi(y)_(y1,.. ,yi-1,0.yi.
.yn-1)ER"
for y = (y1, ... , y"_ 1) E R"-'. For every y E Rn-' we have that p1(a1(y)) _ y, and for any x E Rn we have that
o,[Pi(x)] = (x1,...,xi-1,0,xi+1,...,x") Let G be an open subset of R. We say that a function u: G - R i absolutely continuous on G if it is absolutely continuous on every closed segment [a, fl] c G. Let U be an open subset of R", and let U1 = pi (U). The set U; is open in R"-' . For y E U; let Uy denote the set of all t E R such that the point Q; (y) + te1 belongs to U. The set U , is open and hence a union of at most countably many disjoint open intervals. A function u : U - R will be said to be absolutely continuous in the Tonelli sense, written U c ACT(U), if it satisfies the following conditions: A) For each i = I--, n the partial derivative (8u/8x,)(x), understood as the limit li o u(x + h 1) - u(x) h
exists for almost all x E U, and the function 8u/8x1 is locally integrable on U. B) For each i = 1..... n and almost all y e U; the function u;(y. t) u(o;(y) + te;) of the real variable t is absolutely continuous on Uy. THEOREM 2.5 [1101. For every open set U C R" and any function u E W11,1.c(U) there exists a function u' c ACT(U) such that u(x) = u'(x) almost everywhere on U. Further, the derivatives of u', which exist almost everywhere on U, are the generalized derivatives on u. The class of functions absolutely continuous in the Tonelli sense and with derivatives 8u/8x; in Lp,10(U) are often denoted by ACTp(U) in the literature.
1. INTRODUCTION
20
We shall also need the following theorem on removal of singularities for functions in Wi W.
THEOREM 2.6 [173]. Let U be a bounded open subset of R", and E c U a set closed with respect to U and with projections E, = pi (E) of measure
zero in R"- I. Let v : U\E -. R be a given function. If v is in W PI (U\E), then v belongs also to W PI (U).
PROOF. Let U\E = V, U1 = p; (U), and V; = p; (V ). Then it is clear that
U, D V1 D U1 \ E. The sets U, and V, are open in R"-I. By Theorem 2.5, it can be assumed without loss of generality that v E ACT(U \ E). By hypothesis the projections E; have measure zero in R"- I and so E is a set of measure zero. The derivative Ov/8x;, understood as the limit
lim[v(x + he;) - v(x)]/h, is defined almost everywhere on V, and hence almost everywhere on U. This derivative is integrable to the power p on V, and thus also on U. For y f E, the sets U , and Vy coincide. Since V E ACT(V), the function t i - v[a,(y) + lei] is absolutely continuous on Vy for almost all y E V, by Theorem 2.5. Since U, \ V1 c E;, E, is a set of measure zero, and Uy = Vy for y 0 Ei, we thus get that the function v[a; (y) + lei] of the variable t is absolutely continuous on Uy for almost all y E U,. Hence, v(x) satisfies the two conditions A) and B) in the definition of a function absolutely continuous in the Tonelli sense. Further, its derivatives are integrable to the power p on U, and hence v E W PI (U). The theorem is proved. REMARK. The condition of the theorem concerning E holds, in particular, if E is a set whose (n - 1)-dimensional Hausdorff measure is zero. THEOREM 2.7. Every function u: U -i R of class C°-I (U) belongs to
WW(U). PROOF. The theorem can be obtained as a corollary to Theorem 2.5. However, the discussion of certain details involving measurability questions turns out to be somewhat cumbersome. We give a derivation based on Theorem 2.4. Let U E C°- I (U). This means that u is continuous and satisfies a Lipschitz condition on every compact subset of U. Let Xh, where h > 0, be the function defined in Lemma 1.1, and let vh(x) = Xh(x)(cvh * u)(x) for x E Oh, and vh(x) = 0 for x f U. Define um(x) = v(I/m)(x). Then u," E COO(U) for each m = 1, 2, ... , and um - u in LI,Ia(U) as m -+ oo (the convergence is even locally uniform in U). Let x0 be an arbitrary point in U, and let r > 0 be such that B(x°, r) c U.
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
21
Take an mo with 1 Imo < r/2. Form > mo we have that B(xo, r/2) c `U2k, where h = 1 /m, and thus for all x E B(xo, r/2) u,,,(x) = (W1/m * u)(x) = J
u(x+Y/m)W(Y)dy.
y I51
The ball W(xo, r) is in U; hence there exists an L < oo such that
lu(x') - u(x")l < Lix' - x"I for any points x', x" E W(xo, r). If X E B(xo, r/2) and lyI < 1, then x + y/m lies in W(xo, r), and hence for any x1, x2 E B(xo, r/2) we have
lum(x1)-um(x2)I <- f IyI<1
Iu(x1+m) -u(x2+m) I W(Y)dy
< LIx1 - x21 fyI51 W(Y) dy = LIx1 - X21-
The The function u, belongs to the class C°°. It thus follows from this inequality that IVum(x)1 < L for all x E B(xo, r/2) and all m > mo. Let A c U be a compact set. By what has been proved, for every x E A there exist a ball B(x, 8) c U, a constant L < oo, and an index mo such that IV um (y) 1 < L for all y E B(x, 8) when m > mo. By the Borel theorem,
A can be covered by finitely many such balls. Let m be the largest index mo corresponding to the balls forming such a covering, and let L be the largest of the corresponding constants L. For m > m it is obvious that IVum(x)1 < L for all x E A. Since the compact set A C U is arbitrary, this proves that the sequence (um), m = 1.2, .... is bounded in W.10C(U). On the basis of Theorem 2.4, this implies that u E W,10C(U), which is what was to be proved. REMARK. Theorem 2.7 admits a converse. Namely, if u E WW 10c(U), then u E (more precisely, u(x) is equivalent to a function in C°.' in the sense of Lebesgue measure in Rn). This assertion can be obtained by the simple use of a Sobolev averaging. We leave the details of the corresponding arguments to the reader. 2.4. Transformations of functions with generalized derivatives. For all the functions of class W1 10c in Theorems 2.8 and 2.9 below, as well as for their derivatives, we assume that their values are equal to the natural value at each point where it exists. The results in this subsection will be needed below in connection with the study of functions with generalized derivatives on manifolds.
THEOREM 2.8. Let U and V be bounded open subsets of R. and let of class C' such that a(U) = V and all c: U -+ R" be a
I. INTRODUCTION
22
the first-order derivatives of a are bounded in U. Then for every function v E WIII,C(V) the function u = v o a belongs to W11 I.c(U), and its generalized derivatives can be expressed in terms of the derivatives of v(x) by the formulas au = ax;
o
11
a)
001
/ ax;
j- i \ ayi
(2 . 10)
.
PROOF. Equality (2.10) is true when v E CI. Let v be an arbitrary function of class W,'1( V), and let u; =
" () i=I
ay;
oa
x;
It is required to prove that u; = tIu/ax; for every i = I__, n. We construct a sequence (vm ), m = 1, 2,.... of C°° -functions converging to v in WIII0C(U), and let um = vm o a. Let A c U be an arbitrary compact set. Since a E CI (U), the functions 0a; /8x; are bounded on A. Because a is a diffeomorphism, det a' (x) 0 0 for all x E U, and hence there exists a 8 > 0 such that I det a'(x)I > 6 for all x E A. From this we get
(J-. gym
<_
aCP
P
dx
u;
ri-
=Cn
) I /P
(f I a"' o or A
1
I/P o
al
det a'(x) I
dx)
IDd
I.
J°('I ) ayi
j=1
a!
ayi
y
IIC7t1m
ayi
P,,I
-
0v
ayi
I
P.oca)
The right-hand side of the last inequality tends to zero as m -+ oo, and hence I
I
exi
-uill
asm-+oo. P P.
Since the compact set A c U is arbitrary, this proves that aum/ax; u; in LP,Ia(U) as m - oo. It is also obvious that um -' u in LD,I0c(U) as m -+ oc. This implies that u E Wv I0C(U) and u; = au/ax; for i = 1, ... , n. The theorem is proved.
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
23
THEOREM 2.9. Let U be an open subset of R", V an open subset of Rk, and f : U -+ V a continuous mapping in Wp ,a(U). Then for any C'function v : V - R the composition u = v of is a function of class Wp,,. (U), and its generalized derivatives are expressed in terms of the derivatives of v and f by the same formulas as in the case of smooth functions, i.e., k
ax, -
I j=1
afj (a ° f ) ax,
for PROOF. Let k
u,=E i=1
av
of
aye
af; 8x;
.
n. It must be proved that u, = au/ax, for all i = 1, ... , n. Let to be an arbitrary Sobolev averaging kernel, and let fh = (Oh * f = (coh * fl,..., (oh * fk) Then fh - f in Wv,0C(U) ash - 0. Further, since f is continuous, the convergence fh - f is also locally uniform as h - 0. Let A c U be an arbitrary compact set, and let B = f (A); B c V is compact. Take an rt > 0 such that U,r(B) C V, and let H = U,7(B); H is compact. We have that fh -, f uniformly on A as h - 0; hence there exists a 8 such that I fh(x) - f (x)I < n for all x E A if 0 < h < J. Also, fh(x) E H for x E A when 0 < h < J. Since H is compact, the function &10y; is uniformly continuous on H. For 0 < h < >, and x E A
a [fh(x)] -
e
[f(x)]I 5 w;(Ifh(x) -f(x)I),
(2.11)
where w; is the modulus of continuity of av/ay; on H. This gives us that av
ay;
ofh
Of ay; Of
uniformly on A as h -' 0. We have that
lh
BV,°fha
al
Cay;°f)ax,l `
h
_ ale
I ay;°fhjax,) N-j'i I av
+ ay ; °
fh _
ay; °
f
aaxf;
,
(2.12)
Further,
a [fh(x)]I < M < 00
1. INTRODUCTION
24
for X E A and 0 < h < 6, where M = cont. We conclude from (2.11) and
(2.12) that for0
Iaxf (vo
lh) - u'll
axill p A
pA
+w(Ilfh - .f IIoo.A
A .
afi ll P. !-1 ll axi
The right-hand side of the inequality tends to zero as h -- 0, and thus the derivatives of the functions v o fh converge to the functions u; in LP.1, (U). Since the functions v o fh themselves converge locally uniformly
to the function u = v of, this implies that u; E Lp.10(U) and u; = au/axi. The theorem is proved. 2.5. Dependence of the coefficients in the Imbedding theorems on the size of the domain. In applications of Theorems 2.1, 2.2, and 2.3 it is useful for certain standard domains (for example, for a cube, ball, etc.) to know how
the constants in (2.6), (2.8), and (2.9) depend on the sizes of the domain. This question can be answered by a simple change of variables. We confine ourselves to the case of a ball, since all the arguments go through similarly for a cube.
Let B = B(a, r) be the ball about a with radius r > 0 in R", let BI = B(0,1), and let a be the transformation x a + rx of R". Obviously, a (BI) = B. The Jacobian of a is equal to r". If U E WP (B), then v = U o a E W,(BI ), as follows from Theorem 2.8. Further,
a (x) = for every i = 1, ... , n, and thus
a
(a + rx)r
-(x)IP
I
J B, ax;
dx =
J B,
12au
8Y;
(a+ rx) rp dx au P I- (a+rx)I r"dx=r"I -) au
=rP'" lB1
P
lR(a,,) 8Yi
8Yi
dx.
From this,
Ilvllr.;(B,) = rI -"'DII ull c;(B)
(2.12)
Further, \I /q
IIvIIL,(B,) _ (r j, (u(a + rx)IQr" dx)
= r-"/9IIuIIL,(B).
(2.13)
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
25
Next, let uB
iBl
fu(y)dy.
Obviously,
v(x) dx = uB.
V BI IBl
I JB,
Let p < n, and let q be such that I < q and (n - p)q < n. Choosing the L in (2.6) to be the linear functional v -+ vB,, we get that Ilv - VB1II L0(B) < COIIVIIL,(B,)
In view of (2.12) and (2.13), this implies that for the given p and q II u - uBll L (B) <-
Crl+nlq-nlpII
uII L;(B)
(2.14)
where the constant C depends only on p, q, and r. In the case p > n we get the estimate esssup l u(x) - uBI < Cr'-n/pllull L;(B) xEB(a.r)
(2.15)
from (2.6) in a similar way. Finally, for arbitrary p > 1 inequality (2.8) allows us to conclude that O for every u E W I (B (a, r)) II uII L,(B(a.r)) <- Crll uII L;(B(a.r)).
(2.16)
Similar estimates hold also in the case when a cube Q(a, r) is taken instead of a ball in R. 2.6. A theorem on differentiability of Wp 1' 0C functions almost everywhere.
Let U be an open subset of R". A mapping f : U - R"' is said to be differentiable at a point a E U if there exist a linear mapping L : R" - R"' and a function a: U --. R" with a(x) -' 0 as x -+ a such that
f (x) = f (a) + L(x - a) + a(x)lx - al for all x E U. In this case L is called the derivative off at a and denoted
by f'(a). A mapping f : U - R" is said to be 1 -fold differentiable at a point a E U
if there exist functions P : R" - R'" and a/: R" - R"', with P a polynomial of degree at most I representable in the form P(x) = >)x)« A2X2, the A2 being vectors in R", and with a1(x) --+ 0 as x - a, such that
f (x) = P(x - a) + Ix - al'ai(x)
for all xEU.
26
I. INTRODUCTION
One of the topics in the classical theory of functions of a real variable concerns theorems on differentiability almost everywhere of functions satisfying diverse conditions. A classical example is the famous theorem of Lebesgue on differentiability almost everywhere of a monotone function of one variable. Of other results in this direction we mention the theorem of Aleksandrov [I I] on two-fold differentiability almost everywhere of a convex function of n variables, the theorem of CalderGn and Zygmund on 1-fold differentiability almost everywhere of the functions in WP 1.10when
l p > n, and the theorem of Stepanov [ 163] establishing necessary and sufficient conditions for differentiability almost everywhere of a function f : U - R, where U c R". In the author's paper [131] it is proved that if differentiability is understood in a certain generalized sense, then every function in WP I« is 1-fold differentiable almost everywhere. As shown in [131], this result can serve as a means for establishing various theorems on differentiability almost everywhere (now in the sense of the definition given above) of functions in diverse classes. In particular, the Aleksandrov theorem and the Calderon-Zygmund theorem can be proved in this way. Below we shall need a special case of a theorem in [ 131 ]. Let us state it and note some corollaries. The proof will be given in Chapter III. Take an arbitrary open set U in R". Let G be a bounded measurable subset of R" containing 0 as an interior point, and let 91 be a metric space whose elements are measurable functions on G with values in R"'. Let f : U - R'" be a given function and L: R" -+
R"' a given linear mapping. Take an arbitrary point x E U. Since G is bounded, there exists an t, > 0 such that if 0 < h < 1, then x + hX E U for any X E G. The linear mapping L is called the differential off at x in the sense of convergence in 91 if for 0 < h < >I the function
f(x) -L(X) Ah: X" f(x+hX)h is in 91 and converges in the sense of the metric of 91 to the function identically equal to zero as h 0. If the given condition holds, then we also say that f is differentiable at x in the sense of convergence in 91. Let 9t be the collection of all bounded measurable functions F : G - R'", and let a metric be defined in 91 by the norm IIFIIMCa = sup IF(x)I. xEG
Then L is the differential of a function f : U - R"' in the sense of convergence in M(G) if and only if
f(y) - f(x) - L(y - x) = o(IY - xl)
§2. FUNCTIONS WITH GENERALIZED DERIVATIVES
27
as y -- x. Indeed, let (f (x + hx) - f (x))/h = Lh(x). Assume that L is the differential of f at x in the sense of convergence in R. Let r and R be such that B(0, r) C G C B(0, R). Define A(h) = II Lh - L II M(G) = sup I Lh (X) - L(X)I. XEG
Leth=ly-xl/randX=r(y-x)/ly-xl. Then Then X E B(0, r), and hence If(x+hX)- f(x)-hL(X)I
If(Y)-f(x)-L(Y-x)I <- rIY - xlA(IY rxll and thus
If(Y)-f(x)-L(Y-x)I =o(IY-xI) Conversely, letIf(y)-f(x)-L(y-x)I that If (y)
This means
- f(x) - L(y - x)1:5 IY - xIJ(IY - xl ),
where 6(r) -i 0 as r -, 0. The function 6(r) can be assumed to be nondecreasing. (This can always be made the case by replacing 6(r) by SI (r) = sup0<1<, 6(t) if necessary; it is clear that 6(r) < 61(r) for all r > 0, 61 is nondecreasing, and 51(r) -+ 0 as r - 0 if 6(r) 0 as r - 0.) Setting y = x + hX, where X E G, we get that ILh(X) - L(X)I < 8(hR), which implies that IILh - LIIM(G) -y 0 as h
0.
THEOREM 2.10. Let f : U -+ R"' be a function in W.' (U), where U C R" is an open set and G is a bounded open set such that 0 E G. Then for almost
all x E U the linear mapping
xaz xi i=1
is the differential off at x in the sense of convergence in Wp (G). A proof of Theorem 2.10 will be given in Chapter III, §4.1. We remark that the case of arbitrary G can be reduced trivially to the case when G is a ball. We prove the following assertion as an example showing how Theorem 2.10 can be used in proving differentiability almost everywhere of functions in certain concrete classes.
I. INTRODUCTION
28
COROLLARY 1 [22]. Everyfunction f : U - R'" of class WP(U), where p > n (U an open subset of R"), is difereptiable almost everywhere in U. REMARK. Here and below, every function in W. 1-10c, where p > n, is assumed to be continuous. According to Theorem 2.2, this can always be made the case by changing the values of a function on a set of measure zero if necessary. PROOF. Suppose that x E U is such that the function
Bf(x)Xi
i=l
axi
is the differential off at x in the sense of convergence in W, (B(0,1)). By the theorem, almost all points x E U are such points. This means that as
h-.0
IILh
where Lh(X) =
- Lllw;[e(o.I)I,
(x + hX) - f (x)]. By Theorem 2.2, this implies that IILh - LIIM[e(o.I)) - 0
as h - 0,
and hence L is the differential of f at x in view of the remarks above. The corollary is proved. COROLLARY 2. Suppose that U is an open subset of R. Every mapping f : U - R" of class C°-I is differentiable in U almost everywhere.
PROOF. Let f : U -, R"' be a mapping of class C°-I. Then, using Theorem 2.7, we get that f E W.' Ia(U), and hence f E W 1 (U) for every p > n. The assertion to be proved thus follows immediately from Corollary 1. §3. Mfibius transformations
3.1. Motions and similarity transformations of a Euclidean space. Below, E denotes a Euclidean vector space, i.e., a vector space in which each
pair of vectors x, y is assigned a number (x, y) called the inner product of x and y and satisfying well-known axioms. (Namely: 1) (x, y) = (y, x)
3) (x, x) > 0 Vx E E, and (x, x) = 0 a x = 0.) It is assumed that the dimension of E is at least 2; in other respects it can be arbitrary. In particular, E can also be infinite. In what follows we deal only with mappings of finite-dimensional spaces, but the assumption of finite-dimensionality does not lead to any essential simplifications here. For any x, y E E we have that I(x, y)I <_ Ilxllllyll, with equality if and only if x and y are linearly dependent.
§3. MOBIUS TRANSFORMATIONS
29
A subset P of E will be called a hyperplane if P = {x E El (a, x) - k = 0) for some vector a 0 and number k. Here the vector a and the number k are uniquely determined to within a constant factor, i.e., if a, E E and k, E R are such that P = {x E El (a i , x) - k, = 0}, then there exists a A. E R, A 0 0, such that a, = Aa and k, = Ak. Of course, this assertion is hardly worth stating explicitly in the case of finite-dimensional E, but since the context here is "dimensionless", some proof must be given. Accordingly,
let (a, x) - k = 0 be the equation of the given plane P. Assume that the nonzero vector a, and the number k, are such that all the points of P satisfy the equation (a,, x) - k, = 0. Let x0 = ak/lal2. Then xo E P, and hence (a,, xo) - k, = 0. Let yo = xo + a, -
(a, . a) Ja12
a.
Then (a. yo) - k = 0, i.e., yo E P, and thus (a,, yo) - k, = 0. From this we get after obvious transformations that lal2la,12 = ((a. a,))2, and, consequently, a, = Aa. Further, k, = (a,, xo) = A(a, xo) = 1k, which is what was required. A subset S of E is called a sphere if there exist a point a E E and a
number r > 0 such that S = {x E Ellx - al = r}. The point a and the number r are uniquely determined by the specification of the sphere S,
i.e., if a, E E and r > 0 are such that S = {x E El lx - a, I = r,), then a, = a and r, = r. Indeed, assume that a # a,. Then h = la -a, l iA 0, and the points x, = a + (r/h)(a, - a) and x2 = a - (r/h)(al - a) belong to S. This implies that Ix, - a, I = 1x2 - a, l = rl. We have that lx, - a, I = lr - h l and lx2 -all = I r+ h l , and we get a contradiction, since I r- h l# l r+ h l in view of the fact that h 34 0 and r > 0. Thus, a, = a. For every x E S we have that r, = Ix - a, I = Ix - al = r, i.e., r, = r. A linear transformation P : E -+ E is said to be orthogonal if P(E) = E and l Pxl = lxl for any x E E. (In the case of finite-dimensional E the condition P(E) = E automatically follows from the other conditions imposed on P.) If P: E - E is an orthogonal transformation, then (Px, Py) = (x, y) for any x, y E E.
A mapping _T: E -+ E is called a motion if it is bijective and l9x -
9Y1=lx-yl for anyx,yEE. of E admits the representation Dx = THEOREM 3.1. Every motion a + Px, where a E E and P is an orthogonal transformation of E. .
PROOF. A parallel translation, i.e., a transformation x '-- a + x, is a motion. A composition of two motions is clearly a motion. Let q: E - E
be an arbitrary motion of E, and let a = 9r(0) and P(x) = .x - a. The
I. INTRODUCTION
30
mapping P is a motion. We prove that P is an orthogonal transformation of E. For this it suffices to establish that P is linear. First of all, note that P(0) = 0. Take vectors u, v E E such that u 0 0, v 96 0, and J u + vl = Jul + IvI Then v = Au, where A > 0. Let xo, xI, and x2 be three different points of E lying on a single line. Define x; = P(xi ), i = 0, 1, 2. We prove that the points xo, x'j, and x2' then also lie on a single line and are in the same order as xo, xI, and x2. Assume for definiteness that xI is in the segment [xo, x2]. Let u = x1 - xo
and v = x2 - x'. Then u + v =X2 - xo.
IuI = Ix'1 - xol = lxI - xol,
IvI=1X2-xil=Ix2-xII and hence IuI + IvI = IxI - xoI + Ix2 -xII = Ix2 - xoI = lu + VIFrom this, v = Au, where A > 0. This allows us to conclude that x1 belongs to [xp, x2].
Take any x E E and A E R. We prove first that P(Ax) = AP(x). This
holds when A = 0, or A = 1, or x = 0. Assume that x 34 0, A 0 0, and A # 1. Let Ax = y, x' = P (x), and y' = P (y). We have that ly'I = ly' - 01= ly - 01= lAllyl. The points x, 0, and y lie on a single line. Hence, the points x', 0, and y' also lie on a single line. If A < 0, then 0 lies interior to the segment [x, y], and thus in this case 0 also lies interior to [x', y']. This enables us to conclude that y' = µx', where p < 0. Since ly'l = lAllxl = IAIIx'I, it follows that Iµl = IAI, and hence p = A. If A > 0, then either x lies between 0 and y (when A > 1) or y lies between 0 and x (for A < 1). This implies that the points 0, x', and y' are arranged in the same way, which allows us to conclude that y' = Ax' in this case. Let x and y be two distinct points of E, z the midpoint of the segment
[x, y], z = (x + y)/2, x' = P(x), y' = P(y), and z' = P(z). Then z' lies on [x', y'], and Iz' - x'I = Iy' - z'I, i.e., z' is the midpoint of [x', y'), z' = (x' + y')/2. We thus get that P
(x 2 y
'[P(X) + P(y)J.
By what was proved, this implies that P(x + y) = 2P((x + y)/2) = P(x) + P(y). The linearity of P is established. Since P is bijective and I P(x)l = IxI for every x E E, this establishes that P is orthogonal. For every x E E we have that (x) = a + P(x), and the theorem is proved.
§3. MOBIUS TRANSFORMATIONS
3!
Let El and E2 be arbitrary vector spaces. A mapping f : E! - E2 is said to be affine if there exist a vector k E E2 and a linear mapping L: E! - E2 such that f (x) = k + L(x) for every x E E!. The mapping L is called the linear part of the mapping f . For every x E EI we have that L(x) = f (x) - f (0); hence L is uniquely determined by f . According to Theorem 3.1, every motion of a Euclidean space E is an affine transformation of E. The collection of all orthogonal transformations of E forms a group,
which we denote by O(E). In the case E = R" the symbol 0,, is used to denote the group. The determinant of every transformation P E 0 is equal to ± 1, and the collection of P E 0,, with det P = 1 is denoted by 0; . Obviously, 0n is a subgroup of On. A mapping F of a Euclidean space E into itself is called a general orthogonal transformation if F is representable in the form F(x) = AP(x), where A > 0, A E R, and P is an orthogonal transformation of E. A mapping 0: E - E is called a similarity transformation of E or a similarity if 4) is representable in the form 4)(x) = a + F(x), where F is a general orthogonal transformation. The collection of all general orthogonal transformations of E, as well as the collection of all similarities and all motions
of E, form groups. We do not introduce any special notation for these groups. A special case of similarity transformations is a homothety, i.e., a mapping of the form h2 : x a + A(x - a), where A # 0. The point a is called
the center of the homothety. A homothety for which A = -1 is called a symmetry with respect to the point a. A symmetry with respect to a point is a motion. Obviously, the homotheties form a subgroup of the group of motions of E. An arbitrary mapping 4): E -, E is a similarity if and only if 4) is bijective and there exists a number A > 0 such that 14)(x) -4)(y)I = Alx- yl for any x, y E E. The fact that every similarity satisfies this condition is obvious. Conversely, let 0 be a mapping satisfying this condition. Then the transformation 9' = (1 /i)4) is obviously a motion, and hence 9'(x) =
k+P(x), Vx E E, where P E O(E). From this, 4)(x) = 2k+AP(x) Vx E E, and thus 4) is a similarity. Let x, y, and z be three different points of E, and let (x, y, z) = Ix - yI/ly - z1.
We call this quantity the simple ratio of the points x, y, and z.
1. INTRODUCTION
32
THEOREM 3.2. Let S : E -- i E be a bijective mapping. If for any three distinct points x, y, z E E (x, Y, z) = (S(x), S(Y), S(z)),
then S is a similarity. PROOF. Assume that S : E E satisfies the condition of the theorem, and take any three distinct points x, y, z E E. By assumption, (x, y, z) _ (S(x), S(y), S(z)), i.e.,
Ix - YI _ IS(x) - S(Y)l IS(Y) - S(z)I ' !T!-- T1
which implies that IS(x) - S(Y)I IS(y) - S(z)I ly - zi 1X-y1 As we now show, this equality gives us that the ratio
(3 . 1)
S(x) - S(y)1/1X - YI
is constant. Fix any distinct points a, b E E and let S(a) - S(b)I ll a - bI = A.
Let (p, q) be an arbitrary pair of points in E with p # q. We show that
IS(p) - S(q)lllp - ql = A.
(3.2)
If p = a and q = b, this is obvious. In the case p = a and q 96 b equality (3.2) f o l l o w s f r o m (3.1) if we set x = b, y = a, and z = q in it. The validity of (3.2) in the case when q = a is obtained analogously. Assume that p . 9a and q a. Setting y = a,x = b, and z =pin (3. 1), we get that IS(p) - S(a)I = IS(b) - S(a)I = 2 Ip-al lb - al Setting y = p, x = a, and z = q in (3.1), we find that
IS(p) - S(a)j Ip - al
_-
IS(p) - S(g)l Ip - ql
which implies (3.2). Consequently, we get that
IS(x) - S(Y)I/Ix - yI = const > 0, and hence S is a similarity. The theorem is proved.
§3. MOBIUS TRANSFORMATIONS
33
3.2. MSbius transformations. Definitions. We extend the space E by adjoining to it a certain "improper" element denoted by the symbol oo. The extended space is denoted by E, E = E U {oo}. A topology is introduced
in E as follows. A neighborhood of oo is taken to be any set U whose complement is a bounded subset of E. If x 0 oo, then x E E, and in this case any set containing a ball about x is taken as a neighborhood of x. A sequence (x, ), m = 1, 2, ... , of points in E converges to oo if and only if Ixml -+ oo as m - oo. (If x = oo, then lxl is regarded as equal to oo.) In the case when the dimension of E is finite and equal to n the topological space E is compact. The space E constructed in the way described here will be called the MObius space generated by E. We construct a mapping of the unit sphere S(0, 1) of E x R onto E. The structure of a Euclidean vector space is introduced in E x R in the natural
way; namely, for arbitrary ul = (x1, yl) and u2 = (x2, y2) in E x R (here XI , X2, Y1, y2 E R) we let
Au1 +µu2 = (Ax1 +µx2,Ay1 +µy2),(ul,u2) = (xl,x2)+yly2 The point (x, 0) E E x R will henceforth be identified with the point x in E.
Consider the sphere SE = S(0, 1) = {u E E x RI Jul = 1} in E x R. Let N = (0,1) and S = (0, -1). The respective points N and S will be called the north and south poles of the sphere SE. We construct a certain special mapping a of E into SE. Let a(oo) = N. For x E E, x 96 oo, we join the point x = (x, 0) E E x R to the point N in E x R by a straight line. Let us show that this line intersects SE in some point different from N. The line admits the parametric representation u(t) = tx + (1 - t)N, where t E R.
t Ixl + (1 - t)
We have that u(t) = (tx, 1 - t) and lu(l)l =
.
Setting
lu(l)l = 1, we get that
121x12+1-2t+t2=1. This equation has two different solutions: tl = 0 and t2 = 2/(Ix12 + 1). Let a(x) = u(12) =
2
(\ Ix12 + 1X ,
1Ixx112
2
- ) 1
+1
(3.3)
The mapping a of E into SE so defined is called the stereographic projection.
We mention certain properties of a. First of all we show that a is one-to-one. Indeed, let x1, x2 E E, X1 # x2. If one of the given points x, and x2 is oo, then its image under a is N, and, as is clear from the construction, the image of the other point is different from N; hence in this case a(x1) 0 a(x2). If x1, x2 96 oo, then the lines x1 N and x2N
34
I. INTRODUCTION
have a unique common point-the point N. From this it follows that a(xi) 54 a(x2) in this case. We show that or is a mapping of E onto SE. Indeed, take an arbitrary point u E SE. If u = N, then u = a(oo). Assume that u 96 N. We have that u = (y, z) and IUI2 = 1y12 + z2 = 1, and since u 96 N, it follows that z < 1. The line passing through u and N intersects the hyperplane E in the point
x= 1z Y.
(3.4)
1
Obviously, u = a(x). Each of the mappings or and a-I is continuous. The continuity of or at a point x oo follows immediately from (3.3), and the continuity of a- I at a point u 96 N follows from (3.4). We leave it to the reader to prove that a and a-' are continuous also at the respective points 0o and N. Stereographic projection is thus a homeomorphism of E onto SE. We endow E with the structure of a differentiable manifold by agreeing that the stereographic projection is a diffeomorphism of class CC* from the manifold E onto SE. The equalities (3.3) allow us to conclude that on E the mapping a is a diffeomorphism of class COO from E onto SE \ {N}. This shows that the structure of a differentiable manifold introduced on E is consistent in the natural way with the structure on E. Let S(a, r) be the sphere about a with radius r in E. The inversion with respect to S(a, r) is defined to be the following transformation of E. First of all, let j(a) = oo and j(oo) = a. But if x 96 a and x # oo, then y = j(x) is determined as follows: y lies on the ray from the point a passing through the point x, and is such that Ix-al ly-al = r2. These conditions determine y uniquely. The first condition implies that y - a = ).(x - a), where A. > 0, and the second implies that a. = r2/Ix - a12; hence r2 (x
(3.5) y = j(x) =a + Ix - aa) It is obvious from the definition of j that if y = j(x), then j(y) = x; therefore, j- ' = j, i.e., j is a so-called involution. If x E S(a, r), then 12
j(Y) = X. The inversion j with respect to S(a, r) is a homeomorphism of E onto
itself. Indeed, it follows from the definition that j is bijective. Equality (3.5) enables us to establish that j is continuous at a point x # a, x # oo. The ball B(a, e) is mapped under j onto the complement in E of the ball B(a, r2/e). This clearly implies that j is continuous also at each of the points a and oo.
§3. MOBIUS TRANSFORMATIONS
35
The space E has been equipped with the structure of a C°°-differentiable manifold. It is easy to establish that inversion with respect to S(a, r) is a C°°-transformation. This is a simple consequence of the definitions above. (It suffices to note that or o j o or- I is a C°°-mapping of the sphere S(0, 1) in Ex R.) Let CD be an arbitrary similarity of E. We extend CD to E by setting 1D(oo) = Co.
A mapping (p: E -+ E is called a Mobius transformation if it can be represented as a composition of finitely many inversions and similarities. If rP and lv are Mobius mappings, then VP o Vi- I is obviously also a Mobius transformation, and hence the collection of all Mobius transformations of E forms a group, which we denote by the symbol M(E). This group will
be denoted by M in the case E = R". Every Mobius transformation is a C°°-diffeomorphism of the manifold E.
We mention some properties of Mobius transformations.
Let j be inversion with respect to the sphere S(a, r), j(x) = a + r2(x - a)/lx - a12. Denote by h the homothety x --- a + rx. Obviously, S(a, r) = h[S(0, 1)]. Denote by jo the inversion with respect to S(0, 1). We have that jo(x) = IXIZ
Vx E E \ {0}.
It is not hard to see that j = h o jo o h - I . Each of the transformations h and h` I is a similarity. In view of the remark made above, every Mobius transformation admits a representation rp = ipI o o cDm, where each of the transformations (pi is either a similarity or the transformation jo.
Let U be an open subset of E, and f : U - E a CI-mapping. The mapping f is said to be conformal at a point xo E U if the derivative f'(xo)
of f at x0 is a general orthogonal transformation. A mapping f : U - E is said to be conformal if it is conformal at each point x E E. We show that every mapping rp E M(E) is conformal at each point x e E \ Sp-I(oo). This is clearly true if {p is a similarity. Since a composition of general orthogonal transformations is a transformation of the same type, and every Mobius transformation is a composition of finitely many inversions and similarity transformations, it suffices to see that an inversion is a conformal mapping. Accordingly, let
f (x) = a +
r2(x - a)
Ix-a12
1. INTRODUCTION
36
Simple computations show that at every point x # a and for any vector h 2
f'(x)h =
lx r a12 (h - 2(e, h)e),
where e = (x-a)/lx-al. The mapping P: h .- h-2(e,h)e is a symmetry with respect to the plane (e, x) = 0, and hence is an orthogonal t r a n s f o r m a t i o n . (The f a c t that I P (h) l = I h l is also easy to establish directly.) This
implies that f'(x) is a general orthogonal transformation, as was required to prove. 3.3. MBblus transformations and cross ratios. Construction of MBblus transformations. Let x1, x2, x3, x4 be an arbitrary quadruple of points in E such that xI # x2, x2 96 x3, x3 96 x4, and x4 96 x1. The quantity (XI,x2,x3,x4) =
IxI -x2I Ix3 -x41
(3.6)
IXI - x41 1x2 - X31
is called the cross ratio of the points x1, x2, x3, x4. We join x1 and x2, x2 and x3, x3 and x4, and x4 and x1 by successive segments, obtaining a quadrangle. The numerator of the quotient defining (xl, x2, x3, x4) contains the product of the lengths of one pair of its opposite sides, and the denominator contains the product of the lengths of the other pair of opposite sides. Obviously,
(xl,x2,x3,x4) = l/(x2,x3,x4,x1) = (x3,x4,x1 ,x2) = 1/(x4,XI,x2,X3) = (x2,x1,x4,x3).
(3.7)
Let x1, x2, x3, x4 be an arbitrary quadruple of points in E satisfying the same conditions x1 96 x2, x2 34 x3, x3 x4, and x4 96 x1, with one of the points x; equal to oo. In this case the quantity (x1, x2, x3, x4) is defined as the limit lim(x; , x2, x3, x4), where x, = x; if x; is finite, and x,1 - oo if x; = oo, i = 1, 2, 3, 4. It is not hard to show that this limit always exists. In particular, we have that (XI, x2, x3,
00) = lim IxI - x2I Ix3 - XI x-»°° 1x2 - x31Ix1 - x)
= IxI - x21 = (x1, Ix2 - x31
x2, x3).
(3.8)
It is frequently useful to give the quantity (XI, x2, x3, x4) a definite meaning
also when the conditions x1 # x2, x2 # x3, x3 # x4, and x4 # x, are not satisfied. In this case we set (x1,x2,x3,x4) = lim(x',x2,x3.x4),
§3. MOBIUS TRANSFORMATIONS
37
where x' # xz, x2 # x3, x3 96 x4, x4 96 x;, and the limit is taken under the condition that x, - x1 for each i = 1, ... , 4 under the assumption that this limit exists. Here the values 0 and oo are allowed for (xi, x2, x3, x4). If XI, x2, x3i and x4 are all different from oo, then (x1, x2, x3, x4) is defined if and only if the quotient on the right-hand side of (3.6) is not an expression of the form 0/0, and (3.6) remains valid if we set (XI, x2, x3, x4) = oo when the denominator of (3.6) is zero. THEOREM 3.3. Let x1, x2, x3, x4 be an arbitrary quadruple of points in E for which (xI , x2, x3, x4) is defined. Then for every 9 E M(E) (3.9) (9(x1). 9(x2), 9(x3), 9(x4)) = (XI , x2, x3, x4). PROOF. This equality obviously holds when f is a similarity transfor-
mation. Since an arbitrary Mobius transformation is a composition of finitely many inversions and similarity transformations, it suffices to consider the case when 9 is an inversion. Further, we can confine ourselves to the case when 9 is inversion with respect to the unit sphere S(0, 1), 9 = jo. Assume first that x1 96 x2, x2 # X3, x3 91 x4, and x4 Y6 x1, and all these points are different from 0 and oo. Let e; = x; / I xt 1, i = 1, 2, 3, 4. Then
jo(xt) = etllxtl and (jo(x1). jo(x2), jo(x3), jo(x4))
_
eI
e2
Ix11
IX21
11
e3
e4
eI
e4
e2
e3
IX31
IX41
Ix11
1x41
IX21
IX31
= IIx2Ie1 - Ix21e21 IIx41e3 -1x31e41/IIx4IeI - Ix1Ie4111x31e2 - Ix21e31
We have that IX, 12
IX,
- xjI2. which clearly implies (3.9). The required result can be derived from the proven passage to the limit for the case when the conditions x1 .4 x2, x2 0 X3, x3 # x4, and x4 0 xI are not satisfied, as well as for the case when some of the given points coincide with 0 or with oo. The theorem is Ilxilei - Ixxletl2 =
- 2lx;IIxxI (et,ei) + Ixxi2 =
proved. COROLLARY 1. Every transformation 9 E M(E) such that 9(oo) = oo is a similarity.
PROOF. Let 9 E M(E) be such that (p(oo) = oo. Take three arbitrary distinct points XI , X2, x3 E E. We have that (x1,x2,x3) = (xI,x2,x3.oo) = (9(x1), 9(x2) (91(x1) 9(x2) 9(x3))
The required result follows immediately from this, in view of Theorem 3.2.
1. INTRODUCTION
38
COROLLARY 2. Every transformation ip E M(E) either is a similarity or admits a representation of the form Sp = a o fl, where a is an inversion and ft a motion of E. Such a representation of ip is unique.
PROOF. Let rp E M(E). If ip(oo) = oo, then , is a similarity transformation. Assume that gr(oo) = a 96 oo. Let a1 be an inversion with respect to the sphere S(a, 1), and let f, = a, o 47. We have that 8(°°) = a1 [q (oo)] = a1 (a) = oo, and hence f, is a similarity. Let h be the dilation coefficient of the mapping fl,, R1,IR1(x)-P1(Y)I
=h,
Ix - YI
for any x, y E E. Let r = I / f, and let a be inversion with respect to the sphere S(a. r). A simple computation shows that the mapping y = a o at is a homothety with respect to a with dilation coefficient r2 = 1/h. We have that a, = a-1 o y = a o y and 9= a, ' o.81 = a, o YI = a o (y o Y1)
(Here we have used the fact that a-' = a and a,
a1.) Let ft _ y o f 1. The mapping ft is a similarity. Its dilation coefficient is obviously
equal to (1 /h) h = 1, i.e., 8 is a motion. The existence of the required representation is proved. We prove its uniqueness. Let ip = a, o fl, = C92 0,82, where a, and a2 are inversions and f, and #2 are motions. From this, a1 o a2 = f2 0.81 1. The mapping on the right-hand side is a motion and thus maps oo into itself. Let a be the center of the inversion a2. Then a2(oo) = a, and hence aI(a) = oo, i.e., a is also the center of the inversion
a,. Thus, a, and a2 are inversions with a common center. Suppose that a, is inversion with respect to the sphere of radius r1, and a2 inversion with respect to the sphere of radius r2. Then a 1 o a2 is a homothety with center a and coefficient h = r2 /ri . Since, on the other hand, a1 0 a2 is a motion, it follows that h = 1, i.e., r1 = r2, and hence a, 0 a2 ='E, which implies that a, = a2 and f1 = P2, as was to be proved. COROLLARY 3. Let ip E M(E). Then rp either is a similarity or admits a representation 9 = ft o a, where a is an inversion and ft a motion of E. Such a representation of q1 is unique. PROOF. Suppose that ip E M(E), and that qr is not a similarity. Then g, ' is also not a similarity, and hence cp-' admits a representation ip-' = a1 oft,
by Corollary 2, where at is an inversion and fl, a motion. From this, ip = fl, o a , 1= fl o a, where ft= f i' is a motion and a = alat an 1
inversion. Conversely, if 9 = fioa, where ,8 is a motion and a an inversion, then lp - ' = a o f - '. Since such a representation of the transformation
9-' is unique by Corollary 2, this implies the uniqueness of the desired representation of gyp.
§3. MOBIUS TRANSFORMATIONS
39
COROLLARY 4. Every MObius transformation q of E either is a similarity or is representable in the form 99 = (DI o jo o 472, where jo is inversion with respect to the sphere S(O, 1), and (p, and 92 are similarities.
PROOF. Let (p = a o fi, where a is inversion with respect to a sphere S(a, r) and fi is a motion. We have that a = h o jo o h-I, where h is the homothety x i-s a + rx. From this it follows that 9 = 47I o jo o (P2, where (p I = h and 92 = h- o fi. The corollary is proved. COROLLARY S. Suppose that (p : E E is a bijective mapping such that (3.9) holds for any quadruple X1, x2, x3, x4 of points in E with xI # x2, X2:0 x3, x3 # x4, and x4 # xI. Then (p is a Mobius transformation.
PROOF. Let V: E E satisfy the condition of the corollary. Assume first that 9(oo) = oo. Then for every triple X1, x2, x3 E E we have ('P(xi ), (P(x2), 90(x3)) = (99(x1). 99(x2), P(x3), oo)
= (9(x1 ). 9P(x2). 9P(x3). (P(oo)) = (xl , x2, x3, 00) = (XI, x2, x3).
On the basis of Theorem 3.2 this implies that the restriction of 47 to E is a similarity. Suppose that 99(oo) = a # oo. Let a be an inversion with center a, and let yr = a o (p. The mapping Vi is bijective, and clearly (W(x1), W(x2), -y'(x3), W(x4)) = (XI,x2,x3,x4)
for any XI, x2, x3, x4. Further, yr(oo) = oo, and hence W is a similarity by what was proved. We have that ' = a o yi, and thus (p is a Mobius transformation, which is what was to be proved.
3.4. Mobius transformations and spheres. A subset H of E is called a sphere in E if H is the closure in the topology of E of a set
H' = where the vector
(3.10)
in E and the numbers A, p E R are such that
112-Aµ>0.
(3.11)
For brevity we say in this case that the sphere H in E is determined by the equation Alxl2 x) + p = 0. Some term such as "generalized" sphere, etc., would probably be more
-
appropriate. We say "sphere in E". Here the term "sphere in E" is understood in the same sense as before. We explain what such a sphere in E is geometrically, and at the same time we establish also the meaning of condition (3.11). If A = 0, then # 0 in view of (3.11), and the set
H' = {xEEl (,x)-p/2=0}
40
1. INTRODUCTION
is a hyperplane in E. The only limit point of H' in E which is not in H' is the point oo, and hence H = H' U {oo} in this case. Let A # 0. We have that 2Ix12
-
x) + p = 2(1x12 - 2(c/2, x) +p12)
= 2(Ix - /212 - (II2 - 2µ)/22). In this case the set H' is the sphere in E with radius r = (1 /2) N11- 2µ about the point /2. Obviously, the closure of H' in this case coincides with H'. We thus get that if H is a sphere in E, then either H = If U {oo}, where H' is a hyperplane in E, or H is an ordinary sphere in E. It is clear from the computations that the condition I I2 - 2µ > 0 says that H' is nonempty and does not degenerate into a single point. The remarks in §3.1 about the arbitrariness with which the equation of a hyperplane, as well as the center and radius of a sphere, are determined imply that if H' = {x E E12Ix12 x) + µ} x) + µl = 01, and I4I2 - 2u > 0, then there exists a k 0 0 such that 21 = k2, 1 = k , and = {x E E121 Ix12 -
At 1 = kµ.
THEOREM 3.4. The image of a sphere in E under any Mobius transformation is a sphere in E.
PROOF. Suppose that ip c M(E). If 9 is a similarity, then it transforms
every hyperplane in E into a hyperplane and every sphere in E into a sphere, and this implies the theorem for the given case. Assume that c is not a similarity. Then q, = 9100 o 502, where 9 1 and 992 are similarities, and jo is inversion with respect to S(0,1). Therefore, it suffices to establish that the image of every sphere in E under jo is a sphere. Accordingly, let H be a sphere in E. Assume that H is given by the equation 21x 12 - 2 (4, x) +,u = 0,
where 1412 - 2p > 0. Denote by H' the set of points x E H such that 0 < IxI < oo. The sphere H is the closure in E of the set H', and since jo is a homeomorphism of E onto E, it follows that jo(H) is the closure
in E of the set jo(H'). Then x E E belongs to jo(H') if and only if jo 1(x) = jo(x) E H', i.e., the condition x E jo(H') is equivalent to the condition 21x12
1x2)+p=0ax960&2-2((.x)+µ1i12=0.
The equation A proves the theorem.
x) + µ1x12 = 0 determines a sphere in E, and this
§3. MOBIUS TRANSFORMATIONS
41
THEOREM 3.5. Let f : E , E be a bijective mapping such that for any sphere H in E the set So (H) is a sphere in E. Then rp is a Mobius transformation. PROOF. Assume that ip satisfies the conditions of the theorem. Suppose first that 9(oo) = oo. We prove that in this case 9 is a similarity.
Let P be any hyperplane in E. The set H = P U {oo} is a sphere in E. By a condition of the theorem, p(H) is also a sphere in E. Since t (oo) = oo, it follows that oo E ip(H), and hence ip(H) = Q U {oo}, where
Q is a hyperplane. Obviously, ip maps P onto Q. It follows next from the condition of the theorem that every sphere in E is transformed under 9 into a sphere. Since go is bijective, for any two sets A, B c E with An B = 0 we have that 9(A) n i p (B) = 0. This gives us, in particular, that if the hyperplanes P and Q are parallel, then so are the planes {p(P) and 9(Q).
Let x and y be arbitrary points in E such that x 0 y, and let S be
the sphere constructed with the segment [x, y] as a diameter. The center
of S is the point (x + y)/2, and its radius is Ix - yI/2. Let P and Q be hyperplanes tangent to S and passing through x and y, respectively. The planes P and Q are parallel, and each has a single point in common with S. This implies that the hyperplanes 9(P) and 9(Q) are also parallel, and each has a single point in common with 9(S). The common point of 9(P) and the sphere 9(S) is 9(x), and the common point of 9(Q) and p(S) is 9(y). We see that the sphere 9(S) lies between two parallel hyperplanes and is tangent to them at the points 9(x) and 9(y). From this it follows that rp(x) and 9(y) are the endpoints of a diameter of 9(S). We thus get that if the points x, y E S are the endpoints of a diameter of the sphere S in E, then 9(x) and 9(y) are the endpoints of a diameter of 9(S). Let us prove that for any x, y E E 19(x) - 9(Y)I = MIX - YD. where A is a function on [0, oo). It clearly suffices to show that 19(XI) - 9(YI)I = I9(x2) - f(Y2)1
(3.12)
(3.13)
for any pairs xI, yi and x2, Y2 of points in E such that IxI - Yi I = Ix2 - y21. If IxI - yII = Ix2 - Y21 = 0, then (3.13) obviously holds. Assume that
xI 96 yI and x2 96 Y2. Let SI and S2 be the spheres constructed with [xi, y I] and [x2, Y2] as diameters. Since IxI - yII = Ix2 - y2 1, the radii of SI and S2 are equal. This implies that there exist two parallel hyperplanes P and Q, each tangent to both spheres SI and S2, such that these spheres
lie in the strip bounded by the planes. The planes p(P) and 9(Q) are
1. INTRODUCTION
42
also parallel, and the spheres cp(SI) and ip(S2) are tangent to each of them. This lets us conclude that the radii of #P(S1) and ip(S2) are equal. By what
was proved, the points ip(x1) and p(y1) are the endpoints of a diameter of 9(S2). Similarly, (#(x2) and 9(Y2) are the endpoints of a diameter of 9(S2). Consequently, we get that ISo(xI) - q(y1)I = Ic'(x2) - So(Y2)I, which proves (3.13), and with it (3.12).
Let us now show that the function A is linear, 2(r) = kr, where k > 0. Note first of all that A is nonnegative. We prove that for any r1, r2 > 0 2(r1 + r2) = A(r1) + 2.(r2).
(3.14)
Take arbitrary r1, r2 > 0, let x, y, and z be points on a single line such that Ix - yl = r1, ly - zl = r2, and Ix - z) = r1 + r2, and let x' = ip(x), Y' = 9(y), and z' = cp(z). Then Ix' - y'I = 2(r1), IY' - z') = 2(r2), and lx' - z'l = 2(r1 + r2). We prove that the points x', y', and z' lie on a single line, with y' between x' and z'. Let S1 and S2 be the spheres constructed with [x, y] and [y, z) as diameters. Then [x', y'] is a diameter of rp(S1), and [y', z'] is a diameter of Sp(S2). The spheres Sp(SI) and V(S2) have a unique common point, namely, y'. This implies that x', y', and z' all lie on a single line. To finish the proof it is necessary to see that neither of the spheres ip(S1) or V(S2) lies interior to the other. Indeed, since S1 and S2 lie on different sides of a plane tangent to both, there exists a hyperplane
P tangent to S1 at some point a and to S2 at a point b a. Then the hyperplane c(P) is tangent to 9(S1) at Sp(a), and to go(S2) at the point p(b) 0 Sp(a). If one of the spheres ip(S1) or p(S2) were interior to the other, then this would be impossible. Consequently, y' lies between x' and z', and thus Ix'-Z'I=Ix'-y'I+IY'-z'l,
i.e., 2(r1 + r2) = 2(r1) +2(r2), and (3.14) is established. Since 2(r) is nonnegative, (3.14) implies that the function A. is linear,
2(r) = kr for all r, where k > 0. Since 2(r) > 0 for r > 0, it follows that
k>0. Hence there exists a number k > 0 such that I cP(x) - (v(Y)I lI x - yl = k
for any x, y E E, x # y. As noted in §3.1, this implies that is a similarity, and hence Sp E M(E). In our arguments we assumed that V(oo) = oo. Suppose that 9(oo) _ a A oo. Let a be the inversion transformation with respect to the sphere S(a, 1), and set W = a o Sp. Since each of 97 and a maps every sphere in E into a sphere, while w is bijective and satisfies yi(oo) = oo, it follows from
§3. MOBIUS TRANSFORMATIONS
43
what was proved that v is a similarity. We have that (p = a o v, and thus Io E M(E). This proves the theorem. THEOREM 3.6. Let H be a sphere in E, and let (p E M(E) be such that ip(x) = x for every x E H. Assume that q' is not the identity mapping. Then cP is inversion with respect to H as a sphere in E in the case when oo f H. But if oo E H, then ip is the mirror symmetry with respect to H.
PROOF. Let us first consider the case when H is the closure in E of a hyperplane P = {x E EI(e,x) = 0}. Without loss of generality it can be assumed that e is a unit vector. Let (p E M(E) be such that p(x) = x for all x E H. In particular, we get that (p(oo) = oo, and hence Sp is a similarity, ip(x) = k + Q(x), where Q is a general orthogonal transformation. Since 0 E P, it follows that (p(0) = 0, which leads to the conclusion that k = 0, so that V(x) = Q(x) for all x E E. Let A > 0 be the similarity coefficient with respect to (p. For any xI, x2 E P we have that Ixi - x21= Ic'(xi) - So(x2)I = AIxi - x21, and hence A = 1. If X E P, then 0 = (e, x) = (Q (e), Q(x)) _ (Q (e), x).
From this, Q(e) = µe. Since Q is an orthogonal transformation, µ = ±1. Let X E E be arbitrary. We have that y = x - (x,e)e E P; hence (p (y) =
y = x - (x,e)e. On the other hand, p(y) = Q(y) = Q(x) - (x,e)Q(e), and we get that V(x) = Q(x) = x - (x, e)e + µ(x, e)e = x - (x, e)(1 - µ)e. It follows from this that q' is either the identity mapping (in the case µ = 1) or the symmetry mapping x - x - 2(x, e)e with respect to the plane P. Let H be an arbitrary sphere in E. Two different Mobius transforma-
tions are known which leave the points of H fixed: one is the identity mapping, and the other is inversion with respect to the sphere when H is a sphere in E, and the symmetry with respect to H' when H is the closure of a hyperplane H'. It is required to prove that there is no third mapping having the same property. This has been established in the case when H is the closure of a hyperplane {xI (x, e) = 0}. Let a E M(E) be a Mobius transformation carrying H into the sphere Ho which is the closure of {xI (x, e) = 0}, and let (p E M(E) be such that 9p(x) = x /x E H. Define w = a 0 9P o a-I. It is then obvious that {fo(x) = x for all x E Ho. We
know two mappings ip such that (p(x) = x for all x E H. If there were a third mapping with the same property, then we would get three different mappings w such that y/(x) = x for all x E Ho. As follows from what was proved above, this is impossible, and the theorem is proved.
1. INTRODUCTION
44
THEOREM 3.7. Let 9 E M(E). Assume that there exists a sphere S(a, r) such that its image is a sphere S(b, p), and the center of S(a, r) is carried by c into the center of its image. Then ip is a similarity.
PROOF. Assume that the conditions of the theorem hold. It is required to show that 9(oo) = oo. Suppose, on the contrary, that q (oo) = c 96 oo. Let p and q be the points where S(b, p) intersects the line bc, and let s and t be the points of S(a, r) such that ip(s) = p and ip(t) = q. We have that (s, a, t, oo)
=
Is
-
al
=
1.
On the other hand, (s, a, t, oo) _ (9; (s), S (a), q)(t), c(oo)) = (p, b, q, c)
Ip - bllg - el = Iq - c) Ip - clIq - bl Ip-cI
# l'
since the lengths of the segments [q, c] and [p, c] are clearly different. Thus,
the assumption that 97(oo) # oo leads to a contradiction. The theorem is proved. 3.5. The hypersphere bundle and linear representations of Mobius transformations. In the study of Mobius transformations a certain geometric
construction which we call the hypersphere bundle turns out to be useful. The use of it enables us to describe Mobius transformations in the language of linear algebra.
As above, E will stand for a Euclidean vector space. Also as before, no restrictions are placed on the dimension of E; in particular, it can be infinite-dimensional. Consider the direct product E x R2. An arbitrary point u E E x R2 will be understood as a triple (C, rj, {), where C E E, and q and C are real numbers. For uI = (41, ill, Cl) and u2 = (42,, 2, C2) let (U 1, u2)1 = (01,42) +,11172 - cIC2
A linear mapping L : E x R2 -, E x R2 is said to be pseudo-orthogonal if it is bijective and (Lu1,Lu2)1 = (ut,u2)1 for any u I, u2 E E x R2. The collection of all pseudo-orthogonal transformations of the space E x R2 will be denoted by 01(E x R2). Obviously, 01(E x R2) is a group.
Denote by KE the set of all vectors u E E x R2 such that u 0 0 and (u, u) I = 0. This set is a cone in E x R2 with vertex at 0. If I is the line passing through the points 0 and u, where u E KE, then I \ {0} c KE. In
§3. MOBIUS TRANSFORMATIONS
43
this case we call I \ (0) a generator of the cone KE. Denote by KE the collection of all vectors u = ri, {) E KE for which Z > 0, and let KE be the set of all u = q, C) E KE for which C < 0. The closure of the set KE is the collection of all u {) such that I4I2 + n2 - C2 = 0 and C > 0, and, similarly, KE
={(4,.1.C)EExR2IRI2+q2-{2=0&Z<-0}.
If u = (, q, C) belongs to each of the sets KE and KE, then I412 +1,12 - `r2 = 0
and C = 0, which implies that q = 0 and C = 0, so that the intersection KE n KE consists of a single element-the point 0. If u = (C,17, C) E KE, then C 0 0, for otherwise we would get that q = 0 and = 0. However, the point 0 is excluded from the cone KE. Consequently, KE = KE E n K-
Further, KE n KE = 0 and KE n KE = {0}.
If U E KE, then -u E KE, and if u E KE, then -u E K. We introduce some notation. Let el and e2 be the vectors el = (0, 1, 0) and e2 = (0, 0, 1) in E x R2.
We single out a certain subset Oi (E x R2) of 01(E x R2). Let L 01(E x R2) and u = (C, q, C) = Lee. We have that q2
C2
- = (u,u)I = (e2.e2)1 = - 1. From this, C2 = I I2 + 1,12 + 1, and hence ICI > 1. Denote by O; (E x R2) the ICI2 +
collection of all L E 0 1 (E x R2) such that the coordinate C of the vector u = L(e2) is positive.
LEMMA 3.1. If L E O (E x R2), then L E 01(E x R2) is such that L(KE) C KE, then L E Ot (E x R2).
PROOF. Let h = (p, q, r) = L(e2) and v = (a, fi, y) = L(u), where u = (, ,1, C) E KE. Then
(v, v)1 = (u, u)1 = 0 and (v, h)1 = (u, e2)1 = -C < 0. Further, (v, v)1 = IaI2 + R2 / - y2, and hence Iyi = (h,h)1
Ial + ft . We have that
IpI2+q2-r2=(e2,e2)I =-1;
thus, 1p12 + q2 = r2 - 1. It is required to prove that y > 0. Assume, on the
contrary, that y < 0. We have that
0> (v,h)I IYIr - Ialipi - IPIIgl > IYIr = IYIr - IYI
r2 - I = IYI (r -
VjCgj2
+ p2 Ip12 + q2
r2 - 1) > 0.
1. INTRODUCTION
46
The assumption that y < 0 thus leads to a contradiction, and hence y > 0. Accordingly, we get that if u E KE, then also v = L(e) E K. Assume that L E O(E x R2) is such that L(KE) C K. The vectors a = (0, 1, 1) and b = (0, -1, 1) belong to KE; hence L(a) = 71, C1) E KE and Ljbj = (2.72, C2) E K. We have that e2 = (a + b). Hence,
(p, q, r) = L(e) = 2 (L(a) + L(b)). In view of the condition L(KE) C KE we have CI, C2 > 0, which give us that r = IM + 12) > 0, i.e., L E 0'(E x R2). The lemma is proved. Let L E O1(E x R2). It follows from the definition that L(KE) C K. We show that, actually, L(KE) = KE. Let U E KE be arbitrary. Since L is bijective, there exists a v E E such that L(v) = u. We have that
(v, v) = (Lv, Lv)1 = (u, u)1 = 0, i.e., v E KE. Accordingly, for every u E KE there exists a v E KE such that L(v) = u, and hence L(KE) D KE. If L E Oi (E x R2), then L(KE) C K. Further, if u E KE, then -u EKE,
which implies that -L(u) = L(-u) E KE, and thus L(u) E Kj, i.e., we get that L(KE) C K. Since L(KE) = KE and KE = KE U KE, and since KE n KE = 0, it follows from what has been proved that L(KE) = K,+ and LIKE) = KE, i.e., a transformation of the class 01(E x R2) carries each of the halves KE and KE of the cone KE into itself. We now construct certain special mappings of KE onto E. 7, r) E E x R2. In the space E consider the equation Let u =
k(x, x) -
x) + I = 0,
(3.15)
where k = -7 + C and I = 7 + C. We determine conditions under which the set of solutions of (3.15) consists of a unique point. Obviously, for this it is necessary that k be nonzero. Let k # 0. Then `\
k(x,x)-2(4,x)-I=k((x,x)-2(k,x)+ =k
((x `th`at
k, x
k
(4, k2 Ik
)
\ (3.16)
From this it is clear (3.15) is uniquely solvable if and only if k 96 0 lk = 0. We have that and consequently, (3.15) is uniquely solvable in E if and only if u E KE, and 17 0 C.
§3. M6BIUS TRANSFORMATIONS
47
We now define a certain mapping r : KE --* E. Let u = q,1) E KE. If '1 = 1, then the condition (u, u) I = 0 implies that = 0, and hence u = (0, q, q). In this case we set r(u) = 00. Suppose that q 36 C. Then the equation
has a unique solution x E E, which is taken to be the point r(u). It is not hard to see from (3.16) that
r(u) = k = (1/(-'1 + 1)) We show that r is a mapping of KE onto E. Take an arbitrary point xo E E. It is required to prove that there is then a point u'E KE such that r(u) = x0. If xo = oo, then the existence of the desired u follows immediately from the definition of r. Suppose that xo 54 oo. We construct an equation of the form (3.15) whose solution set consists of the single point xo. Obviously,
(x - xo, x - xo) = (x, x) - 2(xo, x) + (xo, xo) = 0
is such an equation. We let
= xo and determine '1 and C from the
equations
-q+1= 1,
17 +1=(xo,xo).
From this, j7 = ((xo,xo) - 1)/2 and 1 = ((xo,xo) + 1)/2. Let (3.17) u = (i, q,1) = (xo, ((xo, xo) - 1)/2, ((xo, xo) + 1)/2). It is easily verified that u E KE and r(u) = x0, which is what was required to prove.
If x = r(u1) = r(u2), then u2 = aul, where a 0 0. Indeed, if x = oo, then uI = (0,11,11) and u2 = (0,12,'12) , where II,12 96 0, and hence U2 = au1 for a = J72/q1- Suppose that x 96 oo. Then
x=
from which 2 =
11
12
-11+CI -q2+12' a = (12 -'12)(11 - 11) It then follows from
the equalities (I,11) + (q1 - 11)('11 + 11) = 0,
(2,12)+(12-12)('12+12)=0 that also 12 + 12 = a(q1 + 11), which gives us that 072 = all and 12 = a1i, as required. We get that for every x E E the set r- I (x) is a generator of the cone KE. Every point u E r- I (x), u = q, 1), will be called a vector of hyperspherical coordinates of the point x, and we say correspondingly that , q,1 are hyperspherical coordinates of x.
I. INTRODUCTION
48
The triple (KE, T, E) is called the hypersphere bundle over E; E is called
the base of the bundle, KE its space, and the mapping T the projection. We remark that in older references the phrase "system of coordinates" is used instead of the word "bundle". However, according to the latest ideas a system of coordinates is always a one-to-one mapping. On the other hand, there is the general concept of a bundle, and a hypersphere bundle is a special case of this. In cases when the dimension n of E is finite, the given bundle is called a quadrisphere bundle or a pentasphere bundle if n is equal to 2 or 3, respectively, so in the general case we should call it an (n + 2)-sphere bundle.
Let H be a sphere in E. We determine the set T-I (H). Assume that H is given by k1xl2
- 2(a, x) + 1= 0,
(3.18)
where JaJ2 - k1 > 0, and k # 0. Let u =
q, C) be a point of the cone KE such that x = T(u) E H. Then g12 + q2 - C2 = 0, and since oo 0 H, it follows that T(u) 4 oo, and hence q # C and r(u) Substituting the value x = r(u) into (3.18), we get that
0=klCl
1
2 1l2
-2'a, C
'l')+l =kC+1 -2 \a, C
1l)+l.
From this, (a, 4) + bq - c{ = 0,
(3.19)
where b
-k2+1,
c= k 1, Ja12+b2-c2=Jal2-k1 >0.
(3.20)
= Conversely, if (3.19) holds for the point u = (, q, C) E KE, then in view of the conditions k 96 0, b - c 0 0, and q we get by reversing the computations that T(u) E H. This allows us to conclude that T-1(H) is the section of the cone KE by the plane (3.19) in this case. Let us now consider the case when H is the closure of the hyperplane
{xEE;(a,x)-1/2=0}, where a 4 0. Let the point u = q, Z) E KE be such that T(u) E H. If q 0 {, then, as in the preceding case, we get that C, q, and C satisfy (3.19) with values of b and c determined according to (3.20) with k = 0, and, conversely, if (C, q, C) E KE satisfies condition (3.19) and ?10 C, then
T(u) E H. But if q = C, then C = 0, and T(u) = 00 E H. In the given case b - c = 0, and the point u(0, q, q) satisfies (3.19).
§3. MOBIUS TRANSFORMATIONS
49
Accordingly, if H is a sphere in E, then the set z- I (H) is the section of KE by the plane given by (3.19), where the vectors a E E and the numbers b, c E R are such that 1a12+b2-c2>0. (3.21) Conversely, suppose that a section of KE by a plane of the form (3.19) is given, where a, b, and c satisfy (3.21). Let k = -(b+c) and I = b-c, and let H be the sphere in E determined by the equation kIxI2-2(a, x) +1 = 0. Then, as follows from the preceding arguments, the set z- I (H) coincides with the given section of KE. LEMMA 3.2. If cp E Ot (E x R2) carries every generator of the cone KE into itself, then c is the identity transformation.
PROOF. Assume that rp satisfies the condition of the lemma. Then for every vector u E KE we have that cp(u) = A(u)u, where A(u) E R. Further,
it is obvious that A(u) > 0. Let p = (0, 1, 1) and q = (0, -1, 1). The vectors p and q belong to the cone K. By assumption, 9P(p) = AI p and ,p(q) = A2q, where A1, A2 > 0. Take an arbitrary vector u E KE which is
not collinear to one of the vectors p and q. We have that u = q, C), where E E and # 0. Without loss of generality can be assumed to be a unit vector. Then q2 - C2 = -1. Let q1 and CI be another pair of numbers such that q2 - C2 _ -1, and suppose that q1 + CI 96 q + C. The vector v = ( , q1, CI) also belongs to KE and is a linear combination of the vectors p, q, and u, namely, v = ap + flq + u, where a = Z [q1 + CI - (q + C)].
P = Z [C1 - 11 - (C - q)].
Since q2 - Z2 = qi - Ci 34 0, the fact that q1 + ZI 0 q + C implies that also
qI - CI # q - r, and hence a, f 94 0. Let V(u) = A3U. Since V EKE, it follows that ip(v) =µv. We have that ,p(v) = rp(ap + flq + u) = ap(p) + f9(q) + 9(u)
=aAlp+fA2q+A3u. On the other hand, 47 (v) = it(ap+ fiq+u). Since the vectors p, q, and u are linearly independent, this implies that ua = aA1, of = PA2, and a = A3. And since a, f 96 0, this implies that AI = A2 = A3 = A. In particular, qp(p) = Ap, q9(q) = Aq, and ip(u) = Au. Consequently, the arbitrariness
of u E KE gives us that 9(u) = Au for all u E KE, where A = const. Let x = q, C) be an arbitrary vector in E x R2. The vector u = 0, ICI)
s0
1. INTRODUCTION
belongs to KE, and the vector x can be represented as a linear combination of the vectors p, q, and u: x = a p + flq + yu. Since 9p is linear, this implies that 9(x) = a9(p) + fi9(q) + y9(u) = 2(ap + /Jq + yu) = Ax.
Thus, q'(x) = Ax for all x E E x R2. It follows from the condition (x, x), _ ((p (x), 9,(x))1 for any x E E x R2 that A2 = 1, and hence A = 1, because A > 0. We thus get that 9 is the identity mapping. The lemma is proved. THEOREM 3.8. For every ip E M(E) there exists a transformation To E 01(E x R2) such that the mapping diagram T'
KE
KE
TI
!t
EoE
(3.22)
is commutative, i.e., {p(r(u)) = r[T,,(u)] for every u EKE. Such a mapping T,, is unique, and the correspondence
is an isomorphism of the groups M(E) and 0'(E x R2). REMARK 1. Intuitively, the commutativity of the diagram (3.22) means the following. Let x E E and let u = q, C) be a vector of polyspherical coordinates of x. Applying to u the transformation T4, we get a vector of hyperspherical coordinates of the point q,(x). REMARK 2. The isomorphism ip - T,, of M(E) and Ot (E x R2) to be
established in Theorem 3.8 will be called the canonical isomorphism in what follows. PROOF. We first establish the uniqueness of a transformation Tv satis-
fying the condition in the theorem if it exists. Let gv E M(E) and let TI and T2 be transformations in Ot (E x R2) such that 9 o r = r o TI and 9 o r = r o T2. Take an arbitrary generator of the cone KE, let u be a point on this generator, and let x = q(u). Then p(x) = r(TI(u)) = r(T2(u)), and hence T, (u) and T2(u) lie on a single generator of KE. This implies that TI and T2 carry the generator of KE passing through a point u E KE into the same generator, which allows us to conclude that the mapping T2 o T, carries every generator of KE into itself, and thus T2 1 o T, = 1, by Lemma 3.2, i.e., T, = T2. To construct the required transformation T,, we consider first the case when 9' is one of the following transformations:
§3. MOBIUS TRANSFORMATIONS
x+a;
51
2)h2:xi- x, Aj6 0;
4)j: x " x/Ix12. Suppose that rp is one of the transformations Ra, hi, or {PA. Take an arbitrary vector u E KE, u = q, C). Assume first that q 96 { and let k = -q + Z, l = q + r, and x = (1/k) = r(u). The point x is the unique solution of the equation 3)rp4 : x i- Ax,
A E OE;
k(x, x) x) + 1= 0. (3.23) Let y = 9(x). Then x = rp-I (y). Substituting the expression for x into (3.22), we get after the obvious transformations that k1 (y, y) -
y) + 11 = 0,
(3.24)
and 11 are expressed in terms of k, , and l as follows. If
where 9P = Ra, then
k1 = k,
if
I = + ka,
11 = kIa12 + 2((, a) + 1,
(3.25)
= hx, then
k1=
J.
'i=',
11=A1
(3.26)
11 =1.
(3.27)
and, finally, if rp = 9'A, then k1 = k,
I=
The transformation T. is now constructed as follows. Let u = (, 11, C) E E x R2 (u is not necessarily in KE). For u we define a vector I E E and numbers k1 and 11 by using (3.25) if rp = ire, (3.26) if ? = hA, and (3.27) if SD = QA.
Let qI = (-k1 + 11)/2 and CI = (k1 + 11)/2. We get that
T,(u) _ (41, qI . C1). The mapping To is linear. In all cases equalities (3.25)-(3.27) allow us to conclude that g ,1 2 +q1
- C= 14112-11k1 =I4I2-1k=I I2+172-C2.
which gives us that Tq, E 01 (E x R2). Further, if u = (, q, C) E KE is such that C > 0, then in view of the equality I4I2 = C2 - q2 we get that C2 - q2 > 0, and hence C > IqI. By using this inequality it is not hard to show that in each of the three cases rp = na, go = hA, and rp = IPA the component C of the vector 9,,u) is positive. Accordingly, Tq, transforms the upper half of the cone KE into itself, and thus To, E O+(E x R2). We show that the mapping Tq, we have constructed satisfies the condition t o TI, = ? o t. Indeed, (3.24) is uniquely solvable, since (3.23) is uniquely
solvable, and its solution is y = t(u1) = r[T,,(u)]. On the other hand, y = 91(x) = co[r(u)], i.e., t[T,(x)] _ ip[r(u)]. This argument is valid only if q 96 C for the vector u. But if q = {, then u = (0, q, q), and T(u) = oo.
1. INTRODUCTION
52
In this case ip(oo) = oo. It is easy to verify that T,(u) = u for the given u if 9 is any one of the mappings na, hA, or 9A; hence r[T,(u)] = Sp[T(u)] also in this case. Let Sp = jo. For a vector u E E x R2 let T,(u) -q, Z). Then T, E Oi (E x R2). It is easy to verify that 9 o r = r o T,.
Let 9 be an arbitrary Mdbius transformation. Then {p can be represented in the form
9=91 where ipi is a transformation of one of the following types for each is 9a, hA, 9A, or j. Let T, = T,, o oT,,,,. It is easy to verify that Spot = roT,; hence T, is the desired mapping. We now prove that the correspondence 9 E M(E) t- T. is a homomorphism. Let 9 E M(E) be arbitrary. For every u E KE we have that
cp[r(u)] = r[T,(u)]. From this, r(u) = 9-'[r(T,(u))]. Replacing u by Tv ' (u) here, we get that for any u E KE
9-' [r(u)] = r[T;-' (u)], and hence, since T, is unique, Ty ' = T,-, .
(3.28)
Let 47, w E M(E). We have that q [r(u)] = r[T,(u)], and so
V[F(r(u))] =1v(r[T,(u)]) = r[T,(T,,(u))l, i.e., (W o q,) o T = r o (T,, o T,). On the other hand,
(Sv o 9)[r(u)] = r[TW,,(u)l Comparing the last two equalities, we conclude that TWO, = TV o T,.
(3.29)
Equalities (3.28) and (3.29) mean that the correspondence 9 i-+ T. is a homomorphism. We now prove that the correspondence 9 - T, is one-to-one. Let 91 E M(E) and 92 E M(E) be such that T,, = T,, = T. Let X E be arbitrary, and let u E KE be such that r(u) = x. Then 9i (x) = 9 i [r(u)] = r[T,, (u)] = r[T (u)] and
92(X) = 92[r(U)] = r[T,2(u)] = r[T(u)], and hence 91(x) = 92(x). Thus, 91(x) = 9'2(x) for all x E E, i.e., 91 = 92 Accordingly, if T,, = T,,, then 91 = 92, and it is proved that the mapping
9 -- T, is one-to-one.
§4. DEFINITION OF A MAPPING WITH BOUNDED DISTORTION
53
We show that lO E M(E) -+ TI, E O; (E x R2) is "onto". Let T E Oi (E x R2) be arbitrary and let x E E. We find the generator T- I (x) of the cone KE. This generator is carried by T into some other generator A of the cone. The set r(A) consists of a unique point y which we denote by 9,(x).
Since X E E was arbitrary, this defines a mapping c : E - E. Further, as follows from the construction of gyp, qi(T(u)) = T[T(u)] for every u E KE.
We show that q, is a Mobius mapping. If xl # x2, then the generators al = T-I (x) and a2 = T-' (X) of KE are distinct, and hence so are the generators 81 = T(a I) and fl2 = T (a2)- We have that ip(x1) = T(f1) and 9,(x2) = r(#2), which implies that c(x1) 96 rp(x2). Accordingly, 9 is injective. Let us prove that ((E) = E. Indeed, take any y E E and let ft = T'I (y) be the generator of KE corresponding to y; let a = T-I($). If x = r(a), then 9 (x) = y. Thus, ip is a bijective mapping, and ip o T = r o T. We prove that 9) E M(E). For this it suffices to establish that 41 satisfies
the conditions of Theorem 3.5. Let H be an arbitrary sphere in E. We have that H = T(KE n S), where S is the subspace of E x R2 determined by the equation (p, u) I = 0, and the vector p E E x R2 is such that (p, p) I > 0. Let q = Tu. We have that
UET(S)e4T-'UESa(p,T-IU)I =0. But (p. T - I u) I = (T- I q, T -' u) I = (q, u), and hence
T(S) = {u E E x R2(q, u) = 0}.
For the vector q we have that (q, q) I = (Tp, Tp) I = (p. p) I > 0.
Let I= T(S) n KE. The set r(F) is a sphere in E. It is easy to verify that T(r) = f(H). Thus, rp is a bijective mapping of E into E which transforms every sphere into a sphere. By Theorem 3.5, this implies that 9p is a Mobius transformation. The mapping T E 0+ (E x R2) was arbitrary, and we have
that T = T,; hence it is established that c '-. T,, is a mapping of M(E) onto O; (E x R2). The theorem is proved. §4. Definition of a mapping with bounded distortion
4.1. Orthogonal invariants of linear mappings of Euclidean spaces. A measure of nonorthogonality for a linear mapping. The symbols E and F below denote arbitrary finite-dimensional Euclidean spaces. The inner
product of arbitrary vectors x, y E E is denoted by (x, y), and Ixj = (x, x) is the length of a vector x E E. Analogous notation is used for the space F. The symbol IE denotes the identity mapping of E onto itself. (The index E is omitted whenever no confusion is possible.)
1. INTRODUCTION
54
Denote by L(E, F) the collection of all linear mappings of E into F. Let 9: E - E be a general orthogonal transformation. Then 9 carries every sphere in E into another sphere. The purpose of this section is to construct certain quantities characterizing the degree of nonorthogonality of an arbitrary linear mapping. Let A : E F be a linear mapping, and let IIAII = sup I A(x)I 1xI
If A E L(E, E), then det A denotes the determinant of A.
Let A : E - F be a given linear mapping. Then there exists a unique linear mapping B: F - E such that (Ax, Y)F = (x, By)E for any x E E and Y E F. This mapping B is called the adjoint of A and denoted by A. If Cartesian orthogonal coordinate systems are given in E and F, then the matrix of A' with respect to these coordinate systems is the transposed matrix of A. As above, O(E) denotes the collection of all orthogonal transformations
of E. For every P E O(E) we have that detP = ±1, IIPII = 1, and
P' = P-'.
For an arbitrary linear mapping A: E -i F let Im A = A(E) and KerA = A-1(0).
Mappings A E L(E, F) and B E L(E, F) are said to be orthogonally equivalent if there exist transformations' E O(E) and tp E O(F) such that B = WoAoip. It is easy to verify that the relation of orthogonal equivalence is reflexive, symmetric, and transitive. Denote the Kronecker symbol by &ij, where i, j = 1, ... , s; that is, 8ii = 1
for i = j, and Sip = 0 for i s x s identity matrix.
j. The numbers Sid are the elements of the
A system of vectors u, , ... , uk in E is said to be orthogonal if (u1, u j)
J for any i, j = 1, , k.
_
LEMMA 4.1. Let E and F be finite-dimensional Euclidean spaces, and let A E L(E, F), A 0. Then the subspaces KerA and Im A* are completely orthogonal, and their direct sum coincides with E. There exist orthonormal s y s t e m s o f v e c t o r s U 1 ,--- , uk in E and v1,... , vk in F and numbers Ai > 0
(i = 1,...,k) such that (u,,...,uk) is a basis in ImA', (v1,...,vk) is a basis in Im A, and Au; = Aivi,
A'vi = Aiui
for each i = 1,...,k. PROOF. Consider the quadratic form Q(x) = (Ax, Ax) = (A* Ax, x)
(4.1)
§4. DEFINITION OF A MAPPING WITH BOUNDED DISTORTION
55
on E. This quadratic form is nonnegative, and is not identically zero because A 9k 0 by assumption. Let u1,... , u,, be an orthogonal basis in E such that Q(x) can be expressed in terms of the coordinates of x with respect to this basis as follows:
Q(X)=µix Since Q is nonnegative, ,ui > 0 for each i = 1,... , n, and since Q 0 0, at least one of the coefficients µi is nonzero. Without loss of generality it can
be assumed that µi>0fori
k. Let Ai =
ui, and let
i = 1,2,...,k.
vi = (1/2i)Aui, For arbitrary x, y E E we have that
(Ax, Ay) = 2 [Q(X + Y) - Q(X) - Q(Y)] _AtXIYI
For any i, j = 1, ... , k, (vi, v1) =
(4.3)
+AZX2Y2+...+A XkYk
r
J
(Aui, Au1).
Using (4.3), we get from this that (vi, v1) = 8 ,, i, j = 1, 2, ... , k; that is, the system of vectors v1, ... , Vk is orthogonal. The set KerA coincides with the set {x E ElQ(x) = 0}, and hence KerA is the subspace of E spanned by the vectors Uk+l, ... , u,,. In particular,
Aui=0 for i> k.
Let us prove that the vectors v1,... , vk form a basis for Im A. Take an
arbitrary vector y E ImA. Then y = Ax for some x E E, x = xiui. From this, y = E? xiAui = Ei Aixivi, and hence y belongs to the linear En
span of the system of vectors v1, ... , Vk. Since y E Im A was arbitrary and
vi E Im A for each i = 1,... , k, this implies that the linear span of the system of vectors VI ,- , Vk coincides with Im A. We now prove that the vectors u1 , ... , Uk form a basis for Im A*. Indeed, let Vk+ i , ... , vm be such that v1 , ... , Vk , Vk+I, ... , Vm is an orthonormal
basis for F. For any i = 1, ... , n and j = 1, ... , m we have that (ui, A'v1) = (Aui, v1).
(4.4)
For j > k the vector v1 is orthogonal to ImA, and hence (Aui,v1) = 0 in this case. By (4.4), (ui, A'v1) = 0 for each i = 1,... , n for j > k; that is, A'v1 = 0. For j < k it follows from (4.4) that (A'v1, ui) = 0 in the case i > k, since then Aui = 0. For i < k, (A'vj, ui) = (v1, Aui) = 2i(vi, v1) = didil.
56
1. INTRODUCTION
This implies that A'v j = A j up. Therefore, the vectors u I , ... , uk belong to Im A*. For an arbitrary . E Im A' we have that x = A* y for some From this, Y E F, y= We get that every vector y E Im Am is a linear combination of the vectors U1 , ... , uk; hence they form a basis for Im A'. It remains to observe that Im A' is the subspace spanned by u1, ... , uk, and Ker A is the subspace spanned by Uk+1, ... , u,,. This implies that these subspaces are completely orthogonal and their direct sum is E. The lemma is proved. The next lemma enables us to give a precise description of the a r b i t r a r i n e s s with which the systems of vectors U I, ... , uk and v1 , ... , Vk in
Lemma 4.1 are determined, and establishes that the collection of numbers Al, - . )k is uniquely determined by A. LEMMA 4.2. Let A E L(E, F), and let {UI , ... , uk} and {V1, ... , Vk} be orthogonal systems of vectors in E and F, respectively, such that {u1, ... , uk }
is a basis for Im A` c E, {v1, ... , Vk } is a basis for Im A, and for every
i = 1,...,k Au; = .I;v;, where ).; > 0. Then A' Au; = 1? u1 and AA'v1 = .?v, for each i = 1, ... , k, and all the remaining eigenvalues of the mappings A'A and AA' are equal to zero.
PROOF. Suppose that the systems of vectors u1, ... , Uk and vi..... vk and the numbers 1,, i = 1, ... , k, satisfy all the conditions of the lemma. To each of these systems of vectors we add new vectors in order to get an orthonormal basis in the corresponding space. Let u1,... , u and v1,..., vm be bases constructed in this way. Let j > k; we have that A'v j E Im A'. The vectors uk+1, ... , u are orthogonal to Im A', and hence (A*v j , u;) = 0
for i > k. For i < k we get that (A'vj, u;) = (vj, Au;) = )t(vj, v;) = 0. Thus (A'v j, ur) = 0 for all i = 1, ... , k, and hence A'v j = 0 for j > k. Let 1 :5j:5 k. We have that A'v j = al u1 + + akuk. From this,
a; = ui, A vi = Au1, vj) = 2,(v,, vj) = A, ,j, and hence A*v j = A j u j. This leads us to conclude that
AA'vj = A(A'vj) = 0.
§4. DEFINITION OF A MAPPING WITH BOUNDED DISTORTION
57
This, v1, ... , Vk are eigenvectors of the mapping AA', A? is the eigenvalue of this mapping corresponding to vi, and the remaining eigenvalues of AA' are equal to zero.
For i > k and j > k, i 0 j, the quantity (vj. Au,) vanishes in view of the fact that vj is orthogonal to Im A. For i < k and j < k we have that
(vj,Au;) = (A'vj,u;) =Aj(uj,u;) =0, and this leads to the conclusion that Au; = 0 for i = k + I_-, n. By assumption, Au; = A;v; for i < k. In view of what has been proved, this implies that A'Au; = A?u; for i = I,-, k, and A'Au; = 0 for i > k. The lemma is proved. Let A E L(E, F), A 9& 0, let A, , ... , Ak, A; > 0 Vi, be the numbers determined f r o m A according to Lemma 4.1, and let u, , ... , uk be the orthonormal system of vectors indicated in Lemma 4.1. The quantities A I , ... , Ak are called the principal dilation coefficients or singular numbers of A. We assume that they are numbered in such a way that 0 < A, < < Ak. The vectors u; , i = I,-, k, are called principal vectors of A2 < A. According to the definition, the vectors v; = (1 /A; )Au; , i = 1, ... , k, are principal vectors of A'. If A = 0, then it is convenient to assume that A does not have dilation coefficients, so that in this case the collection of dilation coefficients of A is empty. Lemmas 4.1 and 4.2 yield a test for orthogonal equivalence of linear mappings.
COROLLARY. Mappings A, B E L(E, F) are orthogonally equivalent if and only if they have the same sets of principal dilation coefficients.
The proof is left to the reader, since it is not used in what follows. Let A : E - F be a linear mapping. Then II A II is equal to the largest singular number of A. Indeed, let Ak > ... _> A2 > A, > 0 be all the nonzero singular numbers of A, and let u I, ... , u be an orthonormal basis in E such that UI, ... , uk are principal vectors of A. Let x E E be such that jxl < 1 and x = xlul + + Then Ax = Alx,v, + +AkxkVk, principal vectors of A', and we have that , Vk are where V I , IAxI = (Ax, Ax) = A2x2 +
+ Akxk
Ak(Xj +...+x.2) =AkIXl2
which was to be proved.
1. INTRODUCTION
38
In particular, consider the case when E = F and the mapping A E 0. Then dim Im A = dim E = L(E, E) is nonsingular, that is, det A n, and thus all n principal dilation coefficients of A are nonzero. Let > A, > 0 be the set of singular numbers of A, u, , ... , u An > An_, > principal vectors of A, and v; = (1 /A; )Au;. Then v1,... . v are principal vectors of A', and Au; = Aiv; for each i. Let 47 E O(E) be such that rp(vi) = ui. Then the mapping B = Asp is such that By, = Aiv;. Obviously, det B = A, A2
... A,,. On the other hand, det B = det to det A, and we get that 1,A2...A,,=IdetAl
(4.5)
because det rp = f 1 and A, An > 0. If Al An = A, then the mapping B constructed above is such that By, _ Av, for all i = 1,... , n. From this, Bx = Ax for all x E E. We have that A = Bop-', and we get that if Al = ... = An, then A is the composition of an orthogonal transformation and a homothety; that is, A is a general orthogonal transformation in this case. It follows from the equalities Au; = .ti v; , i = 1, ... , n, that
A-Iv,=u;/A,,
i= 1....,n.
On the basis of Lemma 4.2 this enables us to conclude that the singular numbers of A-' are I IA, v, , ... , vn, are principal vectors of A-', and u,,... , un are principal vectors of (A*)-'. For every general orthogonal transformation A : E - E we have that Let us determine the image of the sphere S(0, 1) in E under a mapping A E L(E, E) with det A 96 0. Let u, , ... , un and v1, ... , v be principal vectors of A and A', respectively, and let A.n > An_, > > Al > 0 be the singular numbers of A. A point x = x, v, + + xnvn belongs to A[S(0, 1)) if and only if
A-'(x)= x' u, A,
+x2u2+...+xnvnES(0.1); A2
An
that is, if and only if 2
2
2
A2+22+...+.12=1.
We get that the set A[S(0, 1)] is given by equation (4.6) in the Cartesian orthogonal coordinate system with basis v,,... , v,,; that is, A[S(0, 1)] is an An. ellipsoid with semiaxes of lengths We now introduce some quantities characterizing the measure of how much an arbitrary nonsingular linear mapping A : E -. E differs from a general orthogonal mapping. Let 0
§4. DEFINITION OF A MAPPING WITH BOUNDED DISTORTION
59
of the mapping. Then, as shown above, An = IIAII and 2,22 ...2n = I det Al. In particular, this implies that (4.7)
I det 1:5 11A 11n
always, with equality if and only if 2, = . . . = 2n; that is, if and only if A is a general orthogonal transformation. Let IIAII"/I det Al = 2n/2122 ...2n = K(A).
(4.8)
For every nonsingular linear mapping A we have that K(A) > 1, and K(A) = 1 if and only if A is a general orthogonal transformation. We introduce some other characterizations. Let A : E - E be a nonsingular linear mapping, and let
Ko(A) = K(A- ).
(4.9)
Obviously,
Ko(A) = 2,22 ...2n/2; . (4.10) where 2, < < An are the principal dilation coefficients of A. For every A we have that K0(A) > 1, and K0(A) = 1 if and only if A-, hence also A, is a general orthogonal transformation. The quantities K(A) and Ko(A) have the following geometric meaning. Let 9 be the ellipsoid into which the sphere S(0,1) is transformed by A, let B, be the ball of smallest radius containing 9, and let B0 be the ball of largest radius inscribed in 0. Then
K(A) = IB,I/I`,
Ko(A) = 1.
I/IBoI.
The quantity K(A) is called the outer dilation, and Ko(A) the inner dilation of the linear mapping A. (In the literature the notation K(A) = Ko(A) (0 for "outer") and Ko(A) = KI(A) (I for "inner") is used; to avoid errors we use here the notation in the Russian text of the book.) We estimate K(A) and Ko(A) in terms of each other. Let ai = 2i/(2, ...An)I/". Then Ko(A) = 1/a"I, K(A) = an, an, a, ... an = 1 . Hence, l/a, = a2a3 . . .an < an - '; that is, K0(A) < [K(A)]"-I.
(4.11)
Replacing A by A-I here and considering that Ko(A-I) = K(A), we get that (4.12) K(A) S Let A and B be arbitrary nonsingular linear mappings of the n-dimensional Euclidean space E. Then [Ko(A)J"-I.
det(AB) = det A det B.
1. INTRODUCTION
60
Further, for any x E E with Ixi < 1 IA(Bx)I <_ IIAIIIBxI< IIAIIIIBII.
From this we see that IIABII <_ IIAIIIIBII,
(4.13)
K(AB) < K(A)K(B).
(4.14)
and hence
Applying the same argument to the mappings A' I and B-', we get that
Ko(AB) = K(B-' A-') < K(B- I)K(A-') = Ko(A)Ko(B).
