De Gruyter Expositions in Mathematics 55 Editors Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York, USA Markus J. Pflaum, Boulder, USA Dierk Schleicher, Bremen, Germany Raymond O. Wells, Bremen, Germany
Rainer Picard Des McGhee
Partial Differential Equations A Unified Hilbert Space Approach
De Gruyter
Mathematical Subject Classification 2010: 35-01, 35-02, 34G10, 44A10, 47A60, 47D06, 47E05, 47F05, 47G10.
ISBN 978-3-11-025026-8 e-ISBN 978-3-11-025027-5 ISSN 0938-6572 Library of Congress Cataloging-in-Publication Data Picard, R. H. (Rainer H.) Partial differential equations : a unified Hilbert space approach / by Rainer Picard, Des McGhee. p. cm. ⫺ (De Gruyter expositions in mathematics ; 55) Includes bibliographical references and index. ISBN 978-3-11-025026-8 (alk. paper) 1. Hillbert space. 2. Differential equations, Partial. I. McGhee, D. F. II. Title. QA322.4.P53 2011 5151.733⫺dc22 2011004423
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ⬁ Printed on acid-free paper Printed in Germany www.degruyter.com
Dedication We dedicate this book to the memory of Cliff Bartlett (1923–2010), a mathematician, artist and singer and a member of the Department of Mathematics (as it then was), University of Strathclyde for over thirty years, retiring in the mid-1980s. Cliff offered great encouragement and support to both of us in the early parts of our careers, to DM as a newly appointed lecturer and to RP during a year at Strathclyde as a Visiting Lecturer. Rainer Picard, Des McGhee
Preface
Given that there is a multitude of well-written and useful textbooks and monographs on partial differential equations, one of the obvious concerns of prospective authors might be why there should be another one written. So, it is appropriate to explain at the outset some of the particular features which make this book different from other texts. It will be obvious to the reader that the present book is not a run-off-the-mill text book on linear partial differential equations. First of all, the whole approach – although (with some additional work) extendable to a more general Banach space setting – is established in a Hilbert space setting, as the title of this monograph indicates. Of course, a Banach space setting is more general and sometimes more appropriate, but usually the core results rely nevertheless on a Hilbert space solution theory, a fact sometimes only tacitly acknowledged. We hope to show that, for presenting core ideas, our focus on a Hilbert space setting is not a constraint, but rather a highly suitable approach for providing a more transparent and even fairly elementary framework for presenting the main issues in the discussion of a solution theory for partial differential equations. The reader may also find many topics, dealt with elsewhere, presented here in a slightly different flavor. Indeed, the building blocks (such as extrapolation and interpolation spaces, sums of operators, vector-valued Laplace transform) are largely well known with some of the ideas dating back to the early 1960s, see e.g. [35], [26] for the idea of interpolation/extrapolation spaces. Therefore, it has been and still is surprising to us that the full power of these concepts, which we utilize in this approach, has not been previously exploited to the extent we have found so useful. The differences are somewhat subtle and a more superficial reader may fail to appreciate how different our perspective on the theory of linear partial differential equations is. Indeed, it is this perspective on our approach which may be considered the most innovative feature of this monograph. In contrast to many other books, which are either focusing on specific types of partial differential equations or on a collection of tools for solving a variety of problems associated with various specific linear partial differential equations, we are attempting to assume a more global point of view on the issues involved. Our approach can be classified as a functional analytical one, but this says very little, since nowadays it is the accepted standard to employ functional analytic language to formulate PDE problems. It may, however, come as a surprise that a Hilbert space setting is sufficiently general to cover the core issues of solving PDE problems. We focus on the case of linear partial differential equations, which is of course in one way a severe constraint, but given that by a rule of thumb non-linear problems, if they are at all well-posed, are frequently solvable by using a priori estimates and a fixed
viii
Preface
point argument based on perturbations of the linear theory, we see the restriction to linear partial differential equations more as foundation laying rather than an exclusion of non-linear issues. A natural guideline for approaching problem solving is provided by Hadamard’s celebrated criteria for well-posedness:1 I Uniqueness: there is at most one solution, I Existence: there is at least one solution (at least for a dense set of data), I Continuous Dependence: the dependence of the solution on the data is locally (or weakly locally) uniformly continuous. Compared to these fundamental requirements, qualitative properties of a solution are a secondary consideration. This remark applies in particular to the issue of regularity in connection with solving linear partial differential equations. The historical focus on regularity issues has fostered a number of guiding ideas, which in the light of our approach appear as occasionally misleading. Among these are the notions that I partial differential equations are best classified according to the regularizing properties of their solution operators, I one type of space should be used for solving “all” problems associated with partial differential equations, I problems involving elliptic partial differential equations are “easier” than those involving parabolic partial differential equations, which are again “easier” than those involving hyperbolic partial differential equations. The systematic approach presented here will shed a different and hopefully more illuminating light on these and other issues by I proposing a different classification scheme, I advocating the construction of “tailor-made” distribution spaces adapted to the particular equation at hand, 1
These requirements are straightforwardly illustrated, if we consider the “problem” of finding a solution of the equation F .x/ D f , where F denotes a mapping between – say – metric spaces. The properties are that F must be injective, with dense range and with F 1 being (at least weakly) locally uniformly continuous. We note in particular that the (weakly) local uniform continuity of F 1 allows the extension of F to its closure F given by F .x/ WD lim F .y/ for any sequence y converging to x such that F .y/ also converges. Indeed, for two sequences y .k/ , k D 0; 1, converging to x such that F .y .k/ /, k D 0; 1, are also convergent, we have that F .y .0/ /, F .y .1/ / must converge to the same limit, so that F is well-defined, i.e. F is closable. Weakly local uniform continuity is indeed characterized by mapping Cauchy sequences to Cauchy sequences.
Preface
ix
I showing that from our point of view parabolic and hyperbolic partial differential equations are, in a way, “easier” than elliptic partial differential equations. It will also be seen that our general framework is sufficiently powerful to be applied to general evolution equations. However, to flesh out the ideas presented, we shall illustrate by examples and consider particular systems of partial differential equations from mathematical physics. Accordingly, the book is divided into several natural parts. In Chapter 1 we supply some additional material on functional analysis2 in Hilbert space which may be difficult to find elsewhere. Chapter 2 introduces the idea of what we shall call Sobolev lattices. In Chapter 3, as a first application, we consider partial differential equations with constant coefficients in RnC1 , n 2 N. The results are extended to tempered distributions (set in a suitably extended Sobolev lattice). Then in Chapter 4 the ideas presented are transferred to a more general framework covering a large class of abstract evolution equations. In Chapter 5 this general setting is exemplified by applications to a variety of initial-boundary value problems from mathematical physics. The concluding Chapter 6 offers a new approach to initial boundary value problems by expanding on the ideas and concepts presented. The material is based on lecture notes developed for introductory and advanced graduate level courses on partial differential equations and functional analysis given by the first author over the past three decades at the Rheinische Friedrich-WilhelmsUniversität at Bonn, Germany, at the University of Wisconsin-Milwaukee, Wisconsin, USA, and at the Technische Universität Dresden, Germany, and on a series of lectures given at the Strathclyde University, Glasgow, Scotland, UK. This development from lecture notes has led to proofs being given in more detail than one might normally expect in a monograph. This may slow the readers progress, but it should, however, make the text not only useful as a resource for courses on the topic but also make it suitable as a text for a reading course or for self-study. Apart from the novel approach the material presented in this monograph may in many ways be considered elementary, however, researchers will nevertheless find new results for particular evolutionary system from mathematical physics in later parts of this monograph as well as a very different perspective on seemingly familiar evolutionary problems. This book has been in preparation for a number of years and many colleagues have contributed to the effort either directly through discussion and comment on research papers or at seminar or indirectly through general support and encouragement. The first of us (RP) would like to particularly acknowledge the hospitality of the Department and Mathematics and Statistics, University of Strathclyde, that he has enjoyed first as a young Visiting Researcher and more recently as a Visiting Professor, while DM would mention all colleagues, past and present, in the Department and Mathematics and Statistics, University of Strathclyde, but particularly Adam McBride, Wilson 2
We have chosen to assume that the reader is familiar with basic functional analysis in Hilbert space.
x
Preface
Lamb and Mike Grinfeld of the Applied Analysis Group. Both of us would like to thank Rolf Leis and Gary Roach for encouragement and support throughout our careers. Finally, we both wish to acknowledge the love and support of our wives, Brigitte and Alison, and families – their patience and understanding (of the effort required if not the mathematics itself!) has been necessary, and we hope sufficient, to see this task to a successful conclusion. Dresden / Glasgow, January 2011
Rainer Picard, Des McGhee
Contents
Preface
vii
Nomenclature
xv
1
. . . . .
1 1 2 3 15 19
2
Sobolev Lattices 2.1 Sobolev Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Sobolev Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Sobolev Lattices from Tensor Products of Sobolev Chains . . . . . . .
30 30 56 65
3
Linear Partial Differential Equations with Constant Coefficients 3.1 Partial Differential Equations in H1 .@ C e/ . . . . . . . . . . . . 3.1.1 First Steps Towards a Solution Theory . . . . . . . . . . . . . 3.1.2 The Tarski–Seidenberg Theorem and some Consequences . . 3.1.3 Regularity Loss .0; : : : ; 0/ . . . . . . . . . . . . . . . . . . . 3.1.4 Classification of Partial Differential Equations . . . . . . . . 3.1.5 The Classical Classification of Partial Differential Equations . 3.1.6 Elliptic, Parabolic, Hyperbolic? . . . . . . . . . . . . . . . . 3.1.7 Evolutionary Expressions in Canonical Form . . . . . . . . . 3.1.8 Functions of @ and Convolutions . . . . . . . . . . . . . . . 3.1.9 Systems and Scalar Equations . . . . . . . . . . . . . . . . . 3.1.10 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.11 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . 3.1.12 Some Applications to Linear Partial Differential Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . 3.1.12.1 Transport Equation . . . . . . . . . . . . . . . . . 3.1.12.2 Acoustics . . . . . . . . . . . . . . . . . . . . . . 3.1.12.3 Thermodynamics . . . . . . . . . . . . . . . . . . 3.1.12.4 Electrodynamics . . . . . . . . . . . . . . . . . . . 3.1.12.5 Elastodynamics . . . . . . . . . . . . . . . . . . .
72 72 72 76 89 90 94 104 107 114 121 125 142
Elements of Hilbert Space Theory 1.1 Hilbert Space . . . . . . . . . . . . . . . . . . 1.2 Some Construction Principles of Hilbert Spaces 1.2.1 Direct Sums of Hilbert Spaces . . . . . 1.2.2 Dual Spaces . . . . . . . . . . . . . . . 1.2.3 Tensor Products of Hilbert Spaces . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
156 156 160 170 171 181
xii
Contents
3.2
4
3.1.12.6 Fluid Dynamics . . . . . . . . . . . . . . . . . . . 3.1.12.7 Quantum Mechanics . . . . . . . . . . . . . . . . . Partial Differential Equations in H1 .jD j/ . . . . . . . . . . . . . 3.2.1 Extension of the Solution Theory to H1 .jD j/. . . . . . . . 3.2.2 Some Applications to Linear Partial Differential Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . 3.2.2.1 Helmholtz Equation in R3 . . . . . . . . . . . . . . 3.2.2.2 Helmholtz Equation in R2 . . . . . . . . . . . . . . 3.2.2.3 Cauchy–Riemann Operator . . . . . . . . . . . . . 3.2.2.4 Wave Equation in R2 (Method of Descent) . . . . . 3.2.2.5 Plane Waves . . . . . . . . . . . . . . . . . . . . . 3.2.2.6 Linearized Navier–Stokes Equations . . . . . . . . 3.2.2.7 Electro- and Magnetostatics . . . . . . . . . . . . . 3.2.2.8 Force-free Magnetic Fields . . . . . . . . . . . . . 3.2.2.9 Beltrami Field Expansions . . . . . . . . . . . . . 3.2.3 Convolutions in H1 .jD j/; 2 RnC1 . . . . . . . . . . . . 3.2.4 Integral Representations of Convolutions with Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4.1 An Integral Representation of the Solution of the Transport Equation . . . . . . . . . . . . . . . . . 3.2.4.2 Potentials, Single and Double Layers in R3 . . . . . 3.2.4.3 Electro- and Magnetostatics (Biot–Savart’s Law) . . 3.2.4.4 Potential Theory in R2 . . . . . . . . . . . . . . . 3.2.4.5 Cauchy’s Integral Formula . . . . . . . . . . . . . 3.2.4.6 Integral Representations of Solutions of the Helmholtz Equation in R3 . . . . . . . . . . . . . . 3.2.4.7 Retarded Potentials . . . . . . . . . . . . . . . . . 3.2.4.8 Integral Representations of Solutions of the TimeHarmonic Maxwell Equations . . . . . . . . . . . .
Linear Evolution Equations 4.1 Linear Operator Equations in Sobolev Lattices . . . . . . . . . . . . . 4.1.1 Polynomials of Commuting Operators . . . . . . . . . . . . . 4.1.2 Polynomials of Commuting, Selfadjoint Operators . . . . . . 4.2 Evolution Equations with Polynomials of Operators as Coefficients . . 4.2.1 Classification of Operator Polynomials with Time Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Causality of Evolutionary Problems . . . . . . . . . . . . . . 4.2.3 Abstract Initial Value Problems . . . . . . . . . . . . . . . . 4.2.4 Systems and Scalar Equations . . . . . . . . . . . . . . . . . 4.2.5 First-Order-in-Time Evolution Equations in Sobolev Lattices .