(4.15)
If at least one of the given mappings A or B is a general orthogonal transformation, then equality holds in (4.13). This implies that here each of the relations (4.14) and (4.15) also becomes an equality. Along with K(A) and K0(A) it is useful to consider various kinds of other characterizations of the nonorthogonality of a linear mapping. Let E be a finite-dimensional Euclidean space, and A : E - E a nonsingular linear mapping. Let A be the smallest principal dilation coefficient of A, and A the largest. Define A/A = q(A). Obviously, q(A) > 1 always, and q(A) = 1 if and only if f is a general orthogonal mapping. The quantity q(A), as a measure of nonorthogonality of a linear map-
ping, is most natural. However, it turns out not to be very convenient analytically. The concept of a conformal norm of a linear mapping is used in (1301 to define a whole class of nonorthogonality measures. Let N be a norm in the vector space L(E, E). The norm N is said to be conformal if there exists a constant KN > 0 such that [N(f )]" >_ KN det
f
for any f E L(E, E) such that det f > 0, with equality if and only if f is a general orthogonal transformation. We give examples of conformal norms. Let f E L(E), and let A(f) be
the largest dilation coefficient of f; this quantity is a norm in L(E). It follows from what was proved above that A is a conformal norm in L(E). The coefficient KN = KA is equal to 1 in this case. Fix a Cartesian orthogonal coordinate system and let A = (aik ), 1. k = 1,
... , n, be the matrix of a linear mapping f E L(E, E) in this coordinate
§4. DEFINITION OF A MAPPING WITH BOUNDED DISTORTION
61
system. Let
(a)12) n
E
MM(f) =
I/p
ik
k=1
n
) p/2
n
1/p
Mp(f)_k=1E i=I E aik 2
where p > 1. Further, let
(a) n
max
i=I.2.....n
1/2
k
(k=1 n
I/2
M,(f)= k=1.2..... max n Eak i=1
By using the classical Hadamard inequality it is not hard to prove that M,,
and Mp, 1 < p < oo, are conformal norms. The constant KN is equal to nn/p for N = MP and N = MD . Let N be a conformal norm in L(E). The ratio
[N(f)]n KN(f) _ KNdetf where f E L(E) is a nonsingular linear mapping such that det f > 0, is called the distortion of f with respect to the conformal norm N. Let N: L(E) -# R be a conformal norm, and let
rI(K;N) = sup KN(f), K(f)
r2(K;N) =
sup KN(f)
K(f).
It follows easily from continuity and compactness considerations that r1(K; N) and r2(K; N) are finite for any K > 1. The functions K r1(K; N) and K - r2(K; N) are nondecreasing. Further, T1(K; N) -+ 1 and T2(K; N) - 1 as K --+ 1 for every conformal norm N. It follows from the definitions of the functions TI and r2 that KN(f) <_ rI[K(f);N]. K(f) <_ T2[KN(f);NJ for every f E L(E) with det f > 0. It is clear from these estimates that if K(f) is close to 1 for a mapping f, then KN (f) is also close to 1, and, conversely, if KN (f) is close to 1, then so is K(f). 4.2. Mappings with bounded distortion. Let U C Rn be an arbitrary open set, and let f : U -, Rn be a given mapping in Wp1a(U), where
I. INTRODUCTION
62
p > 1. The linear mapping fl(x) is defined for almost all x E U. Its determinant det f'(x) is called the Jacobian of f at the point x, and is denoted by f (x, f ). We say that a function u : U -' R does not change sign on a set A C U if either u(x) > 0 almost everywhere in A or u(x) < 0 almost everywhere in A.
A mapping f : U -' R" is called a mapping with bounded distortion if it satisfies the following two conditions: (C 1) f is continuous. (C2) f belongs to the class Wry the function f (x, f) does not change sign in U, and there exists a number K > 1 such that
Ilf'(x)II" < KIf(x. f)I
(4.16)
for almost all x E U. A mapping f is said to be quasiconformal if instead of (Cl) it satisfies the stronger condition (C3) f is a homeomorphism. The smallest constant K such that (1.1) holds almost everywhere in U is called the distortion coefficient of f and denoted by K(f). Condition (C2) is equivalent to the simultaneous satisfaction of the following three conditions:
a) f E
and F(x, f) has constant sign in U. ft) There exists a set E C U of measure zero such that if x E U \ E and f (x, f) = 0, then the linear mapping f'(x) is equal to zero. y) There exists a number K > 1 such that K[f'(x)] < K at each point
xEU\E(Ethesetin fl)) with F(x,f)$0. Quasiconformal mappings have been the subject of many investigations in the case n = 2. At present the theory of planar quasiconformal mappings is an extensive part of the theory of functions of a complex variable and
has many important applications. A large role in the evolution of this theory has been played by the fundamental work of Lavrent'ev. The concept of a quasiconformal mapping in n-space was introduced by Lavrent'ev as far back as the 1930's [80], but the beginning of intensive investigations in this area relates to 1960 (see [171], [33], [119), [1451, [ 157], and [ 158]). Mappings with bounded distortion were considered by Lavrent'ev in the case n = 2. In the n-space case such mappings were introduced by the author in 1966 in [124]. There is a survey of investigations devoted to this subject in [ 175]. Some questions in the theory of quasiconformal mappings in space are considered in [ 15], [43], [74], [82], [83], [90], and [140]-[142]. A survey and an exhaustive bibliography of
§4. DEFINITION OF A MAPPING WITH BOUNDED DISTORTION
63
publications in the theory of n-space quasiconformal mappings up to 1967 is contained in the monograph [25]. 4.3. Examples of mappings with bounded distortion. 1. Every nonsingular affine mapping of R" is trivially a quasiconformal
mapping. Let f : U - R" be a C' -mapping of an open domain U C R". In this case if there exist constants y > 0 and M < oo such that I f'(x)) < M and If (x, f ) I _> y for all x E U, then f is clearly a mapping with bounded distortion. In particular, if f : U - R" is a C'-mapping such that F(x, f) # 0 for all x E U, then the restriction of f to any open set V strictly inside U is a mapping with bounded distortion. 2. Let U be an arbitrary polyhedron in R. A mapping f : U - R" is said to be simplicial if U can be partitioned into finitely many simplexes such that the restriction of f to each of these simplexes is affine. Every simplicial homeomorphism f : U - R" is a quasiconformal mapping. Indeed, let fl,..., fn be the components of the vector-valued function. The restriction of fk to the intersection of U with an arbitrary line in R" is a piecewise affine, hence absolutely continuous, function of one variable. The partial derivatives 8fk/8x; are constant on each simplex in the partition of U, and are hence bounded. This proves that f belongs to W (U). The rest is obvious. 3. We give an example of a mapping having a peculiarity distinguishing an arbitrary mapping with bounded distortion f r o m quasiconformal mappings. Take an arbitrary integer m > 1 . Let x = (XI, ... , xn _2, xn_, , xn) E R". If xn_ I = xn = 0, then let f (x) = x. Suppose that xn_ 1 +xn > 0. Then xn_, = r cos 9P and xn = r sin gyp, where r = xn_ I + xn and 0 < c < 2n. In this case let
f (x) = (x1, ... , xn_2, r cos m(, r sin mip). This mapping is continuous, and all points of the plane Rn-2
= {x E Rnlxn_, = x,, = 0}
are carried into themselves. Every circle r with center in Rn-2 and lying in the two-dimensional plane orthogonal to Rn-2 is also transformed into
itself under f. Further, if x makes a circuit of r in one direction, then f (x) runs through t in the same direction m times. We call f a twisting around an axis. The mapping f clearly belongs to C' on the open set R" \ R"-2. Every (n -1)-dimensional half-plane with boundary R"-2 is transformed isometrically by f into another such half-plane. This implies that the principal
1. INTRODUCTION
64
dilation coefficients of f'(x) in the n -1 directions parallel to the indicated half-plane are equal to 1. The dilation coefficient of f'(x) in the direction orthogonal to this half-plane is equal to m. This implies that at each point
x It R"-2 we have that Ilf'(x)ll = m, detf'(x) = in, and
K[f'(x)] =
m"-I,
Ko[f'(x)] = M.
The derivatives off are bounded. Theorem 2.6 allows us to conclude that f E W,e1 Ia(R"). It follows from the foregoing that f is a mapping with
bounded distortion. Further, K(f) = m" I and K0(f) = m. The main peculiarity of the mapping in this example is that f is a homeomorphism in a sufficiently small neighborhood of every point x 0 R"-2 and is not a homeomorphism in any neighborhood of an arbitrary point x E R"-2. It will be shown below (Chapter II, § 10.4) that a peculiarity of the type considered in this example cannot occur for a mapping f with bounded distortion if K(f) is sufficiently close to 1. 4. We give an example of a mapping with bounded distortion whose derivatives are not bounded in any neighborhood of some point a, in contrast to what happens in the examples already considered. Let U = R" and
a = 0, and take an arbitrary a > 0. Let f (0) = 0, and let f (x) = xIxl°-' for x 96 0. This function f will be called a nonlinear homothety. Obviously, f is continuous on R" and belongs to C°° on R" \ {0}. Every ray emanating from the point 0 is transformed by f into itself. The sphere S(0, r) is transformed by f into the sphere S(0, r°) by a similarity. Take an arbitrary point x # 0, and let r = lxl. The dilation coefficient of the linear mapping f'(x) in the direction of the ray Ox is equal to (dr°/drI = ar°-I. The dilation coefficients of f'(x) in the n - 1 directions orthogonal to this ar(°-I)". The quanray are equal to r°-I. This implies that detf'(x) =
tity Ilf'(x)ll is equal to ar°-I in the case a > 1, and Ilf'(x)ll = r°-I for
0 < a < 1, where r = lxl. Hence K[f'(x)] = a"-I in the case a > 1, and K[f'(x)] = 1/a for 0 < a < 1. We have that IIf'(x)II = IxI°-I. It is clear from this that the derivatives of f (x) are unbounded in a neigh-
borhood of 0 in the case 0 < a < 1. Further, the function x -+ If'(x)l is integrable to any power p < n/(1 - a) in a neighborhood of 0. In particular, f E W,1(R) for any p < n/(1 - a). It follows from what has been proved that the mapping f : R" -' R" constructed is quasiconformal. Further, K(f) = an- I and Ko(f) = a in the case a >_ 1, while K(f) = I /a
andKo(f)=(1/a)"-I in the case 0
{(xl,...,x")(=- R"IxI
§4. DEFINITION OF A MAPPING WITH BOUNDED DISTORTION
65
An arbitrary point x = (x1,... ,
I , xn) E R" will be represented as a pair (y, z), where y = (x1 , ... , x"_ I) and z = xn. Let f (0, z) = (0, e-z) _ e-zen, and for y # 0 let z
.f (Y, z) =
- ( IYI
sin IYl) Y. e-z cos lyl
The mapping f is obviously continuous and has derivatives of all orders at each point x = (y, z) with y 96 0. Consider an arbitrary plane r passing through the axis Ox". We introduce a Cartesian orthogonal coordinate system rl) in it, taking the line along which r intersects the hyperplane x" = 0 as the axis O;, and the line Oxn as the axis O. Here the direction of the axis Oq is the same as for the axis Oxn. The intersection rn Z is the strip consisting of all points x E I' such that -7r/2 < < n/2. The mapping f transforms r n Z into a subset of the plane IF. Further, if x has coordinates >l), then f (x) has coordinates (-e-'I sine, e'n cosh) in the plane F. In complex form the restriction off to F is the mapping C .- e
where C = + iq, and it maps the strip -n/2 < ReC < n/2 conformally onto the upper half-plane Im Z > 0. This implies that f is a homeomorphic mapping of the cylinder Z onto the half-plane x" > 0. We prove that f is a quasiconformal mapping. The symbol K(z, r) below denotes the (n - 2)dimensional sphere consisting of all the points x = (y, z) E R" such that IYI = r, while the z-component is fixed. The center of this sphere lies at the point (0, z). The mapping f implements a similarity transformation of K(e_z K(z, r) into the sphere cos r, a-Z sin r). We take an arbitrary point
x = (y, z) E Z such that y # 0 and, as shown above, construct the twodimensional plane I' passing through x and the axis Oxn. Consider also the (n - 2)-dimensional sphere K(z, I yl ). The tangent plane T of K(z, I yl) at x is completely orthogonal to the two-dimensional plane F. The mapping f'(x) transforms the (n - 2)-dimensional plane T and the plane I" into two mutually orthogonal planes-the (n - 2)-dimensional plane tangent to the K(e_z sphere cos I yl , e_z sin IYI) at f (x), and the plane F. The dilation coefficients of the mapping f'(x) are equal to e-- sin IYI/IYI on the plane
T, and equal to a-z on the plane F. We get that n - 2 of the principal dilation coefficients of fl(x) are equal to e-z(sin IYI)/IYI < e-z, and two are equal to a-z. From this, I det f'(x)I = e-"z(sin IYI/IYI)n-2 IIf'(x)II = and hence IYI)n-2 < K(f'(x)) = (IYI/ sin It follows from the above that e-z,
(lr/2)n-2
Ko(f'(x)) = (IYI/ sin IYI)2 <
(n/2)2.
I. INTRODUCTION
66
We get that the mapping f is quasiconformal. Further,
K(f) =
(n/2r-i
and Ko(f) = (n/2)2.
The composition (a = go h of a Mbbius mapping h carrying B(0,1) into
a half-plane and the mapping g = f -' inverse to f is a quasiconformal mapping of the ball onto the cylinder Z. Moreover, K(q) = (ir/2)2.
Ko(9p) =
(ir/2)"-2
The closure of Z in the Mobius space R" is homeomorphic to a closed ball. We have the following assertion ([ 141], [142], [168)). There exists a number co > 0 such that for every quasiconformal mapping f : B(0, 1) i" with K(f) < 1 + co, the closure of f [B(0,1)] is homeomorphic to the closed ball. This implies that a quasiconformal mapping of the ball onto the cylinder Z cannot have distortion coefficient K(f) arbitrarily close to 1.
The existence of a quasiconformal mapping of a cylinder onto a ball was first shown by P. P. Belinskil; his example (which was different from that given) was not published. One problem in the theory of n-space quasiconformal mappings is to describe domains that are quasiconformally equivalent to a ball, that is,
which admit a quasiconformal mapping onto a ball. The domain with outward-directed peak formed by revolving the figure in Figure 1 in space about the OY-axis is quasiconformally equivalent to a ball. At the same time, this is not true for the domain with an inward-directed peak formed by revolving the figure in Figure 2 about the same axis. Y
0
X
1
FIGURE 1
FIGURE 2
§5. MAPPINGS ON RIEMANNIAN SPACES
67
be a domain in R". If 0 is not quasiconformally equivalent to a ball, then let K(O) = Ko (9) = oo. But if .9 is quasiconformally Let
equivalent to a ball, then let
K(9) = inf K(f ),
Ko(9J) = inf Ko(f ),
where the infimum is over the set of all quasiconformal mappings of 9 onto a ball. It was shown in [60] (for the case n = 3) that for the cylinder
z
1
K(Z) = 1 J
/2 sin u du = 1, 3 ... ,
0
'2
In what follows we have to do mainly with mappings of open subsets of R" or of the Mobius space Rn. However, in some cases it is also convenient in studying mappings of sets in R" to deal with mappings of Riemannian
manifolds (given, for example, as surfaces in a Euclidean space of high dimension). It seems to us that the study of such mappings can itself be the source of many interesting problems.
5.1. Riemannian metrics in domains in R. Let U be an open subset of R". We say that a Riemannian metric is given in U if corresponding to each point x E U there is a positive-definite quadratic form of the variable HER", = n n (G(x)
,
)
EEgim) i. 1=i i=i
(5.1)
Further, it is assumed that the functions g;; (x)-the elements of the matrix G(x)-are continuous, and g;i (x) = gj; (x) for any i, j = 1, ... , n. The quadratic form (5.1) is called the linear element of the given Riemannian metric. Let dx; : R" -+ R be the linear function defined by the condition Then the product dx;dyj is the function whose value at the point _ fin) E R" is equal to Correspondingly, the quadratic form (5.1) can be written in the form g(x) = g;j (x) dx; dxx.
(5.2)
Here repeated indices are understood to be summed from I to n. Our goal is to define the concept of a mapping with bounded distortion for the case of a mapping of one Riemannian space into another. It
1. INTRODUCTION
68
is expedient to consider initially the case when we have a mapping of a domain in R", equipped with some Riemannian metric, into another such domain. We first make some remarks about quadratic forms. (Ac, ) and (Be, 4) are given positive-definite Assume that quadratic forms on R", where A and B are symmetric matrices. Let E1 and E2 be n-dimensional Euclidean spaces, each coinciding with R" as a vector space, but with the inner product (4, OE, = (Ac, j7) in E, and E2 = (Be, ,) in E2. Let L : R" -' R" be a given linear mapping, and let 0 < A 1 < . . < A" be the principal dilation coefficients of L, as a mapping from E1 into E2. Assume that L is a nonsingular mapping. Then Al > 0, and we let K(a, b; L) = An/AIA2...A", Ko(a, b; L) = K(b, a; L-') = A1A2
It is useful to assign definite values to K(a, b; L) and K0(a, b; L) in the case when det L = 0. If det L = 0 and L is identically equal to zero (that is, if L = 0 for all E R"), then we let K(a, b; L) = Ko(a, b; L) = 1. But if det L = 0 and L 0 (that is, there exists a E R" such that L() # 0), then we let K(a, b; L) = Ko(a, b; L) = oo. LEMMA 5.1. Let the numbers ao, al , Ro.,8I be such that 0 < ao < al <
00,0
and
floI
I2 <_ (B) <_ Bi 1
12
for any E R". Then for every linear operator L: R" - R" K(a, b; L)<_
(.!..L)"K(L) ,
/"
K(L) < (01
K(a, b; L).
(5.3)
PROOF. The desired inequalities are obvious if det L = 0, for then (a, b; L) is either 0 or oo. Below we assume that det L 0 0. As shown in §4.1, there exist systems of vectors {u,,... , u"} and {vl , ... , v.) such that the first of them is orthogonal in E,, the second is orthogonal in E2, and Lu1 = A1v1 for each i = I,-, n. For any i, j = 1, ... , n we have that (Lu1, LuJ)E2 = A;A1S,J, where a1J = I for i = j and 8,j = 0 for i 96 j. By definition, (Lu;, LuJ)E2 = (Lu1, BLu j) = (u;, L*BLu j) = (u1, A(A-1 L'BL)u1) = (u1, A-1 L*BLuJ)E,.
Let A L'BL = L'. We see that (Lug, ui)E, = 0 for i # j, and (Lug, ug) _ A2 j. From this, L u J = Aj ug . Accordingly, A , ... , AR are the eigenvalues of
i
§5. MAPPINGS ON RIEMANNIAN SPACES
69
the mapping L = A- I L' BL, and u1,... , u" are eigenvectors. In particular, this implies that
detL= IdetLl (detB)/detA. The conditions of the lemma give us that all the eigenvalues of the matrix A lie between ao and while the eigenvalues of B are between .802 and flI , which implies that A r < detB < fii ",
ao" < det A < a;"
This allows us to conclude that
(Ro/aI)"IdetLl <)iA2...A, <(RI/ao)"IdetLl.
(5.4)
The quantity A" is the norm of L, as a linear mapping of EI into E2; that is, An is the largest of the numbers A such that AIIIIE, for E R. In precisely the same way, IILII is the smallest of the numbers .1 ,II I. In particular, IIL IIE2 <_ 4IRIIE, for every . From such that this, 2"aIRI. and hence (aI/fio)2"ICI, for every E R". Consequently, (5.5)
IILII <_
The inequality IIL IIE2 < RI
Since
RI IILIIII
is arbitrary, this leads to the conclusion that < ao IILII.
(5.6)
The required result follows immediately from (5.4)-(5.6). The lemma is proved. The characteristics of a linear mapping introduced here have the following invariance property. Let lO : R" R" and yr : R" - R" be nonsingular
linear mappings, and let Then
a[c'(C)] and (W.b)(C) = b[w-'(C)]
K(p'a, y/. b, WLqp) = K(a, b, L),
(5.7)
Ko(ip*a. yr.b, IVLS') = Ko(a, b, L).
(5.8)
For a proof we introduce Euclidean spaces Ei and E2, each coinciding with R" as a vector space, with II IIE; = and 11411.1, The mappings c and v, regarded as mappings of the Euclidean spaces El,
1. INTRODUCTION
70
and E2 into E, and E2, respectively, are orthogonal. This implies that the E2 are the same principal dilation coefficients of the mapping vLSp: E1 as those of the mapping L: E, - E2. This proves (5.7) and (5.8). Let U and V be open subsets of R". Assume that U and V have Riemannian metrics given by the linear elements g(x) = g;j (x) dx; dx, and
h(y) = h;;(y) dy; dy;. A mapping f : U - V is called a mapping with bounded distortion with respect to the Riemannian metrics on U and V if the following conditions hold: (A) f is continuous and belongs to the class W, 10c(U). (B) The Jacobian f (x, f) of f has a constant sign in U. (C) There exists a number K, 1 < K < oo, such that
K[g(x), h[f (x)], f'(x)] < K
(5.9)
for almost all x E G.
The smallest number K for which (5.9) holds for almost all x E G is denoted by K(g, h; f ). If f : U - V satisfies conditions (A), (B), and (C), then there exists a K < oo such that Ko[g(x), h(x), f'(x)] < K for almost all x E U, and we denote the smallest such K by K0(g, h; f ).
If f : U - V is a mapping with bounded distortion with respect to Riemannian metrics g(x) and h(y) given on U and V, respectively, then for every open set G strictly inside U the restriction off to G is a mapping with bounded distortion. Indeed, G lies strictly inside U. Then G is compact, and U G. Let M = f (G). The set M is compact, and V , M. It follows from continuity considerations that there exist constants ao, 011, flo, and #I such that 0 < a0:5 aI < oo, 0 < Ro < i, < oo, and aoI I2 <_ gii(x) l
i x'4112
for all x E G, while f0'RRI2 <
fi 112
for all y E M. Lemma 5.1 permits us to conclude that
K[f'(x)] <_ (-.a0fl
-)"K[g(x),h[f(x)],f'(x)] <
"
(afaoffilo)
K(g,h;f)
for almost all x E G. This establishes that the restriction of f to G is a mapping with bounded distortion. R" be a diffeomorphism, and V = V (U). Assume that a Let ip: U Riemannian metric is given on V by the linear element h(y) = (H(y)q, q). The diffeomorphism 9 permits us in a certain sense to carry this metric
§5. MAPPINGS ON RIEMANNIAN SPACES
71
over to U. For an arbitrary x E U let the symbol (rp * h) (x) denote the quadratic form g(x) defined by
where G(x) = [rp'(x)]`H[rp(x)]rp'(x). We have that G(x) = Ilgij(x)Ili,j=l..... " The elements of G(x) are expressed by the equalities
gij(x) =
ha
x )]a`pa(x)a(x). 8xi 8x j
(5.10)
Let U and V be open subsets of R". Assume that U has a Riemannian metric given by the linear element g(x) = g11(x) dxi dxj, and V has a Riemannian metric given by the linear element h(y) = hk,(y) dyk dyi. Let f : U - V be a mapping with bounded distortion with respect to the given Riemannian metrics. Assume now that ip : U, - R" and w : V -- R" are given diffeomorphisms, with {'(UI) = U, and let ip(V) = VI. Then the mapping f = w o f o rp : U, -+ VI is defined. The diffeomorphisms rp and w can be interpreted as different coordinate systems given on the respective
sets U and V. Then f should be understood as the representation of f in these coordinate systems. We define the Riemannian metrics (ip'g)(t) = gij[S (t)]di drpj and (w.h)(Y) = [(W-')*h](u) in the sets U, and V1, and we show that f, is a mapping with bounded distortion with respect to these metrics, and that
K(q'g, w.h;f) = K(g.h;f). (5.11) Ko(9;*g. w.h;f) = Ko(g,h;f). Indeed, the mapping f, = f o rp belongs to the class W1 1.10C in view of Theorem 2.8. Further, f((t) = f'(rp(t)) o rp'(t) for almost all t E U1. We have that 11f,'(011 <- Ilf'(w(t))II - Ilso'(t)II-
On every compact set A C U, we have that IIc'(t)II" < CLT(t, p)I, which leads us to conclude that
ilfl (t)II" dt < C fA
fA
f
IIf'(g'(t))II"If(t,,p)I dt
Ilf'(x)II"dx
,(A)
for any compact set A C U1. Consequently, f, is a mapping in W, toc(UI ). Application of Theorem 2.9 gives us that f = w o f, is a mapping of class WW 10c Further, P(t) = w'(Y) 0 f'(x) o Sp'(t)
I. INTRODUCTION
72
for almost all t, where x = p(t) and y = f (x) = f [9(t)]. Let
b() = Then, by (5.7) and (5.8), K((w'(t))'a, yv(u).b; w'(y) o f'(x) o q'(t)) = K(a, b;f'(x))
and similarly for KO. The equality holds for almost all t, and this proves (5.11). The mappings qr and yi are diffeomorphisms, and hence F (I. q,) and F(u, yr) have constant sign on every connected component of the domain of rp and V. This implies that the Jacobian off has constant sign on each connected component of U1. Consequently, f is a mapping with bounded distortion. 5.2. Mappings with bounded distortion on Riemannian spaces. For the convenience of the reader we define certain concepts.
Let M be a Hausdorff topological space with a countable base. The space M is called an n-dimensional manifold if each point of it has a neighborhood homeomorphic to R. A local coordinate system, or chart, in an n-dimensional manifold M is
defined to be a homeomorphism ip: U - R", where U is an open set in M. For x E U let p(x) = (t, (x),.. -, t"(x)). The numbers t, (x), ... , t"(x) are called the coordinates of x with respect to the chart q,, and the real functions t; are called the coordinate functions corresponding to this chart.
The set G = V(U) is open in R. Let rp : U -i R" and V : V -; R" be two arbitrary charts of M such that U n V 96 0. In this case the charts q, and y/ are said to be overlapping. The sets G = Sp(U n V) and H = q'(U u V) are open in R. The functions 0 = V 0 I r and w= ip o Vi -I implement homeomorphisms of G onto H and H onto G, respectively. Further, 0 = w- I and w = 0-1. The charts ip and yi are said to be C'-compatible, where 1 < r < oo, if each of the functions 0 and w belongs to C. We regard any two charts with disjoint domains as C'-compatible for any r such that 1 < r < oo. The functions 0 = yi o 99- and w = q, o yr- I , which are defined for a pair of overlapping charts rp : U - R" and V : V - R", have the following simple meaning. Let x E U n V, t = qp(x), and u = qv(x). Then u = 9(t) and t = w(u); hence 0 enables us to find the coordinates of the point x in the chart Vi from its coordinates in the chart VP. Conversely, the function w enables us to compute the coordinates of x in rp from its coordinates in V-
§5. MAPPINGS ON RIEMANNIAN SPACES
73
An n-dimensional differentiable manifold of class Cr is an n-dimensional
manifold M with a set 21 = {rpa, Ua - R"}aEA of charts such that the domains of these charts cover M, that is, M = vaEA Ua, and any two charts in this set are C'-compatible. A set 2i of charts satisfying this condition is
called an atlas of class C' on M. A chart qi : U -' R" of M is said to be admissible if it is smoothly compatible with any chart qra : Ua -+ R" of the atlas 21. The collection of all charts is itself an atlas of class Cr. We define the concept of a Riemannian space. For this we first define the general concept of a tensor and a tensor field on a differentiable manifold. In what follows we need only certain special cases of these general concepts, but here it is expedient to define the general concept. Let k > 0 and I > 0 be integers such that k + 1 > 0. One says that a k -fold covariant and 1 -fold contravariant tensor, or, briefly, a tensor of type (k, 1), is given at a point
p of a manifold M if to each admissible chart ip : U -' R" defined in a neighborhood of p there corresponds a collection of nk+t numbers aj"... k, where the indices i i , ... , il and J,... , jk run independently from 1 to n. The numbers ... k are the coordinates of the tensor with respect to the
chart 9. The following condition must hold here. Let ip : U -+ R" and
R" be arbitrary admissible charts, each defined in a neighborhood of p, let t I , ... , t,, be the coordinate functions of 47, and let u 1 . . . . . u" be yi : V
the coordinate functions of yr. Then the coordinates of the tensor with respect to w can be expressed in terms of its coordinates aar .k with respect to ip according to the formula bii...11 JI...1k
- apI...pk at-.,l n' ... 01
,,
,
8tp
&P,
' Otal
OUJt
0Jk
where repeated indices are summed from I to n, and the values of the partial derivatives are taken at the point p. Let g be a tensor of type (2,0) at a point p of the manifold M. This tensor is said to be symmetric if for some admissible chart defined in a neighborhood of p the coordinates of g satisfy the relations g;; = gjj (i, j = 1, ... , n). If this condition holds for some admissible chart in a neighborhood of p, then it holds for any other admissible chart in a neighborhood of p. A symmetric tensor g is said to be positive-definite at j is positive-definite, where the a point if the quadratic form g;j are the coordinates of the tensor with respect to an admissible chart qq defined in a neighborhood of p. A tensor of type (0, 1) at a point p of M is called a vector at this point. The collection of all vectors at a point p of M is denoted by TM (p) and called the tangent space of the manifold at this point.
74
1. INTRODUCTION
One says that a tensor field of class C' and type (k, l) is given on a manifold M of class CS (0 < r < s - 1) if to each point p E M there corresponds a tensor of type (k, 1), and for every admissible chart in M the coordinates of the tensor with respect to this chart are functions of class Cr. In studying functions on a differentiable manifold it proves expedient to endow the manifold with a Riemannian metric. For example, specifying the metric facilitates the introduction of a norm in function spaces analogous to the spaces W and so on. We give the necessary definitions. A Riemannian space of class Cr is a pair (M, g), where M is an ndimensional differentiable manifold of class Cr+I, and g is a tensor field of class Cr and type (2, 0) that is symmetric and positive-definite. The tensor g here is called the metric tensor of the Riemannian space. Let M be an n-dimensional Riemannian space of class C, and let p: U R" and yr: U -' R" be two admissible charts in M that have the same domain U c M. Let G = r9(U) and H = yr(U). Then G and H are open sets in R", and the tensor g on M enables us to define certain Riemannian metrics in G and H. Namely, let t E G and let g;1(t) be the coordinates of the metric tensor g at the point p = tp-I (t) with respect to the chart ip. Similarly, if u E H, then let h;1 (u) be the coordinates of the metric tensor of the space at p = yr- I (u) with respect to gyp. We get the Riemannian metric ds2 = g;1 (t) dt;dtl in G, and the Riemannian metric doe = h;1 (u) du; du1 in H. By the general rule for transforming tensor coordinates we have that g,1 (t) = hap(u(t)) &°
a OUP
that is, ds2 = (u'do)2, where u = yr o 9-I is the function implementing the coordinate change in passing from the chart Sp to the chart yr. A certain measure called the volume of a set is defined in a natural way
on every Riemannian space. Let ip : U - R" be an arbitrary admissible
chart in M, let G = 9(U), and let g11(t),i,j = 1, ...,n,t E G, be the coordinates of the metric tensor of the space in this chart. For an arbitrary Borel set E c U we let
vg(E) = f
det IIg,1(t)II dt. (E)
By what was proved in §5.1, vg(E) does not depend on the choice of the chart tp defined on an open set containing E. Let E be an arbitrary Borel set in M. Then E can be represented in the form 00
E=UE1 ,_I
(5.12)
§5. MAPPINGS ON RIEMANNIAN SPACES
75
where E; n E, = 0 for i 34 j, E; is a Borel set for each i, and E; is contained in the domain of some admissible chart ip; : U; -' R". The number vg(E;) is defined f o r each i = 1, 2, .... The sum E°° I vg (E;) does not depend on the choice of the representation (5.12), and by definition it is set equal to vg(E). Let MI and M2 be Riemannian spaces of class C', and let f : M, - M2 be a continuous mapping. Then we say that f is a mapping with bounded distortion if there exists a constant K, 1 < K < oo, such that the following
condition holds. Let (p: U - R" and y/: V - R" be arbitrary admissible charts in M, and M2, respectively, such that f (U) c V. Then the open sets G = (p (U) and H = Vi (V) are defined in R", and the mapping f' = v o f o
9-' : G - H is the coordinate representation of f by means of the charts to and yr. In the set G we have a Riemannian metric ds2 = g;1(t) dt; dt;the coordinate representation of the metric of the Riemannian space M,
by means of the chart ip-and in H the metric dc2 = h(u) du; du; is defined-the coordinate representation of the metric of the Riemannian space M2 by means of the chart yr. The condition to be satisfied by f is as follows. For any charts ip and yi the mapping f ' : G -+ H is a mapping with bounded distortion with respect to the metrics ds2 in G and dc2 in H, and
K(f;g,h)
tion coefficient off and denoted by K(f, MI, M2). If K(f, MI, M2) = 1, then f is said to be conformal. Let MI and M2 be differentiable manifolds. We say that f : M, - M2 is a mapping of class Cr, where r > 1, if f is continuous, and the mapping V o f o ip- I belongs to the class Cr for any admissible charts gi : U - R" in MI and Vi: V - R" in M2 such that f (U) C V. We give special consideration to the case of manifolds imbedded in
R. Let M be a subset of R. The set M is called a k-dimensional manifold of class Cr in R" if it is a k-dimensional manifold and it has an atlas 2 t = (V,,: U Q - Rk } of class C' such that the following condition holds. For every admissible chart 9: U - Rk the inverse mapping x = 9- 1 : G = 9? (U) -- R" belongs to the class C', and the vectors
f (8x/et;)(t), i = I,-, k, are linearly independent at each point t E G. The mapping x = (p'' is called a parametrization of M. Let M be a kdimensional manifold of class Cr imbedded in R", and let x : G -' M be an arbitrary parametrization of it. Take a p E M, p = x(t). The set Ro
I. INTRODUCTION
76
of all points z E R" of the form k
z = x(t)
(t),
where A, E R, i = I,... , k, is a k-dimensional plane passing through z. The plane Rp does not depend on the choice of the parametrization x with p E x(G), and it is called the tangent plane of M at the point p. Denote by Tp the k-dimensional subspace of R" consisting of all vectors of the form k
The numbers u', i = 1, ... , k, will be called the coordinates of the vector u with respect to the chart ip = x-I. The numbers u' transform like the coordinates of a tensor of type (0, 1) in passing from one chart to another. In view of this, the vector u can be identified naturally with a vector at the
point p of M. Setting
g;;(t) = (a (t). at (t)) 1
(i , j = 1,... , k) for an arbitrary parametrization x : G-' M of M, we get
in M a tensor of type (2, 0) that is symmetric and positive-definite; that is, a Riemannian metric in M. To conclude this section we give some examples of conformal mappings of manifolds. 1. Stereographic projection. The following notation will be used. We Let Z E R"+1, z = consider here the (n + 1)-dimensional space R"+I.
(x1,... , x", x"+ 1). Take (xt, ... , x") = x and x"+I = y, and regard the vector z here as the pair (x, y). Then 1z12 = 1x12 + y2. The space R" is identified with the set of all points z E R"+I of the form z = (x, 0).
Let S" be the sphere (z E R"+I lizi = 1), and let N = (0, 1) (the north pole) and S = (0, -1) (the south pole). The sphere S" is a C°Odifferentiable submanifold of R"+I. We define a mapping or of the sphere S" onto R". Let a(N) = oo. If z = (x, y) 96 N, then y < 1, and in this case we let a(z) = x/(1- y); a(z) is the point where the line passing through N and z intersects the plane R" = {(x, y) E R"+IIy = 0}. The mapping a is called the stereographic projection of S" into R". Let x E R", x # oo. We
find a point z = (u, v) E S" such that a(z) = x. We have the equalities x = u/(1- v) and (u12 + v2 = 1. Solving this system with respect to u and v, we find that U
2x = 1 + 1x12'
v
__
IX12 - 1
+
Ix12
§5. MAPPINGS ON RIEMANNIAN SPACES
77
The point z = (u, v) with u and v determined by (5.13) does belong to S", and a(z) = x. This proves that a is a mapping of S" onto j r. The mapping or is one-to-one. Each of the mappings o: S" \ {N} - R" and Q-I : R" - S" \ {N} belongs to the class C°°. We introduce the structure of a differentiable manifold in jr by agreeing that the mapping a : S" -y R is a diffeomorphism. Let us show that the stereographic projection maps S"\{N} conformally
onto R. Take an arbitrary point z = (u, v) E S", z 0 N, and let x = a(z) _ u/(1 - v). Consider the n-dimensional plane P passing through z parallel
to the plane R. Projection onto R" from the point N is a similarity mapping of P onto R" with similarity coefficient I/ (I - v). It is clear from this that the dilation coefficients of a are equal to 1 /(1 - v) in directions
lying in the section S" n P. We compute the dilation coefficients of a in the directions orthogonal to this section. To do this we construct the two-dimensional plane H through the diameter SN and the point z. We introduce an orthogonal coordinate system in H, taking the line Orr(z) as the axis OX. In this coordinate system the circle H n S" is given by the parametric equations x = cost, y = sin t, and the restriction of a to HnS" is the mapping cost
I -sint We have that ip'(t) = 1 /(1-sin t), from which we conclude that the dilation coefficient of a in a direction lying in H is also equal to 1 /(1 - v), and it is proved that a is conformal. It is also possible to consider the stereographic projection with respect to the south pole S of S". In this case the image of a point z = (u, v) E S" is the point x = u/(1 + v), and the inverse mapping is determined by the formulas I-IXI2 2x U 1+1x12,
v
Ix12+1
2. The Mercator projection. By analogy with the well-known cartographical projection we define the Mercator projection to be a mapping of R" with deleted point 0 onto a right circular cylinder. Denote by C" the set {(xl,...,x", y) E R"+1 Ix, + +x,2, = l} in R"+I . Let x be an arbitrary point in R", x = (x1,... , x") 96 0. We set
f (X) =
II:11
XI'
X2
Ixi'...,
l,ln IXI) .
78
I. INTRODUCTION
Obviously, f is a mapping of Rn \ {0} onto the cylinder Cn. Here every
sphere S(0, r) = {x E R"IIxI = r} is mapped by a similarity onto the unit sphere obtained in the section of C" by the plane y = In 1 /r. The dilation coefficients of f at a point x E S(0, r) in the directions tangent to the sphere are thus equal to 1 /r-the ratio of the radii of the image and inverse image spheres. The rays emanating from 0 are carried under the mapping f into generators of the cylinder Cn, i.e., into curves orthogonal to the images of S(0, r). The dilation coefficient along any such ray is equal to
dr'n MI
r We see that the dilation coefficients of fl(x) in n mutually orthogonal directions are equal to 1/r. Consequently, f is conformal. I
CHAPTER II
Main Facts in the Theory of Mappings with Bounded Distortion §1. Estimates of the moduli of continuity and differentiability almost everywhere of mappings with bounded distortion §1.1. Some auxiliary facts. Let U be an arbitrary open domain in R",
and let f : U - R" be a continuous mapping. The mapping f is said to satisfy Holder condition with exponent a on compact subsets of U, where
0 < a < 1, if for every compact set A c U there is a number M(A), 0 < M(A) < oo, such that for any x1, x2 E A
If(xs)-f(x2)I !5
M(A)Ix1-x21°.
(1.1)
It will be shown below that every mapping with bounded distortion satisfies a Holder condition with exponent a = 1 /K on compact subsets of its domain, where K = K(f). We remark that an estimate of the modulus of continuity of quasiconformal mappings was first established by Kreines [73]. The Holder property was first proved for a mapping with bounded distortion by the author ([ 119], [124]) and simultaneously by Callender [24]. The proof that a mapping with bounded distortion satisfies a Holder condition is based on two facts. The first is a lemma due to Money [102]. The second is an inequality which is a corollary of a known isoperimetric inequality.
Let U be an open set in R". For any point x E U let pu(x) = p(x.8U), with the subscript U omitted whenever no confusion can result. LEMMA 1.1 (Morrey's lemma). Let U c R" be an open set in R", and f : U - Rk a function of the class W.' (U), where 1 < m < n. Assume that there exist numbers a (0 < a < 1), M < oo, and 6 > 0 such that fB(a,r)
If'(x)Imdx < 79
Mr"-m+ma
11. MAPPINGS WITH BOUNDED DISTORTION
80
for every ball B(a, r) c U with radius at most 6. Then there exists a continuous function f such that f (x) = j(x) almost everywhere, and the oscillation of j on any ball B(x, r) c U with r < 6/3 and r < p(x)/3 does not exceed CM I /'" ra, where C < oo is a constant.
A proof of the lemma will be given in Chapter III, §4.2. If a continuous function f satisfies the conditions of Lemma 1.1, then
it satisfies a Hdider condition with exponent a on compact subsets of U. Indeed, let A be a compact subset of U, and let y be the smaller of the numbers 6/3 and d/3, where 6 is the constant in Lemma 1.1 and d = dist(A, 8U). We consider the function h defined as follows on the product AxA: h(x, y) = If(x)-f(y)I/Ix-yI forx 96 y, and h(x,x) = 0. Let H be the set of pairs (x, y) E A x A such that Ix - y j > y, and let G = (A x A)\H. The set H is compact, and thus h is bounded on H by continuity. The conclusion of the lemma enables us to deduce that h is bounded also on G. Consequently, h is bounded on A x A, and thus
If(x)-f(y)I <MIx - yI" for anyx,yEA. LEMMA 1.2. Suppose that U c R" and the mapping f : U - R" is in the class W (U). Then for any a E U and almost all t E (0, p(a))
f
det f'(x) dx <
B(a,t)
tn f
S(a.t)
Ilf'(x)II" dax,
where da is the area element of the sphere S(a, t). PROOF. We get the lemma as a corollary to a known isoperimetric inequality. To formulate the latter we give some necessary definitions. Let A: R" R" be a linear mapping, v a unit vector in R", P the (n - 1)dimensional subspace of R" orthogonal to v, and Q. = A(P,,). The restriction of A to P is a linear mapping of an (n - 1)-dimensional Euclidean space P onto the Euclidean space Q,. The determinant of this mapping is denoted by E,,(A). The norm of the linear mapping AIPV clearly does not exceed IIAII. We have IIAIPIIn-1, which implies that
i(A) <
IIAII"-1
(1.2)
Suppose that U is an arbitrary domain in R", a E U, and f : U -+ R" is a mapping of class W . Let
V(1J) = f a.t) det f'(x) dx,
(1.3)
(
where 0 < t < pe(a), and let
F(t, f) = f
A,(x)[f'(x)l dax. (a.t)
(1.4)
§ 1. MODULI OF CONTINUITY AND DIFFERENTIABILITY
81
Here dax is the area element of the sphere S(a, t), and v(x) is the unit vector normal to S(a, t) at the point x. The integral (1.4) is obviously defined for almost all t E (0, p(a)).
In the case when f is a mapping of class C' and the Jacobian of f is nonzero everywhere, F(t, f) is clearly the area of the (n - 1)-dimensional surface into which S(a, t) is transformed by f , and V (t, f) is the oriented volume bounded by this surface. For a suitable definition of the concept of the area of a surface it is possible to interpret the integral (1.4) as the area of a surface also in the case when f is a mapping of the class W,,'. We remark that, by (1.2), IIf'(x)IIn-' for all x such that f'(x) is defined, and this gives us that
F(t.f) <- f
IIf'(x)II' dax.
(1.5)
(a.t)
For almost all t E (0, p(a))
V(t.f)
where f =
(1.6)
(I/nfn)[F(t,f)]"I(n-1),
I). This is the isoperimetric inequality. It expresses the fact that among all closed surfaces in R with given area a sphere bounds the largest volume. We do not give a proof of (1.6), but refer the reader to the author's paper [ 123]. It follows from (1.5) and (1.6) that for almost all t E (0, p(a))
f
w,1,/('
n/(n-1) (at)
det f'(x) dx <
(ka,l)
IIf'(x)II-dax
.
(1.7)
By Holder's inequality,
i-I/n Ilf'(x)IIn-1 dax < (fs(aa)
II.f'(X)II" doX
S(a.t)
I/n
(fs(a,t)
daX
The second integral on the right-hand side is equal to w to-' I. Substituting the last expression into (1.7), we get that
f as required.
det f'(x) dx < (a,t)
rf n
(aa)
IIf'(x)IIdcx,
II. MAPPINGS WITH BOUNDED DISTORTION
82
g1.2. An estimate of the modulus of continuity of a mapping with bounded distortion.
LEMMA 1.3. Suppose that U c R" is an open set in R", f : U -' R" is a mapping of class W, ,',,,,,(U) whose Jacobian has positive sign in U, and IIf'(x)II" < KI F(x, f )l for almost all x E U, where I < K < oo. For X E U and r < p(x) let
v(x,r)=(1/r"/K) f
B(x.r)
Then the function r
IIf'(x)II"dx.
v(x, r) is nondecreasing.
PROOF. For r < p(x)
f
IIf'(x)II" dx < K
B(x.r)
f
r(x, f) dx
I F(x, f )l dx = K J
B(x.r)
(1.8)
(x.r)
because F(x, f) has constant sign in U. On the basis of Lemma 1.2
f
det f'(x) dx< (x.r)
r f IIf'(x)II" da(1.9)
n
S(a.r)
for almost all r E (0, p(x)). From (1.8) and (1.9) we get Kr IIf'(x)II" dx < fs(x.,) IIf'(x )IIdax. B(x.r)
f
I
Ilf'(x)II" dx = w(r),
(1.10)
IIf'(x)II" dax = s(r).
(x.r)
Applying Fubini's theorem, we get that w(r) = for s(t) dt for all r E (0, p(x)).
This leads us to conclude that the function w is absolutely continuous and w'(r) = s(r) for almost all r E (0, p(x)). From (1.10) we have that w(r) < Krtv'(r)/n for almost all r. Multiplying both sides of this inequality by r-("1K)'1, we get after obvious transformations that (w(r)/r"1K)' > 0. Consequently, the function w(r)/r"/K is nondecreasing, and the lemma is proved.
THEOREM 1.1. Suppose that U c R" is an open set, f : U - R" is a mapping of class W (U) such that the function x - 'r (x, f) has constant sign in U, i.e., is either nonnegative almost everywhere or nonpositive almost everywhere, and IIf'(x)II" <_ KIf(x, f )I for almost all x E U. Let fu IIf'(x)II" dx = Mn. Then the vector-valued function f is equivalent, in the sense of the theory of the integral, to some continuous function f'. Further, for every set V lying strictly inside U the oscillation of f' on any ball
§ 1. MODULI OF CONTINUITY AND DIFFERENTIABILITY
83
B(a, r) of radius r < 2d/3 about an a E V does not exceed (CM/d I IK)rh /K. where d = dist(V,, (9U).
PROOF. Let a E V, and let
w(a, r) = J (a.r)
IIf'(x)IIdx M.
According to Lemma 1.3, the function v(r) = w(a, r)r-"lK is nondecreasing, and thus M"d-n/K w(a, r)r-n/K < v(d) <
for all r E (0, d); hence w(a, r) < (M"/d"IK)r"lK. The required result follows directly from Lemma 1.1.
COROLLARY 1. Let f : U - R" be a mapping with bounded distortion.
Assume that fo Ilf'(x)II" dx = M < oo. Then the function f satisfies a Holder condition with exponent a = 1 /K on compact subsets of U. Further, if V is contained strictly inside U, then for any x, y E V
If(x) - f(y)I <- LIx - y11/K,
(1.11)
where the constant L depends only on V, the distance from V to the boundary of U, and the constant M.
REMARK. The example of the mapping with f (0) = 0 and f : x -+ xIxI°-I for x 0 0, where a = 1/K and K > 1, shows that the exponent 1 /K in (1.11) cannot be decreased. For this f we have that K(f) = K and If(x) -f(o)I = IxI'7. COROLLARY 2. Let U be an open domain in R", and F(U, K. M) the col-
lection of all mappings f with bounded distortion of U such that fu IIf'(x)II" dx < M. Then the set of functions f is equi-uniformly continuous on every compact subset of U. §1.3. Differentiability almost everywhere of mappings with bounded distortion. Let U be an open set in R. As we know, a mapping f : U -+ R"
is said to be differentiable at a point a E U if there exists a linear mapping L: R" --+ R" such that f (x) = f (a) + L(x - a) + a(x)Ix - al for all x E U, where a(x) --+ 0 as x
a. The mapping L is called the differential
of f at the point a. Our immediate goal is to prove that every mapping with bounded distortion is differentiable almost everywhere in its domain. The proof of this is based on a certain general proposition about differentiability almost everywhere of functions with generalized derivatives. We
present formulation of this proposition here. Its proof will be given in Chapter III, §4.1. Let us first define some concepts.
84
11. MAPPINGS WITH BOUNDED DISTORTION
Let R be a topological vector space whose elements are functions with values in a finite-dimensional vector space V and defined on the ball B1 = B(0, 1) c R. (Only the case when V = Rk is considered in what follows.)
Let U C R" be an open set. We say that the function f : U -+ V is differentiable in the sense of convergence in 91 (briefly, 9R-differentiable) at
the point a E U if there exists a linear mapping L : R" -+ V such that the function rh : X E B1 - (1/h)[f (a + hX) - f (a) - hL(X)] is defined on BI and belongs to 91 for sufficiently small h, 0 < h < ho, and rh -+ 0 in the topology of R as h -, 0. The linear mapping L is called the 9R-differential
of f at the point a. Let R be the Banach space L. of bounded functions f : B1 -' V, with the norm in L.. defined by IIf IIL = ess supxEB, If (x)I. Assume that the
function f : U -+ V is differentiable at a point a E U in the sense of convergence in L. Let A(h) = II rh II L Then it is obvious that I rh (X) I < A(h) for every X E B1. Take an arbitrary vector such that ICI < ho. Substituting h = ICI and X = /ICI in the expression for rh(X), we get that If (a + ) - f(n) - A(E)I < A(II). ICI
and hence I f (a + ) - f (a) - A(E)I/Icj -p 0 as 5 -, 0. Thus, LO.differentiability at the point a implies differentiability in the classical sense.
It is not hard to verify that, conversely, if f is differentiable at a in the classical sense, then it is Lo.-differentiable at this point. LEMMA 1.4. Suppose that f : U - Rk is a function of class Wp (U), where U c R" is an open set. Then f is differentiable in the sense of convergence in Wp(U) at almost all points x E U, and its formal differential fl(x) is the Wp-differential almost everywhere in U.
This is a special case of a general theorem in [ 131 ]. A proof of the lemma will be given in Chapter III, §4.1.
THEOREM 1.2. Let f : U -+ R" be a mapping with bounded distortion. Then for almost all x E U the linear mapping f'(x) is the differential of f at the point x.
PROOF. Let f : U - R" be a mapping with bounded distortion. Then f E W,;, and thus, according to Lemma 3, the linear mapping f'(x) is the differential in the sense of convergence in W for almost all x E U. Suppose that the point x E U is such that L = f'(x) is the off at x. Let fh(X) = (1/h)[f(x+hX)- f(x)]. Then Ilfh -L111.".B, 0
as h -, 0. The vector-valued function fh belongs to W,,', like f, and its derivatives can be expressed in terms of the derivatives of f by the
§2. CONTINUOUS MAPPINGS ON R"
85
same formulas as in the case of functions of class C'. This gives us that fh'(X) = f'(x + hX) for almost all X E Bi, which enables us to conclude that fh is a mapping with bounded distortion. Further, it is not hard to see that K(fh) < K(f). Let ho > 0 be such that for 0 < h < hp the mapping fh is defined everywhere on the ball B1. Since fh converges in W,,' (B,) to the linear mapping f'(a) as h 0, it can be assumed that the norms IIfhIII.".s, are bounded for 0 < h < ho, Ilfh 111.".e, < M < oo for all such h. Hence,
by Corollary I to Theorem 1.1, the family of mappings fh, 0 < h < ho, is equi-uniformly continuous on every ball Br, where 0 < r < 1. Since fh -+ L in W (B,) ash -+ 0, fh -p fl(x) in L"(B,) in view of the Sobolev imbedding theorem. Since the family of functions fh is equi-uniformly continuous, this implies that fh --' L uniformly on every ball B where 0 < r < 1. We have the relation fh(X) = 2f2h(X/2). For IXI < 1 the point X/2 is in B112. Since f2h(X) - L(X) uniformly on B,,2, this implies that L(X) uniformly on B,, and hence fh - L uniformly on B,. 2f2h(X/2) This proves that f is differentiable at x and L = f'(x) is the differential of f at x. The theorem is proved. §2. Some facts about continuous mappings on R"
§2.1. The degree of a mapping. The main topological concept used in studying n-space mappings is the degree of a mapping. Its definition does not require a cumbersome topological apparatus. Here we present some necessary facts. A set G c R" is called a compact domain if G is compact, its interior G° is connected, and it is the closure of its interior. Let G C R" be a compact domain, and f : G R" a continuous mapping. A point y E R" is said to be (f G)-admissible if y ¢ f (OG). With every triple (f, G, y) with G a compact domain in R", f : G -- R" a continuous mapping, and y an (f, G)-admissible point we can associate a number u(y, f G) called the topological index or degree of the mapping f : G R" at the point y. We first define the concept of degree for the case when f is sufficiently R" smooth. Let G c R" be a compact domain. The mapping f : G is said to belong to the class Ck, where 1 < k < oo, if the restriction of f to G° belongs to Ck, and each of its derivatives of order at most k is
uniformly continuous on G°. Let f : G - R" be a mapping of class Ck, k > 1, on a compact domain
G c R". We say that f is regular with respect to a point y E R" if the following conditions hold: 1) y is (f, G)-admissible, i.e., y ¢ f (WG). 2) The set f -I (y) is finite.
86
II. MAPPINGS WITH BOUNDED DISTORTION
3) At each point x E f -' (y) the Jacobian of f is nonzero. In particular, f is regular with respect to y if f -' (y) is empty. Let f : G -' R" be a mapping of class Ck which is regular with respect to a point y E R. The total number of points in f-I (y) is denoted by N. Let N+ be the number of points x E f - I (y) at which f (x, f) > 0, and
N the number of points x E f -' (y) at which f (x, f) < 0. Obviously, N = N++N-. The number µ = N+-N- is called the degree or topological index of f with respect to the point y and the domain G, and is denoted by µ(y, f, G). If f -I (y) is empty, then each of the numbers N, N+, and N- is equal to zero, and hence u(y, f, G) = 0 in this case. Let A and B be arbitrary sets in R. They are said to form a pair if A D B. Assume that pairs (A,, B,) and (A2, B2) of sets are given. A mapping of the pair (A,, B,) into the pair (A2, B2) is defined to be a mapping f : A, A2 such that f (B,) C B2. We define the concept of a homotopy of continuous mappings of pairs.
Let (A,. B,) and (A2, B2) be two arbitrary pairs of sets in R", and let f and g be continuous mappings of the first pair into the second; f and g are said to be homotopic if there is a family of mappings 9 I, t E [0, 1], of (A, . B,) into (A2, B2) such that: 1) rpo - f and ip, - g; and 2) the mapping tp: (x, 1) - V,(x) of A, x [0, 1) C R"+' into R" is continuous. The mapping da is called a homotopy of the mappings f and g. If f is homotopic to g, then we write fftg. Homotopy of mappings is an equivalence relation, i.e., the following hold for any continuous mappings
f, g, and h of the pair (A,, B,) into (A2,B2): 1) f,, f ; 2) if fag, then ggj; and 3) if fftg and gqh, then fHh. The proofs of these properties are obvious and are left to the reader. The following simple remark will be used repeatedly below. Let G C R" be a compact domain, f : G -- R" a continuous mapping, and y E R" an (f, G)-admissible point. Let (f,,: G - R"), v = 1.2, ... , be a sequence of continuous mappings uniformly convergent to f. Then there is a vo
such that for v > vo the point y is (f G)-admissible, and f, and f are homotopic as mappings of the pair (G, 8G) into the pair (R",R"\{y}). Indeed, f (8G) is compact and y io f (8G), and thus 6 = p(y, f (8G)) > 0. Let vo be such that If, (x) - f (x) I < 8 for v > vo for all x E 8G. This is the desired vo. The homotopy of f and f, for v > vo can be taken to be the function
pr(x) = (1- t)f(x) +
t E [0,1).
LEMMA 2.1. Let G be a compact domain in R", and let f : G - R" and g: G - R" be mappings of class Ck, k > 1, which are regular with respect
§2. CONTINUOUS MAPPINGS ON R
87
to y. If the pair mappings
f: (G, 8G)
(R", R"\{y}) and g: (G, 8G) - (R", R"\{y})
are homotopic, then p (y, f G) = p (y, g, G).
LEMMA 2.2. Let G be a compact domain in R", and let f : G - R" be a continuous mapping. Then for every (f, G)-admissible point y there exists a sequence of mappings f,,: G -' R" of class C°°, v = 1, 2,..., such that f, converges to f uniformly on G as v -' oo, and each of the mappings f, is regular with respect to y.
Proofs of Lemmas 2.1 and 2.2 are given in Chapter III, §5. Using Lemmas 2.1 and 2.2, we define the concept of degree for continuous mappings. Let G be a compact domain in R", f : G -' R" a continuous mapping, and y an arbitrary (f, G)-admissible point.
We construct a sequence of mappings f,: G -y R" satisfying all the conditions of Lemma 2.2. For each v let p(y, f, . G) = tc,,. Since f, - f uniformly on G, there is a vo such that f, and f are homotopic as mappings of the pair (G, 8G) into the pair (R". R"\{y}) for v > vo. If v' > vo and v" > vo, then fv, and f,,, are also homotopic as mappings of (G, 8G) into (R", R"\{y}). Hence, by Lemma 2.1, µ,,, = p,, for any v' and v" larger than vo. This shows, in particular, that the limit go = limv-,, pv exists. This limit does not depend on the choice of the sequence (f,)vEN Indeed, let (gv) be any other sequence of Ck-mappings of G convergent uniformly to f. Then g, -H f, for sufficiently large v > v1, and hence p(y, f.. G) = p(y, gv , G) for all v > vl. Consequently, lim p(y, fv, G) = lim µ(y, gv, G). V-00
V-00
which is what was to be proved. The limit po = limv-°O p(y, f,, , G) is called the degree of the mapping f : G - R" at the point y, and will be denoted by p(y, f, G). For the case of mappings of class Ck this definition leads to the same
result as the original definition. Indeed, if fv : G -' R", v = 1, 2...., is a sequence of mappings of class C°° which converges uniformly to f : G
R" as v - oo, then fv and f are homotopic as mappings of (G, 00 into (R", R"\{ y}) beginning with some v = vo, and thus p(y, fv, G) =#(y, f G) for v > vo, i.e., #(y, f, G)=lim p(y.fv,G).
We make note of some properties of the degree of a mapping which follow immediately from the definition and the main Lemmas 2.1 and 2.2.
88
II. MAPPINGS WITH BOUNDED DISTORTION
I. Let G be a compact domain in R", and fo: G R", and fl : G R" continuous mappings. Assume that the point y E R" is simultaneously (fo. G) admissible and (fl, G)-admissible. If fo and f, are homotopic as mappings of the pair (G, 8G) into the pair (R", Rn\{y}), then
u(Y,fo,G) =u(Y,f,,G). PROOF. By Lemma 2.2, we construct sequences of mappings f,10): G -
R" and f,('): G - R" of class COO which are uniformly convergent to fo and fl, respectively, as v --+ oo. There is a vo such that f ,(O) is homotopic to fo and f,(1) 1) is homotopic to f, for v > vo. Then for v > vo .u (y,
(o), G)
= u(Y, fo, G),
u(y, fv(' ), G) = u(Y, fl, G).
Since fo and fI are homotopic, f ,(O) is homotopic to f,(') for v > vo, which
implies that uu(y, f'0), G) = lc (y, f"), G) for these values of v, and thus y (y, fo, G) = u (y, fl, G), which is what was to be proved. II (The additive property of the degree of a mapping). Let Go, ... , G," be compact domains such that G; c Go for all i = 1, ... , m and G; and G, have no common interior points for i $ j. Let f : Go - R" be continuous. Assume that the point y E R" is such that f (y) c ur Gi and y is (f, G1)admissible for i = 0, ... , m. Then m
u(Y, f Go) = E u(Y, f G,). i=1
PROOF. Suppose that (f,: Go R") is a sequence of mappings of class COO which are regular with respect to y and converge uniformly to f on Go. Then there is a vo such that u (y, f,, Gi) = u (y, f, Gi) for v > vo and i = 0, ... , m. Further, since f, converges uniformly to f, the set f, ' (y) is contained in u' G° for sufficiently large v, v > v, > vo. It is obvious from the definition of the degree for mappings of class Ck that in this case m
u(Y,fv, Go) = Eu(Y,f
.
Gi).
i=1
Passing to the limit as v - oo, we get the required result. III. Let f : G R" be the restriction of a nonsingular affine mapping of R". Then for every (f, G)-admissible point y E R"
u(Y, f
if y ¢ f (G), sgn(det f) if y E f (G). 0
§2. CONTINUOUS MAPPINGS ON R"
89
This proposition is an obvious consequence of the definition of degree for mappings of class Ck. IV. The degree u(y, f, G) is 0 for every mapping f : G - R" identically constant on G and for any (f G)-admissible point y. PROOF. Suppose that f - const on G, f (x) = b for all x E G. Then a point y is (f. G)-admissible if and only if y ILE b. Then f -' (y) is empty, and hence µ(y, f G) = 0, as required. V. Suppose that f : G R" is continuous and y f (G). Then µ(y, f G)
=0. The assertion is obvious from the definition of degree in the case when f E Ck. In the general case it can easily be established by passing to the limit.
VI. Let f : G - R" be a continuous mapping. Then the function y -> µ(y, f, G) of the variable y is constant on each connected component of R"\f (8G) PROOF. We first make the following simple observation. Let f : G --' R"
be continuous, y an (f G)-admissible point, and a an arbitrary vector in R". Then
p(y+a,f +a,G)=µ(y,f,G).
(2.1)
Indeed, assume that f is in C°° and is regular with respect to y. Then the mapping g = f + a is also in C°°. Further, g- I (y + a) = f -' (y), and at each point x E g-' (y + a) the Jacobian of g is equal to that off at x. Hence, g is regular with respect to y, and µ(y + a, g, G) = µ(y, f, G). In the general case (2.1) can be established by an obvious passage to the limit. Let f : G -' R" be continuous, and A an arbitrary connected component of the open set R"\ f (8G). Let yo, y1 E A. Since A is connected, yo and yi can be joined by a continuous path in A. Let y : [0,1 ] -. A be a continuous path such that y(0) = yo and y(1) = y1. Let p,(x) = f (x)- y(t), t E [0, 1]. For each t E [0, 1 ], ip, is a mapping of the pair (G, 8G) into (R", R11\{0}). Indeed, if f (x) - y(t) = 0, then x 95 8G, since y(t) V 8G for all t E [0, 1], and hence if x E 8G, then ip,(x) 0. The function Sp, is a homotopy of the mappings ipo = f - yo and 1v1 = f - y1, which implies that µ(O, f - yo, G) = p(0, f - y1, G).
We can conclude from the foregoing that µ(yo, f, G) = µ(y1, f G)
Since yo and y1 are arbitrary points of A, this proves the given statement.
11. MAPPINGS WITH BOUNDED DISTORTION
90
VII. Let f : G - R" be continuous and A the unbounded component of the set Rn \ f (OG). Then p (y, f, G) = 0 for every y E A.
Indeed, f (G) is compact, and if A is unbounded, then there is a point
y E A such that y io f (G). Then µ(y, f G) = 0. Since u(y, f, G) is continuous on A, µ(y, f G) = 0 for any y E A. §2.2. The degree of a mapping and exterior differential forms. We describe a certain analytic technique used to study the degree of a mapping. Here we assume that the reader is familiar with concepts involving exterior differential forms in Rn, in particular, concepts such as the product of exterior forms, the differential of an exterior form, transformations of forms under differentiable mappings, etc. Only exterior forms of degrees n - 1 and n will be considered. Every exterior form of degree n - 1 in a domain U in the space Rn has an expression n
w(x) = E(-1)k-'uk(x)dxI...dxk-I dxk+1...dx1
,
(2.2)
n=I
where u1, ... , un are functions defined in U. An exterior differential form of degree n is an expression like
w(x) = u(x) dxI dx2 ... dxn.
(2.3)
The differential dco(x) of the form (2.2) is the form
(!!.( x )' xk
dxI dx2 ... dxn.
(2 . 4)
k=1
Finally, if 9(x) = 91(x) dx1 + . + 9n(x) dxn is a form of the first degree, and w(x) is a form like (2.2), then their exterior product is n
9;(x)u;(x)
9(x) A w(x) _
dx1... dxn.
(2.5)
LEMMA 2.3. Suppose that co is an exterior differential form like (2.2),
where the u; are functions of class Co in an open subset U of R. Then fu dw(x) = 0. PRooF. Let us extend the functions ui to the whole of Rn by setting ui(x) = O for x ¢ U. Then
f
r
dw(x) _ /
f>a.
U
du' (x) dxI dx2... dxn dx,
i=:
Since each u; has compact support, , x1...
.
xn) dx1 = 0,
dx-r which gives us that fu dw(x) = 0, as was to be proved. (x1.
1
00
§2. CONTINUOUS MAPPINGS ON R"
91
Let a be an arbitrary point in R". Denote by 9a the exterior differential form of degree n - 1 given by
9a(x) _ nE(- 1)k - i (xk - ak)dxl ...dxk-I dxk+I ...dxn
(2.6)
Ix - aln k=l
defined in R"\{a}. Simple computations show that dOa(x) = 0. Let G be an arbitrary compact domain in R", and f : G - R" a mapping of class C°°. Suppose that the point y is (f, G)-admissible and the mapping
f is regular with respect to y. Let A = f -I (y). The set A is finite and contained in G°. The exterior differential form 9y, is defined in R"\{y}. We use it to construct a certain exterior form f'9y defined on G°\A by f'ey(x) n
k I[fk(x)-Yk]dfl A...Adfk-I Adfk+l A...Adfn (2.7)
If(x)-YI"
-k=1 k=1
Formally, f `9y is obtained from 9y if in the expression for the latter the quantity x1 is replaced by f, (x), i = 1, ... , n. By known results in the theory of exterior forms, the differential of the form f * 9y is zero. LEMMA 2.4. For everyfunction C E Co (R") equal to I in a neighborhood of A and with support contained in G°,
L\A
dC(x) A f'9y(x) = -cvnµ(y, f G).
(2.8)
PROOF. First of all we show that the integral on the left-hand side of (2.8) does not depend on the choice of the function C. Indeed, let CI and C2 be two functions of class Co (R") equal to 1 in a neighborhood of the set A = f - I (y) and with supports contained in G°. Let C = CI - C2. The function C belongs to Co (R") and has support G°\A. We have that
f
o\A
f
dCl A fey -
o\A
d(Cf'ey).
f
GD\A
d2 A .fey fN\A
11. MAPPINGS WITH BOUNDED DISTORTION
92
The coefficients of the form w = Cf'9y are functions of class Co with supports in G°\A. Hence, on the basis of Lemma 2.3 we have that f0l\A dcv = 0, which implies that dC1 A f'9y =
J
dC2 n f'By.
J
If A = f (y) is empty, then u(y, f, G) = 0. In this case the coefficients of the exterior form f'0;. are functions of class COO with supports in G°, and so by Lemma 2.3
f dAf0,
=fGd(CI''ey)=0,
i.e., the equality to be proved is true in this case. Let us now consider the case when f - I (y) is nonempty. Then f (y ) consists of finitely many points x1,... , x",, and F(x, f) 0 at each of them. There is a 6 > 0 such that the balls B(x,, b) are disjoint and f maps each of them homeomorphically into R", with the inverse mapping also in C°°. Let V1 = f [B(x1, 8)]. There is an c > 0 such that B(y, e) c V1 for all i = 1, ... , m. Take a function q E C°° such that S(q) c B(y, e) and P7(x) = I in a neighborhood of y, and let C(x) = q[f(x)]. Then C E C°°,
and C(x) = 1 in some neighborhood of the set A. The support of C is obviously contained in the union of the balls B(x1, 6). The restriction of C to B(x1, 5) is denoted by C;, and the restriction off itself to B(x1, 6) by . We have that
f
fa,\A
dCAf*0Y = r
=)
f
dC1 Af,'6y.
Let di A 9y = u(x) Al ... dx". Then, by known results in the theory of exterior forms,
dC A f'8y = f'(d?l A By) = u[f (x)]f (x, f) dx1 dx2 ... dx",
and thus
f
B(a,.a)
dCi A *0y = f
B(a,.a)
u[f (x)V (x, f) dx.
On the basis of a classical change of variables formula for multiple integrals, the last integral is equal to ai
f
B(y.e)
u(x) dx = ai
f
drl(x) A 9y(x),
B(y.e)
where o = sgn 7 (x, f ). To compute the last integral, observe that it does not depend on the choice of the function >) (this follows formally from
§2. CONTINUOUS MAPPINGS ON It"
93
what has been proved when G = B(y, s) and f is the identity mapping). We take n(x) = Sp(Ix - yl), where (p is a function of a single variable. Then n
dn(x) = co'(Ix - yl)Ix - yl -I E(xi - yi) dxi. i=I
From this,
dn(x)AO
(x)-(0'(Ix-yl)Ix-yl'-"dxldx2...dxn.
Introducing spherical coordinates with center y in R", we find that 00
d?j A O (x) = wn fo
rp'(r)
dr
fp(0)wn
con.
Finally, m
f'\A
dC(x) A f'By = -w
sgnf(x;,f) = -conp(y,J G).
i=1
The lemma is proved. The proof of Lemma 2.1 is based on the use of Lemma 2.4. We omit this proof, referring the reader to [ 139]. §2.3. Change of variables in a multiple integral. Let U be an open set
in R", and f : U -+ R" a continuous mapping. The mapping f is said to have property N if the image of every set E c U of measure zero is a set of measure zero.
THEOREM 2.1. Suppose that U is an open set in R", and f : U - R" a continuous mapping. If f has property N, then the image under f of every measurable set A c U is a measurable set.
PROOF. Let A be any measurable subset of U. Then there exists a sequence (Am)mEN of compact sets such that A D A. for each m and E = A\ U' Am is a set of measure zero. We have that A = E U U;° Am, which implies that 00
f(A) =f(E)U U f(Am).
(2.14)
m=1
Since f is continuous, each of the sets f (Am) is compact, and hence measurable. Since f has property N, f (E) is a set of measure zero and is thus measurable. By (2.14), this implies that f (A) is measurable. The theorem is proved. Let U be an open set in R. A mapping f : U -» Rm is said to be locally Lipschitz if for every compact set A c U there exists a constant LA < oo
II. MAPPINGS WITH BOUNDED DISTORTION
94
such that
If(x) -f(Y)l <- LAIX - YI for any x,y EA. Every CI-mapping f : U -' R'", where U is an open set in R", is locally Lipschitz. Indeed, let A c U be compact, and let
LA= sup If(x)-f(Y)I x,yEA
Ix - YI
It is required to prove that LA is finite. Let (xv , y,,), v = 1, 2, ... , be a sequence of pairs of points in A such that If (x') - f (Y.) I - LA as v - oo. Ixv -YvI Since A is compact, it can be assumed without loss of generality that xv -' x0 and yv - yo as v --+ oo, where xo, Yo E A. If xo 96 yo, then by passing to the limit we get that LA = If(xo) -f(Yo)I < Ixo - Yo{
Suppose that x0 = yo. Since f E C', it follows that
f (x') - f(x") = f'(xo)(x' - x") + a(x', x")Ix' - x"I , where a(x', x") - 0 as x', x" -. xo. From this, If(X-) - f(Y.)I Ixv - Yvl
< If'(xo)I + Ia(xv. Yv)I.
and we get by passing to the limit that LA < I f'(xo)I < oo, which is what was to be proved. THEOREM 2.2. Let U be an open set in R". Every locally Lipschitz map-
ping f : U - R" has property N. PROOF. Let f : U - R" be a locally Lipschitz mapping, and let A c U be a set of measure zero. It must be proved that If (A) I = 0. Let (Um), m = 1 , 2, ... , be a sequence of open sets such that Um is strictly interior to U for each m and U = Ui° Um. Since Um is compact, there exists a constant L. < oo such that If (x) - f (y) I < Lm Ix - y I for any x, y E Um.
Let A. = A n Um. Then IAmI = 0 and A = U' Am. Let e > 0 be arbitrary. For a there exists an open set V c U. such that Am c V and I V I < e. Let V = Ui° Q. be a cube subdivision of V, and rv the length of an edge of the cube Q,. Then f (Am) c U' f (Qv). The diameter of Qv is
§2. CONTINUOUS MAPPINGS ON R"
95
does not exceed Lm/r,,.
equal to f r,,, and hence the diameter of f This implies that
If(Qv)I 5 an(LmV)"r," = Cry, and thus 00
00
If (Am)I 5 >2 If (Q.)1:5 C>2 rn, < Ce.
Since e > 0 is arbitrary, it thus follows that If (Am) I = 0. It remains to note that f (A) = Ui° f (Am), which gives us that If (A)I = 0. The theorem is proved. COROLLARY. Let U c Rn be an open set. f : U -' RR has property N.
Then every CI-mapping
The following result was established in [147].
THEOREM 2.3. Let U be an open set in R", and f : U - Rn a mapping in the class W, 1,C(U), where p > n. Then f has property N. PROOF. Let f E WI10C(U), and let the set A c U be such that IAI = 0.
It is required to prove that If (A) I = 0. We construct a sequence (U.), m = 1, 2,..., of open sets such that Um is strictly interior to U for each m and U = U0° Um. Let Am = A n Um. It suffices to show that If (Am)I = 0 for each m. Let e > 0 be arbitrary, and for it choose an open set V D Am such that IV I < e and Um D V. Let V = U?° Q be a cube subdivision of
V, and r the length of an edge of Q. We have that 00
f(Am) C U f(Qv) m=1
In view of the estimate (2.15) in §2 of Chapter I, the diameter d of the
set f
does not exceed Cr,!-(n/P) (f
I f'(x)I P dx)
1/P
U.
We have that If
I < Qndn,, from which 00 Cnanr'ncl-(I/P))
00 If(Am)I 5 EIf(Qv)I <_
v=1
vv 11
(J ` Q°
If'(x)IPdx) n/P
II. MAPPINGS WITH BOUNDED DISTORTION
96
Let us apply the Holder inequality to the sum on the right-hand side, obtaining 00
If
1-(n/p)
n/p
00
(Can Erv
If`(x)I'dx
(yj
v=1
v=1
) n /P =C"QnIVI1-(n/p)
< CnQ"E1-(n1p)
(fr If'(x)IPdx ff
n/p
('
JL,.IIS)I p dx
Since e > 0 is arbitrary, this gives us that If (Am)I = 0, and the theorem is proved. In §6.2 of this chapter we consider the question of whether property N holds for mappings in the class WP 1,10C when p:5 n.
LEMMA 2.7. Let Q = Q(a, r) be a closed cube in R", and let f : Q , R" be a continuous mapping. Let Q= = Q(a, rr), where r > 0. Assume that there exists a nonsingular linear mapping L : R" -+ R" such that if x E Q and x # a, then
If (x) - f(a) - L(x - a)I < aIx - aI, where a = const > 0. Let V(x) = f(a)+L(x-a), and let e = of I L-1 I < 1. Then
G1-e = 9(QI-e) c f(Q).
GI _e n f (OQ) = O.
and for every y E GI -e
u(y, f, Q) = sgn det L.
PROOF. Take an arbitrary x E 8Q and let x' E QI _e. Let y = tp(x)
and y' = 9(x'). We have that y - y' = L(x - x'), from which x - x' _ L- I (y - y'), and hence Ix - x'I < IL-1IIy -
y'I
Obviously, Ix - x'I > er/2, which gives us that lip(x) - q,(x')l > er/2IL-II = 8%/n`r/2.
We have that y' = ip(x') E GI_,,. Since X' E QI-e was arbitrary, we get that for every x E OQ the distance from the point ip(x) to the set G1_t is at least 8vln-r/2. For every x E 8Q we have that
If(x) - q(x)) < 6Ix - al < (5/r/2.
§2. CONTINUOUS MAPPINGS ON R"
97
This implies that if x E 8Q, then f (x) ¢ GI _E, and hence GI _e n f (aQ) _
0. Let us now consider the function f (t, x) = 9(x)+t[f (x) - ip(x)], where 0<<-t< 1. For x E .9Q we have that
If(t,x) - sv(x)I <- t1f(x) - 9(x)I < 8/r/2, and thus f (t, x) qi GI _e for x E 8Q. Obviously, f (0, x) = ip(x), f (1, x) _
f (x), and f (t, x) is continuous on [0, 1 ] x Q. We get that f (t, x) is a homotopy between iO and f , as mappings of the pair (Q, aQ) into the pair (R", R"\GI_e). From this,
µ(y. f, Q) = µ(y. 9. Q) = sgn det L for all y E GI _e. In particular, µ (y, f,, Q) 96 0 for any y E GI _e. This implies that GI _e C f (Q). The lemma is proved.
LEMMA 2.8. Let f : U -i R" be a continuous mapping, and G C U a compact region. Assume that the point y E R" is such that y qt f (8G), the set f -' (y) n G is finite, and the mapping f is differentiable at each point x E f - I (y) n G, with f (x, f) 0 0. Let N+ be the number of points x E f -' (y) n G at which F (x, f) > 0, and N- the number of such points at which f (x, f) < 0. Then
.u(y,f, G)=N+-N-. PROOF. Assume all the conditions of the lemma. If the set f (y) n G is empty, then y f (G), and hence u(y, f G) = 0, so there is nothing to prove in this case. Assume that f -' (y) n G 36 0, and let xI, ... , xN be all the elements in the set f -' (y) n G. The mapping f is differentiable at each of the points x;, and thus for each i = 1, ... , N
f(x) -f(xi) -f'(xi)(x -xi) where a;(x) -- 0 as x -+ 0. The linear mappings f'(x;) are nonsingular. By Lemma 2.7, there is an ro such that
µ[y.f.Q(x;.r)] = sgnf(x;,f) for r < ro for each i = 1, ... , N. We choose r < ro such that the cubes Ql = ?7(x1, r) are contained in G° and are disjoint. Then, by property II of the degree of a mapping, N
N
a(y,f.G) = Eu(y,f Ql) = F sgnf(xi,f) i=I
=N+-N-. The lemma is proved.
i=I
II. MAPPINGS WITH BOUNDED DISTORTION
98
LEMMA 2.9. Let U be an open set in R", and f : U - R" a continuous mapping. Assume that f is differentiable at the point a E U, and let Q, _ Q(a, r). Then If(Qr)Ilr" - Idet f'(a) I
as r - 0. PROOF. Let L = f'(a), and for r > 0 and x E R" let F.r(X)=f(a+rX)-f(a)
r Then Fr (X) - L(X) as r 0, and the convergence is uniform on the cube U(0, 1). The mapping X -i a + rX transforms Q(O,1) into the cube Q,. This implies that the set F,[Q(0, 1)] is obtained from f (Q,) by a parallel translation and a similarity dilation with similarity coefficient equal to 1/r, and hence 1)]i = If (Q') I.
Let G = L[U(0,1)]. If det L 0 0, then G is an n-dimensional parallelepiped whose volume is equal to I det LI. But if det L = 0, then G lies in some hyperplane, and thus IGI = 0 = I det LI in this case. Let
6,= sup IF,(X)-L(X)I xEQ(0.1)
Then 6, - 0 as r -' 0. For each r the set Fr[U(0, 1)] is contained in the closed 8,-neighborhood Ua, (G) of G. We have that 1 Ua, (G) I - IGI as
r IFr[Q(0.1)11 <- IV i, (G)I.
which implies that lfm If(Qt)I = lim IF,[Q(0,1)]I < IGI =IdetLI. r-.0 r-0 r"
(2.15)
In particular, this gives us the assertion of the lemma for the case when
det f' (a) = 0. We assume below that det fl (a) 96 0. Let q(x) = f (a) + L(x - a), where L = f'(a) as above. Since f is differentiable at the point a, it follows that If (x) - rc(x)I <_ a(x)Ix - aI, where a(x) - 0 as x - a.
Let 0 < e < 1, and let ro > 0 be such that Ia(x)I < e/v'iIL-1I for all x E Q,o. By Lemma 2.7, if r < r0, then f (Q,) D F[Q(1and hence
If (Qr)I >- (1- e)"IQ,II det LI. From this, lim If(Qr)I T_0
r
> (1-e)"IdetLI.
§2. CONTINUOUS MAPPINGS ON R"
99
Since e > 0 was arbitrary, this implies that lim I f (Q')1 > I det LI. .-o
(2.16)
r
The lemma is proved in view of (2.15). Let U be an open set in R", f : U - R" a continuous mapping, and A a subset of U. Assume that v : U R" is a given nonnegative function. For any point y E R" denote by NA(y, f, v) the sum of the values of v at the points of f -I (x) n A (values NA(y, f oo) = oo are allowed). In the case when A = U the index A is omitted in the notation: N(y, f v) f v). Further, we simply write NA(y, f) in place of NA(y, f, v) when v =_ 1. The quantity NA(y, f) is equal to the number of elements in the
set f-1(y) n A. THEOREM 2.2 [116]. Let U be an open set in R", and f : U - R" a continuous mapping. Assume that f has property N and is differentiable almost everywhere in U, with the function x '- F(x, f) locally integrable on U. Then for every nonnegative measurable function v : U -' R the function y - N(y, f, v) is measurable on R", and
f
N(Y, f v) dy =
RFurther,
Jv(x)If(x.f)Idx.
(2.17)
if G c U is a compact region whose boundary has measure zero, then, for every nonnegative measurable function u: R" R such that the function y NG(y, f)u(y) is integrable, the function
x '- u[f (x)lf(x, f ) is integrable over G. and
fu[f(x)]Lf(x.f)Idx = j u(Y)NG(Y. f) dy.
(2.18)
"
fu[f(x)Jf(x.f)dx = fu(Y),u(Y,f G)dy.
(2.19)
R
PROOF. Suppose that the mapping f : U - R" satisfies the conditions of the theorem. We first consider the case when U is bounded, the function
x'- F(x, f) is integrable, and v(x) = 1, so that N(y, f v) - N(y, f ). Let E' be the set of all points x E U such that f is not differentiable at x, and let E" be the collection of points x E U which are not Lebesgue points off (x, f ). By the conditions of the theorem IE'I = 0 and IE"I = 0. Let E0 = E' U E" and U0 = U\Eo. Obviously, I U0I = I U1.
Let m be an arbitrary positive integer. Let X E U0. Then x ¢ E', and hence f is differentiable at x. On the basis of Lemma 2.9, this implies
11. MAPPINGS WITH BOUNDED DISTORTION
100
that
If (Q(x, r)Il r" -i V (x, f )I as r - 0. In the given case it is also true that x io E", and thus x is a Lebesgue point of F(x, f ), i.e., I
r n Q(x.r) IV as r -, 0. Let rm(x) > 0 be such that rm(x) < 1/m and such that if 0 < r < rm(x), then U(x, r) C U and
If(Q(x.r)I n
rn jQ(x.r)
-If(x.f)I
< 2m'
1
If(t.f) -f(x,f)I dt < 2m
.
(2.20) (2.21)
The set Q,,, of cubes U(x, r) such that x E U0 and 0 < r < rm(x) covers U0 in the Vitali sense. Hence, by Vitali's theorem, there exists a ser,), v = 1, 2, ... , of disjoint cubes in Q. such that quence Qm = I Uo\ U,, Qr" I = 0. Let U,, Qm = Vm. Obviously, U D Vm, and hence Em = U\Vm is a set of measure zero. We carry out this construction for each m = 1, 2, ... , and let E = LJo Em. Then IEI = 0, and then A = f (E) is also a set of measure zero, because f has property N. We define certain functions Fm and Ym for each m = 1, 2..... Let fm(x) = 0 for x E Em, and let /m(x) = If (QY )I /rY if x E Qm,. Take X to be the indicator function of the set f (Qm,), and define .Nm(y) X,, (Y) For each m we obviously have that
fiVm(Y)dY
I f (Q=
f
u
/ (x) dx.
(2.22)
It follows from the definition of the functions fm and from (2.19) and (2.20) that for each v
)I -f(x.,f)II dx f Q"'Vm (x) - I)(x.f)IIdx < fo.". If(Q rv +
If(x1,.f) -f(x.f)I dx <
1 mml.
I'
Summing over v, we get that
JLf(x)_Lf(x.f)IIdx IUI. and hence
Jfmxc1x -' J f(x.f)Idx asm -+oo.
(2.23)
§2. CONTINUOUS MAPPINGS ON R"
101
Let y ¢ f (E) be arbitrary. The quantity X. (y) is clearly equal to the number of cubes Qv' containing points of the set f' (y), and thus ,, (y)
< Mu (y, f ), since these cubes are disjoint. We prove that ...4 (y) -,
.u (y, f) as m - oo. This follows from the preceding inequality in the case when Mu(y, f) = 0. Suppose that XU (y, f) > 0. We specify a K > 0 by setting K = .Nu (y, f) when the latter is finite and taking K arbitrarily otherwise. The set f - I (y) contains K distinct elements x1,... , xK. Let a > 0 be the smallest of the distances I x; - xj 1, i, j = 1,... , K. Take mo such that -,In-/m < 6 for m > mo. Since y ¢ f (Em) c A, it follows that f - I (x) is contained in Vm. The diameter of each of the cubes Qm' is at most f /m < 6, and hence none of them can contain more than one of the chosen points x1,... , xK. Consequently, the cubes Qm containing points from x1, ... , xK are disjoint, which implies that 4,m (y) > K. Accordingly, .'Vm(y) > K for each m > mo, and it is thus proved that Am(y) - V,(Y, f) for the given y. Therefore, we get that lrm (y) 5 Mu (Y, f) and lrm (Y) - Mu (Y, f) as m -' oo for all y not in the set A of measure zero. This implies that JJVm(Y)dY
(2.23)'
--j4tu(.f)d.v
"
as m -' oo. It follows from (2.21), (2.22), (2.23), and (2.23)' that
I/(x,f)Idx=(2.24) The first assertion of the theorem is thus proved, at least under the assumption that U is bounded, the function IY(x, f )I is integrable, and v(x) =- 1. We now prove that under the same assumption on U and J(x, f) the equality (2.17) is true also when v is an arbitrary measurable function. It follows from what has been proved that if U is bounded and If(x, f )I is integrable, then .'Vt(y, f) is finite for almost all y E R".
Let (Am), m = 1, 2,..., be an arbitrary sequence of subsets of U such that Am D Am+1 for each m, and let A= n' Am. We show that YA. (y, f) -+ A (y, f) as m - oo for every y such that ..Vu (y, f) < oo. Indeed, take such a y. If Nu (y, f) = 0, then it is clear that .NE (y, f) = 0
also for every E c U, and there is nothing to prove in this case. Let .ilru(Y, f) > 0 and let f -I(y) = {x1, ... , x.4, where .4' = Xu(Y, f). Suppose that x1, ... , xk are all the elements in f -' (y)nA. If i > k, then x, ¢ A, and hence there is an index m; such that x, V Am for m > mi. Consequently, for m > maxi=k+1..... N m; the number of elements in f - I (y) n Am
11. MAPPINGS WITH BOUNDED DISTORTION
102
is equal to k = ..4 (y, f ). This proves that .NA, (y, f) -' YA(y, f) as m
00.
Let E c U be a measurable set strictly interior to U. Then there is a sequence of open sets Vm D E such that Vm is strictly interior to U and oo. Let E0 = nm 1 V. Vm J Vm+1 for each m, and IVmI IEI as m Then E0 D E and IEo\EI = 0. Each of the sets V. is bounded, and the function F(x, f) is integrable over Vm. Hence,
f
If (x,
f)I dx = fM'm(Y.f)1YVm ^
by what was proved. The sequence (Nvm (y, f )), m = 1, 2, ... , of functions is decreasing, and each of them is finite almost everywhere. Passing to the limit, we get that
L.
If(x,f)Idx = JJVEQ(Y.f)dY.
It is obvious that YE(y, f) = ..40(y, f) for every y ¢ f (Eo\E). Since f has property 1V, it follows that If (Eo\E)I = 0, and hence XA(y, f) _ .NE (y, f) almost everywhere. As a result we get that
L If(x, f)I dx =
j
(2.25)
A'E(Y,f)dy.
A function v : U - R is called a step function if v is a linear combination of the indicator functions of finitely many measurable sets contained strictly interior to U. It follows from what was proved that (2.17) holds if v is a step function. For every nonnegative measurable function v there exists an increasing sequence of step functions such that vm(x) - v(x) for all x. We have that v)
VU(Y.f um)
as m - oo for all y E R". Further, the sequence (y
' - YU (Y' .f vm) ),
m = 1, 2,..., of functions is increasing, which implies that
fJVU(Yf.Vm)dYjJIPu(Yf.v)dY "
^
For each m
fu v(x)I F(x Passing to the limit in this equality as m - oo, we get (2.17). Assume now that G C U is a given closed region whose boundary has measure zero. Since f has f(8G) is a set of measure zero. Assume first that u : R" - R is a continuous function. Then the
§3. CONFORMAL CAPACITY
103
function v (x) = u[f (x) ] is continuous, and hence bounded on G. Since
G is compact, F(x, f) is integrable on G, and thus so is the function u[f (x)]f(x, f ). By what was proved,
jIf(x.f)Idx = jA'G(Y.f)dY; hence the function MG(y, f) of the variable y is integrable, and, in par-
ticular, it is finite for almost all y E R. The function v(x) = u(f(x)) takes the same value u(y) at all points of the set f -' (u); consequently, YG(y. f, v) = YG(Y, f)u(y) for every y with MG(y, f) < oo. We thus get that (2.18) holds when u: R" -' R is a continuous function. The general case is derived from this by a standard passage to the limit. Thus (2.18) is proved.
Choosing the v in (2.17) to be the indicator function of the set S of points x E U at which f is differentiable and F(x, f) = 0, we get that
fArs(Y.f)dy = 0, which implies that f (S) is a set of measure zero. Let G C U t e a compact region such that IBGI = 0, and let u: R" - R be a measurable function such that the function f) is integrable. Suppose that A, = f(OG), A2 = f (S n G), A3 is the set of y such that YG(y, f) = oo, and, finally, A4 is the set of y such that u(y) is not defined or is equal to oo. The measure of each of the sets A; is equal to zero. Let A = AI U A2 U A3 U A4, and let
v(x) = u[f(x)]sgnf(x,f). Then the function v(x)I f (x, f ) I is integrable, and
fv(x)lf(xf)Idx = fM(Y.f.v)d.v. By Lemma 2.8, for the given choice of v and for y ¢ A we have that G(Y, f v) = u(Y)u(Y, f, G). i.e., ./YG(y, f. v) = u(y)u(y, f, G) almost everywhere. This establishes the equality (2.19). The theorem is proved. §3. Conformal capacity
§3.1. The capacity of a capacitor. A capacitor in R" is defined to be any ordered pair H = (A0, A,) of disjoint nonempty closed sets of which one is compact. Let a capacitor H = (Ao, A,) be given. The open set U = R"\(Ao U A,) is called the field of the capacitor H. Denote by i3'p(H) = YPp(A0, A1), where p > 1, the collection of all functions v : R" - R of
104
11. MAPPINGS WITH BOUNDED DISTORTION
class C°O(R") for which there exist open sets Uo and U1 such that A0 C U0,
AIcU1iv(x)=0for xEUo,v(x)=lfor xEU1,0
f
IVv(x)IV dx
(3.1)
is finite. The infimum of the integral in (3.1) on the set Wp(Ao, AI) is called
the p-capacity of the capacitor H = (A0, AI) and denoted by Cp(H) _ Cp(Ao, A1).