195 198 205 205 223 223 226 229 229 232 233 235 236 238 243 248 248 249 275 278 281 285 296 297 303 303 303 304 305 305 308 316 329 334
Contents 5
6
xiii
Some Evolution Equations of Mathematical Physics 5.1 Schrödinger Type Equations . . . . . . . . . . . . . . . . . . . . . 5.1.1 The Selfadjoint Laplace Operator . . . . . . . . . . . . . . 5.1.2 Some Perturbations . . . . . . . . . . . . . . . . . . . . . . 5.1.2.1 Bounded Perturbations . . . . . . . . . . . . . . 5.1.2.2 Relatively Bounded Perturbations (the Coulomb Potential) . . . . . . . . . . . . . . . . . . . . . . 5.2 Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Selfadjoint Operator Case . . . . . . . . . . . . . . . . 5.2.1.1 Prescribed Dirichlet and Neumann Boundary Data 5.2.1.2 Transmission Initial Boundary Value Problem . . 5.2.2 Stefan Boundary Condition . . . . . . . . . . . . . . . . . . 5.2.3 Lower Order Perturbations . . . . . . . . . . . . . . . . . . 5.3 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Dirichlet and Neumann Boundary Condition . . . . . . . . 5.3.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Reversible Heat Transport . . . . . . . . . . . . . . . . . . 5.4 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Electric Boundary Condition . . . . . . . . . . . . . . 5.4.2 Some Decomposition Results . . . . . . . . . . . . . . . . 5.4.3 The Extended Maxwell System . . . . . . . . . . . . . . . 5.4.4 The Vectorial Wave Equation for the Electromagnetic Field . 5.5 Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 The Rigid Boundary Condition . . . . . . . . . . . . . . . . 5.5.2 Free Boundary Condition . . . . . . . . . . . . . . . . . . . 5.5.3 Shear and Pressure Waves . . . . . . . . . . . . . . . . . . 5.6 Plate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Thermo-Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
338 338 338 342 342
. . . . . . . . . . . . . . . . . . . . . .
346 349 350 357 366 369 371 372 374 377 382 385 387 391 394 400 407 409 412 413 415 420
A “Royal Road” to Initial Boundary Value Problems 6.1 A Class of Evolutionary Material Laws . . . . . . . . . . . . . 6.2 Evolutionary Dynamics and Material Laws . . . . . . . . . . . 6.2.1 The Shape of Evolutionary Problems with Material Laws 6.2.2 Some Special Cases . . . . . . . . . . . . . . . . . . . 6.2.3 Material Laws via Differential Equations . . . . . . . . 6.2.4 Coupled Systems . . . . . . . . . . . . . . . . . . . . . 6.2.5 Initial Value Problems . . . . . . . . . . . . . . . . . . 6.2.6 Memory Problems . . . . . . . . . . . . . . . . . . . . 6.3 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Reversible Heat Transfer . . . . . . . . . . . . . . . . . 6.3.2 Models of Thermoelasticity . . . . . . . . . . . . . . . 6.3.3 Thermo-Piezo-Electro-Magnetism . . . . . . . . . . . .
. . . . . . . . . . . .
426 427 431 431 437 440 441 443 447 453 454 455 457
. . . . . . . . . . . .
. . . . . . . . . . . .
xiv
Contents
Conclusion
459
Bibliography
461
Index
465
Nomenclature
!
mapping, as in A ! B
!
strong convergence, as in xn ! x1 for the convergence of a sequence .xn /n2N to its limit x1 in the norm topology
*
weak convergence, as in xn * x1 for the weak convergence of a sequence .xn /n2N to its weak limit x1 , i.e. in the weak topology
^
logical “and”
_ V
logical (non-exclusive) “or”
W
n!1
n!1
:::
for all : : : / for every : : :
:::
there is : : : / there exists : : :
7!
maps to, as in x 7! x 2
AN
closure of A
A
adjoint relation or mapping to a relation or mapping A
AV
closure of a closable operator A restricted to elements in CV 1 ./ for some open subset RnC1 , n 2 N
cof.A/
co-factor matrix associated with a square matrix A
cof.A/>
transposed co-factor matrix associated with a square matrix A, adjunct matrix
z
complex conjugate of a complex number z
L.H; H /
continuous linear mappings on H
CV 1 ./
space of smooth functions with compact support contained in the open subset RnC1 , n 2 N
@ b @
derivatives .@0 ; @1 ; : : : ; @n / in R1Cn , n 2 N
@0 b @
time derivative in R1Cn , n 2 N
b @ b @> ˚
spatial derivatives .@1 ; @2 ; : : : ; @n / in R1Cn with @0 denoting the time derivative, n 2 N same as grad same as curl same as div direct sum, orthogonal sum
xvi L
Nomenclature direct summation sign or orthogonal summation sign as in
L t2M
Ht
Div
divergence of .1; 1/-tensor fields
2
element sign as in x 2 C
2
element function as in x D2 .¹xº/ giving the element of a set containing only one element
L
Fourier–Laplace transform with parameter 2 RnC1
L temporal Fourier–Laplace transform with parameter 2 R b f . i / same as L f Grad
symmetric part of the covariant derivative of a vector field
H;k
short for Hk .@ C /, k 2 Z
H;0;0
short for H;0 ˝ H
HL
Hardy–Lebesgue space
i
imaginary unit
h j iX
inner product of the inner product space X
E 1=2 ŒX
inner product space derived from the inner product space X by modifying the inner product h j i X to h j E i X , where E W X ! X is continuous, linear, symmetric and strictly positive definite
dre
smallest integer greater than or equal to the real number r (roof)
brc T
largest integer less than or equal to the real number r (integer part, floor) T T V big intersection symbol, as in M D ¹yj X2M y 2 X º or X2M X
spatial Laplacian, same as b @2 or jb @j
2 jb @j
negative spatial Laplacian, same as or b @2
b @2
spatial Laplacian, same as or jb @j
dsx
line element at a point x
ds
line element
Lin
linear hull, as in LinK A, the smallest linear space over the field K containing the set A
A
the relation ¹.a; b/j .a; b/ 2 Aº with A X Y
ŒA
the set of all negatives of elements in the set A
Œ¹0ºf
null space or kernel of a mapping or function f
N.f /
null space or kernel of a mapping or function f
C
field or set of complex numbers
Re
real part
2
2
Nomenclature
xvii
Im
imaginary part
K
field or set of numbers (either K D R or K D C)
N
monoid or set of natural numbers ¹0; 1; 2; : : :º
R
field or set of real numbers
Z
group or set of integers
e
vector .1; 1; : : : ; 1/
kAk
operator norm of a linear operator A between normed linear spaces
kAkX!Y
operator norm of a linear operator A W X ! Y , X; Y normed linear spaces
?
orthogonal, as in x ? y
M?
ortho-complement of M ,
AŒM
the post-set or image of M with respect to a mapping or function A
AŒM
the post-set or co-domain of M with respect to the binary relation A
AŒX
the post-set of X of a relation A X Y , co-domain, range or image of the mapping or function A
R.A/
range of the mapping or function A
2B
the power set of B
AB
the set of (left-total) mappings from B into A
ŒM A
the pre-image of M with respect to a mapping or function A
ŒM A
the pre-set of M with respect to a binary relation A
ŒY A
the pre-set of a relation A, the pre-image of Y with respect to a mapping or function A or the domain of A
D.A/
the domain of a mapping or function A
Cartesian product as in X Y or Cartesian multiplication sign as in s2M Hx , where X; Y; M; Hs are sets, t 2 M
vector product in R3 as in x y, where x; y 2 R3
PC
orthogonal projector onto the closed subspace C
%.A/
resolvent set of operator A
%1 .A/ ˇ Aˇ
Sobolev lattice resolvent set of A
M
Aˇrestricted to M for a mapping A W D.A/ X ! Y , i.e. the mapping AˇM W D.A/ \ M X ! Y where x 7! A.x/
RH
Riesz mapping, which unitarily maps H onto H
.A/
spectrum of operator A
C .A/
continuous spectrum of A
xviii
Nomenclature
P .A/
point spectrum of A
R.A/
residual spectrum of A
P 1 .A/ Sobolev lattice point spectrum of A R1 .A/ Sobolev lattice residual spectrum of A 1 .A/
Sobolev lattice spectrum of A
C 1 .A/ Sobolev lattice continuous spectrum of A supp
support
supp0
support in direction 0
supp0
temporal support
dSx
surface element at x
dS
surface element
a
˝
algebraic tensor product
˝ S
tensor product big union symbol, as in
S
M D ¹yj
W X2M
y 2 X º or
grad
vector analytic differential operator grad, gradient
curl
vector analytic differential operator curl, curl
div
vector analytic differential operator div, divergence
d Vx
volume element at x
dV
volume element
S X2M
X
Chapter 1
Elements of Hilbert Space Theory
We shall assume that the reader is familiar with major Hilbert space concepts (see e.g. [46, 6, 51, 2] as general references). For later parts of this monograph this includes the spectral theorem for unbounded selfadjoint operators and even finite commuting families of such operators, [10, 42]. However, in order to introduce particular notations and to refresh the readers memory, we shall start by briefly recapping some of the basic ideas.
1.1
Hilbert Space
Definition 1.1.1. An inner product space is an ordered pair X D .L; h j i/ consisting of a linear space L D .M; C; .˛/˛2K / over the field of real numbers K D R or the field of complex numbers K D C (with .x; y/ 7! x C y being the addition in L and x 7! ˛ x the multiplication by the scalar ˛ 2 K) and a functional h j i W M M ! K; .x; y/ 7! hxjyi called the inner product1 , which satisfies (a) for all x 2 M :
hxjxi 2 R0
.WD ¹˛ 2 R j ˛ 0º/;
(b) hxj i W M ! K, is a linear functional2 for every fixed x 2 M , (c) for all x; y 2 M :
hxjyi D hyjxi ;
where : : : denotes complex conjugation, (d) for all x 2 M : hxjxi D 0 ) x D 0 2 L: It is customary not to distinguish notationally between the set M , the linear space L and the inner-product space X thus allowing to speak of the elements of M as elements of the inner product space X or to say that X is a linear space. For easier 1 If only properties (a)–(c) are required, then X is called a semi-inner-product space and h j i is referred to as a semi-inner-product. 2 Note that thus we have chosen the inner product to be linear in the second factor.
2
Chapter 1 Elements of Hilbert Space Theory
reference the (semi-)inner product of an (semi-)inner-product space X will be denoted by h j iX . So, let X be an inner product space. Then the functional p x 7! hxjxiX defines a norm on X (or semi-norm, if property (d) is not required). The crucial triangle inequality follows from the Cauchy–Schwarz inequality p p jhxjyiX j hxjxiX hyjyiX for all x; y 2 X . The latter can be easily seen from the non-negativity of the determinant of the (non-negative) Gramian matrix function hxjxiX hxjyiX .x; y/ 7! hyjxiX hyjyiX associated with the inner product. As a matter of convenience we shall denote the associated (semi-)norm by j jX . Definition 1.1.2. Let X be an inner product space. If X is complete with respect to the norm j jX , then we say that X is a Hilbert space. A standard procedure of constructing Hilbert spaces is by completion of an innerproduct space. Therefore an inner-product space is also known as a pre-Hilbert space. As a prominent example we may consider L2 .R/ as the completion of the space of step functions, i.e. the linear span LinC ¹I j I bounded interval in Rº of characteristic functions I of bounded, left-closed intervals I in R, with respect to the norm induced by the inner product Z .f; g/ 7!
f .t / g.t / dt: R
Lebesgue integration theory shows that L2 .R/ may be identified with (equivalence) classes of square-integrable functions (coinciding pair-wise almost everywhere, compare e.g. the appendix in [50]).
1.2
Some Construction Principles of Hilbert Spaces
A closed subspace of a Hilbert space is also a Hilbert space. Beyond this trivial observation there are three main constructions for obtaining new Hilbert spaces from given Hilbert spaces, which we shall consider more closely: first the direct sum ˚, then the construction of Hilbert spaces based on duality and finally the tensor product ˝ of Hilbert spaces.
Section 1.2 Some Construction Principles of Hilbert Spaces
3
1.2.1 Direct Sums of Hilbert Spaces Direct sums of Hilbert spaces play an important role for the discussion of systems of equations and in particular for understanding linear operators in Hilbert spaces. Definition 1.2.1. Let H S denote the class of all real or complex Hilbert spaces (that is with scalar field K equal to R or C respectively) and let H. / W M ! H S, x 7! Hx , be a mapping, where M is a given set. Then the Cartesian product3 x2M Hx is also a set and naturally gives rise to a linear space by defining the component-wise linear structure .˛ v C w/.t / D ˛ v.t / C w.t / 2 H t
for all t 2 M; ˛ 2 K; v; w 2 Hx : x2M
Let W be the subset of all w 2 x2M Hx such that w.t / D 0 2 H t for all t 2 M but one. Then LinK W is a linear subspace of the Cartesian product space and equipped with the inner product hvjwi WD
X
hv.t / j w.t /iH t
for all v; w 2 LinK W
(1.2.1)
t2M
is a pre-Hilbert space. The completion of this pre-Hilbert space is called the direct sum of H. / , denoted by M Hx : x2M
If M D ¹0; : : : ; nº; n 2 N, N denoting the set of natural numbers (starting with 0), we also write H0 ˚ ˚ Hn L or tD0;:::;n H t or we use a suggestive column matrix notation 0
1 H0 : @ :: A Hn 3
Recall that the Cartesian product ˇ ^ ± ° [ ˇ Hx ˇ f .x/ 2 Hx : Hx WD f W M ! x2M
x2M
x2M
V W We use here and in the following occasionally the standard quantifiers ::: (“for every : : :”) and ::: (“there is a (at least one) : : :”) with possibly additional constraints. T S We prefer these to the more frequently used 8 : : : and 9 : : : because of their analogy to and and also to the logical symbols ^ (“and”) and _ (“or”), respectively. The latter symbols will also be occasionally used in composite mathematical expressions.
4
Chapter 1 Elements of Hilbert Space Theory
for the direct sum of .H0 ; : : : ; Hn /. The elements .xi /iD0;:::;n .x0 ; : : : ; xn / of such a finite direct sum will also be denoted by x0 ˚ ˚ xn or, in correspondence to the column matrix notation for the direct sum, by a column matrix 0 1 x0 B :: C @ : A: xn Remark 1.2.2. The linearity of the structure is easily checked. Note also that the sum in (1.2.1) is indeed a finite sum, since except for finitely many terms the inner products are zero. If M is a finite set then the pre-Hilbert space constructed here is already complete. The simplicity of the construction ofL such a direct sum is reflected in the simplicity of obtaining an orthonormal set4 in x2M Hx from orthonormal sets in the spaces H t ; t 2 M . Indeed, if ! t is an orthonormal set in Hilbert space H t , t 2 M , then the set Lof all ! 2 x2M !x with !.t / D 0 for all but one t 2 M is an orthonormal set in x2M Hx . Moreover, if all !L t are complete orthonormal sets in Hilbert space H t for t 2 M then ! is complete in x2M Hx . Example 1.1. C nC1 with the Euclidean inner product is a direct sum of n copies of C (as Hilbert space with inner product .x; y/ 7! x y): C nC1 D
M
C D `2 .¹0; : : : ; nº; C/:
tD0;:::;n
The spaces `2 .M; C/ WD
M
C;
M an arbitrary set;
t2M
are frequently used as model spaces. As a first application of the concept of a direct sum we recall the following variant of the projection theorem. 4
An orthonormal set ! is a subset of an inner-product space X with hxjxi D 1
for all x 2 ! and hxjyi D 0 for all x; y 2 ! with x ¤ y. Such an orthonormal set is called complete if its linear span LinK ! is dense in X , i.e. the closure of its linear span equals X : LinK ! D X:
Section 1.2 Some Construction Principles of Hilbert Spaces
5
Theorem 1.2.3. Let H be a real or complex Hilbert space and C a closed subspace of H . Then we have H D C ˚ C?