The concept of the capacity of a ring is a special case of the concept of the capacity of a capacitor. A ring in R" is defined to be an open set U whose complement R"\U consists of two connected components A0 and AI, of which at least one (say A0 for definiteness) is compact. Obviously, the pair (A0, AI) is a capacitor whose field is the set U. The set 8U n Ao is then called the inner boundary and OU n AI the outer boundary of the ring U. The capacity of the capacitor (A0, AI) is called the capacity of the ring U and denoted by Cap p U. In the case when p = n the p-capacity is called the conformal capacity. The concept of conformal capacity in R" was introduced by L, wner [89], who also indicated a number of applications of the concept of conformal capacity to the theory of quasiconformal mappings. In particular, he used the concept of conformal capacity to prove that there is no quasiconformal mapping of R" onto a subset of R" whose boundary contains more than one point. Further progress in the investigation of the concept of the capacity of a ring has been made by Gehring, Vaisala, and others (see [174] and [34-37]). Closely connected with the concept of conformal capacity is the concept of the modulus of a family of curves or surfaces, but we shall not dwell on this. Next, we need the concept of the capacity of a capacitor with respect to an open set U. Let (A0, A1) be a capacitor and U an open set such that A0 and A 1 are contained in the closure of U. If this condition is satisfied,
then we say that the capacitor (Ao, AI) and the open set U are joined. We denote by T$'p(A0,A1,U) the collection of all functions u E C°°(U) such that there exist open sets Go D A0 and G1 D AI with u(x) = 0
for x E Go n U, u(x) = 1 for x E G1 n U, and 0 < u(x) < 1 for all x E U. The infimum of the integral fu Iu'(x)I' dx is called the p-capacity of the capacitor (Ao, AI) with respect to U, and is denoted by C,(Ao, Al. U). Obviously, Cp(Ao, AI) = Cp(Ao, AI , R"). If the function u(x) belongs to
the class W p(Ao, A,), then for every open set U joined with (Ao, AI) the
§3. CONFORMAL CAPACITY
105
restriction of u(x) to U belongs to WP(A0, A1, U). This implies that CP(Ao,A1, U) < fu1 u'(x) P dx <
f Iu'(x)IPdx,
and the arbitrariness of u E WP(A0, AI) gives us that CP(Ao,A,, U) < CP(Ao,A1).
Suppose that the capacitor (Ao, A1) and the set U are joined. If U E WP(Ao, A1, U), then v = 1 - u E WP(A1, Ao, U), and, since IVu(x)I = IVv(x)I for all x E U, this implies that CP(Ao,A1, U) = CP(A1,Ao, U). Suppose that a connected open set U c R" and a capacitor (Ao, A1) joined with U are given. The collection of all functions u E C°°(U) vanishing in a neighborhood of A0 and satisfying I/P
IIu111.P.u =
I
u'(x)IP dx)
< 00
(fu
is a vector space, and the functional u IIulI1.P.u is a norm in this space. The closure of the set i 'P(Ao, A1, U) of functions with respect to the norm IIuII1.p,u is denoted by WP(Ao, AI , U). According to this definition, W,,(A0, A1, U) is the collection of all functions u on U for which there exist sequences (um) of functions in i4'P(A0, A1, U) converning to u in L1.1a(U) and such that II uk - um 111.P.u 0 as k, m -' oo.
These definitions extend in a natural way to the case of the space 1". A capacitor in R is defined to be any pair (Ao, AI) of disjoint closed sets. We say that a capacitor H = (Ao, A1) and an open set U in R are joined if A0 u A I is contained in U. For an arbitrary capacitor (Ao, AI) in r let C"(Ao, A1, U) = C"(Ao\{oo}, AI\{oo}, U\{oo}).
We compute the p-capacity of a certain special capacitor in R" when p > 1. Let a and b be such that 0 < a < b. Denote by H(a, b) the capacitor (Ao, AI ), where A0 = R" \B(0, b) and AI = B(0, a). Let be an arbitrary function in WP[H(a, b)]. We specify in R" a spherical coordinate system with origin at 0. Let e be an arbitrary unit vector in R". We have that 1 = c(ae) - 9(be)
f
b
e > dr <
< ip'(re),
f
b
Igo'(re)I dr.
a
a
H61der's inequality gives us that <
f
a
b
b
I qP'(re)I Pr"-' dr
(f r-(R-1)/cP-I) dr a
(3.2)
II. MAPPINGS WITH BOUNDED DISTORTION
106
From this, in the case p = n b
f Ic'(re)Ir"1 dr > Integrating with respect to e, we find that Ja<Jxl
I q'(x)I" dx > Wnl
[ln(b/a)]"-1
which implies that
C"[H(a, b)]
- m"l [l
n(b/a)]"-1.
(3.3)
In the case p # n we arrive at the following estimate, in which y = (p - n)l (p - 1):
Cp[H(a. b)] ? wn((b'' -a)/y).
(3.4)
We show that equality holds in each of the relations (3.3) and (3.4). To do this it suffices to construct a function f of class Wp(Ao. AI) such that the integral (3.5)
f Ic'(x)IP dx
is equal to the right-hand side of (3.3) in the case p = n and the right-hand side of (3.4) in the case p 96 n. The required function cp will be found
in the form V(x) = yr(IxI), where yr is a function of a single variable, from the condition that equality holds in (3.2). This leads to the following differential equation for Vi: [yi'(r)]Prp-I = Cr-(n-')AP-I ), where C is a constant. From this,
y/(r) =
C1rcp-"ll(p-1)
+ C2
in the case p 0 n, while W(r) = CI ln(1 /r) + C2
in the case p = n. The constants C1 and C2 are chosen so that p(a) = 1 and V(b) = 0. It is easy to verify that the function 9p(x) = w(IxI) belongs to the class Wp[H(a, b)], and for it the integral (3.5) really is equal to the right-hand side of (3.4) in the case p 0 n and the right-hand side of (3.3) in the case p = n. Thus, Cp[H(a, b)] = w [(bl - ay)l for p 96 n, where y = (p - n)/(p - 1), and C"[H(a, b)] = cv"/[ln(b/a)J"-I. Next, we use a certain special sequence of functions on R". Let 0m(x) = y]p-1
1 for IxI < m+1, rym(x) = 2-(lnx)/(ln(m+l)) form+1 < IxI < (m+1)2,
§3. CONFORMAL CAPACITY
107
0 for Ixi > (m + 1)2. Let Ipm(x) be the mean function for and 0 m(x) corresponding to the value h = 1 of the averaging parameter. Then 9m(x) = 1 for IxI < m, 60m(x) = 0 for IxI > (m + 1)2 + I (hence, Spm has compact support), 0:5 fpm(x) < I for all x, and
f I'm(x)I" dx < f I(x)Idx = w/[ln(m + 1)]"
-
0
asm --, 00.
THEOREM 3.1. Let U c R" be an open set, and (A0, A1) a capacitor joined to U. Then
Cn[f(Ao),f(A1),f(U)] = Cn(A0,A1, U) for every Mobius transformation f. PROOF. Let U E W" (A0, A 1, U) be an arbitrary function, and let v(x) _
u[f -'(x)], Bo = f (Ao), B1 = f (A1) and V = f (U). The function v vanishes in a neighborhood of B0 and takes the value 1 in a neighborhood of B1. However, v can fail to belong to the class COO (V) here. This
happens in the case when V contains the point z0 such that f(oo) = zo. To avoid the difficulty arising in this connection we proceed as follows. It will be assumed that oo 0 A 1. Since Cl, (A0, A 1, U) = C" (A1, Ao, U), this clearly involves no loss of generality. The set A 1 is compact. Suppose that the integer k is such that Al c B(0, k). Let um = u4)m+k, where
((pm), m = 1,2,3.... is the sequence constructed above. Obviously, U. E JP" (Ao, A 1, U), and the function um is compactly supported. The latter property of um implies that the function vm = um o f -' belongs to the class C°° (V), and it is zero in a neighborhood of z0. Consequently, Cn(B0,B1, V) < fV iv;n(y)I" dy
fu
Ivn[f(x)]I"If(x.f)I dx
_ Iv (um(x)I" dx.
Here we have used the fact that LF(x, p I = I f' (x) I", and hence
Ivm[f(x)]I"If(x,f)I = IV, If(x)]f"(x)I" = It is easy to show that
f I um(x)I" dx -i £
lu'(x)I" dx as m - oo.
From this, Cn(Bo, B1, V) <
JU
I u'(x) I" dx.
Ium(x)In.
If. MAPPINGS WITH BOUNDED DISTORTION
108
and since u E rn(A0,A1, U) is arbitrary, it is thereby proved that Cm (BO, B1, V) -< Cn(A0.AI, U).
Replacing f by f I in the arguments and interchanging the triples (Ao, A1, U) and (B0, B1, V), we get that
Cn(A0,A1. U) < Cn(Bo,B1, V),
and the theorem is proved. The estimate of the conformal capacity in the next lemma will be needed later. LEMMA 3.1. Let U be an open subset of R", and A0 and A 1 closed sets, constructed as follows:
1) U is the domain in R" consisting of all the points x E R" such that 0 < a < lx-xal < b and the vector x-x0 forms an angle less than 0(lx-xol) with a certain unit vector e0 E R", where 0 < 0(r) < 00 < X. 2) A0 is the segment formed by the points x, = xo + re0, where a < r < b. 3) The set AI is contained in 8U, and the sphere S(xo. r) contains points of AI for each r E (a, b). Then Cn(A0, A1, U) > y"[ln(b/a)]/Bo, where yn > 0 is a constant.
PROOF. Let E, be the spherical segment which is the intersection of U with S(xo, r). Let v be an arbitrary function of class l3(A0, A1, U).
The function v is equal to 0 in a neighborhood of A 1, and to I in a neighborhood of A I. We have that
f Iv'(x)I" dx = J(J a
U
lv'(X)l" dax) dr,
E
where dax is the area element of the sphere S(a, r). We get a lower estimate
of the integral f , lv'(x)ln dx for each r E (a, b). Let , be a point of AI in S(xo, r). Obviously, 4, lies on the boundary of E,. Let x(t), p < t < q, be and an arbitrary smooth curve on S(xo, r) such that x(p) = x, x(q)
X(t)EE, for p < t
f
v
/ (v'[x(t)],x'(t)) dt q
Iv'[x(t)lllx'(t)I dt >
v
p
=
I d (v[x(t)])dt = v[x(q)] - v[x(p)] = 1. p
We introduce a certain coordinate system on S(xo, r). Namely, let a plane P0 orthogonal to the radius going to the point , be drawn through the point x0. A Cartesian system with origin at x0 is introduced in P0. Let a be the projection mapping of S(xo, ro) onto P0 with projection from the
§3. CONFORMAL CAPACITY
109
point r, and let r be the inverse mapping. We take the coordinates of the point a(x) E P0 as the coordinates of an arbitrary point x E S(xo, r), x # ,. The line element of the sphere in this stereographic coordinate system will have the form ds2
= [A(t)]2(dt? +
+ dt,2,-1),
where .(t) = 2r2/(r2 + 1(12). We regard x, as having the coordinates (a, 0, ... , 0), where a > 0. This can be made the case by a suitable rotation of the coordinate axes in the plane P0. It is not hard to see that a = r cot(0/2), where 0 = 0(r), and the segment E, is defined by the inequality t, > r cot 0. We now consider on S(xo, r) the family of circular arcs lying in Er and joining x, and cr,. Let ao = a(x,). Every circle in this family has a parametrization x(p) = r[ao + pel, 0 < p < oo, where e is a unit vector in P0 directed into the half-plane t1 > 0. Let h(t) = Iv'[T(t)]I. Then, using (3.6), we get that 00 1
<
J0
lv'[x(p)]lIx'(p)I dp
=1°° h(ao + pe)A(ao + pe) dp. 0
From this, by Holder's inequality, 1<
(f°°#zao + pe)]n[A(ao +
pe)]n-1 pn-2 dp
(3.7)
n-1
X (fOO[A.(ao +
pe)]II(n-1)p(1/(n-1))-1 dp)
Further, A(ao + pe) = 2r2/(r2 + lao + pelt) < 2r2/(r2 + IaoI2 + p2) = 2r2 sin2(0/2)/[r2 + p2 sin 2 (0/2)].
This gives us that the second integral on the right-hand side in (3.7) does not exceed a
1°O (_2r2sin2(0/2)
p0-1 dp,
r2 + p2 sin2(0/2)
Jo
where a = 1/(n - 1). Performing the change of variable p = ur/sin(0/2) in this integral, we find after obvious transformations that the integral is equal to [ynr sin(0/2)]a, where yn > 0 is a constant. The foregoing gives us
Ynr sin[B(r)/2]
<
°O
0
[h(ao + pe)]n[ .(ao +
pe)Jn-1 pn-2
dP
II. MAPPINGS WITH BOUNDED DISTORTION
110
We integrate both sides of the last inequality with respect to the variable e over the unit (n - 2)-dimensional hemisphere. On the right-hand side we get the integral [h(t)]"[A(t)]n-I dt,
fi >r cot(8/2)
which is equal to the integral of Iv'(x)I" over some segment contained in E,. From this,
JE,
Iv'(x)I" drX > W"/y"r sin[B(r)/2],
and this proves the lemma. §3.2. Sets of zero capacity. Let A be an arbitrary compact subset of R". We say that A is a set of zero p-capacity, where I < p < n, if there exists a closed set B disjoint from A such that the set R"\B is bounded and C. (A, B) = 0. We investigate the properties of sets of zero p-capacity. LEMMA 3.2. For every compactly supported function u(x) of class Coo
on R" and allxER" fx"
u(x)
w"
(Vu(Y),
x - y) Ix dy
.
YI"
PROOF. We fix a point x and a unit vector e in R. Since u(x) is compactly supported,
0
u(x) = - f dt u(x + te) dt = - f
00
(Vu(x + te), e) dt.
The required formula is obtained after obvious transformations by integrating both sides of this equality over the sphere in R" and then setting
x+ re = y. LEMMA 3.3. Let u be a function of class C0 (R") with support in the ball B(0, r). Then IIuIIp < 2rIIuII I.p
PROOF. Using the integral representation of the previous lemma, we get
1f
< Iu(x)I - Wn
yJ
IVu(Y)I
Ix - YI"
dy.
I
By Holder's inequality, this gives us the required estimate. to simplify the expression let 1
IVu(Y)I = v(Y).
W"Ix - yIn-I = K(x, Y). B(0, r) = U.
§3. CONFORMAL CAPACrrY
We have
f K(x, y)v(y) dy = f[K(x. Y)]'IDv(Y)[K(x.
(f
K(x, Y)v(Y) ]D
l
I/D
dy)
y) I/P 111
(JK(xY)dY)
<_
I
P -I
f!u(x)Vdx
<_
L
(fK(xY)dx)[v(Y)P]dY (JK(x.YdY). (3.8)
For every xEU=B(O,r)
fUK(x'Y)dY =
1
wn
dy
fu Ix -
YI"-I
<
1f wn
B(0.2r)
dz IzI"-I'
The last integral is easy to compute; it is equal to 2r, and thus fu K(x. y) dy
< 2r. Similarly, fu K(x, y) dx _< 2r for every y E U. Substituting these estimates into (3.8), we get that fu l u(x)ID dx < (2r)P fu Iv(Y)ID dy,
and the lemma is proved. LEMMA 3.4. Let A be a compact subset of R" of p-capacity zero. Then CD(B, A) = 0 for every nonempty closed set B disjoint from A.
PROOF. By definition, there is a closed set B0 c R" such that R"\Bo is bounded and contains A, and CD (Bo, A) = 0. Take an arbitrary function U E WD(Bo, A). Let I E WD(B, A). Then it is clear that v = uq E WD(B, A).
We have that IVv(x)I <_ q(x)I Vu(x)I + u(x)I vq(x)I <_ Ivu(x)I + Mu(x),
where M = sup, Ivq(x)I < oo. From this, IIVIIIL, <_ IIvuIIL, + MII uII L,
Suppose that r > 0 is such that R"\Bo is contained in B(0, r).
By
the preceding lemma, IIuIIL, <_ 2rIIVuIIL,, which gives us that IIVvIIL, <
(1 + 2rM)IIVuIIL,. This obviously implies that CD(B, A) = 0, since the supremum of the norm IIvulkL, on the set WD(Bo, A) is zero by assumption;
the lemma is proved.
11. MAPPINGS WITH BOUNDED DISTORTION
112
THEOREM 3.2. Let A c R" be a compact set, and U D A a bounded open set in R. The p-capacity of A is zero if and only if there exists a function
vELP(R") such that v(y) = Ofor y 0 U and v(y)
f
dy = 00
Ix - Y I "-'
)RR
forallxEA. PROOF. Necessity. Suppose that A and U satisfy the condition of the theorem. Assume that CP(A) = 0. Let A0 = R"\U. Then Cp(Ao, A1) = 0 by Lemma 3.4. For every m E N we construct a function um of class JPP(Ao, A,) such that Ilumll l.p < co"/2m. By Lemma 3.2, um(x) = 1 (V um(Y). x - y) dy,
f
wn
R
Ix - YI"
which leads to the conclusion that yn(Y)
um(x)I < )R-
Ix-yin-1
dy
for all x E R", where vm(y) = lVum(Y)I/wn. We have that ilvmiip < l/2"' Let v = v1+v2+.... Obviously, V E L p(R"), Ilvllp <- IIv1 IIp+IIv2llp+- - -:5 1,
v is nonnegative, and its support lies in B(0, r). For every x E A and for any m /' Ri,
v(Y)dy > Ix
-
u=1
yin-1
vk(Y)dy >uk(x) >- m. R. Ix - yln-1 - n-,
Since m is arbitrary, this means that v(Y) )R.
Ix - YI"-'
dy=o0
for all x E A. The necessity is proved. Sufficiency. Let A C R" be a compact set, and U D A a bounded open subset of R. Assume that there exists a nonnegative function v E LP(R") such that
u(x) =
f
v(Y)Ix -
YI'-n dy
= 00
R^
for all x E A, and v(y) = 0 for y io U. Let D be the 1-neighborhood of U. It is not hard to see that u is bounded above on R"\D, namely, "(x) <_ )R4 v(y)dy <- IUI'-'1PIIIvIlp < oo.
Let L be the supremum of u on R"\D, and let k > 0 be an integer. Denote by Gk the set of all x E R" such that u(x) > k + L. Obviously,
§3. CONFORMAL CAPACITY
113
A C Gk C D, and Gk is open. Indeed, let x0 be an arbitrary point in Gk, and assume that x0 is not an interior point. Then there exists a sequence (xm), m = 1, 2, ... , such that xm -+ xo as m - oo, and u(xm) < k + L for all m. We have that
v(Y)lxm - YI'-" -p v(Y)lxo - YI'-" as m -p 00 for all y such that the expression on the right-hand side is defined, i.e., for almost all y, and thus, by Fatou's lemma, u(xo) =
JRn
v(Y)Ixo -
YI'-" dy < mi fR"
v(Y)(x," -
Y)1-" dY
= lim u(xm) < k + L. m-0o This contradicts the fact that u(xo) > k + L by assumption. Let Jk > 0 be the smallest of the distances from the points of A to the boundary of Gk+1. Let h < 6k/2 and h > 0. We consider the averaging uh of u with averaging parameter h. Obviously, uh (x) > k + L + I everywhere in an h-neighborhood of A. Then
Uh(X) = f w(z)u(x + hz) dz "
=
f" CO(Z) (fR. v(y)lx + hz -
yl'-" dy i dz
-
yI"-" dy) dz
+ hz) Ix = fR w(z) U11 v(Y
= f vh(Y)Ix -
YI'-" dy.
"
Further, IIvh 11 p <_ IIv 11,p. By a theorem of Calderon and Zygmund (see [23
and 99]), IIDuh(x)Iip <- CIIvhIIP <_ CIIvIlp, where C is a constant.
We now revise Uh with the object of getting a function equal to zero outside D and equal to k in a neighborhood of A. To do this let Zo : R - R be a function of class Coo such that 0 < CC(x) < 1 for all x, [o(x) = 0 for x < L, and Co(x) = k for x > L + k + 1. Such a function can clearly be obtained by smoothing the function Z with {(x) = 0 for x < L + 1/2,
Z(x)=x-L-1/2forL+1/2<x
x >- L + k + 1/2. Let wk(x) = lo[uh(x)). Then Wk E C0 00(R11), and the support of wk is contained in D. For X E Gk+ I we have that wk(x) > k. Further, II V wk II p <_ IIV uh ll p <_ C IIv Il p. The function (1 /k)wk belongs to the class tP(R"\D, A), and II (1 /k)wk it 1,,p <- M/k, where M is a constant.
Since k is arbitrary here, this implies that CC(A) = 0. The theorem is proved.
It. MAPPINGS WITH BOUNDED DISTORTION
114
§3.3. The concept of a Hausdorf measure. Cartan's lemma. We introduce some characteristics of massiveness for subsets of R". Let h be a real function defined on [0, oo) and satisfying the following conditions: 1) h is a nondecreasing continuous function on [0, oo).
2)h(0)=0, and h(r) > 0 for r > 0.
3)h(r)-+ooasr-+oo. This function h is called a gauge function. Let E c R" bean arbitrary set, and let B, = B(x,, r,), B2 = B(x2, r2). ... . The sequence (B,) of balls is said to cover the set E if E c U0° B,. If the radius r of each ball B is < e, where e > 0, then the sequence (B,,) is said
to form an e-covering of E. For each sequence of balls (B = B(xr v = 1, 2, ... , covering E we form the sum
The infimum of this sum over the set of all sequences of balls covering E is called the h-content of E and denoted by yh(E). It should be noted that h-content is a comparatively coarse characteristic
of sets. Therefore, we introduce another characteristic. Let e > 0. The infimum of the sum (s) on the set of all sequences (B,), v = 1.2, ... , forming an e-covering of E is denoted by mh (E, e). It is not hard to see that mh(E, e) is a nonincreasing function of the variable e > 0, and mh(E,e) yh(E) as a - oo. The limit lim,_.omh(E,e) = mh(E) is called the Hausdorfh-measure of E. In the case when h(r) = r°, where a > 0, the quantities yh(E) and mh(E) are denoted by y°(E) and m°(E), respectively.' It is clear from the definition that the h-content and the Hausdorff h-measure always satisfy the inequality (3.9) mh(E) > yh(E). In one important special case the equality sign can be substituted here. Namely, we have the following statement.
LEMMA 3.5. The Hausdorfh-measure of a set E is equal to zero if and only if its h-content is equal to zero. PROOF. The necessity follows at once from (3.9).
Sufciency. Assume that yh(A) = 0. Let v be an arbitrary positive integer, and let p be the largest number r such that h(r) = 1/v. Let B(a, r,), B(a, r2).... be an arbitrary sequence of balls covering E and such Editor's note. For this special choice of h, Hausdorff h-measure is called a-dimensional Hausdorff measure.
§3. CONFORMAL CAPACITY
115
that h(rl) + h(r2) + - < I /v. Then h(rk) < 1/v for all k, and hence rk < p,,, so that the given sequence of balls forms an e-covering of A for e = p,,. This implies that mh(A, Ek h(rk) < I/v. As v - oo we -
have that p - 0; thus, mh(A) = limmh(A, e) = lien mh(A.
0.
The lemma is proved.
LEMMA 3.6. For every integer n > 1 there exists a constant C" < 00 having the property that if u, , ... , um.... is an arbitrary system of nonzero vectors in R" such that the angle formed by any two of them is at least x/3, then the number of vectors in the system does not exceed C".
The proof reduces to the following. On the unit sphere in R" let there be given points A1,... , Am, ... such that the distance between any two of them, measured on the sphere, is at least a/3. The geodesic balls of radius n/3 about A1,... , Am, ... on the sphere are disjoint. Let z" be the area ((n - 1)-dimensional) of such a ball. Obviously, the number of points A,. A2, ... does not exceed the ratio w"/t". The desired constant is C" = w"/r". Estimates concerning the connection between capacity and Hausdorff measures are essentially based on a certain general lemma given by H. Cartan in the case n = 2. The following statement is an immediate extension of Cartan's lemma to the case n > 2. Suppose that p is a measure, i.e., a nonnegative completely additive function on the a-algebra of Borel subsets of R. We define µ(x, r) = p[B(x, r)]. LEMMA 3.7 (Cartan's lemma). Let u be an arbitrary measure on R" such that p(R11) < oo, and let h : [0, oo) -. R be a gauge function. Let A C R be the set of x E R" such that µ(x, r) < h(r) for all r > 0. Then the h-content of R"\A does not exceed C"p(R"), where C" is the constant in Lemma 1.
PROOF. The following arguments amount to a direct extension of the proof given by Ahlfors for n = 2 (see [ 109]) to the general case. We define by induction a finite or infinite sequence (Em), m = 0, 1, ... , of sets. Let Eo = R". Assume that the set Em_ 1 has been defined for some m > 1. Let
2m(r) = sup u(x, r), xEEE_,
where r > 0. Obviously, A.(r) is a monotone nondecreasing function. Furthermore,
lim .m(r) < p(R") < oo.
r-.oo
11. MAPPINGS WITH BOUNDED DISTORTION
116
oo as r
Since, on the other hand, h(r)
oo,
Am(r) < h(r)
(3.10)
for all sufficiently large r. It can turn out that (3.10) holds for all r. In this case the construction is complete. Assume that Am(r) > h(r) for some r > 0, and denote by rm the supre-
mum of such values r. Then 0 < rm < oo. Each interval (rm - e, rm) contains a value r such that Am(r) > h(r), and this gives us by passage to the limit that Am(rm - 0) > h(rm).
(3.11)
On the other hand, Am(r) < h(r) for all r > rm, which implies that Am(rm + 0) < h(rm).
(3.12)
Comparing (3.11) and (3.12), we get that the function Am is continuous at rm, and Am(rm) = h(rm). Let e > 0 be arbitrary, and let xm be a point of
E,,,-, such that /L(xm, rm) > Am(rm) - (R")e12m.
Define Bm = B(xm, rm) and Em = Em\Bm. It is obvious that in this case Em c Em-1 and Em 14 Em-1. The construction just described obviously defines a finite or infinite sequence of balls Bm = B(xm, rm), m = 1, 2, ... , and a sequence of sets
Em. Further, Em = R" \ u m Bk for each m > 1, and the center of Bm is in Em_ I and hence does not belong to any of the balls B1,... , Bm_ 1. Moreover, for each m /L(Bm) > Am(rm) -
e#2Rm")
= h(rm) - eµ28")
(3.13)
Finally, note that, since E. C Em_ 1,
Am (r) = sup u(x, r) < sup u(x, r) _ Am-, (r). xEE,,,
xEEm-i
It follows from the definition of rm that rm < rm_ 1 (rm is the supremum of the set Sm of all values of r such that Am(r) > h(r), and the inequality Am(r) < Am_I(r) clearly implies that Sm C Sm_1, whence rm < rm_1). The sequence (rm) of radii is thus decreasing. Let x be an arbitrary point of R". We prove that it cannot belong to
more than C" balls in the sequence (Bm), where C" is the constant in Lemma 3.6. Indeed, let Bk = B(xk, rk) and B, = B(x,, r,) be two balls in our sequence containing a point x. Assume for definiteness that k > 1. Then rk < r, and xk 15 B,. We have that Ix - xk I < rk and Ix - xi I < r1. Since xk % B1, it follows that x 0 xk and Jxk - x,I > r1. Further, x 0x1,
§3. CONFORMAL CAPACITY
117
for otherwise we would have that I x1 - xk I = Ix - xk < rk < r!, which is impossible. In the triangle with vertices x, xk, and xl the side xkxl is the largest. Consequently, the angle of this triangle at the vertex x is the largest of its angles, and hence has measure greater than n/3. We thus get that the angles between any two vectors emanating from the point x and directed toward the centers of those balls B,,... , Bm , ... containing x is greater than n/3. By Lemma 3.6, this implies that x belongs to at most C" balls in the sequence (Bm). Denoting by Xm the indicator function of Bm, we get that Em Xm(x) < C" for all x E R". From this,
µ(B.) _ m
Xm(x)µ(dx) < Cnµ(R"). m fan
E h(rm) < > µ(Bm) + eµ(R") < (Cn + e)µ(R"). m
(3.14)
m
We now prove that the set R"\A is contained in the union of the balls
B,,... , Bm. .... If this sequence is finite, it follows from the construction that µ(x, r) < h(r) for every x not belonging to any of these balls, and this gives us the required inclusion. Assume that the sequence (Bm) is infinite. 0 as m -, oo. By (3.14), the series E' h(rm) converges, and hence rm Take any
XEE=R"\ U00Bm, m=1
and let r > 0. Choose m such that r,,, < r. The point x is in Em_,, and µ(x, r) < h(r), because r > rm. Since x is an arbitrary point of E and r > 0 is arbitrary, this implies that E C A, and hence the balls B1,... , B, .. . form a covering of R"\A.
Thus a sequence (Bm), m = 1,2,..., covering R"\A has been constructed. The radii of these balls satisfy (3.14), which implies that yh(R"\A) < (Cn + e)µ(R"). Since e > 0 is arbitrary, this proves the lemma.
COROLLARY. Suppose that µ is an arbitrary measure on R" such that µ(R") < oo, and h(r) is an arbitrary gauge function. Denote by AA, where . > 0, the set of all x E R" such that µ(x, r) < h(r)/2 for all r. Then
7h(R"\A,) < CA.µ(R"), where Cn is the constant in Lemma 3.6, Cn < oo.
For a proof it suffices to apply Lemma 3.7 to the measure µ, = Aµ.
11. MAPPINGS WITH BOUNDED DISTORTION
118
§3.4. Capacity and Hausdorff measures.
LEMMA 3.8. Let F : (0, oo) - R be a monotonically decreasing absolutely continuous function such that F(r) - oo as r 0 and F(r) -, 0 as r - oo. Let 0 < a < fl, and define
Fa (r) =
j F(a) for 0 < r < a, l F(r) for a < r < oo;
F(a)-F(fl) for0
FF,p(r) =
fort
0
Let p be a nonnegative measure on R", and define u(x, r) = p[B(x, r)]. Then
f
f" Fa,p(Ix - yI)µ(dy)
0
F'(r),u(x, r) dr,
(3.15)
F(r)µ(x, r) dr,
(3.16)
F'(r)p(x. r) dr.
(3.17)
ra 00
Rf
F x - yI)µ(dy)
/ aa
"
f F(Ix - yl)p(dy)
°°
R"
fo
PROOF. Define X (r, 1) = 1 for r > 1, and X (r, 1) = 0 for r < 1. Then for
every xER" p(x, r) = )R" X(r, Ix - yI)p(dy). Substituting
this in the right-hand side of (3.15) and then using Fubini's theorem, we get
f a
a
F'(r)p(x, r) dr =
-
ff R"
a
F'(r)X(r, Ix - yI) dr u(dy)
After uncomplicated transformations we have that P
F'(r)X (r, 1) dr = Fap(1)
for any 1 > 0. This gives us (3.15). The equalities (3.16) and (3.17) are obtained from (3.15) by passing to the limit as fi -+ oo and as a - , 0 and fi oo, respectively. The lemma is proved. THEOREM 3.3 [132). Let U(X)
_
-
f
(0,r)
v(y) dy. Ix - yI"-x
§3. CONFORMAL CAPACITY
119
where 0 < A < n, v(y) > 0, v is integrable to the power p > I in the ball B(0, R), and Ap < n. Suppose that h(r) is a gauge function such that 1
L
[h(r)]1 /pr-(n/p)+x-1 dr < oo.
Then there exist constants KI and K2 depending only on A, n, p, h, and R such that for any S > 0 the h-content of the set of all x E R" with
u(x) >
al
+ K211v1Ip.B(o.R)
does not exceed C"(J(IvI p)p, where C" is the constant in Lemma 3.6.
PROOF. Denote by 6(v, x, r) the integral of v over B(x, r), and by 0(vp, x, r) the integral of vp over the same ball. It will be assumed that v is defined for all x E R", with v(x) = 0 for x V B(0, R). By Holder's inequality, 9(v, x, r) < a,'- I /prn-n/p[0(vp, x, r)]'/P.
(3.18)
Setting F(r) = r1-" in Lemma 3.7, we get that u(x) _ (n - A) fo ' 0(v, x, r) dr rn-x+1
_ (n - A)
f
R
O(v,
x+1)
dr + (n - A) f°° ern,
1) dr.
Let us estimate separately each of the integrals on the right-hand side. For the first integral
9(v,x,r)
f R rn-x+1 dr < SI-I/p fo R 0(vp,x,r)]1/Pdr r(n/p)-x+l U
by (3.18). To estimate the second integral we note that, since v(x) = 0 for IxI > r, 9(v, x, r) is equal to the integral of v over the intersection of B(0, R) with B(x, r), and hence 9(v, x, r) <_ 9(v, 0, R) < QR-1/pRn-n1P[0(vp, 0, R)]1/p
= QA-'/,PR
n-n/pIIvflp.B(O.R)
Comparing the above inequalities, we get that
u(x) < (n -
[8(v"; x, r)]Ilp
UP fo R
where K2 =
an-11pRn-n1p.
rn/p-x+l
dr + K2IIujIp,
(3.19)
120
11. MAPPINGS WITH BOUNDED DISTORTION
Let 8 > 0 be arbitrary, and denote by Ba the collection of all x E R" such that 0(vp, x, r) < h(r)/8p for any r > 0. By Lemma 3.7, the h-content of Aa = Rn\Ba does not exceed SPC" j [v(X)]P dX = C"(SllviIp.e(0.R))". "
By (3.19), u(x) < +K211VIIp,R" for all x E B5, where the constant K1 is equal to the integral I [h(r)]I/pr-("l P)+x-1 dr.
(n
Jo
The set of all x such that v(x) > K1 /8 + K2II U11 p is contained in A5, and hence its h-content does not exceed C"(BIIulI p.R")p. The theorem is proved. COROLLARY 1. Let A be a compact subset of R". If the p-capacity of A is equal to zero, then for every gauge function h with I
I';
[h(r)]'1"r-"1 dr < oo
(3.20)
the h-content of A is equal to zero.
PROOF. Let R > 0 be such that B(0, R) D A. Suppose that the gauge function h satisfies (3.20). By Theorem 3.3, there exists a function v E Lp(R") such that v(x) = 0 for Ixi > R, and for all x E A
u(x) = f v(y)Ix - yII-" dy = oo. Let Aa be the set of all x E R" such that u(x) > K1 /8 + K2IIvIIp, where K1
and K2 are the constants in Theorem 3.3 corresponding to the case A = 1. The set A is contained in Aa for any 8 > 0; hence yh (A) < yh (A,$) for any 8 > 0. According to the lemma, yh(Ao) < C"(SiivIi.R")' . Since 8 > 0 is arbitrary, this implies that yh(A) = 0, and the corollary is proved. Setting h(r) = ra, we get COROLLARY 2. If A is a compact set of p-capacity zero, then for every
a > n - p the a-dimensional Hausdorff measure of A is equal to zero. In particular, if the conformal capacity of A is equal to zero, then for any a > 0 its a-dimensional Hausdorfineasure is equal to zero.
§3.5. Estimates of the capacity of certain capacitors. We consider some special capacitors in R" and show that they have definite extremal properties. The result in this section is due to Gehring [34], [35].
Let I E (0, 1). Denote by KG(t) the capacitor (A0, A,) with A0 the exterior of the ball B(0,1), A0 = R"\B(0,1), and Al the segment consisting
§3. CONFORMAL CAPACITY
121
of all points x = Ae, where 0 < A < t; KG(t) is called a Grotzsch capacitor. Let t > 0. The symbol KT(t) denotes the capacitor (Ao,A1) with A0 the ray consisting of all points x = Ae, A > t, and AI the segment formed by
the points x = µe", -1 < ,u < 0. This capacitor is called a Teichmuller capacitor. The capacitors KG(t) and KT(t) have extremal properties expressed by the following theorems. THEOREM 3.4. Let (Ao, A1) be a capacitor in R" such that A0 contains the
exterior of some ball B(a, r), while AI has a connected component joining the center a of this ball to a point xo E B(a, r), xo 0 a. Then
C"(A0, AI) ? C"[KG(Ixo - allr)] THEOREM 3.5. Let (A0, A1) be a capacitor in R. Assume that A0 contains an unbounded connected component E0, while AI has a connected component E1 containing two distinct points x0 and yo. Let p be the distance from x0 to E0. Then C. (A0, A1) >_ Cn[KT(p/Ixo - Y013
We introduce some notation. The collection of all capacitors in R" satisfying the conditions of Theorem 3.4 and such that the ratio J x0 - al /r
is equal to some number t E (0, 1) is denoted by Kn(t). The set of all capacitors such that the conditions of Theorem 3.5 hold with p/I xo - Yol = t > 0 is denoted by H. (t).
It is not hard to see that KG(t) E Kn(t) for every t E (0, 1), and hence KG(t) has the smallest capacity in the class Kn(t), by Theorem 3.4. Similarly, it follows from Theorem 3.5 that KT(t) is a capacitor of smallest capacity in H"(t). The proof of Theorems 3.4 and 3.5 is our immediate problem. It is based on a certain transformation of functions on Rn called symmetrization. We give without proof the properties of symmetrization we need. These properties can be found, for example, in [I IS]. We remark that the type of symmetrization used here is simpler than in [35]. The proofs of the necessary properties of symmetrization in our case do not require using the Brunn-Minkowski theorem. Symmetrization of functions of a single variable. Let I denote either the whole set R or an interval [-h, h] with 0 < h < oo. Nonnegative measurable functions f (x) and g(x) defined on I are said to be equimeasurable
if the measures of the sets Ef(t) = {x E II f(x) > t} and Eg(t) = {x E IIg(x) >- t} coincide for every t > 0. In this case f and g are said to be rearrangements of each other.
122
II. MAPPINGS WITH BOUNDED DISTORTION
Let f (x) be a nonnegative integrable function. The symmetrization, or symmetrically decreasing rearrangement off , is defined to be the function of > 0 defined on I and such that the following conditions hold: 1) of is equimeasurable with f.
2) (af)(x) = (of)(-x) for every x E 1. 3) of is a nonincreasing left-continuous function on (0, oo) n I, and (of)(0) = limn-0 of W. A function of satisfying all these conditions exists and is unique, as is easy to show. We mention an important special case. Let A be an arbitrary measurable subset of I with finite measure, and let f = XA be the indicator
function of A, i.e., the function with f (x) = 1 for x E A and f (x) = 0 for x A. Then the symmetrization of f is the indicator function of the closed interval symmetric with respect to 0 whose measure is equal to that of A. We call this interval the symmetrization of A. Symmetrization with respect to a plane in R. We hold to the following notation. Let x = (x1,... , x"_ 1, xn) be a point in Rn. We write x = (y, z), where y = (XI, ... , x"_ 1) E Rn- I and z = xn. The space R"- I is assumed to be imbedded in R" by identifying the point y = (x1, ... , X,- 1) E R"- I with the point (y, 0) E R". Take a pointy E R"-I. We denote by 1(y) the line in R" through y orthogonal to the plane R"-1 and thus consisting of all the points of the form (y, z), where y is the given point.
Denote by 0' a domain in R" which is the whole of Rn, or the strip formed by the points x = (y, z) such that I z 1 < h, or, finally, the half-strip consisting of the points (x1, ... , xn_ 1, xn) such that x,- I > 0 and Ixn ( _< h. Let G denote Gn n R"-1. If 0' is R" or a strip, then Cron is obviously R"-1. If GO is a half-strip, then G is a half-plane. If (i" is a strip or a half-strip, then the number 2h is called the width of C?'.
Let f : (1' - R be a nonnegative integrable function. Then for almost all y E Go" the function fy : z E I -- f (y, z) is integrable. For every such y we construct the symmetrization afy of fy. As a result we get a function
of : P' - R defined for almost all y E P and for almost all z, and for each y the function z -' (af)(y, z) is the symmetrization of the function z - .f (Y, z). We next describe a certain form of symmetrization needed later. For an
arbitrary point x = (xI,...,xn_2,xn_I,xn) E R" we set x = (t,xn_I,xn), where t =---(XI, , xn_2) E R"-2. Every point x of the form x = (t, 0, 0) in R" will be identified with the point t E R11-2, and the plane R"-2 will xn = 0 in R". Finally, Rn 1 denotes be identified with the plane xn_ the set of all points x of the form x = (t, xn_ 1, 0), where x,,_ I > 0. Let Dn 2) E R"-2, r > 0 be the set of all points (t, r, 0) E R" with t = . . . . . x __2)
§3. CONFORMAL CAPACITY
123
and -n<0
We get a certain mapping C of D" into Rn. Obviously, C(D") = R", and Z is one-to-one on the set D° of all points (t, r, 0) E D" such that r > 0 and -ir < 0 < ir. Further, C(D°) = Rn\R+ . For t E Rn-2 and r > 0 denote
by C(t, r) the subset {{(t, r, 0)1 - n < 0 < ir) of R". Obviously, C(t, r) is a circle of radius r about (t, 0, 0) E R"-2 whose plane is completely orthogonal to R"-2. Let f be a nonnegative integrable function on new function to : D" -+ R by setting
Rn. We define from it a
ip(t,r,0) = f (t, r cos 0, r sin 0). Let aip be the symmetrization of (p. Obviously,
(a)(t, r, a) = ((.Tq)(t, r, -a) for any t E Rn-2 and r > 0, and (arp)(t, 0, 0) does not depend on 0. This implies the existence on Rn of a function z f such that (zf ) (t, r cos 0, r sin 0) = (aip)(t,r,0)
for any t E Rn-2, r > 0 and 0 E [-n, n]. The function zf is called the cylindrical symmetrization of f with respect to the plane Rn-2 and the half-plane R+ 1. It is useful to see how to look at the cylindrical symmetrization geometrically. For any t E R11'2 and r > 0 we have the circle C(t, r). Under cylindrical symmetrization the restriction of f to any such circle is replaced
by its symmetrization with respect to the parameter 0, -n < 0:5
sr.
THEOREM 3.6. Let f be a nonnegative compactly supported function of class W.' (G") in the domain G", where p > 1. Then its symmetrization of also belongs to W, (G"), and
f I(of)'(x)I° dx < f. If'(x)I° dx. THEOREM 3.7. Let f be a nonnegative compactly supported function of class W, (R") in R", where p >- 1. Then its cylindrical symmetrization zf also belongs to Wp' (RI), and
f I(zf)'(x)I° dx
R^
See [115] for proofs of Theorems 3.6 and 3.7.
U. MAPPINGS WITH BOUNDED DISTORTION
124
PROOF OF THEOREM 3.4. Let K = (Ao, A,) be a capacitor of class K', (t),
where 0 < t < 1. It will be assumed that A0 D R"\B(0,1), and AI is a closed connected set joining the points 0 and tel. The general case clearly can be reduced to this by the successive application of a similarity dilation and a motion of R. Let Ao = R"\B(0,1). Since A0 D A', it follows that C"(Ao, A1) > C"(Ao, A1). We assume below that A0 = R"\B(0,1). Consider the orthogonal projection of A I on the plane R"-1. The projection is a connected set Ai lying in R"-I and joining the points 0 and tel. Let us prove that (3.21) Cn(Ao,A1) > C"(Ao,A'). Indeed, take an arbitrary function f E J'(Ao, AI ), and let of be its symmetrization. We prove that of (x) is equal to 1 in a neighborhood of Ai
and to 0 in a neighborhood of A0. There is a S > 0 such that f (x) = 1 on the ball B(x, 6) for every x E AI and the support of f is contained in the ball B(0, 1 - 8). Then the support of of is clearly also contained in B(0, 1 - S), since of vanishes in a neighborhood of A0. Denote by ly the line orthogonal to R"-I and intersecting R"-1 at the point (y, 0). If (y, 0) E A', then ly contains points of A 1. Since f (x) < 1 and f (x) is equal to 1 in the 6-neighborhood of every point Xo E A I, this implies that for
(y, 0) E A' the function z -+ f (y, z), and with it also (of)(y, z), is equal to 1 on some segment of length 23. This enables us to conclude that of is equal to 1 in a neighborhood of the set A'. Therefore, if f E '(A0, AI), then of E W(A0, A'). From this,
f1f'(x)Idx >j "
rf'(x)dx>C(A0,Ai).
"
Since f E W(A0, A 1) is arbitrary, this proves (3.21). By considering the invariance of the capacity with respect to motions it is not hard to show that the same thing is valid when A I is replaced by the projection on an arbitrary hyperplane passing through the points 0 and el. Let P1, ... , P"_ I be a system of n - 1 mutually orthogonal hyperplanes intersecting in the line Oe 1. We construct a sequence of sets A(11 ),
... , A ( n-1 ), where A(') is the
orthogonal projection of A, on PI, and A''+I) is the orthogonal projection of A(') on P;+I for each i < n - 1. Obviously, Cn(Ao.A,) > C"(AO,Aq(,')) > ... > C.(Ao,A( n-1)).
It remains to note that A(, n-1) is a connected set in the line passing through
the points 0 and te1 and contains these points, hence it also contains [0, te1 J. This gives us that
C(A0, A,) > C(Ao, [0, lei ]) = y(1),
and Theorem 3.4 is proved.
125
§3. CONFORMAL CAPACITY
PROOF OF THEOREM 3.5. Let (Ao, A1) be a capacitor of class H', (t),
where t > 0, and A0 and A 1 are closed connected sets with A 1 bounded and
A0 unbounded. It will be assumed that Al contains the points 0 and en-1, the distance from 0 to A0 is equal to t, and the point xo of A0 closest to 0 lies in the plane of the vectors a"-1 and e, xo = ae" _ 1 +.8e,,. An arbitrary capacitor of class H" (t) can always be made into a capacitor satisfying these conditions by a motion and a similarity dilation. The capacity of the capacitor is not changed by this. We now construct another capacitor
(A',A'). Let x = (y,x"-lx") Define r(x) = Vx,-, +
C1(x) = (y, r(x), 0), C2(x) = (y, -r(x), 0).
The mappings C,1 and C2 are continuous. Let A'0 = C2(Ao) and A, = C, (A,). The sets Ao and A' are connected and closed, and lie in the plane R"' 1.
Further, A' contains the points 0 and e,,-,. The set Ao lies in the halfplane x" = 0, x"_ 1 < -t, and contains the point -te"-1 = c2(xo) We prove that
C"(Ao,A1) > C"(A',A').
(3.22)
Let f E 13'(A0, A 1) be an arbitrary function, and consider its cylindrical symmetrization z f . For I E Rrt-2 and r > 0 denote by C(t, r) the circle
formed by the points (t, r cos 0, r sin 0), where -ir < 0 < n. A point (t, r, 0) with r > 0 belongs to the set Ai if and only if C(t, r) intersects A1. Similarly, a point (t, -r, 0) with r > 0 belongs to Ao if and only if C(t, r) n A0 is nonempty. Let r,0) = f (t, r cos 0, r sin 0),
fp(t,r,0) = (zf)(t,rcos0,rsin0). Assume that (1, r, 0) E A'. Then the function 0 - (t, r, 0) is equal to 1 on some interval contained in [-a, it]. Since 0 < f (x) < 1 for all x, this implies that 7(t, r, 0) = 1 on some interval -S < 0 < J. and thus Oo(t, r, 0) is equal to 1 on some arc of C(t, r). The middle of this arc is precisely
the point (t, r, 0). We show that (zf)(x) = 1 in some neighborhood of (t, r. 0). Let (xm), m = 1, 2, ... , be an arbitrary sequence of points in R" converging to (t, r, 0), and let xm = (tm, x"_l,m, x",m). The point xm lies
on the circle C(tm, rm), where rm = r(xm) - r as m -+ oo. Since f is equal to 1 in a neighborhood of every point x E A I, the length of the arc of the circle C(tm, rm) on which f (x) = 1 is not less than some e > 0 for sufficiently large m. It is not hard to deduce from this that (z f) (xm) = 1 f o r sufficiently large m, and since (xm ), m = 1, 2, ... , is an arbitrary sequence convergent to x = (t, r, 0), it is thereby proved that zf = 1 in
11. MAPPINGS WITH BOUNDED DISTORTION
126
some neighborhood of the given point x. We now show that (zf)(x) = 0 in a neighborhood of every point (t, -r, 0) E A. Suppose (t, -r, 0) E A. Then the function 8 - V(t, r, 0) vanishes on certain intervals contained in This implies that the function 0 vanishes in certain intervals
[-a, -a + 8] and [n - S. n] symmetric with respect to the point 0. The points of C(t, r) corresponding to the values of 0 in these intervals form an arc whose midpoint is precisely the point (t, -r, 0). The function zf vanishes on this arc. Arguing again as in the preceding case, we get that z f vanishes in a neighborhood of every point x E A. The function z f has compact support. It follows from what has been proved that Z f E P(A,, A; ). By Theorem 3.8,
f If'(x)t" dx > f I (zf)'(x)I" dx > C"(A0, A1). "
^
Since f E W(A0,A1) is arbitrary, inequality (3.22) is established. To conclude the proof we use the usual symmetrization. Assume that (Ao, A1) is a capacitor constructed as follows: A 1 is a bounded connected set containing 0 and a"_ 1; A0 is an unbounded set lying in the half-plane x"-1 < -t; Ao contains the point -ten_ 1, and infA. x"_ 1 = -oo. The collection of all capacitors satisfying these conditions is denoted by H"(1). Take an arbitrary function f E 3'(A0, A,), and let Qf be its symmetrization. Denote by Ao and A' the orthogonal projections of A0 and A 1 on the plane R"-1. The function Qf is equal to zero in a neighborhood of Ao and to I in a neighborhood of A'; hence, Q f E [(Ao, A ). We have that
f
f'(x)dx > "
f I(af)'(x)Idx > R"
Since f E W(Ao, A1) is arbitrary, this proves that C"(Ao, A1) > C"(A', Ai ).
Thus, if (A0, A1) is a capacitor of class H"(t), then the capacity of (A0, A,) is not increased if A0 and A 1 are replaced by their orthogonal projections on R"-1. It is clear that instead of R11-1 we can take any hyperplane passing through 0 and a"-1. The capacitor (A0, A1) is made into the capacitor H(t) by successive projections. This gives us that C"(Ao,A1) > C"[H(t)] = q(t), and the proof of Theorem 3.5 is thus complete.
We introduce some notation. Let C"[KG(t)] =
C"[KT(t)] = W"/[InOn(t)]"
§3. CONFORMAL CAPACITY
127
THEOREM 3.8 [35]. The function `Y"(t)/t is nondecreasing in (1, oo), and
there exists a constant A" > 1 such that t < T, (t) < A"t for all t E (1, oo). The functions `P, and 0,, are connected asfollows. (D,, (t) = ('P"[(t+1)1/2])2
for allt>0. We shall not present the proof of Theorem 3.8. The so-called Thompson principle, which gives means for constructing a lower estimate for the capacity of a capacitor, is well known in potential theory. Here we present a certain modification of this principle that can be used for getting a lower estimate of the p-capacity of a capacitor in R" for arbitrary p > 1. Let H = (A, B) be a given capacitor in R", and let U = R"\(A U B) be its field. Also, let p > I be a given number, and let q = p/ (p - 1). For a given vector-valued function f : U - R" of class L, (U) we say that f is divergence free (solenoidal) if
fu
(o(p(x), f (x)) dx = 0
(3.23)
for every 9; E Co (U). If f satisfies this condition, then we also write div f (x) = 0 in U. It is easy to establish by passing to the limit that if 0 f(x) is divergence-free, then (3.23) holds for any function t EW p(U). Suppose that f : U R" is divergence-free, and 91 and op2 are two different functions of class * p (A, B). Their difference ip I - 972 belongs to O
the class W ,(U), and hence
J(V911(x) - V 92 W, f (x)) dx = 0. From this we conclude that the quantity y(f, A. B) =
fV(x))dx
(3.24)
does not depend on the choice of the function rp E JPP (A. B). We call y(f, A, B) the flow of the divergence-free vector-valued function f (x) in the capacitor H. If A and B are bounded by smooth (n - 1)-dimensional surfaces and f (x) is continuous, then
(f (x), v(x)) do = - fUnA (f (x), (x)) do,
y(f, A. B) = J UnB
(3.25)
B
where v(x) is the outward normal vector at the point x E OU and do is the area element of the surface 8U. Formula (3.25) can have a definite heuristic value.
II. MAPPINGS WITH BOUNDED DISTORTION
128
THEOREM 3.9. Let H = (A, B) be a capacitor, let U = R" \(A u B) be its field, and let f : U R" be a vector-valued function of class LQ (U). Assume that f is divergence free, and y(f, A, B) = 1. Then
I-P. CP(A,B) >_ (fu
(3.26)
If(x)IPI(P-1)dx// f
PROOF. Let 9 E Wp (A, B) be arbitrary. Then 1=
From this,
f(Vc(x)
f (x)) dx < fu
(x)I dx
.
(fu IV9(x)Ip dxl/l
\ I/4
I/P
(fu If (x)I° dx)
.
P/9
fu IVip(x)IP dx >_ (fu If (x)I° dx)
Since 9 E Wp(A, B) was arbitrary, the theorem is proved. REMARK. The estimate of the lemma is sharp in the sense that there is equality for some vector-valued function f : U - R" such that div f = 0. Namely, suppose that u(x) E Wp (A, B) supplies the smallest values of the functional
fu IVu(x)IP dx
(3.27)
in the class W1(A,B). Then u(x) is extremal for the functional (3.27), and hence
fu (V,1(x), IVu(x)Ip-2Vu(x)) dx = 0
(3.28)
0
for every function 17 EW P' (U), with compact support, by Corollary I to Lemma 5.9 of Chapter II. It is easy to establish by passing to the limit that (3.28) holds for every function q EW I (U), and hence the vector-valued function g = I V u V' 2 V u is divergence-free. We have that y(g, A, B) = j(Vl(x). IVu(x)Ip2Vu(x)) dx,
where I is an arbitrary function in WP (A, B). Setting >7 = u here, we get that y(g. A, B) = CP(A, B).
For brevity of notation we set Cp(A, B) = C and let f = (1/C)g. We have that
(fu If(x)IQdx )p-I Cp (fu and so equality holds in (3.26) for this function f.
IVu(x)Ipdx)p-1
= C,
§4. GENERALIZED DIFFERENTIAL OF AN EXTERIOR FORM
129
§4. The concept of the generalized differential of an exterior form
§4.1. General facts about exterior forms. We assume that the reader is familiar with the concept of an exterior differential form defined in a domain in R", along with the main properties of exterior differential forms, in particular, the properties of the operations of addition, exterior multiplication, and differentiation of exterior forms. Expositions of these topics can be found in, for example, [146], [139], and [164]. We introduce some notation. Let r , where I < m < n, be the collection of all m-element finite sequences I = (i1.... ,1,,, ), where i 1, ... , im are < im < n. Let K = (k1..... km) E rm integers such that 1 < it < i2 < be a given m-tuple of indices. The exterior form dxk, ... dxkm of degree m is denoted by dxx. An arbitrary exterior form w(x) of degree m on a set U can be written in the form .
w(x)
.11112 ...lm (x) dx,, dx;2 ... dx,m 15 il
(4.1)
E f, (x) dxj, Er,^
where the fj(x) are real functions defined on U. The functions fl(x) are called the coefficients of the form cv(x). The equality (4.1) is called the canonical representation of the exterior differential form w(x). The degree of a form cv(x) is denoted by deg co. A form co will be said to belong to the class C00(U) (LP,10C(U), Wp10C(U), etc.) if all its coefficients belong to C°°(U) (respectively, to LP,10C(U), Wp10C(U), etc.). A form co
is said to have compact support in U if all its coefficients have compact support in U. We introduce the notation Iwl(x) =
lEI
Ifr(x)I2,
where ff (I E rn) are the coefficients of co. If E c U, then let I/P
IIWIIP.E = [fE(I wI (x))P dxl
If deg w = n, then the form w has an expression w(x) = u(x) dx1... dx,,. If E c U is a measurable set, then by the integral of a form w of degree n over E we mean the quantity fE co = fE u(x) dx. An exterior form of degree equal to zero on an open set U is defined to be any real function defined on U. The product of exterior forms w and
II. MAPPINGS WITH BOUNDED DISTORTION
130
q, is denoted by cv A to, and dcv denotes the differential of an exterior form cv of class C I.
Let lv and w be forms of degrees / and k, respectively, and let 0 = wnsp. Then it is not hard to show that for all x E U 101(x) 5 CImI (x)I wI (x).
(4.2)
where C is a constant depending only on k, 1, and n. Let w E L p., (U) and yr E Lq,1a(U) be exterior forms on U of degrees k and 1, respectively, with k + 1:5 n, and let 1 l p + 1 /q = 1 /r < 1. Then their exterior product w n c is in Lr,10(U), and for every compact set A c U IIw A 9111.A <- CIIwIIp.A,
(4.3)
where C is a constant depending only on k, 1, p, and n. Inequality (4.3) is an obvious consequence of (4.2) and Holder's inequality. Let (wv ), v = 1 , 2, ... , be a sequence of exterior forms of degree k
defined on U. Then the forms co, will be said to converge as v - 00 to an exterior form w in Lp,1a(U)(Wp10c (U), locally uniformly) if all the
coefficients of w converge in the indicated sense to the corresponding coefficients of co. Let w, E L p,10C(U) and V E Lq. c (U), v = 1, 2, ... , be sequences of exterior forms converging to forms co and ip in L p,1a and Lq,1a, respectively,
as v - oo. Assume that 1 l p + 1 /q = 1 /r < 1. We have that
Ilwvnwv - wnwllr.A<- II(wv-w)Acvllr.A+Ilwn(ipv-(P)IIr.A 5 Y(Ilwv - wlip.All spvIIq.A + Il(0llp.All9v - 9'IIq.A),
which implies that w A qq,, - to n 9 in Lr,1a. LEMMA 4.1. Let co be an exterior form of class L, ,10c (U) with degree
r > 0 defined in an open set U. If fu q(x) A w(x) = 0 for every exterior form q E Coo (U) of degree n - r with compact support in U, then w(x) = 0 almost everywhere in U. PROOF. Let
to (X) = E cvy (x) dxy. yEr. Fix arbitrarily an r-tuple A E rn, A = (j,, ... , j, ), of indices, and let 10 =
in_,) E rn"_r be an (n - r)-tuple of indices complementary to
A, i.e., such that (j, , ... , j,, i 1, ... , in-r) is a permutation of (1, 2,..., n). Let Sp E Co (U) be arbitrary, and let q(x) = 9(x) dxco. Then J(x) A w(x) = Qrp(x)co; (x) dx1... dxn,
§4. GENERALIZED DIFFERENTIAL OF AN EXTERIOR FORM
131
where a = ± 1, and we see from the condition of the lemma that fu rp(x)wyo(x) dx = 0 for every function q E Coo with compact support in U. This implies that a, (x) = 0 almost everywhere, and the lemma is proved. COROLLARY. Let to 1 and w2 be r th-degree forms of class L 1.1°c (U) such that for every form ri E Coo of degree n - r with compact support in U
I(x)
(01 (X)
1
A = v i(x) v Then wI (x) = (02 (x) almost everywhere in U.
A (02(4
§4.2. The concept of generalized differential of an exterior form.
Let
U c R" be an arbitrary open set, and let w and 9 be exterior forms of class L 1,1a (U), with deg rp = deg co + 1 and deg w = r > 0. Then we say that cp is the generalized exterior differential of the form w if for every form j 7(x) of class COO with degree n - r - I that is compactly supported
in U fu n n (P _
(-1)n-r
fu d1 A w.
The generalized exterior differential of w is denoted by dw, as in the case of forms with smooth coefficients. By the corollary to Lemma 4.1, the generalized exterior differential of a form is unique if it exists. Let co be an exterior form of class C'. Then its ordinary differential is also the generalized differential. Indeed, let I be a form of class C°°
with degree n - r - 1, where r = deg w, and with compact support in U. Then the form ri A w is compactly supported in U, and d(q A w) = drl A w + (-1)"-r-I q A dw, where the differential on the right-hand side is understood in the classical sense. Since the form 1 A w has compact support by Lemma 2.3, we have that fu d(, A (0) = 0, which implies that
JAdw=(-1)r r vdAw. Since the form q of class C°D here is arbitrary, this means that dw is also the generalized differential of co. LEMMA 4.2. Let w and ip be exterior forms of class L 1,10(U), where U
is an open set in R", and deg Sp = deg w + 1. Assume that for every set of forms such that V lying strictly inside U there exists a sequence to in L I (V) as v - oo, co,, has a generalized differential for each v, co,, and dw - 9 in L I (V). Then rp is the generalized differential of Co. PROOF. Let >y be an arbitrary exterior form of class COO(U) with com-
pact support in U and with degree n - r - 1, where r = deg w. Let V be
11. MAPPINGS WITH BOUNDED DISTORTION
132
an open set lying strictly inside U and such that q(x) = 0 for x 95 V. We construct a sequence (w,,), v = 1, 2,..., of exterior forms of class L 1(V ) such that the forms co., converge in L I (V) to the form co, and the dw converge in L I (V) to the form (p. For each v
1.
71 A dw _ (-1),,-r
Jv
d17 A w,,.
(4.4)
Since q is in C°° and has compact support, the coefficients of the forms q and dq are bounded functions. Passing to the limit as v - oo in (4.4), we get that fAgo=(_l)n_1f d Aw.
Obviously, the equality is not violated if V is replaced by U. Since q is an arbitrary form of class C°° with compact support in U, it follows from what has been proved that 9 = dw by definition. The lemma is proved. We make some more remarks about approximation of exterior forms by smooth functions. Take an arbitrary averaging kernel a, and let w be an arbitrary exterior form of degree r > I and of class L1,10(U), w(x) = >2 oil (x) dx,. I
Let
(ah * w)(x) _ >2(ah * wj)(x)dx,.
Denote by a the function x --+ a(-x). Obviously, a is also an averaging kernel. By using Fubini's theorem it is not hard to show that if u is locally integrable and v is a bounded measurable function with compact support in U (U an open set in Rn), then
fu
(ah * u)(x)v(x) dx =
JU
u(x)(&,, * v)(x) dx.
Applying this result to exterior forms, we get that if w and q are exterior forms of class L,,la(U), q is compactly supported and has bounded functions as coefficients, and deg w + deg q = n, then JU
q(x) A (ah * w)(x) = fu ( ah * q)(x) A w(x).
Note further that d(ah * w)(x) = (ah * dw)(x) for every exterior form co of class C' I.
§4. GENERALIZED DIFFERENTIAL OF AN EXTERIOR FORM
133
LEMMA 4.3. Let w and 9 be forms of class L 1.1oc(U), where U is an open
set in Rn and Sp is the generalized differential of w in U. Then ah * dip _ d(ah * w) for every h>0. PROOF. As above, let Uh be the set of all points x E U whose distance to
the boundary of U is greater than h. Let q E C°°(U) be any exterior form with compact support in Uh and with deg q = n - r - 1, where r = deg co. We have the following chain of equalities: JU
ft/h
=
(-1)n-r f (&h
*
IV
n) A 9' =
(-1)n-r
17 A (ah * 9')
IUh
Sinceq E COO is an arbitrary form with compact support in Uh, this implies that ah * ip = d (al, * w), which is what was required.
COROLLARY. Let U be an open set in Rn and co and to exterior forms
in the respective classes Lp,,1a(U), pl > 1, and LpI,10C(U), p2 > 1, and suppose that Sp is the generalized exterior differential of w. Then there exist
sequences (wm) and (fpm) of forms of class C°° such that cp, = dwm for each m, and for every measurable set A lying strictly inside U the forms wm and cpm are defined on A beginning with some m = mo(A) and satisfy the 9' in L p, (A). convergence relations wm - to in L p, (A) and 9'm
Obviously, if (hm ), m = 1 , 2, ... , is an arbitrary sequence such that oo, then the sequences of exterior 0 as m hm > 0 for all m and hm forms wm = ahm * co and 9m = ahm * fp are the desired sequences.
§4.3. Properties of the generalized differential of an exterior form. The linearity of a generalized differential follows in an obvious way from its definition: if exterior forms co, and W2 of degree k in the class L11(U) have generalized differentials dw1 and dw2, then for any al,a2 E R the form alwl + a2w2 has a generalized differential, and d(alw1 + a2(02) _ a1dw1+a2dco2. LEMMA 4.4. Suppose that w and ip are exterior forms in an open set U c Rn with deg co = r, deg rp = s, r + s < n, W E L1(U) and 4) E LQ,Ia(U), where p and q are such that I/ p + I /q < 1. Assume that the generalized differentials dw and dip exist, and that dw E Lp,10(U) and dci E LQ,10C(U). Then the form CO A 9' has a generalized differential, and
d(WAc) =dwnci+(-1)rwAdip. PROOF. We construct sequences (wm) and (fpm), m = 1,2,..., of exterior forms of class COO such that wm -+ w and dwm -- dw in L p,1a(U), and
II. MAPPINGS WITH BOUNDED DISTORTION
134
ip,,, - 9 and diem -+ dip in Lq Joc(U). Let t > 1 be such that 1 It = 1 l p+ 1 /q. Then Wm A 9m - W A tp in L,,1c(U). For each m we have that d(Wm A 9 m) = dWm A (em + (-1)'Wm A diem.
The right-hand side of this equality converges in L,.,a(U) to the form 0 = dW A 9 + (-1)'W A dip as m oo. It follows from this by Lemma 4.2 that 0 is the generalized differential of W A ip, and the lemma is proved. LEMMA 4.5. Let co be an exterior form of class L 1.10c (U). Assume that Co
is the generalized exterior differential of some form of class L1,1.,(U) with degree n - 1 and with compact support in U. Then fu co = 0.
PROOF. The required result is contained in Lemma 2.3 in the case where co is a form of class C'. Let a be an arbitrary averaging kernel, and let co = d0. We set ah * W = cob and ah * 0 = 0h. The forms Oh are compactly supported in U and Wh = dOh for sufficiently small h. We have that fu cob = 0. Passage to the limit as h -. 0 gives the required result.
LEMMA 4.6. If a form co of class L1,la(U) with degree r > 1 is the generalized differential of some form 0. then the generalized differential of w is equal to zero. PROOF. Let I E COO be an arbitrary form with degree n - r -1 and with compact support in U. The form dq A 0 has a generalized differential, by Lemma 4.4. Further, d(d)7 A 0) = (-1)' dq A W. By the lemma, this means that fu dq A W = 0. Since q E C°° is arbitrary, this means by definition that dW = 0. The lemma is proved.
§4.4. The homomorphism Induced on the algebra of exterior forms by a mapping of the domain. Let U C R" and V C R" be open sets in the corresponding spaces. Assume that f : U - V is a given mapping of class COO. Then for every exterior form ca of class C°° defined on V we can construct a certain exterior form 9? on U denoted by f' w and defined as
follows. Let f (x) = (fl (x)...., fm (x)). If w(y) is a form of zero degree, i.e., co is simply a real function defined on V, then (f'W)(x) = W[f(x)). Assume that degw(y) > 0. Then Wj,12...Ik (y) dy;, dy;_ ... dyik ,
W(y) = 1
WO(x) =
E
Wiii2...A[f (x)l dfn(x)
15jl<j2<...<Jk<m
Adf12 (x) A ... A dfj, (x).
§4. GENERALIZED DIFFERENTIAL OF AN EXTERIOR FORM
135
Here df j, j = 1, ... , m, is an exterior form of the first degree, namely,
dfj = =1
afJ (x) dx;. axi
We present some properties of the operation f', assuming that f E C°° and all the forms considered are of class C°°. 1. f'(w1 + 02) = f ww1 + f'w2 for any forms w1 and W2 of the same degree.
2. f `(uw) = (f *u)(f'(o) for every form co and any function u: V -i R.
3. f-(wn 0) = (f'(0) A(f'0) for any forms co and 0. 4. Let U be an open set in R", and iu the identity mapping of U. Then i,w = co for every exterior form co defined on U.
5. Suppose that U c R", V c Rm, and W c Rk are sets open in the V and g : V -+ W are mapping of
corresponding spaces, and that f : U
class COO, and let h = g o f. Then h' = f' o g', i.e., h`w = f'(g`w) for every form w on W.
6. f'(dw) = d(f'w) for every form to. See, for example, [ 139] for proofs of these properties. Our goal is to extend the operation f' to cases when f satisfies minimal
smoothness conditions. Let U c R" and G c Rm be open sets in R" and R'", respectively, and let f : U G be a continuous mapping of class W 11oc(U), p > 1. Let f (x) = (f1(x), ... , fm(x)). The exterior forms
ax.(x)dxj
df,(x) J=1
J
are defined on U, where the derivatives are understood as generalized derivatives. Let w(y) be an arbitrary exterior form on G. The symbol f'w(x) denotes the form on U defined by the same formulas as in the case when f is a mapping of class COO. Namely, if co is a form of degree
zero, then (f'w)(x) = co[f(x)]. But if degto > 0 and w(y) =
E
wj1j2...jk(y)dyj, dyj2 ...dyjk
15ji<j2<...<jk<m
then we set
f'w(x) =
E
wj1h...jk [.f (x)] dfj, (x)
1 < j l <J2 <... <jk <m
ndfj2 (x) n ... A dfjk (x)
If the degree k of w exceeds n, then clearly f'w(x) = 0.
11. MAPPINGS WITH BOUNDED DISTORTION
136
We remark that certain difficulties can arise almost everywhere in the general case when we consider arbitrary forms with measurable coefficients
defined on G. For example, it can happen that the form f'w(x) is not defined on a set of positive measure, etc. In this connection we consider here only forms with continuous coefficients. Since f is in Wp 10C(U), it follows that df E L p.10C(U). Further, lidf,II p.A
for every compact set A c U. On the basis of (1.3), it is easy to establish from this by induction that the form f = (it'... , jk ), k < p < 11f II 1. p,A
is in L p/k.lc(U) for f' dy f = df j, A set AC U
A dfj,. Further, for every compact
Ilf' dYf 11 p.k.A <_ 7k (II I II I ,p.A
)k'
where 7k is a constant that depends only on n and k.
Let V be an arbitrary open set contained strictly inside U, and let f,,: G - D, v = 1, 2, ... , be a sequence of mappings of class COO such that f, , -4 f in W,1(V) as v -+ oo. Then dj ,,, - df in L, (V) as v oo, and hence for k < r fr dyy = df j, ., A dfj..v A ... A dfjk.,, converges to the form f' dyy in Lp(V) as v -+ oo, where p = r/k. It follows, in particular, that if fh = ah * f, where a is an averaging kernel on R", then the forms fti d y f converge to the form f' dy f in Lp,10C(U) as h - 0, where p = r/k, k = degdyf. We remark that the particular properties 1-4 above follow algebraically from the definition of the operation f' and thus remain true also when f is an arbitrary continuous mapping of class W.I. Property 5 is also an algebraic consequence of the definition of f' and the rule for differentiating a composite function. Therefore, it is true also for the case of mappings determined by functions with generalized derivatives under the condition that f , g, and h = go f belong to the corresponding function classes, and the usual rule for differentiating a composite function remains in force also in this case. LEMMA 4.7. Let U c R" and G c R'" be open sets in the corresponding spaces, and f : U G a continuous mapping of class W,11,,,(U). Then for every exterior form w(y) of degree k, 0 < k < r, and class C on G the form f w is of class C in the case when k = 0, and of class L p,1a(U) in the case when k > 0, where p = r/k. If the form w has a generalized differential dw, the form dw belongs to the class C, and k + 1 < r, then f' dw is the generalized differential of f'w. PROOF. Let w(y) be a form of degree k and class C on G. If k = 0,
then (f'w)(x) = w[f(x)], and hence (f'w)(x) is a continuous function
§4. GENERALIZED DIFFERENTIAL OF AN EXTERIOR FORM
137
on U. We consider the case k > 0. Let
&I (Y) _ E wy(Y) dyy. ,F
Er
Then
(f'w)(x) = E wy[f(x)lf*dy,. ,FEr; The functions wy If (x) ] are all continuous, the form f' d yy belongs to the class L,/k,Ioc(U), and, hence, f'w E Lr/k_Ia(U). Let tP = dw, and suppose that p belongs to the class C. Then, applying the constructions to the form q', we get that f*9 E L,/(k+I).I (U). Take
an arbitrary open set V lying strictly inside U, and let f,: V -+ R"' be a sequence of mappings of class C°° such that f, f uniformly as v - 00 and, moreover, f, - f in WI (V). These conditions are satisfied by any sequence ah, * f , where h -' 0 as v - oo, and V C Oh, for all v, so that the forms ah, * f are defined on V. The set f (V) c G is compact, since f is continuous, and hence there exists a 8 > 0 such that p(y, 80 > 8 for all y E f (V). It will be assumed that If. (x) - f (x) I < 8/2 for all x E V. Since f, - f uniformly on V, this assumption obviously involves no loss of generality. By this condition, is contained in the closed 8/2-neighborhood (denote it by H) of f (V). Let u(8), 8 > 0, be the modulus of continuity of co, on the compact set H. Then for each v
wy[L(x)l-wyIf (x)11:5 a(Ilf., -fIIC(- )). This implies that wy(f,, (x) ]
wy[f (x) ] uniformly on V as v -, oo.
Further, f, dyy --+ f dyy in L,/k(U), and hence (f'w)(x) - (f'w)(x) in L,/k(V) as v - oo. Similarly, (f'q,)(x) -' (f'tp)(x) in L,/(k+,)(V) as v
00.
Assume first that w is a form of class C. Then f,'w is also a form of class Cl*, and d (ff w) = f,, do) = f,' ip. Since f' w f' w and d (f,' (o) = A,* dw - f' dw in L 1.1 (V) as v -4 oo, and the set V lying strictly inside U was taken arbitrarily, this proves that f' dw = d(f *(o). We now consider the case when w and q? are forms of class C, and lp = dw, understood as the generalized differential. Again, let V be an open set lying strictly inside U. Let H be an open set in R" lying strictly inside G and such that f (V) c H. Let a be an arbitrary regularizer* on R"', and let co, = ah1, * co and TV = ah, * q>, where the sequence
v= 1,2,...,is such that
all
'Editor's note. Called an averaging kernel above.
v-oo,and Hc&h.
II. MAPPINGS WITH BOUNDED DISTORTION
138
for all v. For each v the forms co,, and Sp belong to the class C°°, and since w and q' belong to the class C, it follows that co,, w and q' - lp uniformly on H as v - oo. For each v we have that f, = dw,,. Consider Then f'op = for each the forms f'w and f'rp = v, where the differential is understood as the generalized differential. For each v
co ,[f(x)lf*dyy,
(f'wy)(x) =
(f'w)(x) =
jr
w,[f (x)lf * dyy,
and w,,,- [f (x)] -+ wy[f (x)] uniformly as v oo. Since f * dyy E LI(V), this implies that f'w(x) L, (V). We conclude similarly that
(f'9))(x)
(f'c)(x) in LI(V).
The set V lying strictly inside U was chosen arbitrarily. We thus get that for every V lying strictly inside U there is a sequence of forms q converging to f "w in L I (V) such that for each v the generalized differential
dq exists, and dq - f ` (dw) in L I (V). By Lemma 4.2, this implies that f' (dw) = d (f' w). The proof of the lemma is complete. We note the special case when the degree k of the form co in the formulation of Lemma 4.7 is equal to zero. This leads to the following statement. COROLLARY. Suppose that G is an open set in R"', v : G - R is a function of class C', and U is an open set in R". Then for every continuous mapping
j: U
G of class W,II((U), where r > 1, the function u = v o f is in
W,lla(U), and its derivatives can be expressed in terms of the derivatives of v and f according to the same formulas as in the case of sufficiently smooth functions. §4.5. Weak convergence of sequences of exterior forms. Below, U is an
open subset of R. Let j m, m = 0,1, ... , be functions of class L p,, (U),
where p > 1. We say that the sequence f converges to fo weakly in Lp,I0C(U) if it is locally bounded in Lp(U), and for every function ip E C0 (U)
fu fm(x),0(x) dx -
ffo(X)(X)dX
as m oo. Let (w," ), m = 1, 2 , ... , be some sequence of exterior forms in Lp,I0(U) of degree k, where 1 < k < n. Then we say that the sequence of forms to.. converges to a form coo weakly in L p,,0(U) as m -+ oo if the coefficients of the forms wm converge weakly in Lp,,a(U) to the corresponding coefficients of coo.
§4. GENERALIZED DIFFERENTIAL OF AN EXTERIOR FORM
139
LEMMA 4.8. Let (corn) be a sequence of exterior forms in L1(U) of deoo. Assume that each of the exterior forms corn, m = 0, 1, ... , has in U a generalized differential, and that the sequence (doom), m = 1, 2, ... , is locally bounded in 4210(U). Then the sequence dcom converges to dcoo weakly in L P2,la(U) gree k that converges weakly in L p, ,1 (U) to a form coo as m
asm -+oo. PROOF. Let a be an arbitrary exterior form of degree n - k - 1 in the class Co (U). For each m we have fWrn Ada =
(_l)c_l f dwmAa. u
U
Since corn -+ wo in L11oc as m - oo, it follows that fCOm
= (_1)c_I fu dwoAa
Ada
as m - oo. This leads to the conclusion that for any exterior form a E Co
of degree n - k - I lim
f
m-oo U
dwrn A a =
f
dwo A a.
U
The lemma is proved.
LEMMA 4.9. Let U be an open set in R", and let fm = (fm 1 ..... frnk ),
1 < k < n, m = 1, 2, ... , be a sequence of vector-valued functions of class W1. Ia(U),
where p > k, that is locally bounded in WP(U). Assume that frn I converges in L to a vector-valued function fo = (fol. . fok) as m -' 00, and let cum = dfmI A ... A dfmk. Then the sequence of forms wm converges weakly in Lp/k,la(U) to the form coo.
PROOF. We prove the lemma by induction on k. The lemma is an immediate consequence of Lemma 4.8 in the case k = 1. Assume that the lemma has been proved for some k, and let fm : U - Rk+I be a sequence of vector-valued functions of class WP IC(U), p >_ k + 1, which is locally
bounded in Wa and locally convergent in LI to a function fo. By the Sobolev imbedding theorem, fm -, fo also in L p,la. Let fm(x) = (fml (x), .fm2(x), ... , ,fm.k+I (x))
In Rk+l we consider the exterior forms u = dYI dY2...dyk, w = U A dyk+I = dYI
V = (-1)kyk+IU,
... dYk dYk+I
II. MAPPINGS WITH BOUNDED DISTORTION
140
Then w = dv for each k. Let Wm =fmu=dfml Adfm2A...Adfmk wm =fine = (' l )kfm.k+IWm,
Com = fmw.
Then wm = dyrm for each m, by Lemma 4.7. The induction hypothesis gives us that w, - 60 in Lp/k.lc for each k as m - 00. On the basis of the Sobolev imbedding theorem, fm.k+I -' fo.k+1 in L,.1« for s < np/(n - p). Let 0 be an arbitrary COO-form of degree n - k with compact support in U. We have that Co. -' Wo weakly in Lp,k.Ioc as m - oo, and fm.k+10 ` fo.k+10
in L. Let us show that as m - oo fm.k+IWm A 0 --+
1
.f0.k+1Wo A 0.
u
Indeed,
f
U
fm.k+IWm A0-
rU
JJO.k+1WmA0:
CIIWmIIL,,k(A)IIJm.k+l - fO.k+l II L,(A).
where A is the support of the form 8 and s = p/(p - k), 1 < s < p. From this,
A 8J fO,k+Im A 0
0
(4.6)
U
as m - oo. Let f E Co be such that III - fo.k+I IIL, < e. Then
f JO.k+IWmA0-J U
fo.k+IWOA01 U
(fOA+1-f)WmA8I+Ifu .f(6JmA0-6joA8)
fu +I If (f -f0.k+l)WOA0I. U
The first and third integrals on the right-hand side are each less than Ce, and the second tends to 0 as m - oo. Since e > 0 is arbitrary, this implies that
Jf0.k+I0)mA0JfOk+i(0OA0.
(4.7)
It is obvious from (4.6) and (4.7) that (4.5) holds. Since the form 0 E COO(U) is arbitrary, this implies that the sequence of forms Wm = fm.k+ 1 d 0m, m = 1, 2, ... , converges weakly in L11(U) to the form yro =
fo.k+1 Coo as m - oo. By the condition of the lemma, the sequence of forms wm = dyrm is locally bounded in Lp/(k+1) This implies on the basis of Lemma 4.8 that com wO weakly in Lp/(k+l).1oc(U), and the lemma is proved.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
141
COROLLARY. Let fm : U - R" be a sequence of mappings of class Wn
1
(U). Assume the following conditions:
a) (fm) is locally bounded in W (U). b) (fm) converges in L I to some mapping fo as m - 00. Then the limit mapping fo belongs to Wn 10C(U), and for every continuous
real function 9: U - R with compact support in U u as m
9(x)f (x, fm) dx - fu 91(x)f (x, fo) dx
oo.
PROOF. The fact that the limit mapping fo is in Wn 1oc(U) follows immediately from the general properties of functions with generalized derivatives. Let fm = (fm i , ... , fmn ), m = 0, 1, 2,..., and let wm = dfmI A ... A dfmn, m = 0, 1, 2..... Then corn -4 coo weakly in L1.1«(U), by the lemma. This means that fu ipcvrn - fu qicvo for every function 9' E Co(U). It remains to see that cum(X) = F(x, fm) x dxI dx2 ...dXn, and the corollary is proved. §5. Mappings with bounded distortion and elliptic differential equations §5.1. A description of a certain class of functionals of the calculus of variations. The real and imaginary parts of an analytic function of a single complex variables are well known to be solutions of the Laplace equation. Here we establish that an analogous fact holds also for arbitrary mappings with bounded distortion. We first give some facts about differential equa-
tions in the form we need. Proofs of the statements about the relevant equations are also presented in part in Chapter III. Some theorems establishing a connection between mappings with bounded distortion and extremal functionals of the calculus of variations are proved. Let U be an open set in R. We consider functionals defined on the set W1110C(U) of functions and representable in the form IF (u, A) =
jF[x.u'(x)]dx.
(5.1)
where A c U is a measurable set, and F(x, q) is a nonnegative function of the variables (x, q), defined for almost all x E U and any q E R. Some restrictions will be imposed on F(x, q). First of all, we assume F(x, q) is such that f [x, u'(x)] is measurable for any u E W110C(U). For studying
II. MAPPINGS WITH BOUNDED DISTORTION
142
mappings with bounded distortion the most important case is that when F(x, q) has the form
fn
n/2 n
F(x, q) = > > O (x)gigj l t=1 J=1
(5.2)
J
where the quadratic form on the right-hand side satisfies the following conditions: the functions 0;j(x) are defined and finite almost everywhere and are measurable in U, and there exist constants A, and A2 such that
0
n
e,;(x)grg; < A2IgI2
2q2<
(5.3)
i=1 j=I
for any x such that all the quantities B;j(x) are defined and finite, and for
allgERn. Although we shall need only the case when F(x, q) has the special structure (5.2) in what follows, it is more convenient to carry out the arguments concerning the properties of the functional (5.1) in general form. We list
the conditions on F(x, q) which ensure the properties we need for the functional (5.1).
A,. There exists a set E C U with mess E = 0 such that F(x, q) is defined and continuous as a function of q in the whole space Rn for every x E, and F(x, q) > 0 for any x and q with x V E. A2. F(x, q) has partial derivatives Fq, = OF/coq;, i = 1, ... , n, continuous in the variables q1, ... , qn for x ¢ E. Let Fg(x,q) = (F,,(x,q).Fg2(x.q),...,Fq.(x,q)).
A3. For any function u E W11 (U) the functions x - F[x, u'(x)] and x + Fg[x. u'(x)] are measurable in U. B. For all x it E (where E is the set of measure zero indicated in condition A,) the function F(x, q) is essentially convex with respect to the variable q, i.e., F(x, 41 +µq2) <_ AF(x, q1) +,uF(x, q2)
for any q1, q2 E Rn and any numbers A and u such that A, p > 0 and + it = 1, with equality if and only if q1 = q2. C. There exists a number p > 1 such that for x io E and any q a1IgIP < F(x, q) <_ a2Igf° + c(x),
(5.4)
where a, and a2 are positive constants, and c(x) is an integrable function.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
143
D. For x %E IFq(x, q)l <_
a3lgj°-1.
(q, Fq(x, q)) >_
a4IgIP,
(5.5) (5.6)
where p > 1 is the same as in C, and a3 and a4 are positive constants. E. F(x, q) > F(x, 0) for x ¢ E and for any q. In what follows we assume that F satisfies all the conditions A, B, C, D, and E here, though some of the statements below are true for the case when F satisfies only part of these conditions. For brevity a function F satisfying all the conditions A-E simultaneously will be called a normal
kernel. Conditions A-E hold with p = n in the special case when F is defined by (5.2), where the matrix function 0 satisfies (5.3). Let F(x, q) be a normal kernel, and V an open set lying strictly inside U. The number IF (14, V) = J F[x, u'(x)] dx V
is defined for every function u E WP' (V). We consider the following boundary value problem for the functional IF: among all the functions taking given values on the boundary of V, find the one which gives IF the smallest possible value. A function satisfying
this condition is called an extremal function for the functional IF. However, it is first necessary to specify what it means to say that some function u E W,(U) (it can even be discontinuous in general) takes given values on the boundary of V, which can in turn to be a fairly bad set. The necessary precise definitions will be given in §5.2. Let F be an arbitrary normal kernel. A function u E WP 1a(U) is called a generalized solution of the equation
8 j8F[x,u'(x)]}=0 ex,
ai
(5.7)
in U if the equality 1u
En OF`[x, u'(x)] oxj (x) dx = J(Fq[x, u'(x)],,'(x)) dx = 0
(5.8)
-i
holds for every function q E W, ,(U) with compact support in U. Equation (5.7) is none other than the Euler equation for the functional (5.1). As will be shown below, every extremal function for (5.1), where
F is a normal kernel, is a generalized solution of (5.7). The converse assertion is also true. Equation (5.7) will be written briefly as follows: div Fq[x, u'(x)] = 0.
(5.9)
11. MAPPINGS WITH BOUNDED DISTORTION
144
If U E W, (V) is a generalized solution of (5.7), then we say that u is a stationaryfunction for the functional IF. We mention a special case of functionals of the form (5.1) that is important for what follows. By the use of concepts introduced in §4, the condition that u be a generalized solution of (5.7) can be represented as follows. We introduce an exterior form WF(u, x) of degree n -1 by setting "
E(-1)k-IFq,[x,u'(x)]dxl
(OF(u,x) =
(5.10)
k=I
... dxk- I dxk+I ... dx,,. Then (Fq[x, u'(x)], q'(x)) dxl dx2... dx, = d)7(X) A cuF(u, x).
Therefore, satisfaction of (5.8) for every q E Co (U) means that the generalized differential of the form (OF(u, x) is equal to zero. Thus, u is a generalized solution of (5.7) if and only if dcwF(u,x) = 0. We mention a special case of functionals of the form (5.1) that is important for what follows. Let F(x, q) = IqI". All the conditions A-E are satisfied here. Equation (5.7) takes the form
n div[fu'(x)In-2u'(x)] = 0.
(5.11)
We have the following particular solutions of (5.11): ul(x) _
u2(x) = C11n Ix
a;x; + aa,
+ C2,
1
aI i=1
where ao, a1, ... , an, CI, C2 E R and a E R" are constants.
§5.2. Variational properties of mappings with bounded distortion. It is known that if h is a harmonic function on the plane of the complex variable
w and f (z) is a holomorphic function, then h[f (z)] is also harmonic. The goal of this subsection is to establish an analogue of this property for mappings with bounded distortion.
We first present some heuristic considerations. Let f : U - R" be a diffeomorphism, and let V = f (U). Assume that u E W (V) minimizes the functional (5.12)
L I u'(y)I" dy
among all functions coinciding with u(x) on the boundary of V. Let v (x) = u[f (x)]. For each i = l , ... , n
a ;;
"
(x) _ c a=I
a (f(x)]
exi W.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
145
This equality can be written as follows in matrix form:
v'(x) = f"(x)u'[f (x)].
(5.13)
Making the change of variable y = f (x) in (5.12), we get that fV
I u'(y)I" dy = f l u'[f(x)]I"If(x, f)I dx. U
Let us express u'[f (x)] in terms of v'(x) by using (5.13). We get that u'[f (x)] = [f''(x)]-1 v'(x). From this,
u'(f (x))I2 = (u'[f (x)], u'[f (x)]) = ([f''(x)]-'v'(x).
[f"(x)]-'v'(x))
= ([f'(x)]-' (f"(x))-'v'(x), v'(x)). Setting
0(x. f) =
f
[f"(x}]-1
(5.14)
we find that the integral in (5.12) is equal to
fu (0(x, f)v'(x), v'(x))"12 dx.
(5.15)
Assigning to a V E W (V) the function v o f, we get a one-to-one correspondence between the spaces W (V) and W (U) under certain natural assumptions about the diffeomorphism f. (As follows from what was proved earlier, this condition holds if, for example, sup I f'(x)I < oo and inf If (x, f )I > 0; in this case the mapping f is quasiconformal.) Therefore, if u is extremal for the functional (5.12), then v = u o f is extremal for (5.15). Equation (5.9) has the following form for the functional (5.15):
div((O(x, f)v'(x),V'(x))"/2-10(x, f)v'(x)) = 0.
(5.16)
This is a relation between the derivatives, and the supposition naturally arises that it can be obtained by formal transformations, without resorting to the fairly stringent assumptions about f that were made above. It is the purpose of the following arguments to prove this supposition. Let f : U R" be a mapping with bounded distortion, where U is an open subset of R". The matrix f'(x) is defined for almost all x E U, and If'(x) I" < K.,,,' (x, f).
(5.17)
We construct a matrix function 0(x) __ 0(x. f) = 0f (x). Let X E U be such that f'(x) is defined and (5.17) holds. If F(x, f) = 0, then let 0(x) = I (the n x n identity matrix). But if F(x, f) 96 0, then 0(x, f ) is determined by (5.14). The matrix-valued function 0(x) is defined for almost all x E U. Let 0(x) = (01 (x)), i, j = 1, ... , n. The elements 0, of the matrix 0(x) are measurable functions on U.
11. MAPPINGS WITH BOUNDED DISTORTION
146
We establish certain properties of the matrix-valued function 0(x) 0(x; f) constructed from an arbitrary mapping f : U - R" with bounded distortion. First, a simple observation about matrices. Let X be a nonsingular n x n
matrix, and let Y = X''. Then XY = I, which implies that I = (XY)' _
Y'X', and hence Y' = (X')-', so that (X-')' = (X')-'. We show that the matrix 0(x) is symmetric and positive-definite, and we
find bounds for its eigenvalues. If f (x, f) = 0 at x, then 0(x) = I, and in this case 0(x) is a positive-definite symmetric matrix with all eigenvalues equal to 1. Suppose that F(x, f) 96 0, and let [f (x, f )]2/" = A and
[f'(x)]-' = L. Then [f'(x)']-' = L' and 0(x) = ALL'. From this, 0(x)' = A(LL')' = ALL' = 0(x), and the symmetry of 0(x) is proved. For an arbitrary vector # 0
A(L', L')
= AIL`I2.
Let 0 < k1 < ... < k" be the principal dilations of the mapping f'(x). Then the principal dilations of the mapping L' = (f'(x)')-' are equal to 1/k1 > ... > 1/k", and hence ICI/k1 > ICI/k" for every . We have ... k")21". This leads to the conclusion that for every
that A = (k1
AI I2/ki (0(x)5) >_ ? The quadratic form (0(x)i, ) is thus positive-definite. We now remark that AgI2/k.2.
l n/2
CkA'
= (k1k2... kn)Iki < Ko(f).
I (A/kn)n/2
=
(K(f))-'
From this, VV E R"[Ko(f )]2/" I I2 ?
[K(f )]-2/" I
I2.
(5.14)
In particular, the eigenvalues of the matrix 0(x) lie in the interval
[K(f) -2/n , K0(f )2/n] THEOREM 5.1. Suppose that U and G are open subsets of R", and f : U G is a mapping with bounded distortion. Assume that the function v E C' (G) is a generalized solution in G of the equation div[Iv'(Y)In-2v'(Y)] = 0.
(5.15)
Then the function u = v o f belongs to W 1a(U) and is a generalized solution in U of the equation
div[(0(x, f)u'(x). u'(x))"l2-' 0(x, f)u'(x)].
(5.16)
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
147
PROOF. The fact that u = v o f belongs to the class Wp 10C(U) follows from Theorem 2.8 of Chapter I. This theorem gives us that for almost all
XEU n
-axi
k=1
This means that the vectors u'(x) and v'[f(x)] are connected as follows:
u'(x) = [.f'(x)]*v'[f(x)].
(5.17)
the vector (8(x)u'(x), u'(x)),12-18(x)u'(x), and by co the
Denote by exterior form
1)J-%dxI
...dxj-1 dxj+1 ...dxn.
j=1
It is required to prove that the generalized differential form co is equal to zero. Let a(y) = Iv'(Y)In-2V'(Y), a(Y) = (a, (y),.-, an(y)), where
aj(y)
=
Iv'(Y)I"-2
0Yj
(Y).
The form
t(Y) =
r(-1)'-1o'(Y)dY1 ...dY;-1 dY'+1 ...dYn i=1
is defined in G. The condition that v is a generalized solution of (5.15) means that dr = 0. The theorem will be proved if we establish that
w(x) = (f'r)(x).
(5.18)
We have n
(f'Q)(x) _
>(-1)'-'Q'[f(x)]dfl A... Adf-, Adl+1 A... Adfn. (5.19)
Suppose that the point x E U is such that the derivative fl(x) is defined
there and (5.12) holds. If f (x, f) = 0, then f'(x) = 0. It follows from the preceding formula that (f'a)(x) = 0. By (5.17), u'(x) = 0 in this case, and hence w(x) = 0. Thus, (5.18) holds if f (x, f) = 0. Next, assume that f (x, f) 0. Let F be the matrix with element F; j equal to the cofactor of
the element (8f /8xj)(x) of the matrix f'(x). Then F = f(x, f)(f")-1. Let A = (A I , ... , A,,) be an arbitrary vector, and let u be the exterior form
Al dx1 + ... + An dxn.
II. MAPPINGS WITH BOUNDED DISTORTION
148
Then
,U A(f`c)(x) =E(-1)'-'a1[f(x)],U A dfi A...Adf_1 Adf+I A...Adfn i=1 n
a[f(x)]dfi A Adf-I nµndf+I A...Adfn, i=t
(5.20)
We introduce the notation f j = a . Then
df1 A...Adf-I A Adf+I A...Adfn ........................
,f-I.n dx1 dx2...dxn.
A1,A2,...,An
,fn.n
Expanding the determinant here with respect to the minors of the ith row and substituting the result into (5.20), we find that (n
n
µ A (f*a)(x) = L F Fij(x)oi[f (x)]Aj dx1 dx2 ... dxn i=1 j=I
(5.21)
= (F' (x)a[f (x)], A) dx1 dx2... dxn. Let
n
E(-1)'-'ai(x)dx1 ...dxi-1 dxi+I ...dxn
(f*o)(x) = i=1
be the canonical representation of the form (f *a)(x), and let 8(x) _ (a1(x), ... , an(x)). Then µ n (f'Q)(x) = (P (X), A) dx1 dx2... dxn.
(5.22)
Comparing (5.22) and (5.21), we get that
f (x) = F (x)Q[f (x)] = f (x, f)(f')-'a [f (x)] We have
o(Y) = which implies that
Iv'(Y)ln-2v'(Y),
u'(x) = (f')-'v'[f(x)],
u'(x), (x. f)(f')-I (f,.)-lu'(x) = (9(x)u'(x), u'(x))"12-1 9(x)u'(x). This establishes (5.18), and the theorem is proved.
fi(x) =
u'(x))(n-2)/2
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
149
§5.3. The classes Wp (U/A) and W I (U/A). We define what it means (U) to coincide on some closed set for two functions u and v of class WI(U)
A c (l, and also what it means to say that u > v on A. Throughout this subsection U denotes a bounded open set, V (as usual) is the closure of U, and 8U is the boundary of U. Let A c U be a given nonempty closed set, and let u be a function of class Wo (U). We say that u belongs to the class W I (U/A) if there exists an open set V D A such that u(x) = 0 almost everywhere on U n V. The closure of W I (U/A) in the space WP (U) is denoted by W p (U/A). Note that a linear combination of two functions in WI (U/A) is a function in Wp(U/A). This implies that WP (U/A) is a subspace of the vector space W,(U). Let u and v be two functions in Wp (U), and let A c V be a closed set. We say that u = v on A and write u = VIA if the difference u - v belongs to Wp (U/A). We mention an important particular case. Let A = 8U. In O
this case we set Wp (U/A) = WD (U/8U) is denoted by W ,(U).
We remark that if a function u is in W p (U/8U), then there exists a sequence (um), m = 1, 2, ... , of functions in Co (U) convergent to u in W,(U). Indeed, let V be an open set containing 8U and such that u(x) = 0 almost everywhere on U n V, and let G = U\V. The set G is compact. Let b = dist{G, R"\ U} > 0. The averaging uh of u is a function of class C°°(Uh) with compact support in Uh for sufficiently small h. Let us extend
uh to the whole of U by setting uh(x) = 0 for x ¢ U. Let (hm), m = 1,2,..., be a sequence of values of At > 0 such that h", - 0 as m oo. Then the sequence of functions u, = uh,, is obviously the desired sequence.
We thus get that the set Co (U) is dense in WI (U/8U), hence also O
O
in Wp(U/8U) =W ,(U). This lets us conclude that u is in W ,(U) if and only if there exists a sequence (um) of functions in Co (U) such that hjum - ujjl,v,u -+ 0 as m -1 oo. Let U be a bounded open set in R", and u a function of class W ,(U). We extend u to R" by setting uo(x) = 0 for x V U, and uo(x) = u(x) for x E U. The function uo will be called the zero extension of u. We prove that uo is a function in Wp W). Indeed, if u E Co (U), then by setting u(x) = 0 for x t! U we obviously obtain a function in Co (R") whose norm in W, (R") coincides with that of u in WP (U). This implies that if a sequence of functions u in Co (U) converges in W,(U) to a function u, then a sequence of functions converging in WpI (R11) is
obtained by extending the functions u to R" by setting
0 for
150
II. MAPPINGS WITH BOUNDED DISTORTION
x ¢ U. Its limit is some function uo. Clearly, uo(x) = 0 for x ¢ U, and uo(x) = uo(x) for x E U, i.e., uo is the desired extension of u. Let u be a function in Co (R") whose support is contained in a ball of radius r > 0. Then, as shown above, IIuIIP.Rn < 2rIIu'(x)II PRn.
This implies that if the set U c R" is bounded, with U contained in a ball O
of radius r > 0, then for every u EW 'IV (U) II uII p.u < 2rII u'(x)II p.u,
(5.18)
O
which implies that for u EW I (U) IIuII1.p.U <- Cll u'II p.u
(5.19)
where the constant C depends only on the radius of the smallest ball containing U. LEMMA 5.1. Let U be an open subset of R". Then every sequence v = 1, 2, ... , of functions in W.110C(U) which is locally bounded in WP, where p < n, has a subsequence which converges locally in L,, where r < np/(n - p), to some function uo E WP 1c(U).
PROOF. Let (Bk), k = 1 , 2, ... , be a sequence of balls such that the closure of B k is contained in U for each k and U;° Bk = U. We use induction to construct subsequences (Uk,,), v = 1 , 2, ... , of the original sequence (u,), v = 1 , 2..... Let (u0,,, ), v = 1, 2, ... , be the original sequence (u,). Assume that the sequence (Uk,,), v = 1, 2, ... , has been defined. Then the norm sequence (Iluk,, II I.P,B*,, ), v = 1, 2, ... , is bounded, by a condition of the lemma. Hence, by the Sobolev-Kondrashov theorem on compactness
of an imbedding, (uk.,,) has a subsequence convergent in L,(Bk+1) Let (uk+l.,,) be this subsequence. The diagonal sequence u1,1, ... , u,,,,, , ... is a subsequence of v = 1, 2, ... , converging in L, on each of the balls Bk,
and thus on any compact subset of U, because every compact set A c U can be covered by finitely many balls in the sequence Bk, k = 1, 2..... The lemma is proved. The definition of what it means for a function u E Wp to be nonnegative on the boundary of U in the case when this boundary is a fairly "bad" set is based on Lemmas 5.2 and 5.3. LEMMA 5.2. Suppose that u is a function of class WP 1a(U), where p > 1,
and cp : R - R is a CI function with bounded derivative gyp'.
Then the
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
151
function v(x) = q [u(x)] is also in WI 10C(U). Further, the derivatives of v can be expressed in terms of those of u by the formula
Y (x) _ cp'[u(x)Ja
8x;
(5.20)
W.
PROOF. Suppose that u E Wp 1OC(U). We construct a sequence (up), v = 1, 2, ... , of functions of class C°°(U) that converges to u in Wp 1a(U) as v -+ oo. Let V be an arbitrary open set lying strictly inside U. Then 8u" 8u -a
Iluv-uIID.I'-40,
8x; II p, V
I 8x;
0
asv - oo. Let K < oo be such that I9'(u)I < K for all
Define U E R. Then
I9,(ul) - 97(u2)I <- Klu1 - u21
for any uI , U2 E R. This implies that v,,(x) - v(x)I <- KI u,,(x) - u(x)I. oo. Each of the functions v belongs and hence 11v, - v II p v -' 0 as v
to the class C', and for every v
i = 1.2...
a, (x) = 9P'[u,(x)] a (x),
. n.
This leads to the conclusion that
? (x) - l [u(x)]a
(x) <-1 s [u,(x)ll I
a (x) a
(x)I
;
+ 1 [1v'[u.,(x)] - 9, [u(x)ll
I
e (x)
As a result we get the estimate
8v II
8x; -
l (u)
8u 8x; II p. v
:5
K ax; I
8u 8x;
11p
v P
+ (/ 1§9'[U-,(x)1 - co'[u(x)]I p
I/D
(5.21)
I a (x)I dx)
The first term tends to zero as v -. oo. The functions u converge to u in measure on V as v - oo. It follows from this that also 9' o u,, -+ ?' o u 9"'[u(x)]I < 2K for all x E U. This in measure. We have that enables us to conclude that the second term in (5.21) also tends to zero as The set V lying strictly inside U was chosen arbitrarily; hence we have
proved that the functions 8v,,/8x; converge in Lp,io to the function
11. MAPPINGS WITH BOUNDED DISTORTION
152
p'(u)(8u/8x;) as v -' oo. This allows us to deduce that the function (ip' o u)Ou/ex; is the generalized derivative 8v/8x; of v. It is thereby established that the function v = ip o u belongs to W, 1OC(U). Further,
a
(x)
='P'[u(x)]a (x)
,t
for all x E U, and the lemma is proved. LEMMA 5.3. Suppose that u is a function of class Wp 1.(U), p > 1. Let
u+(x) = max{u(x), 0} = 1{ju(x)l + u(x)}. Then u+ belongs to Wp1OC(U), and its partial derivatives can be expressed for almost all x E U in terms of the derivatives of u according to the formula au+
ax; (x) _ =[u(x)]ex, (x).
where r(u)= 1 foru>0,andr(u)=0foru<0. PROOF. Let 0 be a nonnegative C°°-function on R such that 8(x) = 0 for x g [0, 1] and f f. 0(x) dx = 1. Let h > 0, and define
rh(u= hJJru )
1
B
1
00
(!-) h
dt,
a h( h(
)-J
00
T1 h( ) dt.
Obviously, 0 < rh(u) < 1 for all u, rh(u) = 0 for u < 0, Th(u) = 1 for u > h, and the function ah is increasing. We have that rh(u) --- r(u) as h -a 0 for any u, where r is the function indicated in the statement of the
lemma: r(u) = 0 for u < 0, and r(u) = 1 for u > 0. Note, further, that
ah(u)=0for u<0,and 00;ah(u)-.u+ ash-'O. Then u+ - h < ah(u) < u+ and ah(u) > 0 for all u. By Lemma 5.2, the functions ah[u(x)] belongs to Wp1a(U). Further, 8(ax0 u) (x) = Th[u(x)]
(x)
for all x. We have that ah(u) -+ u+ and rh(u) - r(u) as At -, 0 for all u. This implies that the function ah o u converge in LP.1OC(U) to u+, and their derivatives 8(ah o u)/8x; converge in LP.1oc to (r o u)&/8x;. This gives us that u+ E WP 10C(U), and 8x; (x) =
The lemma is proved.
r[u(x)]
a W.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
153
I f U E W'(U), then u- = max{-u, 0} is in W, (U). For any two functions u. v E W, (U) the functions COROLLARY 1 .
max{u, v} = 1(u + v + lu - vi), min{u, v} = '(u + v - iu - vi) belong to W,(U). COROLLARY 2. Suppose that u E W, (U), and let
Ek = {x E Uiu(x) = k},
where k E R Then the partial derivatives 8u18x; of u(x) vanish almost everywhere on Ek. PROOF. Let
v(x) = max{u(x), k} = u(x) + [k - u(x)]+ = k + [u(x) -
k]+.
Then for almost all x E U the partial derivatives (8v/ax;)(x) are defined. Using the formula v(x) = u(x) + [k - u(x)]+, we have on the basis of the lemma that
a
(x) = a (x) -:[k - u(x)]a W.
For X E Ek the quantity k - u(x) is 0, and hence t[k - u(x)] = 0. This implies that
a (x) = a (x) for almost all x E E. Using the formula v(x) = k + [u(x) - k]+, we get that for almost all x E U (x) = z[u(x) - k]a (x). The function T[u(x) - k) is zero on Ek, and hence for almost all x E Ek
e (x) = 0, ,
i.e.,
a a (x) =
(x) = 0,
almost everywhere on E, which is what was required to prove. LEMMA 5.4. Let (um ), m = 1, 2,..., be a sequence of functions of class W, (U) that converges in W, (U) to a function uo E W, (U). Then u+ -+ u+
in W4(U)as m--*oo. PROOF. By the condition of the lemma,
f
ju,,,
m
fuI ax;
ax;I
F
dx --+ 0
II. MAPPINGS WITH BOUNDED DISTORTION
154
asm - oofor all i = I-_ m. Since Ium(x) - uo (x)I : I U. (x) - uo(x)I
for any x, this implies, first of all, that
fIu-uIt'dx--'O
asm-ioo.
Further, for each m 12um
axi
auo 1 < T(UM) aura
axi
axi
au0
axi
+IT(um)-T(uo)II a 00I
From this we conclude that
axiIIp.U
Maxi
+
f
U
axm
axjIIp,v (x)Ip
(IT[U.(x)l - t[uo(x)]I p axio
dx\
I/P
(5.22)
f
I
To prove the lemma it suffices to establish that the integral on the right-
hand side of the last inequality tends to zero as m - oo. The sequence (um) has a subsequence for which this integral tends to its limit superior and then we choose a subsequence (umk) of this subsequence such that umk (x) - uo(x) almost everywhere. Let E0 be the set of all x E U such that uo(x) = 0. Then, by Corollary 2 to Lemma 5.3, auo(x)/axi = 0 almost everywhere on E0, and thus IT[um(X)] - T[uo(x)]I
oa (x)
=0
almost everywhere on E0. Suppose that x
E0 and that umk (x) -, uo(x) for this x. Since x 19 E0, it follows that uo(x) 96 0, and hence T[umk (x)] T[uo(x)] as k - oo. We get that IToumk -Touol (all/axi) tends to zero almost
everywhere in U as k -+ oo, which, by Lebesgue's theorem on passing to the limit, implies that
I`
( Toum-Tool
all
dx - 0
axi
as k cc. It follows from what has been proved that the integral on the right-hand side of (5.22) tends to zero as m -' oo. The lemma is proved.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
155
COROLLARY 1 . Let (um), m = 1, 2 , ... , be a sequence of functions of class W 110C(U) that converges in WP 10C(U) to the function uo. Then um --- uo in
WW.1,(U)asm-oo. COROLLARY 2. Let (u,n ), m = 1, 2, ... , be a sequence of functions of class
W, (U) which converges in W, (U) to a function uo E WP (U) as m - oo. IuoI in WP(U) as m - oo. Further, if (um) Then um - uo and I u.I and (vm) are two sequences of functions of class Wo(U) which converge in W,(U) to the respective functions u and v in WPI (U), then max{u,n, vm} - max{u, v}, min{um,vm} - min{u,v}
in W,(U)as m-oo. COROLLARY 3. Let (u,n), m = 1, 2, ... , be a sequence of functions of class Wp 1a(U) which converges in WP 1a(U) to a function uo E Wp 1c(U) as
m - oo. Then um - uo, um - uo and luml - Iuol in Wp.l.,(U) as m - oo. If um and vm are two sequences of functions converging in W,1( U) to the respective functions u and v of class Wv 10C(U), then max{um, vm} as m - oo. max{u, v} and min{um, vm} - min{u, v} in Wp LEMMA S.S. Suppose that the functions u, v E WpI (U) are such that 0
u(x) < v(x) for almost all x E U. If v E WP(U/A), where A c U and A is closed, then u is also in WP (U/A).
PROOF. Let (vm), m = 1, 2, ... , be a sequence of functions of class I,(U/A) which converges to v in W.' (U). Let um(x) = min{u(x), vn(x)}. For each m the function um is in WpI (U) in view of Corollary I to Lemma 5.3. For every m there is an open set Vm D A such that vm(x) = 0 almost everywhere on Vm f1 U. This implies that u,n(x) = 0 almost everywhere on Vmf1 U, i.e., Um E T,(U/A) for each m. We have that u, - min{ u, v} = u in Wp (U) as m - oo, which gives us that u E WD (U/A), and the lemma is proved.
Let u be an arbitrary function of class WP (U). Then we say that u E W I (U/A), where A C U is closed, if its negative part u- vanishes on A, i.e., u- E Wp(U/A). In this case we also say that u is nonnegative on A and write u > 01A. If u and v are functions in WP (U), then we say that u > v IA if u - v > 01A. In other words, the relation u > v lA means
that (u - V)- E Wp (U/A). The set W P (U/8U) will simply be denoted by
W ,(U) below. The definition implies that, in particular, if u E W'(U)
1S6
11. MAPPINGS WITH BOUNDED DISTORTION
and there exists an open set V D A such that u- (x) = 0 almost everywhere
on VnU,then uEW,(U). LEMMA 5.6. For every A C U the set W+P(U/A) is a closed convex cone in the Banach space Wo (U).
PROOF. We are required to prove the following three facts: a) W ,(U/A) is a closed subset of W,(U).
b) Au EW p(U/A) for any u EWP(U/A) and any A > 0, A E R.
c) The sum of any two functions u and v in W p(U/A) is in W I (U/A). We first prove a). By Corollary 2 to Lemma 5.3, the mapping 0: u
u-
of the space W,(U) into itself is continuous. The set W ,(U/A) is the complete inverse image of Wo (U/A) with respect to 0, and since Wp (U/A)
is closed, WP(U/A) is also closed in WD (U). Let u EW ,(U/A). For every A > 0 the relation (Au)- = Au- holds; hence if u- E WD(U/A), then also
(Au)- E Wp(U/A), i.e., if u EW ,(U/A), then also Au EW ,(U/A).
We now prove c). Suppose that u, v EW ,(U/A), and let w = u + v. Since u = u+-u- and v = v+-v-, it follows that w = (u++v+)-(u-+v-). From this, 0 < w-(x) < u-(x) + v-(x). The function u- + v- obviously belongs to Wp(U/A). By Lemma 5.5, this implies that w- E WP(U/A), and thus w EW ,(U/A), which is what was required to prove. COROLLARY. Let A C U, and let u, v, and w be functions in W PI (U). Then the following statements hold:
1. If u>vjAand v>uIA, then u=vIA. 2. Ifu>vjAandv>wIA, then u>w1A. 3. Ifu>vJA, then Au >AvlAfor allA>0. 4. If u > VIA, then for every function w E WP (U)
u+w>v+wIA. The proof is left to the reader.
LEMMA 5.7. Let U be a bounded open set in R", and let A c r and B c V be closed sets. If U E Wp (U/A) and U E Wp (U/B), then u E Wo(U/(A u B)).
PROOF. Let (v,") and (w,") be sequences of functions in W.' (U) convergent to u in W.' (U) and such that each v. vanishes in a neighborhood
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
157
of A, while each wm vanishes in a neighborhood of B. Let min{vm, wm), 71m - min{vm, wm }.
Then clearly each of the functions m and rim vanishes in a neighborhood of A U B. We have that m -' u+ and rim -, u- in Wp (U) as m -' oo. This implies that u+ and u- are in W1(U/(A U B)), and hence u = u+ - u- E WD(U/(A U B)), which is what was required to prove. Lemma 5.3 can be extended to the case of functions of the form F o u, where F is an arbitrary piecewise smooth function. A function F : R -+ R is said to be piecewise smooth if F is continuous and there exists a finite set E such that the derivative F(u) is defined and continuous for every
point u io E, and the limits F'(u - 0) and F'(u + 0) exists at each point U E E and are finite. Obviously, if u E E, then the derivatives FL'(u) = hlim0
FR(u) = hlimo
flu +
h
- F(u) = F'(u - 0),
F(u + h) - F(u) h
= F'(u + 0)
exist at u.
LEMMA 5.8. Let F : R - R be a piecewise smooth function. Assume that there exists a constant k < oo such that IF'(u)) < k at each point u where the derivative F'(u) is defined. Then for every function u E W110C (U) (where U is an open set in R) the function v = F o u belongs to Wp10C (U),
and for almost all x E U av 8x;
(x) = FL[u(x))
8u
8x, W.
(5.23)
REMARK. Equality (5.23) remains true if the left-hand derivative of F in it is replaced by the right-hand derivative. PROOF. Let F = ah` * F, where a is an arbitrary averaging kernel on
R, and h - 0 as v - oo. Then the sequence of functions F,, converges to F uniformly in R as v - oo, IF;(u)I < k for all v, and F;(u) -+ Fl(u) at each point u E. Let v = F,, o u. By Lemma 5.3, the functions v are all in Wn 1a(U). Further,
a (x) = F,[u(x)] a (x)
almost everywhere in U.; We have that v,, -' v uniformly as v - oo and the sequence v is clearly locally bounded in W'P' 1,C(U). On the basis
of Theorem 1.1 in Chapter I, this implies that v E WD1a(U). Let E = {u1,... , um) be the finite set of points where F is not differentiable, and
158
II. MAPPINGS WITH BOUNDED DISTORTION
let B;= {xEUju(x)=u;}andB=UrB;. If xio B, and u(x) and (8u/8x;)(x) are defined, then ev, (x) ax;
- F'[u(x)]a (x)
as v - oo for this x. If CBI = 0, then this proves that the right-hand side of (5.23) is the limit of (x) as v - oo. Assume that JBI > 0. Then, by (x) = 0 almost everywhere on B. Hence, Corollary 2 to Lemma 5.4,
a
a (x) -' 0 = FL[u(x)]a (x) for all x E B. Thus, the ;-derivatives in U to the function
converge almost everywhere
x - FL[u(x)]a (x) as v - oo. This establishes that
a (x) = FL[u(x)]a (x) ;
almost everywhere in U, and the lemma is proved. §5.4. The Dirichlet problem, extremal functions, and generalized solutions of the Euler equation for functionals of the calculus of variations. Let
U c R" be an arbitrary open set, and let F(x, q) (x E U, q E R") be a function satisfying all or some of the conditions A-E (see §5.1).
THEOREM 5.2 (Semicontinuity of the functional IF). Assume that F satisfies conditions A. B, and C. Let v = 1, 2, ... , be a sequence of functions of class W. (U) converging in L 11 (U) to a function uo E W, (U) as v -' oo. Then for every measurable set A c U IF(UO, A) < lim IF(u,,, A). V -00
Proofs of this theorem and of certain more general results will be given in Chapter III, §3.3.
Using Theorem 5.2 and some results in § 5.3, we can now prove a theorem on the existence of a solution of the Dirichlet problem. For v E W, (U) denote by Wp(v, U) the collection of all functions f E W,(U) 0
such that f = vIOU. Formally, Wp(v, U) = v+ W I (U). THEOREM 5.3. Suppose that F satisfies conditions A, B, and C in §5.1. Then for every function v E Wp(U) there exists a function vo E W. (v, U)
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
159
at which the functional u - IF(u, U) takes its smallest value on the set Wp(v, U), i.e., IF(vo, U):5 IF(U, U) for any function u r: Wp(v, U).
PROOF. Suppose that F and U satisfy all the conditions of the theorem. Let y = inf IF (u, U). uEWv(v.U)
Since IF(u, U) > 0 for all u E WW(v, U) and IF(v, U) < oo, it obviously follows that y > 0 and y < oo. Let U, E Wp(v, U) be such that IF (u, U) < y + (1 /v), and let wv(x) = uv(x) - v(x). For each v the function wv is in the class WP(U). By condition C in §5.1, O
L Iu'(x)Ipdx < (1/a1)JuF[x, u'' (x)]dx
_ (1/al)IF(uv, U) < (1/a1)(y+(1/v)) 5 (1/al)(y+ 1).
From this, (5.24)
II w;II p.u <_ Ilv'Ilp.u + [(y + 1)/a1]11p = M< oo, O
where M is a constant. Since wv is in W ,(U) and U is bounded, it follows from (5.24), in view of (5.19), that the sequence (wv) is bounded in W'(U). This allows us to conclude that (wv) has a subsequence that converges weakly in WpI (U) to some function wo. In order not to complicate O
the notation, we assume that (wv) is this subsequence. Since W p(U) is convex and closed in Wp(U), it is also weakly closed there, which implies O
that wo is in W o(U). Let uo = wo+v. The function uv = wv+v converges weakly in W, (U) to uo as v - oo, and hence uv -, uo in L p.1a(U) in any case. By Theorem 5.2, this gives us that
IF(uo, U) < lim IF (u.,, F) < y. P- 00
On the other hand, IF(uo, U) > y by the definition of y. Consequently, IF(uo, U) = y. This proves the theorem. A function u E Wp(U) is called an extremal of the functional IF on an open set U if u gives IF the smallest possible value on the set WW(u. U), i.e.,
IF(v, U) > IF(u, U) for every function v E W,(U) such that v = uIOU. We now investigate the question of a connection between the extremal functions of IF and stationary functions, i.e., generalized solutions of the equation div[FQ(x, u')] = 0. This investigation is based on the following statement.
11. MAPPINGS WITH BOUNDED DISTORTION
160
LEMMA 5.9. Let V be a bounded open set in R", and let u and v be functions of class WP (V). Assume that the function F : (x, q) -' F (x, q) satisfies conditions A-D in §5.1. Then the function f (t) =IF(u + tyi, V) is convex and differentiable in R. Further, for every t E R
fl(t) =Fq[ x, u'(x) + tyr'(x)]) dx. f(yi'(x)
(5.25)
PROOF. We first prove that f is convex. Let 11, t2 E R and numbers A > 0 and p > 0 with A +,u = 1 be given arbitrarily, and let to = At 1 + 11t2.
Then, by condition B on F in §5.1,
F(x,u'+tow') =F[x,A(u'+tly/')+p(u'+t2yi')]
D for F. COROLLARY 1. If u is an extremal function for the functional IF in a bounded open set V C R", where F satisfies conditions A-D in §5.1, then u is a stationary function for IF on V.
PROOF. Indeed, let u be an extremal function for IF on V C R". Let 0 n EW PI (V) be arbitrary, and define f (t) = IF(u+trl, V). The function f is convex and differentiable on R. For each t E R we have that u+tn = uIBV. Since u is extremal, IF(U+tri, V) > IF(U, V) for all 1. This means that f attains a minimum at t = 0, and hence f'(0) = 0, i.e.,
f(J'(x).FqEx.u'(x)])dx
= 0.
0
Since I EW 10 (V) is arbitrary, the corollary is proved. COROLLARY 2. Let U E W p (V) be an extremal function for the functional
IF on V, where V is a bounded open set in R". If v is a stationary function for IF such that v = uIBV, then v coincides identically with u on V. PROOF. Let V E Wp (V) be a stationary function coinciding with u on the boundary of V. For all x E V we have that for 0 < t < 1
F(x, v'(x) + t[u'(x) - v'(x)]) < (1 - t)F[x, v'(x)] + tF[x, u'(x)].
(5.26)
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
161
Since F is essentially convex (condition B in §5.1), equality at a point x E V for 0 < t < 1 can hold if and only if v'(x) = u'(x). Let f (t) = IF [v + t(u - v), V]. Integrating (5.26) termwise, we get that
f(t) <(1 -t)f(0)+tf(1)= f(0)+t[f(1)- f(0)]
(5.27)
for t E [0, 1 ]. From this, [f (t) - f (0)]/t < f (1) - f (0). Passing to the limit as t -. 0, we get that f'(0) < f (1) - f (0). We have that
f'(0) = f(17'(x),Fq[x,v'(x)])dx, where q = u - v. Since v is a stationary function, it follows that f'(0) = 0, and thus
f(1)-f(0)>0.
(5.28)
Since u is an extremal function,
f (t) = IF[V + t(u - v), V] > f(1) = IF(U, V)
(5.29)
for all t E [0, 1], and, in particular,
f(0) > f(1).
(5.30)
Comparing (5.28) and (5.30), we get that f (1) = f (0). By (5.27), this implies that f (t) < f (1) for I E [0,1 ], and hence f (t) = f (1) = f (0) + t [ f (1) - f (0)] for all t E [0,1 ) on the basis of (5.29). Thus, we get equality
when we integrate (5.26) termwise. This implies that equality holds in (5.26) for almost all x E V. Hence, v'(x) = u'(x) for almost all x E V, and thus v - u is constant on V. Since v - u vanishes on the boundary of V, v - u =- 0, i.e., v = u, which is what was to be proved. COROLLARY 3. Let V be an open set lying strictly inside U, and let V E W,(U). Then the function u E Wp(v, V) for which IF (u, V) attains a minimum on Wp(v, V) is unique. §5.5. The maximum principle for extremals of functionals of the calculus of variations. If a harmonic function is nonnegative at all points of the boundary of a set U, then it is also nonnegative everywhere interior to U by the well-known maximum principle for harmonic functions. If ul and
u2 are two harmonic functions such that u1(x) > u2(x) on the boundary
of U, then u1(x) > u2(x) also everywhere in U. Our immediate goal is to extend this result to functions which are extremals of more general functionals of the calculus of variations. Below, F denotes a function of the variables (x, q), x E U (an open subset of R") and q E Rn, satisfying conditions A-D in §5.1.
162
11. MAPPINGS WITH BOUNDED DISTORTION
THEOREM 5.4 [1341. Suppose that U is a bounded open set in R", and u and v are two stationary functions for the functional IF on U, with u, v E W,(U). If u > V IOU, then u(x) > v(x) almost everywhere in U.
PROOF. Assume, on the contrary, that u(x) < v(x) on a subset A of U
with positive measure. Let ri = (u - v)-. By assumption, u - v EH'D(U). O
This means by definition that q EW I (U). On the basis of Corollary 2 to Lemma 5.2, ni'(x) = 0 for almost all x ¢ A, since it is obvious that j7(x) = 0 for x A. On A we have that n + (u - v) = 0, so the same Corollary 2 to Lemma 5.2 gives us that q'(x) + [u'(x) - v'(x)] = 0
for almost all x E A, i.e., i'(x) = v'(x) - u'(x) almost everywhere on A. We have that
r F (x, u' + ni') dx > f F(x,u')dx, U
(5.31)
U
since u + q = uIBU, and an extremal function with given boundary values is unique. Further,
F(x,u'+I')dx £\A F(x,u'+ni')dx+ J F(x,u'+ii')dx
(5.32)
A
= JU\A F(x,u')dx+J AF(x,v')dx. Combining (5.31) and (5.32), we conclude that
F(x,v')dx> A
f
F(x,u')dx.
(5.33)
A
Let us now consider the function v - ri. We have that
JF(x.v' - n')dx= fU\A F'+
f F(x,u')dx A
< JU\A F(x, v') dx +
f F(x, v') dx = f F(x, v') dx. A
U
However, this contradicts the fact that v is extremal for IF and v - q = vI8U. The contradiction obviously proves the theorem. COROLLARY. Assume that F also satisfies condition E in §5.1, i.e., F(x, q)
> F(x, 0) for any x E U and q E R". Let u E W, '(U) be a stationary func-
tion for IF on U. Suppose that there exists a number L E R such that
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
163
u(x) < LIOU(u(x) > L18U). Then u(x) < L (respectively, u(x) > L) almost everywhere in U.
For a proof it suffices to note that the function v(x) - L is a solution of the equation div Fq[x, v'(x)] = 0, by the condition E in §5.1. §5.6. Harnack's inequality and its corollaries. One of the main properties of extremals of functionals of the calculus of variations (used in an essential way in what follows) is connected with the so-called Harnack inequality. Let F(x, q) be an arbitrary function satisfying conditions A-E in §5.1. Further, let p, a1, a2, a3, and a4 denote the constants in these conditions. THEOREM 5.5 (Harnack's inequality for a ball). There exists a number
C = C(p, a1, a2, a3, a4) > 1 such that if u is a nonnegative generalized solution of the equation div Fq[x, u'(x)] = 0 in the ball B(xo, 2r), then
ess sup u(x) < C essinf u(x). xEB(xo.r)
xEB(xo.r)
The constant C does not depend on the radius of B. This theorem is a special case of a general theorem of Serrin on solutions of nonlinear elliptic equations [ 156]. The proof of Serrin's theorem follows an idea of Moser [ 104]. A proof of Theorem 5.5 based on this idea is also presented in Chapter III. We mention some corollaries to the theorem.
THEOREM 5.6. Let U c R" be an open set, and u(x) a generalized solution in U of the equation div Fq[x, u'(x)] = 0, where F satisfies all the conditions A-E in §5.1. Then u is continuous in U and satisfies a Holder condition with exponent a on compact subsets of U, where 0 < or < 1 and depends only on the constants p, a1, a2, a3, and a4 in conditions A-E and the number n.
It is not hard to show that a result analogous to Theorem 5.5 remains true if instead of B(xo, 2r) one takes an arbitrary bounded domain U in R", and instead of B(xo, r) an arbitrary compact subset of U. We confine ourselves to the following particular result. LEMMA 5.10. There exists a constant C2 = C(2., p, a1, a2, a3, a4) where 0 < . < 1, such that if u is a nonnegative solution of the equation div Fq[x, u'(x)J = 0 in the domain U = B(xo, r)\{xo}, then max
xES(xo.Ar)
u(x) <_ C2
min u(x).
xES(xo,Ar)
11. MAPPINGS WITH BOUNDED DISTORTION
164
THEOREM 5.7 [156] (Liouville's theorem). Let u(x) be a generalized solution of the equation
divFF[x, u'(x)] = 0,
(5.33)
defined on the whole of R. If u(x) is bounded on R", then it is identically constant. PROOF. Let
M(r) = max u(x), jxI
m(r) = min u(x). lxI
For every r > 0 we have that m(r) < u(x) < M(r) on the ball B, = B(0, r). Consider the functions
v1(x) = u(x) - m(2r),
v2(x) = M(2r) - u(x).
The functions vI and v2 on B2r are nonnegative, vI satisfies the equation div Fq[x, v'(x)] = 0, and v2 satisfies div Fq[x, -v'(x)] = 0. The function
F(x, q) = F(x, -q) satisfies conditions A-E in §5.1 with the same constants as F. Therefore, the Harnack constant for F has the same value as for F. We have that max vI (x) = M(r) - m(2r), min vI (x) = m(r) - m(2r), xEB,
xEB,
max v2(x) = M(2r) - m(r),
min v2(x) = M(2r) - M(r).
xEB,
From this,
xEB,
M(r) - m(2r) < C[m(r) - m(2r)], M(2r) - m(r) < C[M(2r) - M(r)],
(5.34)
where C is the constant in Theorem 5.5. Adding the inequalities in (5.34) termwise, we get after simple transformations
M(r) - m(r) < [(C - 1)/(C + 1)][M(2r) - m(2r)]. From this, induction gives us that
M(r) - m(r) < [(C - 1)/(C+ 1)]"[M(2"r) - m(2"r)] (5.35) for all v = 1, 2,.... If the function u(x) is defined for all x E R" and Iu(x)1 < K = const < oo, then M(2"r) - m(2"r) < 2K for all v. Passing to the limit as v oo in (5.35), we get that M(r) - m(r) = 0 for any r > 0, i.e., u(x) _ const in R", which is what was to be proved. §5.7. The concept of the flow of a stationary function in a capacitor.
Let
K = (A, B) be an arbitrary capacitor in R". The symbol U(K) will denote the set R"\(A U B)-the field of the capacitor K. We also introduce the notation BAU(K) = U(K) n A and OBU(K) = U(K) n B.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
165
Assume that the field U of the capacitor K = (A, B) is a bounded subset
of R. This last condition will be assumed throughout the subsection unless a statement to the contrary is made. Let W.' (K) = WP (A, B) be the
collection of all functions u E WP[U(K)] such that u(x) = 0 on 8Au(K) and u(x) = 1 on 8BU(K). Suppose that a stationary function u for the functional IF is defined in the field U of the capacitor (A, B), where, as usual, F is assumed to satisfy all the conditions A-E in §5.1. Let I E WP (A, B) be an arbitrary function, and consider the integral 1(7) = f(J1'(x),Fq[x.u'(x)l)dx.
(5.36)
This integral does not depend on the choice of the function g E Wp (A, B). Indeed, let g1 and 172 be two arbitrary functions in Wp (A. B). Then 1(111) -1(g2)
=fu (gi (x) -1!'2(x). Fq[x. u'(x)]) dx.
We have that OU = 8A U U 8B U. The function q0 = q I - q2 is equal to
zero on each of the sets 8A U and OB U. By Lemma 5.7, this implies that 170 = OIOU, and hence
Fq[x, u'(x)]) dx = 0, fU(q'(x), which leads to the conclusion that 1(g1) = I(12), as required. The quantity I (q), q E WD (A, B), will be called the flow of the extremal function u for the functional IF in the capacitor (A, B), and denoted by A(A. B, u, F). Note that if g E WW (A, B), then clearly 1 - g E Wp(B, A), which gives us that
S2(A,B,u,F) = -fl(B,A,u,F) for every generalized solution of the equation div Fq[x. u'(x)] = 0 in U(K) = R"\(A U B). THEOREM 5.8 [134]. Suppose that K = (A, B) is a capacitor in R" whose
field U(K) = U is a bounded subset of R", and uI and u2 are stationary functions in U(K) for the functional IF, where F satisfies conditions A-E in §5.1. If U1 = u2 on the set 8AU and u1 > u2 on BBU, then Q(A, B, u1, F) > Q(A, B, U2, F).
(5.37)
If u1 > u2 on OAU and u1 =142 on BBU, then
S2(A,B,u1,F) <S2(A,B,u2,F).
(5.38)
II. MAPPINGS WITH BOUNDED DISTORTION
166
PROOF. Suppose that u I = u2 on 8A U and u 1 > U2 on 8B U. Assume
that there exists a number 8 > 0 such that uI - u2 - 8 > 0 on BB U. Let q = min{(u1 - u2/8, 1}. We have that =1-(u1-U2-1
17
The function ((uI - 112) /8 - 1)- is equal to zero on OBU, and hence q = 1 on 8B U. Further, U I - u2 = 0 on 8A U. This implies that q = 0 on 8A U. We thus get that q is equal to 0 on 8AU and 1 on OBU, i.e., q E Wp(A, B). Let
f (t) =
r F[x, u' (x) + tq'(x) ] A. U
According to Lemma 5.9, f is convex and differentiable for all t E R. Further,
f'(t) _
f
(q', Fq(x, u2 + tq')) dx.
U
In particular,
f'(0) =
JU
(q', Fq(x, u2)) dx = Q(A, B, U2, F),
1'(a) = Uf(i'
.
Fq(x, u2 + 6j7')) dx.
We now mention that for almost all x E U (q'(x), Fq[x, U'2 (X) + 3)7'(X) 1) = (q'(x), FQ[x, ui (x)]).
(5.39)
Indeed, let P c U be the collection of all x E U such that [uI (x) - U2(x)]/S > 1.
For almost all x E B we have that q'(x) = 0, and hence (5.39) holds for almost all x E B. But if x E U\P, then [uI (x) - u2(x)]/S < 1. By continuity of uI and u2 on U, the last inequality holds also in some neighborhood of x. In this neighborhood 8q'(x) = ul (x) - u2 (x), and hence Fq[x, u2 (x) + 8q'(x)] = Fq[x, uc(x)]
in a neighborhood of the given point x. Thus, the proof of (5.39) is complete. It follows from the foregoing that
P(J) =
Fq[x, ui (x)]) dx = Q(A, B, u1, F).
Since f is convex, the derivative f' is an increasing function, and thus
f'(0) < f'(8), i.e., S2(A, B, u I, F) > S2(A, B. 112, F).
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
167
We now get rid of the condition that u I - u2 > 6 on aB U. Let vT, where T > 0, be a stationary function for IF equal to u I on OAU and to u I + T on 0B U. By what has been proved, the function of the variable r - f(A, B, v, F) is nondecreasing on (0, oo). For each T O(A, B, VT, F) > 12(A, B, U2, F)
by what has been proved. To conclude the proof it suffices to establish that S2(A, B, vT, F) - S2(A, B, uI , F)
asT-- 0.
The function u I + T is also a stationary function for IF in U. Further, u1 < vT < u1 + T on the boundary of U, which, by Theorem 5.4, implies
that uI(x) < vT(x) _< u1(x) + r for all x E U. Since F is convex with respect to the second argument for any q and q0, we get
F(x, q) ? F(x, qo) + (q - q0, Fq(x, qo)). From this,
fu F[x, v=(x)] dx - fu F[x, u' (x)] dx >
f (vT(x) - u, (x), Fq[x, u, (x)]) dx = TS2(A, B, u1, F), U
fu F[x, u, (x) ] dx - fu F[x, v=(x)] dx >
f
(u, (x) - vT(x), Fq[x, ve(x)]) dx
> -Tf2(A, B, vT, F) > -r (A, B, v1, F). These inequalities obviously give us that
f F[x, vi(x)] dx -+ fF[xx)Jdx as r -. 0. We have that vT -y u 1 in U as r 0. On the basis of Theorem 3.6 in Chapter III, it follows from this that
Q(A,B.vT,F) -i2(A,B,uI,F) asT --- 0. The proof of the first inequality in the theorem is thus complete. The case when u1 >- u2 on A and u1 = u2 on B reduces to that already considered if we note that i2(A, B, u, F) = -S2(B, A, u, F).
By what has been proved, if u1 > u2 on A and u1 = U2 on B, then
i2(B,A,u2,F) > 11(B,A,u2,F),
168
11. MAPPINGS WITH BOUNDED DISTORTION
which implies that Q(A, B, u1, F) < S2(A, B, U2, F),
as was to be proved. §5.8. The set of singular points of stationary functions for functionals of the calculus of variations. THEOREM 5.9 [1281. Let U be an open set in R", A C U a set closed with
respect to U, and F(x, q) a function satisfying all the conditions in §5.1. Assume that U\A is nonempty and that there exists a function v which is a stationary function for IF on U\A and is such that limx-xo v(x) = oo for every point x0 in A that is a limit point for U\A. Then the p-capacity of A is equal to zero.
PROOF. 1) Let xo E A be a point of A which is a limit point for U\A. Let B, be an open ball about x0 lying strictly inside U, and B2 C BI a ball concentric to it such that the open ring domain K = BI \B2 contains points of U\A. Let S, and S2 be the boundary spheres of B, and B2, and let So be a sphere concentric to S, and S2i lying in the ring K, and containing points of U\A. The function v is continuous at each point x E B, not in
A, and v(x) -' oo as x -- y if y E A, which implies that v is bounded below on B,. By assumption, v is a stationary function for IF. This gives us that v + C is also a stationary function for IF for any constant C, and in view of this it can be assumed that v(x) > 0 for all x E Bt. We extend the definition of v by setting v(x) = oo for x E A. Let the number T > 0 be arbitrary, and denote by PT the set of all x E B, such that v (x) > T; PT is obviously closed with respect to B1. Let QT = B2 n PT,
HT = B, \QT and GT = B, \PT. The sets HT and GT are open. Let v'(x) = min{v(x),T}. The function v' belongs to WPJOC(U). Indeed, let x E U. If x io A, then there exists a neighborhood V of x lying strictly inside U\A. The restriction of v to this neighborhood is a function of class WP (V). On the basis of Lemma 5.8, this implies that the restriction
of v' to V is also a function in WP (V). Let X E A. Then there is a neighborhood V of x in which v (x) > T, and hence v' (x) - T on V. Thus, every point x E U has a neighborhood V such that the restriction of v' to V is a function of class WP. From this, v' E WP1.(U). It is not hard to see also that v' is continuous on U. We now construct a stationary function VT for IF equal to 0 on the sphere S, and equal to T on the part of the boundary of HT lying in B2.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
169
To do this we take a function uo E Co (R") such that 0 < uo(x) < T for all x, with uo (x) = 0 in a neighborhood of S1, and uo(x) = T in a neighborhood of 92. Then VT is a stationary function for IF equal to uo on the boundary of HT.
The function f (x) identically equal to T is stationary for IF on U. Obviously, T - uo(x) > 0 everywhere in R", and thus T - VT(X) > 0 on the boundary of HT. This gives us that VT(x) < T everywhere on HT. In exactly the same way we get that VT(X) > 0 on HT.
Let us now prove that v*(x) > VT(x) everywhere on GT. Note that v'(x) = v(x) on GT by definition; hence v` is a stationary function for IF on G. It suffices to establish that v' > VT on the boundary of GT. We construct a sequence (wm), m = 1, 2, ..., of functions of class Co (R") such that wm(x) = 0 in a neighborhood of S1, wm(x) = uo(x) = T in a neighborhood of PT, and Ilwm - VT III . p.H,. - 0 as m - oo. Such a O
sequence exists, since VT - uo EW 1 (HT), uo E C°°(U) n WPI (HT), and O
Co (HT) is dense in W ,(HT). Let wm = min{wm, T}. By Corollary 1 to Lemma 5.4,wmEWo(HT),and wm-4UT =min{vT,T} in WPI (HT) as m - oo. Since vT < T, we get that Wm - VT as m -+ oo. Let e > 0 be arbitrary. Then the function (wm(x) - v'(x) - e)+ = hm(x) is equal to zero in a neighborhood of the boundary of GT. Averaging hm, we get a function of class C°° equal to zero in a neighborhood of the boundary of O GT. This implies that hm EW I (GT). Passing to the limit as m - oo, we O
find that the function h = (VT - v' - e)+ belongs to W I (GT). This means
that v* + e > VT on the boundary of GT, and hence v'(x) + e > vT(x) everywhere on GT. Since e > 0 is arbitrary, this proves that v'(x) > VT (x) everywhere on GT. Let x1 be a point of So in GT. Such points exist by assumption. We have that VT(XI) < v(x1) = v' (x1). The function VT is nonnegative and is an extremal function for IF on K. Using Harnack's inequality for the domain K and the compact set So, we get that VT(X) < L = const for an x E So, and the constant L does not depend on T. This finishes the first step of the proof. In this step we constructed an extremal function VT for IF on the set HT D K. Here VT(X) = 0 on the sphere S, which is the outer boundary of HT, VT(X) = T on the part of the boundary of HT lying in the ball B2, and VT (x) < L = const on So.
Denote by Q(T) the quantity Q(SI, QT, VT, F)-the flow of the extremal VT with respect to the capacitor (S1, QT). The function 17 _ (1/T)VT(x) is in the class Wp(S1,QT) and, hence,
II. MAPPINGS WITH BOUNDED DISTORTION
170
S2(T) =
r
(q'(x), Fq[x, V' (x)]) dx
= (1/T)
L
(vT(x), Fq[x, V' (x)]) dx.
From this, by condition D (see §5.1), T
JIlT 1 v'(x)IP dx < MI(T), w here
k < oo is a constant. Considering that VT = T, , where q E
Wv(S1, QT), we get that CP(SI,QT) <
kS2(T) TP-1
(5.40)
We now show that S2(T) is bounded above by a constant independent
of T. Suppose that µ is an extremal for IF in the ring Ko between the spheres S1 and So, is equal to 0 on S1, and is equal to L + 1 on So. Then VT on S1 and u > VT on So, and hence, by Theorem 5.8, SZo = S2(SI , So, p, F) > S2(SI , So, VT, F).
However, it is not hard to see that CIA, So, VT, F) = S2(SI,QT.VT,F) = S2(T); thus, S2(T) < CIO for every T.
By what has been proved, it follows from (5.40) that CP(SI , QT)
kS2o/TP-I
Since A n B2 C QT, this implies that
Cp(S1,AnW2) < k120/T"-', and, since T > 0 is arbitrary here and p > 1, Cp (S1, A n W Z) = 0.
We have obtained that each point of A that is a boundary point in U has a neighborhood V = B2 such that the p-capacity of A n V is equal to zero. Let us now prove that all the points A are boundary points in U. Assume that this is not so. Then A has interior points. Let xo be a boundary point of A° in U. Then x0 is also a boundary point of A. Hence, by what has been proved, there exists a neighborhood V of xo such that V n A is a set of zero p-capacity, and hence V n A is a set of measure zero. The set V n A has interior points, since x0 is a boundary point of A°, and thus the open set V n A° is nonempty. However, this contradicts the fact that V n A is a set of measure zero. The contradiction proves that A does not have interior points. Every point of A therefore has a neighborhood V such that V n A is a set of capacity zero. Consequently, Co(A) = 0, and the theorem is proved.
§5. ELLIPTIC DIFFERENTIAL EQUATIONS
171
§5.9. Liouville's theorem on conformal mappings in space. A mapping f : U -+ R" of an open domain U c R" into R" will be called a generalized conformal mapping if f is a mapping with bounded distortion that is not identically constant, and its distortion coefficient K(f) is equal to 1. The purpose of this subsection is to prove that every generalized conformal transformation of R" is a Mobius transformation.
If f is a generalized conformal mapping of a domain U C R", then the linear mapping f'(x) is a general orthogonal mapping for almost all x E U. THEOREM 5.10. Every generalized conformal mapping of an open domain U C R" is a Mobius transformation.
PROOF. Let U be an open domain in R", and f : U -' R" a generalized conformal mapping. The matrix-valued function Of (x) is defined almost everywhere in U, where Of (x) = I if f'(x) is a singular mapping, and
6f(x) = Ir(x.f')12I"[f''(x)]-1[ f,(x).]-1 otherwise. Since fl(x) is a general orthogonal transformation, it follows
that [f'(x)J-I[f'(x)']-I = x.21, where A = Idet[f'(x)l-' I'I" = Lf (x. f )I -''". This implies that Of (x) = I in this case. Thus, if f is a generalized conformal mapping, then Of (x) = I almost everywhere, and equation (5.16) for it takes the form div[Iu'(x)In-2u'(x)] = 0.
(5.41)
On the basis of Theorem S.1 we thus establish that if the function u = v
is a solution of (5.41), then the function u = v o f also satisfies this equation. In particular, each of the components fI , ... , f" of the vectorvalued function f is a solution of (5.41). We now use a result of Ural'tseva [77], according to which every solution of (5.41) is a function of class C 1,a for some a > 0. Further, if G is the set
of x E U for which I u'(x)I > 0 (G is open, because u'(x) is continuous), then u(x) belongs to COO on G. If F(x, f) = 0 at a point x E U, then all the derivatives of f are also equal to zero at this point. This implies that if F(x, f) = 0 for all x E U, then the mapping f is identically constant, which is excluded by the definition of a generalized conformal mapping. Let G be the collection of all x E U such that I,." (x, f )I > 0, and let Go be an arbitrary connected component of G. Let u(x) = f (x), i = 1, ... , n, be one of the components of the vector-valued function f. The function u satisfies (5.41), and I u'(x) I = If(x, f ) I II" for all x E Go. From this,
172
II. MAPPINGS WITH BOUNDED DISTORTION
Iu'(x)I > 0 for all x E Go. Consequently u(x) belongs to C°O(Go), and hence f r= C°O(G0). The set Go is open. We prove that it is closed with respect to U. Let xo E U be an arbitrary limit point of Go in U. Let u(x) = I f'(x)I. By Ural'tseva's theorem, the derivatives of a solution of (5.41) are continuous in U. This gives us that the function U(x)
is continuous in U, and hence u(xo) is the limit of µ(x) as x -+ x0 in Go. Since If'(x)I > 0 for all x for every Mobius transformation, this implies that µ(xo) > 0. Consequently, f is a Mobius transformation in some neighborhood of xo. If two Mobius transformations coincide on some open set, then they clearly coincide everywhere. This enables us to conclude that xo E Go. Thus, Go contains all its limit points in U; that is, Go is closed relative to U. Since U is connected, we have that Go = U, and the theorem is proved. REMARK. The theorem was first proved by the author [120] and by Gehring [34J for the case when f is a homeomorphism. In the formulation given here the theorem was proved by the author [126] using a different method. §5.10. The property of quasi-invariance of conformal capacity.
THEOREM 5.11. Let U be an open domain in R", f : U - R" a homeomorphism with bounded distortion, and V = f(U). Suppose that A and B are arbitrary disjoint closed sets contained in Ti, Al = f (A), and BI = f (B). Then
C"(A,B, U) < K(f)C"(A1,BI, V). PROOF. Let V E 1P(A1, B1, V) be arbitrary, and let u = vof. By Lemma
4.7, u is in W, 1a(U), and u'(x) = [f'(x)]'v'[f(x)] for almost all x E U. Hence
U'(x)I" <_ Iif'(x)II"Iv'[f(x)1I" < KI,F (x, f )II v'[f (x)ll".
The function u vanishes in a neighborhood of A and is equal to I in a neighborhood of B, which easily gives us that u E W" (A, B, U). We have
fulu'(x)I"dx <_ K(f) fu Iv'[f(x)ll"If(x,f)I dx
= K(f) f Iv'(y)I" dy,
§6. TOPOLOGICAL PROPERTIES
173
and so C,, (A. B, U) < K(f) fV Iv'(y)I" dy. S ince V E T4'(AI,B1, V) was arbitrary, this proves the desired inequality.
§6. Topological properties of mappings with bounded distortion
Let U c R" be an open set in R", and f : U - R" a continuous mapping of class §6.1. Continuous mappings with nonnegative Jacobian.
W,
(U). Then we say that f is a mapping of monotone type if the Ja-
cobian of f has constant sign in U, i.e., if either f (x, f) > 0 almost everywhere in U or f (x, f) < 0 almost everywhere in U. For a mapping f : U -' R" of monotone type let a(f) = 1 if F(x, f) > 0 almost everywhere in U, and a(f) = -1 if F(x, f) < 0 almost everywhere in U. Every mapping with bounded distortion is a mapping of monotone type. Denote by 6 the mapping (y1 , ... , y,) (-YI, Y2, .... yn) of the space R.
Let f : U - R" be a mapping of monotone type. Then f = 0 o f is also a mapping of monotone type. Further, a(f) = -a(f ). Let G c U be a compact domain, and a an (f, G)-admissible point. Then the point 0(a) is (f, G)-admissible, and µ[0(a), f, G] = µ[a, f, G]. This assertion follows immediately from the definition in §2 of the degree of a mapping.
LEMMA 6.1. Suppose that U is an open set in R" and f : U - R" is a continuous mapping of class W1 such that F(x, f) > 0 for almost all x E U. Then µ(y, f, G) is nonnegative for every compact domain G c U and any (f, G)-admissible point y E R. PROOF. We take an arbitrary compact domain G c U and construct a sequence (f,,: V - R"), v = 1, 2, ... , of C°O-mappings, defined on an open set V containing G and lying strictly inside U, that converges uniformly to f on V and is such that Ilfv - f II 1,".v -i 0 as v - oo. Such a sequence (f,) exists according to Theorem 1.7 in Chapter I. Then as v - 00
fVIf(x,fv)-f(x,f)I dx-+0. Denote by P the collection of all x E V such that f (x, f,) < 0, and by Qv the set of x E V at which F(x, f,) = 0. According to Theorem 2.2, f,(Q,,) is a set of measure zero. The set Pv is open, and since ,17(x, f,) 36 0
for all x E Pv, the set f, (P,) is also open. Theorem 2.2 now allows us to conclude that
Ifv(Pv)I <- 4 1/(x,fv)I dx
If. MAPPINGS WITH BOUNDED DISTORTION
174
for all v. As v -' oo the functions F (x, f,) converge in L 1(V) to the
PV
function F (x, f ), which gives us that as v - 00
f If (x f )I dx = f If (x,
dx - J[f(xf )]dx = 0. I - 0 as v -- oo. Take an arbitrary
This leads to the conclusion that I f,
point yo E R"\f(8G), and let 6 = p(yo, f(8G)) > 0.
Since f, - f
uniformly as v - oo, there is a v1 such that the ball B(yo, 6/2) does not intersect the set f,(8G) for v > v1. Let vo > v1 be such that for v > vo the measure of the set f, (P,) is less than the volume of B(yo,5/2). Then for v > vo. The B(yo, 6/2) contains a point yI such that yI V f, (P,, U set f,- ' (yI) is then not contained in which implies that f is regular with respect to the point y, and the Jacobian of f, is nonnegative at each point x E f,,- I (y). This enables us to conclude that p(yI , f,, G) > 0 for all v > vo. The ball B(yo, 6/2) is contained in a connected component of R1'\f,,(8G), and hence p(yo, f G) = p(y1. f G) for v > vo; in particular, p(yo, f,,, G) > 0. Passing to the limit as v which is what was required to prove.
oo, we get that p(yo, f, G) > 0,
LEMMA 6.2. Suppose that U is an open set in R" and f : U - R" is a continuous mapping of class WP 1.C(U) such that f(x, f) > O for almost all x E U. Let G and G1,... , G,,, be compact domains contained in U and such that no two of GI , ... , Gm have common interior points, and G D Um Gk. Then m
p(y.fG)>_>p(y.fGO
(6.1)
k=I
f (8Gk) for all k = 1, ... , m.
for every point y E R" with y 0 f (8G) and y
PROOF. Consider the set V = Go\ U' Gk. If V is empty, then (6.1) holds with equality, in view of the additivity property of the degree of a mapping (Proposition II, §2.1). Assume that V 0 0. Let { V.+ 1, Vm+2,... } be the collection of all connected components of V, and let Vk = Gk for
k > m + 1. Each of the sets Gk is a compact domain. By assumption, (y) is contained in the union of the sequence of open sets G°, G2, ... , and since f -' (y) is compact, it can be covered by finitely many of these sets. Let f -' (y) be contained in the union of the first 1 sets Gk, 12! m.
f
Then I
m
p(y,f G)=>p(y,fGO ?> k=1
k=1
(y,fGk),
§6. TOPOLOGICAL PROPERTIES
175
by the additivity property of the degree of a mapping, since each of the terms on the right-hand side is nonnegative by Lemma 6.1. The lemma is proved.
THEOREM 6.1. Let f : U - R" (where U is an open set in R") be a mapping with bounded distortion. Take a compact domain G c U and a point y E R. If y is (f, G)-admissible, i.e., y is not in the image of the boundary of G under f, and y belongs to f (G), then lp(y, f, G) I > 1 and sgn µ(Y, f, G) = or (f ) PROOF. Assume first that the Jacobian off is nonnegative in U, suppose
that the compact domain G and the point Yo E R" satisfy the condition of the theorem. Let S = f -1(yo) n G. The set S is closed with respect to G and hence compact, and S does not contain boundary points of G, since yo f (8G) by assumption. Hence, S C Go. Let x0 be an arbitrary boundary point of S. We set d = p(yo, f (0G)) and find a 6 > 0 such that if Ix - xol < S, then I f (x) - Yol = I f (x) - f (xo) I < d. Choose an arbitrary r such that 0 < r < J. It is clear that B(xo, r) C G. Denote by D(r) the collection of all points x E B(xo, r) such that f (x, f) > 0. The set D(r) is measurable, and its measure is nonzero. Indeed, assume that ID(r)I = 0. Then
f (x, f) = 0 for almost all x E B(xo, r), and hence each of the partial derivatives (8f/Ox;)(x) is equal to zero almost everywhere in B(xo, r), by the inequality defining mappings with bounded distortion. This implies that f is constant on B(xo, r) and, in particular, f (x) = f (xo) = y for all x E B(xo, r). This enables us to conclude that B(xo, r) c S, i.e., x0 is an interior point of S, which contradicts the choice of x0. According to Theorem 1.2, the mapping fl(x) constructed formally
from the partial derivatives of f for almost all x E U is the differential of f. Consequently, there is a point x1 E D(r) such that f'(x1) is the differential of f at x1. We have that
f(x)
=f(xl)+f'(xl)(x-xI)+a(x)Ix-x,I,
where a(x) 0 as x - x1. Since det f'(x1) > 0, f'(x1) is a nonsingular mapping, and thus there is a number 2 > 0 such that I f'(x1)(h)I > 2Ihi for any h E R". Since a(x) - 0 as x -' x1i there is a po > 0 such that Ia(x)I < 2/2 for Ix - x1 I < po. For p < po the mapping f on the sphere S(x1, p) is homotopic in R"\{f (x1)} to the affine mapping x - f (x1)+ f'(x,)(x-xo), which implies that for 0 < p < po the degree of f at the point y, = f (x1) with respect to B(x1, p) is equal to 1. For sufficiently small p the ball B(x1, p) is contained in G, and hence µ(Y1.., G) ? µ4Y1, f. B(x, p)) = 1
176
II. MAPPINGS WITH BOUNDED DISTORTION
by Lemma 6.2, i.e., µ(y1, f, G) > 1. We have that
Iy1- yol = If (x1) - f(xo)I < d. since Ix - xoI < J. Consequently, the interval [yo, y1] does not contain points of f (8G) and thus lies in a single connected component of R"\ f (8G). This implies that µ(yo,f, G) =µ(y1,f G) > 1,
and the theorem is proved for the case when 7(x, f) > 0 almost everywhere in U.
The case when a(f) _ -1, i.e., F(x, f) < 0 almost everywhere in U, can obviously be reduced to the case already considered by virtue of the remarks made at the beginning of this section. The theorem is proved. COROLLARY 1. Let f : U -+ R" be a mapping with bounded distortion. Then for every compact domain G c U the exterior component of R" \f (0G) does not contain points y E f (G). REMARK. Continuous mappings having the property in Corollary 1 are called monotone mappings.
COROLLARY 2. Let f : U - R" be a mapping with bounded distortion. Then for any closed ball ff(a, r) c U the function If (x) - f (a) I attains its greatest value on the boundary of B(a, r).
Indeed, let
1=Xmaaxr)If(x)-f(a)I,
11 = max If(x)-f(a)I TEB(a.r)
Assume that l1 > 1. Let x1 be a point of B(a, r) where I f (x)- f (a)I = 11. Obviously, Ix1 - al < r. The set f (8G), where G = B(a, r), is contained in B[ f (a), 1]. The point f (x1) lies outside this ball, and hence belongs to the exterior component of R"\f (8G), which contradicts Corollary 1.
§6.2. Satisfaction of condition N for mappings with bounded distortion. The goal of this subsection is to prove that every mapping with bounded distortion has property N. The proof uses only a few properties of mappings with bounded distortion. In this connection we establish here an assertion more general than indicated in the heading.
We first prove a statement enabling us to get information about the behavior of a mapping of class W on almost all planar sections of its domain.
§6. TOPOLOGICAL PROPERTIES
177
LEMMA 6.3. Let U be an open subset of R"-', and let f : U - R" be a continuous mapping. If f belongs to the class W,' 1OC(U), then f (U) is a set
of measure zero in R.
PROOF. Let f satisfy the conditions of the lemma. For an arbitrary set A c R" let d (A) = diam A. By the Sobolev imbedding theorem (estimate (2.15) in Chapter I), the following inequality holds for every (n -1)dimensional cube Q = Q(a, r) C R"- I and any mapping g: Q(a, r) -, R" of class W (Q): I/n
d [g(Q)] < Mr'1" (f Ig'(x)I" dx) Q
where M is a constant. Let Q(a, r) be an arbitrary (n - 1)-dimensional cube contained in U. We partition it by planes parallel to the coordinate planes into k"- I equal cubes QI , ... , Qm, where m = k"-1. For each i = 1, ... , m
d[f(Q1)] < M (k /'l n (f If'(x)I" dx)1 /" Q, Raising both sides to the nth power and summing over i, we get m
E(d[f(Qi)])" < i=1
k"r
f If(x)l" dx.
(6.2)
Q
The right-hand side of this inequality tends to zero as k - oo. Each set f (Qi) is contained in a ball of radius d[f (Q1)]. It thus follows from (6.2) that for any e > 0 there exists a finite system of balls that covers f (Q) and has sum of volumes less than e. This implies that f (Q) is a set of measure zero. Since U can be represented as the union of a sequence of cubes lying strictly inside U, what has been proved implies that f (U) is the union of a countable family of sets of measure zero, and hence If (U) I = 0. The lemma is proved. We introduce some auxiliary notation. Denote by pi the mapping of R" that assigns to a point x = (xI , ... , x; . ... , x") E R" its ith coordinate xi. If U is an open subset of R", then pi(U) is clearly an open subset of R. Take a number h E R and an integer i with 1 < 1:5 n, and denote by qi,h the mapping of R"-' into R" defined by
gi.h(tl,t2,...,to-1) = 1) E R". If U C R" is an open set and h E pi (U), then the collection of t E Rn-1 with gi,h(t) E U forms an open subset of R"-'. This set will be denoted by Pj.h(U).
11. MAPPINGS WITH BOUNDED DISTORTION
178
LEMMA 6.4. Let U be an open subset of R", and let f : U -+ R" be a mapping of class W I I.C(U), where p > 1. For an integer i between 1 and n let fi(t, h) = f [gi,h(t)]. Then the function t " f,,(t, h) belongs to the class Wv 10C[PI,h (U)] for each i with 1 < i < n and for almost all h E pi(U). Furthermore, the derivatives (0f /8tk)(t, h) are the restrictions of the corresponding derivatives of f. Namely, let Oj f (x) = (Of/Oxj)(x). Then
k (t, h) = (8k f)[q,.h(t)] for k < i,
8 and
ak (t) = Bk+If[gi.h(t)] fork > i. PROOF. Let Q be an arbitrary cube lying strictly inside U. We construct
a sequence (f,), v = 1, 2, ... , of functions of class C°° defined in Q such that Ilfu -f II I.p.Q -- 0 as v -, oo. It will be assumed that 11f, -f II I, p,Q < V for each v = 1, 2, .... Obviously, this can always be achieved by passing to a subsequence. Define f, (x) - f (x) = r,, (x). Let x = (xl, ... , x,"). We regard x as the pair (t, xi), where t is the point in R"- I obtained if the ith coordinate of x is crossed out, so that x = qi,X, (t ). Let n
w(x)(Ir(x)IP+F'I axk(x)I ° 00
v=1
(6.3)
k=1
The integral of each term on the right-hand side does not exceed 1 /2VP; thus, the numerical series obtained by integrating (6.3) converges, and this implies that w is an integrable function. Let (F,), v = 1, 2, ... , be a sequence of Lebesgue-integrable functions. If E', IIF,,IIL, converges, then F,, (x) 0 almost everywhere. In view of this observation, r, (x) - 0 and a (x) -+ 0 almost everywhere in Q. Let A c Q be a set of measure zero on which the quantity -Xk(x)IP
Ir
(x)I°+EI k=I
does not tend to zero as v oo. The set pi (Q) is some interval (ai, bi). Let E'(Q) be the collection of h such that the plane xi = h intersects A in a set of positive (n - 1)-dimensional Lebesgue measure. For such h the set q-,h' (A) is a set of measure zero in R"- '. (The mapping qi,h maps gi.ti (A) onto the indicated section of A.) Further, let E;' (Q) be the set of h E (ai, bi) for which the function t '- wi(t, h) is not integrable on the (n - 1)-dimensional cube Qi = Pi,h(Q). By Fubini's theorem, E;'(Q) is a set of measure zero. Let Ei(Q) = Ei(Q) U E;' (Q).
§6. TOPOLOGICAL PROPERTIES
179
Take an arbitrary h E (a1, b;) not in E; (Q). Then
J
w; (t, h) dt =
"
00 (J'Q .
J=1
I
x (t, 8Xj
h)Idt < oo.
For this h (r,(t, h)IP dt - 0, IQI
fQ
I aX, (t,
h)I di
- 0,
0 and a (t, h) -. 0 for almost all t E Q. Considering and r, (t, h) the expressions for r and its derivatives, we get that if h ¢ E;(Q), then h) converge in L p (Q,) to f(-, h), and their derivatives the functions where j = k h) converge in Lp(Q) to the functions h) for k < i, and j = k + I for k > i. This enables us to conclude that belongs to the class Wp(Q;), and the derivative Y h) coincides with the corresponding generalized derivative of f . To finish the proof we represent U as the union of a sequence (Q,), of zero measure in R is defined v = 1, 2, ... , of open cubes. The set E, is clearly a set of measure zero. for each v. Then the set E; = U E; Let h E; . Then on P;.h the function f, ( , h) belongs to the class Wv . which are open in Rn-1, this Since P;(U) is the union of the sets implies that f, is a function of class Wv 1C(P;(U)). The lemma is proved. Let U c R" be an open set, and f : U -. R" a continuous mapping. We say that f is stable if it satisfies the following condition S. Let G be an arbitrary compact domain contained in U. Then for every point y E f (G)\ f (8G) there is an e > 0 such that, for every continuous mapping q': G - R" with If (x) - 47 (x)I < e, for all x E G the set c (GO) contains y. We present a certain sufficient condition ensuring the stability of a map-
ping. Let f : U - R" be a continuous mapping. Assume that for every compact domain G c U and any point y E f (G) not in f (8G) the quantity µ(y, f, G) is nonzero. Then f is a stable mapping. Indeed, assume that f satisfies the given condition. Take an arbitrary compact domain G c U and a point y E f (G)\f (8G). Let e= p(y, f (8G)). Then e> 0. Let (P: G -. R" be a continuous mapping such that If (x) - (p(x)I < e for all x E G. The mapping Sp is homotopic to f as a mapping of the pair (G, 8G) into the pair (R", R"\{0}). The required homotopy is given by the mapping t E [0, 1]. pt(x) = (l - t) f(x) + hp(x), This implies that y V ip(8G) and µ(y, q , G) = p(y, f, G) 0 0, and allows us to conclude that y E f (G°), which is what was required to prove.
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II. MAPPINGS WITH BOUNDED DISTORTION
To understand the intuitive meaning of VI (F, 6) and V2(I', 6) we con-
sider the case when r is the segment with endpoints x, and x2. Let us transfer the point x, to 0o by inversion with respect to the sphere So about x2 passing through x1. The segment r then does into a ray A emanating from x,. The cross ratio is invariant with respect to Mobius transformations. We have that
(x,y,x,,oo) _ Ix-yl/Ix-x,1, and the condition (x, y, x, , oo) < 8 means that Ly - xI < Six - x, 1; that is, that y belongs to B(x, 61x - x,1). The image of V, (I', 6) is the union of these balls and is a right circular cone whose generators form the angle ip = arc sin 6 with its axis A. This cone is the intersection of the half-spaces
with boundary planes passing through x, and having outward normals at an angle 7t/2 + rp to A. Inversion with respect to So carries each such halfspace into a ball whose boundary passes through x, and x2 and whose radius to x, forms the angle n/2 + rp with the vector x2 - xI. In this case the set vi (F, a) is a "spindle", the intersection of all such balls. In this case it is clear that V2(F, 8) = V, (F, 6). In the case of an arbitrary continuum F the set v (r, 8) is a distinctive kind "curvilinear" spindle joining x, and x2.
Following Martio [92], we say that a domain U c jr has the property U(6) (0 < 6 < 1), and write U E U(8), if for any two points X1, x2 E U there exists a continuum r joining x, and x2 such that V(r, 6) C U. LEMMA 12.4 [92]. Let U be a domain of class U(a, fi) in R". Then there exists a 6 E (0, 1] depending only on a, P, and n such that U E U(8). Conversely, if U E U(8) for some 6 E (0, 1], then U E U(a, fi), where a and fi depend on 6 and n.
The concept of a quasidisk turns out to be useful in many questions of analysis not directly connected with the theory of quasiconformal mappings. In this connection we refer the reader to the monograph [40], where a number of such results are given. In particular, the concept of a homogeneous domain arises naturally in connection with the problem of extension of the classes WP (U), that is, with the problem of conditions under which there exists for every u E WP (U) a function 9u E WP (R") such that
(0u)(x) = u(x) for all x e U. Further, it is required that the operator
u -' 9u be a continuous mapping from W.' (U) into W.' (R"). Many papers have been devoted to the problem of extending the classes W. The reader can find more complete information about this in [48]. (In this connection see also [45], [47], [64], [179], and [ 180].)
§6. TOPOLOGICAL PROPERTIES
181
(Here N(y, Q) is the number of elements in f -I (y)nQ; N(y, fm, Q) > Xm(Y) for all y E RR.) Since Xm(Y) -' X(Y) almost everywhere, it follows that
If(Q)I =
f X(Y)dy < lim JXm(Y)dY
liIfm(Q)IRm-.oo
m-oo
The left-hand side of (6.4) tends to the limit fQ If (x, f )I dx. Passing in (6.4) to the limit, we thus get that if (Q)1:5 fQ If(x. f )I dx.
Inequality (6.5) has been established so far under the assumption that the cube Q is normal. Assume now that Q = U(a, r) is not normal. Since the cube Qh = U(a, h) is normal for almost all h such that U D Qh, there is a decreasing sequence (rm ), m = 1, 2, ... , of values such that rm r as m - oo and each of the cubes Q,,^ is normal. For each m we have that f (Q) c f (Q,.), and hence If(Q)I <_ If(Qrm)1 5
f If (x, f)dx.
Passing to the limit as m - oo, we get that (6.5) holds also in the ca when Q is not assumed to be normal. Let E be an arbitrary set of measure zero contained in U, and let e > 0 be arbitrary. Since the function x - If(x, f )I is locally integrable in U, for the given e > 0 there is an open set V D E with V C U such that
L f(x,f)Idx< C. v The set V is representable as the union of a sequence of disjoint half-open cubes, each of which lies strictly inside V. For each m
If(Q.)I <_
I
Qf.,
If(Q)I <_
00
V=I
If(Q.)I _<
F(x,f)Idx.
00
v=I
f I F(x.f)Idx Q"
=f IF(x,f)ldx<e, and hence If (E)I < e. Since e > 0 is arbitrary, this proves the theorem.
11. MAPPINGS WITH BOUNDED DISTORTION
182
COROLLARY 1 [123]. Every homeomorphism f : U - R" of the class (U) (U an open subset of R") has property N. 10C
W,'
COROLLARY 2 [125]. Every mapping with bounded distortion has property N.
REMARK 1. Let f : U -. R" be a homeomorphism, where U is an open subset of R. Assume that f belongs to WP 1a(U), where 1 < p < n. Then it can happen that f does not have property N. An example for the case n = 2 was constructed by Ponomarev [144]. The construction presented in [ 144] carries over in an obvious way to the case of arbitrary n. REMARK 2. The conclusion of Theorem 6.2 is false in general without
some additional conditions of a topological nature on f. We give an example showing this in the case n = 2. Let G be a domain on the plane with boundary a simple closed curve for which the Lebesgue measure of any arc is nonzero. Let f (z) implement a conformal mapping of the halfdisk {z I I z < 1 and Im z > 0) onto G. The mapping f is continuous and belongs to W21. We extend f to the disk I z 1 < 1 by setting f (z) = f (T) if Im z < 0. The function f is continuous in the closed disk Izi < 1 and belongs to the class W2 on each of the half-disks I z1 < 1, Im z > 0, and
Im z < 0. This implies that f E W21. The mapping f does not have property N: the segment Im z = 0, -1 < Re z < 1, is carried by this mapping into some arc of the curve L. The Lebesgue measure of this arc is nonzero, and hence f does not have property N: the two-dimensional Lebesgue measure of the image under f of some set of measure zero (a segment in this case) is nonzero. The author does not know of any examples of this kind for n > 2. §6.3. Topological properties of mappings with bounded distortion. We first define some concepts relating to the topology of mappings. Let X and Y be arbitrary topological spaces, and let f : X Y be continuous. The mapping f is said to be open if f (U) is open in Y for every open set U in X. If f (A) is closed for every closed set A c X, then f is said to be a closed mapping. The mapping f is said to be zero-dimensional if for every y E Y the set f '' (y) does not have connected components containing more than
one point. If for every y E Y all the points in f ' I (y) are isolated, then f is called an isolated mapping. One says that f is a local homeomorphism if every x E X has a neighborhood U such that the restriction of f to U is a homeomorphism of U into Y. A point a E X is called a branch point of a continuous mapping f : X Y if f is not a homeomorphism in any neighborhood of a; the collection of all branch points of f is denoted by B f.
§6. TOPOLOGICAL PROPERTIES
183
LEMMA 6.5. Suppose that U is a domain in R" and f : U -. R" is a mapping with bounded distortion. If f is not identically constant in U, then the full inverse image of a point under f is a set of zero n-capacity.
PROOF. Let y E R" be an arbitrary point. The set A = f -I (y) is closed with respect to U, and V = U\A is an open set. The function u(x) = ln(1/lx - yl) belongs to the class C°° and is a solution of the equation div[lu'(x)ln-2u'(x)] = 0 in R"\{y}. Hence, by Theorem 5.1, the function
v(x) = u[f(x)] = ln[1/(If(x) - II)] is a solution in the open set V of the equation
div[(0(x)u'(x), u'(x))I"-2j/20(x)u'(x)] = 0, where the matrix 0(x) is positive-definite, and there exist constants AI , A2 > 0 such that the eigenvalues 0(x) lie between AI and A2 for all x. It is obvious
that v(x) - oo as x tends in V to an arbitrary boundary point of A in U. We thus see that all the conditions of Theorem 5.9 are satisfied here, and hence C" (A) = 0. The lemma is proved. LEMMA 6.6. Let A c R. If the conformal capacity of A is equal to zero, then for every point a E R" the sphere S(a, r) is disjoint from A for almost all r > 0.
PROOF. By Corollary 2 to Theorem 3.4, the a-dimensional Hausdorff
measure of A is equal to zero for every a > 0. In particular, the linear Hausdorff measure of A is equal to zero. By Lemma 3.5, this implies that the 1-content of A is equal to zero, i.e., for every e > 0 there is a sequence (B(xk, rk)), k = 1, 2, ... , of balls which covers A and is such that the sum of the radii is less than e. If the sphere S(a, r) contains points of A, then it intersects at least one of the balls B(xk, rk ). The set of r such that S(a. r) intersects at least one of the balls B(xk, rk) is a union of intervals of lengths 2r1 , 2r2, ... , and it has measure less than 2e. Thus, the set of x such that S(a, r)f1A is nonempty has measure less than 2e, and the lemma is thereby proved, because e > 0 is arbitrary. THEOREM 6.3 [125]. Every mapping with bounded distortion that is not identically constant is an isolated mapping.
PROOF. Let f : U -. R" be a mapping with bounded distortion which is not identically constant in U. It will be assumed that a (f) = 1. The general case can obviously be reduced to this one. Let xo E U be arbitrary, let yo = f (xo, and let A = f - I (yo). The set A is closed relative to U and
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II. MAPPINGS WITH BOUNDED DISTORTION
is a set of zero capacity. By Corollary 2 to Theorem 3.4, this implies that the linear Hausdorff measure of A is equal to zero. Let r > 0 be such that B(xo, r) = B lies strictly inside U, and S(xo, r) n
A = 0. Then yo ¢ f (8B), and hence the number µo = µ(yo, f, B) is defined. W e prove that the set A n B consists of at most uo elements. Indeed, let x1,... , xk be an arbitrary finite system of points in AnB. Since A is a set of zero conformal capacity, Lemma 6.6 gives us that there are ball neighborhoods B1,... , Bk of the respective points x1,... , xk such that the boundary of each set B; does not intersect A, B D B; for i = 1, ... , k, and B1, ... , Bk are disjoint. We have that f (x;) = yo and yo f (8B;) and hence u(yo, f, B;) is defined. By Lemma 6.2, k
µ(Yo, f Bi)
µo = µ(Yo, f B) > i=1
Since Yo E f (B;) for each i = I_-, k, Theorem 6.1 gives us that µ(yo, f, B1)
> 1 for all i. From this we have that µo > k, and hence A n B consists of at most µo elements. Thus, A n B is a finite set. We get that every point xo E U has a neighborhood containing finitely many points of f [f (xo)]. The theorem is proved. THEOREM 6.4 [125]. Every mapping f with bounded distortion from an open domain U to the space R" which is not identically constant is an open mapping.
PROOF. It is required to prove that f (V) is open for every open set V c U. Indeed, take any point x E V and let y = f (x). The set f - I (y) has zero capacity, and hence by Lemma 6.6 there is a ball Bo about x such that y ¢ f (8Bo) and Bo c V. Then µ(y, f Bo) is defined, and I# (y, f, Bo)I > I in view of Theorem 6.1, since y E f (Bo). The last inequality will hold
at all points of some neighborhood of y. Hence, all the points in this neighborhood belong to f (Bo), i.e., y is an interior point of f (Bo), hence
also of V. Thus, f (x) is an interior point of f (V) for any x E V, i.e., f (V) is an open set, which is what was required to prove.
THEOREM 6.5 [129]. Suppose that U is an open domain in R" and f:U R" is a mapping with bounded distortion. Then the function y -#(y. f, G) is bounded for every compact domain G c U. PROOF. We assume that F(x, f) > 0 almost everywhere in U. The general case can clearly be reduced to this one. The assertion is true if f is identically constant, so it will be assumed that this is not the case. Let xo E U. There is a 61(xo) > 0 such that f (x) 96 f (xo) for 0 < Ix - x0I < b1(xo). Let Bo = B[xo, 61(xo)] and So = Mo. Then yo = f (xo) f (So).
§6. TOPOLOGICAL PROPERTIES
185
Let 82(xo) > 0, 62(xo) < 61(xo), be such that for r < 82(xo) the image of B(xo, r) under f is contained in a connected component of R"\f (So). Suppose that y E f [B(xo, r)] and y ¢ f [S(xo, r)] where r < 82(xo). Then p[y, f B(xo, ro)]
p(y, f Bo) = p(Yo, f. Be).
This shows that the function y -+ p[y, f, B(xo, r)] is bounded for r < 62 (x0)
Using the theorem of Borel, we construct a finite system of balls
B(x1,p1),....B(xk,Pk) covering the compact domain G, where p; = 82(x;)/2. Let Y E R" be an arbitrary point. For each i = 1, ... , k let r; > 0 be such that pi < r; < 2 pi and the sphere S(x1, ri) does not intersect the set f -' (y). Let B; = B(x1, ri). It is not hard to see that k
p(y, f G)
p(Y, f Bt). r=1
On the other hand, u(y, f, B,) < p,, where p, depends only on the point x; and is equal to the index of f at xi. This gives us that k
pr = const.
p(y, f, G) < po = =1
The theorem is proved. We make some remarks about isolated continuous mappings. Suppose that U is an open domain in R", and f is an isolated continuous mapping. Let a E U be an arbitrary point. The point a is an isolated point of the set f - I [f (a)], and thus there is a 8 > 0 such that the closed ball B6 = B(a, 8)
does not contain points of f - I [f (a)] other than a and is contained in U. Let G be an arbitrary compact domain contained in Ba and such that a E G°. We have that f -I ([f (a)]) n Bd = {a} c G°, and hence p[f (a). f, G] = p[ f (a), f, Ba] in view of Proposition II in §2.1. Thus, IA[ f (a), f, G] has the same value for all compact domains G contained in Ba
and containing a as an interior point. This common value of p[f (a), f, G] is called the index off at the point a and denoted by j (a, f ). This quantity is a kind of analogue of the concept of the multiplicity of a root of an equation. Suppose that U is an open set in R", f : U -+ R" is an isolated mapping, and G c U is a compact domain. Assume that the point b E R" is (f G)admissible. Then it is not hard to see that the set f - I (b) n G is finite. Let
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11. MAPPINGS WITH BOUNDED DISTORTION
a1 , ... , am be its elements. Then m
(6.6)
µ(b, f G) = E j(ak, f). k=1
Indeed, let G1,.... Gm be the closed balls of radius r about the points al , ... , am. If r is sufficiently small, then these balls are contained in G° and are disjoint, and thus, by the additivity property of the degree of a mapping (Proposition II in this chapter, §2.1) m
.u(b,f, G)=>µ(b,fGk). k=1
On the other hand, it is obvious that u(b, f, Gk) = j (ak, f ), and equality (6.6) is thereby proved. Let f : U - R" be a mapping with bounded distortion. Then it follows from Theorem 6.1 that Ij(a, f)I > 1 for every point a E U. Further,
sgnj(a.f) = a(f) COROLLARY TO THEOREM 6.5. Let U be an open domain in R", and f : U -+ R" a mapping with bounded distortion. As before, for A c U and
y E R" let N(y, f, A) be the number of elements in the set f -' (y) n A (N(y, f A) = oo if this set is infinite). Then the function y I., N(y, f A) is bounded for every compact set A c U.
PROOF. Let A c U be compact. By the Borel theorem, there is a finite set of balls B, = B (ai , rk), i = I,-, m, such that A C U k Bi, and for
each i the closed ball G1 = B(ai, 2rk) is contained in U. The function y - µ(y, f, Gi) is bounded, by the theorem: Ii(y, f, G1)I < ui = const < 00 for all y ¢ f (8G, ). Take an arbitrary y E f (A), and let i 1, ... , it be the indices of the balls Bi containing points of the set f -' (y) n A. For each i there is a Si such that ri < ai < 2ri and the sphere S(ai, 6i) does not contain
points of f -' (y ). Let Hi, = B(ai;, 8i; ). Then the number µ(y, f, Hi,) is defined. By (6.6),
N(y..1. Hi,) < Ip(y, f. Hi) I For every y ¢ f (8Gi U OH,) we have
Ip(y,f Hi)I <_ Iµ(y.f G;)I. Now (6.6) allows us to conclude that Ij(y, f, G) I increases as G becomes larger, and this implies that Iu(y, f, Hi;)I < µi,. Obviously, /
m
N(y,f, A),.5 EN(y,f,Hi,) j=1
and the corollary is proved.
i=1
§6. TOPOLOGICAL PROPERTIES
187
THEOREM 6.6 [129]. Suppose that U is an open set in R" and f : U - R" is a mapping with bounded distortion which is not identically constant. Let the point a E U be such that I j (a, f )I = 1. Then f is a homeomorphism in some neighborhood of a.
PROOF. We assume that o(f) = 1; the case o(f) _ -1 can be reduced to this case. To prove the theorem it suffices to establish that there is a neighborhood V of a in which f is one-to-one. Suppose that there is no such neighborhood. Let 60 > 0 be such that x E U and f (x) # f (a) for 0 < Ix - of < 60. By our assumption, for every positive integer m there
are points xm and x, such that Ix.' - aI < 1/m, Ix - a( < 1/m, and f (xm) = f (x,") = ym. The point ym converges to b = f (a) as m - oo. This implies that for sufficiently large m/ p[ym, f. B(a, 60)] = p[b, f, B(a, ao)] = j(a, f) =1.
On the other hand, using (6.6), we get by the nonnegativity of the index of f that for all m > m0
u[ym f B(a, 80)] > j(xm, f) + j(x. f) > 2. This is a contradiction, and the theorem is proved. COROLLARY. Let f : U - R" be a mapping with bounded distortion. If
Jj(x, f)I > 1 at a point x E U, then f is not a homeomorphism in any neighborhood of x.
The existence of branch points (and even of whole branch curves) for mappings with bounded distortion is the peculiarity which distinguishes mappings with bounded distortion from arbitrary quasiconformal mappings. The set of branch points is obviously closed with respect to the domain of the mapping. The dimension of B f does not exceed n - 2, as a consequence of the following theorem of Chernavskii [26], [27]. THEOREM 6.7. Suppose that U C R" is an open domain and f : U - R" is an arbitrary isolated open continuous mapping. Then the set of branch points off is at most (n - 2)-dimensional. A proof of Theorem 6.7 can also be found in Vaisala's paper [ 172]. §6.4. A theorem on removable singularities. Many investigations have
dealt with the problem of removable singularities, i.e., the question of when a quasiconformal mapping or a mapping with bounded distortion defined on some set can be extended to a larger set with preservation of its properties (see [33], [ 100], and [ 158]). Questions relating especially to the case of arbitrary mappings with bounded distortion were considered
II. MAPPINGS WITH BOUNDED DISTORTION
188
in [ 114]. For lack of space we cannot present here the results obtained in these references despite their importance, and we confine ourselves to the following theorem, which is used in § 10.
THEOREM 6.8 (Vaisala [173]). Suppose that U is an open domain in
R", and E c U is a set closed relative to U with (n - 1)-dimensional Hausdorff measure zero. Then every quasiconformal mapping f : U\E R" admits a unique continuous extension g: U -+ R". Furthermore, g is a quasiconformal mapping of U, K(g) = K(f) and K0(g) = Ko(f ). PROOF. Since E does not have interior points, the required extension is unique if it exists. To prove the theorem it thus suffices to establish that the limit limX-,,b f (x) exists for every xo E E. Since the last assertion is local, it can be assumed that U 96 R", f (U) 0 R", and f is bounded; this can be made true by performing an inversion transformation in addition. We introduce a function h: U -, R" by setting h(x) = f (x) for x ¢ E and h(x) = 0 for x E E. Then h is a mapping of class W,7110C(U) in view of Theorem 1.5 in Chapter I. For almost all x E U\E, and thus for almost
all x c= U, we have that h'(x) = f'(x), and thus Ih'(x)I" < K(f)If(x,h)I almost everywhere in U. On the basis of Theorem 1.1 in this chapter it follows that there exists a continuous mapping h*: U -+ R" such that h' (x)
= h (x) almost everywhere in U. Obviously, h' (x) = f (x) for almost
all x E U. The mapping g = h` is the desired extension of f to U. Obviously, g is a mapping with bounded distortion. Since g'(x) = f'(x) almost everywhere in U, we see that K(g) = K(f) and Ko(g) = Ko(f ). We show that g is one-to-one. Assume, on the contrary, that there are two points x1, x2 E U such that g(x1) = g(x2) = y, x1 96 x2. Let V1 and V2 be disjoint neighborhoods of x1 and x2, and let W, = g(Vi ), i = 1, 2. The sets W, and W2 are open, and y E W, for i = 1, 2. There
is a a > 0 such that B(y.6) c W1 n W2. Since E is a set of measure zero, g(E) is a set of measure zero, and thus there is a point y' E B(y, 5) such that y' ¢ g(E). Suppose that the points x,' E V1 and X2' E V2 are
such that g(xi) = g(x2) 0 y'. Since the neighborhoods V, and V2 are disjoint, xi x2. Further, xl , x2 E, which leads us to conclude that g(x,) = f(x;). This contradicts the fact that f is a homeomorphism, and the theorem is proved. §6.5. On the method of moduli. We present some facts from which the
reader can get an impression of the method of moduli in the theory of spatial mappings. We do not provide proofs for the main facts connected with this method, but refer the reader to the corresponding literature. In the planar case the method of moduli was developed in a well-known paper
§6. TOPOLOGICAL PROPERTIES
189
of Ahifors and Beurling [10]. Applications of the method to the theory of quasiconformal mappings and mappings with bounded distortion are considered in [ 113], [91 ], and [94]-[96]. A curve in R" is defined here to be any continuous mapping y : [0, 11
--+
R" not constant on any interval [a, P] c [0, 1]. Let y: [0, 11 - R" be a given curve. The curve y determines a certain metric 6 in [0, 1], where 8(t1,12) is the diameter of the set y([t1, t2]) for It, t2 E [0, 1]. The onedimensional Hausdorff measure on [0, 1] corresponding to this metric is denoted by sy and called the arc length of the curve y. A curve y is said to be rectifiable if sy([0, 1]) < oo. Let p be an arbitrary Borel-measurable function defined in a domain U
of R", and let y : [0, 1] - U be a curve in this domain. Then the integral fo p[y(t)]dsy(t) (if it makes sense) is called the integral of the function p with respect to arclength along the curve y, and is denoted by fy p(x) dsX.