(1.2.2)
in the sense of unitary equivalence. Here ˇ ^ ° ± ˇ hxjyiH D 0 : C ? WD x 2 H ˇ y2C
The unitary equivalence expressed in (1.2.2) allows us to speak of an orthogonal decomposition of H . Indeed, we have that every x 2 H can be written in a unique way as x D u C v with u 2 C and v 2 C ? . The mapping PC W H ! H; x 7! PC .x/;
(1.2.3)
where PC .x/ is the element of the singleton ¹u 2 C j x u 2 C ? º, i.e.5 PC .x/ WD 2.¹u 2 C j x u 2 C ? º/; is a well-defined linear mapping called the orthogonal projector associated with C . Writing 1H or simply 1 for the identity mapping in H , we see that6 I C is the range7 PC ŒH of PC I C ? is the range .1 PC /ŒH of .1 PC / and I PC ı PC D PC . 5 As for the notation we recall that 2 is a relation between classes. The notation 2. / indicates, however, that 2 should here be understood as a mapping, which is only possible if the arguments are sets with a single element, i.e. a singleton. So, the domain of M 7! 2.M / is the class of all singletons and the value 2.M / is the element of the singleton M :
¹xº 7! 2.¹xº/ WD x: 6
In general an orthogonal projector is defined in terms of these three properties. For a binary relation R (or a binary correspondence .R; X Y /, where R is a relation in X Y , i.e. a subset of X Y ), we define the pre-set of a set M under the relation R as ° ˇ _ ± ˇ ŒM R WD x ˇ y2M 7
.x;y/2R
6
Chapter 1 Elements of Hilbert Space Theory
Example 1.2. Consider the direct sum of Hilbert spaces M Ht : H D t2M
Then the mappings Hs W H ! H; x 7! Hs .x/ with
´ x.s/ .Hs .x//.t / WD 0
for t D s; otherwise
are orthogonal projectors for all s 2 M . Recalling that .Hs .x//.s/ D x.s/ 2 Hs , we see that Hs ŒH can (and occasionally will) be identified with the (tagged) Hilbert space Hs , s 2 M . We note the following important consequence of the projection Theorem 1.2.3. Theorem 1.2.4. Let H be a Hilbert space with scalar field K and M H a subset. Then LinK M D M ?? : and the post-set of a set N under the relation R as ° ˇ ˇ RŒN WD y ˇ
_
± x2N :
.x;y/2R
V
We recall that if R is right-unique, i.e. x;y;z ..x; y/; .x; z/ 2 R H) y D z/, then R is called a function. A correspondence .R; X Y / with R right-unique is commonly referred to as a mapping, more commonly written as R W ŒY R X ! Y; x 7! R.x/; where as a consequence R.x/ WD 2.¹yj.x; y/ 2 Rº/ is well-defined. Here ŒY R is called the domain of the mapping R, sometimes also written as D.R/. The post-set RŒX of X under the right-unique R is referred to as the range of the function or mapping. It is customary although occasionally confusing to make little distinction between functions and mappings. A small part of the resulting confusion is that some authors speak of multi-valued mappings, when they should say relations or correspondences. Since mathematics strictly speaking only deals with sets (even “elements” are sets in the framework of the Zermelo–Fraenkel set theory), to speak of set-valued mappings is also not too helpful. What is meant and could easily be said is that in a canonical way every correspondence .R; X Y / gives rise to a mapping ŒY R X ! 2Y ; x 7! RŒ¹xº; which is frequently identified with the correspondence .R; X Y /.
Section 1.2 Some Construction Principles of Hilbert Spaces
7
Moreover, if N M then M ? ˚ .M ?? \ N ? / D N ? : Let .xn /n be a sequence in a Hilbert space H . One says .xn /n converges weakly to n!1
x1 in H , in symbols xn * x1 in H , if ^ hujxn iH ! hujx1 iH
as n ! 1:
u2H
The usual convergence in norm is also referred to as strong convergence. That weak convergence is really a weaker concept than strong convergence is implied by the Cauchy–Schwarz inequality jhujxn x1 iH j jujH jxn x1 jH which clearly shows that strong convergence implies weak convergence. Orthonormal sequences provide examples of not strongly convergent sequences which are, however, weakly convergent to zero. Despite this apparent difference in strength, the above result can be seen to yield the following characterization of the closure of a subspace, which contains the nucleus of the issue of “weak versus strong”. Proposition 1.2.5. Let M be a subspace of the Hilbert space H . Then ˇ ° ± _ n!1 ˇ MN D x1 2 H ˇ xn * x1 : .xn /n 2M N
Remark 1.2.6. This result states in the obvious terminology that the strong closure of M is equal to the weak closure of M . Proof. Since weak convergence follows from strong convergence, we obviously have ˇ ° ± _ n!1 ˇ MN x1 2 H ˇ .xn * x1 / : .xn /n 2M N
So, let x1 2 H be such that there is a sequence8 .xn /n 2 M N with n!1
xn * x1 : Since hujxn iH D 0 for all n 2 N and u 2 M ? , we also see hujx1 iH D 0 for all u 2 M ? , that is x1 2 M ?? D MN : 8 In general, the class of mappings defined on a class A and values in a class B is suggestively written as B A . Sequences are mappings with domain N, the natural numbers, which we assume to begin with zero (as suggested by there construction based on the empty set). Sequences in M are therefore elements of M N .
8
Chapter 1 Elements of Hilbert Space Theory
The decomposition argument employed here can also be utilized to obtain the following weak compactness result. Proposition 1.2.7. Let .xn /n 2 H N be a bounded sequence in a Hilbert space H . Then there is a subsequence .e x n /n 2 H N such that n!1
e x n * x1 for some x1 2 H . Proof. As argued above, for u 2 ¹xn j n 2 Nº? we have hujxn iH D 0 for all n 2 N and so it suffices to consider the subspace H0 WD LinK ¹xn j n 2 Nº in place of the Hilbert space H . Note that .xn /n is a bounded sequence in H0 and if the weak limit x1 exists it must also be in H0 . By virtue of the Weierstrass selection theorem .k/ .k1/ /n such that of calculus we now obtain recursively subsequences .xn /n of .xn .k/ .0/ .k/ .hxk jxn iH /n converges in C, k 2 N, where .xn /n D .xn /n . Obviously, .xn /n is a subsequence of .xn /n for all k 2 N. We also see that by our taking subsequences .k/ consecutively we even have that .hxs jxn iH /n converges for all s k. We claim that .n/ the diagonal sequence .xn /n has the desired property. Clearly, the diagonal sequence .n/ is also a subsequence of .xn /n . Moreover, .xn /nDk;.kC1/;::: yields a subsequence .k/ .n/ of .xn /n2N for all k 2 N. Therefore, .hxs jxn iH /n converges for all s 2 N. By Gram–Schmidt orthonormalisation (ignoring elements of the sequence linearly dependent on earlier elements) we obtain a countable complete orthonormal set o0 for H0 . Let .un /n be an enumeration of o0 . By construction, uN is just a linear .n/ combination of .xn /nD1;:::;N for all N 2 N, so we have that .hus jxn iH /n converges for all s 2 N. We are therefore led to define x1 WD
1 X sD1
lim hus jxn.n/ iH us
n!1
which is well-defined because v u1 uX .n/ t jhus jxn iH j2 D jxn.n/ jH sup¹:jxn jH jn 2 Nº DW c0 < 1: sD1
Note that this also implies jx1 jH c0 . Moreover, we find for any uD
1 X sD1
hus juiH us 2 H
Section 1.2 Some Construction Principles of Hilbert Spaces
9
that N ˇD X ˇ E ˇ ˇ ˇ ˇ jhu j xn.n/ x1 iH j ˇ hus juiH us ˇ xn.n/ x1 ˇ H
sD1
1 ˇ ˇD X E ˇ ˇ ˇ ˇ hus juiH us ˇ xn.n/ x1 ˇ Cˇ sDN C1
H
N ˇ ˇX ˇ ˇ hus juiH h us j xn.n/ x1 iH ˇ ˇ sD1
v u 1 u X jhus juiH j2 ; C 2 c0 t sDN C1
and we see that the last term can be made small independently of n by choosing N sufficiently large. With fixed N the first term then can be made small by making n sufficiently large. This is the desired convergence statement. Remark 1.2.8. The result can be restated as saying that closed, bounded sets are weakly (sequentially) compact, in the sense that every sequence in such a set contains a weakly convergent subsequence with weak limit in the set. In contrast, all closed, bounded sets in a Hilbert space H are compact if and only if the closed unit ball is compact, which in turn is equivalent to H being finite-dimensional. We have already mentioned the construction of Hilbert spaces as closed subspaces of Hilbert spaces. In connection with the direct sum of Hilbert spaces and in accordance with earlier remarks we make the following definition about subspaces of a direct sum of Hilbert spaces. It will often be convenient and intuitive to think of mappings, and in particular linear mappings between Hilbert spaces H0 and H1 in the way they were originally made precise in the framework of set theory, namely as particular subsets of the Cartesian product H0 H1 or rather of the direct sum H0 ˚H1 . In this sense, a subset A of H0 ˚ H1 is a (binary) relation between H0 and H1 . If in addition A is right-unique, i.e. ^ .x ˚ y0 ; x ˚ y1 2 A ) y0 D y1 /; x2H0 ; y0 ;y1 2H1
then A defines a mapping9 A W D.A/ H0 ! H1 ; x 7! A.x/; 9 The original subset A is then occasionally referred to as the graph of this mapping for which we use the same name A.
10
Chapter 1 Elements of Hilbert Space Theory
where
ˇ _ ° ± ˇ D.A/ WD x 2 H0 ˇ x˚y 2A y2H1
and A.x/ is the element of the singleton ¹y 2 H1 j x ˚ y 2 Aº, in symbols A.x/ WD 2.¹y 2 H1 j x ˚ y 2 Aº/: If a relation A is a linear subspace of H0 ˚ H1 , i.e. ^ .x0 ˚ x1 C ˛ y0 ˚ y1 D .x0 C ˛ y0 / ˚ .x1 C ˛ y1 / 2 A/; x0 ˚x1 ;y0 ˚y1 2A; ˛2K
then A is also called a linear relation. For a relation A in H0 ˚ H1 we write ˇ _ ° ± ˇ AŒ WD y 2 H1 ˇ x˚y 2A ; x2
ˇ _ ± ˇ Œ‚A WD x 2 H0 ˇ x ˚ y 2 A D A1 Œ‚; °
y2‚
where A1 denotes the inverse relation A1 WD ¹y ˚ x 2 H1 ˚ H0 j x ˚ y 2 Aº to A. AŒ is called the post-set of under the relation A and Œ‚A is called the pre-set of ‚ under the relation A. AŒH0 is simply called the post-set of A and ŒH1 A the pre-set of A. If A is a mapping then R.A/ WD AŒH0 is called the range of A and D.A/ D ŒH1 A is the domain of A. The set N.A/ WD Œ¹0ºA D A1 Œ¹0º is the null space of A. Note that A.x/ D 2.AŒ¹xº/: A right-unique linear relations is of course nothing but a linear mapping in the usual sense. Lemma 1.2.9. A right-unique relation A H0 ˚ H1 , Hi ; i D 0; 1, Hilbert spaces, is linear if and only if A W D.A/ H0 ! H1 is linear. Proof. The only thing to check is that linearity of a mapping is indeed the same as linearity of the subspace A. This becomes clear by observing that A.˛ x C y/ D ˛ Ax C Ay is the same as saying ˛ .x; Ax/ C .y; Ay/ D .˛ x C y; ˛ Ax C Ay/ 2 A:
Section 1.2 Some Construction Principles of Hilbert Spaces
11
This lemma yields the justification for calling a right-unique linear relation A a linear mapping or – synonymously – a linear operator. Remark 1.2.10. For linear mappings A we use the simplified multiplicative notation Ax WD A.x/ for x 2 D.A/: It should be noted that scalar multiplication ˛ as a mapping in a Hilbert space H is also a linear operator. With little risk of confusion we will therefore also in many instances simplify notation by letting ˛ x WD ˛ x
for all ˛ 2 K; x 2 H;
where K denotes the scalar field of H . Recall that for a mapping A W D.A/ H0 ! H1 the subset A H0 ˚H1 is often referred to as the graph of A. The restriction of the norm in H0 ˚H1 to A H0 ˚H1 is therefore also known as the graph norm j jD.A/ associated with the mapping A: ^
jxjD.A/ WD
q q 2 2 2 2 jxjH C jyj D jxjH C jA.x/jH D j.x; y/jH0 ˚H1 : H1 0 0 1
.x;y/2A
For a linear operator A the domain D.A/ of A is a linear subspace of H0 . A linear operator A is called a closed linear operator if A is a closed subspace of H0 ˚ H1 . A linear operator A is called a closable linear operator if A is a closed subspace of H0 ˚ H1 and a right-unique relation. In this case A is a closed linear operator. If a linear operator A is closed it is a Hilbert space (as a closed subspace of H0 ˚ H1 ). With the graph norm the domain D.A/ is also a Hilbert space, which clearly is unitarily equivalent via the bijective isometry D.A/ ! A; x 7! .x ˚ A.x// to A, where D.A/ is equipped with the graph norm j jD.A/ and A with the norm of H0 ˚ H1 . To establish a Hilbert space as the completion of D.A/ of some closable linear mapping A, is one of the standard procedures for constructing Hilbert spaces. Any dense subset of D.A/ in the sense of the graph norm of A is known as a core of A. In this sense, if A is closable that D.A/ is naturally a core for the closed linear N operator A. Sometimes we will have occasion to consider linear operators between finite direct sums of Hilbert spaces. In these casesLa matrix notation L is appropriate and convee niently suggestive. Let A W D.A/ H ! 7 k kD0;:::;m j D0;:::;n H j be a linear
12
Chapter 1 Elements of Hilbert Space Theory
operator. Define Aj k WD H ej A Hk ; j D 0; : : : ; n and k D 0; : : : ; m, with the notation of example (1.2). With the identification introduced there we can consider Aj k as the mapping ej ; Aj k W D.Aj k / Hk ! H x 7! Aj k .x/ with D.Aj k / WD Hk .D.A//, j D 0; : : : ; n and k D 0; : : : ; m. Now, if \ M D.Aj k / D.A/ D kD0;:::;m j D0;:::;n
holds, then for
we may write
0
1 x0 M B C x D @ ::: A 2 D.A/ Hk kD0;:::;m xm 0
10 1 A00 A0m x0 B :: : : : CB : C @ : : :: A @ :: A WD Ax: An0 Anm xm
Conversely, an operator matrix
0
1 A00 A0m B :: : : : C @ : : :: A An0 Anm L L e j with domain defines an operator from kD0;:::;m Hk to j D0;:::;n H M \ D.Aj k /: kD0;:::;m j D0;:::;n
In some cases it may actually be feasible to expand the matrix notation and to also arrange the argument in a more general matrix form to insinuate action of an operator matrix on a matrix of arguments of suitable size. Since this notation takes its intuition from elementary matrix algebra, we shall freely use this extension of notation without further explanation. As a first, simple application of this matrix notation we will recall the concept of complexification. Definition 1.2.11. Let H be a real Hilbert space. Then the complexification HC of H can be defined as H H ˚H D H
Section 1.2 Some Construction Principles of Hilbert Spaces but equipped with a modified scalar multiplication by ˛ 2 C defined by H H ! ; H H x0 x0 7! ˛ x1 x1 with
˛
x0 x1
WD
Re ˛ I m ˛ I m ˛ Re ˛
for
x0 x1
x0 x1
2
D
H H
.Re ˛/ x0 .I m ˛/ x1 .I m ˛/ x0 C .Re ˛/ x1
13
(1.2.4)
:
Remark 1.2.12. It is straightforward to confirm that the complexification HC is a complex Hilbert space. In this context it should be noted that any complex Hilbert space H can also be considered as a real Hilbert space by restricting the scalar field to R and using Re.hjiH / as inner product. Thus henceforth we restrict our attention to the case of complex Hilbert spaces. The corresponding reasoning for real Hilbert space is usually just a simplification of the arguments for complex Hilbert space. Returning to our view of a linear mapping A W D.A/ H0 ! H1 as a particular pre-Hilbert (sub)space of H0 ˚ H1 , we note that, by the projection theorem, we have the orthogonal decomposition H0 ˚ H1 D A ˚ A? for any closed linear relation A H0 ˚ H1 . The closed linear relation A? is of particular interest. Definition 1.2.13. Let A be a relation in H0 ˚ H1 . Then A WD . A? /1 will be called the adjoint relation. Here A D .1/A WD ¹.x; y/ 2 H0 ˚ H1 j .x; y/ 2 Aº: If A is a linear mapping, it is called the adjoint operator.