Let r be an arbitrary set of curves in R". A Borel-measurable function p > 0 on R" is said to be admissible with respect to the family r, or, briefly, I, admissible, if fy p(x) dsX > I for every curve y E r. The infimum of the
integrals f .[p(x)]" dx on the set of all I'-admissible functions p is called the modulus of the family r and denoted by M(r). Let r be an arbitrary family of curves lying in an open subset U of R", and let f : U - R" be a continuous mapping. The image of the family r under f is defined to be the collection f (r) of all curves f o y : [0, 1] --+ R", where y : [0, 1] -. R" is an arbitrary curve in r. If r is an arbitrary family
of curves in the set V = f (U), then let f -1(r) be the collection of all curves y : [0, 1] -+ U such that f o y E r. It is natural to call the family f - I (F) the inverse image of r under f. Application of the method of moduli to the theory of mappings with bounded distortion is based on the following two theorems. THEOREM 6.9 [113). Suppose that U is an open set in R" and f : U - R" is a mapping with bounded distortion. Then for every family r of curves in U
M(r) > (1 /K(f ))M[f (r)] THEOREM 6.10 [113]. Suppose that f : U -, R" is a mapping with bounded distortion, and G c U is a compact domain contained in U and such that f (OG) coincides with the boundary of the set H = f (G). Let r be a family of curves in H, and m > i an integer such that Ii(y, f, G)I = m for all y E H°. Then
m(r) < (Ko(f )l m)M[f -' (r)]
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11. MAPPINGS WITH BOUNDED DISTORTION
We mention that Vaisala [ 174] has obtained certain further strengthenings and refinements of these theorems. It follows from Theorems 6.9 and 6.10 that if f is a homeomorphism with bounded distortion, then for every family of curves in U
(1 /K(f ))M[f (r)] < M(r) < K0(f)M[f(r)]. This result was established by Shabat [ 157], [158], and Vaisala [ 174]. An essential role in applications of the method of moduli is played by
estimates for the moduli of certain families of curves. The required estimates can be obtained from estimates of the capacities of certain capacitors in view of the following theorem of Ziemer [ 181 ]. THEOREM 6.11. Suppose that K = (A0,A1) is a capacitor in R", where Al is compact, and let r be the collection of all curves y: [0, 1] -+ R" such that y(0) E A0 and y(1) E A1, and y(t) belongs to the field U of the capacitor (Ao, A1) for 0 < t < 1. Then M(r) = C. (A0, A,).
We remark that it is possible to use the method of moduli only after establishing that every mapping with bounded distortion is an isolated open mapping. A proof of this fact without the use of the apparatus of differential equations is not known. §6.6. Bi-Lipschitz mappings. We first make some remarks about mappings of class Wn 1a(U) defined on open of R.
LEMMA 6.7. Let U be an open subset of R", and f : U -+ R" a local homeomorphism of class W, 1.(U). Then f has property N, f is differentiable in U almost everywhere, and the Jacobian off has constant sign on every connected component of U. Further, for every open V c U the measure of the set of x E V with f (x, f) > 0 is nonzero. PROOF. Let f : U -+ R" satisfy all the conditions of the lemma. Then it is easy to see that the conditions of Theorem 6.2 hold for f, and this gives us that f has property N. Fix an arbitrary point x0 E U and a number 8 > 0 such that the ball B(xo, 6) is strictly inside U and f is a homeomorphism on B(xo, 6). We use a result in §4.3 of Chapter III. On B(xo, 6) the mapping f satisfies the condition E(T) indicated there, with constant T = 1. Further, f belongs to the class W (B(xo, S)) on B(xo, 6), and is hence differentiable almost everywhere in B(xo, 6) according to Theorem 4.3 in Chapter III. Since xo E U was arbitrary, this proves that f is differentiable on U almost everywhere.
§6. TOPOLOGICAL PROPERTIES
191
Let N(xo, a) = G, H = f (G), and W = f [B(xo, J)] = H°. Take an = B(x, r) C G and let E = f (0). We use Theorem 2.2 in this chapter, taking the indicator XE of E as u(y); that is, u(y) = 1 for y E E and u(y) = 0 for y¢ E. Since f I G is a homeomorphism, NG(y, f) = I for all y E W, and the function y -- µ(y, f, G) is constant arbitrary ball
on W. Further, Iµ(y, f, G)I = 1, and hence µ(y, f, G) = µo, where µo = ±1.
In this case u[f (x)] = 1 for x E 0, and u[f (x)] = 0 for x
-12F. Using
Theorem 2.2, we get that
Ze If (x, f )I dx = fE
NG(y,
f) dy = IEI.
ff(xf)dx = f ,u(y,fG)dy = uolEI. E
Since f is a homeomorphism, IEI > 0, and hence the set of x E f (x, f) # 0 has nonzero measure. Moreover,
ff(x.f)dx = uo f
with
If(x,f)I dx.
Since 0 is an arbitrary ball in B(xo, S), this implies that
f(x,f) = µolf(x.f)I for almost all x E B(xo, 8). In particular, either f (x, f) > 0 almost everywhere in B(xo, 8), or f (x, f) < 0 almost everywhere in B(xo. S). In particular, the measure of the set of x E B(xo. 6) with f (x, f) 96 0 is nonzero. Since xo E U is arbitrary, this gives us that the measure of the set {x E VI f (x, f) 0 0) is nonzero for any open set V C U.
Let U+ be the set of x E U such that f (t, f) > 0 for almost all t E B(x, 6) for some number 8 > 0, and let U_ be the set of x E U such that f (t, f) < 0 almost everywhere in B(x, 6) for some 8 > 0. In view of what has been proved, U+ U U_ = U, U+ n U_ = 0, and each of the sets U+ and U_ is open. This implies that every connected component of U is contained in one of the sets U1. or U_, and the proof of the lemma is complete.
Let U_ be an open subset of R". A mapping f : U - R" is said to be bi-Lipschitz if there exists a constant L, 1 < L < oo, such that for every xo E U there is a number 6 > 0 for which the ball B(xo, 6) is in U and
Ix' - x"I
Llx'-x"I < If (x') L for any x', x" E B(xo, 6). The infimum of the numbers L satisfying this condition will be denoted by L(f).
II. MAPPINGS WITH BOUNDED DISTORTION
192
If f : U -i R" is a bi-Lipschitz mapping, then it is a local homeomorphism. By Theorem 2.7 in Chapter I, f belongs to the class W ,10,(U), and, in particular, f E W 10(U). This gives us that f is differentiable almost everywhere. Further, by Lemma 6.7, the Jacobian off has constant sign for x E U. If the mapping f : U - R" is bi-Lipschitz and U contains the segment joining x, and x2, then
If (x2)-f(xI)I<-L(f)Ix2-x11.
(2)
Indeed, assume that [x1, x2] is contained in U, and let L > L(f) be arbitrary. By definition, every point x E U has a neighborhood B(x, 6) such that (1) holds for any x1 and x2 in this neighborhood. Covering [x1, x2] by finitely many such neighborhoods, we find a finite sequence of points ao = x1, a2, ... , am = x2 arranged in order on [x1, x2] such that for each m
If(ai) - f(ai-1)15 Llai - ai- II Summing these inequalities, we get that m
If(ai) -f(ai-)I
If (X2) - f(xi)I 5 i=1
m
Since L > L(f) is arbitrary, this yields (2). Let f : U -' R" be a homeomorphism. In this case if f is bi-Lipschitz, then so is f -' . Further, L(f) = L(f -1). We show that if f : U - R" is a bi-Lipschitz mapping, then for every xo E U there is a 6> 0 such that L(f)Ix2-x11 :5 If(x2)-f(x1)I :5 L(f)Ix2-x11
(3)
for any x1, x2 E B(xo, 6); that is, the constant L in (1) can be taken equal to L(f). Let B(xo, p) be an arbitrary ball about x0 on which f is a home-
omorphism, and let V = f [B(xo, p)] and y = f (xo). There is an e > 0 such that B(yo, e) c V; let 6, 0 < 8 < p, be such that If (x) - yol < e for lx - xol < J. By what was proved above,
if(x2) -f(xi)I <- L(f)lx2 -x11 for any x1, x2 E B(xx, p). Further, in precisely the same way we conclude that
If-'(Y2) - f-'(YI)I 5 L(f-')IY2 - YII = L(f)IY2 - YII
§6. TOPOLOGICAL PROPERTIES
193
for any y1, y2 E B(yo, e). Setting yI = f (x1) and y2 = f (x2) here, where x1, x2 E B(xo, 6), we get that
Ix2 -x11 <- L(f)If(x2) -f(x1)I. and hence both the inequalities in (3) hold for any x1, x2 E B(xo, S). Let us show that every bi-Lipschitz mapping has bounded distortion. With this goal we first establish that at any point x E U where f is differentiable the linear mapping F = f'(x) is also bi-Lipschitz, and L(F) < L(f ). For an arbitrary t E R, t 0 0, and X E R" let f(x+tX)- f(x)
Ft(X) =
I
For any XI, X2 E R" we have that
IF,(X1)-F,(X2)1 =
f(x+1X1)-f(x+tX2)
I
t
Let d > 0 be such that the inequalities in (3) hold for any x1 and x2 in B(x, S). There is a to > 0 such that if 0 < t < to, then 1 tX1 I < 6 and
1tX21 <6. For0
L(f)
J X2 - X1 I <- I Ft(X2)
- F,(XI )I < L(f )I X2 - XI 1
If f is differentiable at x, then F1(X) - F(X) as t - 0, where F = f'(x), and we thus get that for any X1, X2 E R"
L(f)1
x2 - X1I < IF(X2) - F(X1)I <- L(f)IX2
- X11.
This means that F is bi-Lipschitz, and L(F) < L(f). In particular, it follows from what has been proved that at each point x E U the principal dilation coefficients of the linear mapping fl(x) lie in [(1 /L(f )), L(f )], which implies that f(x) is nonsingular and K[f'(x)] < [L(f )]2"-2 for almost all x E U. This establishes that f has bounded distortion. Further,
K(f) s
[L(f)]24-2
It is not hard to see that in this case we also have that Ko(f) 5 [L(f )]2a-2.
II. MAPPINGS WITH BOUNDED DISTORTION
194
§7. Local structure of mappings with bounded distortion §7.1. Preliminary remarks. Let f : U R" be a mapping with bounded distortion, where U is an open set in R. Let xo E U be an arbitrary point,
and let yo = f (xo) and S = f - I (yo). All the points in S are isolated, by what was proved in §6. The function
v(y) = ln(l/IY - Yol) belongs to the class COO in R"\{yo} and is a solution of the equation div[lv'(Y)ln-2v'(Y)]
= 0.
By Theorem 5.1, this implies that the function u = v o f is a generalized solution on U\S of the equation u'(x))("-2)/29(x)u'(x)]
div[(0(x)u'(x),
= 0,
where the matrix 9(x) is determined from the condition that ([f'(x)]')-I
9(x) = I F(x, f )I2/"[f'(x)]-I if 7(x, f) 96 0, and 0(x) = I otherwise. Further, if q is the differential form
=
n
n(Y)
E(-1)k-llv'(Y)I"-2
(Y)dYl
k=1
...dYk-I dYk+I ...dyn,
and the form C is defined by C(x) = (f'j)(x), then dC(x) = 0, and for every function io E C' d(p n C = sgn f (x, f)(9u', u')("-2)"2 (0u', gyp') dxl, dx2 ... dxn. We have that
(7.1)
Yr - Yoi
ayt(Y)_-1y-Yo12. From this, n -Y01")F(-1)k-I(Yk
DI(Y) = (-1/IY
-Y0k)dYI
k=1
...dYk-I dYk+1 ...dyn,
and hence ->7 is the same exterior form used for computing the degree of a mapping. Let to > 0 be such that the ball Bo = B(xo, to) is contained in U and does not contain other points of S. Then for any x with 0 < Ix - xol < to
§7. LOCAL STRUCTURE
195
the point f (x) is different from yo = f (xo). For every compact domain G C No with xo in its interior we have that .u (Yo,f G) = u(Yo,f Bo) =j(xo,f). Let V be an arbitrary compactly supported C'-function with support in Bo that is equal to 1 in a neighborhood of xo. According to Lemma 2.4 in this chapter, 'B0
d (x) n C(x) = Wn.U[Yo, f Bo] = wj(xo, f ).
Considering (7.1), we conclude from this that
sgn f (x, f)
f(Ou'. u')"2/2(0u', p,') dx = cvj(xo, f ). o
Here u(x) = - In If (x) - yol, and 0(x) is the matrix-valued function determined from f in §5.2 of this chapter. Observe now that j(xo. f) > 0 if f preserves orientation, i.e., f (x, f) > 0 almost everywhere, and j(xo. f) < 0 otherwise, i.e., ifF(x, f) < 0. Because of this, the formula just obtained can be rewritten as u')cn-2)/2(0u',
(7.2) S,') dx = wnlj(xo, f)I. fa. The function u is a stationary function for the functional f F[x, u'(x)] dx on U\S, where F(x, q) = (0(x)q, q)"l2. In R" we consider the capacitor K = (A, B), where A = R"\B(xo, to) and B = {xo}, and we denote by S2(xo, f) the flow function of u in this capacitor. By definition,
(9u',
S2(xo, f) = f (F. [x, u'(x)] q'(x)) dx ^
=n
f
(8u',
uYn-20 (Ou', 4p') dx,
Bo
and thus, by (7.2), (7.3) S2(xo,f) = nanj(xo,f). Let f : U - R" (where U is an open set in R") be a mapping with
bounded distortion that is not identically constant in U. Let xo E U be an arbitrary point. We introduce some quantities characterizing the behavior of f in a neighborhood of this point. By results in the preceding section, there exists a number 8 > 0 such that f (x) 96 f (xo) for 0 < Ix - xoI <_ bo. The supremum of such numbers 8 is denoted by 8 j(xo). (The index f in this notation will be omitted in what follows whenever this does not lead to confusion.)
196
11. MAPPINGS WITH BOUNDED DISTORTION
Let O < T < &(xo). Define L f(xo, T) = Ixmax T If (x) - f (xo)I. If (X0, T) = IXmint If (x) - f (xo) I.
Obviously, L f(xo. T) and If (xo, T) are continuous functions of T; moreover,
0 < 1f(xo, T) < L f(xo, T), and L f(xo, T) - 0 as T -' 0. The quantity L f(xo, T) is a nondecreasing function of T. Indeed, if this were not so, then there would be numbers TI and T2 such that TI < T2 and
Lf(xo,TI)>Lf(xo,T2).
(7.4)
The image of the sphere S(xo, T2) is contained in the ball B(yo, r), where yo = f (xo) and r = L f(xo, TI ). It follows from (7.4) that there is an x E B(xo, T2) such that the point y = f (x) lies outside B(yo, r). It is not hard to see that p(y, f, B(xo, TI )) = 0. However, this contradicts Theorem
6.1. Hence for each T E (0, 6(xo)) there exists a number rI = a(r) < T such that L f(xo, r j) = If (X0, T). (If such a number TI is not unique, we choose the largest of them as a (T).) §7.2. Some estimates of a solution of an elliptic equation having one singular point. Let U be an arbitrary bounded domain in R. Assume that a function F(x, q) which is a normal kernel in the sense of §5.1 is defined
for almost all x E U, i.e., F(x, q) satisfies all the conditions A-E in §5.1, and the exponent p in conditions C and D is equal to n.
Choose any point a E U, and let U' = U\{a}. For 6 > 0 let Ua = U\B(a, 6). We consider the generalized solutions of the equation
div F.[x.u'(x)] = 0,
(7.5)
in U' which satisfy the condition that u(x) -+ oo as x - a. By a theorem of Moser proved in Chapter III, every such solution u(x) is continuous in
U. If u(x) is a solution of (7.5) in U', then the conditions u(x) = 0 on 8U, u(x) >_ 0 on 8U, and u(x) < 0 on 8U will mean that u is in W (U6 ) for all sufficiently small & and vanishes (is nonnegative or nonpositive, respectively) on the component 8U of the boundary of Ua in the sense of the definitions in §5.3 of the present chapter. Let u(x) be a generalized solution of (7.5) in U' = U\{a}. Let Ao = R"\U and AI = {xo}. Denote by i?.o the flow of the solution u(x) of (7.5) with respect to the capacitor (Ao, AI). Our goal is to get estimates for the solution u(x) of (7.5). Let us first recall some facts established in §3. Denote by AT(l), where I > 0, the
§7. LOCAL STRUCTURE
197
capacitor (A0, A,) with A0 the interval -1 < xn !5 0 of the Ox,-axis and A 1 the ray x > I of this axis. Define CC[AT(l)] =
wn/[lnc(l)]n-1.
(7.6)
As shown in §3, there exists a constant do > 1 such that for all 1 > 0
1+1
(7.7)
The capacitor AT(1) has the following extremal property. Assume that (E0, E1) is a given capacitor in R" such that E0 contains two distinct points x0 and yo in a single connected component of E0, and E, has an unbounded connected component containing a point z0 with I = Izo - xo1/1yo - xoI. Then (7.8) Cn(E0.E1) ? Cn[AT(l)] _
LEMMA 7.1. Suppose that U is a bounded open domain in Rn, a E U, U' = U\{a}, and u(x) is a function defined and continuous on U\{a} = U' u 8U which is a generalized solution of equation (7.5) in U. Assume
that u(x) -' oo as x - a, and u(x) < 0 for x E OU. Let h > 0. Denote by Ah the collection of all points x E U\{a} such that u(x) < h, and let Mh = U\Ah. Then Mh is a connected open set, the set Ah is closed, and if the boundary of U is connected, then Ah is also connected.
PROOF. Since u(x) is continuous in U\{a} and u(x) -, oo as x -, a, the set Ah is closed. This implies that Mh is open. Assume that Mh is not connected. Then it has at least two different connected components: M' and M". Since u(x) oo as x - a, there is a neighborhood V of
a such that u(x) > h for all x E V, x 0 a. Obviously, V c Mh, and thus a c Mh. It will be assumed that a E M". At each boundary point of M' we have that u(x) < h. We show that u(x) < h on the boundary of M' also in the generalized sense defined in §5.3 above. Indeed, let Vm(x) = (h + 1 /m - u(x))-, where m = 1, 2,... is an integer. Obviously, vm vanishes in a neighborhood of the boundary of M'. As m -, oo we have
that v,,, - v = (h - u) - in W (M'), and thus (h - u)- E W (M'/8M'), i.e., h - u EW ,1,(M'), which is what was to be proved. From this and from the particular case of the maximum principle for equations of the form (7.5) (the corollary to Theorem 5.4) we get that h - u(x) > 0 in M'. This contradicts the fact that u(x) > It for all x E M' by assumption. The contradiction proves the connectedness of Mh. Assume now that the boundary of U is connected. We prove that Ah
is connected in this case. Obviously, OU c Ah. The point a does not belong to Ah, and hence there is a 6 > 0 such that the closed ball B(a, 6)
198
11. MAPPINGS WITH BOUNDED DISTORTION
does not contain points of A. Let Ua = U\B(a, 6). Assume, contrary to what is to be proved, that Ah is disconnected. Since u(x) is continuous in U\{a}, Ah is obviously closed, and it follows from our assumption that there exist nonempty closed sets A' and A" such that A' U A" = A and A' n A" = 0. One of the sets A' or A" contains the connected set 8U. We assume that A" D 8U. Since A' and A" do not have common points, there exist open sets Cl and C' such that A' C C, All c G", and
the closures of C and G" are disjoint. Note that A' c U\B(a, S), and since A' n 8U = 0, it follows that A' c U\B(a, S) = U;. In view of this it can be assumed that C c U;, since this inclusion can always be achieved by replacing C by the intersection G n Q* if necessary. The boundary of C lies in the closure of Ua, and since (' is disjoint from 8U (because 8U C A" C G'), all the points of 8C are interior points of U, and a V 8G. Since 8C n Ah = 0, u(x) is continuous and u(x) > h at each point x E 8G. By continuity, there is a number y > 0 such that u(x) > h + y at each point x E 8C. Again by continuity, u(x) > h + y on 8C also in the sense of the generalized definition in §5.3. Indeed, for
each m = 1, 2.... the function vm(x) = (u(x) - h - y + 1/m)- vanishes in a neighborhood of 8C, and um(x) -+ v(x) _ (u(x) - h - y)- in W (G') as m - oo; consequently, v E W n(G'), and this means that u(x) > h + y on 8C in the sense of the definition in §5.3. By the minimum principle O
for a solution of an equation of the form (2.1) (the corollary to Theorem 5.4), this implies that u(x) > h + y for all x E G, which contradicts the fact that u(x) < h for all x E A' C C by assumption. The contradiction proves that Ah is connected, and the lemma is proved. THEOREM 7.1. Suppose that U is a bounded open domain in R", a is a point in U, U' = U\{a}, and u(x) is a function defined and continuous on U\{a} which is a generalized solution of equation (7.5) in U'. Assume that
u(x) < 0 on 8U and u(x) -1 co as x
a. Let ro be the distance from a
to the unbounded connected component of R" \-U, and let Qo be the flow of the solution of (7.5) in the capacitor (R"\U, {a}). Then f4 > 0, and for all
x E U\{a} (S2o/a4(0n)i/"-1 InOn(ro/Ix - al), (7.9) u(x) !5 where On is the function defined by (7.6), and a4 is the constant of inequality (5.6) in §5.1.
PROOF. Suppose that u(x) satisfies all the conditions of the theorem. Take an arbitrary point xo E U, x0 j4 a, and let h = u(xo). Assume first
that h > 0. Let A be the set of all x E U such that u(x) < h, and let M = U\A. By Lemma 7.2, M is connected. This implies that M is also
§7. LOCAL STRUCTURE
199
connected. Obviously, a E M and xo E M. Since u(x) is continuous on U\{a} and u(x) < 0 on OU, M does not intersect )U. Let V = U\M. The boundary of V consists of two parts. One of them is eU, and the other (the intersection vnM) is denoted by 8M V. Let uo(x) be a generalized solution of (7.5) in V satisfying the following boundary conditions: uo(x) = 0 on
8U and uo(x) = h = u(x) on 8MV. Since u(x) is continuous on U\{a}, u(x) < 0 on OU also in the generalized sense of §5.3. Indeed, the function Vm = [1/m - u(x)]- vanishes in a neighborhood of OU by the continuity of u(x). As m oo we have that vm -+ (-u)- = u+ in W,,' (V). From this, u+ = 0 on aU in the sense of §5.3, i.e., u(x) < 0 on OU. By Theorem 5.8, this implies that
Q(R"\U, M, uo, F) < f(R"\U, M, u, F) = 120. Let C(x) = uo(x)/h. Then { E W (V), C(x) = 0 on aU, C(x) = 1 on aM V, and, hence,
S2(R"\U M, uo, F) = f (C'(x), Fq[x. uo(x)1) dx ^
h
f(u(x). Fq[x. uo(x)]) dx.
We have that (q, Fq(x, q)) >_ g4IgI" (inequality (5.6) in §5.1). Therefore,
S2o > C (R"\U, M, uo, F) > h J ^ (uo(x)I" dx R
(7.10)
=a4h"-1 fVIC'(x)I"dx.
Since the function u(x) takes positive values in U', what has been proved gives us that SZo > 0. The last integral on the right-hand side of (7.10) can be estimated as follows:
f I{'(x)I" dx > C"(R"\U, M).
(7.11)
The set M contains the points a and x0. The set R"\U has an unbounded connected component, and the distance from a to this component is equal to ro. Using (7.8), we get that C"(R"\U, M) > ao"/[ln4(1)]"-1.
where l = ro/Ixo - al. On the basis of (7.10) and (7.11), this gives us that a4cv"h"-1 /[ln
D(1)]"-1 < flo, and hence
u(xo) = h < (0)01a4(0,,) 11(n- 1) In 0(1).
Since this inequality obviously holds at all points x where u(x) < 0, the theorem is proved.
§7. LOCAL STRUCTURE
201
u(x) > 0, and thus x E V. Using the estimate in Theorem 7.1, we get from this that In 1 ((a,r)) = u(x) <-
[K(f)Ij(a,f)I]'1"-' 1n4"(l).
which clearly implies (7.12).
Let us now apply the estimate in Theorem 7.1 to an arbitrary point x E B(a, r) different from a. We get that u(x) = in
L
If(x)-f(a)I
<
[K(f)Ij(a.f)IJ'A"-1)
x lnD"(r/lx - al). From this,
If (x) - f(a)I > al)l-Q,(a.f) [0"(rllx L where aI (a, f) _ [K(f ) Ij (a, f)I]'l{"-'). Let us use (7.7). We have that [1/A"llui(a.f).
Setting l = r/Ix - al here and considering that Ix - al < r, we have that
If(x) - f(a)I ?
y,,(a,f)(Ix
-
aI/r)°1(a.f),
where y" = 112A, is a constant. The proof of the theorem is complete.
§7.4. Behavior of a mapping with bounded distortion near an arbitrary point of the domain.
THEOREM 7.3. Suppose that U is an open set in R", and f : U - R" is a mapping with bounded distortion that is not identically constant in U. Then for every point a E U there is an ro > 0 such that if 0 < r < ro, then If (x) - f(a)1 <- Lf(a,r)M1(Ix -
al/r)a(a.f)
for Ix - a1 < r, where MI is the constant on the right-hand side of (7.12)
a(a,f) =
[Ij(a,f)I/Ko(f)]'i("-'),
and ro can be determined as in Theorem 7.2.
PROOF. Take an arbitrary point a E U and determine from it numbers
ro and r1 as indicated in Theorem 7.2. If 0 < Ix - al < r1, then x E U, f (x) 0- f (a), and If (a. r1) = L f(a, ro). Let u(x) = In[ 1 /If (x) - f (a)I]
The function u(x) is a generalized solution of the equation div F. [x, u'(x)] = 0
(7.14)
§7. LOCAL STRUCTURE
201
u(x) > 0, and thus x E V. Using the estimate in Theorem 7.1, we get from this that In
Lf(a,r) If(a,r)
= u(x) - < [K(f)Ij(a,Inc"(1).
which clearly implies (7.12).
Let us now apply the estimate in Theorem 7.1 to an arbitrary point x E B(a, r) different from a. We get that L u(x) = in < [K(f)Ij(a, f)1l11("-1)
If(x) - f(a)I -
x Inc"(r/lx - al). From this,
If(x) -f(a)I
,-
-al)]-ai(a.f)
L
where a1(a, f) = [K(f )I j(a, f )I ]'l("-'). Let us use (7.7). We have that [I/fin(!)]at(a.f) , [1/A"l]Q1(a.f).
Setting I = r/ Ix - al here and considering that Ix - al < r, we have that If (x) -f(a)I Y"i(a.f)(Ix-al/r)«i(a.I), where y" = 1/2A,, is a constant. The proof of the theorem is complete. §7.4. Behavior of a mapping with bounded distortion near an arbitrary point of the domain.
THEOREM 7.3. Suppose that U is an open set in R", and f : U - R" is a mapping with bounded distortion that is not identically constant in U. Then for every point a E U there is an ro > 0 such that if 0 < r < ro, then
If(x) - f(a)1 <- Lf(a,r)MI(Ix - aI/r)°(a.f) for Ix - al < r, where MI is the constant on the right-hand side of (7.12)
a(a,f) =
[If(a.f)I/K0(f)]'l("-1),
and ro can be determined as in Theorem 7.2.
PROOF. Take an arbitrary point a E U and determine from it numbers
ro and r1 as indicated in Theorem 7.2. If 0 < Ix - al < r1, then x E U, f (x) 0 f (a), and If (a, r1) = L f(a, ro). Let u(x) = In[ I /If (x) - f (a)ll.
The function u(x) is a generalized solution of the equation
divF.[x, u'(x)) = 0
(7.14)
202
II. MAPPINGS WITH BOUNDED DISTORTION
in G = B(a, r1)\{a}, where F(x,q) = (O(x)q, q)"/2, and u(x) -' oo as x -' a. For r E (0, ro) let A(r) = In[111f (a, r)],
2(r) = ln[l/L f(a, r) ].
The inequalities 2(r) < u(x) < A(r) hold at each point of S(a, r), u(x) _ 2(r) for at least one point x E S(a, r), and u(x) = A(r) for at least one point x E S(a, r). Fix an arbitrary r < ro, r > 0, and let uo(x) = u(x) -2(r). The function uo is also a solution of (7.14); uo is continuous on B(a, r1)\{a}, and uo(x) > 0 at each point x E S(a, r). Since uo(x) is continuous, uo(x) > 0 on S(a, r) also in the generalized sense defined in §5.3. Take an arbitrary
point xo E B(a, r), xo 0 a. Let Ixo - al = t. We have that 0 < t < r. Let v, and v2 be solutions of (7.14) in the ring domain H = B(a, r)\B(a, t) which satisfy the following boundary conditions:
v1(x) = 0 on S(a, r), V2(X)=VI(X) on S(a, r),
v1(x) = uo(x) on S(a, t), V2(X) = A(t) - 2(r) on S(a, 1).
Let A0 = Rn\B(a, r) and A, = B(a, t). We have that v, > uo on S(a, r) = BA0H, V2=v1 on8AOH,
V1=V2 on S(a, t) = 84, H. v22: vI
on 0.4,H.
On the basis of Theorem 5.8 of this chapter, this implies that fo = 12(Ao, A,, uo, F) < S1(Ao, A, , v1, F) < S2(Ao, A, , v2, F), i.e.,
nw"IJ(a,f)I
S2(Ao,AI,v2,F).
(7.15)
Let A(t) - 2(t) = h. The function A is decreasing, and hence A(t) > A(r) > 2(r); thus, h > 0. The equality h = 0 is impossible here, since v2 would otherwise be identically zero, and then ((A0, A,, v2, F) would be equal to zero, which is impossible in view of (7.15). Thus, h > 0. Let C(x) = v2(x)/h. Then C(x) = 0 on S(a, r), C(x) = 1 on S(a, t), and, hence, S2(Ao, A1, V2, F) =
f (C'(x). Fq[x, v2(x)]) dx H
(7.16)
n T fy (0(x)vi(x), vi(x))"/2 dx. Let w(x) be an arbitrary function of class W (H) such that w(x) = 0 on S(a, r) and w(x) = 1 on S(a, t). Since v2 minimizes the integral
fo(xu'(x. u'(x))"/2 dx
§7. LOCAL STRUCTURE
203
on the set of functions equal to 0 on S(a, r) and equal to h on S(a, 1), it follows that
L (9(x)v2(x), v2(x))n12 dx < h" f (O(x)w'(x), w'(x))"I2 dx < Ko(f)h" 1 1W, (X) In dx, H
which, by (7.15) and (7.16), gives us that ncvnlj (a, f )I 5 nKo(f )h"-' JH w'(x)I" dX.
Since w is an arbitrary function equal to 0 on S(a, r) and I on S(a, t), this implies that ncoJj(a, f ) I < nKo(f )h"-' CC(Ao, A1). As established in §3 of this chapter,
Cn(Ao,A1) = cvn/(lnr/t)"-t, and as a result we arrive at the inequality Ij(a,r)I <_ K0(f){[A(t)
-A.(r)]/(lnr/t)}"-i,
whence A(t) - A(r) > [Ij(a,f)I/Ko(f)]'"("-`)In(r/t), i.e., In
L f(a, r) 1f (a, t)
> a(a, f) In(r/t).
From this, If (a, t) < (t/r)e(a.f).
Lf(a,r)
(7.17)
On the basis of Theorem 7.2,
Lf(a,t) <M.
(7.18)
1f (a. r)
We get from (7.17) and (7.18) that
If(x)-f(a)I L f(a, r)
<
Lf(a,t) <M If (a, t)
(Ix-a \ r
The theorem is proved.
The results in this section are a refinement of results in the author's paper [ 134]. We remark that estimates analogous to those established here were also obtained by Poletskil [ 113] and Martio [91 ] by another method.
204
11. MAPPINGS WITH BOUNDED DISTORTION
§8. Characterization of mappings with bounded distortion by the property of quasiconformality
§8.1. The concept of a mapping which is quasiconformal at a point and in a domain. The purpose of this section is to prove that the properties of mappings with bounded distortion established in the preceding sections can be generalized in a particular sense. Further, a geometric characterization of mappings with bounded distortion will also be obtained. Suppose that U C R" is an open set and f : U -- R" is an isolated open
mapping. Then for every point x E U the integer j(x, f) (the index of f) is defined. We say that f is of constant sign if either j(x, f) > 0 for all x E U, or j(x, f) < 0 for all x E U. In the first case f will be called positive, and in the second negative. An isolated open mapping of constant sign will be called a mapping of type T, or, briefly, a T-mapping. By results in §6, every mapping with bounded distortion is a T-mapping.
Let x0 be an arbitrary point in R. Assume that some closed neighborhood G,(xo) of x0 is defined for every t E (0, 1]. We say that the set {G,(xo)It E (0, 1]} of neighborhoods forms a normal system if there exists a continuous function v : R" -' R such that v(xo) = 0, v(x) > 0 for x It x0,
G,(xo)={xER"Iv(x)
inf Ix - xol,
(8.1)
RG(XO, t) = sup Ix - xol.
(8.2)
rG(xo, t) =
XEr,(x0)
XEr1(X0)
The limit inferior a = lim[rG(x, t)]/RG(x, t)
(8.3)
is called the regularity parameter of the family {Gt(xo)10 < t < 1 }. The system {G,(xo)10 < t < l} of neighborhoods is said to be regular if a > 0. Suppose that f : U - R" is a mapping of type T, xo E U and {G,(xo)10 < t < 11 is a normal system of neighborhoods of x0. Let rG(xo, t, f) = XEr,fa) If(x) - f(xo)I,
(8.4)
RG(xo, t, f) = sup 11(x) - f (xo)I.
(8.5)
XEr,(x0)
The mapping f is said to be quasiconformal at x0 if there exists a normal regular system { Gt (xo )10 < t < 1} of neighborhoods of x0 such that
f = lim rG(xo. t. f) > 0 1-0 RG(xo, t, f )
§8. CHARACTERIZATION BY QUASICONFORMALITY
205
The quantity q(xo, f) = 1/af, where a is the regularity parameter of the system G,(xo), is called the coefficient of quasiconformality off at x0.
A mapping f : U - R" of type T is said to be quasiconformal in a domain U c R" if f is quasiconformal at each point x E U, and there exists a constant Q, 1 < Q < oo, such that q(x, f) < Q for all x E U. The smallest constant Q satisfying this condition is called the quasiconformality
coefficient off in U and denoted by q(U, f ).
Let f : U - R" be a mapping with bounded distortion. For every domain V lying strictly inside U there exists a constant ko < oo such that I j (x, f) I < ko for all x E V. By Theorem 7.3, this implies that f is quasiconformal on V. In the notation used below, the index G indicating the given normal system of neighborhoods will be omitted whenever no confusion is possible. §8.2. Differentiability almost everywhere of quasiconformal T-mappings.
Let U C R" be an arbitrary open domain, and let f : U - R" be a Tmapping. We assume that f is a positive mapping. For an arbitrary set A c U and any y E R" let N(y, f A) be the number of elements in f -' (y) tl A. The function y --i N(y, f, A) is the multiplicity function of f considered in §2.4. Lemma 2.4 in §2.4 allows us to conclude that this function is the limit of a sequence of Borel-measurable functions, and thus it is measurable.
LEMMA 8.1. Suppose that f : U - R" is a positive T-mapping. If a set A lies strictly inside U, then the function y - N(y, f, A) is bounded. PROOF. Suppose that A lies strictly inside U. Let Y E R" be an arbitrary point. By the condition of the lemma, all the points off -' (y) are isolated. This implies that for every point x E A there is a ball about x lying strictly
inside U whose boundary does not contain points of f '' (y). The set A is compact. Covering A by finitely many such balls, we get an open set G such that A c G, G is compact and contained in U, and 8G does not contain points of f (y). We have that
N(y, f A) 5 N(y, f, G) 5 E j(x, f) = µ(y. f G). xEf -' (y)nG
The function y - µ(y, f, G) is constant on each connected component of R"\ f (8G). This implies that each point y E R" has a neighborhood in which N(y, f A) is bounded. By the compactness of f (A), this gives us that N(y, f, A) is bounded on f (A). For y ¢ f (A) we have that N(y, f A) = 0. It is thereby established that N(y, f, A) is bounded, and the lemma is proved.
11. MAPPINGS WITH BOUNDED DISTORTION
206
For a T-mapping f : U - R" we let If (A)
= JR N(y, f A) dy, where A is a Borel set contained in U. Obviously, l f is a measure on the
a-algebra of Borel sets contained in U. For every compact set M c U there exists a constant k = k(M) such that If (A) < kl If (A) I for any Borel
set A c M. Fix a domain U c R" and a T-mapping f : U -+ R" that is quasiconformal in U. Let Q > q (U, f) be an arbitrary number. For each point x E U let { G, (x) 10 < t < 1 } be a fixed normal regular system of neighborhoods
such that a(x)f(x) > 1/Q, where a(x) is the regularity parameter of the system GI(x) and
fl(x) =1im r(x, t, f)/R(x, 1, f ). t- .0
Here r(x, t), R(x, t), r(x, t, f ), and R(x, t, f) are defined by (8.1)-(8.5). Obviously, a(x),f(x) > 1/Q for all x E U. It will be assumed (without loss of generality) that
r(x, t, f)/R(x, t, f) > 1/Q
r(x, t)/R(x, t) > 1/Q, for any x E U and any t E (0, 1].
LEMMA 8.2. Let f : U - R" be a quasiconformal T-mapping. For X0 E U let
k(xo) = Xim Xo If(x) - f(xo)I/Ix - xol. Then k(xo) < oo for almost all xo E U, and f4[k(x)]" dx < Q2"If (A) for every Bore! set A c U. PROOF. By the classical Lebesgue theorem on differentiating set functions, the limit lim If [G_(x)] = Y(x)
t-0
I G1(x)I
exists and is finite for almost all x E U. Further, for every Borel set A c U
L We show that if y(x) < oo at a point x E U, then also k(x) < oo; moreover, k(x) < Q2[y(x)]I /". Fix an arbitrary point x0 E U such that Y(xo) < oo.
Let (x,,), v = 1, 2, ... , be a sequence of points of U such that x as v -' oo, x 96 xo for all v, and
If (x.) - f (xo)I/Ix,, - xoI
k(xo)
xo
§8. CHARACTERIZATION BY QUASICONFORMALITY
207
as v - oo. Let v be the generating function for the system of neighborhoods G, = G, (x0). Let t = v and y,, = f (x,,). Obviously, x is a boundary point of G,, which implies that
Ix - xoI > r(xo, Further, since x E 8G,,,, it clearly follows that
IYv - YoI: R (xo.t,,,f) From this, R(xo.
Qr(xo, t.) <- QI x - xoI
(8.8)
and
r(xo,t,,.f) >(I IQ)R(xo, I., f) ? (1/Q)IY,. -yol.
(8.9)
We have that (8.10)
If(G, )I <- If(G,j, whence
If (GIJIIG, I ? If (G,.)I/IG, I. hence The set G,. is conained in the ball about x0 with radius R(xo, also in the ball of radius QIx - xoI about xo, by (8.8). From this,
IGt.I < anQ"Ixv - xol".
(8.11)
The set f (G,.) clearly contains the ball of radius r(xo, t,,, f ), hence also the ball of radius I y,, - y01 1Q, by (8.9). From this,
If(G,JI ? (a IQ")IY. -YoI"
(8.12)
Inequalities (8.10)-(8.12) allow us to conclude that If (G,,,)
1
IYv - YoI"
I - - Q2n Ix
-xoI".
Passing to the limit as v -' oo, we get that ?
2n
(Y(xo) IY.'- YOI" X ) = [k(o)]"/Q2"
Ixv -xoI Inequality (8.7) allows us to conclude that Q
f[k(xo)]'1 dx < Q2nlf(A) for every Borel set A c U. The lemma is proved. COROLLARY. If f : U -+ R" is a quasiconformal T-mapping, then f is differentiable almost everywhere in the domain U. Indeed, we have the following classical result due to Stepanov.
208
ll. MAPPINGS WITH BOUNDED DISTORTION
THEOREM 8.1 [163]. Let U be an open set in Rn, and f : U -+ R" a continuous function. If for almost all x E U lim r-.X
If(y)-f(x)I Iy-xI
then f is differentiable almost everywhere in Rn. By Lemma 8.2, every quasiconformal T-mapping obviously satisfies the conditions of Stepanov's theorem, and this leads to the corollary.
§8.3. The condition of absolute continuity for a real function of a single variable. We make some remarks about continuous functions of a single real variable. Let f : [a, b] -' R be a given function. The function f is said to satisfy condition N if the image f (E) of every closed set E C [a, b] with measure zero is a set of measure zero. Let f : [a, b] - R be a continuous function. For x E [a, b] - f (x) l . Df (x) = hi o l f(x + h) h
The function Df is measurable. Indeed, let an be the set of all rational numbers h 96 0 such that Ih I < 1 In, and let
Dn f(x) =
f (x + h) - f(x)
sup
h
O
Then D f (x)
Df (x) for all x. By continuity,
=hu)
(x+h)-f(x)
f
h
From this it is clear that Dn f is measurable, being the upper envelope of a countable set of measurable functions, and hence Df = limn. Dn f is measurable. LEMMA 8.3. Let f : [a, b] - R be a continuous function. If f has property N and fQ Df (x) dx < oo, then f is absolutely continuous.
PROOF. Suppose that f : [a, b] - R satisfies all the conditions of the lemma. Let lo (x) = fo Df (t) dt, and let A be the collection of all points x E [a, b] at which Df (x) < oo, 9 is differentiable, and to'(x) = Df (x). Then [a, b]\A is clearly a set of measure zero. Take an arbitrary number e > 0. Let [p, q] C [a, b]. We say that [p, q] is an interval of type H if for all x E [p, q] 1P(x) - q(p) + e(x - p) ? f (x) - f (p) > -[w(x) - So(p) + e(x - P)]
(8.13)
§8. CHARACTERIZATION BY QUASICONFORMALITY
209
If [p, q] is an interval of type H, then the measure of the set f ([p, qJ) is clearly at most 2[(P(q) - ap(p)] + 2e(q - p). Define w, (x) = ip(x) - f (x) and W2(x) = rp(x) + f (x). Suppose that x E A and x < b. Then
-X
w,(x') -
lim x'--x+o
x
-
lim W2(x') - W2(x) > 0, X X x'-.x+0
> 0,
which implies that there exists a 8(x) > 0 such that if x < x' < x + 5(x), then wl (x') - wl (x) > -e, W2(x') - W2(x) > -e.
x'-x
x'-x
This leads to the conclusion that if x < x' < x + S(x), then
9P(x') - P(x) - If (x') - f (x)l = wi (x') - wl (x) > -e(x' - x), IP(x') - q'(x) + If (x') - f (x)l = W2(x') - W2(x) > -e(x' - x). It obviously follows from these inequalities that every interval [x, x'] with
x < x' < x + 8(x) is an interval of type H. Take arbitrary points X1, x2 E [a, b] such that x, < x2. We prove that
If([xl,x2])I < 2
f
I
Df(t)dt.
(8.14)
x,
Let A0 = A n [xl, x2]. On the basis of the Vitali covering theorem, there
exists an at most countable set {[pm, qm], m = 1, 2, ... } of disjoint intervals of type H contained in [x,, x2] such that if U, _ U[pm, q,], then A0/U, is a set of measure zero. Denote by U the union of the open intervals (pm, qm). Obviously, I U, I = I UI = x2 - x,, and thus E0 = [x,, x2]\U is a closed set of measure zero. We have that [XI, X21 =
(U(Pm.m))
UEo,
m
f ([x,-x2]) =
Uf([Pmgml) Uf(EO) M
Since E0 is a closed set of measure zero, and f has the property N by assumption, it follows that If (Eo)I = 0, which gives us that
If([xl.x2l)I :5 EIf[(Pm,gm)1I. m
Since each of the intervals [ pm , qm ] is an interval of type H,
edt
If [(Pm, gm)1I < 2[9(qm) - ip(Pm)l + 2e(gm - Pm)
= 2 f-(Df)(t) m
+ 2e(qm - Pm).
(8.15)
11. MAPPINGS WITH BOUNDED DISTORTION
210
Summing these inequalities termwise, we get by (8.15) that
If([xl,x2])I <2 fU Df(t)dt+gelUl 2
r
x2
Df (t) dt + 2e(x2 - x1).
X)
Since e > 0 is arbitrary, this proves (8.14). Since If (x2) - f (x1)1 < f ([x1, x2]), it follows from what was proved that for any x1, x2 E [a, b], x1 < x2,
-2 If(x2) - f(xI)I <
f
x2
fl
Df(t)dt.
By well-known properties of the Lebesgue integral, the last inequality implies that f is absolutely continuous, and the lemma is proved.
§8.4. The analytic nature of quasiconformal T-mappings. We now establish that every Q-quasiconformal mapping of type T belongs to the class WW.10C
(U).
The proof of this is split into several steps and is based on an idea in Men'shov's classical paper [97].
Let U be an open domain in R", and f : U R" a quasiconformal mapping of type T. Take an arbitrary integer m such that q(U, f) < m. For each point x E U fix a regular normal system {G,(x), 0 < t < 1} of neighborhoods such that if a(x) is the regularity parameter of this family and
B(x) = lim r(x, t, f)/R(x, t, f 1-0
then a(x)fl(x) > 1/q(U, f) > 1/m. We assume that 1 r(x, t) r(x, t, f) and > m R(x, t) R(x, t, f) > m 1
for all t E (0, 1]. Obviously, there is no loss of generality in this.
Let a = (a,_., a") be an arbitrary point in U, and let h > 0 be such that the cube Q = Q(a, h) is in U. Denote by Ck (1 < k < n) the section of U by the plane Pk(a) _ {x E R" Ixk = ak }. If X E Ck, then let p(x) be the interval Ixk - ak I h/2 of the line passing through x parallel to the kth coordinate axis of R". If A is a subset of Ck, then let p(A) be the union of all the intervals p(x) with x E A. For an arbitrary set E lying in some 1-dimensional plane P, IEI, denotes the 1-dimensional Lebesgue outer measure of E.
§8. CHARACTERIZATION BY QUASICONFORMALITY
211
LEMMA 8.4. The function f has property N on the interval p(x) for almost all points x E Ck in the sense of (n - 1)-dimensional Lebesgue measure in the plane Pk(a).
PROOF. We define in Ck a measure u by setting u(E) = If [p(E)l for a Borel set E c Ck. The set p(E) is obviously also a Borel set, so p(E) is defined.
Let K(x, r) be the (n - 1)-dimensional ball in Pk(a) about x with radius r. Define D(x) = li my[K(x, r)]/I K(x, 1
By the classical results of Lebesgue on differentiation of set functions, D is finite for almost all x E Ck. Suppose that the point x E Ck is such that f does not have property N on p(x). We show that D(x) = oo in this case. By the preceding remark, this will prove the lemma. According to the definition of property N, there is a closed set E C p(x) such that IEII = 0 and m1[f (EI)] > 0. (Here ml, as earlier, denotes the one-dimensional Hausdorff measure on R".) Let ES be the collection of all x E E such that r(x, 1) > I Is, where s > 0 is an integer. Obviously,
0 E U Es,
I00
f (E) = U f (Es).
which implies that 00
mi[f(E)] <_ >2 m1[f(Es)]. s=1
and m1[f(Es)] > 0 for at least one s, because m1[f(E)] > 0. Fix an arbitrary such value of s, and let y = y1 [f (Es)] be the 1-content of f (Es). By the choice of s we have that y > 0, and for every system of balls covering f (Es) the sum of the radii is at least y/2.
Let e > 0 be arbitrary, and let N > 0 be an integer such that h/2N(m + 1) < 1 Is. We divide the interval p(x) into 2N(m + 1) equal intervals. Let V be the union of all the intervals in the partition which contain points of E. Since IEl I = 0 and E is closed, it follows that I Vn I I - 0
as N - oo. Let No be such that I Vv I < e for N > No. Fix an arbitrary N > No. From the intervals forming VN choose all those which contain points of Es, and take any point of Es in each of them. Let , ... , be all the points obtained in this way. Let p = h/2N(m + 1). For each i= 1, ... , I there is a 1>0 with t; < 1 such that r(4; , t;) = p.
212
II. MAPPINGS WITH BOUNDED DISTORTION
The required number ti can be found as follows. Let vi be the generating function of the system { G, ('i) 1 0 < t < 1) of neighborhoods. The largest
p) is the desired value ti. Let G. The sets value of vi in GI, ... , G, cover the set E, and the sets Hi = f (G) cover f (ES). The ball ti, f) about qi = f contains Hi. This gives us that of radius li, f) > yl2m. Ei Let us divide the intervals in the partition of p(x) constructed above into 2m + 2 groups by putting into a single group the intervals obtainable one from another by parallel translation by a multiple of h/N. Corresponding to this, the points I , ... , , are distributed into 2m + 2 classes by putting the points belonging to the intervals of a single group into a single ti, f) is class. For at least one of these classes the sum of the values r at least y/4(m + 2)m = 0. We assume that l , ... ,q are all the points 4i in this class. By construction
Iii - j I > h/N - 2h/(2m + 2)N = mh/(m + 1)N ti) _ mp = mh/N(m + 1). The balls Bi with i = 1, ... , q are disjoint, and for any i, j < q, i 0 j. Denote by Bi the ball about i of radius
Bi D Gi, which implies that the neighborhoods Gi are disjoint. The number q does not exceed the number of intervals forming the set VN; therefore, the sum of the lengths of these intervals, which equals qp, does not exceed Vn < e. Consider the ball K(x, m p). Let Z. = p[K(x, p)] be the cylinder
constructed on this ball. The set f (G,) contains the n-dimensional ball about qi with radius r(4i, ti, f ). We have that Gi C Z,, and since the Gi are disjoint, If (ZP) is not less than the sum of the volumes of the balls ti, f ), i.e., about qi with radii q
If (Zp)
ti, f )l"
Qn i=I
By the elementary inequality u + I the ui are nonnegative, we get that
If (Zp) > a ql-n
+ uq > q 1-"(u I +
[r(i.tif)]
+ uq)", where
Qn(y/2m)ng1-n
>_
i=1
On the other hand, we have that qp < e, which implies that 1 /q < p/e, and hence q1-" > p"-1/en-1. From this, If(Zp) > and, consequently, If (Zp)IK(x, mp) > C/e"-1, where C > 0 does not depend Qn(y/2m)"p"-1/en-I
on N. We have that p = h/2N(m + 1). Since N > No here is arbitrary, this implies that D(x) > C/en-1, and thus the arbitrariness of e > 0 gives us that D(x) = oo, as was to be proved.
§8. CHARACTERIZATION BY QUASICONFORMALITY
213
LEMMA 8.5. Let f : U R" be a mapping of type T, where U is an open domain in R". If f is quasiconformal in U, then f belongs to the class W,
1
(U).
PROOF. Let Q = Q(xo, h) be an arbitrary cube such that Q c U, and let Ck be the section of Q by the plane xk = xok. By Lemma 8.4, the vector-valued function f has property N on p(x) for almost all X E Ck. Let
Df (x) = 1im If (x') - f(x)I
IX - xI
The function Df is integrable to the power n on Q in view of Lemma 8.2. This gives us that Df is integrable over p(x) for almost all x E Ck. By Lemma 7.4, f is absolutely continuous on p(x) for every x E Ck such that f has property N on p(x) and Df is integrable on p(x). We have that
azk(x)i = hi m(I/h)Ifi(x+hek)-f,(x)I lImm
If(x+h) - f(x)I =Df(x).
h-.o
Ihl
The function Df is integrable to the power n. Thus, we get that for each k = 1.2, ... the function f is absolutely continuous on almost all lines parallel to the kth coordinate axis, and its derivative Of/8xk is integrable to the power n on Q. By the test in § 1 of Chapter I for a function to belong
to W,;, this implies that f is in W,' (Q). Since the cube Q c U was taken arbitrarily, this establishes that f E Wry 1a(U), and the lemma is proved. §8.5. Main result. LEMMA 8.6. Suppose that U is an open domain in R", and f : U -- R" is a continuous mapping of type T. Assume that f is differentiable at a point
a, with dfa 0 0, and that there exists a normal system {GG(a), 0 < t < 1} of neighborhoods of a with regularity coefficient a > 0 such that lim
r(a,t,f)
r-.oR(a,t,f)
> 0.
Then
,F (a. f) = det dfa # 0, sgnf (a. f) = j(a, f). where j(a, f) is the index off at a, and the quasiconformality coefficient of the linear mapping L = dfa does not exceed 1 /aft. PROOF.
If F(a, f)
0, then sgn f (a, f) = j(a, f) in view of the
properties of the index indicated in §2. Suppose that the linear mapping L= is nonzero, u1,... , u" is an orthonormal system of principal
II. MAPPINGS WITH BOUNDED DISTORTION
214
. . . . v, are orthogonal unit vectors such that Lu; = u;v;, where uI > . . . > un are the principal dilations of L.
direction vectors of L, and v 1 .
To simplify the notation we assume that a = 0 and b = f (a) = 0. Clearly this can always be arranged by suitably choosing the coordinate sys-
tem. Let r(t) = r(0, 1), R(t) = R(0, t), rf(t) = r(0,1, f), R1(z) = R(0, t, f), At = BG,, and B, = f (A,). Since f is differentiable by assumption at the point a = 0, we have that f (x) = L(x)+e(x)Ix1, where e(x) --4 0 as x -+ 0.
Suppose that a(t) > 0 and b(t) > 0 are such that a(t)uI E A, and b(t)un E At. Then rf(t) <_ If[b(t)unll,
Rf(t) >_ If[a(t)ul]l
From this,
rf(t) < If[b(t)uR]I Rf(t) - If[a(t)ut]I < If[b(t)unll
b(t)
We have that
_<
b(t) b(t) If[a(t)uill a(t) a(t) . R(t) a(t)
(8.16)
If[a(t)ul]I r(t)
f[b(t)un] = unb(t)un +e2(t)b(t),
f[a(t)u1] = ula(t)u1 +e1(t)a(t), where eI(t),e2(t) -- 0 as t -+ 0. Hence,
If[b(t)un]l b(t)
_ un.
If[a(t)ul]I a(t)
__, ul
as t - 0. Further,
limR(t)/r(t) = 1/a, and, passing in (8.16) to the limit as t --+ 0, we get that f < un/uI 1 /a. This gives us that An > 0, and thus L is nonsingular, i.e., det L = f (a, f) # 0. Further, U I /un < 1 /aft, and the lemma is proved. THEOREM 8.2 [134). Every mapping f : U - Rn of type T which is quasiconformal in a domain U c Rn is a mapping with bounded distortion.
PROOF. Suppose that f : U - Rn satisfies all the conditions of the theorem. Then, by Lemma 8.5, f belongs to Wn 10C(U). By the corollary
to Lemma 8.2, f is differentiable almost everywhere in U. Lemma 8.7 now allows us to conclude that the Jacobian of f has constant sign in U. Finally, by the same Lemma 8.7, the distortion coefficient of the linear mapping f'(x) does not exceed q(U, f) < oo. Thus, f satisfies all the conditions in the definition of a mapping with bounded distortion, and the theorem is proved.
§8. CHARACTERIZATION BY QUASICONFORMALITY
215
COROLLARY. Let U be an open subset of Rn, and f : U - Rn an isolated open mapping. Assume that f is quasiconformal in every domain 0 lying
strictly inside U, and there exists a constant Q > 1, Q < oo, such that q(x, f) < Q at each point x E U\B f. Then f is a mapping with bounded distortion.
PROOF. It follows from Theorem 8.2 that f is a mapping with bounded distortion on every domain lying strictly inside U. For almost all x E B f
we have that f (x, f) = 0, and hence f'(x) = 0. The inequality q(x, f) < Q holds at each point x ¢ Bf. This implies that K[f'(x)] < Qn at each point x l! B f such that f is differentiable and f(x, f) 96 0. Consequently, for almost all x E U
IIf'(x)IIn < Qnf(x.f), and hence f is a mapping with bounded distortion. §8.6. Homeomorphic quasiconformal mappings. As an application of the above results, we prove the following assertion.
THEOREM 8.3. Let U be an open set in Rn, and f : U - Rn a homeomorphism. If f is quasiconformal, then the inverse mapping g = f -' is also quasiconformal. Moreover, q(V, g) = q(U, f ), where V = f(U).
PROOF. Let e > 0 be arbitrary. For every point x E U we construct a normal system { G, (x) I0 < t < 1) of neighborhoods such that lim
rG(x, t)
,moo RG(X, t)
a,
rG(x, t. f) = lim P ,.o RG(x, t, f) ,
-
with
of>k=1/[q(U,f)+e]. Let Ht(y) = f [G,(x)], where y = f (x). The sets H,(x) clearly form a normal family of neighborhoods of the point y = f (x). Further, if v is the generating function of the family of neighborhoods G,(x), then w(x) = v[g(x)] is the generating function for the family of neighborhoods H, (x ). We have that
rG(x, t, f) = inf IY' - YI = rH(Y, t), y'EBH,
Ro(x, t , f) = sup IY' - YI = RH(Y.1). y'EBH,
rG(x, t) = inf I g(Y') - g(Y)I = rH(Y, t. g), y'EBH, RG(x,1) = sup I g(Y') - g(Y)I = RH(Y, t, g) y'EBH,
11. MAPPINGS WITH BOUNDED DISTORTION
216
It is clear from this that g is quasiconformal at the point y, and the quasiconformality coefficient for it at y does not exceed I /a.8 < q(U, f)+e. Since Y E V and e > 0 are arbitrary, the proof of the theorem is complete. We prove the following assertion as an application of the last theorem.
THEOREM 8.4. Suppose that U c R" is an open set, and f : U -+ R" is a homeomorphic mapping with bounded distortion. Then F(x, f) 0 for almost all x E U. PROOF. Suppose that f : U -' R" satisfies the condition of the theorem. Let V = f (U) and g = f -1. The mapping g is quasiconformal by the theorem. Let E1 be the set of all points x E U at which f is not differentiable. Then IE1I = 0. Let E2 denote the collection of all points x E U at which
f is differential and F(x, f) = 0. Since f has property N, If (E1)I = 0. By Theorem 2.2, If(EA)I =
f
f(x,f)Idx = 0.
E2
Consequently, A = f (E1 u E2) is a set of measure zero. Since g is a quasiconformal mapping and thus has property N, Ig(A)I = 0. We have that g(A) = E1 U E2, and this proves the theorem. §9. Sequences of mappings with bounded distortion
§9.1. A theorem on local boundedness of sequences of mappings with bounded distortion.
THEOREM 9.1. Suppose that U is an open domain in R", and (fm : U R"), m = 1, 2, ... , is a sequence of mappings with bounded distortion such that the sequence (K(fm)) of their distortion coefficients is bounded. If (fm) is locally bounded in L"(U), then it is locally bounded also in the sense of W (U).
PROOF. Let fm = (fm1,... , fm,,). We take an arbitrary value of i such
that 1 < i < n and let um = fmi. For each m the function um is a generalized solution in U of the differential equation div((Om(x)u'(x),
u'(x))(""2)-10,n(x)u'(x))
= 0, where Or(x) = 8(x, fm) is the matrix defined from fm as indicated in §5. This means that for every V E W (U) with compact support in U
1 (0mum, u)'2 -1(Omum, q,') dx = 0.
(9.1)
The matrix 8m is symmetric. Since the sequence K(fm) is bounded by an assumption of the theorem, there is a constant a, 0 < a < 1, such that
§9. SEQUENCES OF MAPPINGS WITH BOUNDED DISTORTION
217
for every m and for almost all x E U (9.2)
1/a2ICI2
a2112 <-
for any vector E R". The constant a here depends only on the supremum K of the sequence (K(fm)), m = 1, 2. ... .
Let A c U be an arbitrary compact set, and let 6 > 0 be such that the closed J-neighborhood Uo(A) of A is contained in U. We set G = U5(A) and construct a function C E Co (U) such that C(x) = 1 for x E A, {(x) = 0 for x G, and 0 < C(x) < 1 for all x. Let la = umC" in (9.1). We transform the integrand. We have that rp' = u;,,C" + nun,{"-I C'. Further, ' n/2- I , n/2 n uj
+
(8mum'
(em m' u m)
um)
+n(9nu'm' u'm
)"/2-I m (e u' m
n
The inequalities (9.2), along with the inequalities I(9. ' )I <- IOmCII nI <- l/aI Ilni.
give us that (emU
,
U 1)(n-2)12(0mum, (p') > a"Iu,,I"C"
- an 1IU,
In-1Cn-1IumIIC,I
Let us use Young's inequality:
ab < (an/n) + (n - 1)b("-1)/"/n for any a > 0 and b > 0. Setting a = I um I I Z' I/t and b = I u;" I Cn-1 t, where
t > 0 will be chosen later, we get that i (n-2)/2 i l mu in' um) (emum,' (ei
(a" -
nan 1
t(n - 2)/n) Iu'
I"cn
- umI r'I nl tn
The number t is now chosen from the condition
an - (n -
l)/a"t("-1)1n = a"/2.
As a result (em U'
u )(n-2)/2(emurn 9') >- a"/2 lu;n1%n - Un IC,I"lt".
(9.3)
Integrating inequality (9.3) termwise and considering (9.1), we get that 0>
an f 2 u
dx - t1
f
IuminIC'In dx,
U
whence
f I U, (x)I"[C(X)]" dX < c fu Ium(X)I"IC'(X)I" dx, U
218
II. MAPPINGS WITH BOUNDED DISTORTION
where C = 2/(at)" is a constant, and C depends only on the constant K (the supremum of the sequence (K(fm)), m = 1, 2.... ). We have that C(x) = 1 for x E A and C'(x) = 0 for x
f
G. From this,
Ium(x)l" dx < CI JG I um(x)I" dx.
(9.4)
Since G C U and G is compact, and since the sequence (fm ), m = 1, 2,..., is locally bounded in L"(U) by hypothesis, (9.4) gives us that the sequence (fA Iu'm(x)I" dx), m = 1, 2, ... , is bounded. Since the compact set A C U was taken arbitrarily, the theorem is proved.
§9.2. A theorem on the limit of a sequence of mappings with bounded distortion.
THEOREM 9.2. Suppose that (f,,,: U - R"), m = 1, 2,..., is an arbitrary sequence of mappings with bounded distortion that converges locally
in L"(U) to a mapping fo: U - R. Assume that the sequence (K(fm)), m = 1, 2, ... , of distortion coefficients is bounded. Then the limit mapping fo is a mapping with bounded distortion, and
K(fo) _< lim K(fm). M-00
PROOF. It obviously follows from the conditions of the theorem that the sequence (f", ), m = 1, 2, ... , is bounded in L",1,,c(U). Hence, it is locally
bounded also in W (U), by Theorem 9.1. According to the corollary to Lemma 4.9 of this chapter, the conditions of the theorem imply that the functions ,&: x F(x, fm) converge weakly in L1,1a(U) to the function A : X- F(x, fo) as m oo, i.e., for every continuous function rp with compact support in U
L 9 (x)f (x, fm) dx -
fu
47(x)f (x, fo) dx.
(9.5)
For each of the mappings fm the Jacobian has constant sign in U. We say that fm is a positive mapping if .7(x, f",) > 0 almost everywhere in U, and a negative mapping if,(x, fm) < 0 almost everywhere in U. The relation (9.5) allows us to conclude that if there exist positive mappings with arbitrarily large indices, then f,, g'(x)f (x, fo) dx > 0 for every nonnegative function 91 E Co (U), and hence, (x, fo) > 0 almost everywhere in U. Similarly, if there exist negative mappings with arbitrarily large indices, then 7(x, fo) < 0 almost everywhere in U. It follows from the foregoing that if there exist both positive and negative mappings with arbitrarily large indices, then F(x, fo) = 0 almost everywhere in U, and the mapping
§9. SEQUENCES OF MAPPINGS WITH BOUNDED DISTORTION
219
fo is identically constant. In this case all the assertions of the theorem are obviously satisfied. We assume in what follows that the limit mapping fo is not equivalent (in the sense of the theory of the integral) to a constant mapping. Then there exists an index mo such that the mappings fm with m > mo are either all positive or all negative. For simplicity it will be assumed that the fm are positive for all m. Then F(x, fo) > 0 almost everywhere in U. Let Ko = lim,"_,, K(fm). Take an arbitrary number e > 0 and choose from the sequence (fm ), m = 1, 2, ... , a subsequence (fmk ), m 1 < m2 < ... , such that K(fmk) < Ko + e for all k. For each k and for almost all x, Ifmk (x)I" < (Ko + e)f (x, fmk )
Let ip be an arbitrary nonnegative function in CO(U). Multiplying both sides of the last inequality by 9(x) and integrating with respect to x, we get that
f
(Ko+e)J
9(x)J(x,fma)dx.
(9.6)
u
u
Ask -. oo
f
U
99(x)f (x,
f
iv(x).f (x, fo) dx.
U
Further, by the theorem proved in §3 of Chapter III on semicontinuity for functionals of the calculus of variations,
f f
c(x)Ifo(x)I"dx < lim
f f
rp(x)IJ;,,k(x)I"dx.
k-oo ft On the basis of the foregoing, it follows from (9.6) that U
cv(x)Ifo(x)I"dx < (K0+e)
U
U
co(x)f(x,fo)dx,
and I fo(x)I" < (Ko+e),r(x, fo) almost everywhere in U, because here ' is an arbitrary nonnegative function of class Co(U). The mapping fo belongs to the class Wry 10(U) in view of Theorem 9.1. It is thereby established that fo is a mapping with bounded distortion. Further, we get that K(fo) < Ko+
e. Since e > 0 is arbitrary, it follows that K(fo) < Ko = lim_0" K(fm), and this completes the proof of the theorem. COROLLARY. Let (fm ), m = 1, 2, ... , be an arbitrary sequence of quasiconformal mappings ofan open domain U c R" that converges in L",1a(U) to some mapping fo which is not identically constant in U. Then the limit mapping fo is also quasiconformal.
PROOF. Let G be an arbitrary compact domain contained in U. Then for every point y ¢ fo(8G) the degree p(y, fm, G) of fm with respect to G
11. MAPPINGS WITH BOUNDED DISTORTION
220
is defined for sufficiently large m. Further, u(y, fm, G) = µ(y, fo, G) for all sufficiently large m. Since each of the mappings fm is quasiconformal, I µ(y, fm, G) I < 1 for all m. Consequently, I#(y, fo, G) I < 1 for every point
y ¢ fo(8G). The set fo '(y) n G is finite, and µ(y, fo,G) is equal to the sum of the indices of f at the points of fo ' (y). By Theorem 6.1 in §6, all these indices have the same sign, and each of them is a nonzero integer. It follows from the inequality II (yo , f, G) 1 < 1 that f - I (yo) n G consists of at most one point. Since y ¢ fo(80 was taken arbitrarily, this permits us to conclude that fo is one-to-one on Go. The compact domain G c U was taken arbitrarily, so this implies that fo is one-to-one in U. (Since U is connected, for any two points xi, x2 E U there is clearly a compact domain G c U such that x, and x2 belong to the interior of G.) Consequently, f is a homeomorphic mapping of U, and this proves the corollary. Theorems 9.1 and 9.2 were established by the author in [ 130]. §9.3. A sufficient condition for relative compactness of a family of mappings with bounded distortion.
THEOREM 9.3. Let U be an open subset of R", and 0 a family of mappings with bounded distortion from U to R". Assume the following conditions: for every closed ball G = B(xo, r) c U there exist numbers K(G) < oo and L(G) < oo such that for any f e di the distortion coefficient of f'(x) does not exceed K(G) almost everywhere in G. and
f I f (x) I' dx < L(G). Then the family 4) is uniformly equicontinuous on every compact set A c U.
PROOF. Take an arbitrary point xo E U and let r > 0 be such that G = B(xo, r) c U. We prove that there exists a constant M(G) < oo such that 11f
f
BIf'(x)Idx + f(xo.,/2)
I f (x)Idx < M(G).
Indeed, if not, then there exists a sequence (fm ), m = 1, 2,..., of functions in 4) such that IIfm II w (B(xo.,/2)1 - oo as m -{ oo. By a condition of the theorem, this sequence is locally bounded in L"(B(xo, r)), and K(fm) < K(G) = const < oo. Hence, by Theorem 9.1, this sequence is locally bounded in W [B(xo, r)], and we clearly get a contradiction. This proves the existence of the required constant M(G). The family 0 is thus bounded in W [B(xo, r/2)). The estimates in § 1 allow us to conclude that it is uniformly equicontinuous in B(xo, r/4). We get that every point x0 E U has a neighborhood B(xo, S) in which 0 is
§10. THE SET OF BRANCH POINTS
221
uniformly equicontinuous. Let A c U be a compact set. Covering A by finitely many neighborhoods of the indicated kind, we get that (D is uniformly equicontinuous on A, and the theorem is proved. §10. The set of branch points of a mapping with bounded distortion and locally homeomorphic mappings
§10.1. The measure of the set of branch points. The main result in this subsection is the following statement. THEOREM 10.1 [ 137). The intersection of the set B f of branch points of a mapping with bounded distortion with any (n - I)-dimensional plane is a set whose (n - 1)-dimensional Lebesgue measure is equal to zero.
The proof is based on certain auxiliary constructions, to which we now proceed. Let f : U - R" be an arbitrary mapping with bounded distortion, where U is an open set in R". As usual, B f denotes the set of branch points of f. A disk in R" is defined to be any (n -1)-dimensional open ball in R".
The disk about a with radius r > 0 and lying in an (n - 1)-dimensional plane P will be denoted by K(a, r, P). In cases when knowledge of the plane of a disk is not necessary we use the simpler notation K(a, r). A disk K(a, r) is said to be clean if it does not contain branch points of f. Let K(ao, r, P) c U be an arbitrary disk in R". We consider the collection of all clean disks K(b, p) contained in K(ao, r, P). Since B f is closed with respect to U and is at most (n - 2)-dimensional, it does not contain any disk, which implies that the indicated set of disks is nonempty. The
supremum of the ratios p/r on the set of all these disks K(b, p) is denoted by a(a, r, P) and will be called the measure of cleanness of the disk K(a, r, P). The infimum of a(a, r, P) on the collection of all numbers r < 0 with r < p(a, 8U) and all planes P passing through a is denoted by a(a). LEMMA 10.1. a(a) is positive for every point a E U.
PROOF. Assume, on the contrary, that a(a) = 0 f o r some point a c -
Then there exist a sequence (r,,), v = 1 , 2,..., of values and a sev = 1, 2, ... , of planes passing through a such that a = a(a, r,, 0 as v - oo. It can obviously be assumed without loss of generality that the numbers r tend to some limit ro (0 < ro < oo) as U.
quence
v -+ oo, and that the planes P converge to some plane P0 passing through a. Two cases are possible: 1) ro > 0, and 2) ro = 0. Let us take the first case. We have the disk K(a, ro, Po). In the case ro = oo it clearly coincides with P0. Let x0 be an arbitrary point of K(a, ro, Po),
and let x be the orthogonal projection of it on P. Then x - xo as
11. MAPPINGS WITH BOUNDED DISTORTION
222
V - oo, aoI - Ixo - aol < ro, and hence lx - aol < r, for sufficiently large v, i.e., x, E X, E K(ao, r,,, By assumption, a,, - 0 as v -+ oo. Let K(b,,, c K be a disk that contains x and such that p< a,,r,, + (1 /v). By the definition of the number a,, = a(a, r,, P,) the ball contains points of the set B1. Let yv be one of these. Obviously, 1 Iv), and hence Ix,, - y,,l - 0 as v -- oo. This implies that y,, -. xo as v - oo. Thus, x0 is the limit of a sequence of branch points of f. Since B f is closed relative to U, this gives us that xo E B f. Because K(b,, ,
Ix - y.1 <
xo E K(a, ro, Po) was taken arbitrarily, we get that K(ao, ro, Po) c B1, which is impossible, since B f is at most (n - 2)-dimensional. Thus, the case ro > 0 leads to a contradiction. Let us now consider the case ro = 0. There is a b, > 0 such that a1 < p(a, 8u) and f (x) # f (a) for 0 < lx - al < d1. Let
L1 =Ix-al=a, min If(x) -f(a)I. There is a 62 > 0 such that 0 < 82 < 81 and
1= max If(x)-f(a)I <-L1. lxal
Assume that r,, < 62 for all v, and let
A = L f(a, r,,) = max If(x) - f(a)I. lx-al =r,, A,. =1 f(a,
lrm nrV If(x) - f (a)l
We now define a sequence of mappings F,,, v = 1, 2, ... , of the unit ball B1 = B(0, 1) by setting F,,(X) = r,,X) - f (a)] for X E BI
and v = 1, 2..... Obviously, F is a mapping with bounded distortion. Further, we have that
K(f) and IF, (X) 1 < 1 for all v. For each point y E R" µ(y, F,,. B(0. 1)) = u(y,.. f B(a, r,,)),
where y = f (a)+ A, y. In view of the condition that the index is bounded (Theorem 6.5) there exists a constant uo < oo such that l y[ y, f, W (O, r,,)] 1 <
µo for all v. The sequence of mappings F, v = 1, 2, ... , satisfies all the conditions of Theorem 9.2, and hence the sequence of mappings is equi-uniformly continuous on every ball B(0, t), where 0 < t < 1. From the sequence we choose a subsequence converging to some mapping FO: BI - R" uniformly on every ball B(0, t). In order not to complicate the notation we assume that F,, is the required subsequence. By Theorems 7.2 and 7.3, for each v
A,M2(lx - al
If (x) - f (a)l
<-
(Ix - al /r)°:
223
§10. THE SET OF BRANCH POINTS
if 0:5 Ix - al < r, where MI and M2 are constants, and a1 =
Setting x = a +
K(f)Ij(a,f)l111n-1),
Ij(a,f)I1n1-1)/Ko(f).
a2 =
here, we get that for each v M21XI°' < IFF(X)I < M11XIQ2
for all X E B(0, 1). Passing to the limit as v
oo, we conclude that also
M21XI°' <- IFo(X)I 5 M1IXI°2
in B(0, 1). In particular, this implies that F0 is not identically constant in B(0, 1) = B1. By Theorem 9.1, F0 is a mapping with bounded distortion. Denote by Q the (n - 1)-dimensional plane passing through the point
0 and parallel to P. The planes Q converge to a plane Qo parallel to Po as v - oo. Let K be the disk obtained by intersecting B1 with the plane Q,,, v = 0, 1, .... The mapping X - a + rX obviously establishes a one-to-one correspondence between the set of branch points of F and the set B f n B (a, r,,). This implies that the measure of cleanness of the is equal to the measure of cleanness of the disk K with disk K(a, r , 0 as v - oo. respect to the mapping F,,, and a(0, 1, Take an arbitrary point x0 of K0, and let e > 0 be such that IxoI < 1 - e and Fo(x) 0 Fo(xo) for any x with 0 < Ix - xol < z. Let Yo = Fo(xo).
a
Then yo V Fo(S(xo, e)), and thus the quantity u(yo, F0, B(x, e)) is defined.
Let x be the point of K,, closest to x0. Obviously, x x0 as v - oo. It Ixol < 1 - e. By assumption, a - 0 as v - oo. is also clear that In the plane Q we construct a disk K(b, p) contained in K and such
that a < p < a + 1 /v. By the definition of the number a,,, this disk contains branch points of F. Let be one of these. Then it is clear that 2(a, + 1/v), and hence - x0 as v -i oo. Let yY = -+ F0 uniformly on every compact subset of B(0,1) and, in particular, on B(xo, e) as v -' oo. This gives us that y - yo = Fo(xo) as
I-
F
v -+ oo. For sufficiently large v
AY, F,,,r(xo,e)) =,u(Yo.Fo.B(xo,e)) For sufficiently large v (namely, for v such that y E B(xo, e) and y F [S(xo, e)]) the inequality (µ[Y, F,,, B(xo, e)]I >
2
holds since , is a branch point of F. We conclude from this that I µ(Yo, F0. B(xo, e))I > 2 also holds. The number e > 0 with e < I - IxoI was arbitrary, so it follows that I j(xo, Fo)I > 2, i.e., x0 is a branch point of F0. Since x0 E Ko was taken arbitrarily, we get that the set of branch
224
II. MAPPINGS WITH BOUNDED DISTORTION
points of some mapping with bounded distortion contains a disk, which is impossible. Thus the case ro = 0 leads to a contradiction, and the lemma is proved. PROOF OF THEOREM 10.1. Let P be an arbitrary (n - 1)-dimensional plane, and let E = P n B1. For a set A C P the symbol IAI here denotes its (n - 1)-dimensional Lebesgue measure. Take an arbitrary point x E E and consider the ratio A(x, r) = IE n K(x, r, P)I/IK(x, r, P)I.
For every e > 0 each disk K(x, r) contains a disk of radius at least [a(x) -e]r which does not contain points of B1. This implies that %(x, r) < 1 - [a(x)]"-I for each r. Let A(x) = lim,.o(x, r). The number A(x)
is the so-called upper density of E at the point x. We get that A(x) < 1 - [a(x)]"-I < 1. Thus, the set E has upper density less than 1 at each of its points x. By classical theorems (see, for example, [149]) on differentiation of set functions, this implies that E is a set of measure zero in P. The theorem is proved. COROLLARY 1. The set of branch points of any mapping with bounded distortion is a set of measure zero.
COROLLARY 2. Let f : U -p R", where U C R" is an open set. Then the differential off at a point x E U is a nonsingular linear mapping for almost all x E H. PROOF. The set B f is closed. Suppose that x ¢ B f. Then there exists a neighborhood V of x such that f is a homeomorphism on V. By Theorem
8.3, fl(x) is nonsingular almost everywhere on V. Covering U\B f by countably many neighborhoods on each of which f is a homeomorphism, we get the required result.
Let U C R" be an open set, and f : U -, R" an open mapping. The mapping f is called §10.2. Some lemmas on local homeomorphisms.
proper if f - I (A) is compact for every compact set A C f (U). LEMMA 10.2. Let U be a domain in R", f : U --+ R" a continuous open mapping, G a domain lying strictly inside f (U), and V a connected component off - I (G). If V lies strictly inside U, then f (V) = G. G does not contain images of boundary points of V, and the restriction off to V is a proper mapping. PROOF. Assume, on the contrary, that there is a point yo E G such that yo E f (8V). Suppose that the point xo E 8V is such that f (xo) = yo. Since
§10. THE SET OF BRANCH POINTS
225
f is continuous, there is an e > 0 such that B(xo, e) c U and f [B(xo, e)] c G. Since x0 is a boundary point of V, B(xo, e) fl V is nonempty. Hence, V' = B(xo, e) n V is connected. Obviously, f (V') c G. At the same time, V C V' and V 0 V', since x0 E V' but x0 it V. We thus get a contradiction to the fact that V is a connected component of f -I (G).
Let us now prove that f (V) = G. The set f (V) is open relative to G. Since G is connected, for the proof it suffices to establish that f (V) is closed relative to G. Let yo E G be a limit point of f (V). Then there is a sequence (y,), v = 1, 2, ..., of points in f (V) such that y - y as v -- oo. Let y = f where x E V. It will be assumed that x tends
to some point x0 E V as v - oo. Then yo = f (xo). Since G does not contain images of boundary points of V, it now follows that xo E V, and yo E f (V). Thus, f (V) contains all its limit points in G, and hence f (V) is closed relative to G. We now prove that f I v is a proper mapping. Let Q c G be compact. The set P = f - I (Q) n V is closed relative to V. We prove that it is closed in R". Assume that this is not so. Then there is a point xo E P such that x0 0 V. Then X0 E 8V, and hence yo = f (xo) E 8G. By the continuity of f, yo E Q, and we get a contradiction to the fact that Q C G. The lemma is proved. LEMMA 10.3. Let G be an open domain in R", and f : G - R" a continuous local homeomorphism. If f is one-to-one on some compact set K c G, then f is a homeomorphism of some neighborhood U C G of K.
PROOF. Let UU(K) be the S-neighborhood of K, where 8 > 0. We prove that f is one-to-one on U5(K) for some d > 0. Assume that this is not so. Then for every m there is a pair of points x,,, x,',', E G such that p(x'",, K) < 1 /m, p(xm", K) < 1 /m, x,' 96 xm" , and f f (x"). Without loss of generality it can be assumed that the sequences and (xm") converge to points x', x" E K as m -, oo. By the continuity of f,
f(x') = ii m f (Xm') = lim f(Xm) =
f(x").
Since f is one-to-one on K, the equality f (x') = f (x") gives us that x' = x", and thus the sequences and (x,;,) converge to the same point xo E K. However, this contradicts the fact that f is a local homeomorphism. Since every continuous one-to-one mapping of a domain in R" is a homeomorphism, this proves the lemma. We say that a domain G c R" can be exhausted simply connectedly if there exists a sequence (Gm), m = 1,2,..., of simply connected domains such that Gm C Gm+1 for each m and U Gm = G.
11. MAPPINGS WITH BOUNDED DISTORTION
226
Let X be an arbitrary topological space. A path in X is defined to be X. Here the point x = a (O) is called the start of the path a, y = a(1) is called the end, and the path or is said to join the points x and y. A path or is said to be closed if a(0) = a(1). A topological space X is said to be arcwise connected if for any two points x and y in X there is a path joining x and y. any continuous mapping a : [0, 1]
An arcwise connected topological space X is said to be simply connected
if for any closed path a : [0, 11 - X there exists a continuous mapping C : [0,1 ] x [0,1 ] - X such that C(t, 0) = a(t), C(0, u) = C(1, u) = a(0) = a(l) for all u, and C(t, 1) = a(0) for all t E [0, 1]. The mapping { is called a homotopy of the path a into the point a(0). Let X and Y be connected topological spaces, and f : X Y a continuous mapping. Such an f is called a covering map if the following conditions hold:
a) f maps X onto Y. b) Every point y E Y has a connected neighborhood V such that every connected component of f -' (V) is mapped homeomorphically onto V by f.
We have the following statement [55].
THEOREM 10.2. Let f : X -' Y be a continuous mapping. Assume that X is connected and Y simply connected. If f is a covering map, then f is a homeomorphism. LEMMA 10.4. A proper local homeomorphism rp of a bounded domain
G C R" onto a domain G that can be simply connectedly exhausted is a homeomorphism.
PROOF. Let (G,), m = 1, 2, ... , be a simply connected exhaustion of G. We construct a sequence (Gm), m = 1, 2, ... , of subsets of G by induction.
Let GI be a connected component of ip-'(G). If Gm has already been constructed, then Gm+I is the connected component of rp-' (Om+,) which contains G'm. Let com be the restriction of rp to Gm. On the basis of Lemma 10.2, rpm maps Gm onto G'm and is a proper mapping for each m. Take an arbitrary point y E 7m. The set 9m' (y) is compact, since rp is a proper mapping. Because rpm is a local homeomorphism, all the points of rpm' (y) are isolated. This implies that rpm' (y) is a finite set. Let xi , ... , xk be all the elements of Spm' (y). Denote by U; a neighborhood of x; such that U1 C Gm and SPm is a homeomorphism on Ur1, and let v = nki-1 VPm Uk) Finally, let U; = Spm' (V) r1 U. Then rpm maps U, onto V for each i, and rpm'(V) is the union of the sets U,. We thus get that every point y E G, has a neighborhood V such that each connected component of rpm' ( V) lies
§ 10. THE SET OF BRANCH POINTS
227
strictly inside G,,, and is mapped homeomorphically onto V. This means that corn is a covering map. On the basis of Theorem 10.2, this implies that rpm is a homeomorphism of Gm onto Gm. Let G = 1J0° Gm. Then G c G, and q.' is clearly a homeomorphism on G. We prove that qp is a homeomorphism of the whole domain G. To do
this it suffices to establish that G = G. Obviously, G is open. We show that G is closed with respect to G. Indeed, let xo be a limit point of G, and let yo = q'(xo). There is an m1 such that yo E G. Let U be a connected neighborhood of x0 such that rp(U) c Gm. There is an m2 such that U contains points of G,"2; let m > m, and m > m2. The set Gm2 U U is connected, and Sp(Gm2 U U) c This implies that Gm2 u U c G'm, and thus xo E Gm; hence xo E G. Therefore, & contains all its limit points, and
is thus closed relative to G. Since G is connected, what has been proved implies that G = G. The lemma is proved. § 10.3. The measure of the image of the set of branch points for mappings with bounded distortion. Let a and b be arbitrary points in R", with a 0 b. Denote by Pa (b) the ray in R" formed by the points x = a + t (b -a), where
t > 1. Let E be an arbitrary set in R", and a a point not in E. The union of the rays Pa(x) with x E E is denoted by Pa(E). LEMMA 10.5. Let E be a set in R" such that m0(E) = 0 for some a > 0, and let a be a point not in E. Then ma+1 [Pa(E)J = 0. PROOF. Take an arbitrary integer k > 1, and let Tk be the ring consisting
of all points x E R" with Ilk < Ix - al < k. Let Ek = E n Tk and Pk = Pa(Ek) n Tk. Obviously, 0 for each k. Further, P4(E) = U' Pk, and the lemma will be proved if we establish that mQ+I (Pk) = 0 for each k, because the Hausdorff measure is countably additive. Let e > 0 be arbitrary. By the definition of a Hausdorff measure, there is a sequence
of balls B; = B(x,, ri) covering Ek such that r; < 114k for each i and F,, r° < E. We can obviously assume that each B, contains points of Ek, since the balls for which this is not true can be excluded from the sequence (B;), i = 1, 2,..., without detriment to the rest of its properties. Under this condition, p(a, B,) > 1/2k for all i, and hence a ¢ B, for all i. Let H; = Pa(Bi) n Tk. The sets HI, H2,... obviously cover Pk. Let v; = (x, - a)/Ix; - al. Denote by Bj.m the ball about a + (2m - 1)r;v;
with radius r;,,,, = (2k2 + 1)r;, where m > 1 is an integer. The balls Bi,m cover H1 for m = 1, ... , m;, where m; is the largest integer m such that (2m - 1)r; < k. Consequently, we get a set {B;,m, 2,..., 1 < m < m;} of balls that covers Pk. We have that
228
11. MAPPINGS WITH BOUNDED DISTORTION
rl?m < (2k2 +
1)a+1 k+ l
2r;
r°+1 =M F r° < Me,
where M is a constant. Since e > 0 is arbitrary, this implies that mQ+1(Pk ) = 0, and the lemma is proved.
THEOREM 10.3 [44]. Let U be a domain in R", and f : U - R" a mapping with bounded distortion. If its set Bf of branch points is nonempty, then (n - 2)-dimensional Hausdorfineasure of f (B f) is positive. PROOF. Let f : U -+ R" satisfy the condition of the theorem. Assume, contrary to what is to be proved, that M, -2[f (Bf)] = 0. Take an arbitrary point xo E B1, and let yo = f (xo). Let 8 > 0 be such that B(xo, 6) C U and f (x) 0 yo for 0 < Ix - xoI < J. Such a 6 exists since f is an isolated mapping. Let y > 0 be the smallest value taken by If (x)-yoI on the sphere S(xo, 8). Let G = B(yo, y/2). Obviously, G C f [B(xo, 8)]. Let V be the connected component off -1(G) containing xo. Since V is connected, and If (x) - .vol < y/2 for all x E V, it follows that V C B(xo, 8), because
otherwise V would contain points of S(xo, 6), and If (x) - yoI > y for x E S(xo, 6). Further, it is also clear that V c B(xo, 6), since otherwise the supremum of If (x) - yoI on V would be at least y. On the basis of Lemma 10.2 we get that f (V) C G and f (8V) c 8G. We show that f (B f n V) is a set closed relative to G. Indeed, let z E G be
a limit point of f (B f n V). Then there is a sequence (t ), v = 1, 2, ... , of z. Without loss of generality it can be points in B f n V such that f assumed that t - to E V as v -. oo. Since B f is closed relative to U and V c U, it follows that to E B1. Obviously, f (to) = z. Since G does not contain boundary points of V, we have that to E V, hence z E f (B f n V), which is what was to be proved. Let f (B f n V) = A. By assumption, the (n - 2)-dimensional Hausdorff measure of A is equal to zero. Hence, there is a point b E G such that b io A. We construct the set Pb (A) and let H = G\Pb(A). Then the (n -1)dimensional Lebesgue measure of Pb(A) is equal to zero on the basis of Lemma 10.5. Since A is closed relative to G, Gn Pb(A) is also closed, and hence H is open. The set H is simply connected. Indeed, since b E A, every closed curve in H is shrunk to the point b without leaving H by the deformation defined by the formula f (t, x) = b + t(x - b). Let W be one of the connected components of f -1(H) contained in V. Then W c B(xo, 8), and Lemma 10.2 gives us that f (W) = H, f (8W) c 8H, and the restriction of f to W is a proper mapping. On the basis of
§ 10. THE SET OF BRANCH POINTS
229
Lemma 10.4, for every connected component of f -' (H) lying in V the restriction of f to this component is a homeomorphism.
Let W,,..., W,... be all the connected components of f -' (H) n v, be the mapping inverse to the restriction off to W. Then f,,,- I and let is quasiconformal on the set H = G\Pb(A), and since m"_1[Pb(A)] = 0, it follows by Theorem 6.8 that f,, ' extends to a homeomorphism of G into
V. Let Wo = f7 ' (G). Then Wo c V. Obviously, Wo is both open and closed relative to V, and since V is connected, Wo = V. Consequently, f is a homeomorphism of V into R", and this contradicts the fact that V contains branch points of f. The theorem is proved. §10.4. A local homeomorphism theorem. The main result in this section is the proof of a theorem of V. M. Gol'dshteln which says that a mapping with bounded distortion whose distortion coefficient is close to 1 is a local homeomorphism.
LEMMA 10.6. Suppose that U c R" is an open set, f : U - R" is a mapping with bounded distortion, k > 1 is a constant, and Bk is the set of all points x E U such that Ij(x, f )I > k. Then Bk is closed relative to U.
PROOF. It will be assumed that j(x, f) > 0 for all x E U. The general case can obviously be reduced to this one. Let X0 E U be an arbitrary limit point of Bk. Take an r > 0 such that if G = B(xo, r), then yo = f (xo) is an
(f, G)-admissible point, and µ(yo, f, G) = j (x, f ). Let (x,), v = 1, 2, ... , be an arbitrary sequence of points in Bk convergent to x0, and let y = Then yL - yo as v - oo. Hence, there is a vo such that if v > vo, f, G) = y(yo, f G). We have then y,, is also (f, G)-admissible and that µ(y,,, f, G) > j(xv, f) > k, and hence j(xo, f) = µ(yo, f, G) > k, i.e., x0 E Bk, which is what was to be proved.
f
LEMMA 10.7. Suppose that U c R" is an open set, f : U -' R" is a mapping with bounded distortion, A c U is a compact set, and jo = maxXEA Ij(x, f )I. For X E U let T(x) be the distance to the element of f -' [f (x)] closest to x and different from x. Let B be the collection of all points x E A such that I j(x, f )I = jo. Then there is a number 6 > 0 such for al/xE B. that
PROOF. Assume, on the contrary, that the required 8 > 0 does not exist. Then there is a sequence of points (x,), v = 1, 2, ... , in B such that 0 as v - oo. Without loss of generality it can be assumed that v = 1, 2, ... , converges to some x0 E A as v - oo. the sequence Let k = jo - 1, and let Bk = {x E UI I j (x, f) I > k). By the theorem, Bk is closed relative to U, which implies that B = A n Bk is compact, and
11. MAPPINGS WITH BOUNDED DISTORTION
230
hence xo E B. There is an r > 0 such that G = B(xo, r) c U, the point yo = f (xo) is (f, G)-admissible, and p(Yo, f G) _ j(xo, f). Let yY = f (x.,)
Then y - yo as v
oo, and hence there is a v1 such that y is (f G)-
admissible for v > v1, and ac(y,,, f, G) = µ(yo, f, G). Let x,, be the closest
point to x such that f
f
and x,, 0 x,,. Then Ix,' - x I - 0 as
v -- oo in view of our assumption, and thus there exists a v2 > v, such that x' and x belong to B(xo, r) for v > v2. For v > v2 we have
lu(y,f,G)I >- I.i(x,,,f)I+I1(x,,f)I > o+ 1. However, this contradicts the fact that I.U(Y,,, f G)I = I µ(yo, f. G)I =1o
for v > v,. Accordingly, the assumption that the lemma is false leads to a contradiction, which is what was to be proved. We next use Theorems 7.2 and 7.3. The conditions of the latter theorem contain constants r, and ro whose choice depends on the given point.