14
Chapter 1 Elements of Hilbert Space Theory
Remark 1.2.14. The reason for defining this peculiar looking combination of operations lies in the property that hxjviH0 D hyjuiH1
(1.2.5)
for all .x; y/ 2 A and .u; v/ 2 A . In this form, the ortho-complement is closer to the construction of adjoint operators (see later). Lemma 1.2.15. Let A be a relation in H0 ˚ H1 . Then .A1 /? D .A1 /? D .A? /1 D ..A/? /1 D .A? /1 :
(1.2.6)
In other words, the three operations involved in the definition of A commute. Proof. Let .x; y/ 2 .A1 /? H1 ˚ H0 , i.e. ^ h.x; y/j.u; v/iH1 ˚H0 D 0: .u;v/2A1
Then we have the following chain of equivalences .x; y/ 2 .A1 /? ^ , h.x; y/j.v; u/iH1 ˚H0 D h.x; y/j.v; u/iH1 ˚H0 D 0 .u;v/2A
,
^
h.y; x/j.u; v/iH0 ˚H1 D 0
.u;v/2A
, .y; x/ 2 A? , .x; y/ 2 .A? /1 : As a consequence we obtain the following result directly from Theorem 1.2.4. Theorem 1.2.16. Let H0 ˚ H1 be a direct sum of two (complex) Hilbert spaces and let A H0 ˚ H1 be a relation. Then LinC A D A : Proof. According to (1.2.6) and by definition of the adjoint relation we have A D ...A1 /? /1 /? D ...A?? /1 /1 / D A?? D LinC A: Here we have used .1/.1/ D 1 and that Theorem 1.2.4 holds.
Section 1.2 Some Construction Principles of Hilbert Spaces
15
Remark 1.2.17. Note that A is indeed a double ortho-complement. Therefore Theorem 1.2.4 applies. So, if A is a linear relation then its (strong) closure AN is also equal to its weak closure. If AN is a linear mapping then it can be also characterized as the N adjoint relation of A . Since the latter is referred to as a weak characterization of A, we have established that weak (closure) equals strong (closure). The projection theorem now gives the following useful decomposition result. Theorem 1.2.18. Let H0 ˚ H1 be a direct sum of two (complex) Hilbert spaces and A H0 ˚ H1 a closed, linear relation. Then we have the orthogonal decompositions H0 D Œ¹0ºA ˚ A ŒH1 and
H1 D Œ¹0ºA ˚ AŒH0 :
Proof. By a straightforward calculation we have ^ y ? AŒH0 , 0 D hyjviH1 D h.0; y/j.u; v/iH0 ˚H1 .u;v/2A
, .0; y/ 2 A? , .y; 0/ 2 A , y 2 Œ¹0ºA : This shows that AŒH0 ? D Œ¹0ºA and the second decomposition follows from the projection theorem in H0 ˚ H1 . The first decomposition now follows by replacing A by A and using Theorem 1.2.16. Remark 1.2.19. With A D PC as the orthogonal projector defined in (1.2.3) and H D H0 D H1 we recover from Theorem 1.2.18 the earlier Theorem 1.2.3. It can be shown that PC D PC , i.e. PC is selfadjoint10 , and Œ¹0ºPC D .1 PC /ŒH .
1.2.2 Dual Spaces As an important application of Theorem 1.2.18 we shall now recall the Riesz representation theorem, which provides a construction of another Hilbert space H as the 10 A relation satisfying A D A is called selfadjoint. A relation is called skew-selfadjoint if A D A , which can be simply stated as A1 D A? :
If A is an operator, then we speak of a selfadjoint or a skew-selfadjoint operator, respectively. If for a linear relation A we have A A , we say A is Hermitian. A Hermitian operator A W D.A/ H0 ! H0 is characterized by the property hAxj yiH0 D hxj AyiH0 for all x; y 2 D.A/. If A is a Hermitian operator and D.A/ is dense in H0 , i.e. A is densely defined, then A is called symmetric.
16
Chapter 1 Elements of Hilbert Space Theory
dual of a given Hilbert space H . For this construction, let first H WD ¹f 2 C H jf linear ^ f Lipschitz continuousº: Since any mapping into the numbers is called a functional, the elements of H are referred to as continuous11 , linear functionals. On H we do not use the linear structure induced by C, but rather define the following modified12 linear structure: .˛f C g/.t / WD ˛ f .t / C g.t / for all ˛ 2 C; f; g 2 H ; t 2 H:
(1.2.7)
Thus, H becomes a linear space. We choose this particular linear structure in order that the Riesz mapping (see Definition 1.2.21 below) becomes linear. We state without proof the Riesz representation theorem. Theorem 1.2.20 (Riesz representation theorem). Let H be a complex Hilbert space. For any f 2 H there is a unique w 2 H such that f .x/ D hwjxiH
for all x 2 H:
Definition 1.2.21. Let H be a complex Hilbert space. The mapping RH W H ! H; f 7! RH .f /; V where RH .f / WD 2.¹w 2 H j x2H f .x/ D hwjxiH :º/, is – according to the Riesz representation theorem – a well-defined mapping called the Riesz operator or the Riesz mapping. The Riesz mapping enjoys the following properties: Lemma 1.2.22. Let H be a complex Hilbert space. The Riesz mapping RH W H ! H is a linear mapping and norm-preserving13 in the sense that jRH f jH D jf jLip 11 12
for all f 2 H :
Recall that for linear mappings Lipschitz continuity is equivalent to continuity. With the imaginary unit i we have in particular that for f 2 H 1 f D f .i /: i
13
The functional j jLip given by ² jf jLip WD sup
ˇ ³ jf .x/ f .y/jB1 ˇˇ x; y 2 D; x ¤ y ˇ jx yjB0
(1.2.8)
Section 1.2 Some Construction Principles of Hilbert Spaces
17
Proof. The linearity14 follows straightforwardly from the definition of RH and the properties of the inner product. For all f; g 2 H ; y 2 H and ˛ 2 C we calculate hRH .˛ f C g/ j yiH D .˛ f C g/.y/ D ˛ f .y/ C g.y/ D ˛ hRH .f / j yiH C hRH .g/ j yiH D h˛ RH .f / C RH .g/ j yiH I that is RH .˛f C g/ D ˛ RH f C RH g: Moreover, by the Cauchy–Schwarz inequality we have for all y 2 H jf .y/j D jhRH f j yiH j jRH f jH jyjH
(1.2.9)
and so jf jLip jRH f jH : Since equality can occur in (1.2.9) (e.g. for y D RH f ), we see that jRH f jH is also the best possible constant, i.e. (1.2.8) holds. From (1.2.8) we also see that j jLip actually is a pre-Hilbert space norm, since the left-hand side is clearly a pre-Hilbert space norm associated with the inner product .x; y/ 7! hRH x j RH yiH : Thus, we have that H equipped with this inner product is a pre-Hilbert space. Lemma 1.2.23. Let H be a complex Hilbert space. The Riesz mapping RH W H ! H is onto. Proof. Let y 2 H be arbitrary. The linear functional fy W H ! C; x 7! hy j xiH , is clearly linear and by the Cauchy–Schwarz inequality also Lipschitz continuous. Indeed, as in the proof of the previous lemma we have jhy j iH jLip D jyjH : for the linear space of Lipschitz continuous functions f W D B0 ! B1 , B0 ; B1 normed linear spaces, constitutes a semi-norm. For a fixed argument x0 2 D the functional f 7! jf .x0 /jB1 C jf jLip defines a norm for such functions. For the subspace of linear mappings we may choose x0 D 0 to see that for these j jLip is already a norm. In the latter case we also write kf k WD jf jLip and f 7! kf k is referred to as the operator norm of the linear mapping f . In some cases, where domain and range spaces may be unclear, we shall also write k kX!Y if linear mappings from normed linear space X to normed linear space Y are concerned. 14 The particular choice of linear structure is designed to make the Riesz mapping a linear operator.
18
Chapter 1 Elements of Hilbert Space Theory
Thus, we have fy 2 H . Moreover, by definition fy .x/ D hy j xiH D hRH fy j xiH for all x 2 H and therefore RH fy D y: (1.2.10) As a consequence we have Theorem 1.2.24. Let H be a complex Hilbert space. Then H is also a Hilbert space (the so-called dual space to H ) and the Riesz mapping RH W H ! H is unitary. Proof. Given our previous results we only need to show completeness of H . So, let .fn /n be a Cauchy sequence in H . Then, since RH is norm-preserving, we also have that .RH fn /n is a Cauchy sequence in H . By the completeness of H we have y WD lim RH fn 2 H: n!1
According to (1.2.10) we have RH fy D y for fy W H ! C; x 7! hy j xiH . Thus, (again by norm-preservation) from RH fn ! y as n ! 1 we obtain fn ! fy
as n ! 1:
Since RH is norm-preserving it is clearly one-to-one and by the above indeed a unitary mapping. Remark 1.2.25. It is worth comparing the adjoint to the dual relation. Let A H0 ˚ H1 be a linear relation, then the dual relation is given by A} WD .Aa /1 H1 ˚ H0 ; where the so-called annihilator Aa of A is defined as Aa WD ¹f 2 H0 ˚ H1 j f .x/ D 0 for all x 2 Aº: The connection between A and A} is given by composition with the corresponding Riesz mappings RHi ; i D 0; 1, : A D RH0 A} RH 1
(1.2.11)
The dual relation is the more general concept, since it makes sense in a Banach space setting, whereas the adjoint uses the concept of orthogonality. In case of rightuniqueness one speaks of the dual mapping or dual operator. As an example, given 1 the Riesz mapping RH W H ! H , then RH D RH W H ! H . However, } W H ! H is such that RH } g.RH f / D .RH g/.f /
Section 1.2 Some Construction Principles of Hilbert Spaces
19
for all f; g 2 H . According to (1.2.11) we have } 1 D RH D RH RH RH RH
and it follows that } : 1H D RH RH
Thus we have } 1 D RH RH D RH :
In the following we shall always make use of the implied identification of H with H via the unitary mapping } RH RH W H ! H :
In some instances it may also be useful to even identify H with H .