Recall how r, and ro were chosen. Let f : U -+ R" be a mapping with bounded distortion, where U is an open set in R", and let a E U. Then r, is such that 0 < r, < p(a, 8U), and f (x) 96 f (a) for 0 < Ix - al < r1. As ro we can take any number r > 0 such that L f(a, r) = max I f (x) - f (a)I < l f(a, r1) Ix-al=r
min I f (x) - f (a)I.
Ix-al=r,
LEMMA 10.8. Let f : U
R" be a mapping with bounded distortion,
where U is an open set in R", let A c U be a compact set, and let S = dist(A, 8U) > 0. Assume that there exists a constant r, = r1 (A) > 0 such that f (x) is different from f (a) for every a E A and any x E U with 0 < Ix - al < r1. Then there is a number ro = ro(A) > 0 such that the estimates in Theorems 7.2 and 7.3 hold ifa E A, 0 < r < ro, and Ix -a l < r, x E U.
PROOF. Take an arbitrary r' < min{b, ro} and denote by p(x) the dis-
tance from f (x) to the image under f of the sphere S(x, r'), p(x) = If (x, r'). It is not hard to see that p is continuous on A, and p(x) > 0 for all x E A. Hence, there is a number po > 0 such that p(x) > po for all XEA. Let
L,(x) = Lf(x,t) = max If (y) -f(x)l, 1Y-xl=,
where t > 0. The function L, is continuous on A, and L,(x) > 0 for all x E A. Further, Lt(x) is a nondecreasing function of t for each fixed x. As
§ 10. THE SET OF BRANCH POINTS
231
t -. 0 we have that Lt (x) - 0 for all x E A. Since A is compact, L,(x) -+ 0 uniformly on A as t , 0, by the classical theorem of Dini. Hence, there is an ro > 0 such that La(x) < po for t < ro. Since po < p(x) = If (x, r'), ro is clearly the desired number.
THEOREM 10.4 [44]. Suppose that U is a domain in R", f : U -+ R" is a mapping with bounded distortion, and N > 2 is an integer. Let BN be the collection of all points x E U such that I j(x, f ) I > N. Then for every y > (Ko(f)/N)1/(n-1) the Hausdorff measure myIf(BN)] of the set f(BN) is zero.
PROOF. Denote by B),, the collection of all points x c- U such that I j (x, f )1 = N, and let BN be the collection of all points x E U such that I j (x, f) I > N+ 1. According to Lemma 10.6, BN is closed relative to U, and
thus U, = U\BN is an open set. Obviously, BN c U1. Take an arbitrary open set V strictly inside U1. For all x E UI we have that I j (x, f) I < N; hence BI, is the collection of all x E U, such that I j (x, f )I > N - 1. On the basis of Lemma 10.6, BI is closed relative to U. Let A = V n BN. The set A is compact, and I j(x, f )I = N for every x E A. By Lemmas 10.7 and 10.8, there exist numbers r1 > 0 and ro > 0
such that L f(a, ro) < If (a, r,) and f (x) 0 f (a) for 0 < Ix - al < r, if
aEA. By Theorem 7.3, for every a E A and for Ix - a 1 < ro
If(x) - f(a)I <- Lf (a, ro)MI (Ix - aI/ro)°2 where
a2 = and MI, is a constant. Since A is compact,
[N/KO(f)]11(1-1)
If(x) - f(a)I <- M(Ix - aI/ro)°2
(10.1)
for all a E A, where M < oo is a constant. According to the corollary to Theorem 10.1, BN is a set of measure zero. Let e > 0 be arbitrary. Then there is a sequence of balls B,,, v = 1, 2, ... , covering A such that the radius r of B does not exceed ro/2, and E r," < e. Moreover, it will be assumed that B contains points of A. We consider the set f (B,). Let a E B, n A. For all x E B we have that Ix - a 1 < 2r < ro, and hence, by (10.1),
If (x) - f (a)1:5 M(Ix - aI/ro)°2 <- (2°2M/ro)r,°=.
Let (2°2M/ro)r'2 = p,,. The set is thus contained in some ball B,, of radius pt,. This gives us a sequence of balls By which covers f (A). We
H. MAPPINGS WITH BOUNDED DISTORTION
232
have that 2"(M/ro)"1°2 E r," < Le, V
V
where L = 21(M/ro)"12 < 00. Since e > 0 is arbitrary, this implies that the
y-content of the set f (A) = f (BI n A) is zero, where y = n/a2. Consequently, m,, [f (BI n Y)] = 0. Covering U, by countably many balls lying strictly inside U1, we get from what has been proved that m, [f (B).,)] = 0 for y = nKo(f)/NI i("_,).
For M > N let BM be the collection of all points x E U such that I j (x, f )I = M. On the basis of what has been proved, my[f (BIM )] = 0
for y = yy = nKo(f)/MI/c"-I). From this, my[f(Bk)] = 0 also for y = yN. We have that f(BN) = UM=N f (BM). By the countably additivity of a Hausdorff measure, this implies that my[f (BN)] = 0 for any y > nKo(f)/NI/("-I). The theorem is proved. COROLLARY. Let f : U -' R" be a mapping with bounded distortion such that B f is nonempty. Then there exists at least one point x E B f at which
Ij(x,f)I <- [n/(n -
2)]"-I[Ko(f)]"-I.
(10.2)
Indeed, let N be the smallest integer such that nKo(f )/N II (n-1) < n - 2. Assume that I j(x, f )I > N for all x e B1. Then it follows from the theorem that m"_2[F(Bf)] = 0, which is impossible in view of Theorem 10.3.
THEOREM 10.5 [44]. There exists a Ko > I such that every mapping f : U - R" with bounded distortion K(f) < Ko is a local homeomorphism.
Assume, on the contrary, that no such constant Ko exists. Then for each integer v > 1 there is a mapping with bounded distortion fV : U -. R" such that K(f,,) < 1 + 1 /v and f is not a local homeomorphism; hence, the set of branch points of f, is nonempty for each v. By the corollary to Theorem 10.4, there exists a constant N < oo such that for each v there is an x E B with I j (x , f,) l < N. On the basis of Theorems 7.2 and 7.3, for each v there exists a number r > 0 such that if Ix - x I < r,,, then PROOF.
MIA
(Ix-xvI r,
)
< Ifv(x) - fv(xv)I
Ix - xvIJ < M2AV r
(10.3) 21'
,
§ 10. THE SET OF BRANCH POINTS
where
A = Lr.(x,,, r.,) =
at.v = K(fv)If(xv,
f')Il/(n-I)
If(x) -f(x.)I, <
Ij(xv,fv)11/(n-I)/K0(f.,)
a2.v =
From this,
max
233
2No/(n-I)
= al,
> 122 > 0.
MIA,Ix (1i' < If. (x) -fv(xv)I rv
II
<M2A, (Ix
)
a2
J
Let BI = B(0, 1) be the unit ball in Rn. For X E B let
F.,(X) = (l/A..)[f,,(XV+rEX) -f,,(x.,)]. Obviously, F is a mapping with bounded distortion. Further, 1 as v - oo, and the point 0 is a 1 + (1 Iv), so that branch point of F,,. By (10.3), MAXI°" < Fi(x) < M2IXI°2
for v = 1, 2,.... In particular, it follows from this that the sequence of mappings is bounded in L,(BI). Hence, by Theorem 9.1, this sequence is uniformly continuous on every interior ball B(0, t), where has a subsequence converging in Ln,10(BI) 0 < t < 1. Hence, to some mapping F0. In order not to complicate the notation, we assume that (F,), v = 1, 2, ... , is this subsequence. By Theorem 9.2,
K(Fo) S 1im.,,
1, and hence K(Fo) = 1. It clearly follows from (4.3) that MAXI-' < IFo(X)I < M2IXI°2, and thus F0 is an identically constant mapping. On the basis of Theorem 5.10, F0 is a Mobius mapping, 2. and hence a local homeomorphism. On the other hand, B(0, t)1 >- 2 for every ball B(0, t), 0 < t < 1, which From this, Iµ(0, oo, that Iµ(0, F0), B(0, t)I > 2 gives us, after passage to the limit as v
for all t E (0, 1), and hence I j (0, Fo) I > 2. This is a contradiction, and the theorem is proved. Theorems 10.4 and 10.5 were also obtained independently by Martio, Rickman, and Vaisala [96]. On the whole, their proof coincides with the proof of Gol'dshtein given here. The precise value of the constant Ko in Theorem 10.5 is unknown. In this connection we mention the following result of Poletskii [ 113]. Suppose, as in Theorem 10.4, that BIN denotes
the collection of all points x E B f such that I j(x, f) I = N. If n = 3, and the set BN is a rectifiable curve, then K(f) > N. The estimate is sharp; equality holds when f is a twist about an axis.
11. MAPPINGS WITH BOUNDED DISTORTION
234
§10.5. A theorem on the radius of injectivity. The purpose of this subsection is to prove Theorem 10.6, due to Martio, Rickman, and Vaisala [96]. The proof given here (written out at the author's request by V. M. Gol'dshtein) coincides on the whole with that in [96]. The only difference is that we do not use the concept of the modulus of a family of curves. Theorem 10.6 implies the well-known theorem of Zorich [182] (Corollary 2 to Theorem 10.6). The proof of Theorem 10.6 follows a scheme close to that used in [182). In particular, Lemmas 10.3 and 10.9, which play an important role in the proof of Theorem 10.6, are due to Zorich. We note that the theorem in [182] was formulated earlier by Lavrent'ev [80], but it was first proved only in [182].
LEMMA 10.9. Suppose that f : U - R" is a local homeomorphism of a
domain U in R", and A and B are subsets of U such that f is a homeomorphism on each of them. If A n B is nonempty and f (A) n f (B) is connected, then f is one-to-one on A u B. If each of the sets f (A)\ f (B) and f (B)\ f (A) is open relative to f (A) u f (B), then f is a homeomorphism on
AUB.
PROOF. The restrictions of f to A and B are denoted by f1 and f2, respectively. Let f (A)n f (B) = Q, and let C1 = fi ' (Q) and C2 = fz ' (Q). To prove that f is one-to-one on A U B it suffices to establish that C1 = C2. We first prove that C1 n C2 = A n B. It will follow from this, in particular, that C1 n C2 is nonempty. Since C1 c A and C2 c B, we have that C1 nC2 c An B. Take an arbitrary point x E An B. Then f (X) E Q. Since x E A, it follows that x E f1 ' (Q) = C1, and X E f2' (Q) = C2 because x E B, i.e., x E C1 nC2. This leads to the conclusion that A n B c C1 nC2, i.e., A n B = C1 nC2. We now prove that f (Cl n C2) is open relative to Q. Take an arbitrary point x E C1 n C2, and let y = f (x). Since f is a local homeomorphism, there exists a neighborhood Vo of x such that f maps Vo homeomorphically onto some neighborhood Wo of y. Since f is homeomorphic on each of the sets A and B, there exists a neighborhood W1 c Wo of y in R" such that f; '(WI n A) C Vo and f2'(WI n B) c Vo. Let V1 = f-I (WI) n Vo. Since f is a homeomorphism on V0, it follows that f(V1 nA) = W1 n f(A) and f (VI n B) = WI n f (B), and hence f (A n B n VI) = Q n W1. Since A n B = C1 c C2, we have that f (Cl n C2 n VI) = Q n W1. This proves that f (Cl n C2) is open relative to Q. We now prove that f (Cl nC2) is closed relative to Q. Suppose that (x,), v = 1, 2, ... , is a sequence of points in C1 n C2 such that the sequence (y = f (x.,)), v = 1, 2, ... , converges to some point yo E Q. Let I = f i ' (yo) E C1 and 2 = f2 1 (yo) E C2. Since f is a homeomorphism on
235
§ 10. THE SET OF BRANCH POINTS
each of C, and C2, the sequence (xv) must converge to each of the points
, and 2, i.e., , = 2 = o E C, n C2, and hence yo E f (C, n C2), which establishes that f (Cl n C2) is closed. Since Q is connected by assumption, what has been proved implies that f (Cl n C2) = Q, and thus Q, = Q2. It is proved that f is one-to-one on A U B. To establish that f is a homeomorphism on A U B it suffices to verify that the inverse f - I of the restriction of f to A u B is a homeomorphism on the set P = f (A)u f (B). If y E Q = f (A)n f (B), then the continuity of f - I at y follows from the fact that f is continuous on each of f (A) and f (B). Suppose that y ¢ Q. Then either y E f (A)\ f (B) or y E f (B)\ f (A). In both cases there is a neighborhood W of y such that W n If (A) u f (B)] coincides either with W n f (A) or with W n f (B). Since f - I is continuous on each of f (A) and f (B), this implies that the restriction of f -I to W n If (A) u f (B)] is continuous, and hence f is continuous on f (A) U f (B). The proof of the lemma is complete. LEMMA 10. 10. Suppose that f : U -- R" is a quasiconformal mapping, B(xo, r) c U, yo = f (xo), and Lo = L f(xo, r) = Ixmax r If (x) - Yol, to =If (xo, r) = IXm on If (x) - Yol
If B(yo, Lo) c f (U), then Lo/1o <
exp[(y"K(f))I/("-1)]
where y" > 0 is a constant. PROOF. Let
G = f -I [B(Yo, Lo)].
u(x) = ln(L/If(x) - f (a)I)
The function u is a generalized solution of the equation u'(x))("/2)-19 f(x)u'(x)]
div[(Of(x)u'(x),
(*) =0 in U. Further, u(x) = 0 on the boundary of G. The complement R"\G in R" is obviously an unbounded connected set whose boundary coincides with that of G. The boundary of G contains points of the sphere
S(xo, r). The flow S2o of the solution u(x) of (*) with respect to the capacitor (Ao, A,) with A0 = R"\Go and A I = {xo} is equal to n( On. By Theorem 7.1, u(x) <-
[K(f)]'I("-')
al)
for all x E G. Let x, be a point of S(xo, r) such that If (x) - f (xo)I = 1o. For this point u(x) = ln(L/1), and we get that ln(L/1) This proves the lemma.
[K(f)]'1("-1)ln(D"(1).
11. MAPPINGS WITH BOUNDED DISTORTION
236
THEOREM 10.6 [96). Let n > 3. Then there exists a function p, (K) on
the interval 1 < K < oo such that if f is a local homeomorphism with bounded distortion on the ball B(0,1) c R" with K(f) < K, then f is a homeomorphism on the ball B(0, pR(K)).
PROOF. Let f : B(0,1) - R" be a local homeomorphism with bounded
distortion. Take an arbitrary number r with 0 < r < 1. Then f is continuous in the closed ball W(0, T). We assume that f (0) = 0; this clearly involves no loss of generality.
For an arbitrary connected set A C R" such that 0 E A let V(A) be the connected component of f -I (A) containing 0. Let B, = B(0, r) and let G, = yi(B,). Since f is a local homeomorphism, it follows that G, c B(0, T) for sufficiently small r, and the restriction of f to G, is a proper mapping. By Lemma 10.4, the restriction off to G, is a homeomorphism for each such r. Let ro be the supremum of the values of r such that f is a homeomorphism on G,. By Lemma 10.4, f is a homeomorphism also on Gro.
We show that G,o contains points of S(0, r). Assume that this is not so. We prove, first, that f is one-to-one on G,,. Indeed, assume, for example, that there exist points x, , x2 E G,o such that x1 34 x2, while f (x,) = f (x2) = yo. The mapping f is a homeomorphism on G,,,, and hence at least one of x1 and x2 (say x1) must be a boundary point of G,o; consequently, yo E S(0, ro). Since f is open, x2 cannot be an interior point of G,o, for otherwise the point f (x2) = yo would be an interior point of B(0, ro). Let V1 and V2 be disjoint neighborhoods of x1 and x2 such
that f is a homeomorphism on each of V1 and V2. Let b > 0 be such that B(yo, 6) c f (VI) n f (V2). Denote by f the restriction of f to Vi,
and let U1 = f-1[B(yo,6)], i = 1,2. Let D = B(0,ro)nB(yo,6) and
f-I(D)=QiCUi. We prove that Qi = U; n G,o. Indeed, if x E U; n G,o, then f (x) E B(yo, 6), while f (x) E B(0, ro), and hence f (x) E D. This leads to the conclusion that x E Qi; therefore, Ui n Gro C Q. Assume that Qi contains points not in Ui n Gro. Since xi is a boundary point of G,o, Ui n Gro is nonempty. Hence, Q U G,0 is connected. Since f (Qi U G,o) c f (Qi) u f (G,o) = B(0, ro), this gives us that Qi u G,o c G,0, and, in particular, Q. c G,o. From this, Qi c G,o n Ui, i.e., Qi = G,o n Ui. The sets Q1 and Q2 do not have common elements, and their images coincide with D. We get a contradiction to the fact that f is one-to-one on G,o. By Lemma 10.2, this implies that f is a homeomorphism on some neighborhood H of Gro. Denote by fo the restriction off to H. Then for r = ro + S, where 8 > 0 is
§ 10. THE SET OF BRANCH POINTS
237
sufficiently small, fo-1 maps the ball B(0, ro + 6) homeomorphically onto a set lying strictly inside B(0, r), and we get a contradiction to the definition of ro.
Thus we have a set Gro c B(0, r) such that f is a homeomorphism on Gro. Further, Gro turns out to be sufficiently "long": it contains the point O and points of S(0, T). To prove the theorem it is necessary to see that Gro is also a sufficiently "thick" set.
Let h > 0 be the distance from 0 to the boundary of Gro. We construct the ball B(0, h) and let r, = mini, ,=h If (x)I. Next, maxlxl=h If (x)I is obviously equal to ro. By Theorem 7.2, ro/r, < It follows
from the definition of ro that there is a point yo E S(0, ro) which is the image of some point xo E Gro belonging to S(0, r). For simplicity of the notation it will be assumed that yo = The general case can obviously be reduced to this one. Denote by St, where ItI < r,/2, the sphere about yo passing through the point x, = tea. The symbol K,,,,, where Its < r, and t < u < 2ro - t, denotes the spherical segment consisting of all points x = (x1,... , in S, for which x < u. Let fo be the restriction off to G,o. Denote by M,.. the connected component of f -1(K,,u) containing fo 1(x, ). For t E (-r1, r, ) let u, be the supremum of the numbers u E (t, 2ro - t) such that f is a homeomorphism on M,,,,. The union of all segments K1,u such that
-r1 0 such that t lies in (-r, , r,) and K,.,, is contained in W' for It -toy < 8 and u < uo+S. By this, the mapping f is a homeomorphism on M,,u. This implies that the union of all the segments K,,,, with It-tol < a and u < uo + & is contained in V. The point is an interior point of this
union. Hence, it is an interior point of V, and so it is proved that V is open, because
E V was arbitrary.
Let K,,,, be the open spherical segment consisting of the points x = x < u. Let M,,,,, denote the union of the (x1,... , 1, onto K,,u,. The sets M1.,, such that t < u < u,. The function f maps mapping is a homeomorphism, by Lemma 10.4. does not lie strictly inside B(0, r) for any t. AsWe show that sume, on the contrary, that Mto,u,o lies strictly inside B(0, T) for some
238
U. MAPPINGS WITH BOUNDED DISTORTION
to E (-r1, r,), and let Mo be the closure of By the assumption, Mo is contained in B(0, T). The mapping f is one-to-one on Mo. Indeed, assume, on the contrary, that there are points XI, x2 E Mo such that x, x2 and f (x1) = f (x2) = yo. Let (x,,,) and (x2,,), v = 1, 2,..., be sequences of points in Mto,,,,, such that x1,, - x, and x2,, x2 as v - oo. Let y;,, = f (x,,,). Then y,,, - yo and y2v - yo as v oo. Let us join the points x,,, and y2,, by an arc y lying in K,0.,,,0 in such a way that this arc shrinks to the point yo as v oo. (We remark that here we are using the
fact that n > 3.) Let y = fo
where fo is the restriction of f to
The ends of the arc y converge to x1 and x2. The upper topological limit of the sequence of arcs y is a connected set whose points are all mapped into xo. We get a contradiction to the assumption that f is a local homeomorphism. This proves that f is one-to-one on Mo. By the compactness of Mo, f maps Mo homeomorphically onto M,,.y,0. On the basis of Lemma 10.3, this implies that f extends to a homeomorphism of some neighborhood V of Mo. Let W = f(V). If K,,.,,,0 does not coincide with S,,, then f is a homeomorphism on the set M,,,u,o +S for some S > 0, and we arrive at a contradiction to the definition of u,,. Therefore, K,0.,,,0 = S,,, and f maps Mo homeomorphically onto S,,. The bounded component P of R"\Mo is contained in B(0, T). Since Mo passes through interior points of G,,, P intersects G,,. The boundary of P is mapped homeomorphically onto the sphere Sy,. This implies that li(y, f, P) = I for every y E B(yo, to), and µ(y, f, P) = 0 for y V B(yo, to), which allows us to conclude that f maps P onto B(yo, to). The mapping f is a homeomorphism on each of the sets G,, and P, and the intersection f (G,,) n f (P) = B,, n B(yo, to) is connected. By Lemma 10.9, this implies that f is a homeomorphism on G,0 U P. In G,0 there is a sequence
v = 1, 2,..., converging to some point xo E S(x, r) and such that y,, =
f
yo as v - oo. For sufficiently large v the points y are in B(yo, to).
This implies that the points x belong to P for sufficiently large v, and limx is an interior point of P c B(0, T). We thereby get a contradiction, and hence the given case is also impossible. Thus, for every t E (-r1, r1) the set M,,,,, does not lie strictly inside B(0, r), and hence the closure of M,.,,, contains points of S(0, T). Let C, be some limit point of M,,,,, on S(0, r), and let z, = f (1,). The closure of the set formed by the points zt is denoted by F1. Finally, let Fo be the interval consisting of all the points x = te,, with 111 < r1. Let M be the union of all the sets M,.,,, with It I < r1. The mapping f is one-to-one from M onto V. The inverse mapping f -I is continuous on V. This implies that M is open and f is a homeomorphism on M. The theorem will be proved if we establish that
§ 10. THE SET OF BRANCH POINTS
239
M is also a sufficiently "thick" set. The required result will be obtained by certain estimates for the capacity. Since M,,,,, has limit points on S(0, T), the spherical segment k,,,,, cannot
lie strictly inside Bro. This implies that for every t E (-r, /2, r, /2) the vectors y-yo, where y E K,,,,,, form an angle 0(1) at least 00 = arc cos(3/4) with the vector -en. On the basis of Lemma 3.1, this allows us to deduce that
C"(F0,F1,V) > C In r0+r,/2 r0-r,/2 >where C" > 0 is a constant. Let A0 = f y' (F0) and A, = fM' (F1), where fm denotes the restriction
off to M. Then, by Theorem 5.11, Cn(F0,F1, V) < K(f)C"(Ao.A,,M) < K(f)CC(Ao, A,, B(0, T)).
The set A0 is contained in S(0, T), and A, lies in B(0, 8) c Gro. From this, C"(A0, A 1 , B(0,
r)) < G = C"(R"\B(0, T), B(0, 6)) = &), In 1-n (T/3);
hence wnK(f)In1-n(r/8)
> C"r1/ro > A(n,K),
and passage to the limit as r - 0 gives us that 8 > Sp(K, n) > 0, which is what was to be proved. COROLLARY 1. Let f : U -' R" be a mapping with bounded distortion. If f is a local homeomorphism and B(x0, r) c U, then f is a homeomorphism in the ball B[xo, rp"(K)], where p"(K) is the constant in the theorem, and
K = K(f ).
This is obtained by applying the theorem to the mapping f (x) = f (x0 + rX ), which is defined in B(0, 1) and is a homeomorphism on B(0, p" (K)) by the theorem. Hence, f is a homeomorphism inB[xo,rpn(K)], which is what was to be proved. COROLLARY 2 [182). Let f : R" R" be a local homeomorphism with bounded distortion. Then f is a homeomorphism.
Indeed, by Corollary 1, f is a homeomorphism in the ball B(0, p"(K)r) for any r > 0. Therefore, f is a homeomorphism of the whole space R", which is what was required.
11. MAPPINGS WITH BOUNDED DISTORTION
240
§11. Extremal properties of mappings with bounded distortion § 11.1. The homomorphism generated on the algebra of exterior forms by a mapping with bounded distortion. Let U and G be open sets in R", and f : U G a continuous mapping of class W,'1a(U). Then, as shown in §4, a certain exterior form f * co can be defined for every exterior C' -form w
of degree k (0 < k < r) on G. The requirements imposed on the exterior forms in this result can be weakened in the case when f is a mapping with bounded distortion. It is the purpose of this subsection to prove the last fact.
LEMMA 11.1. Let U be an open set in R", f : U -+ R" a mapping with bounded distortion that is not identically constant in U, and u(y) an integrable function on R. Then the function u[f (x)]f(x, f) = v(x) is locally integrable in U, and for every compact domain G contained in U with boundary a set of measure zero
jv(x)f(xf)dx = JR
u(y)µ(y, f G) dy.
PROOF. By Theorem 2.2 of this chapter, v is locally integrable in U if u is such that for every chapter domain G c U the function u(.)N(., f, G) is integrable. What was proved in §6 gives us that N(y, f, G) <_ 10(y, f G)I f, G) is bounded and has for any compact domain G. The function compact support in R", by what was proved in §6. From this, the function y - u(y)N(y, f, G) is integrable for every function u E L1,10c(R"), and the lemma is proved. COROLLARY. Let f : U - R" be a mapping with bounded distortion, and
E an arbitrary set of measure zero in R. If f is not identically zero, then f -' (E) is a set of measure zero. This proposition can be proved by applying the lemma to the indicator
function of E and using the fact that if f is a mapping with bounded distortion, then f (x, f) # 0 for almost all x E U. LEMMA 11.2. Suppose that U and G are open sets in R", and f : U
G
is a mapping with bounded distortion. Let w(y) be an exterior form of degree k > 1 defined on G and belonging to LP,10C(G), where p = n/k. Then f'w is a form of class LP,10C (A), and for every compact set A c U there is a constant L(A) < oo such that IIf'wIIp.A <_ L(A)II wII p.f(A) If ? is a form of degree k + 1 in the class LP, ,1oc(G), where pi = n/(k + 1), such that rp = dw, then also f' co = d (f * w).
§ 1 1. EXTREMAL PROPERTIES
241
PROOF. Suppose that w E Lp,10C(G) and co(y) = E, al(y) dyj. Then, by definition,
(f *(0)(x) _ >a,[f(x)]df,(x), where d f , (x) = d l , (x) A
A d f k (x) f o r I = (i 1, ... , 1k). Using inequality
(4.2) in §4, we get that
Idf,(x)I <- Cidf;,(x)IIdf2(x)I...Idfk(x)I
< CI F(x, f )Ikl" for all x, where C is a constant depending only on k and n. From this, If *w(x)I <- CI wI [f (x)]If(x, f
)Ikj".
Let B0 = B(xo, r) be an arbitrary closed ball in U. Then
J If 'w(x)I p dx < C; o
< C;
f
f [I wI(f (x))]PV (x. f )I dx Bo
[I wI (y)]°µ(y. f BO) dy _< [L(B0)]PII wII p.f(B0)
f(Bo)
and the first assertion of the lemma is clearly proved. We prove the second assertion. Let ? E Lp,,1a(U). Replacing w by 9 in the preceding arguments, we get that f* (P E Lp, .1a(U). It remains to show that f'rp = d (f'w). This follows from Lemma 4.7 in the case when
9 and co are forms of class C. We consider the general case. Take an arbitrary compact set A C U, and let B = f (A). Let wh and 9'h be Sobolev averagings of the forms co and gyp. Then Sph = dcoh for each h. As h - 0 we 0. By inequality (* ), this have that Il wh col l p.B --' 0 and 119h - 9II p, .B
-
gives us that f'9'h f'co in Lp,,10C(U), and f*wh - f'w in Lp,1oc(U) as h - 0. By Lemma 4.7 in this chapter, f*wh = d(f'wh) for each h. This implies that f'9 = d(f'w) in view of Lemma 4.2, and the lemma is proved.
LEMMA 11.3. Suppose that U and G are open sets in Rn, f : U - G is a mapping with bounded distortion, and u: G - R is a function of class W11,10C(G). Then the function v(x) = u[f (x)] is a function of class W(U). Further,
v'(x) = [f'(x)]'u'[f(x)]. PROOF. We construct a sequence u,,, v = 1, 2, ... , of functions of du class C10(G) such that u - u almost everywhere in G and du,, everywhere. Each of in L,,,1a(U). Then (f'u)(x) the functions f'u is continuous. The forms r'du belong to L,,,1a(U),
II. MAPPINGS WITH BOUNDED DISTORTION
242
and they converge to f' du in L".1,,,(U) as v - oo. By Lemma 4.7,
f' du =
for each v. The coefficients of the forms f' du', are thus generalized derivatives of the functions f' u, and they converge in L" 1 (U) to the corresponding coefficients of the form f' du as v - oo. The functions f ' u converge almost everywhere to f' u. This implies that the coefficients of the form f' du are generalized derivatives of the function v = f' u. It follows immediately from the definition of f* u that v'(x) = [f'(x)]'u'[f (x)], and the lemma is proved. THEOREM 11.1. Let U and V be open sets in R", and let f : U - V and g: V -- R" be mappings with bounded distortion. Then the composition h = g o f : U - R^ is also a mapping with bounded distortion. Further,
K(g of) < K(g)K(f ),
Ko(g of) S Ko(g)Ko(f )
PROOF. On the basis of Lemma 11.3 each of the real functions gi[f (x)] belongs to the class W,' 1OC(U). Let E c V be a set of measure zero such
that g is differentiable at each point y E, and let A = f - I (E). By the corollary to Lemma 11.1, A is a set of measure zero. Let B be the set of points where f is not differentiable, and let H = A u B. Then H is a set of measure zero, g o f is differentiable at each point x E U\H, and (g o f)'(x) = g'(y)f'(x), where y = f (x). From this it follows that f (x, g o f) has constant sign in U, and II(g o f)'(x)II" < Ig'(Y)I1"11f'(x)11"
< K(g)K(f)If(Y. g)11f(x, f)1
= K(g)K(f)If(x. go f )l at each point x E U\H. This proves that g o f is a mapping with bounded distortion, and K(g o f) < K(g)K(f ). The estimate for Ko(g o f) can be established similarly by considering the linear mappings (go f)'(x), g'(y), and f'(x). The theorem is proved. §11.2. Main theorem. Suppose that G is an open set in R. Assume that for almost all y E G a symmetric matrix a(y) is defined with elements which are Borel-measurable functions of y, and that there exist constants
a and fl, 0 < a < f < oo, such that for every
E R"
(11.1) a112 < (a(Yg. ) <_ fR12. Let u be a function of class W,, I,,c(G). We introduce the exterior form "
w(y, a, u) = (a(Y)u'(Y),
u'(Y))n12-I E(_ I) i-1
i=I
x (a(Y)u'(Y))i dY1 ... dYi-I dYi+1 ... dY".
11.2)
§ 1 I . EXTREMAL PROPERTIES
243
For every function I E C°° (D) u'(Y))"12-I
dIl n w(Y, a, u) = (a(Y)u'(Y).
(11.3)
x (a(Y)u'(Y),17'(Y)) dYI dY2...On-
Let us consider the differential equation
div{(a(y)u'(y), u'(y)){"/2)-1 a(y)u'(y)} = 0.
(11.4)
Comparing Lemma 4.8 and the definition of a generalized solution of (11.3), we get that a function u(y) is a generalized solution of (11.4) if and only if the generalized differential of the form co(y, u, a) is equal to zero.
Let f : U -+ D be an arbitrary mapping with bounded distortion, where U is an open set in R". For almost all x E U the linear mapping f'(x) is defined and is nonsingular. If f'(x) is defined and nonsingular at a point x E U, let z(x) be the matrix defined by
r(x) = [f'(x)]-I
a[f(x)J([f'(x)J*)-11f (x. f )12/".
(11.5)
If f'(x) = 0, let r(x) = I. By what was proved in §5.2 of this chapter, the matrix t(x) is symmetric and positive-definite, and its eigenvalues lie in the interval [a[K(f )]-2In , f [K0(f )]2/"J
THEOREM 11.2. Let U and G be open sets in R", and f : U -+ G a mapping with bounded distortion. If u(y) is a solution on G of equation (11.4), then w(x) = u[f (x)] is a solution of the equation div{(r(x)w'(x), w'(x))("I2)-1 r(x)w'(x)} = 0,
(11.6)
where the matrix-valued function T(x) is determined from a(y) according to (11.5).
PROOF. The function u belongs to W 10C(G). By Lemma 11.3, this (x)] implies that w belongs to W110C(U). Further, w'(x) = for all x E U. We introduce the exterior form "
C(x) =
(r(x)w'(x),w'(x))("")-I E(_1)'-I(r(x)w'(x)); i=1
x dx1... dxr- 1 dxi+1 ... dx".
The condition that w be a solution of (11.6) is equivalent to the conditions that dC(x) = 0. Since dw = 0, where w is the form in (11.2), to prove that dC(x) = 0 it suffices by Lemma 11.2 to establish that C(x) = f'w(x).
II. MAPPINGS WITH BOUNDED DISTORTION
244
For almost all x E U, ,(x, f) # 0. We have n
E(-1)i-Iwi(y)dy1 ...dyi-I dyi+I ...dyn.
w(y) = i=1
Let n
qp(x) = (f *w)(x) = E(-1)'- I pi(x) dx1 ... dxi-I dxi+I ... dxn. i=1
We consider the vectors (*co)(y) _ (cv1(y),...,con(y)) and (*c9)(x) _ (r9I (x),.... (p, (x)). Obviously,
(*w)(y) = (a(y)v'(y),
v'(y))(n/2)-Ia(y)v'(y)
By (5.13) in §5,
*V(x) = I/(x,f)I2In[f'(x)]-I
(*(O)[f(x)](f'(x)*)-I.
After obvious transformations, we get from this that *r(x) = *9(x), and hence C = ip, which is what was to be proved. COROLLARY [ 1331. Let f : R" - R" be a mapping with bounded distortion of R" into itself that is not identically constant, and let A = Rn\f (R"). Then A is a set of zero n-capacity.
PROOF. The set f (R") is open, and hence A is closed. Assume that the n-capacity of A is positive. A point xo E A is said to
be essential if the n-capacity of A n B(xo, r) is positive for every r > 0. The collection of all nonessential points of A is denoted by E. If xo E E, then there is an r > 0 such that the n-capacity of A n B(xo, r) is equal to zero. For every point x e A n B(xo, r) the n-capacity of A n B(x. 6) is equal to zero in this case, where a = r - Ix - xol > 0, and hence all the points x E AnB(xo, r) are nonessential. We thus get that AnB(xo, r) c E. This proves that E is open relative to A. Let x0 E E and r > 0 be such that the n-capacity of A n B(xo, r) is equal to zero. Then there exist a point y with rational coordinates and a rational number p > 0 such that x0 E B(y, p) c B(xo, r). Obviously, B(y, p) n A is a set of zero n-capacity. Since xo E E is taken arbitrarily and the set of balls B(y, p) of the indicated
type is countable, it follows that E is the union of a countable collection of sets of zero n-capacity, and hence has zero n-capacity. The set B = A\E is closed, and its n-capacity is nonzero. Let a and b be any two boundary points of B, a 0 b. Since E clearly does not have interior points, a and b are boundary points also for A. Let p be an arbitrary number such that 0 < 2p < I b - al, and let BI = B n B(a, p) and
§11. EXTREMAL PROPERTIES
245
B2 = B n $(b, p). Obviously, BI and B2 are sets of positive n-capacity. Let W (BI, B2) be the set of all functions in W (R") equal to zero on BI and equal to I on B2. Since BI and B2 have nonzero capacity, the capacity of the capacitor (BI, B2) is nonzero, and hence there is a function u(x) E W (BI, B2) minimizing the functional fR, l u'(x)l" dx. The function u(x) is a solution of the equation div(Iu'(x)jn-2u'(x)) = 0
on R"\(BI U B2), hence also on f (R"). The function u(x) is not constant in f(R"). Indeed, since a is a boundary point of A, any neighborhood of a contains points of f (R"), and hence, by continuity, u(x) takes values arbitrarily close to 0 on f (R"). Considering the behavior of u(x) near b, we get in exactly the same way that u(x) takes values arbitrarily close to 1 on f (R"). It follows that u(x) is not constant on f (R"). Obviously, 0 < u(x) < 1 for all x. By Theorem 11.2, v (x) = u[f(x)] is a solution on R" of the equation
div[(O(x)v'(x),v'(x))n12-IO(x)v'(x)] = 0.
(11.6)
For all x E R" we have that 0 < v (x) < 1, and hence it is identically constant in view of the Liouville theorem for generalized solutions of (11.6) (Theorem 5.7 in this chapter). However, this contradicts the fact that v(x) is not constant on R" by construction. Thus, the assumption that the n-capacity of A is nonzero leads to a contradiction, and the corollary is proved. This result is given here as an example of the application of Theorem 11.2. Picard's theorem for complex functions of a single variable is well known. If f (z) is an analytic function of a single variable defined on the whole complex plane C, then C\ f (C) is finite and consists of at most two elements. It is natural to ask whether there is an analogue of Picard's
theorem for mappings with bounded distortion on R" for an arbitrary n > 2. There is indeed such an analogue, as established by Rickman [ 147]. Namely, he showed that for every n > 2 there exists an integer q = q(n, K) such that for every mapping f : R" -' R" with bounded distortion that is not identically constant the set R"\f (Rn) is finite, and the number of elements in it does not exceed q. The proof of Rickman's theorem requires the use of a very refined technique which is an extension of the Nevanlinna theory of the distribution of values of entire functions to the case of mappings with bounded distortion. Certain estimates of solutions of equations of elliptic type are also used. In particular, Theorem 11.2 is used in [147].
246
11. MAPPINGS WITH BOUNDED DISTORTION
§12. Some further results
Below we give a survey of some investigations in the theory of n-space mappings with bounded distortion, and on closely related questions not reflected in the main text. This survey of materials is mainly determined by the personal interests of the author and does not pretend to be complete. There are many excellent publications on the theory of quasiconformal mappings. Here we should mention in the first place the remarkable investigations of Ahlfors, Gehring, and my Finnish colleagues. Space does not permit a discussion of all these publications.
§12.1. Classes of domains in R. We describe some classes of domains that arise naturally in the study of mappings with bounded distortion. These classes can be useful in other questions when the requirement of a smooth boundary is too restrictive and at the same time it is known that an arbitrary domain in R" can fail to have the needed properties. 1. Domains of John type [60]. Let r and R be numbers such that
0 < r < R < oo, and let U be an open domain in R. We say that U belongs to the class f (r, R) if there is a point a E U such that any point x E U can be joined to a in U by a rectifiable curve x(s), 0< s< 1 (the parameter s is the arclength), for which x(0) = x, x(1) = a, the length 1 of the curve does not exceed R, and for all s E [0,1] p[x(s), 8U] > rs/1.
(12.1)
The point a in this definition is called the marked point of U. We say that U is a domain of class F, or, in other words, a domain of John type, if U E F(r, R) for some r and R. It follows from the definition that if U E F(r, R), then U is contained in the ball B(a, R). Further, setting s =1 in (12.1), we get that p(a, 8U) > r, and hence B(a, r) is contained in U. We set r/1= y in (12.1) and let B(s) = B[x(s), sy] and r(x) = U B,.. By
(12.1), B(s) c U for each s, and thus r(x) c U. In the case when x(s), 0 < s < 1, is a segment the set r(x) is the ordinary circular cone consisting of all possible segments joining x to points of B(a, r). In the general case r(x) can be regarded as a distinctive kind of curvilinear cone with vertex
x. Intuitively, a domain U E f (r, R) can be characterized as a domain such that any point x in it can be reached by the vertex of a cone made from some elastic material by bending it in such a way that it does not go out of U. Domains of the class F were first used in the theory of space mappings with bounded distortion in the author's paper [ 140] (if one disregards the
§ 12. SOME FURTHER RESULTS
247
work of John himself in which he studied a question that also can be related to the theory of mappings with bounded distortion.) Domains that are star-like with respect to a ball (see §2.2 in Chapter I) are a special case of the domains of class 7. Every bounded domain whose boundary is a smooth manifold of class C' belongs to f. Every domain that is a union of finitely many domains of class F belongs to F (see [48]). In particular, every domain belonging to the class S defined in §2.2 of Chapter I is a domain of John type. The Sobolev imbedding theorems presented in §2.2 of Chapter I are true also in the case when U is a domain of John type (see [48]). Besov introduced a certain class of domains, which he called domains satisfying the flexible horn condition, that is a generalization of the class
,F. In particular, domains with the flexible horn condition can be unbounded. Function classes generalizing the classes Wv are considered in [ 17] and [ 16], and Sobolev-type imbedding theorems are established for them. We present some results of a technical nature that concern domains of John type. First of all, we give two modifications of the original definition.
LEMMA 12.1 [168). An open domain U in R" is a domain of class F if and only if there exist numbers a and R and a point a E U such that 0 < a < 1, 0 < R < oo, for every x E U there is a rectifiable curve x(s), 0 < s <_ 1 (the parameter s is the arclength), with length l at most R such that x(O) = x, x(l) = a, and p[x(s) ''9u] > as
(12.2)
for all s E [0,1].
(There is also a proof of the lemma in [144].) The fact that a domain of class F satisfies the condition of the lemma is obvious. The hard part is to prove the converse assertion. (In principle, the factor multiplying s in (12.1) can be arbitrarily large; in (12.2) this factor is a certain fixed constant.)
Let 6 be a number such that 0 < 6 < 1. We say that U belongs to the class %(6) if it is bounded and there is a point a E U such that for every point x E U we can construct an x : [0, 1] - U with x(0) = x and x(l) = a for which Ix(t2) - x(ti)I < 3P[x(t2),OU] for any t I and t2 with 0 < t 1 < t2 < 1. We say that U E WB if U E AB (6) for some 6 E (0, 1].
II. MAPPINGS WITH BOUNDED DISTORTION
248
LEMMA 12.2 [93]. Every domain U of class F(r, R) belongs to the class WB(r/R). Conversely, if U E FB(8) forsome8 E (0, 1], then U E f (1/jp2, cl) depends on n and 6; here 1= diam U and Sp > 1.
We note one more useful property of domains of John type. Let U be a bounded domain in R. Then we say that U is a domain of type I., where
0
r
r=1- ln(pInn+ a)'
0
in polar coordinates. It is easy to verify that the integral (12.3) is equal to 0o for the domain U = V for any y E (0, 1]. The product V x BRn-2(0, 2) is an example of a domain not having the property I. in the case n > 2. The following statement was proved by Trotsenko [168].
LEMMA 12.3 ([144], [168]). If U is a domain of class ,(r, R) in R", then U has the property I., where y = y1(r/R)", 0 < y1 < 1, and y1 depends only on n. The estimate
< C R k R"_y fu [p(x, 8U)]' - ( r is valid, where C and k depend only on n, k > 0, and 0 < C < oo. dx
The proof of Lemma 12.3 is based on an estimate of the number of cubes in a Whitney decomposition of the domain that have distance to the boundary of U lying between h and 2h, where h > 0. As shown in [168], the condition "U is a domain of John type" in the statement of the lemma
can be replaced by a weaker condition. Namely, let 0 < a < 1. We say that a domain U in R" has the property Aa if there is a a > 0 such that for any x E 8U and any t E (0, a] there is a point y E U such that Ix - y1 < t and p(y, 8U) > at. If a domain U in R" is bounded and has the property A,,, then U has the property I,. for some y E (0, 1).
2. Uniform domains. A domain U in R" is called a domain of class U(a, fl), where 0 < a < ft < oo, if for any two points x1, x2 E U there exists a domain G c U belonging to the class f (aIx1 - x2j, fl(x1 - x21) such that x1, x2 E G. A domain U is said to be uniform if U is a domain of class U(a, f) for some a and P. The class of uniform domains was
§12. SOME FURTHER RESULTS
249
introduced in [93]. The interest in and significance of this class for the theory of quasiconformal mappings has to do with the following result of Martio and Sarvas [93]. THEOREM 12.2. Let f : R" - R" be a quasiconformal mapping, and U a domain of class U(a, f). Then f (U) is a domain of class U(e2,1 /e), where 8 E (0, 1) depends only on n, K(f ), a, and ft. In particular, it follows from Theorem 12.2 that the image of a ball with respect to a quasiconformal mapping of R" onto itself is a homogeneous domain.
Let n = 2. A domain U c R" is called a quasidisk if there exists a quasiconformal mapping f : R2 -+ R2 such that U = f (B(0, 1)). A planar curve r is called quasicircle if it is the boundary of some quasidisk. Ahlfors obtained a geometric characterization of a quasicircle in 1963 [2]. Let G be a domain in R" whose boundary is a simple closed curve L, and let x and y be two arbitrary points of L. These points divide L into two arcs. Denote by d (x, y) the smallest of the diameters of these arcs. One says that G satisfies the arc condition if there exists a constant c such that 1 < c < oo and d (x, y) < c l x - y J for any x, y E L.
THEOREM 12.3 [2]. Let G be a domain in R. Then G is a quasidisk if and only if its boundary is a simple closed curve and satisfies the arc condition.
It follows from Theorems 12.2 and 12.3 that if a planar domain G satisfies the arc condition (G is bounded by a simple closed curve), then G is a homogeneous domain. The converse is also true: if a planar domain G bounded by a simple closed curve is homogeneous in the sense of the definition given here, then it satisfies the arc condition, as shown in [93). We mention further a certain characterization obtained in [92] for homogeneous domains.
Let x, and x2 be two different points in R = R" U {oo}, and let r be a continuum joining them; that is, a continuum containing x, and x2. Assume that S E (0,1 ]. Denote by v, (r, j) the set of all points y E in for which
(x,y,x1,x2) = Ix - YI Ixl -x21
Ix-x1I.IY-x2I
for some x E r. Similarly, let V2(r, S) be the set of all points y E R with (x, y, X2, x,) < d
for some x E r. Let v (r, a) = v1(r, 6) n v2(r, 6).
H. MAPPINGS WITH BOUNDED DISTORTION
250
To understand the intuitive meaning of V, (T, S) and V2(T, S) we con-
sider the case when r is the segment with endpoints x, and x2. Let us transfer the point x, to 0o by inversion with respect to the sphere So about x2 passing through x1. The segment r then does into a ray A emanating from x1. The cross ratio is invariant with respect to MSbius transformations. We have that
(x,y.x,,o0) = Ix-yl/Ix-x,I, and the condition (x, y, x1, oo) < S means that Iy - xI < Six - x, 1; that is, that y belongs to B(x, Six - x, 1). The image of v, (T, j) is the union of these balls and is a right circular cone whose generators form the angle Sp = arc sin 6 with its axis A. This cone is the intersection of the half-spaces
with boundary planes passing through x, and having outward normals at an angle it/2 + rP to A. Inversion with respect to So carries each such halfspace into a ball whose boundary passes through x, and x2 and whose radius to x, forms the angle 7r/2 + q with the vector x2 - x1. In this case the set vi (T, S) is a "spindle", the intersection of all such balls. In this case it is clear that V2(T, S) = V, (T, j). In the case of an arbitrary continuum r the set V(T, j) is a distinctive kind "curvilinear" spindle joining x, and X2.
Following Martio [92], we say that a domain U c R" has the property U(S) (0 < S < 1), and write U E U(S), if for any two points X1, x2 E U there exists a continuum T joining x, and x2 such that V(T, 6) c U. LEMMA 12.4 [92]. Let U be a domain of class U(a, fi) in R n. Then there exists a S E (0, 1] depending only on a, fl, and n such that U E U(S). Conversely, if U E U(S) for some S E (0, 11, then U E U(a, fi), where a and fi depend on S and n.
The concept of a quasidisk turns out to be useful in many questions of analysis not directly connected with the theory of quasiconformal mappings. In this connection we refer the reader to the monograph [40], where a number of such results are given. In particular, the concept of a homogeneous domain arises naturally in connection with the problem of extension of the classes WW(U), that is, with the problem of conditions under which there exists for every u E WP (U) a function 8u E Wp(R") such that
(6u)(x) = u(x) for all x E U. Further, it is required that the operator u " 8u be a continuous mapping from WL(U) into Wp(R"). Many papers have been devoted to the problem of extending the classes Wo. The reader can find more complete information about this in [48]. (In this connection see also [45], [47], [64], [179), and [ 180].)
§12. SOME FURTHER RESULTS
251
In the case n = 2 the concept of a uniform domain enables us to give an exhaustive characterization of domains that are images of a ball un-
der a quasiconformal mapping of R". For n > 3 this is not the case. Tukia' showed that a cylinder constructed on any locally nonrectifiable unbounded quasicircle (that is, a curve that is the image of a line under a quasiconformal mapping of R2) separates the space into two homogeneous domains homeomorphic to a ball that cannot be transformed into a ball by a quasiconformal mapping of in. Another example (due
to Trotsenko): Let U, and U2 be the domains in in, n > 3, given by U, = {(x1,...,xn)Ixn < Ixl1} and U2 = {(x1,...,x")Ixn > Ixll}. The common boundary of U, and U2 is the surface F in R" defined by the equation x" = ix, I. Assume also that 00 E F. It is easy to verify the U, is a homogeneous domain, while U2 is not (because the angle at the point 0 is zero). This gives us that there does not exist a quasiconformal mapping f : R" _ R" such that U, is the image of some ball under this mapping. Indeed, if there were such a mapping, then by replacing f by a composition with a Mobius mapping we would get that U2 is a quasiconformal image of a ball, hence a homogeneous domain in R", according to Theorem 12.2. §12.2. Stability in the Liouville theorem on conformal mappings of a space and related questions. 1. According to Liouville's theorem, proved in §5.9, every mapping f
with bounded distortion not identically constant and such that K(f) = 1 is a Mobius mapping. In [81 ] Lavrent'ev posed the question of whether a mapping with K(f) close to 1 is close to being a Mobius mapping. He also obtained the first results in this direction (see below for a history of the question). The study of mappings with distortion coefficient K(f) close to I led to proofs that such mappings have many properties not holding in the general case. Some results of this kind will be presented below. 2. First of all note that as K(f) 1 the differentiability properties of f improve in the sense that the power p to which the derivatives of f are locally integrable increases without bound. Namely, the following statement is valid. THEOREM 12.4. There exist constants go > 0 and C < oo such that every
mapping f : U -- R" with bounded distortion and with K(f) < 1 + Jo belongs to the class Wp 10C(U) for any
p:5 C/(K(f) - 1).
(12.4)
' Pekka Tukia, A quasiconformal group not isomorphic to a Mobius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149-160.
II. MAPPINGS WITH BOUNDED DISTORTION
252
For K > 1 denote by pn(K) the supremum of the numbers p such that every mapping f : U -+ Rn with K(f) < K belongs to Wp 1 (U).
Let f : U -' R" be the nonlinear homothety f (x) = IxI°-Ix for all x r= Rn, where 0 < a < 1. As shown above, K(f) = 1 /a and f E Wp [a(R")
for every p < n/(1 - a) = nK(f)/(K(f) - 1). This gives an upper bound for pn (K):
pn(K) < nK/(K - 1). In particular, it follows from (12.4) and (12.5) that
O(1/(K - 1))
(12.5)
(12.6)
asK - 1. The estimate in Theorem 12.1 holds only under the assumption that K(f) is less than some value Ko = 1 + 60. For K(f) > 1 + 60 the theorem does not give any information about the value of pn(K). The fact that p(K) > n was established earlier for quasiconformal mappings by Gehring [38], and for mappings with bounded distortion by Meyers and Elcrat [98].
The results in [38] and [98] do not imply that pn(K) - oo as K - 1. Comparing Theorem 12.4 and the result in [98], we get that there exists
a function pn(K), 1 < K < oo, such that pn(K) > n for all K, pn(K) = O(1/(K - 1)) as K -' 1, and every mapping f with bounded distortion in Rn belongs to the class W. 1,10C for any p < pn[K(f )].
For the case n = 2 Boyarski in [19] and [20] established that p(K) > 2. The estimate p(K) = 0(11(K - 1)) as K -+ 1 was obtained by Lehto for
the case n = 2 [84]. The fact that pn(K) - oo as K - 1 was proved also by Iwaniec [58]. The estimate pn(K) = O(ln(1/(K - 1))) as K --+ 1 was established for pn(K) in [58].
The question of the exact value of pn(K) remains open. There is a conjecture that pn (K) = nK/(K-1), so that the worst mapping with respect to its differentiability properties is precisely the nonlinear homothety x
xlxla-1, a = 1/K. 3. The next statement is the main result in the problem of stability in the Liouville theorem on conformal mappings. THEOREM 12.5. Let U be a domain of class r(r, R) in R. where n > 3. Then for every p > n there exists a number b0 = a(p), depending only on n and p, such that the following assertions hold. If f : U -, Rn is a mapping with bounded distortion for which K(f) < Jo(r/R)2 + 1, then there exists a Mobius mapping rp such that 3
IV-'[f(x)]-xl <- C172 (K(f)-1)
(12.7)
§ 12. SOME FURTHER RESULTS
253
for all x E U, and if g(x) = (p-1 [f (x)], then
r 1u
Rnt2p+k
Ig'(x) - IIp dx < C2 r2p+k [K(f) - llp.
(12.8)
where I denotes the identity matrix, the numbers C1 and C2 depend only on p and n, and k is the constant in Lemma 12.3.
The theorem shows that if K(f) is sufficiently close to 1, then there exists a Mobius mapping q close to f in the sense that the mapping g(x) = ip-1 If (x)] is almost the identity. The estimate (12.8) shows that the derivatives of g(x) are close to the corresponding derivatives of the identity mapping in the sense of the La-metric. THEOREM 12.6 [169]. Let U E U(a, fl) be a domain in Rn with oo E U. Then there exist constants co > 0 and c < oo, depending only on a, P, and
n, such that any mapping f : U - jr with bounded distortion carrying o0 into 0o has the following property. If K f = 1 + e < 1 + co. then for any ball B(xo, r) there is a similarity T such that
ITf(x)-xl
In Theorems 12.5 and 12.6 the closeness of a mapping to a Mobius mapping is measured by the extent to which the composition q) o f, c a Mobius mapping, differs from the identity mapping. It would be more natural to consider the difference f (x) - Sp(x), where qr is an arbitrary Mobius mapping. Dairbekov [28] established the following statement. THEOREM 12.7. Let U c R be a domain of class U(a, fl). Then there exist numbers eo > 0 and C < oo depending only on n, a, and 8 such that for any quasiconformal mapping f : U -+ Rn with K(f) < 1 +eo there exists a Mobius mapping rp such that
If(x) - 9(x)1 <- C- r(f(U)l(K(f) - 1), where
.
[ f (U)] denotes the diameter of the domain
0[f (U)] =
sup xi EU.x2EU
If (X2) -f(xl)I
4. We say a few words about the history of the question. Lavrent'ev [811
established the stability of Mobius mappings in the class Cl-*. Namely,
II. MAPPINGS WITH BOUNDED DISTORTION
254
he proved that if f : B(0, 1) -+ R° is a quasiconformal mapping of B(0,1) into itself such that I f'(x1) - f'(x2)I S HI x1 - x21
for any x1, x2 E B(0, 1), where 0 < H < oo and 0 < a < 1 are constants, then for every ball B(0, r) with 0 < r < 1 there exists a Mobius mapping c such that If (x) - c'(x)1 <- u[K(f) -1, r, H, a] for all x E B(0, r), where µ(e, r, H, a) depends only on e, r, H, and a, and It was shown by the author in [ 121 ] and [ 120] that there exists a function
µ(e, r) of the variables e > 0 and r E (0, 1) such that µ(e, r) -' 0 for each r, and for every quasiconformal mapping f : B(0,1) -' B(0,1) there exists a Mobius mapping q such that for all x E B(0, r) If (x) - so(x)1 < 4u(K(f) -1,r).
(12.9)
The proof of this in [ 121 ] was based on the following considerations. We first define a certain measure of the deviation of a mapping f from a Mobius mapping. For an arbitrary Mobius mapping tp let
a(f,rp,r)= sup If(x)-rp(x)I, and 6(f,r)=inf6(f,(p,r) q
xEB(0.r)
(the infimum is over the set of all Mobius mappings of B(0, 1)). The quantity W. r) measures the difference between the mapping f : B(0,1) B(0,1) and a Mobius mapping. Now let uo(e, r) =
sup
K(f)
a(f, r).
It is required to prove that po(e, r) - 0 as a -- 0. Obviously, 00(c, r) is a nondecreasing function of e; hence the limit lime_o po(e, r) ='i exists and is nonnegative. Assume that q > 0. Then there is a sequence (fm ), m = 1, 2, ... , of quasiconformal mappings of B(0, 1) into itself such that K(fm) -+ 1 as m - oo, and J(fm, r) > 1/2 for each m. Since the sequence of integrals
f
(0.1)
If,,(x)I"dx,
m= 1,2,...
is bounded, the sequence (fm) is uniformly equicontinuous on each ball
B(0, r) with 0 < r < 1. In view of this it can be assumed that (fm) converges to some mapping fo: B(0, 1) -+ B(0, 1), uniformly on every ball B(0, r) (this can obviously always be achieved by passing to a subsequence). With regard to the limit mapping fo it is established below that
§ 12. SOME FURTHER RESULTS
255
fo is either a Mobius mapping or identically constant. In the case when fo, r) - 0 as fo is a Mobius mapping we get that, on the one hand, m - oo. On the other hand, a(fm, fo, r) > 8(fm, r) >
>0 2 for each m, and we thus get a contradiction. If fo is identically constant, then for each m there is a Mobius mapping rpm : B(0, 1) -. B(0, 1) such that I Spm(x) - fo(x)I < m for all x E B(0, 1). Obviously,
6(fm,com,r):6(fm,fo,r)+
,
and hence 8(fm, rpm, r) - 0 as m -' oo. On the other hand,
a(fm, 9m, r) > a(fm, r) > 2 > 0,
and we again arrive at a contradiction. (The author's paper [121] was subjected to the criticism when it appeared that the case fo - const was not considered in it; as is clear from the foregoing, this is trivial.) The fact that the function µ(e, r) in (12.9) can be replaced by µ(e), where µ(e) - 0 and µ(e) does not depend on r, was established by Belinskii [12) under the assumption that the given mapping f is normalized in a definite way. It was later proved in [ 14) that µ(e) can be taken equal to Ce, where C is a constant. The final result, contained in Theorem 12.5, was obtained by the author in [ 140]-[142] (see also the author's book [ 144]).
5. We can consider different kinds of integral characteristics of the nonconformality of a mapping. If such a characteristic is given, then the question arises as to whether a mapping is close to a Mobius mapping if this characteristic is small for it. (It is assumed that the zero value of the characteristic corresponds to a Mobius mapping.) Here we present a result of this kind. Let f : U R" be a mapping of class W (U), f (x) = (f, (x), ... , f, (x)), where U is a domain in R. For an arbitrary measurable set A C U let
.(f, A) =
f > If'(x)I" dx. A i=1
For each x E U such that the linear mapping f'(x) is defined we have that
> n det f'(x).
11. MAPPINGS WITH BOUNDED DISTORTION
256
Equality holds here if and only if f'(x) is a general orthogonal transformation and det f'(x) > 0. This implies that if for f
2(f U)=n,fu .(x.f)dx. then 7(x, f) > 0 and f'(x) is a general orthogonal transformation for almost all x E U; consequently, f is a MObius transformation by Liouville's theorem.
Assume that U is bounded. Let the constants M and y be such that 0 < u < M < oo, and let V be a set of positive measure strictly inside U. Denote by W(µ, M, V, e) the collection of all mappings f : U -+ R" in W (U) such that: (I) IIfIIw., (u) < M;
(II)9'(f,V)>µ;and (III) fuE If'(x)I"dx < n(I +e)fuf(x,f)dx. I
Let A be a set lying strictly inside U. For f E W (U) denote by 6(f. A) the infimum of 11f - c II w (A) when ip runs through the set of all Mdbius transformations. Let µ(e, A) =
sup
6(f, A).
fEW(p.M,V,e)
THEOREM 12.8. The relation µ(e, A) = O(&I l") as e -' 0 holds for every measurable set A lying strictly inside U and such that V c A.
6. As consequences of the stability result (Theorem 12.5) we present some theorems on smoothness of quasiconformal mappings and conformal mappings of Riemannian spaces.
Assume that the distortion coefficient for the mapping f : U - R" at the point x tends to 1 as x -+ a E U. It turns out that if this convergence is sufficiently fast, then f is differentiable at a. We give the precise formulations. Let U be an open subset of R", and f : U
R" a mapping with bounded
distortion. Then f is differentiable in U almost everywhere. If f is dif-
ferentiable at a point x E U, then let K f(x) = 1 if f (x, f) = 0, and Kf(x) = K[f'(x)] otherwise. Let w: [0, oo) -+ R be a given function. We say that w satisfies the Dini condition if it is a nondecreasing function and
f
co(t)dt
for some k > 0. (In this case the indicated integral is clearly finite for every k > 0.)
§ 12. SOME FURTHER RESULTS
257
THEOREM 12.9. Let f : U -i R" be a mapping with bounded distortion, a a point in U, and co: [0, oo) - R" a function satisfying the Dini condition.
If K f(x) - 1 < co(lx - al), (12.10) then f is differentiable at a, and f'(x) is a nonsingular general orthogonal transformation. Further, there exists an ro > 0, depending only on cv and n, such that for 0 < r < ro the estimate
If(x)-f(a)-f'(a)(x-a)l <0C(r,lx-al)jx-a1
(12.11)
holds in the ball B(a, r), where si = SUPIx_aI ,,/4 If (X) - f (a)I, and C(r, h) depends only on n, co, r, and h; C(r, h) - 0 as h -' 0. It is possible to take
C(r,h) = Co
h
+cv(4h)+h f r ,2t) dt Jh
+j4h !It)
dt
,
(12.12)
0
where Co = const > 0.
Theorem 12.9 yields a theorem on regularity of conformal mappings of Riemannian spaces. Let B, = B(0, 1). Assume that Riemannian metrics are given in B, by positive-definite quadratic differential forms:
dsI =gk, (x)dxkdxt,
ds2 =h;;(y)dy'dyf
(12.13)
with continuous coefficients gw and hi j. The mapping f : B, - B, is said to be conformal with respect to the Riemannian metrics ds, and ds2 if f E W, 1,,c, f (x, f) has constant sign in B1, and there exists a function A(x) > 0 such that for almost all x E B, hri[f (x) ) gfk (x) e i (x) _ [A(x)) gkt(x).
(12.14)
(Repeated indices are understood to be summed from 1 to n.) The mapping f is said to be isometric with respect to the Riemannian metrics ds2 and ds2 in the case when (12.14) holds for f with the function A(x) = 1. If f is an isometric mapping, then the derivatives 8f /8x f are bounded on every compact subset of B1. THEOREM 12.10. Assume that Riemannian metrics are given in B, by the quadratic differential forms (12.13), and let f : B1 - B, be conformal with respect to these metrics. If for every r E (0, 1) there exists a function cv, : [0, oo) --+ R satisfying the Dini condition that is the modulus of continuity for each of the functions gk1 and h;j, k, 1, i, j = I--, n, on the ball B(0, r), then f belongs to the class C1, det f'(x) # 0 for all x E U. and on every ball B(0, r) with 0 < r < 1 the derivatives (8f l8xl)(x) have modulus
11. MAPPINGS WITH BOUNDED DISTORTION
258
of continuity Or admitting the estimate 0,(h) < C, (1, h) for 0 < h < q, where C,(q, h) is obtained by replacing w by w, in (12.12).
In particular, if gki and hip belong to the class C°, where 0 < a < 1, then f belongs to C I See [ 144] and [ 143] for the proofs of Theorems 12.9 and 12.10. A result analogous to Theorem 12.10 was stated by Calabi and Hartman [21] for mappings. f : B, -, B1 isometric with respect to the Riemannian metrics ds2 and ds2. However, an unfortunate mistake crept into their
arguments (the authors conclude from the inequalities 0 < x < it and I sin x -1 I < e that Ix - a/21 < Q). If the correct estimate Ix - it/2I < C,/ is used, then it turns out that the arguments in [21] are valid only when instead of the Dini condition w satisfies the stronger condition rk 0
This error does not affect the other results in [21 ]. If under the conditions of Theorem 12.9 the function co with K f(x) < w(Ix - al) does not satisfy the Dini condition, then the conclusion of the theorem can turn out to be false. An example is given in [1441. The proof of Theorem 12.9 is based on the estimate (12.7) in Theorem 12.5. A sequence of balls Bm = B(a, pm) is constructed with pm = 2-m-3r and r > 0 such that B(a, r) C U. For each m the MSbius mapping corn is such that qp-I (f (x)) - xI < Cw(4pm)pm
in the ball Bm_2 = B(a, 4pm). The difference Iip'(0) is majorized by the quantity Cw(pm). The Dini condition is equivalent to convergence of E w(pm). This implies that the sequence (1pm(0)) of linear mappings converges, which allows us to conclude finally that f is differentiable at a. Theorem 12. 10 is obtained by using Theorem 12.9. If the functions gki and h,j satisfy stronger regularity conditions than in Theorem 12.10, then f is also regular in a stronger sense than follows from Theorem 12.10. Namely, we have the following statement. THEOREM 12.11 ([1601,[59]). If the functions gkr and hi; in (12.13) be-
long to the class C'-°, where 0 < a < 1, then every mapping f : B1 --' B1 conformal with respect to the Riemannian metrics dsi and ds2 belongs to the class C'1
Theorem 12.11 was proved in [87] for the case r = oo. In [159] it was established under the additional assumption that the function A in
§12. SOME FURTHER RESULTS
259
(12.14) satisfies a Lipschitz condition. The proof of Theorem 12.11 given by Shefel' [160] is based on some geometric considerations not connected with Theorem 12.6. Iwaniec [59] derived the desired result from estimates of solutions of elliptic equations. A result analogous to Theorem 12.11 was obtained by Calabi and Hartman [21 ] for mappings isometric with respect to Riemannian metrics ds2 and ds2. The case of a metric with variable sign is also considered in [21 ] (that is the forms ds? and ds2 are not required to be positive definite). 7. The following injectivity theorem of Martio and Sarvas is another application of Theorem 12.5. THEOREM 12.12 ([93]). Let U be a domain of class U(a,,8) in R", n > 3. There exists a constant c > 0 depending only on n, a, and ft such that every mapping f : U -' R" with bounded distortion for which 1 < K(f) 1 + c is a homeomorphism of U into R".
R" be a given mapping, and take an arbitrary 8. Let f : B(0, 1) MObius transformation p. We find the quantity sup I ci[f (x)] - xl = a(V. f )
xE8(0.1)
and let
3(f) = inf a(go, f) Denote by µ" (e) the supremum of 8(f) on the set of all mappings f : B(0,1)
-* R" with bounded distortion such that K(f) < 1 + e. Theorem 12.5 allows us to conclude that µ(e) <- Cc for 0 < e < co, where C = const. The exact value of the constant C, like that of co, is unknown. In this connection we mention the following result of Semenov [ 153], [170). THEOREM 12.13.
O p(ee) <2 +n 4 2 - n
K(n) - ii
4
2n
3(n - 1)B(1
1
"-1 1
( n+1
1
+
11
n-2
=2+O 7= In particular, for n = 3, 4, 5, \\and 6,
3 + 56 ,
K(3) <- 11. 5,
K(4) <
K(5) <- 5.71,
K(6) < 2.8+
242 342
11. MAPPINGS WITH BOUNDED DISTORTION
260
§12.3. Stability of isometric and Lorentz transformations. 1. Here we present results from John's important paper [60] on stability (in the sense of the present book) of isometric mappings. First we make some remarks about the definition of quasi-isometric mappings.
Let U be an arbitrary open domain in W. Let f : U -' Rn be a given mapping and L > 1 a given number. Then we say that f belongs to the class 11 (L) if its Jacobian F(x, f) is either nonnegative almost everywhere or nonpositive almost everywhere, and for almost all x E U the principal dilation coefficients of the linear mapping fl(x) either are all equal to zero or lie in the interval (L- I, L). If the mapping f : U -' Rn belongs to the class II (L), then, as follows
from the definition, the function x " I f'(x)I is bounded, and hence f belongs to the class WP 1,10C for any p > 0. Further, it follows from the definition that for each x outside a set of measure zero either f(x) = 0 or the principal dilation coefficients Al < In the last case
< ,1n of f'(x) lie in
K[f '(x)] = An-'I)1 ... An-1 <
(L-I
,
L).
L2n-2
and we get that for almost all x E U Ilf'(x)IIn < L2n-2ldet f'(x)I.
Thus, if f E 11(L), then f is a mapping with bounded distortion. Further,
K(f) <
L2n-2.
An arbitrary mapping of class II (L) can fail to be a local homeomorphism. The mapping f in Example 3 of §4.3 in Chapter I, which is an m-fold twisting about the (n - 2)-dimensional plane Rn-2 = {xIxn_1 =
xn = 0), belongs to 11(L) with L = m. At the same time, f is not a homeomorphism in any neighborhood of an arbitrary point x E Rn-2. Theorem 10.5 of the present chapter yields the following proposition.
LEMMA 12.5. There exists a number Lo > 1 such that if 1 < L < Lo, then every mapping f : U -. Rn of class I1 (L) is a local homeomorphism. PROOF. By Theorem 10.5, there exists a Ko > I such that if f : U -+ Rn
is a mapping with bounded distortion and K(f) < Ko, then f is a local homeomorphism. The number Lo = K 2n-2) clearly works. We say that f : U - Rn is a mapping of class 12(L), where L > 1, if f is a bi-Lipschitz mapping and L(f) < L, that is, every point x E U has a neighborhood G such that for any y, z E G with y 0 z <
If(Y)-f(z)I
(12.15)
§ 12. SOME FURTHER RESULTS
261
LEMMA 12.6. For every L > 1
(12.16) 12(L) c I,(L), and there exists an Lo > 1 such that the classes 11(L) and 12(L) coincide for 1 < L < Lo.
PROOF. The inclusion (12.16) is a consequence of results in §6.6 of this chapter. Let Lo > 1 be the constant in Theorem 12.14, and let f E 1, (L), where
I < L < Lo. Then f is a local homeomorphism by Theorem 12.4. Take an arbitrary point xo E U, and let 6 > 0 be such that B(xo, 6) is contained in U and f is a homeomorphism on B(xo, 6). For almost all x E B(xo, 6) we have that I f'(x)I < L, which implies (see the remark after Theorem
2.7 in Chapter 1) that If (y) - f (z)I < LI y - zI for any y, z E B(xo, d). Let H = f [B(xo, 6)] and yo = f (xo). The set H is open. There is an e > 0 such that B(yo, e) c H; let G = f -' [B(yo, e)] n B(xo, 6). Let B c B(yo, e) be the set of y E B(yo, e) at which either f -' is nondifferentiable
or f (y, f -1) = 0, and let A be the set of x E G at which either f is nondifferentiable or f (x, f) = 0. Then E = AU f -' (B) is a set of measure zero, and If (E) I = 0. At each point y E B(yo, e) not in f (E) the mapping
f -' is differentiable, and df -' (y) = (df (x))-', where x = f (y). This implies that I df -' (y) I < L, and hence If-`(u)
- f-'(v)I < LIu - vI
for any u, v E B(yo, e). Choosing y, z E G arbitrarily and setting u = f (y) and v = f (z) in the last inequality, we get that
L IY - zI <- If (Y) - f(z)I, and hence (12.15) holds for any y, z E G. Since X0 E U was arbitrary, f E 12(L). In view of (12.16), the lemma is proved. Let U be an open domain in R. A mapping f : U R"is said to be quasi-isometric if it belongs to the class 12(L) for some L > 1. The smallest number L > 1 such that f E 12(L) (it is not hard to show that such a number exists) is denoted by L(f). THEOREM 12.14 (JOHN [60]. Let U be a domain of class f (r, R) in R",
and f : U - R" a mapping of class I2(L). Then there exist a point xo E U and an orthogonal transformation y such that for all x E U 2
If(x) - f(xo) - y(x -x0) I where C depends only on n.
C[L(f) - I]R .
II. MAPPINGS WITH BOUNDED DISTORTION
262
The theorem shows that if the difference L -1 is sufficiently small, then the mapping f E 12(L) is uniformly close to some isometric mapping of R. The next theorem shows that the derivatives off E I2(L) are close to those of an orthogonal transformation if L is close to 1. THEOREM 12.15 ([60]). Assume that the domain U c R" is a cube and f : U - R" is a mapping of class 12 (L). Then there is an orthogonal matrix y such that for every p > 1 UP
1
C TV I
fU If'(x) - yI dx I
where the constant C depends only on n.
Theorem 12.15 extends easily to an arbitrary domain of class f (r, R). By Lemma 12.6, Theorems 12.14 and 12.15 remain true if 12(L) is replaced by I, (L) in their statements. The proof of Theorem 12.15 makes essential use of a theorem of John and Nirenberg [63] on functions with bounded mean oscillation. The formulation and proof of the latter theorem will be given below (Theorem 1.1 in Chapter III). A certain analogue of the theorem of John and Nirenberg is used in the proof of Theorem 12.5. The problem arises of the exact value of the constants in Theorems
12.14 and 12.15. Let U be a domain in R", and f : U - R" a quasiisometric mapping. For an arbitrary isometric transformation ip: R" - R" let
J(f,cv) = sup I V -' [f (x)] - xl and 6(f) = inf8(f, so). XEU 1P
where the infimum is over and set of all isometrics ? of R". The supremum
of 6(f) over the collection of all mappings f : U - R" in 12(L) is denoted by µu (L) THEOREM 12.16 [153]. If U is a ball in R", then
firm- L(Ll <2n+1. L-i 2. We mention some other properties of quasi-isometric mappings.
LEMMA 12.7 ([60]). Let U be a domain in R", f : U -' R" a mapping of class 12(L), and x, and x2 two arbitrary points of U. If the ellipsoid of revolution
S={xllx-x,I+Ix-x21
§12. SOME FURTHER RESULTS
263
is contained in U, then
LIx1 -x21:If (x1)-f(x2)I
and hence f is a homeomorphism on V.
For a domain c R" let L(O) be the supremum of the numbers L > 1 such that every mapping f :. -' R" of class 12(L) is one-to-one. If = R", then, by a theorem of Zorich (Corollary 2 to Theorem 10.6), every mapping f : 9 - R" is a homeomorphism of R" onto itself, and hence L(9') = oo in this case. It is easy to construct examples of domains with L(O) = 1. Any planar domain whose boundary has a cusp point directed inward (see Figure 2 in Chapter I, §4.3) is such a domain. John [62] showed that if .9 is a ball or half-space, then L(O) > Let n = 3, and let z, r, i be cylindrical coordinates in R3, -oo < z < oo, r > 0. Let f be the mapping (z, r, 9) i- (z/ 2 + e, r/ 2 + e. (2 + e)(,). It is easy to verify that f is quasi-isometric, L(f) = 2 -+c, and f is not one-to-one on the half-space 0 _< 9 < it. This shows that for a half-space we have L(O) < 2 _+z for any e > 0; hence L(O) < 2 in this case. .
21/4.
The construction of the last example extends easily to the case of arbitrary n, and we thus get that L(O) is finite for a half-space. Following John, we call a domain in R" rigid if L(O) > 1. By the theorem of Martio and Sarvas (Theorem 12.12), every uniform domain in R" is rigid. The problem arises of determining necessary and
sufficient conditions for a domain in R" to be rigid. This problem was solved by Gehring [39] for the case n = 2. He showed that a domain U in R2 is rigid if and only if every connected component of its boundary with more than one point is a quasicircle. 3. Gurov established an analogue of Theorems 12.14 and 12.15 for Lorentz transformations [50]-[53]. Here we state his results. Denote by A" an n x n diagonal matrix with the first n - 1 diagonal elements equal to 1 and with last diagonal element equal to -1: 0 t
.=
(0
1_.
11. MAPPINGS WITH BOUNDED DISTORTION
264
Let U be an open domain in R", and f (x) = (f, (x), ... , f, (x)) a map-
ping of class W,1(,(U) from U into R". We say that f is an c-quasiLorentz mapping if f satisfies the following conditions: a) The generalized derivative (8f"/8x")(x) is nonnegative in U. b) For almost all x E U
f'(x)'ff'(x) =F. + A(x),
(12.17)
where the matrix A(x) is such that IA(x)j < e. Let A(x) = (aiJ(x)), i, j = 1, ... , n. Then (12.17) is equivalent to the following system of equalities:
fi.i(x)fi.j(x) + ... + fn-1.i(x)Jn-1.!(x)
- fe.i(x)f,'., (x)
= aid + aid,
where aid = 1 for i A j, aii = 1 for i < n, and a"" _ -1. (As earlier, we set fk.i(x) _ Mappings for which
f'(x)ff'(x) = f,
ax (x) > 0.
are Lorentz mappings. A proof of this assertion when f is sufficiently smooth can be found in any fairly complete guide to relativity theory. Theorem 12.17 below gives us the assertion under the assumptions made here about the smoothness of f. The collection of all e-quasi-Lorentz mappings of an open domain U c R" is denoted by QL(e, U). THEOREM 12.17 (GUROV [50], [511). There exist constants 8" > 0 and
A,, < oo such that if U is a domain of class F(r, R) and f : U - R" an e-quasi-Lorentz mapping with e < b", then there exists a Lorentz transformation ip such that for all x E U
IiP[f (x)] - xI < A"e(R2/r).
Certain additional assumptions about U ensure that the derivatives of an e-quasi-Lorentz mapping are close to the derivatives of a Lorentz mapping.
Let U be a domain in R", and denote by 8h U (h > 0) the set of all points x E U with p(x, R"\U) < h. Define act = sup X 81, UIl h°
where a > 0.
§ 12. SOME FURTHER RESULTS
265
THEOREM 12.18 (GuRov [52], [53]). Let U be a domain of class f (r, R)
such that Q,, is finite for some 0 < a < 1. Let
0<e
1
If f E QL(e, U), then the derivatives off are p-integrable, and there is a Lorentz transformation ip such that (;.R)1/p
III(w 0 f)'- IIII L,(v) <- CntpR In R I
Here A, B, and C are constants depending only on n. The proof of Theorem 12.17 in [53] makes essential use of an analogue of the John-Nirenberg theorem on functions with bounded mean oscillation. THEOREM 12.19 ([52], [53]). Suppose that the vector-valued function u(x) = (u1(x), ... , is integrable in the cube Qo = Q(a, r) c Rn. Assume that there is a constant f < (1/2)n+3 such that for every cube Q c Qo there exists a vector UQ E R"' for which fQ I u(x) - uQl dx 5 RI uQl - IQI
For a > 0 let E (a) denote the Lebesgue set {x E QoII u(x) - UQoI > aI UQol }.
If a> of I uQ0I, then IEu(a)I <- Bn(1
+a)-bt/PIQo1,
where B. and b,, are constants and depend only on n. COROLLARY. If the function u : Qo -+ Rn satisfies the conditions of the theorem, then it is integrable to any power p < bn/ f .
We remark that the class of quasi-Lorentz mappings has received little attention on the whole. The mapping inverse to a quasi-Lorentz mapping can fail to be quasi-Lorentz. The composition of two quasi-Lorentz mappings is not in general quasi-Lorentz. In this connection one might think that the class itself is badly defined. However, attempts to modify the definition in such a way that the mapping inverse to a quasi-Lorentz mapping and the composition of two quasi-Lorentz mappings are also quasiLorentz mappings lead to loss of the stability property in Theorems 12.17
and 12.18. A generalization of Theorem 12.19 is used in the proof of Theorem 12.5 on stability in the Liouville theorem.
II. MAPPINGS WITH BOUNDED DISTORTION
266
§ 12.4. Quasiconformal and quasi-isometric deformations. Semigroups of quasiconformal transformations.
1. In the study of quasiconformal and quasi-isometric mappings it is useful to consider certain special differential operators well known in continuum mechanics.
Let U be an open subset of R. For a mapping v : U - R" of class WII1 (U) let
[V'(x) +V'(x)']. (Q2v)(x) = 1[v'(x) +v'(x)'] - (n-I divv(x))1, (I is the n x n identity matrix). For each x E U the quantities (QIv)(x) and (Q2v)(x) are n x n matrices whose elements are expressed in terms of the derivatives of the components of the vector-valued function v as follows:
(QIV)ri =
1
avj
8x;)
8xj 1
(Q2V )'J =
Ov;
+a(I-_
Su;
L
8x
divv,
where S;j is the Kronecker symbol. If v(x) is interpreted as the velocity field of a deformable body (in the
case n = 3), then the matrix-valued function (Q,v)(x) will be identified with the velocity affinor of the deformation of the body, and (Q2v)(x) will be the part of this affinor responsible for the interior forces (see [117]). In connection with the theory of n-space mappings the operators Q, and Q2 were first considered by Ahlfors [3] and the author [135]. We present arguments for the likelihood that Q, and Q2 can be useful in the study of quasiconformal and quasi-isometric mappings. Below, U denotes an open domain in R". Let f : U --+ R" be a quasiisometric mapping, i.e. a mapping of class 12(L). Assume that L is suf-
ficiently close to 1, while f (x) is close to the identity mapping f (x) = x + u(x), with not only u(x) but also its derivatives small. Let L = I + e. For almost all x E U 1
(1
where
+02 - (f'(x) f'(x)) S (1 + e)2 .
,
(12.18)
is an arbitrary unit vector. We have that
(f'(x) . f'(x)) = (f'(x)'f'(x)
.
).
It follows from (12.18) that the eigenvalues of the symmetric matrix f'(x)' f'(x) lie in the interval (1 /(1 + e)2, (1 + e)2). We have that
(f'(x)')f'(x) = I + u'(x) + u'(x) + (u'(x))*u'(x).
§ 12. SOME FURTHER RESULTS
267
Throwing out terms of higher than second order with respect to the
derivatives of u(x) in the expression for f'(x)`f'(x), we get that the eigenvalues of the matrix u'(x) + u'(x)* = 2(Q1 u)(x) lie in the interval ((1 + e)-2 - 1, (1 + e)2 - 1) (to within quantities of higher than second order of smallness compared with e). A vector-valued function v : U R" will be called a quasi-isometric deformation if it belongs to the class W1110C(U) and for almost all x E U the eigenvalues of the matrix (Q1 u)(x) lie in (-M, M), where m = const is finite. The operator Q2 arises similarly as a result of linearization of the quasiconformality condition.
Let f : U -' R" be a mapping with bounded distortion. Assume that K(f) is close to 1, K(f) = 1 + e, where a is small, and f (x) is close to the identity mapping, f (x) = x + u(x), where the vector-valued function u(x) is small. It will be assumed that the derivatives of u(x) are also small. Consider the matrix-valued function
Of(x) = (.1''(x))-1(f'(x)*)-1If (x.f)I2/" for almost all x. We have that f(x) = I + u'(x). This gives us that, to within quantities of higher than first order of smallness,
(fi(x))-' -- I - u'(x),
(fi(x)`)-' -- I - u'(x)
I.F(x.f)121" ;Ze 1 + 2
tru'(x),
and
Of (x) -- I - [u'(x) + u'(x)` - n tru'(x)I] = I - 2(Q2u)(x). The eigenvalues of the matrix Of (x) lie in the interval (K(f)-2, Ko(f )2).
We have that K0(f) < K(f)"-'. This leads to the conclusion that, to within quantities of higher than second order of smallness compared with e, the eigenvalues of Of (x) lie in (1 - 2e, 1 + 2(n - 1)e), and hence the eigenvalues of (Q2u)(x) are in (-(n - 1)e, e). Following Ahlfors [3], we say that a vector-valued function v : U R" in a quasiconformal deformation if v E W110(U) and there exists a constant M < oo such that for almost all x E U the eigenvalues of (Q2u)(x)
lie in (-M, M). If v : U R" is a quasi-isometric deformation, then f is a quasiconformal deformation. Indeed, in this case each of the functions Ov
1
8v1
2
8xj + ex,
268
II. MAPPINGS WITH BOUNDED DISTORTION
is bounded in U. In particular, the function &v,/0x, is bounded, and this implies that 8vj I 8v, (Q2e)ri = 2 (Oxj + 0x,
0,j
-n
dive
is a bounded function on U. Above we obtained an interval nonsymmetric with respect to 0 for the eigenvalues of (Q2u)(x), namely, the interval(-(n - 1)e, e). To avoid this it is expedient instead of K(f) and Ko(f) to consider the quantity
K(f) = max{K(f ), Ko(f )}. If K(f) and Ko(f) are replaced by k(f) in the above arguments, then for k(f) = 1 + e we get that the eigenvalues of (Q2u)(x) lie in the interval (-e, e). For any two mappings f and g with bounded distortion on R" we have
K(f o g) 5 k(f)K(g). 2. Let In be the group of motions of R", and let Mn, as above, denote the collection of all Mobius transformations of R ; I" and Mn are Lie groups,
and, as such, each is endowed with the structure of a C°°-differentiable manifold. A vector-valued function u: R" -* R" is said to be an infinitesimal motion if there exists a smooth path tp : [0, 1] - In in In such that 9(0) = I (here and below, I is the identity mapping) and u(x) = dt 97(t. x)I r=o
for all x E R. A vector-valued function u: R" - R" is called an infinitesimal Mobius transformation if there exists a smooth path w : [0, 1] -' Mn such that w(0) = I and u(x) = jw(t.x)I:=o for all x E R". Denote by Ei (n) the collection of all infinitesimal motions of R", and by L2(n) the set of all infinitesimal Mobius transformations of R"
If f is a motion of R", then f'(x)' f'(x) is the identity matrix. Let f (x) = 9(t, x), where gyp: [0, 1] --+ In is a path in In emanating from the x) = I identity of the group In. Differentiating the equality (47''(t, with respect to t and setting t = 0, we get after obvious transformations that the vector-valued function u(x) = (09/0t)(0, x) satisfies the equation (Ql u)(x) = 0.
(12.19)
§ 12. SOME FURTHER RESULTS
269
It can be shown that every solution of the system (12.19) is a vectorvalued function u(x) of the form
u(x) = a + Kx,
(12.20)
where a = const is a vector in R", and K is a skew-symmetric matrix. It is easy to verify that every function of the form (12.20) satisfies (12.19) and belongs to 11 (n). Thus, E1 (n) coincides with the set of all functions of the form (12.20). This also implies that E, (n) coincides with the collection of all solutions of (12.19).
Let f : R R be a Mobius transformation. Then for all x E R" with f (x) 0 oo the matrix
Of(x) = (fl (x))-`(fI (x)*)-' LF(x,f)I2t" is the identity, Of (x) = I. Consequently, also
(of(x))-' =
f'(x)'f1(x)If(x,f)I-21n = I.
Setting f (x) = V(t, x), where V/: [0, 1] Mn is a smooth path in M. with V(0, x) x, differentiating the resulting equality with respect to 1, and setting t = 0, we find after obvious transformations that the vectorvalued function u(x) = (8V//&)(0, x) satisfies the system of equations Q2u(x) = 0.
(12.21)
Every solution of (12.21) has the form
u(x) = a + (K + al)x + 2(b, x)x - IxI2b,
(12.22)
where a E R, a and b are vectors in R", and K is a skew-symmetric matrix. It is easy to verify that every function u(x) representable by (12.22)
satisfies (12.21) and belongs to E2(n). The set E2(n) of functions thus coincides with the set of solutions of (12.21) and, at the same time, with the collection of all functions of the form (12.22). 3. One of the main results in the theory of the operators Q1 and Q2 involves estimates of the norm of u(x) in Wp in terms of the La-norm of the matrix-valued function Qiu(x), i = 1, 2. The estimates of this kind for the operator Q1 are called Korn's inequalities. We give the necessary definitions. Let X be a vector space. A linear
mapping P: X - X is called a projection if P(Px) = Px for every x, or, what is the same, P(y) = y for every y E P(X). Recall that a domain U in R" is called a domain of class S if U is a union of finitely many domains that are star-like with respect to a ball.
270
II. MAPPINGS WITH BOUNDED DISTORTION
THEOREM 12.20. Let U be a domain of class S in R", and let p > 1 be given. Then for every continuous projection n: Wp (U) --- E, (n) there exists a constant C, = C, (n, p, n) such that for every function u E WP (U) IIu - nullw;(v) < CI IIQIUIIL,(U)-
THEOREM 12.21. Let U be a domain of class S in R", and let a number p > 1 be given. Then for every continuous projection n: W,(U) -' 2:2 (n) there exists a constant C2 = C2(n, p, n) such that
llu - null wo(u)
< C2II Q2uII L,(u)
for every u E WD (U).
It was apparently Friedrichs who first proved Theorem 12.20 rigorously for the case n = 2 [32]. Theorem 12.20 was proved in the general case formulated here independently by the author [135], Necas [108], and Mosolov and Myasnikov [105]. Theorem 12.21 was proved by the author [135].
The proofs of Theorems 12.20 and 12.21 are based on the following considerations. Assume first that U is bounded and star-like with respect to some ball B(xo, r) c U. Then
u(x) =
L K(') (x, Y)u(Y) dY +i.1=I > fu Kjj) (x,Y)(Q1u)ij (Y)dy
for any function u E W PI (U), where I = 1, 2, and Koi) (x, y) is an n x n matrix whose elements are bounded C°°-functions on U x U for any x and y. As a function of x for fixed y, Kol)(x, y) belongs to 11 (n). The function y) takes values in R", belongs to the class CO, on the set of all (x, y) E U X U for which x $ y, and admits the representation (lXx
- Y) Ix where the function H t (x, y) is bounded and continuous and has bounded derivatives that are continuous for x 0 y, and co;j(z) is a function of class C°° in the domain R11\{0}. The integral representations (12.23) can be established for domains starlike with respect to a ball. Theorem 12.20 and 12.21 can be obtained for such domains by using a theorem of Calderon. The case when U is an arbitrary domain of type S is easily reduced to this case. The integral representation (12.23) is nonunique for each of the operators Q, and Q2. The technique used in [135] to get (12.23) is described in
§12. SOME FURTHER RESULTS
271
general form in [ 138]. Ahlfors [5] also obtained an integral representation for Q2v analogous to (12.23) for the case when the domain is a ball.
THEOREM 12.23 (AHLFORS [6]). Let f : U -. R" be a quasiconformal deformation, and B(a, r) an arbitrary ball in U. Then for any x, y E B(a, r) 2r C1+log lx If(x)-f(y)I - YI In particular, it follows from the theorem that if f is a quasiconformal deformation, then f satisfies a Holder condition with exponent a for all a E (0, 1). Since, as mentioned above, a quasi-isometric deformation is automatically quasiconformal, Theorem 12.22 automatically remains valid if Q2 is replaced by Q, and quasi-isometric deformations are used instead of quasiconformal deformations in its formulation. .
Of other results touching on the operators Ql u and Q2u proper, we mention criteria for solvability of the system of equations Q, u = F and Q2u = F. Properly speaking, one criterion for solvability of the system Ql u = F is the Saint-Venant condition well known in mechanics. Let F(x) = (F;j(x))j.j=1....,,, and F;j(x) = Fj,(x) for any i and j. If the functions F; j belong to the class C2 (U), then the system of equations
j _ F;j, i,j = 1,...,n, axj + (this is clearly the system Q, u = F in expanded notation) is solvable if and only if au;
auax;
a2F,j + a2Fk, axkUXI ax; ax,
- axkax a2F`, - a2Fkf j ax; axl
=0
(12. 24)
for any i, j, k, I = 1, ... , n. If the derivatives are understood as distributions, then this criterion for solvability of the system Q, u = F is valid without any assumptions about the differentiability of the functions F; j. Namely, we have the following assertion. THEOREM 12.23. Suppose that U is an open domain in R", and F(x) _ (F; j (x)), i, j = 1,... , n, is a matrix-valued function whose elements are functions of class LP.1a(U) (1 < p < oo) with F, j(x) = Fj;(x) for any i and j. If (12.24) holds for any i, j, k, 1, and the derivatives there are understood as distributions, then the system of equations Q, u = F is solvable in U. REMARK. Under the conditions of the theorem, (12.24) is equivalent to the following: for every function V E C0(U) 82 IP
1U
(Fr'axkax,
+ Fk1
02 41
ax axj
- F;laxkaxj - Fkjax,ax,) dx = 0.
Ii. MAPPINGS WITH BOUNDED DISTORTION
272
Ahlfors [4] established an analogous criterion for solvability of the system Q2u = F. Some further questions about Q2 are considered in [5], [7], and [8]. In particular, some special differential operators connected in a natural way with Q2 are introduced there, and their properties are investigated. 4. Flows of quasiconformal mappings. The following is a well-known problem in the theory of spatial quasiconformal mappings (see [83]). Let f : U - R" be a quasiconformal mapping. Is it possible for every e > 0 to construct a finite sequence of quasiconformal mappings fl..... fm such that K(f;) < 1 + e for each i = 1, ... , m and f = fm o fm- I o . . . o fl? This question is of interest because quasiconformal mappings with K(f) close to 1 are considerably more convenient to study than general quasiconformal mappings, due to the stability in the Liouville theorem (Theorem 12.5). The question posed is usually called the composition problem. An analogous composition problem can also be posed for quasi-isometric mappings. In connection with this question the thought occurs to give special consideration to mappings that can be represented for any e > 0 as a composi-
tion of finitely many quasiconformal mappings with K(f) < 1 +e. Among such mappings it is natural to single out those that can be included in a quasiconformal flow. We give the necessary definitions. Below, as above, we let K(f) = max{K(f ), Ko(f ) } for any quasiconformal mapping f.
Let X be an arbitrary set. A mapping F : R x X - X is called a flow
ifF(0,x)=xandF(t+s,x)=F(t,F(s2x))forall xEXandt,sER. The concept that arises when the half-line R+ = [0. oo) is used in this definition instead of the set R of all real numbers is called a semiflow. (That is, a semiflow is a mapping F : R+ x X - X such that F(0, x) = x and F(t + s, x) = F(t, F(s, x) for any t, s > 0.) In the case when X is a topological space and F a continuous mapping, one says that F is a topological flow (respectively, a topological semiflow). Instead of F(t, x)
we also write Ft(x). For an arbitrary mapping f : X - X we let f 0 = id and fk = f o fk-1 for any integer k > 0. If f is a bijection, and let fk = (fJkl)-1 for an integer k < 0. A fractional (rational) power of f : X -' X is defined to be any mapping g : X -, X such that gi = f k for some integers k and 1, where k is not a multiple of 1. We say that a mapping f : X -+ X is included in a flow F : R x X - X
(a semiflow F : R+ x X -' X) if f = Fro for some to E R (respectively, to E R+).
§12. SOME FURTHER RESULTS
273
Let A c R" be a given set. It is assumed that every point x E A is a limit point for the set of interior points of A. A quasiconformal flow (semiflow) on A is defined to be a topological flow (semiflow) in A such that for each t the restriction of F, to A° is a quasiconformal mapping, and there exists a constant A > 0 such that K(F,) < exp Al t i for all t. A quasi-isometric flow (semiflow) on A is defined to be a topological flow (semiflow) F on A such that for each t the restriction of F, to A° is a quasi-isometric mapping, and there exists a constant A > 0 such that L(F,) < expAlt) for all t. Assume
that the limit u(x) = lim
F(t, x) - x =
OF (0, x)
at exists for all x E A. The function u(x) thus defined is called the infinitesimal generator of the flow (semiflow) F. t
THEOREM 12.24. Let F, be a quasiconformal flow (semiflow) in B =
B(0, 1) c R". Then F has an infinitesimal generator u(x) in B. The vector-valued function u(x) is a quasiconformal deformation, and if F is
a flow, then (x, u(x)) = 0 for all x E S(0, 1); if F is a semiflow, then (x, u(x)) < O for all x E S(0,1). Conversely, let u(x) be a quasiconformal deformation in B. In this case if (x, u(x)) < 0 for all x E S(O, 1), then there exists a quasiconformal semiflow (F,) whose infinitesimal generator is u(x). Such a semi flow is unique. If a quasiconformal deformation u: B - R" is such that (x, u(x)) = 0 for all x E S(0,1), then there exists a quasiconformal flow such that u(x) is the infinitesimal generator. Theorem 12.24 was proved independently by Semenov [ 150], Ahlfors [6], and Riemann [118]. If u(x) is the infinitesimal generator of the quasiconformal flow F,, then IIQ2uIIL can be taken as a constant A such that k(Ft) < expAt. The coefficient K(F,) in the definition of a quasiconformal flow can be replaced by q(F,), as Semenov did. The concept arising as a result of this change is equivalent to the preceding concept. For a quasiconformal flow F the distortion coefficient q(Ft) admits the estimate q(Ft) < exp(u ti), where
µ= min ess sup j 1: 1 i<j
a`` + a i
l+
I
i=l
li a
t
lk ax
l
If P B - R" can be included in a quasiconformal flow, then it has every fractional power, and the composition problem is solved positively
for it in the obvious way. In the general case it is impossible to get a
11. MAPPINGS WITH BOUNDED DISTORTION
274
solution of the composition problem by using Theorem 12.24 directly, as shown by the example (constructed by Ivanov [57]) described below. We present the result in [57]. Let cp : [0, 1) -i [0, 1] be a continuous bijection such that cp(x) > x for all x E (0, 1). A point a E (0, 1) is said to be a relatively attracting point for the mapping (p if for all x E [a, (p(a)] (Pm(x) - 9m(a) = 0. lim (pm+I moo (a) - ,m(a)
Let .Y denote the class of bi-Lipschitz mappings 9 of [0, 1] onto itself that have relatively attracting points.
Let U be an open set in R", let a E U, and let r > 0 be such that B(a, r) c U. Take an arbitrary function c E K and define a mapping
f:U - Uby
f (x) = a + lx r
al rp
(x-r a) (x - a)
for 0 < Ix - xol < r and f (x) = x for x = a and for Ix - al > r. Clearly f is a bi-Lipschitz mapping, and hence a quasiconformal mapping of U onto itself. The mapping f preserves orientation. The collection of
all mappings f that can be obtained by this construction is denoted by
.7(U). THEOREM 12.25 [57]. No mapping in the class _T (U) has a bi-Lipschitz fractional power, and for n > 2 none has a quasiconformal fractional power. Every mapping of class .7(U) can be included in a topological flow on U. Every CI -neighborhood of the identity transformation on U contains a CI -
diffeomorphism in .7(U). On the other hand, _T (U) does not contain a mapping of class C2.
A result analogous to Theorem 12.24 is true also for quasi-isometric mappings, as Semenov has shown [ 151 ]. The statement of the corresponding theorem is obtained by replacing the word "quasiconformal" in Theorem 12.24 by the word "quasi-isometric". Semenov also obtained an analogue of Theorem 12.24 for quasi-Lorentz transformations [ 151 ]. Here the quasi-Lorentz property in [ 151 ] is understood in a broader sense than in the work of Gurov cited above. 5. Domains UI and U2 in R" are said to be quasiconformally equivalent
if there exists a quasiconformal homeomorphism f : UI - R" such that U2 = f (Ul ). Let x(f) denote one of the distortion coefficients of f (that is, K(f) is K(f), or Ko(f ), or q(f), or some other distortion coefficient). The infimum of K(f) over the set of all quasiconformal homeomorphisms f
§12. SOME FURTHER RESULTS
275
from U, onto U2 is denoted by 8K(Ul, U2). If U, and U2 are not quasiconformally equivalent, then we let vK (U, , U2) = oo. The quantity vK (U, , U2) is called the quasiconformality coefficient of the pair (U1, U2) of domains corresponding to the given distortion coefficient K. The problem arises of determining 8K(U,, U2), at least for certain concrete pairs of domains U, and U2 in R". Only scattered results are known in this direction (see [42], [41 ], and [ 166]). Here we consider a closely related problem. Assume that
U, and U2 are domains with smooth boundary extremely close to each other. It is required to investigate the asymptotic behavior of &(U], U2) for various choices of distortion coefficient K. This leads to a certain extremal problem for the operator Q2, and it is in this connection that we consider the question here. The results below are due to Vasil'chik [ 176]. We consider bounded domains in R" whose boundaries are (n - 1)dimensional manifolds of class C2. If a domain U satisfies this condition, then we say that U is a domain of class C2. Let U be a domain of class C2 in R. A regular deformation of U is defined to be a family (U, ),E(o.a) of domains of class C2 such that Ti, = d)(t, U) for every t E [0, 6), where fi: [0, oo] x U -' R" is a mapping of class C2 with C(0, x) = x for all x E U, so that U = U0. Denote by v(s) the outward unit normal vector at a point s E 8U. For S E 8U let w(s) =
a-(0, x). (P (S) = (v(s), w(s)).
We call cp the velocity function of the boundary of U with respect to the deformation (U,),Elo.a) of it. Let (U,),E[o.o) be a given regular deformation of a domain U of class C2. As t - 0 the function CD, converges uniformly on Ti together with the first and second derivatives to the identity mapping and its corresponding derivatives. Therefore, CD, is a homeomorphism of U onto U, for sufficiently small t c- [0, 6) (it is assumed below that CD, is a homeomorphism
of U onto U, for all t E [0,8)). Further, K((V,) - 1 for any distortion coefficient K, and, moreover, L(C,) also goes to 1 as t - 0. We have that 6K(U, U,) < K(CD,), which leads to the conclusion that dK(U, U,) I as 0, where is valid as t t -. 0. The relation 8K(U, U,) = I + t'RK + r is either 1 or 2 (r is I for the basic distortion coefficients K, K0, and q, while r is 2 for a certain special distortion coefficient defined below). The determination of the coefficient RK in the asymptotic representation of 6K (U, U,) reduces to the solution of a certain special extremal problem.
11. MAPPINGS WITH BOUNDED DISTORTION
276
For an arbitrary symmetric matrix X we set M(x) = max(Xt, t), m(X) = min(Xt, t), 1,1=1
1:1=1
µ(X) = max I(Xt,1)I = max{IM(X)I, Im(X)I}.
IXI = trX'X =
F
(Obviously, M(X) is the largest and m(X) the smallest eigenvalue of X.) In addition to the distortion coefficients considered earlier we introduce the quantity
H(f) = esssup xEU
I.lt(x)In
nnI2 f(x,f)
Let (Ul)IE[o.a) be a regular deformation of a bounded domain U, and so(s), s E 8U, the velocity function of the deformation. Denote by W(9) the collection of all vector-valued functions v E W1 (U) such that (v(s), v(s))
_ cp(s) for all s E 8U. We consider the functionals 1(v) = ess sup M[(Q2v)(x)], xEU
lo(v) ess sup{-m(Q2v(x))}, xEU
IQ(v) = esssup{M[Q2v(x)] - m[Q2V(x)]}, xE U
IF(v) = esssup{IQ2v(x)I2}, xEU
IL(v) = ess supM[(Q1v)(x)] xEU
on the set W(9). The infima of these functionals on the set of functions v E W(rp) will be denoted by A(U, c),.o(U, 9)2q(U, c), )F(U, ip), and 2L(U, gyp), respectively.
THEOREM 12.26 [176]. Let (U,)(E(o.o) be a regular deformation of a bounded domain U of class C2, and let ip(s) be the velocity vector of the deformation on the boundary of U. Then A(U.SO)=1i m Aq
8K(U. u,) - 1 t
(U,9 )=lim
Ao(U,47)=lim AF
1-0
(U 9
8Ko(U. U,)
)1i m
-I
t H(U, U,)
-I
t2
t
(Here 6L(U, U,) is the infimum of L(f) over the set of all quasi-isometric mappings of U onto U,.)
§ 12. SOME FURTHER RESULTS
277
§12.5. Mappings with distortion coefficient close to 1. If for a mapping
f : U - R" with bounded distortion the quantity K(f) - I is sufficiently small, then it has properties that do not hold in general for mappings with
arbitrary K(f). Some results establishing properties of this kind were indicated above (Theorems 10.5 and 12.5). We shall present other results of the same type. 1. We consider the problem of approximating an arbitrary mapping with bounded distortion by mappings of a simpler nature. Let U be a domain in R". A mapping ip : U R" is said to be simplicial if there exists a simplicial complex (infinite in general) such that every compact subset of U is covered by finitely many simplexes of the complex, c is continuous, and f is affine on every simplex on the complex. We say that the simplicial approximation problem is solvable for a map-
ping f : U -+ R" with bounded distortion if for every e > 0 there exists a simplicial mapping ip : U - R" with bounded distortion such that I9(x) - f (x) I < e for all x E U, and the distortion coefficient of 1P can be estimated in terms of the distortion coefficient of f :
K(,p) 5 F,[K(f)], R is a function bounded on every interval [ 1, a], 1 < where F, : [ 1, oo) a < oo. (The function F, can obviously be assumed to be nondecreasingthis can be achieved by replacing F, (t) by sup,<,,<, F, (u).) Assume that for every e > 0 it is possible to construct a mapping V : U
R" of class CI such that F(x, ip) # 0 for all x E U, if (x) - (p(x)I < e for all x E U, and K(qi) < F2[K(f)], where F2: [ 1, oo) R is bounded on each interval [ 1, a], I < a < oo. In this case we say that the smooth approximation problem is solvable for f. The smooth approximation problem is easy to solve in the case n = 2 with the help of known existence theorems in the theory of planar quasiconformal mappings. The solution of the simplicial approximation problem is easily obtained from this. There are no analogous existence theorems for the case n _> 3. The question of solvability of the simplicial approximation problem for
an arbitrary mapping with bounded distortion remains open. However, if the distortion coefficient of f is close to 1, then this question admits a positive solution, as does the question of smooth approximation. The corresponding results were obtained by Kopylov in [66] and [67]. We give their formulations. R" is a given Suppose that U is a given domain in R", and f : U mapping. For h > 0 let Oh = {x E R"lp(x, CU) > h}. Let ho be such that
11. MAPPINGS WITH BOUNDED DISTORTION
278
if 0 < h < ho, then (jh is nonempty. Assume that f belongs to L,,,a(U). Then for every x E Uh the number fh (x) =
f
6h" "1
f (x + y) dy
(12.25)
YI
is defined. The function fh : Oh R" is the average of f in the Steklov sense. The function fh is continuous. If f is continuous, then fh belongs
to C'. THEOREM 12.27 (KoPYLOV [67]). There exists a number co > 0 such
that if f : R" - R" is a quasiconformal mapping with K(f) < I + Co, then for every h > 0 the mapping fh(x) defined by (12.25) is quasiconformal,
f(x,fh)00forallx,and
Kh(f) <_ F[K(f)], R is nondecreasing, and F (t)
where the function F : [ 1, 1 + co) t
1 as
1
THEOREM 12.28 (KOPYLOV (67]). Let f : U -+ R" be a quasiconformal
mapping. If K(f) < 1 +
where e, = const > 0, then fh is quasiconformal on the domain UA _ {xl p(x, CU) > A}, if h is sufficiently small, and F (x, fh) 96 0 for all x E &A. The quasiconformality coefficient of fh tends to 1 as K(f) - 1. More precisely, there exist a constant S > 0 and a function G(t, u) of the variables t E (0, oo) and u E [ 1, 1 + et) such that lim r-0.u
G(t, u) = 1, I
and if h < SA and K(f) < I + e,, then fh is quasiconformal on &A, with K(fh) <_ G
( K(f)) ,
.
Theorems 12.27 and 12.28 remain true if the Steklov average in their formulations is replaced by an averaging in the sense of Sobolev. 2. The class of uniform domains in jr was defined in part 2 of § 12.1. It coincides with the class U(J) of domains, where 0 < S < 1. By a ball in R" we shall mean a set which is a ball B(xo, r) in R", or the complement R"\B(xo, r) of such a ball, or an open half-space in R"; a k-dimensional sphere in R" is either an ordinary k-dimensional sphere in R" or a set of the form P u {oo}, where P is a k-dimensional plane in R". In studying quasiconformal mappings close to conformal mappings it is expedient to investigate the structure of domains of class U(S) with coefficient S close to 1. Here we present results of Trotsenko [1691, [1701
§ 12. SOME FURTHER RESULTS
279
about the structure of the domains U(8) for 6 close to 1. In particular, he established various characterizations of domains of this class that are equivalent as 6
1.
First of all we note that along with the class U(J) it is expedient to consider also a certain other class U(S). Let x1 and x2 be two arbitrary points in R n, and T an arbitrary continuum in R" joining them. For X E r denote by B(x, x1, x2, a) (6 > 0) the set of all points y E R" such that (x. y, xl , x2) = Ix
- x1 I - lY -x21
The set B(x, x1, x2, 8) is a ball in R". Let VI (t, 6) be the union of the balls B(x, x1, x2, 6) as x runs through r, let V2(r, 6) be the union of all the balls B(x, x2, x1, a) (the order of x1 and x2 has been changed), and let V(r, 8) = VI (I", 8) u V2(I', 8). We say that a domain U belongs to the class U(8) if for any points X1, x2 E U there exists a continuum I' c U such
that x1, x2 E r and VI (I', S) c U. (For comparison we remark that the definition of the class U(5) requires that V(r, 6) c U.) If U E U(1), then U also belongs to the class U(1), and U is either a ball in R" or such that E = R"\U is a closed subset of an (n - 2)-dimensional sphere in R". Obviously, if U E U(8), then also U E U(S). Trotsenko [170] showed that for every 6 E (0, 1 ] U(a) C U [ 1
-
(1 - 8)/(l + a),
.
Let x1, x2 E R" be given points, and let B be a ball in F. Define (xi, x2, B) = sup inf{xl , x2, x, y}. xEB )'qB
If x1 = 0, x2 = oo, B = B(a, r), and x1 V B, then (XI, X2, B) = rllal <- 1.
as is easy to verify directly. If X1 E B, then (xi, x2, B) = oo, and if x2 E B and x1 V B, then (x1, X2, B) = 1. In particular, this makes it clear that if (xl , x2, B) < 1, then x1 and x2 do not belong to the closure of B. Let x1 = 0, x2 = oo and B = B(a, r), where lal > r. The sphere S(0, 1al2 -_71 is orthogonal to the ball B(a, r). Inversion with respect to this sphere transforms B(a, r) into itself, the point 0 into oo, and oo into 0. Since the cross ratio (XI, x2, x, y) is invariant with respect to Mobius transformations, this implies that in this case (xi, x2, B) = (x2, x1, B), and hence (x1, x2, B) = (x2, x1, B) whenever (x1, x2, B) < 1.
Ii. MAPPINGS WITH BOUNDED DISTORTION
280
A family (Br), I E [0, 1), of balls in R" is said to be continuous if in some
spherical metric introduced in in their centers and radii are continuous functions of t. We say that Br - xo as t to if x(t) - xo and r(t) - 0 as to (x(t) is the center of B, and r(t) is its radius). t We introduce some classes of domains in in. Let e > 0, e < 1, be given, and let U C R" be a domain such that R"\U is nonempty and oo e 8U. Then we can make the following definitions: (I) U belongs to Vo(e) if U E U(1 - e). (II) U belongs to the class VI (e) if for any XI, x2 E OU there is a continuous family (B, C U), t E (0, 1), of balls such that (xI, x2, B,) > 1 - e
for all t, B, - x, ast- 0, andB,-.x2ast- 1. (III) U belongs to the class V2(e) if for any x E 9U\{oo) and any r > 0 there exists balls B(y, r(1 - e)) C U and B(z, r(1 - e)) c R"\U such that
Ix-yI=Ix-zI=r.
(IV) U belongs to the class V3(e) if for each point x E OU\{oo} and any
r > 0 there is a hyperplane P = P(x, r) passing through x and such that p(y, P) < er for every y E OU f1 B(x, r). We say that U satisfies condition Vi(c) if it belongs to the class Vi(e). Condition Vo is an interior condition (for interior points of the domain), VI is a one-sided boundary condition, and V2 and V3 are two-sided boundary conditions. We remark that condition V3(e) means that the boundary of the domain is flat to within e. Obviously, if U E V2(e), then R"\U also belongs to V2(e), and if U E V3(e), then 1" \_U E V3(e).
THEOREM 12.29 (TROTSENKO [ 169], [170]). For sufficiently small e > 0
the classes V, (i = 0, 1, 2, 3) are equivalent; namely, for any i. j = 0. 1, 2, 3
there exist an e,j > 0 and a function g,j(e) such that gij(e) - 0 as e - 0,
and for0<e<e,j Vi(e) C Vj[gij(e)1
Trotsenko obtained the following values of the functions gi j and the constants ei j: g01(e) = e, eoi = 1;
823 = 0.1;
g12(e) = 43c, 812 = 10-4;
g32(e) = 2e, e32 = 1;
g23(e) =
8e,
g21(e) = 230VE, 921 = 10-5;
g2o(e) = 690, e20 = 2. 10-6.
The g,j and eij can be estimated for the other values of i and j in terms of these six.
§ 12. SOME FURTHER RESULTS
281
COROLLARY. There exist a number co > 0 and a function µ(e) defined on
[0, e) such that µ(e) - 0 as e -' 0, and if a domain U in R" belongs to the R"\U is nonempty, then U' E U(1- µ(e)). class U(1- e) and the set U' = From the values of the functions g,, and the constants e,, given above it is not hard to get explicit values for the constant co and the function µ(e) in the corollary. As 6 -' I the domains of the class U(8) become arbitrarily close to a sphere. This fact is expressed by the next assertion. THEOREM 12.30 [170]. Let U be a bounded domain in R". Assume that U = So (V), where lp E Mn, and V is a domain of class V1(e). If e < 0.01, then there exist an a E U and an r > 0 such that
B(a, r) C U C B (a, r (1 + 3.3vrc-)).
THEOREM 12.31 ([169], [170]). There exist an co > 0 and a function 6(e), 0 < e < co, such that 6(e) 1 as e -- 0, and for every quasiconformal
mapping f : R -. R with K(f) = 1 + e < I + co the image of a ball is a domain of class U(6(e)).
The proof is obtained by using the special case of Theorem 12.6 corresponding to U = R". is Let U be a domain in R" such that oo E W. A mapping f : R" called a reflection with respect to the boundary of U if f (x) = x for every point x E 8U and f is an involution, i.e. f (f (x)) = x for any x E R". Let f : R" - R" be a reflection with respect to the boundary of U. Then we say that f is a (1 + a)-quasi-isometric reflection if
If(x1)-f(x2)I :5 (1+a)Ix1 -x21 for any x1, x2 0 oo. Obviously, in this case it is also true that
(1 +a)-11x1 -x21 S If(x1)-f(x2)1 for any x1, x2 96 00-
THEOREM 12.32 ([169], [170]). Let U be a domain in the class V3(e),
where 0 < e < 5. 10-5n-2. Then there exists a (1 + a)-quasi-isometric reflection f : R -» R" with respect to the boundary of U, where or < 70n f. THEOREM 12.33 ([169], [170]). Let U be a domain of class U(a, f) and
let oo E U; then there exist an a and a 8 > 0, depending on a, P. and R" with bounded distortion for which n, such that any mapping f : U
II. MAPPINGS WITH BOUNDED DISTORTION
282
f (oo) = oo and K(f) < 1 + 6 has the following property: if the points xo, XI, z1, z2 belong to U and Ixo - x11 > Ixo - z1 , i = 1, 2, then
-cie)
Iz1 - zol l
I+c,t
< If(z1) - f(z2)I < (1 +cie) rizl - zoll
I-c2e
\ IxI - xoI \ IxI - xoI) - If (xi) - f (xo)I Here c1 and c2 are positive constants that depend only on n.
3. Now we give some results of Trotsenko relating to the problem of extending mappings with bounded distortion. THEOREM 12.34 [169]. Let U C R" be a domain in the class II (d ). Then there exist constants to > 0 and c < oo, depending only on 6 and n, such that
for every mapping f : U - R' with bounded distortion K(f) = 1 +e < 1 +eo there is a quasiconformal mapping F: R RR with F(x) = f (x) for all
xEUandK(F)
The following theorems give a complete characterization of the image of a line under a quasiconformal mapping of R" with distortion coefficient close to 1. Let R = RU{oo}, and let y: R -+ Rt be a continuous curve with y(oo) _ 00.
If a and b are points in R", then [a, b] denotes the segment with endpoints a and b; that is, the set of points x of the form x = Aa + (1 - A)b, where 0 < A < 1. We say that a curve y satisfies condition TI (e), where e = const > 0, if P(Y(t), [Y(t1),Y(t2)]) 5 elY(tI) - Y(t2)1
for every t E [t1, t2] for any ti, t2 E R with t1 < t2. The curve y is said to satisfy condition T2(e) if (1 + e2)IY(t)
- I(v)I <- IY(t) - Y(u)I + IY(u) - Y(v)I for any t,u,v ER with t< u
THEOREM 12.35 ([169], [170]). There exist constants to > 0 and co < oo. depending only on n > 3, such that for any quasiconformal mapping
f : R R with f (oo) = oo and K(f) = 1 + e < 1 + co, the image f (R) of a line satisfies conditions TI (coe) and T2(coe).
This theorem can be established by a simple application of Theorem 12.33. It is considerably harder to prove the converse. Namely, we have the following assertion.
§ 12. SOME FURTHER RESULTS
283
THEOREM 12.36 ([169], [170]). There exist numbers co > 0 and c < 00, depending only on n > 2. such that for any curve y in in satisfying condition TI (e), where e < co, it is possible to construct a quasiconformal mapping f : R" R" for which
1) f (R) = y(R);
2) K(f) < l + ce.
§12.6. The general concept of stability classes.
1. Here we describe a certain general notion of stability for classes of mappings by developing on a conceptual level the results in § 12.2 on stability in Liouville's theorem on conformal mappings. This concept was worked out by Kopylov in the series of papers [65], [69]-[72], [29], and [30]. These papers contain a new direction in the metric theory of spatial mappings. (For simplicity some of the special questions in the theory discussed here are presented in a less general form than was done by Kopylov.) Let n, m > 1 be integers, and let 0 be a set of mappings defined on open subsets of R" and taking values in R'". It will be assumed that the class 0 satisfies some of the conditions below (the numbering differs from that in the original papers of Kopylov).
A homothety of R" is defined to be a transformation f : R" - R" of the
form f(x)=2x+a,where A>0,AER, andaER". (KI ). The class 0 is invariant under homothety transformations in the spaces R" and R"'; that is, for every f E 0 and homotheties q : R" R" R"' the mapping Vi o f o 9 belongs to 0. and w : R" (K2). Every mapping f E 0 is continuous, and for any U C R" every set of functions G c 0 such that If (x) 1 < M = const < oo for all x E U is uniformly equicontinuous on every compact subset of U. (K3). The class 0 is closed with respect to locally uniform convergence;
that is, if for an f : U -+ R'" there exists a sequence (f,,: U -, R"'), v = 1, 2, ... , of functions in 0 such that f, f uniformly on every compact subset of U as v -' oo, then f E 0. (K4).
If f : U -. R"' is in 0, then for every open set V C U the
restriction of f to V belongs to 0. If f : U R'" is such that every point x E U has a neighborhood U. C U such that the restriction of f to U., is in 0, then f is in 6. If the class 0 of functions satisfies all these conditions (KI)-(K4), then we call it K-normal. In the sense of restrictions on the structure of the class 0, the most stringent of these conditions is (KI ). This is clear, for example,
from the fact that if 0 is a K-normal class of mappings, then every f E 0 satisfies a Holder condition locally. (This assertion follows from a general
284
11. MAPPINGS WITH BOUNDED DISTORTION
result of Kopylov [69], but is easy to establish directly.) At the same time, it is easy to construct classes of mappings satisfying conditions (K2), (K3), and (K4) such that the Holder condition is not satisfied in general.
Basic in the Kopylov theory is the class X,," of holomorphic mappings of the complex space C" into the complex space C'" (here C" is identified in the natural way with R2" ). This class is obviously K-normal. Another example of a K-normal class is the collection M. of all Mobius mappings f : U -a R", where U is an arbitrary open domain in R".
The collection of all identically constant mappings is clearly also a Knormal class. 2. In broad outline, the basic content of the Kopylov theory of stability
of classes of mappings is as follows. Assume that 0 is a given class of mappings defined on open domains of R" and taking values in R'". Certain
other classes are constructed from 0 by a kind of perturbation. For this purpose Kopylov introduces functionals characterizing how far a given mapping f : U - R'" (U an open domain in R") is from the given class 0. Functionals of two types are introduced. A functional of the first type characterizes the closeness of the given mapping f to the class 0 globally. A functional of the second type characterizes the deviation of f from 0 in an infinitesimal neighborhood of an arbitrary point of the domain. The class 0 is stable if the mapping is close to 0 globally whenever the value of the local closeness functional is small (that is, the value of the functional of the first type is also small for f).
First, there arises the problem of finding classes of mappings stable in the Kopylov sense, and, second, for classes satisfying the stability condition there arises the problem of describing the classes of mappings close to it. Kopylov has obtained far-reaching results in the solution of each of these problems. Some of them are formulated below.
We note at once that the whole theory proves to be substantive only if the class 0 satisfies definite conditions. We assume that it is K-normal. Accordingly, let 0 be a K-normal class of mappings defined on domains in R" and taking values in R' .
Assume that f : U - R'" is a given continuous function, where U is a domain in R", and let p E (0, 1] be arbitrary. Take a ball B = B(x, r) c U, and denote by Bo the ball B(x, pr) concentric with it and by A(f. B) the diameter of the set f (B). We define a certain number p B(f, 0) from B and f. If A(f B) is either 0 or oo, then let D,B(f 0) = 0. But if 0 < A(f, B) < oo, we first put f; = (1 /A(f, B)) f, and then define P.e(f 18)
§12. SOME FURTHER RESULTS
285
to be the infimum of the quantity IIfB - gII c(B,,) =
sup I fB (t) - 9(0 1
I(-xl<pr
over the set of all mappings g E 0 defined in B. Since for every vector h E R'" the mapping g = h belongs to 0, it is clear that always 4p.B(f 0) < 1 (to see this it suffices to take g - f; (x), where x is the center of B). Denote by cp(f, 0) the supremum of p,B(f, 0) over the set of all balls B c U. This the first closeness functional introduced by Kopylov. Another closeness functional is defined as follows. For every ball B c U
the quantity p.B(f 0), as a function of p, is nondecreasing. Let 1
B(f 0) = f
.B(f ) dp
and, further,
(f ) = SUP B(f BCU
For this point all the constructions are natural enough. To some extent the appearance of the integral may seem to have little motivation. The
introduction of B(f, 0), and with it (f 0), was dictated by the desire to have a characteristic of the deviation of f from 0 independent of the choice of a concrete value of p. On the other hand, the functionals (f, 0) and p(f, 0), where 0 < p < 1, are equivalent in the sense that for every e > 0 there is a 6 > 0 such that if (f, 0) < 6, then p(f 0) < e; conversely, if p(f, 0) < 6, then (f, 0) < e. Intuitively, if one of the functionals 4(f, 0) and 4p(f 0) tends to zero, then so does the other. The functional 0) is invariant with respect to homothety transformations; that is, if (p: R" - R" and w : R'" - R" are homotheties, then 4p(W o f o cp, 0) = 4p(f. 0)
for every f : U - Rm. This implies that the functional 4(., 0) has the same property. If f : U - R"' is a continuous function and p(f. 0) = 0 for some p E (0, 1], then f belongs to the class 0. If 4p(f. 0) (0 < p < 1) is small, then f is close to the class 0 in the sense of the uniform norm. More precisely, the following assertion holds. Let U c R" be a given domain, A c U a compact set, and V an open domain strictly inside U and containing A. Then for every c > 0 there is a 6 > 0 such that if p(f, 0) < & for an f : U - R1, then there is a g: V -. R'" in 0 such that If (x) - g(x)I < e for all x E A. The number S depends on c, p, the ratio dist(A, R"\V )/d (A), and the given class 0.
II. MAPPINGS WITH BOUNDED DISTORTION
286
An analogous assertion remains true if (f 0) is considered instead of
P(f 0)
-
The functional (f 0) is the main global closeness functional considered here.
We now define a certain functional characterizing the closeness of a mapping to a given class 0 locally. Thus, let 0 be an arbitrary K-normal class of mappings of domains in R" into R", and let f : U - Ilm be a given continuous mapping. Then for any point x E U let urn e(x.r)(f 0) and
(f 0) =
xEU
f 0)
(our notation is somewhat different from Kopylov's). Obviously,
=r(f 0):5 (f 0) for every f : U Rm. Denote by W' (n, m) the set of all mappings f : U -+ Rm, where U is an arbitrary domain in R", that belong to one of the classes WP 10C(U) for some p > n.
0 as a - 0, and if f : U - Cm is a mapping
THEOREM 12.37. There exist a number co > 0 and a function a : [0, co)
[0, oo) such that a(e)
in W' (2n, 2m) on an open domain U in C" and E(f, X,m) < e, where 0 < e < co, then
(f.m) < a(e).
This theorem shows that the class of holomorphic mapping is stable in the sense that smallness for a given mapping f of the functional measuring local closeness to the class Zn.m enables us to conclude that the global functional is also small for f. The next problem arising here is to give a complete description of the class of mappings f E W' (2n, 2m) for which E(f, A,' m) is small. For every mapping f : U - Cm in W' (2n, 2m), where U is a domain in C", the generalized derivatives (8f/&;)(z) and (8f/82j)(z), i = l , ... , n are defined. Let f I , ... , fm be the components of f . A multidimensional system of Beltrami equations is defined to be a system of equations of the form n m P
_
QkP(Z) (1=I P=I
.
(12.26)
§ 12. SOME FURTHER RESULTS
287
The collections of functions Qk;(z), i, p = 1, ... , m, k, a= 1, ... , n, will be denoted by the symbol Q(z), and system (12.26) will be written briefly as
= Q(z) L a For an arbitrary n x m matrix Z = (Z(,'k), p = 1, ... , m, a =
(12.27) 1,
... , n, we
set m IZI =
n
E E IZPI2 P=1
=1
and let Q(z)Z denote the n x m matrix W = (Wk) with n
nl
WR = EEQk,,(z)Z', R= 1
i=1 2 .
,
. .
,m,k = 1,2,.. ,n.
v= 1
The supremum of IQ(z)ZI over the set of all Z with IZI < I is denoted by IIQ(z)II. System (12.27) is elliptic if and only if IIQI100 = sup IIQ(z)II < 1.
THEOREM 12.38. Suppose that U is an open domain in Cn and f : U
C'" a mapping of class W1(2n,2m). IfE(f,,,,) <e < 1/16vi, then f is a solution of some system of the form (12.27) with IIQIIoo 5 16,/ne < 1.
Kopylov also considered other closeness functionals, and established theorems analogous to Theorem 12.37 for other classes of mappings. In particular, a certain analogue of Theorem 12.37 holds for MObius mappings, which is a corollary to the stability theorem given in § 12.2. Kopylov (together with Dairbekov) also investigated the questions of stability of solutions of systems of first-order elliptic equations (see [29] and [30]). Use of the closeness functionals introduced by Kopylov has also led to the determination of a characterization of the boundary values of mappings with bounded distortion on a ball that have distortion coefficient close to 1 (see [65] and [72]). §12.7. A characterization of quasiconformal mappings as mappings preserving the space W,1. Let U and V be open sets in R", and f : U -' V a quasiconformal mapping. Then for every function u E W (V) the function u o f also belongs to the class W (U). We thus get a linear mapping
f': uEW,,'(V)-uof EW,'(U). This mapping is also bounded. The idea naturally arises of trying to characterize all the linear mappings of W (V) into W (U) that can be obtained
11. MAPPINGS WITH BOUNDED DISTORTION
288
in this way. It may be supposed that the mappings having this property should be sought among the mappings that preserve some additional structure in W. I. This idea was realized in [ 107], [88], and [86] by a generalization of the concept of a Royden algebra. Here only continuous functions
in the class W are considered, so that we are concerned with a certain proper subset of the space W,,'. In [177] Vodop'yanov and Gol'dshtein obtained the required characterization by using the order structure in W . We present a formulation of their result. Let U be an arbitrary open domain in R". Instead of W (U) it is convenient to work with the space L,I, (U); that is, with the set W,,' (U) of functions endowed with the seminorm IIUIIL.'(u)
=
(LI(x)'dxY'
Let U and V be open domains in R". A linear operator A: L,I,(V) L,I,(U) is called a structural isomorphism of the spaces L,I,(V) and L (U) if A is a bijective mapping, and the following conditions hold: (a) A* (v) > 0 if and only if v > 0. (fi) If v - 1, then A(v) is also the function identically equal to 1.
(y) If v", - 0 almost everywhere as m -' oo, then A(v,,,) - 0 almost everywhere.
(6) If (wm E L,I,(U)), m = 1, 2,..., tends to zero almost everywhere in U, then A- I (wm) tends to zero almost everywhere in V. Two domains U, and U2 are said to be (1, n)-equivalent if the mappings 0j: W (U, U U2) - W (U1), where 0; (v) = v I u,, i = 1, 2, are isometric isomorphisms. THEOREM 12.39. For every structural isomorphism A : L,I, (V) - L,I, (U)
there exists a unique quasiconformal homeomorphism of U into R" that satisfies the following conditions. 1) The domain cp(U) is (1, n)-equivalent to V.
2) (Af)(x) = v[ip(x)] almost everywhere in U for every function v E L,I,(V).
This theorem is only one of a cycle of investigations carried out by Gol'dshtein and Vodop'yanov. The reader can obtain more detailed information about this work in [48], where, in particular, there is a detailed bibliography.
CHAPTER III
Some Results in the Theory of Functions of a Real Variable and the Theory of Partial Differential Equations §1.
Functions with bounded mean oscillation
Let a number p > 1 be given. A function f : U -' Rd, where U is an open set in R", is called a function with bounded mean oscillation in the L p-sense if there exists a number to > 0 such that for every cube Q C U (only cubes with faces parallel to the coordinate planes are considered) there is a vector CQ E Rd such that If(x)-cQIpdx
1
IQI
Q
The smallest number co satisfying the above condition is called the mean
oscillation of f in L p. Intuitively, the condition that a function f has bounded mean oscillation means that it can be approximated by a constant on every cube Q in a certain sense, uniformly with respect to the choice of cube. The concept of a function with bounded mean oscillation was introduced by John and Nirenberg [63]. Our immediate problem is to derive the estimates they obtained for the measures of Lebesgue sets of an arbitrary function with bounded mean oscillation. These estimates will be used below in 92 in proving the Serrin-Moser theorem in connection with the Harnack inequality for elliptic equations. THEOREM 1.1. Let U C R" be an open set, and f : U - Rd a function with bounded mean oscillation equal to cc. Let Q be an arbitrary cube contained in U. Denote by EQ(t) the set of points x E Q such that If (x) - cQI > 1. Then for every t > co the measure of EQ(t) admits the estimate IEQ(t)I <- [(1/cot') jQ
If(x) - cQIpdx] exp[y(1- t/w)], 289
(1.1)
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
290
where y > 0 is the constant y = 1/(1
(1.2)
+2'lPe).
PROOF. By a condition of the theorem, for every cube Q C U
w"IQl > which implies that t°IEQ(t)I < fQ If(x) - cQIpdx.
(1.3)
The smallest number L such that IEQ(t)I 5 (Llwp) - fQ
If(x) -
cQIpdx
for every cube Q C U is denoted by F(t). It is clear from (1.3) that
F(t)_<(w/t)" for allt>0. Take an arbitrary cube Qo c U, and a number k > 1. (The specific value of k will be chosen later.) For brevity we write cQo = co. By assumption, If(x) - colpdx< w"IQoI < kwpIQoI. JQ
Let us partition Qo into 2" equal cubes by planes parallel to the coord nate planes, and single out those of the cubes for which the inequality If(x)-colpdx
(1.4)
fails to hold. Each of the remaining cubes is again divided into 2" equal parts, and again those for which (1.4) fails to hold are singled out, and so on. Continuing this process, we get a finite or infinite sequence of cubes Q", for which
f
If(x) - colpdx > kco"IQml.
(1.5)
QM
The cube Qm is obtained by subdivision of some cube Q = Q; satisfying (1.4). We have that
f
If(x)-colpdx< f If(x)-colpdx
Qm
Qm
Moreover, I Q; I = 2" I Qm I and, consequently,
f If (x)-colpdx<2"kcv"IQml Qm
(1.6)
§ 1.
FUNCTIONS WITH BOUNDED MEAN OSCILLATION
291
for each m. The sequence of cubes Q. can turn out to be empty, of course. Let H = Um Qm. For every point x E Qo\H there exists a sequence of cubes containing x such that the diameter of Q tends to zero as v oo, and (1.4) holds for each v. As v -- oo, 1
IQv
f If(x) - coIPdx -- If (x) -coIP I
for almost all x E Qo\H, and so for almost all x E Qo\H If (x) - coIP < k'/Pw.
(1.7)
Choose a number t > k'/Pw arbitrarily, and consider the set EQ0(t). Since (1.7) holds almost everywhere on Qo\H, EQ(t) is contained in H to within points forming a set of measure zero. If IHI = 0, then IEQ0(t)l = 0, and (1.1) holds for the given t. Assume that IHI > 0. We have EQo(t)I = > IEQ0(t) fl QmI m
Consider the set EQ0(t) n Q,". According to the definition of a function with bounded mean oscillation, to the cube Q.. there corresponds a vector (denote it by cm) such that
I If(x) - CmIPdX < CUPIQmI For every point x E EQ0(t) f1 Q.. we have that If (x) - col > I > k'1 cv. Also,
If(x) -COI <- If(x) - CmI + icm -Col.
(1.8)
Let us estimate I cm - col. Considering (1.6), we get
ICmcol
(
Icm-CoIPdX1 I /P J
< (jjf 1
JJJ
11/P
mICm-f(X)IPdx)
1
-ColPdx\\)
I/P
+(Qm11(x)
<w+2"/Pk'/Pw.
and, thus, Icm - cQI < (2"/Pk'/P + 1)w.
By (1.8), this entails that for all x E EQ0(t) f1 Qm
If(x) - CmI 2: If(x) - Col - (2"/Pk'/P+ 1)w> t - h.
(1.9)
292
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
where
h = (2"/Pk'/P + 1)w.
(1.10)
This means that if x E EQ,(t) n Qm, then x E EQ,,,(t - h) for t > h. Consequently, IEQ0(t) n Qm1 <
I
Q.
If(x)-cmlpdx< IQmIF(t-h).
Summing over m, we get IEQO(t)I 5 IHJF(t - h).
(1.11)
Inequality (1.5) implies the following estimate for the measure of H:
I HI < (1/kwp)> f If(x) - colpdx m
Qm
<(1/k(,,jp) 40 If(x)-cQo$ dx,
which, by (1.11), gives us
IEQ0(t)I <-[F(t-h)/kwp]
f Jf(x)-colpdx.
(1.12)
Qo
The cube Qo c U was taken arbitrarily, and it thus follows from (1.12) that
F(t) < (1/k)F(t - h).
(1.13)
The functional equation 9(1) = (1/k)q (t - h) has a special solution 9(t) = where 2 = (Ink)/h. If the inequality F(t) < V(t) holds on [w, co + h], then it is not hard to see that F(t) < ?(t) for all t > co. This reduces the matter to choosing C such that F(t) < Ce-z' for all t E [co, co + h]. We have that F(t) _< (w/t)P for all t. The function g(t) = wPeu/tP has the unique extremum point t = p/A on (0, oo), and it is a minimum point. This implies that g(t) takes its largest value in each interval [a, b] c R at an endpoint. Comparing g(w) and g(w + h), we get
that g(w) > g(w + h). Thus, wPe'/tP < g(w) = e l and F(t) < eA(I-') for all 12! co.
The estimate obtained is better, the larger A is. We have that A = (Ink)/w[2"/PkI/P + 1]. The function of k on the right-hand side attains a maximum on [ 1, oo) when k is the root of the equation 1 + 1 /2""Pk = In k. Approximately, k = e + 2-11/p +.... The estimate in the theorem is obtained if we take k = e.
§1. FUNCTIONS WITH BOUNDED MEAN OSCILLATION
293
COROLLARY. Let f : U -' Rd (U c R" an open set) be a function with bounded mean oscillation in L p equal to w. Then for any a < y/&) the function elf (x)1 is integrable on every cube Q c U, and for each such cube JQ
exp[aIf (x) - cQI ] dx +(ae'l[y-aw]wp-')fQ
If(x)-cQIpdx.
(1.14)
The function f is integrable to any power q > p. For any q, and 42 with
q1>q2>P.
If
If
(x)-cQlQ,dx
JQ
(x)-cQIQ2dx,
(1.15)
where C > 0 is a constant depending only on q1, q2, and p.
PROOF. We first make some simple remarks about the Lebesgue integral. Let u(x) be a nonnegative function integrable in the cube Q, and let A(t) (t > 0) be the Lebesgue set {x E QIu(x) > t}. Then 00
u(x) dx =
fIA(1)I dt.
The condition e' "(x) > t, where t > 0, holds for all x E Q when 0 < t < 1, and is equivalent to u(x) > (In t)/a when t > 1. This allows us to conclude that the Lebesgue set {xIea"(x) > t} of the function a"" coincides with Q for 0 < t < 1 and with A[(In 1)/a] for t > 1. From this we get
f
e«"(x)dx
Q
f f
00
= IQI +
IA[(1/a)Int]Idt
I
00
= IQI +a
IA(t)Ie"'dt.
(1.16)
0
Further, the Lebesgue set {xl[u(x)]p > t} coincides with A(t'1v), which leads to the conclusion that 00
00
/ [u(x)]pdx =
JQ
f
IA(t1p)Idt = p
IA(u)Iup-'du.
(1.17)
fo
Let u(x) = If (x) - cQI and 1p = fQ[u(x)]p dx. By (1.1),
I A(t)I < (1p/wp) exp[y(1 - t/w)].
Applying this estimate to the right-hand side of (1.16), we get that expa[u(x)] is integrable in Q if a < y/w. Further
[expalf(x) -cQI dx < IQI + Q
,
(Y-aw)wp-11p
M. FUNCTIONS OF A REAL VARIABLE AND PDE'S
294
and this establishes (1.14). Let q > p be given. Using (1.17), we find that 00
f f(x) - CQ dx = qf IEQ(t)Itq-'dt = q f IEQ(t)ltq-'dt +q f IEQ(t)Itq-Idt. 00 w
o
The first integral on the right-hand side can be estimated as follows. For 0 < I < co we have that tq-I < Wq-ptp-I, which implies that q
f
0IEQ(t)Itq-'dt
f
IEQ(t)Itp-'dt
0 oo
<
qWq-p f
I EQ(t)I tp-'dt =
(q/p)Wq-pJp
0
By the estimate in the theorem, q
f
00
I EQ(t)I
tq-Idt
< (qJp/Wp) f
tq-I
exp[y(l - t/W)]dt.
The change t/W = u of the variable of integration gives us that q
f
oo
m
I EQ(t)I tq-I dt <
qWq-piper
f.
tq-1e-Y'dt.
1
From this,
If (x) - CQlg dx < C(q)Wq-p1p.
(1.18)
This is the second inequality of the corollary for the case when q2 = p. Since f is a function with bounded mean oscillation, Ip < WPIQI. We get from this that If(x) - CQlgdx < CgcvglQl JQ
for every q > p for any cube Q c U. This means that f is a function with bounded mean oscillation in the Lq-sense for any q > p. Further, the mean oscillation off in Lg is equal to W i = CW, where C = const. Setting p = Q2 and q = qI in the above arguments, we get the desired inequality fQ If(x) - CQIq, dx <
The corollary is proved.
CWq,-qz fQ if (X)
- CQIg=dx.
§2. HARNACK'S INEQUALITY
295
§2. Harnack's inequality for quasilinear elliptic equations
2.1. Preliminary remarks. Let U be an open set in R", and A(x, q) an R"-valued function such that for almost all x E U the vector A(x, q) is defined for any vector q E R", and the following conditions hold: a) For every measurable function q: U -' R" the function x -' A[x, q(x)] is measurable.
b) There exist constants a,, a2 > 0 and a number p > 1 such that for almost all x E U and any q E R" IA(x,q)I
(2.1)
<-a,IgIP-I.
(2.2)
(q. A(x, q)) ? a2I qI P.
Let u be an arbitrary vector-valued function of class Wp10C(U). By condition a), the function x -+ A[x, u'(x)] is measurable. In view of (2.1) it admits the estimate I A[x, u'(x)]I S ai I
u'(x)IP-
I
and, consequently, the function A[x, u'(x)] is locally integrable to the power p' = pl (p - 1) in U. The function u is called a generalized solution of the equation div A[x, u'(x)] = 0
(2.3)
if for every function (p E C0(U)
f(co'(x),A[x, u'(x)])dx = 0.
(2.4)
Note that if u is a function of class WPII0C(U), then, since x A[x, u'(x)] is locally integrable to the power p', it is easy to get from (2.4) by passing to the limit that (2.4) holds also for every function ' E WP (U) with compact support in U.
Let F(x, q) be a normal kernel in the sense of §5 in Chapter II, i.e., a function satisfying all the conditions A-E in §5.1. Then the vector-valued function A(x, q) = Fq(x, q) =
/ OF
(x, q), Oft (x, q),.... OOq, F (x, q)
is defined. It follows from A-E in §5.1 of Chapter II that the function
satisfies conditions a) and b) just introduced. A function u E WPIIoc is stationary for the functional IF if and only if u is a generalized solution of (2.3), where A = Fq. Our purpose is to prove Harnack's inequality (Theorem 5.7 in §5 of Chapter II). It gives us as a corollary that generalized solutions of (2.3)
296
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
have the Holder property. The following remark is useful below. Let
f : R" - R" be a given similarity transformation. Then l = f'(x) is a general orthogonal transformation. Let k = 111. Assume that A(x, q) is a given vector-valued function defined for almost all x E U and any q in
R" and satisfying conditions a) and b). Let V = f (U). For X E V and q E R" let
A(x, q) = (1/0- I )A[f (x), I(q)] It is easy to verify that A also satisfies a) and b), and with the same values of the constants a1 and a2 as A. If u(x) is a generalized solution of the equation div A[x, u'(x)] = 0 in U, then the function v(x) = u[f (x)] is a generalized solution of the equation div A[x, v'(x)] = 0. There are two competing methods for investigating the question of an estimate for the moduli of continuity of solutions of an equation with the form (2.3) and of certain more general equations. DeGiorgi's wellknown article [31] is fundamental for one of them. The second method is based on an idea of Moser [103], [104]. A far-reaching development of DeGiorgi's method has been given by Ladyzhenskaya and her students (see, for example, the monographs [78] and [77]). Moser's method in its original form was applied to the investigation of an equation of the form i=1
a (aj(x)!!_(x)) I I=1
= 0.
(2.5)
An extension of Moser's method to the case of quasilinear equations of the form divA(x, u, u') = 0 was given by Serrin in [155] and [156]. The arguments presented below mainly follow Moser's method. The results obtained here make up a special case of results of Serrin. We do not touch on the general case, since it is not needed for the purposes of this book. On the other hand, the arguments simplify considerably in the special case we need. We remark that, in contrast to Moser and Serrin, we do not use the concept of a subsolution of equation (2.3). 2.2. Main inequalities. The proof that generalized solutions of (2.3) are continuous and the proof of Harnack's inequality for these equations are based on certain integral inequalities, the derivation of which is our immediate task. Recall that a function F: R - R is said to be piecewise smooth if there exists a finite set E c R such that F is continuously differentiable at each point not in E, and for all u E E the finite limits F'(u - 0) and F'(u + 0) exist. If F is a piecewise smooth function, then at each point u E R the derivatives Ft(u) and FR(u) exist and are finite. If u(x) is a function of class Wo10C(U) and F is a piecewise smooth function such that the derivative FL(u) is bounded in R, then, by Lemma
§2. HARNACK'S INEQUALITY
297
5.8 in Chapter II, the function v = F o u also belongs to WP I0c. Further, v'(x) = FL[u(x)]u'(x) for almost all x E U. The right-hand derivative can be taken here instead of the left-hand derivative. LEMMA 2.1. Suppose that F is a piecewise smooth nondecreasing func-
tion on R such that if F(u) 96 0, then FL(u) 96 0 and FR(u) 0 0. Assume that FL(u) is a bounded function on R. Define the auxiliary function 4) by setting c(u) = 0 if F(u) = 0, and (D(u) = [F(u)]P/[FL(u)]P-' if F(u) 76 0. Let M be an arbitrary compact set in R", let G D M be an open set lying strictly inside U, and let 6 = 6(M, 8G) be the infimum of the distances of the points in M from the boundary of G. Then every generalized solution u(x) of (2.3) on U satisfies
f F'[u(x)]l u'(x)l p dx <
S'' JCD[u(x)]dx.
(2.6)
M
where the constant C, depends only on n, p, and the constants al and a2 in conditions a) and b) of §2.1. PROOF. Let u(x) be an arbitrary generalized solution of (2.3) on U, and
let the function F: R - R and the sets M and G satisfy all the conditions of the lemma. The function v = F o u belongs to the class WP ia(U), and v'(x) = FL[u(x)]u'(x) for almost all x E U. We construct an auxiliary function r. Let ri > 0 be any number such that 3'i < 8. Let A,, = U,,(A) be the n-neighborhood of a set A. Define
t(x) = [I - p(x, M,,)/(J - 2ri)]+.
Obviously, t(x) = I for X E M,,, and t(x) = 0 if p(x, M,,) > 8 - 2q. For any x, , X2 E R" we have that
IP(xi,11'jn)-p(x2,Mn)I <- Ix, -x21, which leads us to conclude that It(x0 - t(x2)1 <- Ix, - x21/(8 - 2il) for any x,, x2 E R". Applying to the function t the operation of averaging by means of some nonnegative kernel with averaging parameter h < 4, we
get a function r such that r(x) = 1 for x E M, r(x) = 0 for x 0 G, and lr'(x)I < l /(J - 21) for all x. For every function Sp of class WP (U) with compact support in U,
f(co1(x) ,A[x, u'(x)])dx = 0. u
(2.7)
298
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
Here let V(x) = [T(x)]PF[u(x)]. The given function rp is in W,(U), and its support is in the open set G. Further, (o'(x) = p[T(x)]p-1 r'(x)F[u(x)] + [T(x)]PF'[u(x)]u'(x). (Here and below, we write F' instead of FL.) For A = A(x, u') (rp', A) = TPF'(u)(u', A) + By (2.2) and (2.1),
PTP-1 F(u)(t', A).
(u', A(x, u')) > a2Iu'IP, I (T'. A(x, u'))I <- a1Ir'I Iu'IP-1.
which leads us to conclude that (2.8)
(P', A) > a2TPF'(u)I u'I P - a1PTP-' I F(u)I Iu'IP-1IT'I.
Let x be such that F[u(x)] # 0. Then F'[u(x)] 0 0. We have Young's inequality XY < XP/p + Y(P-1)/P(p - 1)/p, where X, Y > 0. Let x be such that F'[u(x)] 0 0. Setting l[F'(u)](P-1)/P.
X = tir'I I F(u)I Y=
I
[F'(u)](P-I )1PIu1IP-I tP-1
where t > 0 will be chosen later, we get that a1 pTP-IIF(u)I
IVuIp-1IVTI
< a1 1Pl t'I PI F(u)I P/(F'(u))P-1
+ al (P - 1)[F'(u)]I Substituting this into (2.8), we arrive at
uIPTPt(P-Il/P
1)1t(P-1)1P]F'(u)I u'IPTP - a,tPIT'IP(I>'[u(x)]. (2.9) (q', A) > [a2 - a1(P The last inequality was derived under the assumption that F[u(x)] # 0. It is not hard to see that it remains true also in the case when F[u(x)] = 0. The quantity t is determined from the condition 1)/r(P-1)/P = a2/2. a2 - a1(P -
Integrating (2.9) termwise, with (2.7) in view, we get that 0 > (a2/2)
f
dx.
F'(u)I u'IPTP dx - a1tP J
U
U
Since T(x) = 1 for x E M and IT'(x)I < 1/(8 - 2q), this gives us that JM
[F'(u)]Iu'IPdx <
(a C2i1)P
[D[u(x)J c dx,
where CI depends only on a1, a2, and p. The number q > 0 is arbitrary. Passing to the limit as >' -' 0 in the displayed inequality, we arrive at the desired inequality. The lemma is proved.
§2. HARNACK'S INEQUALITY
299
COROLLARY I. Suppose that u(x) is a generalized solution of (2.3) on U. M is a compact set, and G D M is an open set lying strictly inside U, and let & = 8(M, 6G). Then for every m > p fM 1 u(x)I m-PI u'(x)1 P dx <- [(m - p + 1)a-]p
fIu(x)Imdx,
PROOF. Let v > 0 be an arbitrary integer, and define
for 0 < u < v,
(2.10) um-P+I
(m - p + 1)vm-Pu - (m - p)vm-P+I for u > v
and The function FF, is piecewise smooth, and its (u)]P-I. Then derivative is bounded. Let
um/(m - p + 1)P-I for 0 < u < v,
vm-P(m - p + 1)(u - yv)P, Setting F = F and
where y = (m - p)/(m - p + 1), and
0 = 0 in (2.6), we get that fM FY[u(x)]I u'(x)IP dx <
f
cD,[u(x)]dx.
(2.11)
As v - oo we have that FF[u'(x)]I u'(x)I -- (m - p + 1)I u(x)I m-PI u'(x)I P
for all x. From this, (m - p + 1)
Iu(x)Im-PIu'(x)IPdx JM
f
lim M F;[u(x)]Iu'(x)I"dx. y_.Qp
(2.12)
Inequality (2.10) is true if its right-hand side is equal to oo. Assume that the integral on the right-hand side of (2.10) is finite. As v -. oo we have that 40.,[u(x)] - l u(x)I
m/(m
-p+
1)P-I
and 0 <-
Iu(x)Im(m - p + 1)
for each v for all x E U. In this case the functions fi, o u are majorized by an integrable function, and thus, by Lebesgue's theorem, 1c(D,,[u(x)]dx
(m- p+ 1)P-I
fGIu'(x)Imdx.
(2.13)
Passing to the limit in (2.11) and considering (2.12) and (2.13), we get (2.10).
300
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
COROLLARY 2. Let u(x) be a generalized solution of (2.3) on an open set U. Assume that u(x) is nonnegative. Then, for any sets M and G such that M c G, M is compact, G is open, and G lies strictly inside U, and for any numbers m o p - 1 and h > 0 the inequality fM[u(x) + h]'"-PIu'(x)I P dx :5 alm
-p + 1I j[u(x) +
h]"' dx
(2.14)
holds, where G, and 6 have the same meaning as in the lemma.
PROOF. Three cases are possible: 1) m < p - 1; 2) p - 1 < m < p; 3) m > p. In case 1) inequality (2.14) is obtained at once if we substitute in (2.6) the function F defined as follows: F(u) = u - h'"-P+1 for u < 0, and F(u) = -(u + h)"'-p+1 for u > 0. This function F clearly satisfies all the conditions of the lemma. In the case p - 1 < m < p we define F(u) as follows: F(u) = u + h'"-P+1 for u < 0, and F(u) = (u + h)'"-P+I for u > 0. Clearly F satisfies all the conditions of the lemma in this case. In the case m > p inequality (2.14) is a consequence of (2.10), since u(x) is a generalized solution of (2.3), then u(x) + h is also a generalized solution of this equation for any h E R. We remark that DeGiorgi's method for investigating generalized solutions of (2.3) is based on the use of an inequality which can be obtained from (2.6) by setting F(u) = (u - k) + or F(u) _ -(u - k) -, where k is an arbitrary real number. 2.3. Consequences of the integral inequalities in §2.2. Let a E R" be an arbitrary point. For r > 0 denote the open cube Q(a, r) by Q,. For an arbitrary measurable set A with 0 < Al I< oo and any function f integrable on A let
ff(x)dx = (1/IAI) f f(x)dx. A
A
We mention another corollary to the Sobolev imbedding theorem. Let f be a function of class W, , where p:5 n, in the cube Q, = Q(a, r) c R". Let s be such that 1 < s < n/(n - p). Then, by Lemma 2.1 in Chapter I, IIf - J IIPS.Q. < CrII If'I IIP.Q, Here
IIIII'.A = ( IAI
f If
(x)I' dx)
1/r
t > 1. We have that IIfIIPs.Q, <-
1 If -f1ip,.Q, +If1 :5 1if - fIIPsQ + IIfIIPQ,'
(2.15)
§2. HARNACK'S INEQUALITY
301
From this, by (2.15), IIfIIPS.Q, < CrIlf'IIP.Q. + IIfIIP.Q,.
Finally, using the inequality (a + b)P <_ 2P-I (aP + bP) for a, b > 0 and p > 1, we get that (IIfIIPs.Q,)P < C(rPIIf'II p.Q. + IIfIIp.Q,).
C
If(x)IPsdx)
If'(x)IPdx+ f If(x)IPdx).
Q,
(2.16)
Q,
Q,
LEMMA 2.2. Let v > 0 be a function of class Wp on the cube Qro = Q(a, ro) in R". Assume that there exist constants C1 > 0, C1 < oo, and I E It I # p, such that for any r1 and r2 with 0 < r2 < r1 < ro J [v(x)]m-PI v'(x)I P dx < Q,,
[vxm dx.
(Im_II(ri -r2)) P JQ()1 C1
(2.17)
,
T hen v is bounded in any cube Qr, where 0 < r < ro. Further,
esssupv(x) < C2yr XEG
ro
(2.18)
1(
(JQ,0IV(X)JP
I /P
where C2 depends only on the constants C1 and I in (2.17), and yo(u) _ u)s/(s-1), where y = (n + p)/p. uY/(ln PROOF. Let r1 and r2 be arbitrary numbers with 0 < r1 < r2 < ro. By the condition of the lemma, (2.17) holds for every m > p. Let m > p be such that for any m the right-hand side of (2.17) is finite. Let w(x) = [v(x)]'"/P. We show that w E W, (U), and
w'(x) = To do this we introduce the auxiliary function Wk(X) = Fk[v(x)], where (m/P)[v(x)]m/P-I V'(x).
Fk(v)=0for v<0,Fk(v)=vm/P for 0<-v
21
p
km/P-1(v - k) + km/P
for v > k. The function Fk is piecewise smooth, and its derivative is bounded. By Lemma 5.8 in Chapter II, this implies that wk E WD (U). Fur-
ther, w' (x) = Fk[v(x)]v'(x). We have that FF(v) = 0 for v < 0, Fk(v) _ (m/p)v('"/P)-I for 0 < v < k, and FF(v) = (m/p)k(m/P)-I for v > k. The inequalities 0 < Wk(X) < w(x) and Iwk(x)I < (m/p)[v(x)](m/P)-IIv'(x)I hold for each k. Since the right-hand side of (2.17) is assumed to be finite,
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
302
this implies that the sequence (Wk) is bounded in W I (Q,2). As k -, 00 we have that Wk (x) - w(x) for all x. From this, w E WP (Q,). Further,
w'(x) = lim wk(x) =
(m/p)[v(x)](m/P)-Iv'(x).
k-oo In view of the foregoing, (2.17) gives us after obvious transformations that
f
Q
lPfr Q'I
Iw'(x)IPdx <
(pim_lIo)
w(x)IPdx, I
where 6 = rl - r2. For s > I let y(s) be the smallest value of I psk -11, where k runs through all the nonnegative integers. Let yo be the supremum of the numbers y(s) for s in the interval (1, so), where so = n/(n - p). Obviously,
yo > 0. We choose an arbitrary value of s such that 1 < s < so and y(s) > yo/2. Then 1 psk -11 > yo/2 for all integers k > 0. Denote by H the set of all numbers m of the form m = psk, k > 0, k an integer. As m - oo
we have that mP/Im -11P -. 1, and Im -11 > 7o/2 for all m E H. This implies that the ratio mP/I m -11 P is bounded on the set H. Consequently,
f
Iw'(x)I P dx <
Q.±
Cf TP
1w(x)IP dx
Q.,
for every m E H for w = vm/P. Using inequality (2.16) for the chosen value of s E (1. so), we get that I/s
(
Iw(x)IPsdx)
C"' IQ,ZI
<
C"(rfQ
tiw'(x)IP + Iw(x)IP]dx)
rP
(rl ?r2)P 4,1 [w(x)]P
dx +
f
[w(x)]P dx
Q'2
In the last term on the right-hand side we replace the domain of integration by Q,, (this does not decrease the integral), and get as a result that I /S
[w(x)]s dx)
q,(rll r2)
(J,2
f
[w(x)]dx, Q.,
where Cp"[1 + 1/(p- 1)P] = (p (p). Considering that w(x) = [v(x)]m/P, we get the inequality I/S
(7Q, [v(x)]"'s dx)
< ip(r1 /r2)
f
[v(x)]m dx.
(2.19)
Here it is assumed that m E H, i.e., m = psk, where k > 0 is an integer. Fix an arbitrary value r, 0 < r < ro. For 1 < p < ro/r we have that co(p)(p - I)P = Cp"[(p - 1)P + 1] < C(ro/r)"+P,
§2. HARNACK'S INEQUALITY
303
where C =const. Let h = ln(ro/r), and define q" = ps" and p" = r exp(h/21), v = 0, 1, 2, .... Then po = ro, p" > for each v, and p" , r as v oo. Let us write out (3.5), setting r, = p, r2 = p.+,, and m = q" in it. We have that h --E!L-
Pv+1
-exp
<
P" P1,+1
h
h = exp 29,+1
(2V - 2"+1
C(rolr)"+p
C(rolr)"+P
[(P,,/p"+1) - I ]P
[exp(h2-v-1) - l]P
2(v+1)pC
n+P
r0
h° This leads to the conclusion that for each v dx
)
C2"rJQ[tJ(x)r ((f[v(x)] )n+PA.(2.20)
S
\
m
Let Ilq,
Cf
[ v(x)Jq°
dx)
= P.
Q1,
Then it follows from (2.20) that for each v = 0, 1, 2....
C(
p"+
r)
I /q.
n+p
2("+ I
hP
)v/q, p,,
From this we conclude by induction that for all v
/
v.,
n+p
P" :5 C \r Lo- )
20- Po.
where °°
l
1
1
q0
q1
4"-1
v=0
s
I
Ps"
p(s - 1).
00
q0
L s" v=0
q1
From this, for all v,
P' < C2r
7
(2.21)
where C2 = const and y = (n/p) + 1. We now observe that
lira P = lira IJ-00 V-00
/
1
I
1Q1"'
f
a".
I /q.
[v(x)]q°
dx)
= esssupv(). XEQ,
304
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
Passing in (2.21) to the limit as v -' oo, we thus get that I/P
esssupv(x) < C2V(ro/r)(f xEQ,
\\
[v(x)]Pdx)
/
Q,o
where V(u) = uY/(Inu)s/(s-1) and y = (n + p)/p. This proves the lemma. 2.4. Boundedness of generalized solutions of equation (2.3). Harnack's inequality. It follows immediately from results in §§2.2 and 2.3 that every generalized solution of (2.3) is bounded. Namely, we have LEMMA 2.3. Let U be an open set in R' , and u(x) a generalized solution
of the equation div A[x. u'(x)] = 0 in U, where the function A satisfies conditions a) and b) in §2.1. Then u(x) is equivalent in the Lebesgue sense to a function f4 (x) bounded on every set M lying strictly inside U. Further.
for any cubes Q(a, r) and Q(a, ro) with 0 < r < ro and Q(a. ro) strictly inside U the estimate
esssupIu(x)I
\I/P
xEQ(a.r)
(u(x)IPdx/ )
(2.22)
(a.ro)
is valid, where C depends on n, p, and the constants al and a2 in conditions a) and b), and y/(u) depends only on n and p.
PROOF. Let u(x) be a generalized solution of (2.3) on U. Then, by Corollary 1 to Lemma 2.1, u(x) satisfies the inequality
M r m dx, [(m - p + 1)8]P JG[u(x)] M where m > p is arbitrary, M c G, M is closed, G is open, and
f lu(x)I'"-Plu'(x)IP dx <
6
inf
xEA
p(x,
R"\G).
Setting M = Q(a, r2) and G = Q(a, r1), where 0 < r2 < r1 < ro, we get that the function v = Jul satisfies the conditions of Lemma 2.2, which imply the boundedness of u(x) in Q(a, r) and (2.22). The set U can be represented as a union of countably many cubes lying
strictly inside U. By modifying the values of u in each of these cubes on a set of measure zero, we can clearly get a function bounded on every compact subset M c U, and the lemma is proved. LEMMA 2.4. There exists a real function C(A) that is bounded in (0, 1) and has the following property. Let u(x) be an arbitrary nonnegative solution of (2.3) in the cube Q(a, ro) c R", where the vector-valued function A
§2. HARNACK'S INEQUALITY
305
satisfies conditions a) and b) of §2.1. For A E (0, 1) let
u(x), n M(A) = esssup u(x) m(A)
xEQ( ess
o)
xEQ(a.A.ro)
Then M(A) < C(A)m(A). For fixed A E (0, 1) the quantity C(A) depends only on n, p, and the constants a, and a2 in conditions a) and b).
PROOF. Take a cube Q(a, ro) c R" and a generalized solution u(x) of (2.3) on this cube, with u(x) > 0 almost everywhere on Q(a, ro). We introduce the following notation. If A E (0, 1), then let Ha = Q(a, Aro). Take an arbitrary number A with 0 < A < 1, and let A, = (1 +A)/2. Then A < A, < 1. For any cube Q = Q(x, r) denote by Q the cube Q(x, r/A1) concentric with Q. Note that if Q C HA,, then Q C HI = Q(a, ro). Take an arbitrary cube Q = Q(x, r) C HA,. We use the inequality (2.8) with
M = Q and G = Q. For this pair of sets M and G we have that S = (r/A1) - r = r(1 -.1)(I + 2) = r5,, and we get that for any m < p with
mop-1 andforanyh>0 / fo
l u'(x)l o
[u(x) + h]P-m
dx <
C I [u(x) + h]"' dx. [(p - m - 1)SI r]P
(2.23)
Setting m = 0 here, we arrive at the inequality Iu'(x)I
Q
P
(UX+)
dx
C
IQI CI IQI
[(p - 1)S, ]P rp
ro
(2.24)
where CI = C/,ii (p - 1)PS°. Lemma 5.8 in Chapter II allows us to conclude that the function w(x) _ ln[u(x) + h] is in Wpl (Q), and
w'(x) = u'(x)/[u(x) + h]. In view of this, (2.24) can be rewritten as follows:
Iw'(x)IPdx 5 C,IQI/r". JQ
Let wQ = f Qw(x) dx. Then, by Lemma 2.1 in Chapter I,
JQI() - WQI"dX i.e.,
f
CrP fQlw'(x)IPdx5C21Q1.
QIw(x) - wQIP dx < C2,
(2.25)
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
306
where the constant C2 depends only on p, n, a,, a2, and A, and C2 admits
the estimate C2 < C/(1 - A)P, where C now depends only on p, n, a,, and a2. Inequality (2.25) means that w is a function with bounded mean oscillation in the cube HA,. By the corollary to Theorem 1.1 in § 1 of this
chapter, there exists a constant y > 0 such that for every a < y/M2 the function v(x) = expaIw(x) - wQI is integrable for any cube Q c HA,, and
f v(x)dx< 1+ Q
ey
y-aC2 J
Q
1w(x)-wH,,I Pdx.
Fix the value a = min{y/2C2, (p - 1)/2} here. Then, considering (2.25), we get
JQV(X)dX < C3,
(2.26)
where C3 depends only on n, p, 2, and the constants a, and a2 in conditions a) and b) of §2.1. In particular, (2.26) gives us
fQ exp[aw(x) - awQ] dx < C3,
f(()
+ awQ] dx < C3.
Multiplying these inequalities termwise, we get that
f
Q
exp[-aw(x)] dx
f
Q
exp[-aw(x)] dx < C32,
i.e.,
fQ [u(x) + h]" dc f [u(x) + h]-" dx < C.
(2.27)
Q
This inequality holds for any h > 0 for any cube Q c HA,. Now let r, and r2 be arbitrary numbers with 0 < r2 < r1 < r21, and use (2.8) once again, with M = Q,, = Q(a, r2) and G = Q,, = Q(a, r,). Then for every m 54 p - 1
f
Q.,
[u(x) + h]m-PI u'(x)to dx <
C
[(rl - r2)(p - 1 - m)]P x
f
[u(x) + h]'" dx.
(2.28)
,
We introduce an auxiliary function z defined by z(x) = [u(x) + h1-1P. h)-,'IP has a bounded conSince u(x) is nonnegative, while F(u) = (u + tinuous derivative on [0, oo), it follows that z E WPI and
z'(x) = -(a/p)[u(x) +
h]-("IP)-1
§2. HARNACK'S INEQUALITY
307
Substituting in (4.7) the expression for u(x) + h in terms of z(x), after setting -am/ p = q and making obvious transformations we get
f [z(x)]Q-Plz'(x)iP dx
[(r1 - r2) 9 + 1)]P
Q,.
f lz(x)]Q dx,
(2.29)
where ! = a(p - 1)/p. By (2.27), the function z is integrable to the power p on Q,. On the basis of Lemma 2.3 this implies that z is bounded on every cube Q,. with 0 < r < r0). I. Further, 1/P
[z(x)]P dx)
sup z(x) <- Ctp (s Il() (r
eXsc
,
(2.30)
H,
where yo (u) = O[ 1 /u -
1)31(s- 1)] as u - 1 + 0, and C depends only on the
constants C and I in (2.29). Let r = r0A in (2.30). Note further that ess sup z(x) = [ess inf u(x) + h]-"IP, xEQ,
xEQ,
which, by (2.30), gives us that
/r
\1/r1
ess i f u(x) + h > C(A)/ I J H [u(x) + h]-" dx f XEQ, C2 (A)
\I [u(x) + h]" dx I
.
(2.31)
(71111
We again use Corollary 2 to Lemma 2.1. For every m 36 p - 1 and any concentric cubes Q,, and Qr.,, where 0 < r2 < r1, we have that I.
JQ,1
[u(x) + h]'"-Pl u'(x)VP dx
C
- AM - p + I l )P x
rQ [u(x) + h]' dx.
(2.32)
Let (u(x) + h)" 1P = w(x). Since a < p, a/p < I. The function F(u) _ (u + h)"1P is continuously differentiable in [0. oo), and its derivative is bounded. This implies that w E Wp (Q,, ), and h]1n/P)-1
u'(x). w'(x) = (al p)[u(x) + Substituting in (2.32) the expression for u(x) + h in terms of w(x), we get after obvious transformations that
f [w(x)]''-Pl w'(x)l P dx < Q,
C
(alk - (p - I )p/ al )P x 412 [zu(x)]A dx,
308
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
where k = pm/a. The last inequality holds for every k > p. On the basis of Lemma 2.2, this implies that w is bounded on every cube Qr with
0
(f /
ess Hjp w(x) < Cw C ?
\I/p
[w(x)]p dx I H;
` I/p
=C
`J \J
H[u(x) + h]° dx f
/
,
.
(2.33)
We now observe that ess sup w(x) = xEHH
( ess sup[u(x) + h]
a/P
\ xEH,r
Substituting the estimate in (2.33) into (2.31) and setting r = Rr in (2.31), we get that h + ess sup u(x) < C(t) I ess sup u(x) + h xEH;
L xEH;
The number h > 0 is arbitrary. Letting h go to zero, we arrive at the inequality of the lemma. COROLLARY 1. Every generalized solution u of (2.3) is continuous and satisfies a Holder condition on every set V lying strictly inside its domain U.
PROOF. Let u be a generalized solution of (2.3) in an open set U C R". Let a E U be arbitrary, and let S > 0 be such that Q(a, r) c U for 0 < r < S. By Lemma 2.3, u(x) is bounded on every such cube. Let
M(r) = esssupu(x), m(r) = esssupu(x). xEQ(a,r)
xEQ(a.r)
The functions u(x) - m(r) and M(r) - u(x) are generalized solutions of (2.3). In view of the lemma, this implies that
M(r/2) - m(r) < C[m(r/2) - m(r)], M(r) - m(r/2) < C[M(r) - M(r/2)], where C > 1. Multiplying these inequalities termwise, we get that
(C + 1)[M(r/2) - m(r/2)] < (C - 1)[M(r) - m(r)], which implies that
M (2)
- m (r) < [(C - 1)/(C + 1)][M(r) - m(r)].
§2. HARNACK'S INEQUALITY
309
Let w(r) denote the oscillation M(r) - m(r) of u(x) in the cube Q(a, r). We have that w(r/2) < Ow(r), where 0 = (C - 1)/(C + 1) < 1. This gives us by induction that
w(r2-") < 0"w(r),
v = 1,2,....
The last inequality allows us to conclude that
w(p) < 0-'(p/r)`tw(r), where a = (In 1/0)/ In 2, 0 < p < r, and it is thereby proved that ever generalized solution of (2.3) satisfies a Holder condition. COROLLARY 2. Let U c R" be an open set. Then for every compact set M c U there exists a constant C > I such that for any nonnegative solution u(x) of (2.3) on U (2.34) max u(x) < C min u(x).
The proof of the corollary is based on the Borel theorem, and is left to the reader because it is clear. The Harnack constant of the pair (M, U) of sets is defined to be the smallest number C > I such that (2.34) holds for any equation of the form (2.3) with A satisfying conditions a) and b) of §2.1 and any nonnegative generalized solution of (2.3) on U. By definition, this constant C depends only on the constants p > I and a, , a2 > 0 in conditions a) and b) of §2.1 and on the choice of the pair (M, U). The next statement is used in the study of mappings with bounded distortion. LEMMA 2.5. Let the pairs (MI, U,) and (M2, U2) be given, where U, and
U2 are open, M, and M2 are compact, and U, D M, for i = 1, 2. If there exists a similarity f of R" such that f (U,) = U2 and f (M,) = M2, then the Harnack constants of the pairs (MI, U,) and (M2, U2) coincide. PROOF.
Let f'(x) = / and k = 111. Assume that the equation
div A[x, u'(x)] = 0 is given in U2, where the vector-valued function A satisfies conditions a) and b) of §2.1. Let A(x,q) =
.A[x,l(q)]. The vector-valued function A(x, q) is defined for almost all x E U, and any q E R", and satisfies conditions a) and b) of §2.1 with the same values of the constants a, and a2. If u(x) is a solution of the equation div A[x. u'(x)] = 0, then f4(x) = u[f (x)) is a solution of div A[x. u'(x)] = 0. The conclusion of the lemma follows in an obvious way from these remarks.
1/ko-i
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
310
§3. Theorems on semicontinuity and convergence with a functional for functionals of the calculus of variations 3.1. Weak convergence of sequences of functions in measure spaces.
Let
us consider functions defined on a locally compact metric space 91 and taking values in R'. It is assumed that a measure p is given on the Qalgebra of Bore] subsets of 91, with p(f1) < oo. A function u: 91 -, Rl is said to belong to the class LP(91, p) (p > 1) if I/p
IIuII,.91 _ f u(x)I pp(dx))
< 00.
OR
We introduce some more notation. Let Co(9t) denote the collection of all compactly supported real functions on 91, and S(91) the set of indicator functions of Borel subsets of 91 with compact closures. Let p > 1 and let (u,,,: 9t -+ RI), m = 1, 2, ... , be an arbitrary sequence of functions in Lp(91, At). The sequence (ur) is said to converge weakly in
L(9t) to a function uo E L(91) if f co(x)um(x)p(dx)
f rp(x)uo(x)p(dx)
as m -+ oo for every ip:91 - R of class Lq(91), where q = p/(p - 1). Assume that (u,,,: 9t -- RI), m = 1, 2, ... , is a sequence of functions in L, (91). Then the sequence (u,), v = 1 , 2, ... , is said to converge weakly to uo in L I (91) if
f
(x)p(dx) -,
f uo(x)9(x)p(dx)
as v oo for every function (P E Co(tB). We describe a general situation that encompasses various types of weak convergence. Let K be a class of B-measurable bounded real functions on 91. For U E L 1(91), u: 91 -- R1, and fp E K we set (u, rp) =
f
(p(x)u(x)p(dx).
rn
Let (u,,: 9 1 - RI), v = 1 , 2, ... , be a sequence of functions of class L I (3t, p). The sequence (u,), v = 1, 2, ... , is said to converge in the WKsense to a function uo E L 1(91, p) if: 1) the norm sequence (I I u, I I p.9I) , v = 1, 2, ... , is bounded, and 2)
(p) -+ (uo, gyp) as m
oo for every 97 E K.
We mention some simple properties of convergence in the WK-sense. Denote by K' the collection of all linear combinations of functions of the class K. Let Z p(K) (1 < p < oo) be the closure of the set K' in the space Lp(91, p). Denote by Z ,,.(K) the collection of all functions that are limits of uniformly convergent sequences of functions in K.
§3. SEMICONTINUITY AND CONVERGENCE
311
LEMMA 3.1. Let (u,: 9 - RI), v = 1, 2, ... , be the sequence of L I (91,,u)functions converging in the wK-sense to a function uo E L I (9R, p). Then u -+ uo also in the wZ,0(K)-sense. Suppose that p > I and q = pl(p - 1) > 1. If the functions u are all in L(91, p) and the sequence of norms I I u I I p.9q is bounded, then u, -' uo in the wZ,(K)-sense.
PROOF. Suppose that u - uo in the WK-sense, i.e., (u,,, (p) --+ (uo, () for every p E K. Then it is obvious that (u,,, () -. (uo, so) also for every function that is a linear combination of functions in K, i.e., u - uo in the WK.-sense. Let (p E Z,,(K), and let M < oo be such that IIuj1.'R <
M for all v. Take an arbitrary e > 0 and choose Ip, E K' such that Iso, (x) - rp(x)I <.e/3M for all x. Then (u", (v) - (uo, P)I <_ I (u., 4p) - (u" 4v1)I + I (uY. (P 1) - (uo. q'I)I + I(uo, (P I) - (uo. 47)I.
It remains to observe that I (u, 4P) - (u,,, pv1)1 < Me/3M = e/3,1(uo, (p) - (uo, (p 1)I < Me/3M,
and (u., (#I) --+ (uo, (p 1) as v -+ oc, since cp I E K*. This leads to the conclusion that I(u,,, (p) - (uo, ip) < e for sufficiently large v, i.e., (u,,, (P) (uo.ip)asv--+oo. Assume that the norm sequence (I I u I I p.9t) is bounded, where p > 1, and
let q = p/(p - 1). Suppose that
M for all v. Let 4P E Zq(K)
be arbitrary, and for a given e > 0 let (PI E K` be such that 11(p, e/3M. Arguing as in the preceding case, we then get that
- fo I Iq.9R <
(U., ''P) - (uo, (P)I <_ I (u", (P) - (u,,. qiI) I
+I(u,,,(P ) - (uo,4vl)I+I(uo,(P t) = (uo,(P)I I - (uo. v1)I + IIUolip.9II, - VI 1lq.9R < (2/3)e + I (uy. (pl) - (uo, 91)1.
<- IIuv11,-IIso - 4,I Ilq.9 + I ( U , , ( P
We have that (u , gP I) - (uo, ip I)
0 as v
oo, and this implies that there
exists a vo such that I (u,, , (p) - (uo, gyp) I < e for v > vo. This establishes that
(u,,, 4p) -+ (uo, 4p) as v - oo, and the lemma is proved.
v = 1, 2,..., be a sequence of LI()t, p) functions converging in the WK-sense to a function uo, and let (4y), v = 1, 2,..., be a uniformly convergent sequence of functions in Z,,,, (K), with ipo = (uo, ipo) as v - oo. rp,,. Then (u,,, LEMMA 3.2. Let
PROOF. Let
8 = sup I4,,(x) - (00 (x)I xEgR
Ill. FUNCTIONS OF A REAL VARIABLE AND PDE'S
312
Then
- 0 as v (u,,, g")
oo. For each v,
- (uo, 90)I <-1(u,., 9" - go) +I(u,.,9o) - (uo,too)I
Then 8 -+ 0 and (u,,, S'o) -' (uo, go) as v -4 oo, because go E Z.,, (K). -i (uo, go) as v - oo, which is This leads to the conclusion that (u,,, what was required to prove.
We remark that if K = Co(at), and also if K = S(at), then Zq(K) _ Lq(at, p) for q > 1. This yields the following assertion. COROLLARY. Let (u,,: at - RI), v = 1, 2, ..., be a sequence of L p(at, p)functions such that the norm sequence (J u I I p.9j) is bounded. If converges weakly in L I to a function uo e L p(at, p), then u -, uo weakly in L p. If the sequence v = 1, 2, ... , converges to uo in the w.,-sense, then converges weakly to uo in Lp(at, p).
3.2. Some lemmas about convex functions
LEMMA 3.3. Let U be a convex open subset of R', and let (Fm), m = 1, 2, ... , be a sequence of convex functions on U such that there is a finite
limit Iimm.... F,,, (x) = Fo(x) for all x E U. The function Fo defined in this way is convex, and Fm -, Fo locally uniformly in U. PROOF. Obviously Fo is convex. Let Q = Q(a, t) be an arbitrary closed cube contained in U, with vertices u1 , ... , u,. (r = 21). Every point x E Q can clearly be represented in the form x = Al u1 + + 2,u,, where 2i is nonnegative and Al + + A, = 1. Using Jensen's inequality, we get that r
Fm(x) < j:2,Fm(ui). i=1
The sequences (Fm (ui)) are bounded. Let Fm (ui) < M for all m and any i = 1. ... , r. It is then obvious that Fm(x) _< M for all x E Q. We show that the functions Fm are also bounded below on Q. The sequence (Fm(a)) (a is the center of Q) is clearly bounded. Suppose that IFm(a)I < MI for all m. Let Lm be an affine function such that Lm(a) _ Fm(a) and Lm(x) < Fm(x) < M for all x E Q. Then Lm(X) = (pm, x - a) + Fm(a).
We prove that Ipm1 < 2(M + MI)/t. Indeed, if I 0, then this is obvious. Let I pmI 0 0. Setting x = a + to/2, where e = pm/Ipml, we get that Lm(X) = (t/2)(pm. e) < M - Fm(a) < M + MI ,
§3. SEMICONTINUITY AND CONVERGENCE
313
which implies that t 1 pm 112 < M + M1, as required. It is obvious that Lm(X) > -M1 - IPm) IX
- al,
for all X E Q, which leads to the conclusion that Lm(x) > Mo for all x E Q(a, r), where Mo = const, and hence Mo < Fm(x) < M for all
m= 1,2,... andallxEQ. We now consider the cube Ql = Q(a, 1/2) concentric with Q and having sides half as long as those of Q. Let x1, x2 E Ql be two arbitrary points in Q1, and let Km be an affine function such that Km(xl) = Fm(xl), and Km(x) < Fm(x) for all x E U. We have that Km(x) _ (qm, x - XI) + Fm(xl ),
Fm(XI) > Mo.
Suppose that qm 0 0. Let x = x 1 + to/4, where e = (1 / I qm I )qm. Obviously,
x E Q, and this gives us that
tlgm1/4= (qm,x-Xl) <M-Fm(Xo), which implies that Igml < 4(M - Mo)/t. This inequality is true also when qm = 0. We have that Fm(X2) - Fm(Xi) > Km(X2) - Km(XI) = (9m, x2 - x1) > -LIx2 - x11,
where L = 4(M - Mo)/t. Interchanging the roles of x1 and x2, we get in exactly the same way that Fm(X1) - Fm(X2) > -L1X1 -xi1, which leads to the conclusion that I Fm(XI) - Fm(X2)I :5 LIx1 - x2l
for any points x1, x2 E Q1.
It follows from what has been proved that the sequence (Fm) of functions is uniformly equicontinuous in Q1i and hence Fm -' Fo uniformly in Q1.
Thus, every point a E U has a neighborhood in which Fm - F0 uniformly. This obviously implies that Fm -- Fo uniformly on every compact set A c U. The lemma is proved. We now describe a construction for approximating convex functions on R' by affine functions.
Let F be an arbitrary function on a set E c R'. Denote by AF the set of all points (x, z) E R1+1 such that x E E and z > F(x). The set AF is called the supergraph of F. Let E = R'. If the function F: RI R is continuous, then AF is closed. The function F: R1 -' R is convex if and only if AF is a convex set.
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
314
Let H be an arbitrary closed convex set in R"', and let a be a point not in H. Denote by b the point in H closest to a. Such a point exists because H is closed. The point b closest to a is unique in view of the convexity of H. Let p = b - a and c = (a + b), and let L(x) = (x - c, p). i P in R'+1. This plane passes The equation L(x) = 0 determines a plane through the midpoint c of [a, b] and is perpendicular to [a, b]. For all x E H we have that L(x) > 0, while L(a) < 0. Intuitively, this means that P separates a from the set H, with a on one side of this plane and H on the other. Let F: R' R be a convex function. Take an arbitrary point xo E R1 and a number e > 0. Consider the points b1 = (xo, F(xo)) and a1 = (xo,F(xo) - e) in Obviously, a1 AF. Let b2 be the point of AF closest to a1. Let P be the plane through the midpoint of [a,, b2] and perpendicular to this segment. The plane P is defined in R'+1 by an R"+1.
equation L(x, z) = (h, x) + kz + B = 0 with h E R' and B, k E R, and L(x, z) > 0 for all (x, z) E AF, while L(xo, zo - e) < 0, where zo = F(xo). This implies that k < 0, and the equation of P can be rewritten in the form z = (p,x) + c. The condition that L(x, z) > 0 for all (x, z) E AF and the
condition that L(x, z) < 0 for (x, z) = a1 give us that F(x) > (p, x) + c for all x E R', and F(x0) - e < (p, xo) + c. The vector p and the number c are uniquely determined by F, the point x0, and the number e > 0. The notation p = pF(xo, e) and c = CF(XO, e) will be used in this connection. LEMMA 3.4. Let (Fm), m = 1, 2, ... , be a sequence of convex functions
on RI that converges to a convex function F0 on R1 as m - oo, and let (xm), m = 1,2,..., be a sequence of points in R' converging to x0 as m - oo. Then PF,,,(Xm, e) -' PF0, (X0, e), CF,,,(Xm, e) ~ CFO(X0, e)
asm -moo.