1.2.3 Tensor Products of Hilbert Spaces A final construction principle for new Hilbert spaces we shall explore is the so-called tensor product of Hilbert spaces. It can be rooted on the concept of multi-linear forms on Hilbert spaces. We shall be more detailed here since this concept – although extremely useful for the discussion of partial differential equations – is rarely included in elementary functional analysis texts. Let .Hi /iD0;:::;n be a family of .n C 1/ (complex) Hilbert spaces. A mapping W H0 Hn ! C is called continuous .n C 1/-linear form on H0 Hn if y 7! .: : : ; xi1 ; y; xiC1 ; : : :/ 2 Hi for xk 2 Hk for k D 0; : : : ; n, k ¤ i , i D 0; : : : ; n. The implied linear structure for such .n C 1/-linear forms f; g is given by ^
.˛ f C g/.x/ WD ˛ f .x/ C g.x/:
˛2C; x2H0 Hn
Given xi 2 Hi , i D 0; : : : ; n, a continuous .n C 1/-linear form x0 ˝ ˝ xn on H0 Hn can be defined by x0 ˝ ˝ xn .u0 ; : : : ; un / WD hx0 ju0 iH0 hxn jun iHn
20
Chapter 1 Elements of Hilbert Space Theory
for ui 2 Hi , i D 0; : : : ; n. Such continuous .n C 1/-linear forms are referred to as decomposable continuous .n C 1/-linear form. Hence, we can generate a linear space called the algebraic tensor product of .Hi /iD0;:::;n W˝ WD
a O
Hk WD LinC ¹x0 ˝ ˝ xn j xi 2 Hi ; i D 0; : : : ; nº:
kD0;:::;n
Defining hx0 ˝ ˝ xn j u0 ˝ ˝ un iH0 ˝˝Hn WD hx0 ju0 iH0 hxn jun iHn
(1.2.12)
for all xi ; ui 2 Hi ; i D 0; : : : ; n, we obtain a candidate for an inner product for W˝ by sesqui-linear extension. Since the values may depend on the representation, this process may a priori merely create a sesqui-linear relation, i.e. a relation, which is linear w.r.t. the second component and conjugate linear w.r.t. the first component. More precisely, we have a relation between W˝ W˝ and C, i.e. a subset B of .W˝ W˝ / C given by the set of all ordered pairs of the form .a; b/ with X X aD ˛i x0;i ˝ ˝ xn;i ; ˇj u0;j ˝ ˝ un;j i2N
j 2N
and bD
X X
˛i ˇj hx0;i ˝ ˝ xn;i j u0;j ˝ ˝ un;j iH0 ˝˝Hn ;
i2N j 2N
where ˛; ˇ 2 C N ;
ŒC n ¹0º˛; ŒC n ¹0ºˇ finite;
xk ; uk 2 HkN ;
k D 0; : : : ; n;
such that, as we shall demonstrate, if ..x; y/; !/; ..x; z/; / 2 B then also ..x; "y C z/; "! C / 2 B and ..y; x/; ! / 2 B for all !; ; " 2 C; x; y; z 2 W˝ . From these properties we shall derive that we have actually defined a sesqui-linear functional on W˝ . P x D i2N ˛i x0;i ˝ ˝ xn;i , y D PSo, consider arbitrary linear combinations P j 2N ˇj y0;j ˝ ˝ yn;j and z D k2N k z0;k ˝ ˝ zn;k . Then we see "y C z D "
X j 2N
ˇj y0;j ˝ ˝ yn;j C
X k2N
k z0;k ˝ ˝ zn;k
Section 1.2 Some Construction Principles of Hilbert Spaces and
X X
21
˛i "ˇj hx0;i ˝ ˝ xn;i j y0;j ˝ ˝ yn;j iH0 ˝˝Hn
i2N j 2N
X X
C
˛i k hx0;i ˝ ˝ xn;i j z0;k ˝ ˝ zn;k iH0 ˝˝Hn
i2N k2N
X X
D"
˛i ˇj hx0;i ˝ ˝ xn;i j y0;j ˝ ˝ yn;j iH0 ˝˝Hn
i2N j 2N
C
X X
˛i k hx0;i ˝ ˝ xn;i j z0;k ˝ ˝ zn;k iH0 ˝˝Hn
i2N k2N
showing the desired linearity property. Using (1.2.12) we also obtain X X ˛i ˇj hx0;i ˝ ˝ xn;i j y0;j ˝ ˝ yn;j iH0 ˝˝Hn i2N j 2N
D
X X
˛i ˇj hx0;i j y0;j iH0 hxn;i j yn;j iHn
i2N j 2N
D
X X
˛i ˇj hy0;j j x0;i iH0 hyn:j j xn;i iHn
i2N j 2N
D
X X
˛i ˇj hy0;j ˝ ˝ yn;j j x0;i ˝ ˝ xn;i iH0 ˝˝Hn
: (1.2.13)
i2N j 2N
This shows the desired conjugate symmetry. From (1.2.13) we also get X X ˛i ˛j hx0;i ˝ ˝ xn;i j x0;j ˝ ˝ xn;j iH0 ˝˝Hn i2N j 2N
D
X X
˛i ˛j hx0;i j x0;j iH0 hxn;i j xn;j iHn :
(1.2.14)
i2N j 2N .k/
Let .Aij /i;j be a non-negative root of the Gramian .hxk;i j xk;j iHk /i;j , k D 0; : : : ; n. Then we obtain from (1.2.14) X X ˛i ˛j hx0;i ˝ ˝ xn;i j x0;j ˝ ˝ xn;j iH0 ˝˝Hn i2N j 2N
D
X X X
i2N j 2N s0
D
X s0
X
X X sn
i
sn
.0/
.0/
.n/
.n/
˛i ˛j Ais0 As0 j Aisn Asn j .0/
.n/
˛i As0 i Asn i
X j
.0/ .n/ ˛j As0 j Asn j :
22
Chapter 1 Elements of Hilbert Space Theory
The last term is, however, the square of the norm of w in `2 .N nC1 /, where X .0/ .n/ ˛i As0 i Asn i ws0 sn WD i2N
for s D .s0 ; : : : ; sn / 2 N nC1 , and so X X ˛i ˛j hx0;i ˝ ˝ xn;i j x0;j ˝ ˝ xn;j iH0 ˝˝Hn 0: i2N j 2N
Next, we want to show definiteness of our candidate for a semi-inner product. So let X X ˛i ˛j hx0;i ˝ ˝ xn;i j x0;j ˝ ˝ xn;j iH0 ˝˝Hn D 0: i2N j 2N
Then, since 2 Re
X
˛i hx0;i ˝ ˝ xn;i j u0 ˝ ˝ un iH0 ˝˝Hn
i2N
C hu0 ˝ ˝ un j u0 ˝ ˝ un iH0 ˝˝Hn X X D ˛i ˛j hx0;i ˝ ˝ xn;i j x0;j ˝ ˝ xn;j iH0 ˝˝Hn i2N j 2N
C 2 Re
X
˛i hx0;i ˝ ˝ xn;i j u0 ˝ ˝ un iH0 ˝˝Hn
i2N
C hu0 ˝ ˝ un j u0 ˝ ˝ un iH0 ˝˝Hn ˇX ˇ2 ˇ ˇ Dˇ ˛i x0;i ˝ ˝ xn;i C u0 ˝ ˝ un ˇ
H0 ˝˝Hn
i2N
0
for all ui 2 Hi ; i D 0; : : : ; n, this implies by a standard argument on quadratic forms that we must also have X ˛i hx0;i ˝ ˝ xn;i j u0 ˝ ˝ un iH0 ˝˝Hn D 0; i2N
for all ui 2 Hi ; i D 0; : : : ; n. This is, however, by definition X X ˛i hx0;i ju0 iH0 hxn;i jun iHn D ˛i x0;i ˝ ˝ xn;i .u0 ; : : : ; un / 0D i2N
i2N
P for all ui 2 Hi ; i D 0; : : : ; n. In other words the .n C 1/-linear form i2N ˛i x0;i ˝ ˝ xn;i is the zero functional. We may finally conclude independence of the chosen representation. Indeed, if X X ˛i x0;i ˝ ˝ xn;i D ˇj y0;j ˝ ˝ yn;j i2N
j 2N
Section 1.2 Some Construction Principles of Hilbert Spaces then
X
˛i hx0;i ˝ ˝ xn;i j u0 ˝ ˝ un iH0 ˝˝Hn
i2N
X
j 2N
D
23
X
ˇj hy0;j ˝ ˝ yn;j j u0 ˝ ˝ un iH0 ˝˝Hn ˛i x0;i ˝ ˝ xn;i .u0 ; : : : ; un /
i2N
X
ˇj y0;j ˝ ˝ yn;j .u0 ; : : : ; un /
j 2N
D 0: This shows that DX
ˇ E ˇ ˛i x0;i ˝ ˝ xn;i ˇ u0 ˝ ˝ un
H0 ˝˝Hn
i2N
WD
X
˛i hx0;i ˝ ˝ xn;i j u0 ˝ ˝ un iH0 ˝˝Hn
i2N
defines a conjugate linear functional of the first factor. By conjugate symmetry, the analogous result follows for the second factor, i.e. ˇX DX E ˇ ˛i x0;i ˝ ˝ xn;i ˇ ˇj u0;j ˝ ˝ un;j i2N
WD
X X
i2N
H0 ˝˝Hn
˛i ˇj hx0;i ˝ ˝ xn;i j u0;j ˝ ˝ un;j iH0 ˝˝Hn
i2N j 2N
defines a sesqui-linear functional on W˝ with the properties of an inner product. With this we can now define the tensor product of Hilbert spaces. Definition 1.2.26. Let .Hi /iD0;:::;n be a family N of .n C 1/ (complex) Hilbert spaces. The completion of the pre-Hilbert space akD0;:::;n Hk with respect to the induced norm j jH0 ˝˝Hn is called the tensor product of .Hi /iD0;:::;n and denoted by N iD0;:::;n Hi or by NH0 ˝ ˝ Hn . Let Vi be a subspace of Hi , i D 0; : : : ; n. The pre-Hilbert space akD0;:::;n Vk is called the algebraic tensor product of .Vi /iD0;:::;n a
a
and also denoted by V0 ˝ ˝ Vn . The concept of tensor products simplifies dramatically if the Hilbert spaces are a priori constructed as completions of pre-Hilbert spaces of functions. Proposition 1.2.27. Let .Hi /iD0;:::;n be a family of .n C 1/ (complex) Hilbert spaces and .Vi /iD0;:::;n a corresponding family of pre-Hilbert spaces of functions, i.e. Vi
24
Chapter 1 Elements of Hilbert Space Theory
C Mi ; Mi sets, such that Hi is the completion of Vi , i D 0; : : : ; n. Then the subspace V0 Vn WD LinC ¹.t0 ; : : : ; tn / 7! f0 .t0 / fn .tn / j .f0 ; : : : ; fn / 2 V0 Vn º of C M0 Mn (with the linear structure induced by C) equipped with the inner product hji a a defined by
6
V0 ˝˝Vn
hFf j Fg i
a
WD hf0 jg0 iH0 hfn jgn iHn
a
V0 ˝˝Vn
(1.2.15)
for all f D .f0 ; : : : ; fn /; g D .g0 ; : : : ; gn / in V0 Vn where Ff abbreviates the mapping .t0 ; : : : ; tn / 7! f0 .t0 / fn .tn /, is a pre-Hilbert space. The mapping F.f0 ;:::;fn / 7! f0 ˝ ˝ fn extends by linearity to an isometry JH0 ˝˝Hn W
6 D
6
V0 Vn ! H0 ˝ ˝ Hn . The completion of V0 Vn is then unitarily equivalent to H0 ˝ ˝ Hn , where the unitary mapping is given by the continuous extension JH0 ˝˝Hn of JH0 ˝˝Hn to this completion. Proof. Although the result is just a special case of a general result, the implicit claim that hji a a is actually an inner product (rather than a semi-inner prodV0 ˝˝Vn
6
uct) needs to be confirmed. To see this definiteness, we consider a linear combination P i ˛i Ffi 2 V0 Vn satisfying ˇX DX E ˇ ˛i Ffi ˇ ˛j Ffj D 0: a a i
V0 ˝˝Vn
j
From the Cauchy–Schwarz inequality we obtain that D ˇX E ˇ ˛i Ffi D0 Fg ˇ a a V0 ˝˝Vn
i
for all g 2 V0 Vn . This yields D ˇX E ˇ 0 D Fg ˇ ˛i Ffi a
a
V0 ˝˝Vn
i
D
X i
˛i hFg j Ffi i
a
a
V0 ˝˝Vn
and using definition (1.2.15) (and letting fi D .fi0 ; : : : ; fin /; g D .g0 ; : : : ; gn /), we obtain X 0D ˛i hg0 j fi0 iV0 hgn j fin iVn i
D ˇX E ˇ D g0 ˇ ˛i hg1 j fi1 iV1 hgn j fin iVn fi0 : i
V0
Since V0 is a pre-Hilbert space and g0 2 V0 was arbitrary, this implies X ˛i fi0 .t0 / hg1 j fi1 iV1 hgn j fin iVn D 0 i
Section 1.2 Some Construction Principles of Hilbert Spaces
25
for all t0 2 M0 . By induction we get as desired X ˛i fi0 .t1 / fin .tn / D 0 for all ti 2 Mi ; i D 0; : : : ; n: i
This tensor product construction is extremely useful for our context, since it allows for the transition from one variable to multi-variable function spaces. N Example 1.3. `2 .ZnC1 / D kD0;:::;n `2 .Z/ Recalling that the elements of Vi WD `2 .Z/ are complex number sequences, we realize that with Mi D Z we encounter an instance of the last proposition. The correspondence .zi0 zin /.i
nC1 0 ;:::;in /2Z
7! .zi0 /i0 2Z ˝ ˝ .zin /in 2Z
extends to a linear isometry I . Since the elements generating `2 .ZnC1 / (according to its definition) have product form ¹.i0 ;:::;in /º .j0 ; : : : ; jn / D ¹i0 º .j0 / ¹in º .jn / : : ; jn / 2 ZnC1 , we see that I extends to a unitary mapping IN for .i0 ; : : : ; in /; .j0 ; :N 2 nC1 from ` .Z / onto kD1;:::;n `2 .Z/. N Example 1.4. L2 .RnC1 / D kD0;:::;n L2 .R/ This result follows by the same arguments as the previous example. For this we first need to notice that it is sufficient to use characteristic functions I0 In with Ik D Œak ; bk Œ; ak < bk ; k D 0; : : : ; n, in order to generate L2 .RnC1 /, since jI0 In J0 Jn jL2 .RnC1 / D 0 for Jk D ak ; bk Œ, Jk D ak ; bk or Jk D Œak ; bk , k D 0; : : : ; n. The linear span of these restricted characteristic functions is however a function space. Moreover, we have I0 In .t0 ; : : : ; tn / D I0 .t0 / In .tn /
for .t0 ; : : : ; tn / 2 RnC1 ;
from which the claim follows in the same way as for the previous example. Example 1.5. Similar to the previous examples, it can be seen that in general the completion L2 .R; H / of the space of H -valued step functions LinC ¹t 7! Œa;b Œ .t / h j h 2 H ^ a; b 2 Rº; is just L2 .R/ ˝ H for every Hilbert space H . The next proposition gives the details in a more general setting.
26
Chapter 1 Elements of Hilbert Space Theory As a final variant of the same reasoning we record the next proposition for later use.
Proposition 1.2.28. Let H0 and H1 be (complex) Hilbert spaces and V0 a preHilbert space of functions, i.e. V0 C M ; M a set, such that H0 is the completion of V0 . Then the subspace V0 H1 WD LinC ¹t 7! f .t / w j f0 2 V0 ^ w 2 H1 º of H1M (with the linear structure induced by H1 ) equipped with the inner product hji a
3
V0 ˝H1
defined by hf w j g vi
a
V0 ˝H1
WD hf jgiH0 hwjviH1
for all f; g 2 V0 ; w; v 2 H1
3
where f w denotes the mapping t 7! f .t / w is a pre-Hilbert space. The mapping f w 7! f ˝w extends by linearity to an isometry JH0 ˝H1 W V0 H1 ! H0 ˝H1 . The completion of V0 H1 is then unitarily equivalent to H0 ˝H1 , where the unitary mapping is the continuous extension JH0 ˝H1 of JH0 ˝H1 to this completion.
3
C
3
Proof. The result follows the same way as the previous P one. To see the definiteness of hji a we consider again a linear combination i ˛i fi wi 2 V0 H1 satisfying V0 ˝H1
DX
ˇX E ˇ ˛i fi wi ˇ ˛i fi wi
i
a
V0 ˝H1
i
D 0:
From the Cauchy–Schwarz inequality we obtain ˇX E D X ˇ ˛i fi wi a D ˛i hf j fi iV0 hw j wi iH1 f wˇ V0 ˝H1
i
i
D ˇX E ˇ D f ˇ ˛i fi hw j wi iH1
V0
i
D0
for all f 2 V0 ; w 2 H1 . Since V0 is a pre-Hilbert space and f 2 V0 is arbitrary, we get X ˛i fi .t / hw j wi iH1 D 0 (1.2.16) i
for all t 2 M . But (1.2.16) implies D ˇX E ˇ ˛i fi .t / wi wˇ i
which in turn yields
X
H1
D 0;
˛i fi .t / wi D 0
i
for all t 2 M , since once again H1 is a pre-Hilbert space and w 2 H1 is arbitrary.
Section 1.2 Some Construction Principles of Hilbert Spaces
27
This result shows that tensor products can be employed to manage the transition from scalar-valued function spaces to spaces of Hilbert space-valued functions. We also have the following useful general density result: Lemma 1.2.29. Let .Hi /iD0;:::;n be a family of .n C 1/ (complex) Hilbert spaces and .Si /iD0;:::;n a corresponding family of respective subsets. If Vi WD LinC Si is dense in Hi for i D 0; : : : ; n, then a
a
V0 ˝ ˝ Vn D LinC ¹x0 ˝ ˝ xn j xi 2 Si for i D 0; : : : ; nº is dense in H0 ˝ ˝ Hn . Proof. It suffices to approximate decomposable elements, i.e. elements of the form x0 ˝ ˝ xn 2 H0 ˝ ˝ Hn . Let vi WD .vi;j /j be a sequence approximating xi 2 Hi , i D 0; : : : ; n. Then we can estimate jx0 ˝ ˝ xn v0;j ˝ ˝ vn;j jH0 ˝˝Hn
n X
jv0;j ˝ ˝ vk1;j ˝ .xk vk;j / ˝ xkC1 ˝ ˝ xn jH0 ˝˝Hn
kD0
n X
jv0;j j jvk1;j jjxk vk;j jHk jxkC1 jHkC1 jxn jHn :
kD0
Since the last term is a finite sum of products of norms of a difference tending to zero and bounded factors, the right-hand side goes to zero. Thus we have v0;j ˝ ˝ vn;j ! x0 ˝ ˝ xn
as j ! 1:
Based on this lemma we can now address the issue of generating orthonormal sets in tensor product spaces from orthonormal sets in the “factor” spaces. Proposition 1.2.30. Let .Hi /iD0;:::;n be a family of n (complex) Hilbert spaces and .oi /iD0;:::;n a corresponding family of respective orthonormal sets. Then15 Œo0 ˝ ˝ Œon WD ¹x0 ˝ ˝ xn j xi 2 oi for i D 0; : : : ; nº 15 Here we utilize a general notational scheme, which we shall use throughout. Generating for a mapping or rather an operation f W M0 Mn ! N a new set by applying f to all elements of certain subsets 0 M0 ; : : : ; n Mn , will be indicated by using square brackets rather than round brackets as one does when f is applied in the normal way to elements m0 2 M0 ; : : : ; mn 2 Mn , i.e. we write f .m0 ; : : : ; mn / briefly for f ..m0 ; : : : ; mn // for the evaluation of f at .m0 ; : : : ; mn / 2 M0 Mn , but f Œ 0 n for the set ¹f .m0 ; : : : ; mn /j .m0 ; : : : ; mn / 2 0 n º. Since frequently the operation symbol is written between the arguments, we arrive at notations such as
Œ¹1; 2; 3º C Œ¹3; 4º D ¹4; 5; 6; 7º or Œ¹u; vº ˝ Œ¹x; yº D ¹u ˝ x; u ˝ y; v ˝ x; v ˝ yº or the one used here for applying ˝ repeatedly.
28
Chapter 1 Elements of Hilbert Space Theory
is an orthonormal set in H0 ˝ ˝ Hn . Moreover, if oi is complete in Hi for i D 0; : : : ; n, then Œo0 ˝ ˝ Œon is a complete orthonormal set in H0 ˝ ˝ Hn . Proof. Let x0 ˝ ˝ xn ; y0 ˝ ˝ yn 2 Œo0 ˝ ˝ Œon . Then hx0 ˝ ˝ xn jy0 ˝ ˝ yn i D hx0 jy0 iH0 hxn jyn iHn ´ 1 if x0 D y0 ; : : : ; xn D yn ; D 0 otherwise; which shows that indeed Œo0 ˝ ˝ Œon is an orthonormal set in H0 ˝ ˝ Hn . Recalling that for an orthonormal set ok to be complete means that Vk WD LinC ok is dense in Hk , k D 0; : : : n, the rest of the proposition follows from the previous lemma. Since closed linear operators between Hilbert spaces are just (right-unique) closed linear subspaces of the direct sum of the Hilbert spaces, it would be natural to define the tensor product of linear operators as tensor products of these subspaces. There is, however, a more useful (algebraic) concept of a tensor product for linear operators Ak H0k ˚ H1k ; k D 0; : : : ; n, defined as the linear relation given by16 a
a
.A0 ˝ ˝An /.x0 ˝ ˝ xn / WD A0 x0 ˝ ˝ An xn
(1.2.17)
for all xk 2 D.Ak /; k D 0; : : : ; n. That (1.2.17) defines a linear operator a
a
a
a
.A0 ˝ ˝An / W D.A0 / ˝ ˝ D.An / H00 ˝ ˝ H0n ! H10 ˝ ˝ H1n and not just a linear relation is not obvious and needs a careful Nconsideration. We must confirm that different representations of an element x 2 akD0;:::;n D.Ak / do a
a
not lead to let .0; w/ 2 A0 ˝ ˝An , Pdifferent ‘images’. To show right-uniqueness P i.e. w D i ˛i A0 x0;i ˝ ˝ An xn;i and i ˛i x0;i ˝ ˝ xn;i D 0. The latter is the same as X 0D ˛i hx0;i jv0 iH00 hxn;i jvn iH0n i
for all vk 2 H0k ; k D 0; : : : ; n. If .xk;i /i is linearly independent for k D 0; : : : ; n, then all coefficients ˛i must be zero. In this case w D 0. Otherwise, let .yk;j /j be a maximal linearly independent subfamily of .xk;i /i so that for all i X xk;i D ak;ij yk;j j 16
a
We shall have to live with the ambiguity that for linear mappings A; B the mapping A˝B is not the tensor product of A; B considered as linear spaces,
Section 1.2 Some Construction Principles of Hilbert Spaces
29
for suitable coefficients ak;ij . Then X
˛i x0;i ˝ ˝ xn;i D
XX
i
i
j0
X
˛i a0;ij0 an;ijn y0;j0 ˝ ˝ yn;jn
jn
D0 and due to linear independence X
˛i a0;ij0 an;ijn D 0
i
for all multi-indices .j0 ; : : : ; jn / appearing in the sum. Trivially this yields XX X ˛i a0;ij0 an;ijn A0 y0;j0 ˝ ˝ An yn;jn D 0 i
j0
jn
and by the linearity of the operators .Ak /kD0;:::;n 0D
X
˛i A0
X
i
D
X
a0;ij0 y0;j0 ˝ ˝ An
j0
X
an;ijn yn;jn
jn
˛i A0 x0;i ˝ ˝ An xn;i D w:
i a
a
Thus right-uniqueness of A0 ˝ ˝An – as defined above – is shown. a
a
If A0 ˝ ˝An is a closable, linear operator then we define a
a
A0 ˝ ˝ An D A0 ˝ ˝An : We have
a
a
a
a
a
a
A˝B ˝C D .A˝B/˝C D A˝.B ˝C / and in the closable case A ˝ B ˝ C D .A ˝ B/ ˝ C D A ˝ .B ˝ C /; so that we may focus our attention on the case of two factors with the general case following by induction. With this rather brief review of Hilbert space concepts, we are ready to consider the functional analytical frame work utilized in our discussion of partial differential equations.
Chapter 2
Sobolev Lattices
In order to make the concept of a Sobolev (or extrapolation) lattice more easily digestible, we first consider a slightly simpler and more commonly discussed concept, namely that of a chain of Hilbert spaces.
2.1
Sobolev Chains
The starting point of our discussion is the idea of a Gelfand triple. We begin with a first observation formulated in the following lemma. Lemma 2.1.1. Let H0 ; H1 be Hilbert spaces such that j
H1 , ! H0 (in the sense of j being a dense and continuous embedding1 ). Then j}
H0 ,! H1 : Here j } denotes the dual mapping associated with j . Proof. By assumption we have that the range j ŒH1 is dense in H0 and the null space Œ¹0º j D ¹0º. We need to show that j } is one-to-one and that j } ŒH0 is dense 1
A continuous, linear mapping j W H1 ! H0 from a Hilbert space H1 into a Hilbert space H0 is in this context called an embedding if it is injective. If j has dense range j ŒH1 in H0 , then we speak of a dense embedding. We may identify j ŒH1 with H1 and so think of H1 as a subspace of H0 and j as given by the identity j W H1 ! H0 ; x 7! x: Thus, linearity and injectivity become obvious. This particular j has dense range if and only if H1 is a dense subspace of H0 . The continuity of j can here be expressed by the existence of a constant C0 such that jxjH0 C0 jxjH1 holds for all x 2 H1 . To underscore that the j is generated by x 7! x we frequently write 1H1 !H0 or 1H1 ,!H0 for this embedding (as we write 1H0 1H0 !H0 – or even shorter just 1 – for the identity mapping in H0 ).