PROOF. Let am = (xm, Fm(xm) - e), and let b,, be the point of AFm closest to am. It obviously suffices to establish that am -+ a = (xo, Fo(xo) -
e) and bm -. b as in -p oo, where b is the point of AFo closest to a. In view of Lemma 2.3, Fm converges to F0 uniformly on every compact subset E c R. This implies that am oo. Let b' _ a as m b' as m oo. For each m we have that l bry, - am l > I bm - am l , because b.. is the point of AFm closest to am; lb,,, - aml -' lb - at as m - oo, and hence the sequence (l b' - am l) , m = 1, 2, ... , of distances is bounded. This (xo, F(xo)),
Fm(xo)). This it is clear that b,;,
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implies that the sequence (I b,,, - am 1), m = 1, 2, ... , is also bounded, and hence (bm), m = 1, 2...., is a bounded sequence. Let (bm, ), m1 < m2 < ..., be an arbitrary convergent subsequence of this sequence, and let b' = limk_ bmk. Since Fm -, F uniformly on every bounded subset of R', it follows that b* = (y, F(y)) for some y E R'. This implies that lb' - al > lb - al. On the other hand, Ibmk -al
for each k, and -amkl
al
as k - oc; by passage to the limit this implies that lb* - al < lb - al. Hence, lb* - al = lb - al, and since the point of AF closest to a is unique, it follows that b' = b. We thus get that any convergent subsequence of (bm) converges to b. Hence, bm -. b as m -- oo. This obviously proves the lemma. 3.3. Theorems about semicontinuity of functionals of the calculus of variations. Below, 3t denotes a locally compact metric space, and µ a measure on the a-ring of all Borel subsets of 31 such that µ(91) < oo. THEOREM 3.1 [27]. Let (Fm:91 x RI --+ R), m = 0, 1, 2,..., be a sequence
of nonnegative continuous functions such that Fm - Fo locally uniformly in 33 x RI as m - oo. If for each m the function q Fm(t, q) is convex for every t E 9t, then every sequence (um: 91 --+ Rt), m = 1, 2, ... , of functions of the class L I (3t,µ) that converges in the wca-sense to a function uo: 33 R', uo E L, (3't,µ), satisfies the inequality
f
Fo[x,
t
Fm[x, um(x)]u(dx). uo(x)]µ(dx) < ral 00 4
PROOF. For any set A c 91 let
L Let (um,Fm,91)=1m, m=0,1,2,.... Given an arbitrary e > 0, we construct a compact set K c 91 such that uo is continuous on K and (uo, F0, K) > to - e in the case when Io < oo, and (u, Fo, K) > 1 /e when to = oo. We construct a decreasing sequence of open sets U1 D U2 D ... such that the closure of each is compact, and their intersection coincides with K. R be a continuous function such that (p, (x) = 0 for x U,,, Let /p,,:91 1 for x E K, and 0 < (p,, (x) < 1 for all x. Let uo =(u01 . . uot), where the u01 are real functions. For each i we construct according to the
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
316
Urysohn extension theorem a continuous function vi: 91 - R such that vi(x) = uo,(x) for x E K and Ivi (x) 1 <_xEK max Iuoi(x)I.
and let v(x) = (vi (x), ... , v, (x)). Then 1v(x)I < l maxc Iu(x)I
for allxE91. For each m = 0, L... and each x E 91 we have a convex function Fm,x: q - Fm(x, q) of the variable q E R' and a point v = v(x). Given an and a number c = cF,,,, arbitrary q > 0, we construct a vector p = corresponding to the function Fm,x, the point v = v(x), and the number n in accord with the constructions of §2.1. Let pm(x) = pF and cm(x) cFm ,.. Then Fm(x, q) > (pm(x), q) + cm(x) for all m, x, and q, and Fo[x, v(x)] - q < (Po(x), v(x)) + co(x).
(3.1)
Each of the functions pm and cm is continuous. Indeed, let (xk), k = 1, 2, ... , be an arbitrary convergent sequence of points of 31, with limit x0. We have that
v(xk) - v(xo),
Fm (xk,q) -. F.(xo,q)
as k - oo, and this implies in view of Lemma 3.4 that pm(xk) Pm(xo) and cm(xk) - cm(xo) as k - oo. This proves that pm and cm are continuous.
By arguing similarly it can be proved that lim Pm(xm) = P0(xo). M-00
lim cm(km) = co(xo) m-oo
for every convergent sequence (xm), m = 1, 2,.... with xm - xo as m oo. This implies that pm and cm converge as m - oo to functions po and co uniformly on every compact subset of 9t. By the foregoing, and dm.,,(x) = Let qm.,,(x) = the functions qm,,, and dm.,, are continuous and compactly supported, and qm.v - qo.v and dm,,, -+ do.,, uniformly on 9% as m - oo. It is obvious that for each m Fm(x, u) >- (gm..,(x), u) + d,,,., (x).
Note that qo.,,(x) = po(x) for x E K. Integrating (3.1) termwise, we get that (uo, Fo. K) < r {(po(x), uo(x)) + co(x)} u(dx) + 1µ(K). K
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317
The number q > 0 is arbitrary. It will be assumed that q is chosen so that (uo, Fo, K) - qµ(K) > Io - e when to < oo, and (uo, Fo, K) - qµ(K) > 1 /e when lo = oo. (The quantity (uo, F0, K) is clearly finite, because K is compact, F is continuous, and uo is continuous on K.) As v -- oo the integral
f{(qo.(x). uo(x)) + do.Y(x)}µ(dx) = f ipv(X){(po(X), uo(X)) + co(x)}µ(dx)
tends to a limit equal to
j{(Po(x). uo(X)) + co(x)}µ(dx), 1 for x E K, 0 < {p(x) < 1 for since 0 for x 0 K, all x, the functions po(x) and co(x) are continuous, hence bounded on U, (because the closure of U, is compact), and uo is integrable. We fix a vo such that
f9l{(go.w(k), uo(X)) + do.0(x)}p(dx)
is larger than 10 - e in the case when to < oo, and larger than 1 /e when to = ce. Let qm = gm.rro and dm = dm,,,e , m = 0, 1, 2,.... For each m IM
J{(m(x). um(X)) + dm(X)}µ(dX).
(3.2)
We have that qm - qo and dm - do uniformly as m - oo. The functions and d,,, are all compactly supported and vanish outside the compact set U1. The functions um converge to uo in the we-sense as m - oo. In view of Lemma 3.2, this implies that the right-hand side of (3.2) tends to the limit {(qo(x), uo(x)) + do(x)}.u(dx)
as m -+ oo. Passing in (3.2) to the limit as m -' oo, we get from this that lim I, > to - e if l0 < oo, m-oo
and
lim Im > 1 /e if 10 = oo.
m-"O
Since e > 0 is arbitrary, this establishes that
lim Im > J.
m-.oo
which is what was required.
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318
The continuity conditions imposed on F in Theorem 3.1 can be essentially weakened. Here it is obvious that convergence in the wCa-sense must be replaced by convergence of a stronger type. We present the corresponding formulations. Let F: 93x R' R be given. We say that F is a function of type E, and write F E E, if for every e > 0 there is a compact set A c 9t such that u (91\A) < e and the restriction of F to A x R1 is continuous on the metric space A x R'.
Let F E E. Then we say that F E IV if for almost all (in the sense of the measure u) points x e 31 the function q F(x, q) is convex on R'. By using the known C-property of measurable functions it is not hard
to show that if F is a function of type E, then for every p-measurable R' the function x - F[x, u(x)] is measurable. function u: 91 Let Fm: 9l x R' - R, m = 1, 2, ... , be an arbitrary sequence of functions of type E. We say that this sequence E-converges to a function Fo E E as
m - oo (notation: Fm - FoIE as m - oo) if every subsequence of it has a subsequence (Fmk) , MI < m2 < .... such that for any c > 0 there is a compact set A c 91 such that µ(9t\A) < e, each of the functions Fmk and F0 is continuous on A x R', and Fmk
Fo locally uniformly on A x R1 as
k - oo. THEOREM 3.2 [127]. Let Fm: 91 x R' -+ R, be a sequence of functions of class EV that E-converges to a function FO: 31 x R' R as m -* oo, and let all the functions Fm be nonnegative. Then every sequence (um: 9R RI) of L 1 (91, p) functions convergent in the ws-sense to a function uo: 91 R' as
m - oo satisfies the inequality Fo[x, uo(x)],u(dx) < nlimp I Fm[x, U,n(x)]P(dx).
f9l PROOF. For an arbitrary set A /C 9t let
(um, Fm, A) = J Fm[X, um(X)]µ(dX). A
Further, let (um Fm , 9t) = I,,,, m = 0, 1, 2. .... Take a subsequence of (Fm) such that the values Im tend to limn-. 1m. Given an arbitrary positive integer v, we find a compact set K C 9t with 1 /v and a subsequence (Fmk), m1 < m2 < ..., of the constructed subsequence such that Fm, - F0 locally uniformly on K x R'. For every continuous vector-valued function ry defined on K we have that ,
fK
( 9(x), um(X))y(dX)
-f
uo(x)),u(dx),
as follows from the remarks in §3.1. This means that um(x) -+ uo(x) on Hence, we can apply Theorem 3.1 the subspace K in the
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319
to the space K and the sequences (F,",) and (um4 ), k = 1, 2..... As a result we get that
lim f F,,,[x, um(x)]P(dx) = lim f F., [x, u,,,, (x)]µ(dx)
M-00
k-'oo
t
lim J Fm, [x, u,A (x)]µ(dx) ?
t
f
Fo[x, uo(x)]p(dx).
It remains to see that the integral on the right-hand side of the last inequality tends to
f Fo[x, uo(x)]u(dx) as v - oo, and the theorem is proved. 3.4. Corollaries to Theorems 3.1 and 3.2. Let U c R" be an arbitrary bounded open set, and let u: U R'" be a mapping of class W1 (U). The matrix (8u,/0xe), i = 1, ... , m, j = 1, ... , n, is denoted here by u'(x). The vector space of m x n matrices is identified with the space R'". Let F(x, u, w) be a function such that for almost all x E U the expression F(x, u, w) is defined for every u E R"' and w E R'"". We say that F belongs to the class LV(U) if for almost all x E U and any u E Rm the
function w - F(x, u, w) is convex on R"'" and for any e > 0 there is a compact set A c U such that I U\AI < e and F(x, u, w) is continuous on
AxR"'xR"'". LEMMA 3.5. Let (f,,,: U
R"') be a sequence of /unction of class W, (U),
where U is an open set in R", and p > 1. Assume that the norm sequence (I If.II I.P.t,), v = 1, 2, ... , is bounded and the functions f, converge in LI to a function fo: U - R"' as v - oo. Then fo E W, (U), and the functions f converge weakly in L(U) to fo. PROOF. The fact that the limit function fo belongs to W, (U) follows from Theorem 1.1 in Chapter 1. Let K be the collection of all continuously differentiable functions cp: U R with compact support in U. Then the class Z,,,(K) contains all the
continuous functions with compact support in U, and the class Zq(K) coincides with Lq(U) when q > 1. Suppose that the vector-valued function f has components Then for any V E K
f JL
,,'
(x)ip(x) dx = - f
f,.,(x)U- (x) dx.
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320
Since ,,,j -' fo.i in L 1(U), it follows that
-f fv,i(x)az (x) dx - - f fo.i(x).axj- (x) dx u
(x)9D(x) dx. = f uefo'' axi
Thus,
fu as v
as' (x)c,(x) dx - fu aaz,i (x)9(x) dx
oo. This means that af".i
alai
axi
axi
in the WK-sense. By Lemma 3.1, this implies the statement of the lemma. THEOREM 3.3 [127]. Let F: (x, u, w) -, F (x, u, w) (x E U, V E R n, w E R'"") be a function that is defined and continuous on the set U x R" x Rmn,
and convex in w for any x E U and u E Rm. Let (f,), v = 1, 2, ... , be a sequence of vector-valued functions of class WI such that the functions f, are all continuous, f, -, fo locally uniformly on U as v -+ oo, and the norm sequence (I If.II1,I.u). v = 1, 2, ... , is bounded. Then
f F[x.fo(x),fo(x)]dx < lim f F[x,fv(x),f,,(x)]dx. v-'oo U U PROOF. The theorem is a corollary to Theorem 3.1. Let
f,'(x) = wv(x),
F. (x. w) = F[x, f., (x), w].
Then F,, (x, w) -+ F0 - (x, w) for all x E U and W E Rmn as v
oo.
Since F is convex and continuous and since f, --+ fo locally uniformly as w) - Fo(x, w) locally uniformly on U x R. v - oo, we have that By Lemma 3.5, the functions f converge to fo in the woo-sense. Thus, we have shown that all the conditions of Theorem 3.1 hold here, and the theorem is thus proved. THEOREM 3.4 [127]. Let (x, u, w) - F(x, u, w) be a nonnegative function of class I V (U) defined for all x E U, U E R"' and W E Rm", where U
is an open set in R", and let (f,: U - R"'), v = 1, 2, ... , be a sequence of vector-valued functions of class Wp (U) (p > 1) that is bounded in Wp (U).
Assume that the functions f, converge in LI(U) to a function fo: U - R" as v oo. Then
f
U
F[x, fo(x), fo(x)]dx < v-+oo lim
F[x, f. (x), f,,(x)]dx. fu
§3. SEMICONTINUITY AND CONVERGENCE
321
PROOF. Take a subsequence of (f,) for which the integrals
JU
F[x, f, (x),
converge to their limit inferior, and then take a subsequence of that sequence such that f, -' fo almost everywhere in U. In order not to complicate the notation it will be assumed that (f,), v = 1, 2, ... , is this second subsequence. Let F,, (x, w) = F[x, f,(x), w]. Take an arbitrary
c > 0, and for it take a compact set K c U such that F is continuous on K x R'" x R"'" , I U\K) < e, the functions f, are all continuous on K, and f, - fo uniformly on K. The existence of such a compact set K follows from Lusin's theorem on the C-property of a measurable function, Egorov's theorem on sequences of measurable functions converging almost everywhere, and the fact that F is a function of class X V. Then it is clear that F,, (x, w) -' Fo(x, w) locally uniformly on K x R'"". Since e > 0 is of functions Z-converges to arbitrary, this proves that the sequence Fo(x, w) as v -+ oo. In view of Lemma 3.5, the functions w = converge to wo = fo weakly in Lp(U) as v -p oo, and hence w wo in the ws(u)-sense. Now Theorem 3.4 is an immediate consequence of Theorem 3.2. 3.5. The convex envelope of a function.
Let c = {F} be an arbitrary
nonempty set of functions on R1. For X E RI let Fo(x) = supFEO F(x), a real function defined on R' with values in the extended real line R = R U {-oo, oo}. This function is called the upper envelope of the set D of functions. If all the functions F E 0 are convex and F,(x) < oo for all x E R', then the function F0 is easily shown also to be convex. Let F: R1 - R be an arbitrary function. Denote by VF the collection of all convex functions G such that G(x) < F(x) for all x. Assume that VF is nonempty. Then supGE VF G(x) < oo for any x E R', and Fv, is called the convex envelope of F in this case. A function F: R' - R is said to be essentially convex if F is convex and
F[Axo+(1 -A)x1]
0
G: R1 -' R be a continuous function such that IG(u)1 < kF(u) + M, where k > 0 and M are constants. Let FA be the convex envelope of the function AF(u) + G(u), and let HA(u) = AF(u) + G(u) - FA(u). The function FA is
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
322
defined for any A > k, HA(u)
0 as A - oo for all u E R', and
0:5 H1(u) < 2kF(u) + 2M
(3.3)
PROOF. For all U E Rl we have
AF(u) + G(u) > (A
k)F(u) - M,
by a condition of the lemma. The right-hand side of this inequality is a convex function for A > k; hence the convex envelope of 2F + G exists for every A > k. Further, FA(u) > (A - k)F(u) - M for all u. This implies that HA(u) _ AF(u) + G(u) - FA(u)
< (A + k)F(u) + M - (A - k)F(u) + M = 2kF(u) + 2M
for A > k, and thus it is established that HA(u) < 2kF(u) + 2M. It is obvious by definition that HA is nonnegative, and (3.3) is proved. It remains to prove that HA(u) -- 0 as A - oo for all u E R'. Let uo E Rl
be an arbitrary point. Since F is a convex function, there is an affine function L such that F(uo) = L(uo) and F(u) > L(u) for all u. Take an arbitrary e > 0. Since G is continuous at uo, for the given e there exists a d > 0 such that if Iu - uoI < 6, then IG(u) - G(uo)I < e. Let M1 = M + I G(uo) - &I. Then for all u E R'
kF(u)+M1 +G(u) > G(uo)-e. We now prove that there is an a > 0 such that for every u E R' with Iu - uoI > S
F(u) - L(u) > a(kF(u) + M1). (3.4) Assume on the contrary that there is no such a. Then for every m = L2.... there is a point u,,, E R' for which Iu,n - uol > 6, while F(u,,,) - L(u,n) < (1/m)kF(um) + (1 /m)M1.
(3.5)
The ray [uo, um] intersects the sphere S(uo, 6) at some point vm. We have that vm = uo + Am(u,n - u0), where 2m = a/Ium - u0I. By (3.5),
(1 - (k/m))[F(u,n) - L(u,n)] < (k/m)L(u,n) + (M1 /m).
(3.6)
Since L is an affine function, I L(u) I < Al u - uoJ + B, where A and B are constants. By (3.6), this tells us that
kk+m1k. 0
(3.7)
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323
The convexity of F - L implies that F(v,m) - L(vm) < Am[F(um) - L(um)],
which, by (3.7), gives us that F.(v..
L(vm) <
Ak6 + Bk + MI
m-k
m-k
(3.8)
Choose a subsequence of (vm) that converges to some point vo E S(uo, 6).
Then, since F - L is nonnegative, (3.8) implies that F(vo) - L(vo) = 0. Thus, the nonnegative convex function F - L vanishes at two different points uo and vo. This implies that F - L vanishes on the whole segment [uo,`vo], obviously contradicting the essential convexity of F. Thus, a contradiction has resulted from the assumption that there is no a > 0 such that (3.4) holds when J u - uol > S.
Accordingly, let a > 0 be such that F(u) - L(v) > a(kF(u) + MI] for all u with Ju - uol > 6. Let A > 1 /a. Then for j u - uol > d AF(u) + G(u) = A[F(u) - L(u)] + G(u) + AL(u) > Ao[kF(u) + M,] + G(u) + AL(u) > kF(u) + M, + G(u) + AL(u) > AL(u) + G(uo) - e.
ForIu-uol<8 AF(u) + G(u) _ ).[F(u) - L(u)] + G(u) + AL(u) > G(u) + 2L(u) > AL(u) + G(uo) - e.
Thus, AF(u) + G(u) > AL(u) + G(uo) - e for all u E R'. The function u - AL(u) + G(uo) - e is convex. Hence, FA(u) > AL(u) + G(uo) - e for all u. In particular, setting u = uo and noting that L(uo) = F(uo), we get that 0 < HA(uo) < e. Therefore, FF(uo) > AF(uo) + G(uo) - e. Since e > 0 is arbitrary and the last inequality requires only that A be greater than I /a, this establishes that HA(uo) -' 0 as A - oo. The lemma is proved.
LEMMA 3.7. Let (F,,), m = 1, 2, ... , be a sequence of functions on RI such that the following conditions hold:
Fm(u) = Fo(u) is finite for any u E R'. and F, -' Fo in R' a) locally uniformly. b) There exists a function 0: RI - R that is bounded below and such that Fm(u) > 0(u) f o r all u E R', m = 1, 2, ... , and 0(u)l(Iul + 1) oo as Jul - oo. Let Gm and Go be the convex envelopes of the functions F, and F0. Then
Gm - Go as m -' oo locally uniformly in R'.
324
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
PROOF. All the functions Gm and Go are clearly convex, and to prove that Gm - Go locally uniformly it suffices by Lemma 3.3 to establish that
for all uERi. Take an arbitrary point uo E R1, and let L be an affine function such that G(uo) = L(uo) and G(u) > L(u) for all u E R1; L(u) = (p, u) + A, where p E R1 and A is a real number. Let e > 0 be arbitrary, and let K
be the collection of all u E R' such that 0(u) < L(u) - e. The set K is bounded, because 0 (u) / (I u l + 1) - oo as J ul - oo. For all u E K we have that Fo(u) > L(u), and since Fm converges to F locally uniformly, there is an index mo such that Fm(u) > L(u) - e for m > mo and for all u E K.
Since Fm(u) > 0(u) for all u, the inequality Fm(u) > L(u) - e holds also for u 0 K when m > mo. From this, Gm(u) > L(u) - e for all u when m > mo, and, in particular, Gm(uo) > L(uo) - e = G(uo) - e for m > mo. Since e > 0 is arbitrary, what has been proved implies that lim Gm(uo) > G(uo).
m-.oo
We now prove that lim Gm(uo) < G(uo).
m-roo
(3.9)
For each m we construct an affine function Lm such that Gm(uo) = Lm(uo) and Gm(uo) >- Lm(u) for all u. Then
Lm(u) = (Pm, u - u0) + Gm(u0) For each m we have that Gm (uo) < Fm (uo ), which implies that the sequence (Gm(uo)), m = 1, 2, ... , is bounded above. Let 0(u) > M for all
u, where M E R. Then Gm(uo) > M for all m, and hence the sequence (Gm(uo). m = 1, 2, ... , is also bounded below. Let Pm = (Pm1, ... , NO and Pmi = (Pm, ei). We have that Lm(u0 + e,) = Pmi + Gm(u0) < Fm(u0 + ef), Lm(u0 - ei) _ -Pmi + Gm(u0) : Fm(u0 - ei). This leads to the conclusion that each of the sequences (pm, ), m is bounded. Let us first construct a strictly increasing sequence of numbers m such that Gm(uo) tends to the limit superior limm.... Gm(uo), and then
choose a subsequence (denoted by (mk), MI < m2 < ...) of it such that Gmk (uo) and pmk i tend to finite limits as k -+ oo. Then Fmk (u) > Lmk (u)
for each k and each u, and the functions Lmk converge to some affine function Lo as k - oo. Passing to the limit, we get that Fo(u) > Lo(u) for all u, and hence Gm(uo) > Lo(uo). But LO(uo) = lim Gmk (u0) = lim Gm (uo)
by construction, and this establishes (3.9). The lemma is proved.
§3. SEMICONTINUITY AND CONVERGENCE
325
3.6. A theorem on convergence with a functional. Corresponding to the two variants of the semicontinuity theorem we have proved, there are two possible variants of a theorem on convergence with a functional; to avoid repeating awkward formulations we introduce concepts allowing us to unify the different formulations. We distinguish two cases, Situation I and Situation II. As above, let 9% be a locally compact metric space, and µ a measure defined on the a-algebra of Borel subsets of 31 such that µ(31) < oo. Consider a sequence (Fm: R x R1 - R), m = 1, 2, .... of functions. In Situation I it is assumed that all the functions Fm are continuous on 31 x R', nonnegative for all (x, u), and convex in U for each x E 91, and that they converge locally uniformly as m - oo to a function Fo(x, u) that is essentially convex in u for almost all values of x E R in the sense of u. In Situation II it is assumed that Fm E EV(31), the functions Fm are nonnegative, Fm -+ F0 in the sense of E, and for each e > 0 there exists a compact set K c 31 such that µ(3't\K) < e and Fm is continuous on K x 31I and essentially convex in u for x E K. In both situations it is assumed that there exists a continuous function
0:R1-Rsuch that 0(u)/(IuI+1)-ooas Jul--'oo,0(u)>0for all u,and Fm(x, u) > 0(u)
(3.10)
for any x and u. Finally, let (Km:3t x R1 - R) be another sequence of functions such that for m = 1, 2, .. . IKm(x, u)I < kFm(x, u) + Mm(x),
(3.11)
Mo as m -+ oo. In Situation I it where k > 0 is a constant, and Mm will be assumed that the functions Km are all continuous on 91 x R' and converge to Ko locally uniformly on 9t x 911, M, = M2 = .... and all the functions Mm are constant. In Situation II it is assumed that all the Km belong to the class E and 1-converge to K0, while the functions Mm are µ-integrable and converge in L 1 (91, p) to the function Mo. Finally, let (um), m = 1, 2, ... , be a sequence of L1(91, µ)-functions uo in the wc (ryl)-sense in Situation I as with values in R' such that u,,, uo in the ws(gi)-sense in Situation II. m oo, and um THEOREM 3.5 [1271. Let (Fm), (Km), and (um) be sequences of junctions
satisfying all the above conditions. If
f
Fm[x, Um(x)]µ(dx)
-.f Fo[x, uo(x)]µ(dx)
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
326
as m - oo, and the integrals here are all finite, then
f
Km[x, um(x)]i(dx)
f Ko[x, uo(x)],u(dx)
asm -+oo.
PROOF. We assume that Mm(x) - Mo(x) almost everywhere in Situation II. Let Fm (x, u, A), where A > k, be the convex envelope of the function
AFm(x, u) + Km(x, u). We prove that in Situation I the function (x, u) -' Fo(x, u, A) Fm(x, u, A) is continuous on 91 x Rr and that Fm(x, u, A) locally uniformly on R x R' as m oo. In Situation II, Fm(x, u, A) is a 7.(9R)-function that 1-converges to Fo(x, u, A) as m -" oo. Take an arbitrary e > 0. Let A c R be a set such that the functions Fm and K,,,, m = 0, 1, ... , are continuous on A x R1 and locally uniformly convergent to F0 and K0, respectively, and the M,,,, m = 0, 1, ... , are continuous on A and converge
oo. In Situation I we let A = R, while in to Mo uniformly on A as m Situation 11 the set A is compact and µ(R\A) < e. For A > k we have for x E A that A.Fm(x, u) + Km(x, u) > (A - k)00
- q,,,
where 00 = inf 0(u) and q,,, = SUPXEA M,,, (x). The functions AFm +Km are
nonnegative on A for A > (q,,,/0o) + k. of points in A the functions F, (u) _ For every convergent sequence AFm(x,,, u) + Km(x,,, u) converge to Fo(u) = AFm(xo, u) + Km(x, u), where x0 = x,,. By (3.10), the sequence F,, satisfies all the conditions of Lemma 3.7, and this implies that the convex envelopes of the F converge
to the convex envelope of F0. This proves that the function (x, u) -, Fm(x, u, 2) is continuous on A x 3It. The sequence (qm), m = 1, 2, ... , is bounded. Let q0 = sup q, Using
Lemma 3.7 once more for an arbitrary convergent sequence (xm), m = 1, 2, ... , of points in A, we get that F,,,(xm, u, A) F0(x0, u, A) as m oo, where x0 = limm, xm. This proves that the functions (x, u) Fm(x, u, A) converge locally uniformly to (x, u)
Fo(x, u, A) on A x R1
asm -oo. Let
P. = f
F,,,[x, um(x)]/L(dx),
QM =
IA
f
oo, by a condition of the theorem. Let be a sequence of positive integers such that
Then Pm + Qm -+ PO + Q0 as m
(mk), MI < m2 <
Fm[x, um(x)]it(dx).
A
...,
lim PmA = lim P,,.
k-oo
m-oo
§3. SEMICONTINUITY AND CONVERGENCE
327
Since Pm and Q. are nonnegative and Po + Qo < oo, this implies that limk_. Pm, < oo, and the sequence (Qm,) has a limit. We get that
li mPm=Po+Qo- lim Qmx. k-oo
However, Theorems 3.1 and 3.2 give us that limk_, QmA > Qo, which leads us to conclude that limm_.oo Pm < P0. On the other hand, since limm... Pm > Po, it thus follows that Pm --- Po as m - oo. This implies that Qm Qo as - oo. For each m m J{AFm[X. um(X)] + Km[X, um(X)]}L(dx)
fFm[X.Um(X).AJ/2(dX).
(3.12)
For A > k + qo/9o all the functions x Fm[x, um (x). A] are nonnegative. Passing to the limit in (3.12), we get that M-00
T1 -M f Fm(x, um, A)µ(dX).
m-oo A
In view of Theorem 3.1, fir-m-
f
m-+oo A
f
Fo(x, uo, A)p(dx).
(3.13)
Fm(x, um)P(dx) f Af Fo(X, uo)p(dx)
(3.14)
Fm(X, um, A),u(dx) >
A
Further, by what has been proved,
Af A
A
as m - oo. Comparing (3.12)-(3.14), we conclude that {2F0(x, uo) + K(x, uo) - Fo(x, uo, A)}p(dx) JA
>
f Ko(x, uo)p(dx) -M-00 lim f Kr(x, um)p(dx). A
(3.15)
A
By Lemma 3.6, the absolute value of the integrand on the left-hand side here does not exceed the function 2kFo[x, uo(x)] + 2Mo(x), which is integrable over A, and it tends to zero as A -, oo. This implies that the integral on the left-hand side of (3.15) tends to zero as A - oo. Passing to the limit in (3.15) as A oo, we find that lim
f K,,,[x. um(x)]p(dx) > f Ko[x, uo(x)]p(dx).
m__oo A
A
(3.16)
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
328
Replacing Km by -Km (m = 1, 2,...) and Ko by -K0 in these arguments, we get
lim
m-oo
f f
f
Km[x, um(x)],u(dx) <
Ko[x, uo(x)]µ(dx).
(3.17)
A
A
It obviously follows from (3.16) and (3.17) that Km[x, um(x)]/L(dx) -' f Ko[x, uo(x)]µ(dx) A
A
as m - oo. The proof of the theorem is complete in Situation I, since A = 91 in this case. Consider Situation II. Let
> 0 be arbitrary, and choose A such that
Fo[x, uo(x)]µ(dx) < ,l,
Mo(x)1u(dx) < ,.
f9t\A
J9t\A
Asm -oo, Fm[x, um(x)]p(dx) = Qm -' f9 t\A
f
J
t\A
Fo[x. uo(x)]1(dx) = Qo,
M.(x)µ(dx) - f\A 9
From this, there is an m, such that for m > m 1 f9
t\A
f
Fm[x, um(x)]µ(dx) < pl.
Mm(x),u(dx) < q.
(3.18)
t \A
It is obvious that for m > m, (k + 1),.
(3.19)
Ko[x, uo(x)].u(dx) <(k+1),7.
(3.20)
Km[x, L\A
It is also clear that
L\A
Let m2 > m, be such that for m > m2
I f K,,,[x, u,,,(x)]u(dx) A
f Ko[x, uo(x)]µ(dx)I < IT A
By (3.18)-(3.20),
I f Km[x, um(x)]/L(dx) - f K0[x. uo(x)],u(dx)I
<,7+2(k+ 1)q
§3. SEMICONTINUITY AND CONVERGENCE
329
for m > m2. Since q > 0 is arbitrary, this proves that
J Ko[x, uo(x)]u(dx) = m-+oo lim f Km[x, um(x)]u(dx). t
(3.21)
9l
Thus, the theorem is proved in Situation II under the temporary assumption that Mm(x) - Mo(x) almost everywhere. We take an arbitrary sequence of positive integers and select from it a subsequence (Mk), MI < m2 < ..., such that Mmk (x) - Mo(x) almost everywhere. In view of what has been proved, (3.21) holds for this sequence. Since the original sequence of positive integers was arbitrary, this establishes that
lim f Km[x, um(x)]u(dx)
m-'oo
fK0[x, uo(x)]u(dx).
The proof of the theorem is complete. 3.7. Corollaries to the theorem on convergence with a functional. Below,
U denotes a bounded open set in R. We consider functions of the class Wp (p > 1) defined on U and with values in Rm. For every such function f let f' denote the Jacobi matrix of the mapping f. The space of m x n matrices is denoted by Rm". Let F : R'"" - R be an arbitrary function satisfying the following conditions: a) F is an essentially convex function on R"'" b) There exist constants k1, k2 > 0 such that kllvlp < F(v) < k2lvlp for every v E Rm".
THEOREM 3.6. Let (f,: U --' RI), v = 1, 2, ... , be a sequence of Wp (U)functions converging in L1 to some function fo E Wp. Assume that
r
F[fo(x)]dx
u
F satisfies conditions a) and b). Then f - fo in Wn (U). of functions satisfies the conditions in the theorem. By Lemma 3.5, the functions converge weakly to PROOF. Assume that the sequence
fo in L,(U) as v - oo. For V E Rm" let K(x, v) = Iv - fo(x)Ip. For any x and u, Iv - fo(x)I" <- (Ivl + Ifo(x)I)p < 2p-i(IvIp + Ifo(x)Ip) < (2p-' /k1)F(v) + 2p-' Ifo(x)Ip. Let Fm(x, v) = F(v) and Km(x, v) = K(x, v). The functions Km and Fm satisfy all the conditions in Theorem 3.5 relating to Situation II; hence
f uK[x,
fu
K[x, fo(x)]dx = 0,
as v - oo, i.e., Ilf,' - follo.u - 0 as v - oo. This proves the theorem.
M. FUNCTIONS OF A REAL VARIABLE AND PDE'S
330
§4. Some properties of functions with generalized derivatives 4.1. A theorem on differentiability almost everywhere. The concept of
a function differentiable in the sense of convergence in R was defined in Chapter II, § 1.3, where 9t is a topological vector space. By analogy it is possible to introduce the concept of the limit of a function in the sense of convergence in 91 and the concept of continuity in the sense of convergence in 91. These concepts are needed below in studying the question of differentiability almost everywhere. We present the definitions. Let U c R" be an arbitrary open set whose elements are real functions defined in the unit ball B(0, 1) of R". A number k E R is called the limit of a function f: U R at a point a in the sense of convergence in 91, or, briefly, the 9t-limit of f as x a, if there exists a number ro > 0 such that for 0 < r < ro the function
F,: X EB(0, 1)- f(a+rX)-k belongs to the class 91, and F, tends to the function identically equal to zero as r 0, in the sense of the topology in R. (We assume that the zero function belongs to 91.) If, in particular, the value f (a) of f at a is the limit of f in the sense of convergence in 91, then we say that f is continuous at a in the sense of convergence in 9t. By choosing 91 in different ways we get different variants of the concept of limit and continuity at a point for a function. Suppose, for example, that 91 is the space M of bounded functions (p: B(0, 1) R and the topology in M is defined by the norm 114IIM =
sup Icv(x)I.
xEB(O. I )
It is not hard to show that the limit in the sense of convergence in M coincides with the limit in the usual sense. The concept of a function continuous at a point a E U in the sense of convergence in M is equivalent to the usual concept of a function continuous at a. A function f : U -+ R is said to be L p-continuous at a point a E U if lim
h-.0
f
If(a+hX)- f(a)IPdx=0.
(4.1)
(0.1)
If f is Lp-continuous at a, then we say also that a is a Lebesgue L p-point
of f. Performing a change of the variable of integration in the integral in (4.1) according to the formula a + rX = 1, we get that (4.1) is equivalent
§4. FUNCTIONS WITH GENERALIZED DERIVATIVES
331
to the following condition:
f
lim
r-o B (1 a, r )I I
(
a,.)
f(t) - f (a)I p dt = 0.
Let U C R" be an open set. Denote by'23o(U) the collection of all Borel sets lying strictly inside U. A measure on U is defined to be any countably additive function p : 'B0(U) -+ R. Let f be a function of class L 1.1oc(U). Then a certain measure p f is defined by
p f(A) =
J
f (x)dx,
A E 9Ro(U).
A
The measure p f is called the indefinite integral of f. Let X E U, and let p be a measure on U. The limit µ[B(x, r)] = (Dp)(x), lim
-o I B(x, r)I if it exists, is called the density of p at the point x. The measure p is said to be differentiable at a point a if its density (Du)(x) at this point is defined and finite. THEOREM 4.1. Every measure p in an open set U C R" is differentiable almost everywhere in U. If y is the indefinite integral off E L1,10C(U), then
Dp(x) = f (x) for almost all x E U. (See, for example, [149] for a proof of the theorem.) Let E C U be a measurable set, and let PE(A) = IA n El for A E %o(U).
The function pE is a measure. Obviously, PE is the indefinite integral of the indicator function XE of E. It follows from Theorem 4.1 that for almost all x E E the density of the measure pE is equal to 1, and hence the set of x E E such that DPE(x) < 1 is a set of measure zero. In particular, if DpE(x) < 1 for all x E E, then E itself has measure zero; this fact was used in proving Theorem 11.1 of Chapter II. We have the following theorem, also due to Lebesgue.
THEOREM 4.2. Suppose that U is an open subset of R", and f : U R" is a function of class Lp,10C(U). Then f is continuous in the sense of convergence in L p almost everywhere in U. In other words, if f E L1( U), then for almost all x E U B(0.1)
ash- 0.
If(x+hX)-f(x)IpdX,0
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
332
Note that Theorem 4.1 implies only that for almost all x E U
f (x) = lim
1J
h-0 Qn
f (x + hX)dX.
B(0,1)
For the proof of Theorem 4.2 see, for example, [ 183]. LEMMA 4.1. Let u : U -+ Rn (U an open subset of Rn) be a function in lo,1a(U). If X E U is a Lebesgue L y-point of u, then
- iu u(z)W(zhx)dz-iu(x) f
B(o.1)
W(y)dy
for every function W E C°° compactly supported in Rn inside the unit ball
B(0,1)ash-+0. Performing a change of variable in the integral by setting (z - x)/h = y, we get that PROOF.
f
(z h x) dz = f u(x + hy)W(y)dy hU(Z), B(o.1) u Assume that x is a Lebesgue LP-point of f. Then
Iu(x + hy) - u(x)IP dy - 0 as h -+ 0; this clearly gives us the statement of the lemma. LEMMA 4.2. Suppose that u is a function of class W110c (U), where U is
an open subset of Rn, and p > 0 is a function of class C00 with compact support in B(0. 1). Then for almost all x E U
4
u(x + hy) - u(x) h
(0.1)
fh
"
lf
in
o
i=1
u
ip(y)dy
a
(z) z; 1
x,,p(z - x) dz dt
(4.2)
when 0 < h < p(x, 8U). PROOF. Let u E Wv 1oc (U ), and let
9(h)
=
18(0,1) u(x + hy)ip(y)dy = hn
f
e(X
X
r)
For h < p(x, 8U) the closed ball B(x, h) is entirely contained in U, and the function z ip((z - x)/h) vanishes outside this ball; hence 0(h) = hn
Juzo (z x) d
§4. FUNCTIONS WITH GENERALIZED DERIVATIVES
333
From this it is clear, in particular, that 0 is a C°O-function in the interval (0, p(x, 8U)). Differentiating with respect to h, we get that h"9'(h)+nh"-I0(h)=
_
2fUE00 -
I
`Czhx)(zj-xi) J
x u(z)dz.
Let 2(z) = ip((z - x)/h). Then 02
0z;
z-x
1
hay;
At
and hence
(zhx)(z;-x;)=hjzt-h4,(zhx) [(zi-X09(ZMl
We substitute this into (4.3). The function
(Z_X) belongs to C°°, and is compactly supported in U for h < p(x, 8U). Transferring the differentiation operator 0/,9z; in the well-known way to the function f, we get from this that z, nhn- I0(h) = f h"9'(h) + At 8z;
x; au ()(z x)
dt_
+hJuu(z)IP(zhx)dz.
nh"-10(h); thus,
The last term on the right-hand side is equal to
9,(h)_
8u(z)z;-x; z -xl dz.
h"fuz;
(
h
h
From this we find by integration that 8(h) - 0(e)
jhEif axaJ (z) z-x; "
t
rp
(z_x)d)dl
(4.4)
Further,
0(e) = J
u(x + ey)So(y)dy ro .I)
By Lemma 4.1, at each Lebesgue point x of u (almost all x E U are such points)
40. 1)
u(x + ey)c(y)dy - u(x) fo.l) 9)(y)dy. e(
Passing to the limit in (4.4), we get that if x is a Lebesgue point of u, then (4.2) holds for this x. The lemma is proved.
M. FUNCTIONS OF A REAL VARIABLE AND PDE'S
334
THEOREM 4.3. Let f : U - R be a function of class WP 10C(U). Then the
linear mapping f'(x) is the differential off at the point x E U in the sense of convergence in Wp for almost all x E U.
This is a special case of a certain general theorem established by the author in [ 131 ].
PROOF. For x E U and h > 0 let Rh,.x be the function Y E B(0,
1
-
f (x + hY) - f (x)
)
h
of (x)Y
1.
=l ax'
The theorem asserts that for almost all x E U the quantity IIRh.x II wp[B(o.I )]
tends to zero as h -' 0. Let q' be an arbitrary Col-function with compact support in B(0, 1) and such that fR,,rp(y)d y = 1. For U E W I [B(0, 1)] let IIuIIwp =
rp(z)u(x)dx
+ t-I
Bro. l)
lla
(4.5) llL,[B(o.l)]
This defines a norm which, in view of Sobolev's classical results, is equivalent to IIuIII.p.B(o.I). To prove the theorem it thus suffices to establish the following statements: A. I I y Rh.x I I L,[B(o. 1 ) , -' 0 as h - ' 0 for almost all x E U for each i = ,
n.
B. For almost all x E U
I
Rh.x(y)(p(y)dy - 0
B(O,I)
ash-'0.
Let us prove A. For each i = 1, ... , n aye
Rhx(y) = ax (x + hy)
- ax (x),
and, hence, P
ayiRh.xllc.
(frno.i)Iax,(x+hy)-ax (x)I
dy
In view of Theorem 4.2 the integral on the right-hand side tends to zero as h -. 0 for almost all x E U, and statement A is proved. We now prove B. The proof is based on the equality in Lemma 4.2. Let x be a Lebesgue LP-point of f and each of its derivatives. Let 0 < h < p(x, 8U). Then, by Lemma 4.2,
§4. FUNCTIONS WITH GENERALIZED DERIVATIVES
335
(I/h)[f (x + hy) - f (X)]cD(y)dy (I 1h)
II
of(Z)Z,-X,Z-X)dz
R
/h
In
p
ax;
1=1
r
r
J
dr.
(4.6)
Let us now use Lemma 4.1, with u
We get that as r 1
r"
L,
8x,
IV(Y) = y,co(Y)
0
of (Z)z, -x rz-x1 dZ r
r
Lf (x
BIO.I, 49x,
/J
7as
This implies that the right-hand side of (4.6) tends to the limit
J
"
f O. I)
Cdr
(X)Yr(P(Y)d3'
h -. 0. Thus, for the given x E C' lim
f(x+hy) -f(x)(D(v)dv
h-040.1)
h
= f ( o.1) i=1 ar(x)v,co(v)dv. ax,
This establishes B, and the theorem is proved.
4.2. Proof of Lemma 1.1 in Chapter II. The lemma can be proved by means of a certain integral representation due to Sobolev. We present the corresponding formula here. A derivation of it can be found, for example, in [ 155] or [ 156]. Let us first consider the case of functions on the unit ball.
Take an arbitrary function rp > 0 of class Col(RI) such that S((p) is contained in the ball B(0, 1/3) and fR, (p(y)d), = I. For every vectorvalued function _1: B(0, 1) - RA of class CI we have
f(X) = f
w(y)f(y)dy + ( O.I)
f
K, (x. ( 0.1)
y)/f (y)dy.
(4.7)
where the functions K, are such that IK,(x,y)I < L/Ix - yl"-I for any x, y E B(0, 1), where L < oo is a constant, and K, belongs to C°° in the subdomain of R-" consisting of all pairs (x. y) with x, y E B(0. 1) and x y. (See, for example, [ 138], [1611, or [ 162] for a proof of (4.7).) It is usual to consider real functions in deriving (4.7). The result can be
111. FUNCTIONS OF A REAL VARIABLE AND PDE'S
336
obtained for vector-valued functions by applying (4.7) to the individual components of the function. Let f be an arbitrary function of class C' on the ball B(a, r). Applying (4.7) to the function f(X) = f (a + rX), we get after obvious transformations that
f(x) = (1/r") f(a.r) 9[(y -a)lr]f(Y)dy B
n
+
i=1
where Hi (x,
f
B(a.r)
Hi(x. Y)
"1(x-a K,
of
axi (Y)dY.
Y-a`
,
r-1
y)
r
1
r
The function H; is a function of class COO on the set of all pairs (x, y) E RZ"
such that x, y E B(a, r) and x
y, and it admits the estimate IH;(x, y)I <
L/Ix - yIn-I. Let U be an open domain in R" and f : U -, Rk a function of class
W.' (U), where 1 < m < n. Assume that there exist numbers a > 0, S > 0, and M < oo such that for every ball B(a, r) C U with radius less than 8
IIf'(x)II'" dx <
Mrn-m+mR
(4.9)
Assume first that f is a C°°-function. Let a E U be an arbitrary point, and let r < p(a)/3, where p(a) = p(a, 8U). We use the integral representation (4.8). Define
(1/r") fa.r) ip[(y - a)lr]f(Y)dy = k B(
Then
If(x) - kI 5 E t=1
IH1(x.Y)I JB(a,r)
I
ax (Y)I dy.
We have that I
H1(x,Y)I 5 LIx-
yln-I,
(Y)I
5 IIf'(Y)II
az
From this,
If(x) - kI < f
B(a.r)
(dLIIf'(Y)IIdy/Ix
- YIn-1)
§4. FUNCTIONS WITH GENERALIZED DERIVATIVES
337
For X E B(a, r) the inclusion B(x, 2r) D B(a, r) holds, and at the same time B(x, 2r) c B(a, 3r) c U. Hence, If (x) - kI < nL
f f
II f'(Y)II (dy/I x - yl"-' )
B(.Y.2r)
= nL
(0.2r)
II f'(x +
z)Ildz/IzI"-I.
(4.10)
Let
v(t) = J
A(t) = f0.r) Ilf'(x + z)IIda2,
S(IIf'(x + z)lldz, ( 0.r)
where do is the area element of the sphere. We have that v(t) = fo A(t)dt.
In our notation, (4.10) gives us that
If (x) - kl < nL
f
2r (A(t)/t"-1)d1
=
nLf
2r
v'(t)dt/tn-1.
(4.11)
0
0
Let us estimate v(t). By the Holder inequality,
v(t)= f
(
0,1)
Ilf'(x+z)Ildz I/m
llf'(x + z)IIm dz
<-
anI/mtn-n/m
B(O.t)
< (anM)I /mtn/m-I+"tn-n/m
(anM)I/mtn-I+".
(4.12)
We transform the integral on the right-hand side of (4.11) by integration by parts. The use of (4.12) gives us f 2r
`I
di = L(1
+ (n - 1) f 2r v(tin)dt < - CI M1 /mr".
I2r 10
From this,
(4.13) If(x) - kI < C2M1 /"'r,,, where C2 = 2"nLQ,I,/m(l + (n - 1)/a) is a constant. Since the point x E B(a, r) was arbitrary, it follows from (4.13) that the oscillation of f on the set B(a, r)\E does not exceed CMI /mr".
We now get rid of the condition that f E C°°. Let f : U - Rk be an arbitrary function satisfying the conditions of Lemma 4.1. Take an arbitrary averaging kernel a > 0, and let fh (h > 0) be the mean function for f constructed from this kernel. For every x E Ch m
f'(x + hz)a(z)dz lll V8(0.1)
<
f
0.1)
Ilf'(x + hz)II"a(z)dz.
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
338
Integrating both sides over the ball B(a, r) C flh, we get that
f
II fn(x)II mdx < (a.r)
f
IIf'(x + hz)IImdx a(z)dz. 0.1)
(ka.r)
If X E B(a, r), then x + hz E U for Izl < 1. Estimating the inside integral, we find from (4.9) that Mr*'-'"+m'.
II fh(x)II m dx <
(4.14)
Thus, fh satisfies the same conditions as f. By what has been proved, this gives us that the oscillation off on every ball B(a, r) with a E Uh and r < p(a, 8U)/3 does not exceed Cr", where C is a constant. This enables us to conclude that the family (fh) of functions is uniformly equicontinuous on every set lying strictly inside U. Ash - 0 we have that Ilfh -f II L, (A) -' 0
for every set A lying strictly inside U. This implies that the family of functions fh converges locally uniformly to some function f as h - 0. The function f is continuous in U, and f (x) = f (x) almost everywhere in U. Passing to the limit, we get that the oscillation of f on B(a, r) does not exceed CMI /mr° for r < p(a)/3. The proof of the lemma is complete. 4.3. An estimate of the modulus of continuity of a monotone function of
class W . Let U be an open subset of R", and f : U ---' Rm a continuous mapping. We say that f satisfies condition E(T), where T > 1 is a constant, if d[f (B(x, r)]:5 Tdf [S(x, r)] for every closed ball B(x, r) c U. In particular, if f is a homeomorphism, then f satisfies condition E(T) with the constant T = 1. LEMMA 4.3. Let f : S(xo, r) - Rm be a continuous function of class W on the sphere S(xo, r) in R". Then
d(f[S(xo, r)]):5 Cnr'l"
IIf'(x)II" do (JS(XO.,)
where C" is a constant, 0 < C" < oo. PROOF. Since n > dim S(xo, r) = n - 1, the Sobolev imbedding theorem gives us the estimate d (f [S(xo. r)]) <- C(r)
f
I/n
IIf'(x)II" do
S(x0.r)
It is not hard to conclude from similarity considerations that C(r) _ C"rIl", which is what was required to prove (see also [48]).
§4. FUNCTIONS WITH GENERALIZED DERIVATIVES
339
THEOREM 4.4. Let U be an open subset of R", K c U a compact set, and V D K an open set strictly interior to U. Then there is a constant 6 > 0 such that If (XI) - f (x2)I <- C"(M)IIn 719(IxI - x21) for every mapping f : U - R'" of class W, I.c (U) satisfying condition E(T) and for any two points xI and x2 in K with IxI - x21 < 8, where C" is the constant in Lemma 4.3,
M=
f
IIf'(x)II" dx,
V
and
9(t) = (2/ln(2/t))I/";
9(t)-0ast--+0. PROOF. Let p = dist(K, R"\ V). Denote by a the largest number t such
that t < 2 and (t/2) + 112 < p. Let xI and x2 be two arbitrary points in K such that IxI - x21 = 2h < J.
Then h < 6/2 < 1. Let x0 ='(xI + x2). We construct the balls B(xo, h) and B(xo, v*rh-) For every x E B(xo, vrh-) we have that
Ix-x11
Let (f,), v = 1, 2, ... , be a sequence of C°°-functions such that f (x) uniformly on V as v - oo and
J I f,'(x) - f'(x)I" dx < 2-" for every v. We have that
fB(x.v) f(x) - f'(x)dx = 0f
S
If.(x) - f'(x)IdQ
Let
f'(x)I" dx = JS(x0.r)
For each v = 1. 2.... VIT
fo
and hence
Ff M=I
o
F,,(r)dr < oo.
dr.
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
340
0 as v -. oo for almost all r E (0, V ri). For each This implies that v such that this holds, the restriction of f to S(xo, r) belongs to the class W,,l (S(xo, r)). For r E (0, f) let
y(r) = d (f [S(xo, r)]) By the estimate of Lemma 4.3, [Y(r)]" < Cn
J
r
(xo.r)
If'(x)Ido
for almost all r E (0, Oh_). Integrating with respect to r, we get that
1
f
[Y(r )l" dr < Cn
r
<- Cn f
'f
If'(x)I" do dr
S(xo,r)
h
B(xo,f)
If'(X)I" dX < CnM
Let A be the set of t E [h, v1h_] such that [Y(t)l
h
[Y(r)]" dr.
dr -< f
(4.15)
r
The measure of A is positive. Indeed, assume not. Then for almost all t
yt) t
f
h
dr > r
` f V [Y(r)l" dr. h
Integrating termwise, we get a contradiction. Fix an arbitrary t E [h, vrh-] for which (4.15) holds. For this t [y(t)l"Invh
which implies that Y(t) <_
[l"( M
n )I/,/n
= C"M1
1"e(Ix1
- x21)
We have that f (x1) E f [B(xo, t)] and f (x2) E f (B(xo, t)); hence If (x1) f (x2)I does not exceed the diameter of f [B(xo, 1)]. Since f satisfies condition E(T), the diameter of f [B(xo, 1)] does not exceed Td(f [S(xo, t)]) _ Ty(t). We get finally that If (XI) - f (X2)I < Ty(t) < C"M11"T19(Ix1 -
which is what was required to prove.
X21),
§5. DEGREE OF A MAPPING
341
COROLLARY. Let U be an open set in R. Assume that the mapping f : U - R'" has property E(T). If f c- Wnl 1a(U), then f is differentiable almost everywhere in U.
PROOF. ForxEU,h>0and YER" let F'h(Y)=f(x+hY)-f(x)
h
On the basis of Theorem 4.3 the functions Fh converge in W [B(0, 1)] to the function
f'(x)(Y) _
7i (x)Yi i=1
for almost all x E U as h --+ 0. A simple change of variables enables us to conclude that the functions Fh(Y) converge to f(x) (Y) in W,,' [B (0, r) ]
for any r > 0; in particular, Fh(Y) -' f'(x)(Y) in W [B(0, 3)] ash -' 0. The function Fh has property E(T) for each h, and there are constants ho > 0 and M < oo such that for 0 < h < ho IF,,(Y)IdY < M
We use the theorem, setting K = B(O,1), V = B(0, 2), and U = B(0, 3) in it; we get that the family (Fh), 0 < h < ho, of functions is uniformly equicontinuous on B(0, 1). Consequently, Fh(Y) - f'(x)(Y) uniformly in B(0, 1) as h - 0. However, this means that f'(x) is the usual differential of f at x. The corollary is proved. §5. On the degree of a mapping
We present proofs of Lemmas 2.1 and 2.2 in §2 of Chapter II. All references below relate to that section. Let G be a compact domain in R", and let fo: G R" and fl: G - R"
be mappings of class Ck, k > 1. The mappings fo and f, are said to be smoothly homotopic as mappings of the pair (G, OG) to the pair (R", R"\{y}) if there exists a homotopy c of fo and f, which is a Ckmapping on G x [0, 1). LEMMA 5. 1. If the Ck-mappings (k > 1) fo : G - R" and g, : G - R" are homotopic as mappings of the pair (G, OG) to the pair (R", R"\{y}), where y E R", then they are also smoothly homotopic.
R" be an arbitrary homotopy of fo and PROOF. Let f : G x [0, 1] f, as mappings of the pair (G. OG) to the pair (R". R"\{ y}). For each t E [0, 1], f,: x " f (x, t) is a mapping of (G, 8G) to (R", R"\{y}). Let H = f (8G x [0, 1 ]). The set H is compact, and y 0 H. This implies that there
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
342
exists a 6> 0 such that I z - y I < 6 for all z E H. Let ? : G x [0,1 ] --+R" be a COO-mapping such that I W (x, t) - f (x, t) I <612 for (x, t) E G x [0,1 ]. A mapping w satisfying this condition can be constructed by extending f to Rn+I with preservation of continuity and applying the averaging operation to the resulting extension. Let 'P(x, 1) = W (X, t) + (1 - t)[fo(x) - w(x, 0)] + t[fi (x) - W (x,1)] Obviously, ip is a mapping of the same smoothness class as fo and fl, and also q'(x, 0) = fo(x) and Sp(x, 1) = f,(x). We let q t(x) = 9(x, t) and show that q, is a mapping of (G, 8G) to (R", R"\{y}). Take an arbitrary point xo E 8G. Let zo = f,(x) and z, = Spt(xo). Take an arbitrary point xo E 8G. Let zo = fr(x) and z, = tot(xo). Then
Izi - zol < I u(xo, t) - f (xo, t)I + (1- t)lfo(xo) - w(xo. 0)I + tIf, (xo) - v(xo, t) I < (6/2) + (1 - t)6/2 + to/2 = a
The point zo belongs to H, and hence Izo - y j > J. Since I z, - zol < 6, this implies that rpt(xo) 0 y. The point xo E 8G was chosen arbitrarily, so this proves that qP, is a mapping of (G, 8G) to (R", R"\{y}), and thus ip is a homotopy of fo and fl. The lemma is proved. LEMMA 5.2 (SARD'S THEOREM). Let U c R" be an open set, f : U - R" a CI -mapping, and E c U the set of all points x E U at which the Jacobian
of f vanishes. Then f (E) is a set of measure zero in R.
PROOF. Let a be an arbitrary point of E, and let r > 0 be such that the closed cube Qo = U(a, r) is contained in U. For x, y E Qo let y(t) _ (1 - t)x + ty. Then
f(y) - f(x) = f[y(1)l - f[y(0)l
=1 dt(f[y(t)])dt = f' E CV [y(t)](y1-x;)dt. (5.1) '&Xi i=1
Further,
f (y) - f (x) - f'(r)(y - x)
(,Lf [y(t)] - 8x (x)) (y; - x,)dt.
(5.2)
The functions 8f/8x; are continuous. Since Qo is compact, this implies that they are bounded and uniformly continuous on Qo. Let I(8f/8x;)(x)I < M; for all i = 1, ... , n and x E Qo, and let e; be the modulus of continuity of 8f/8x;. Then (5.1) gives us that
§5. DEGREE OF A MAPPING
343
n
If (Y) - f(x)I 5 EMiIY1 - 4:5 MY - xl 1=1
where M =
Ei M?. We conclude similarly from (5.2) that If (Y) - f (x) - f'(x)(Y - x)l <- COY - xl)I Y - xl.
(5.4)
where e
Let v be an arbitrary positive integer, and divide the cube Qo into v" equal closed cubes by planes parallel to the coordinate planes. The diameter of each cube in the subdivision is equal to r f /v, and its volume is (r/v)". Let Q be a cube in this subdivision such that Q n E is nonempty. We estimate the measure of the set f (Q). Take an arbitrary point y E Q n E and let v = f (y). Then V E f (Q), and
If(x) - vI< MIx- y1:5 Mfr/v for every x E Q.
According to the definition of E, J(y, f) = 0. Let Y be the set of all
points u E R" of the form u = f (y) + f'(y)(h), where h E R. Since J(y, f) = 0, Y is a k-dimensional plane passing through the point v = f (y), where 0 < k < n - 1. We construct an arbitrary (n -1)-dimensional plane X containing Y. For X E Q we have that If (x) - f (Y) - f'(Y)(x - Y)1:5 CO X - YI )I x - YI > e(r f /v)(rv1'n_)/v.
The point f (y) + f'(y)(x - y) belongs to X; hence the distance from f (x)
to this plane is at most e[r f /v](r f /v). Let K = B(v, Mr f /v) n X. It follows from what has been proved that for x E Q the point f (x) lies in the right circular cylinder of height 2e(r f /v)(rV'n/v) whose middle section is the (n - 1)-dimensional disk K. This implies that If (Q)I <-
v)(r/v)" = Ct(rvil v)I QI.
(5.5)
Let Qi, ... , Q," be the cubes in our subdivision of Qo which contain points of E. Obviously,
344
III. FUNCTIONS OF A REAL VARIABLE AND PDE'S
Applying (5.5) to each cube Q;, we get that If (E n Qo)I <- Ce(rvrn_/v)IQoI
The right-hand side of this inequality tends to zero as v - oo. This tells
us that If(EnQ0)I=0. Every point a E E thus has a neighborhood Qo such that f (Qo n E) is a set of measure zero. Covering E by countably many such neighborhoods, we get that f (E) is a set of measure zero. The lema is proved. PROOF OF LEMMA 2.1. Let G be a compact domain, and f : G -. R" a mapping of class Ck, where k > 1 and a is an (f, G)-admissible point. Let A I = f - I (a), and let A0 = R" \G°. Take an arbitrary function C E W(A0, A1). The function C is equal to 1 in some neighborhood U1 of the set A 1. Let P = f (G\U1). The set P is compact, and a 0 P. Hence, there is a 6 > 0 such that the ball B(a, 6) does not contain points of P. According to (2.6), for every y E R" the exterior form 9y is defined. We construct
the exterior form f'B,, from it. For y r= B(a.8) the form dC A f'9 is defined on G°\U1. On U1 we have that dC - 0. We extend the definition of the form dC A f' By by setting it equal to zero on U1. As a result we get a form defined and continuous everywhere in G°. The coefficients of this form are continuous functions of x and y. Let
u'(y, f,G) = -
1f co,
d A fOy.
Q)
The integral on the right-hand side does not depend on the choice of C, as can be shown by arguments repeating the beginning of the proof of Lemma 2.4. By the same Lemma 2.4, if f is regular with respect to y, then
u' (y. f G) = u(y. f G). On the basis of classical theorems on integrals depending on a parameter, u' (y, f, G) is continuous. Lemma 5.2 allows us to conclude that the set
of y E R" such that y 0 f (8G) and f is regular with respect to y is a set of measure zero. This implies, in particular, that u* (y, f, G) takes integer values on a dense subset of B(a, 6), which, since u' is continuous, is possible only if u' is constant and equal to an integer for y r= B(a, 6). Thus, #* (y. f, G) is an integer for y E B(a,6). Let fo and f1 be two homotopic Ck-mappings (k > I) of the pair (G, (9G) to the pair (R", R"\{ a}). By Lemma 5.1, there exists a smooth homotopy
f of fo and f, as mappings of pairs. Let A 1 = f -I (a), and let A, be the image of A 1 under the mapping (x, t) E R"+ I
x. The set A I is compact,
and G° D A1. Take an arbitrary C E W(A0. A1), where A0 = R"\G°. For
§5. DEGREE OF A MAPPING
345
t E [0, 1] let f, be the mapping x 1-- f (x, t), and let
u(t)_- 1 to, f
ddnf,-0..
"\Ai The coefficients of the form dd A f,* 9Q depend continuously on t, and this
implies that µ is continuous. For each t E [0, 1] the quantity µ(t) is an integer, and hence p(t) -const in [0, 1]. If fo and f, are regular with respect to a, then µ(0) = u(a, fo, G),
ju(l) =#(a. fl, G).
and thus u(a, fo, G) = µ(a. fi G). ,
The lemma is proved. PROOF OF LEMMA 2.2. Suppose that f : G -i R" is continuous, and a is an (f, G)-admissible point. We construct a sequence of C°°-functions (p,: G -. R" converging uniformly to f. By Lemma 5.2, for each v there
is a point y such that l y, - al < l 1v and f, is regular with respect to y,,. Let h = y - a, and let f, = rp + h,,. As v -1 oo the mappings f, converge to f uniformly, and each of them is regular, with respect to a. The lemma is proved.
Bibliography
ROBERT A. ADAMS
1. Sobolev spaces, Academic Press, 1975. LARS V. AHLFORS
2. Quasiconformal reflections, Acta Math. 109 (1963), 291-301.
3. Kleinsche Gruppen in der Ebene and im Raum, Festband 70. Geburtstag R. Nevanlinna, Springer-Verlag, 1966, pp. 7-15. 4. Conditions for quasiconformal deformations in several variables, Contributions to Analysis (Collection Dedicated to Lipman Bers), Academic Press, 1974, p. 19-25. 5. Invariant operators and integral representations in hyperbolic space, Math. Scand. 36 (1975), 27-43. 6. Quasiconformal deformations and mappings in R", J. Analyse Math. 30 (1976), 74-97. 7. A singular integral equation connected with quasiconformal mappings in space, Enseign. Math. (2) 24 (1978), 225-236. 8. A singular operator in hyperbolic space, Complex Analysis and Its Applications (I. N. Vekua Seventieth Birthday Vol.), "Nauka", Moscow, 1978, pp. 40-44. 9. The Holder continuity of quasiconformal deformations, Amer. J. Math. 101 (1979), 1-9. LARS AHLFORS AND ARNE BEURLING
10. Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101-129. A. D. ALEKSANDROV
11. The existence almost everywhere of the second differential of a convex function and some properties of convex surfaces connected with
it, Leningrad Gos. Univ. Uchen. Zap. Ser. Mat. 6 (1939), 3-35. (Russian) 347
BIBLIOGRAPHY
348
P. P. BELINSKII
12. On the continuity of spatial quasiconformal mappings and Liouville's theorem, Dokl. Akad. Nauk SSSR 147 (1962), 1003-1004; English transl. in Soviet Math. Dokl. 3 (1962).
13. Stability in the Liouville theorem on spatial quasiconformal mappings, Some Problems in Mathematics and Mechanics (M. A. Lavrent'ev Seventieth Birthday Vol.), "Nauka", Leningrad, 1970, pp. 88-102; English transl. in Amer. Math. Soc. Transl. (2) 104 (1976). 14. On the order of proximity of a spatial quasiconformal mapping to a conformal mapping, Sibirsk. Mat. Zh. 14 (1973), 475-483; English transl. in Siberian Math. J. 14 (1973). P. P. BELINSKII AND M. A. LAVRENTIEV [LAVRENTIEV]
15. On locally quasiconformal mappings in n-space (n = 3), Contribution to Analysis (Collection Dedicated to Lipman Bers), Academic Press, 1974, pp. 27-30. O. V. BESOV
16. Integral representations of functions in a domain with the flexible horn condition, and imbedding theorems, Dokl. Akad. Nauk SSSR 273 (1983), 1294-1297; English transl. in Soviet Math. Dokl. 28 (1983). 17. Estimates of the L p-moduli of continuity and imbedding theorems for
domains with the flexible horn condition, Dokl. Akad. Nauk SSSR 275 (1984), 1036-1041; English transl. in Soviet Math. Dokl. 29 (1984). O. V. BESOV, V. P. WIN, AND S. M. NIKOL!SKII
18. Integral representations of functions and imbedding theorems, "Nauka", Moscow, 1975; English transl., Vols. I, II, Wiley, 1979. B. V. BOYARSKII [BOGDAN BOJARSKI]
19. Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk SSSR. 102 (1955), 661-664. (Russian) 20. Generalized solutions of a system of first-order differential equations of elliptic type with discontinuous coefficients, Mat. Sb. 43 (85) (1957),
451-503. (Russian) EUGENIO CALABI AND PHILIP HARTMAN
21. On the smoothness of isometries, Duke Math. J. 37 (1970), 741-750. ALBERTO P. CALDER6N
22. On the differentiability of absolutely continuous functions, Riv. Mat. Univ. Parma 1951, 203-213.
BIBLIOGRAPHY
349
A. P. CALDERON AND A. ZYGMUND
23. Singular operators and differential equations, Amer. J. Math. 79 (1957), 901-921. E. DAVID CALLENDER
24. Holder continuity of n-dimensional quasi-conformal mappings, Pacific J. Math. 10 (1960), 499-515. PETRU CARAMAN
25. n-dimensional quasiconformal mappings, Ed. Acad. Rep. Soc. Romania, Bucharest, 1968; rev. English transl., Ed. Acad. Roman, Bucharest, Abacus Press, Tunbridge Wells, and Haessner, Newfound-
land, N. J., 1974. A. V. CHERNAVSKII
26. Finite-to-one open mappings of manifolds, Mat. Sb. 65 (107) (1964), 357-369; English transl. in Amer. Math. Soc. Transl. (2) 100 (1972).
27. Addendum to [26], Mat. Sb. 66 (108) (1964), 471-472; English transl. in Amer. Math. Soc. Transl. (2) 100 (1972). N. S. DAIRBEKOV
28. On the stability of classes of conformal mappings on the plane and in space, Sibirsk. Mat. Zh. 27 (1986), no. 5, 188-191. (Russian) N. S. DAIRBEKOV AND A. P. KOPYLOV
29. On stability in the C-norm of classes of solutions of systems of linear partial differential equations, Ninth School Theory of Operators in Function Spaces (September 1984), Abstracts of Reports, Ternopol', 1984, p. 137. (Russian) 30. c-stability of classes of mappings, and systems of linear partial differential equations, Sibirsk. Mat. Zh. 26 (1985), no. 2, 73-90; English transl. in Siberian Math. J. 26 (1985). ENNIO DE GIORGI
31. Sulla dz erenziabilitb e I'analiticitb delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25-43. K. O. FRIEDRICHS
32. On the boundary-value problems of the theory of elasticity and Korn's
inequality, Ann. of Math. (2) 48 (1947), 441-471. F. W. GEHRING
33. The definitions and exceptional sets for quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 281 (1960).
BIBLIOGRAPHY
350
34. Rings and quasiconformal mappings in space, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 98-105.
35. Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499-519. 36. A remark on the moduli of rings, Comment. Math. Helv. 36 (1961), 42-46. 37. Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1963), 353-393. 38. The L P -integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. 39. Injectivity of local quasi-isometries, Comment. Math. Helv. 57 (1982), 202-220. 40. Characteristic properties of quasidisks, Sem. Math. Sup., vol. 84, Presses Univ. Montreal, Montreal, 1982. F. W. GEHRING AND F. HUCKEMANN
41. Quasiconformal mappings of a cylinder, Proc. Romanian-Finnish Sem. Teichmiiller Spaces and Quasiconformal Mappings (Bra$ov, 1969), Publ. House Acad. Socialist Republic Romania, Bucharest, 1971, pp. 171-186. F. W. GEHRING AND J. VAISALA
42. The coefficients of quasiconformality of domains in space, Acta Math. 114 (1965), 1-70. V. M. GOL'DSHTEIN
43. A homotopy property of mappings with bounded distortion, Sibirsk. Mat. Zh. 11 (1970), 999-1008; English transi. in Siberian Math. J. 11 (1970). 44. On the behavior of mappings with bounded distortion when the distortion coefficient is close to 1, Sibirsk. Mat. Zh. 12 (1971), 1250-1258; English transl. in Siberian Math. J. 12 (1971).
45. Extension of functions in the classes B,q across quasiconformal boundaries, Theory of Cubature Formulas and Applications of Functional Analysis to Problems of Mathematical Physics, (Proc. Sem. S. L. Sobolev, 1979, vol. 1), Inst. Mat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1979, pp. 12-32. VLADIMIR MIKAILOVITCH GOL'DS`TEIN [V. M. GOL'DSHTEIN] AND SERGEI KONSTANTINOVITCH VODOP'JANOV [S. K. VODOP'YANOV]
46. Prolongement des fonction de classe PpI et applications quasi conforme, C. R. Acad. Sci. Paris Ser. A-B 290 (1980), A453-A456.
BIBLIOGRAPHY
351
V. M. GOL'DSHTEIN
47. Extension of functions with first generalized derivatives from plane domains, Dokl. Akad. Nauk SSSR 257 (1981), 268-271; English transl. in Soviet Math. Dokl. 23 (1981). V. M. GOL'DSHTEIN AND Yu. G. RESHETNYAK
48. Introduction to the theory of functions with generalized derivatives and quasiconformal mappings, "Nauka", Novosibirsk, 1983; English transl., Foundations of the theory offunctions with generalized derivatives and space quasiconformal mappings, Reidel (to appear). V. M. GOL'DSHTEIN, V. I. Kuz'MINOV, AND I. A. SHVEDOV
49. Classes of differential forms analogous to classes of Sobolev functions,
Preprint, Inst. Mat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1981. (Russian) R. Zh. Mat. 1982, 2A655. L. G. GUROV
50. On the stability of Lorentz mappings, Dokl. Akad. Nauk SSSR 213 (1973), 267-269; English transl. in Soviet Math. Dokl. 14 (1973). 51. Stability estimates for Lorentz mappings, Sibirsk. Mat. Zh. 15 (1974), 498-515; English transi. in Siberian Math. J. 15 (1974). 52. On the stability of Lorentz transformations. Estimates for the derivatives, Dokl. Akad. Nauk SSSR 220 (1975), 273-276; English transi. in Soviet Math. Dokl. 16 (1975). 53. On the stability of Lorentz transformations in space, Sibirsk. Mat. Zh. 21 (1980), no. 2, 51-60; English transl. in Siberian Math. J. 21 (1980). L. G. GUROV AND Yu. G. RESHETNYAK
54. On an analogue of the concept of a function with bounded mean oscillation, Sibirsk. Mat. Zh. 17 (1976), 540-546; English transl. in Siberian Math. J. 17 (1976). SZE-TSEN Hu
55. Homotopy theory, Academic Press, 1959. V. P. WIN 56. On an imbedding theorem for a limiting exponent, Dokl. Akad. Nauk SSSR 96 (1954), 905-908. (Russian) V. V. IVANOV
57. Smooth isomorphisms not having quasiconformal fractional powers, Sibirsk. Mat. Zh. 27 (1986), no. 3, 103-11 1; English transl. in Siberian Math. J. 27 (1986).
BIBLIOGRAPHY
352
TADEUSZ IWANIEC
58. On Lo-integrability in PDEs and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 301-322. 59. Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions, Dissertation, Polish Acad. of Sci., Warsaw, 1982. FRITZ JOHN
60. Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391-413. 61. On quasi-isometric mappings. I, Comm. Pure Appl. Math. 21 (1968), 77-110. 62. On quasi-isometric mappings. II, Comm. Pure Appl. Math. 22 (1969), 265-278. F. JOHN AND L. NIRENBERG
63. On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. PETER W. JONES
64. Quasiconformal mappings and extendibility of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88. A. P. KoPYLov 65. On the behavior of quasiconformal space mappings close to conformal mappings on hyperplanes, Dokl. Akad. Nauk SSSR 209 (1973), 1278-1280; English transl. in Soviet Math. Dokl. 14 (1973). 66. On approximation of quasiconformal space mappings close to conformal mappings by smooth quasiconformal mappings, Sibirsk. Mat. Zh.13 (1972), 94-106; English transl. in Siberian Math. J. 13 (1972). 67. Integral averagings and quasiconformal mappings, Dokl. Akad. Nauk SSSR 231 (1976), 289-291; English transl. in Soviet Math. Dokl. 17 (1976). 68. On the removability of a ball for space mappings close to conformal
mappings, Dokl. Akad. Nauk SSSR 234 (1977), 525-527; English transl. in Soviet Math. Dokl. 18 (1977). 69. Stability of classes of multidimensional holomorphic mappings. I: The concept of stability. Liouville's theorem, Sibirsk. Mat. Zh. 23 (1982), no. 2, 83-111; English transl. in Siberian Math. J. 23 (1982). 70. Stability of classes of multidimensional holomorphic mappings. II: Stability of classes of holomorphic mappings, Sibirsk. Mat. Zh. 23 (1982), no. 4, 65-89; English transl. in Siberian Math. J. 23 (1982).
BIBLIOGRAPHY
353
71. Stability of classes of multidimensional holomorphic mappings. III: Properties of mappings close to holomorphic mappings, Sibirsk. Mat. Zh. 24 (1983), no. 3, 70-91; English transl. in Siberian Math. J. 24 (1983). 72. On boundary values of mappings of the half-space that are close to being conformal, Sibirsk. Mat. Zh. 24 (1983), no. 5, 76-93; English transi. in Siberian Math. J. 24 (1983). M. KREINES
73. Sur une classe de fonctions de plusieurs variables, Mat. Sb. 9 (51) (1941), 713-720. S. L. KRUSHKAL'
74. On the absolute continuity and differentiability of certain classes of mappings of multidimensional domains, Sibirsk. Mat. Zh. 6 (1965), 692-696. (Russian) S. L. KRUSHKAL', B. N. APANASOV, AND N. A. GUSEVSKII
75. Kleinian groups and uniformization in examples and problems, "Nauka", Novosibirsk, 1981; English transl., Amer. Math. Soc., Providence, R. I., 1986. K. KuRATOWSKI
76. Topology. Vol. 2, 4th ed., PWN, Warsaw, and Academic Press, New York, 1968. O. A. LADYZHENSKAYA AND N. N. URAL'TSEVA
77. Linear and quasilinear elliptic equations, 2nd ed., "Nauka", Moscow, 1973; English transl. of 1st ed., Academic Press, 1968. O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, AND N. N. URAL'TSEVA
78. Linear and quasilinear equations of parabolic type, "Nauka", Moscow, 1967; English transl., Amer. Math. Soc., Providence, R. 1., 1968. M. LAVRENTIEFF [M. A. LAVRENT'EV)
79. Sur une classe de representations continues, Mat. Sb. 42 (1935), 407424. 80. Sur un crittre differentiel des transformations homeomorphes des domaines h trois dimensions, C. R. (Dokl.) Acad. Sci. URSS 20 (1938), 241-242. 81. Stability in Liouville's theorem, Dokl. Akad. Nauk SSSR 95 (1954), 925-926. (Russian)
BIBLIOGRAPHY
354
82. On the theory of quasi-conformal mappings of three-dimensional domains, J. Analyse Math. 19 (1967), 217-225. M. A. LAVRENT'EV AND P. P. BELINSKII
83. Some problems of geometric function theory, Trudy Mat. Inst. Steklov. 128 (1972), 34-40; English transl. in Proc. Steklov Inst. Mat. 128 (1972). OLLI LEHTO
84. Remarks on the integrability of the derivatives of quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 371 (1965). 85. Univalent functions and Teichmuller spaces, Springer-Verlag, 1987. JACQUELINE LELONG-FERRAND
86. Interpretation fonctionnelle des homeomorphismes quasi-conformes, C. R. Acad. Sci. Paris Ser. A-B 272 (1971), A 1390-A 1392. 87. Geometrical interpretations of scalar curvature and regularity of conformal homeomorphisms, Differential Geometry and Relativity (A. Lichnerowicz Sixtieth Birthday Vol.), Reidel, 1976, pp. 91-105. LAWRENCE G. LEWIS
88. Quasiconformal mappings and Royden algebras in space, Trans. Amer. Math. Soc. 158 (1971), 481-492. CHARLES LOEWNER
89. On the conformal capacity in space, J. Math. and Mech. 8 (1959), 411-414. A. MARKOUCHEVITCH [A. I. MARKUSHEVICH]
90. Sur certaines classes de transformations continues, C. R. (Dokl.) Acad. Sci. URSS 28 (1940), 301-304. 0. MARTIO
91. A capacity inequality for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 474 (1970). 92. Definitions for uniform domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 197-205. 0. MARTIO AND J. SARVAS
93. Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), 383-401. 0. MARTIO, S. RICKMAN, AND J. VAISALA
94. Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 448 (1969).
BIBLIOGRAPHY
355
95. Distortion and singularities for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 465 (1970). 96. Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 488 (1971). D. MENCHOFF [D. E. MEN'SHOV]
97. Sur une generalisation dun theortme de M. H. Bohr, Mat. Sb. 2 (44) (1937), 339-356. NORMAN G. MEYERS AND ALAN ELCRAT
98. Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J. 42 (1975), 121-136. S. G. MIKHLIN
99. Multidimensional singular integrals and integral equations, Fizmatgiz, Moscow, 1962; English transl., Pergamon Press, 1965. V. M. MIKLYUKOV
100. On removable singularities of quasiconformal mappings in space, Dokl. Akad. Nauk SSSR 188 (1969), 525-527; English transl. in Soviet Math. Dokl. 10 (1969). 101. On the asymptotic properties of subsolutions of quasilinear elliptic equations and mappings with bounded distortion, Mat. Sb. 111 (153) (1980), 42-66; English transl. in Math. USSR Sb. 39 (1981). CHARLES B. MORREY, JR.
102. Multiple integral problems in the calculus of variations and related topics, Univ. California Publ. Math. (N.S.) 1 (1943), 1-130. JORGEN MOSER
103. A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457-468. 104. On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577-591. P. P. MOSOLOV AND V. P. MYASNIKOV
105. A proof of Korn's inequality, Dokl. Akad. Nauk SSSR 201 (1971), 36-39; English transl. in Soviet Math. Dokl. 12 (1971). G. D. MosTOw 106. Quasi-conformal mappings in n-space and the rigidity of hyperbolic
space forms, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968), 53-104.
BIBLIOGRAPHY
356
MITSURU NAKAI
107. Algebraic criterion on quasiconformal equivalence of Riemann surfaces, Nagoya Math. J. 16 (1960), 157-184. JINDIkICH NEZ`AS
108. Sur les normes equivalentes dans Wpk)(S2) et sur la coercitivitts des
formes formellement positives, Equations aux Derivees Partielles (1965), Sem. Math. Sup., vol. 19, Presses Univ. Montreal, Montreal, 1966, pp. 102-128. ROLF NEVANLINNA
109. Eindeutige analytische Funktionen, Springer-Verlag, 1936. S. M. NIKOL'SKII
110. Properties of certain classes of functions of several variables on differentiable manifolds, Mat. Sb. 33 (75) (1953), 261-326; English transi.
in Amer. Math. Soc. Transl. (2) 80 (1969). LOUIS NIRENBERG
111. On nonlinear elliptic partial differential equations and Holder continuity, Comm. Pure Appl. 6 (1953), 103-156, 395. I. S. OVCHINNIKOV AND G. D. SUVOROV
112. Transformations of the Dirichlet integral and space mappings, Sibirsk.
Mat. Zh. 6 (1965), 1292-1314. (Russian) E. A. POLETSKII
113. The modulus method for nonhomeomorphic quasiconformal mappings, Mat. Sb. 83 (125) (1970), 261-272; English transl. in Math. USSR Sb. 12 (1970). 114. On the removal of singularities of quasiconformal mappings, Mat. Sb. 92 (134) (1973), 242-256; English transl. in Math. USSR Sb. 21 (1973). G. POLYA AND G. SZEGO
115. Isoperimetric inequalities in mathematical physics, Princeton Univ. Press, Princeton, N. J., 1951. T. RADO AND P. V. REICHELDERFER
116. Continuous transformations in analysis. With an introduction to algebraic topology, Springer-Verlag, 1955. P. K. RASHEVSKII
117. Riemannian geometry and tensor analysis, 3rd ed., "Nauka", Moscow, 1967; German transl. of 1st ed., VEB Deutscher Verlag Wiss., Berlin, 1959.
BIBLIOGRAPHY
357
H. M. REIMANN
118. Ordinary differential equations and quasiconformal mappings, Invent. Math. 33 (1976), 247-270. Yu. G. RESHETNYAK
119. On a sufficient condition for Holder continuity of a mapping, Dokl. Akad. Nauk SSSR 130 (1960), 507-509; English transl. in Soviet Math. Dokl. 1 (1960). 120. On conformal mappings of space, Dokl. Akad. Nauk SSSR 130 (1960),
1196-1198; English transl. in Soviet Math. Dokl. 1 (1960). 121. Stability in the Liouville theorem on conformal mappings, Some Problems in Mathematics and Mechanics (M. A. Lavrent'ev, Sixtieth Birthday Vol.), Izdat. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1961, pp. 219-223. (Russian) 122. On stability in Liouville's theorem on conformal mappings of space, Dokl. Akad. Nauk SSSR 152 (1963), 286-287; English transl. in Soviet Math. Dokl. 4 (1963). 123. Some geometric properties of functions and mappings with generalized derivatives, Sibirsk. Mat. Zh. 7 (1966), 886-919; English transl. in Siberian Math. J. 7 (1966). 124. Estimates of the modulus of continuity for certain mappings, Sibirsk. Mat. Zh. 7 (1966), 1106-1114; English transl. in Siberian Math. J.
7(1966). 125. Space mappings with bounded distortion, Sibirsk. Mat. Zh. 8 (1967), 629-658; English transl. in Siberian Math. J. 8 (1967). 126. Liouville's conformal mapping theorem under minimal regularity as-
sumptions, Sibirsk. Mat. Zh. 8 (1967), 835-840; English transl. in Siberian Math. J. 8 (1967). 127. General theorems on semicontinuity and convergence with functionals, Sibirsk. Mat. Zh. 8 (1967), 1051-1069; English transl. in Siberian Math. J. 8 (1967). 128. On the set of singular points of solutions of certain nonlinear elliptic equations, Sibirsk. Mat. Zh. 9 (1968), 354-367; English transi. in Siberian Math. J. 9 (1968). 129. On a condition for boundedness of the index for mappings with bounded distortion, Sibirsk. Mat. Zh. 9 (1968), 368-374; English transi. in Siberian Math. J. 9 (1968). 130. Mappings with bounded distortion as extremals of integrals of Dirich-
let type, Sibirsk. Mat. Zh. 9 (1968), 652-666; English transl. in Siberian Math. J. 9 (1968).
358
BIBLIOGRAPHY
131. Generalized derivatives and differentiability almost everywhere, Mat.
Sb. 75 (117) (1968), 323-334; English transl. in Math. USSR Sb. 4 (1968). 132. On the concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Zh. 10 (1969), 1109-1138; English transl. in Siberian Math. J. 10 (1969). 133. On extremal properties of mappings with bounded distortion, Sibirsk. Mat. Zh. 10 (1969), 1300-1310; English transl. in Siberian Math. J. 10 (1969).
134. The local structure of mappings with bounded distortion, Sibirsk. Mat. Zh. 10 (1969), 1311-1333; English transl. in Siberian Math. J. 10 (1969). 135. Estimates for certain differential operators with finite-dimensional kernels, Sibirsk. Mat. Zh. 11 (1970), 414-428; English transl. in Siberian Math. J. 11 (1970). 136. On a stability estimate in Liouville's theorem on conformal mappings of multidimensional spaces, Sibirsk. Mat. Zh. 11 (1970), 1112-1139; English transi. in Siberian Math. J. 11 (1970). 137. On the set of branch points of mappings with bounded distortion, Sibirsk. Mat. Zh. 11 (1970), 1333-1339; English transl. in Siberian Math. J. 11 (1970). 138. Certain integral representations of differentiable functions, Sibirsk. Mat. Zh. 12 (1971), 420-432; English transl. in Siberian Math. J. 12 (1971). 139. Lectures on mathematical analysis. Mathematical analysis on manifolds, Novosibirsk. Gos. Univ., Novosibirsk, 1972. (Russian) 140. Stability in Liouville's theorem on conformal mappings of a space for domains with nonsmooth boundary, Sibirsk. Mat. Zh. 17 (1976), 361-369; English transl. in Siberian Math. J. 17 (1976). 141. Stability estimates in Liouville's theorem and L p-integrability of the derivatives of quasiconformal mappings, Sibirsk. Mat. Zh. 17 (1976), 868-896; English transl. in Siberian Math. J. 17 (1976). 142. Stability estimates in the class Wp for Liouville's theorem on conformal mappings for a closed domain, Sibirsk. Mat. Zh. 17 (1976), 1382-1394; English transl. in Siberian Math. J. 17 (1976). 143. Differentiability properties of quasiconformal mappings and conformal mappings of Riemannian spaces, Sibirsk. Mat. Zh. 19 (1978), 1166-1183; English transl. in Siberian Math. J. 19 (1978). 144. Stability theorems in geometry and analysis, "Nauka", Novosibirsk, 1982. (Russian)
BIBLIOGRAPHY
359
Yu. G. RESHETNYAK AND B. V. SHABAT
145. On quasiconformal mappings in space, Proc. Fourth All-Union Math.
Congr. (Leningrad, 1961), Vol. II, "Nauka", Leningrad, 1964, pp. 672-680. (Russian) GEORGES DE RHAM
146. Vari0tes difErentiables. Formes, courants, formes harmoniques, Actualites Sci. Indust., no. 1222, Hermann, Paris, 1955. SEPPO RICKMAN
147. On the number of omitted values of entire quasiregular mappings, J. Analyse Math. 37 (1980), 100-117. FRDERIC RIESZ AND BOLA Sz. NAGY
148. Lecons d'analyse fonctionnelle, 2nd ed., Akad. Kiado, Budapest, 1953; English transl., Ungar, New York, 1955. STANISLAW SAKS
149. Theory of the integral, 2nd rev. ed., PWN, Warsaw, 1937; reprint, Dover, New York, 1964. V. I. SEMENOV
150. On one-parameter groups of quasiconformal homeomorphisms in a Euclidean space, Sibirsk. Mat. Zh.17 (1976),177-193; English transi. in Siberian Math. J. 17 (1976). 151. Semigroups of quasi-Lorentz mappings, Sibirsk. Mat. Zh. 18 (1977), 713; English transl. in Siberian Math. J. 18 (1977). 152. Uniform estimates of stable isometries, Sibirsk. Mat. Zh. 27 (1986), no. 3, 193-199; English transl. in Siberian Math. J. 27 (1986). 153. On uniform stability estimates for quasiconformal and quasi-isometric mappings in space, Dokl. Akad. Nauk SSSR 286 (1986), 295-297; English transl. in Soviet Math. Dokl. 33 (1986). 154. Stability estimates for quasiconformal space mappings of a starshaped domain, Sibirsk. Mat. Zh. 28 (1987), no. 6, 102-118; English transl. in Siberian Math. J. 28 (1987). JAMES SERRIN
155. A Harnack inequality for nonlinear equations, Bull. Amer. Math. Soc.
69 (1963), 481-486. 156. Local behavior of solutions of quasi-linear equations, Acta. Math. 111
(1964), 247-302. B. V. SHABAT
157. The modulus method in space, Dokl. Akad. Nauk SSSR 130 (1960), 1210-1213; English transl. in Soviet Math. Dokl. 1 (1960).
360
BIBLIOGRAPHY
158. On the theory of quasiconformal mappings in space, Dokl. Akad. Nauk SSSR 132 (1960), 1045-1048; English transl. in Soviet Math. Dokl. 1 (1960). S. Z. SHEFEL'
159. Conformal correspondence of metrics, and the smoothness of isometric immersions, Sibirsk. Mat. Zh. 20 (1979), 397-401; English transl. in Siberian Math. J. 20 (1979). 160. Smoothness of a conformal mapping of Riemannian spaces, Sibirsk. Mat. Zh. 23 (1982), no. 1, 153-159; English transl. in Siberian Math. J. 23 (1982). V. 1. SMIRNOV
161. A course in higher mathematics, Vol. 5, rev. ed., Fizmatgiz, Moscow, 1959; English transl., Pergamon Press, Oxford, and Addison-Wesley, Reading, Mass., 1964. S. L. SOBOLEV
162. Applications of functional analysis in mathematical physics, Izdat. Leningrad. Gos. Univ., Leningrad, 1950; English transl., Amer. Math. Soc., Providence, R. 1., 1963. W. STEPANOFF [V. V. STEPANOV]
163. Sur les conditions de /'existence de la 32 (1924/25), 511-527.
totale, Mat. Sb.
SHLOMO STERNBERG
164. Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N. J., 1964. G. D. SUVOROV
165. Extensions of topological structures and metric properties of mappings, Contributions to Extension Theory of Topological Structures (Proc. Sympos., Berlin, 1967; J. Flachsmeyer et al., editors), VEB Deutscher Verlag Wiss., Berlin, 1969, pp. 257-273; English transl. in Amer. Math. Soc. Transl. (2) 134 (1987). A. V. SYCHEV
166. Moduli and n-dimensional quasiconformal mappings, "Nauka", Novosibirsk, 1983. (Russian) HANS TRIEBEL
167. Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag Wiss., Berlin, 1977, and North-Holland, Amsterdam, 1978.
BIBLIOGRAPHY
361
D. A. TROTSENKO
168. Properties of domains with nonsmooth boundary, Sibirsk. Mat. Zh. 22 (1981), no. 4, 221-224. (Russian) 169. Extension from a domain and approximation of spatial quasiconformal mappings with small distortion coefficient, Dokl. Akad. Nauk SSSR 270 (1983), 1331-1333; English transi. in Soviet Math. Dokl. 27 (1983). 170. Extension of spatial quasiconformal mappings that are close to conformal mappings, Sibirsk. Mat. Zh. 28 (1987), no. 6, 126-133; English trans]. in Siberian Math. J. 28 (1987). JUSSI VAISALA
171. On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I No. 298 (1961). 172. Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I No. 392 (1966). 173. Removable sets for quasiconformal mappings, J. Math. and Mech. 19
(1969/70), 49-51. 174. Modulus and capacity inequalities for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 509 (1972). 175. A survey of quasiregular maps in R", Proc. Internat. Congr. Math. (Helsinki 1978), vol. 2, Acad. Sci. Fenn., Helsinki, 1980, pp. 685691. M. YU. VASIL'CHIK
176. On a lower estimate of the distortion coefficient for infinitely close domains, Sibirsk. Mat. Zh. 19 (1978), 547-554; English trans]. in Siberian Math. J. 19 (1978). S. K. VODOP'YANOV AND V. M. GOL'DSHTEIN
177. Lattice isomorphisms of the spaces W and quasiconformal mappings, Sibirsk. Mat. Zh. 16 (1975), 224-246; English transl. in Siberian Math. J. 16 (1975). 178. Quasiconformal mappings and spaces of functions with first generalized derivatives, Sibirsk. Mat. Zh.17 (1976), 515-531; English transi. in Siberian Math. J. 17 (1976). 179. Sobolev spaces and special classes of mappings, Novosibirsk. Gos. Univ., Novosibirsk, 1981. (Russian) S. K. VODOP'YANOV, V. M. GOL'DSHTEIN, AND T. G. LATFULLIN
180. A criterion for extendibility of LZ functions from unbounded plane
domains, Sibirsk. Mat. Zh. 20 (1979) 416-420; English transi. in Siberian Math. J. 20 (1979).
BIBLIOGRAPHY
362
WILLIAM P. ZIEMER
181. Extremal length and p-capacity, Michigan Math. J. 16 (1969), 4351. V. A. ZORICH
182. A theorem of M. A. Lavrent'ev on quasiconformal space maps, Mat. Sb. 74 (116) (1967), 417-433; English transl. in Math. USSR Sb. 3 (1967). A. ZYGMUND
183. Trigonometric series, 2nd rev. ed., Vol. I, Cambridge Univ. Press, 1959.
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