Section 2.1 Sobolev Chains
31
in H1 . By the unitary character of the Riesz mapping, it suffices to establish the . We have by the projection theorem analogous properties for j D RH1 j } RH 0 H1 D j ŒH0 ˚ Œ¹0º j
and
H0 D j ŒH1 ˚ Œ¹0º j :
Thus, the density of j ŒH0 in H1 and Œ¹0º j D ¹0º follow from Œ¹0º j D ¹0º and the density of j ŒH1 in H0 , respectively. The best embedding constant for both embeddings is the same, namely the operator norm kj k D kj k D kj } k:
(2.1.1)
It is customary to identify H1 with the subspace j ŒH1 H0 and we shall do so henceforth. In this case, the embedding j W H1 ! H0 becomes the canonical embedding 1H1 ,!H0 2 L.H1 ; H0 / given by 1H1 ,!H0 W H1 ! H0 ; x 7! x: } Then, its dual 1H gives the corresponding canonical embedding of H0 into H1 . 1 ,!H0 We have by definition of the dual mapping that } .1H f /.x/ D f .1H1 ,!H0 x/ 1 ,!H0
for all f 2 H0 and x 2 H1 . Consequently, } 1H f D f ı 1H1 ,!H0 1 ,!H0
for all f 2 H0 . Combining the two continuous and dense embeddings with the Riesz mapping RH0 gives 1H1 ,!H0
} 1H
RH0
1 ,!H0
* H1 ,! H0 ) H0 ,! H1 : RH
0
If we choose to identify H0 and H0 , then RH0 turns into the identity 1H0 and we may glue these embeddings even closer together } 1H
1H1 ,!H0
1 ,!H0
H1 ,! H0 D H0 ,! H1 or more concisely 1H1 ,!H0
} 1H
1 ,!H0
H1 ,! H0 ,! H1 : As a consequence of identifying H0 and H0 we have f .x/ D hf j xiH0
(2.1.2)
32
Chapter 2 Sobolev Lattices
for all f 2 H0 D H0 and all x 2 H0 D H0 . For f 2 H0 and g 2 H1 , we have by definition of h j iH1 and RH1 } } h1H f j giH1 D hRH1 .1H f / j RH1 giH1 1 ,!H0 1 ,!H0 } D .1H f /.RH1 g/ D f .1H1 ,!H0 RH1 g/: 1 ,!H0
With (2.1.2) we have f .1H1 ,!H0 RH1 g/ D hf j 1H1 ,!H0 RH1 giH0 and so } hf j giH1 D h1H f j giH1 D hf j 1H1 ,!H0 RH1 giH0 D hf j RH1 giH0 1 ,!H0 (2.1.3)
for all f 2 H0 and g 2 H1 . A triple .H1 ; H0 ; H1 / with H1 H0 and H1 densely and continuously embedded in H0 via x 7! x is known as a Gelfand triple2 or a rigged Hilbert space. Realizing that the canonical dense and continuous embedding ‘,!’ may be considered as an order relation, we see that ¹H1 ; H0 ; H1 º is a totally ordered set, i.e. a chain. We shall therefore refer to the triple .H1 ; H0 ; H1 / also as a Gelfand chain. Since 1H1 ,!H0 is a bounded operator on H1 , the unbounded “identity” mapping given by 1H1 H0 !H1 WD .1H1 ,!H0 /1 is a closed operator and so A WD .1H1 H0 !H1 / 1H1 H0 !H1 is a non-negative3 , selfadjoint operator A H0 ˚ H0 satisfying D.A/ H1 . We p recall that for the selfadjoint, strictly positive operator square root4 A of A we have 2 By making the appropriate natural identifications, we may and shall – as a matter of convenience – always assume that H0 H1 : 3
A is actually a (strictly) positive operator. Indeed, we find jxjH0 jAxjH0 hxjAxiH0 D hxj.1H1 H0 !H1 / 1H1 H0 !H1 xiH0 D h1H1 H0 !H1 xj1H1 H0 !H1 xiH1 D hxjxiH1
1 hxjxiH0 k1H1 ,!H0 k2
for all x 2 D.A/ and so A
1 : k1H1 ,!H0 k2
4 A convenient way to think of the square root of A is in the framework of the function calculus associated with A via the spectral theorem, compare e.g. [6]. It follows that p 1 A : k1H1 ,!H0 k
Section 2.1 Sobolev Chains that
33
p j AxjH0 D jxjH1
p for all x 2 D. A/ D H1 :
Now, since H1 H1 is dense in H1 , we may consider the densely-defined operator e .1/ W H1 H1 ! H1 ; C p x 7! Ax: e / if there is an f 2 H such that Then y 2 D.C 1 .1/ p e .1/ x j yiH D h Ax j yiH D hx j f iH hC 1 1 1 e .1/ / D H1 and utilizing (2.1.3), the definition of the Riesz map and for all x 2 D.C the identification of H0 with H0 we find hx j f iH1 D hRH1 x j RH1 f iH1 D x.RH1 f / D hx j RH1 f iH0 and on the other hand p hx j f iH1 D h Ax j yiH1 p D hRH1 Ax j RH1 yiH1 p D Ax.RH1 y/ p D h Ax j RH1 yiH0 : Thus p h Ax j RH1 yiH0 D hx j RH1 f iH0 e .1/ / D H1 and so for y 2 D.C e / we read off that RH1 y 2 for all x 2 D.C .1/ p D. A/ and p A RH1 y D RH1 f so that
p e C .1/ RH1 A RH1 : p ŒD. A/ we obtain the reverse inclusion. This shows that Similarly, for y 2 RH 1 p e C .1/ is unitarily equivalent to A and therefore also selfadjoint with the same specp e is strictly positive and selfadjoint and consequently trum as A. In particular, C .1/
e D C e D C e .1/ DW C.1/ C .1/ .1/
34
Chapter 2 Sobolev Lattices
is a strictly positive, selfadjoint p operator in H1 . Moreover, for x 2 H0 H1 we recall (2.1.2) and, noting that A is onto H0 , we get ˇ ˇ ² ³ ² ³ jx.y/j ˇˇ jhx j yiH0 j ˇˇ jxjH1 D sup y 2 H1 n ¹0º D sup y 2 H1 n ¹0º jyjH1 ˇ jyjH1 ˇ ˇ ² p 1 p ³ jh A x j A yiH0 j ˇˇ D sup p ˇ y 2 H1 n ¹0º j A yjH0 ˇ ² p 1 ³ jh A x j ziH0 j ˇˇ D sup ˇ z 2 H0 n ¹0º jzjH0 p 1 D j A xjH0 :
We see in particular that p e .1/ C.1/ H1 ˚ H1 : A H0 ˚ H0 H1 ˚ H1 and C p This suggests that we consider A as a linear operator in H1 ˚ H1 and then by definition we have p A C.1/ or C.0/ WD
p
A D C.1/ jH0 ;
1 C.0/
p
1
A
1 D C.1/ jH0 :
This association between the Gelfand triple and the operator C.0/ motivates the following definition. Definition 2.1.2. Let C H ˚ H be a densely defined, closed linear operator in a Hilbert space H with 0 in the resolvent set %.C /5 . Then the triple .H1 .C /; H; H1 .C // 5 We recall the definition of the resolvent set associated with a linear operator B acting in a Hilbert space H :
%.B/ WD ¹ 2 C j . B/ is injective, . B/ŒH is dense in H ; . B/1 is boundedº: The resolvent set associated with a closed linear operator B can be shown to be always open, its complement is therefore closed in C and known as the spectrum .B/ of the linear operator B. We take the opportunity to also recall the major parts of the spectrum. The point spectrum P .B/ is defined as the set P .B/ WD ¹ 2 C j . B/ is not injectiveº; the continuous spectrum C .B/ is defined as the set C .B/ WD ¹ 2 C j . B/ is injective, ŒH . B/1 D D.. B/1 / is dense in H , . B/1 is not uniformly continuousº:
Section 2.1 Sobolev Chains
35
with H1 .C / denoting D.C / equipped with the norm jC jH and H1 .C / the dual space of H1 .C / (denoting analogously D.C / equipped with the norm jC jH ) will be called a short (abstract) Sobolev chain. We need to verify that this is well-defined, i.e. that .H1 .C /; H; H1 .C // is a chain. That H1 .C / is a Hilbert space follows since C was assumed to be closed and jxjH D jC 1 C xjH kC 1 k jC xjH D kC 1 k jxjH1 .C / :
(2.1.4)
Indeed, hC j C iH is an inner product in H1 .C / (with definiteness following from the invertibility of C ). Moreover, according to (2.1.4) we have a continuous mapping of H1 .C / into H given by the identity and therefore an embedding, and since C is densely defined we also have H1 .C / dense in H . Thus, we have established the canonical embedding H1 .C / ,! H: The analogous results hold for C in place of C so that H1 .C / ,! H: Therefore we also have the canonical dual embedding H ,! H1 .C / DW H1 .C /: Identifying H with H , we get H1 .C / ,! H D H ,! H1 .C / and so .H1 .C /; H; H1 .C // is totally ordered with respect to the canonical dense and continuous embeddings, i.e. indeed a chain. We see that, by definition, a short Sobolev chain associated with a selfadjoint operator C is a Gelfand chain. Conversely, we already noticed that every Gelfand chain may be reformulated as a short Sobolev chain with the role of C played by the selfadp joint operator A, where A is the selfadjoint operator associated with the canonical embedding of H1 in H0 . As our terminology suggests, there are also long Sobolev chains extending short ones. Before they can be properly defined we need some supporting results. and the residual spectrum R .B/ is defined as the set R .B/ WD ¹ 2 C j . B/ is injective, ŒH . B/1 D D.. B/1 / is not dense in H º: These spectral parts are clearly disjoint and together make up all of the spectrum: .B/ D P .B/ [ R .B/ [ C .B/:
36
Chapter 2 Sobolev Lattices
Lemma 2.1.3. Let C H ˚ H be a densely defined, closed linear operator in a Hilbert space H with 0 2 %.C /. Then C n is also closed and densely defined for every n 2 N. Moreover, we have 0 2 %.C n / and jC n xjH kC 1 kn jxjH
(2.1.5)
for all x 2 D.C n /. In particular, D.C n / equipped with the norm jC n jH is a Hilbert space denoted by Hn .C /, n 2 N. The mapping CkC1;k W HkC1 .C / ! Hk .C /; x 7! C x is unitary, k 2 N. Proof. We prove this result by induction. For n D 0; 1 the result is obviously true by assumption. So let C k be densely defined and closed for all k 2 N; k n. We first show that C nC1 is closed. Let x D .xk /k be a sequence in D.C nC1 / with xk ! u and C nC1 xk ! y as k ! 1. If we can show that u 2 D.C nC1 / and C u D y, we have confirmed closedness of C nC1 . Since 0 2 %.C /, we get C n xk D C 1 C nC1 xk ! C 1 y for k ! 1. By the assumed closedness of C n we get C 1 y D C n u and consequently C n u 2 D.C /, i.e. u 2 D.C nC1 /. Applying C to both sides we see that y D C nC1 u. Now we want to show that C nC1 is densely defined. Let x be a sequence in D.C n / approximating Cy; y 2 D.C /. Then we have C 1 xk ! y
in H
and C n xk D C nC1 C 1 xk , i.e. .C 1 xk /k is a sequence in D.C nC1 / converging to y 2 D.C /. Since on the other hand D.C / is dense in H , the desired denseness of D.C nC1 / in H follows. Next we show estimate (2.1.5). Clearly, jC n yjH kC 1 kn jyjH for all y 2 H . Letting y D C n x with x 2 D.C n / gives jxjH D jC n C n xjH kC 1 kn jC n xjH which yields estimate (2.1.5). If we can show that C n is onto, then we have 0 2 %.C n /. But this is clear, since C n C n y D y
Section 2.1 Sobolev Chains
37
and C n y 2 D.C n / for every y 2 H . That Hn .C / is a Hilbert space is now also clear. Indeed, hC n j C n iH is clearly a semi-inner product and from (2.1.5) definiteness follows. Therefore only completeness remains to be shown. Again by (2.1.5) we have, with ˛ WD 1CkC11 kn 2 0; 1Œ, kC 1 kn 1 C kC 1 kn
q
2 2 jxjH C jC n xjH ˛ kC 1 kn jxjH C .1 ˛/ jC n xjH
jC n xjH q 2 2 jxjH C jC n xjH for all x 2 D.C n / and completeness follows from the completeness of the domain of a closed operator with respect to the graph norm. Finally, we have jCkC1;k yjHk .C / D jC yjHk .C / D jC kC1 yjH D jyjHkC1 .C / for all y 2 D.C kC1 /; k 2 N. Thus, CkC1;k is isometric. Moreover, CkC1;k is also onto: if y 2 D.C k / then x D C 1 y 2 D.C kC1 / and CkC1;k x D C x D C C 1 y D y. So, CkC1;k is indeed unitary. Recalling C also satisfies the assumptions imposed on C , i.e. C is a densely defined, closed linear operator with 0 2 %.C / D %.C / , we also get Hilbert spaces Hn .C /;
n 2 N:
We introduce their dual spaces and define Hn .C / WD Hn .C / for n 2 N. Realizing that Hn .C / ,! H ,! Hn .C / D Hn .C / constitutes a Gelfand chain .Hn .C /; H; Hn .C / / we have that H H0 .C / D H0 .C / is densely and continuously embedded in Hn .C / via the corresponding canonical embedding. Definition 2.1.4. Let C H ˚ H be a densely defined, closed linear operator in Hilbert space H with 0 2 %.C /. Then the family of Hilbert spaces .Hn .C //n2N is called the positive Sobolev chain associated with C , the family of Hilbert spaces .Hn .C //n2N is called the negative Sobolev chain associated with C . The family .Hn .C //n2Z is called the (long) Sobolev chain associated with C .
38
Chapter 2 Sobolev Lattices
Remark 2.1.5. We see that a long Sobolev chain joins the positive Sobolev chain associated with C and the dual of the positive Sobolev chain associated with C . Thus, in general Hn .C / is not the dual space of Hn .C /. The purpose of the construction of a long Sobolev chain is to produce a continuous extension of the operator C to all spaces of the chain. Speaking of a Sobolev chain is justified by the follow result. Theorem 2.1.6. Let .Hn .C //n2Z be the Sobolev chain associated with the operator C . Then we have for all n 2 Z and k 2 N HnCk .C / ,! Hn .C / (in the sense of the canonical embeddings being dense and continuous). The mappings e nC1;n W D.C jnjC1 / HnC1 .C / ! Hn .C /; C x 7! C x e nC1;n for all n 2 Z. have unitary closures CnC1;n WD C Proof. That for all n 2 N HnC1 .C / ,! Hn .C /
(2.1.6)
is a consequence of 0 2 %.C /: jyjHn .C / D jC n yjH D jC 1 C nC1 yjH kC 1 k jyjHnC1 .C / for all y 2 HnC1 .C /. Since D.C / is dense in H , we have a sequence x D .xk /k in D.C / such that xk ! C nC1 y for k ! 1 and so we also see that C n C n xk ! C n y for k ! 1 and that .C n xk /k is a sequence in D.C nC1 / converging to y in Hn .C /. Since y 2 D.C nC1 / was arbitrary, this shows the density of the embedding. Also for n 2 Z<0 D Z n N, the canonical, dense and continuous embedding HnC1 .C / ,! Hn .C / follows from Lemma 2.1.1. The best embedding constants are the same as for the case n 2 N. We have k1HnC1 .C /,!Hn .C / k D kC 1 k for n 2 N. Likewise we obtain k1HnC1 .C /,!Hn .C / k D kC 1 k D kC 1 k and so kC 1 k D k1HnC1 .C /,!Hn .C / k D k1H k nC1 .C /,!Hn .C / ˘ D k1H k D k1Hn .C / ,!HnC1 .C / k nC1 .C /,!Hn .C /
D k1Hn .C /,!Hn1 .C / k for n 2 N.
(2.1.7)
Section 2.1 Sobolev Chains
39
e nC1;n Since D.C jnjC1 / D HnC1 .C / for n 2 N, we already know that CnC1;n D C is unitary in this case. e nC1;n is densely defined since D.C e nC1;k /DHjnjC1 .C /D Now let n2 Z<0 . Then C jnjC1 / is densely and continuously embedded in H0 .C / and – by duality – D.C H0 .C / is dense in Hn .C / for all n 2 Z<0 . We also see that e nC1;n ŒHjnjC1 .C / D C ŒHjnjC1 .C / C D CnC1;n ŒHnC1 .C / D Hn .C / D D.C jnj / is dense in Hn .C / for n 2 Z<0 . Since the Riesz mapping RHn .C / W Hn .C / D Hn .C / ! Hn .C / are known to be unitary for n 2 Z<0 , we get for y 2 HjnjC1 .C / and Cn;n1
jhCy j xiHn .C / j jhRHn .C / Cy j RHn .C / xiHn .C / j D jxjHn .C / jRHn .C / xjHn .C / D
jCy.RHn .C / x/j jhCy j ziH0 .C / j D jRHn .C / xjHn .C / jzjHn .C /
D
jhy j C ziH0 .C / j : jzjHn .C /
With z WD RHn .C / x, this yields jhCy j xiHn .C / j jhCy j ziH0 .C / j D jxjHn .C / jzjHn .C / D D
jhy j C ziH0 .C / j jzjHn .C / jhy j Cn;n1 ziH0 .C / j jCn;n1 zjHn1 .C /
Now, abbreviating w WD Cn;n1 z, we obtain
jhCy j xiHn .C / j jhy j wiH0 .C / j D jxjHn .C / jwjHn1 .C / D D
jhy j wiH0 .C / j jwjHn1 .C / jy.w/j jwjHn1 .C /
:
:
40
Chapter 2 Sobolev Lattices
Thus, we have e nC1;n yjH .C / D jC yjH .C / ; jC n n ˇ ² ³ jhCy j xiHn .C / j ˇˇ D sup ˇ x 2 Hn .C / n ¹0º jxjHn .C / ˇ ² ³ ˇ jy.w/j ˇ D sup ˇ w 2 Hn1 .C / n ¹0º jwj Hn1 .C /
D jyjHnC1 .C / ; e nC1;n is a densely defined isometry with dense range and therefore so we see that C its closure CnC1;n W HnC1 .C / ! Hn .C / is unitary, n 2 Z. Remark 2.1.7. The proof of the previous result suggests a convenient notation for the application of a functional in Hn .C / D Hn .C / on an element of Hn .C /, n 2 Z. Indeed, since we have identified H D H we have H D H0 .C / D H D H0 .C / D H0 .C / D H0 .C / D H0 .C / D H0 .C / and consequently f .g/ D hf j giH D hg j f iH D g.f /
for f; g 2 H . We also have, for f 2 Hjnj .C /; g 2 Hjnj .C /, the estimate jf .g/j jf jHn .C / jgjHn .C / : Therefore, by the density of Hjnj .C / in Hn .C / and of Hjnj .C / in Hn .C /, we can continuously extend the mapping hjiH W Hjnj .C / Hjnj .C / Hn .C / Hn .C / ! C; .f; g/ 7! hf j giH : Because of the intuitive benefits, we shall employ the same inner product notation for this continuous extension. Thus, the equality hf j giH D f .g/ D hRHn .C / f j giHn .C / D hg j f iH D g.f / D hRHn .C / g j f iH n .C /
holds (by the extended definition) for all f 2 Hn .C / and g 2 Hn .C /, n 2 Z. The unitary mappings CnC1;n also give rise to the linear operators C.n/ W HnC1 .C / Hn .C / ! Hn .C /; x 7! CnC1;n x for n 2 Z.
Section 2.1 Sobolev Chains
41
For use in the following, we define, for j; k 2 Z, Ck;j WD Cj C1;j Ck;k1
for k > j;
Cj;j WD 1Hj .C / ; Ck;j WD Cj;k
for k < j:
Lemma 2.1.8. Let .Hn .C //n2Z be the Sobolev chain associated with the operator C H ˚ H . Then C.k/ W HkC1 .C / Hk .C / ! Hk .C /; x 7! CkC1;k x is a densely defined, closed linear operator with %.C.k/ / D %.C / for every k 2 Z. Moreover, C.k/ and C C.0/ are unitarily equivalent;
k 2 Z:
Proof. That C.k/ is densely defined follows from the density of HkC1 .C / in Hk .C /. The closedness of C.k/ follows from the unitarity of CkC1;k , since CkC1;k x D C.k/ x
for all x 2 HkC1 .C /:
If we show that C.k/ is unitarily equivalent to C.0/ D C , then equality of resolvent sets, indeed of all spectral parts, follows. Then we have Ck;j D Cj;j 1 CkC1;k D Cj 1;j Ck;kC1
for k < j:
It follows that, as composition of unitary mappings, we have Ck;j W Hk .C / ! Hj .C / is unitary;
j; k 2 Z;
(2.1.8)
and so 1 Cj;k D Ck;j D Ck;j ;
Cj;k Ck;j D 1Hj .C /
for all j; k 2 Z. We shall now show that C.k/ D C0;k C.0/ Ck;0 :
(2.1.9)
42
Chapter 2 Sobolev Lattices
For k 2 N and x 2 HkC1 .C / we have C.k/ x D C x D C k C C k x D C0;k C.0/ Ck;0 x: Thus, equality follows since C0;k ŒD.C.0/ / D C0;k ŒH1 .C / D C k ŒH1 .C / D C k1 ŒH0 .C / D HkC1 .C / and so D.C / D D.C.0/ / D Ck;0 ŒHkC1 .C /: For k 2 Z<0 we may argue in the following way. Equation (2.1.9) still holds at least for x 2 HjkjC1 .C /. Since HjkjC1 .C / is dense in HkC1 .C /, it follows that C.k/ D C.k/ jHjkjC1 .C / D .C0;k C.0/ Ck;0 /jHjkjC1 .C / : On the other hand, D..C0;k C.0/ Ck;0 /jHjkjC1 .C / / D D..C.0/ Ck;0 /jHjkjC1 .C / / and for every y2D..C.0/ Ck;0 /jHjkjC1 .C / / there is a sequence x D .xj /j 2HjkjC1 .C /N such that xj ! y in Hk .C / and kC1 xj ! z1 D .C.0/ Ck;0 /jHjkjC1 .C / y C.0/ Ck;0 xj D C.0/
in H0 .C /:
for j ! 1. Thus, we find that k C.0/ xj ! Ck;0 y
in H0 .C /
and kC1 C.0/ xj ! z1
in H0 .C /;
for j ! 1 which implies that Ck;0 y 2 D.C.0/ / and C.0/ Ck;0 y D z1 . In particular, we have that xj ! C0;kC1 z1 in HkC1 .C / and so xj ! C0;kC1 z1 in Hk .C / as j ! 1. Hence, by uniqueness of limit we have y D C0;kC1 z1 2 HkC1 .C /: Thus, we have shown D..C.0/ Ck;0 /jHjkjC1 .C / / HkC1 .C /: Conversely, let y 2 HkC1 .C /. Then, by the density of HjkjC1 .C / in HkC1 .C /, there is a sequence x D .xj /j in HjkjC1 .C / such that k C.0/ xj ! Ck;0 y
in H0 .C /
Section 2.1 Sobolev Chains
43
and kC1 C.0/ xj ! CkC1;0 y
in H0 .C /;
for j ! 1 which implies Ck;0 y 2 D.C.0/ / and C.0/ Ck;0 y D CkC1;0 y. This shows that HkC1 .C / D..C.0/ Ck;0 /jHjkjC1 .C / /: Thus, we have HkC1 .C / D D.C.0/ Ck;0 / D D..C.0/ Ck;0 /jHjkjC1 .C / / and so C.k/ D C0;k C.0/ Ck;0 for k 2 Z<0 . The operators Cj;k introduced above can be utilized to give an analogue to the Riesz mapping. Between Hk .C / and Hk .C /, we have the unitary mapping Ck;k W Hk .C / ! Hk .C /: Note, however, that for the Riesz mapping we have RHk .C / W Hk .C / D Hk .C / ! Hk .C / for k 2 N: The inverse RH W Hk .C / D Hk .C / ! Hk .C / of RHk .C / is equal to k .C / the Riesz mapping RHk .C / . However, in the selfadjoint case we have the following result.
Proposition 2.1.9. Let .Hn .C //n2Z be the Sobolev chain associated with the selfadjoint operator C H ˚ H . Then RHk .C / D Ck;k
for all k 2 Z:
Proof. Since C is selfadjoint, we have Ck;k W Hk .C / ! Hk .C /;
RHk .C / W Hk .C / ! Hk .C / for k 2 N:
Moreover, hRHk .C / x j yiHk .C / D x.y/ D hx j yiH0 .C / D hC k C 2k x j C k yiH0 .C / D hC 2k x j yiHk .C / D hCk;k x j yiHk .C / ;
44
Chapter 2 Sobolev Lattices
for all x 2 H2jkj .C /; y 2 Hjkj .C /. This implies RHk .C / x D Ck;k x for all x 2 H2jkj .C /. By the density of H2jkj .C / in Hk .C / the desired equality of the unitary mappings follows. As a consequence of our general findings we also see that C.k/ C.k1/
for all k 2 Z:
This suggests the introduction of the linear mapping C1 WD
[
C.k/ W H1 .C / WD
k2Z
[
Hk .C / ! H1 .C /;
k2Z
x 7! C1 x with C1 x WD C.k/ x for x 2 HkC1 .C /; k 2 Z. Definition 2.1.10. Let .Hn .C //n2Z be Sthe Sobolev chain associated with the operator C . A sequence x in H1 .C / WD k2Z Hk .C / is called a Cauchy sequence in H1 .C / if x is a Cauchy sequence in Hk .C / for some k 2 Z. Likewise we shall call the sequence x convergent in H1 .C / if it converges in some Hk .C /; k 2 Z. Lemma 2.1.11. The linear space H1 .C / is complete in the sense that every Cauchy sequence in H1 .C / converges in H1 .C /. The limit element is uniquely determined. Conversely, every sequence x converging in H1 .C / is also a Cauchy sequence in H1 .C /. Proof. The first part of the lemma is obvious. Indeed, if x D .xj /j is a Cauchy sequence in H1 .C / then it is by definition a Cauchy sequence in some Hk .C /, k 2 Z. Since Hk .C / is a Hilbert space we must have that x is convergent in Hk .C / to some yk WD limj !1 xj 2 Hk .C /. The last statement is also clear. It remains to be seen that if x also converges in some other space Hr .C /; r 2 Z, then if xj ! yk in Hk .C / and x ! yr in Hr .C / as j ! 1, we must have yk D yr . By the continuity of the embeddings the assumption implies xj ! yk in Hmin¹k;rº .C / and xj ! yr in Hmin¹k;rº .C / as j ! 1 therefore yk D yr in Hmin¹k;rº .C /. Thus, in particular, yr 2 Hk .C / and yk 2 Hr .C / or yr ; yk 2 Hmax¹k;rº .C / by the dense and continuous canonical embeddings Hmax¹k;rº .C / ,! Hk .C /; Hk .C / ,! Hmin¹k;rº .C /;
Hmax¹k;rº .C / ,! Hr .C /; Hr .C / ,! Hmin¹k;rº .C /:
Section 2.1 Sobolev Chains
45
Now, again by-passing intricate topological issues, we define a convenient concept of continuity for mappings in H1 .C /. Definition 2.1.12. Let .Hn .C //n2Z , .Hn .D//n2Z be Sobolev chains associated with operators C and D, respectively. Then a mapping A W H1 .C / ! H1 .D/ will be called continuous from H1 .C / to H1 .D/ if for every k 2 Z there is an s 2 Z such that Hk .C / ! Hs .D/; x 7! Ax is continuous. A mapping A W H1 .C / ! H1 .D/ will be called continuous of finite order if there is a n 2 Z such that for all k 2 Z we have that Hk .C / ! HkCn .D/; x 7! Ax is continuous. The largest such number n (if it exists) is called the order of A. If n can be arbitrary then we say A is of order 1. Remark 2.1.13. Continuity of a mapping A W H1 .C / ! H1 .D/ simply means that for every y 2 H1 .C / and a sequence x converging to y in H1 .C / we have that the sequence Ax converges to Ay in H1 .D/. Note that by the continuity of the embeddings between spaces in the chain, to show continuity of A it suffices to find a suitable s 2 Z, such that Hk .C / ! Hs .D/; x 7! Ax is continuous for all sufficiently large negative indices k 2 Z. In the sense of this definition we see, in particular, that if C is unbounded then C1 is continuous in H1 .C / of finite order. Indeed, we have that Hk .C / ! Hk1 .C /; x 7! C1 x coincides with Ck;k1 and is even unitary for all k 2 Z. So C1 is continuous in H1 .C / of order 1 if C is unbounded, otherwise of order 1. If C is bounded, the Sobolev chain consists of Hilbert spaces which are equal as linear spaces and equipped with different but equivalent norms. Therefore, the case that C is bounded is of little interest.
46
Chapter 2 Sobolev Lattices
Proposition 2.1.14. Let .Hn .C //n2Z be the Sobolev chain associated with the operator C and C1 defined as above. Then we have k Hk .C / D C1 ŒH0 .C /
for all k 2 Z:
Proof. We have straightforwardly for k 2 N (C0;0 WD 1H0 .C / ) Hk .C / D C0;k ŒH0 .C / D Ck;k1 C1;0 ŒH0 .C / 1 1 C1;0 ŒH0 .C / D Ck;k1 k D C1 ŒH0 .C /:
Similarly, for k 2 Z<0 k Hk .C / D C0;k ŒH0 .C / D C1;0 Ck;k1 ŒH0 .C / D C1 ŒH0 .C /:
Therefore, we can ‘move’ up and down in the Sobolev chain by applying C1 and respectively. Indeed, we have
1 , C1
kj Ck;j C1 :
In particular, the continuous operator C1 conveniently extends the original (unbounded) operator C to the Sobolev chain. This construction mimics in an abstract setting what is done for example in extending differentiation to distributions (see the examples below). It is frequently convenient to refer to a dense subspace of ‘nice’ elements, which in ı
the abstract setting play the role that C 1 -functions have in the theory of distributions. A natural choice is to take \ H1 .C / WD Hn .C / n2Z
with the obvious notion of convergence in H1 .C / as convergence with respect to all norms j jHn .C / ; n 2 Z. Lemma 2.1.15. Let .Hn .C //n2Z be the Sobolev chain associated with the operator C . Then \ H1 .C / WD Hk .C / k2Z
is dense in Hm .C / for every m 2 Z. Moreover, H1 .C / is a core for C.m/ , i.e. H1 .C / is dense in D.C.m/ / with respect to the graph norm, for all m 2 Z.
Section 2.1 Sobolev Chains
47
Proof. Let " 2 R>0 and x 2 Hm .C / be arbitrary and let y0 WD x 2 Hm .C /. Since HmC1 .C / is dense in Hm .C /, we can choose an element y1 2 HmC1 .C / such that " jy1 y0 jHm .C / < : 2 Recursively, we choose ynC1 2 HmCnC1 .C / such that jynC1 yn jHm .C / kC 1 kn jynC1 yn jHmCn .C / " < nC1 for n 2 N: 2 The sequence .yn /n can now be considered as a sequence in Hm .C /. We have jyn y0 jHm .C /
n X
jyk yk1 jHm .C /
kD1
n X
kC 1 kk1 jyk yk1 jHmCk1 .C /
(2.1.10)
kD1
"
n X
2k ";
kD1
and for 0 t < n; s 2 N (note that then t < k 1 for k 2 N>n ) jynCs yn jHmCt .C /
nCs X
jyk yk1 jHmCt .C /
kDnC1
nCs X
kC 1 kk1t jyk yk1 jHmCk1 .C /
kDnC1
" kC 1 kt
nCs X
2k
kDnC1
" n kC 1 kt ! 0 as n ! 1: 2 This shows that .yn /n is (eventually) a Cauchy sequence in every space HmCk .C /, k 2 N. Let zk be the limit in the space HmCk .C /, k 2 N. Then we have, by the canonical embedding HmCkC1 .C / ,! HmCk .C / and the uniqueness of limits, that zk D zs for all k; s 2 N, and so, defining z WD zk
for all k 2 N;
48
Chapter 2 Sobolev Lattices
we have z 2 H1 .C /. Letting n ! 1 in (2.1.10), we obtain jz xjHm .C / ": Since x 2 Hm .C /; m 2 Z and " 2 R>0 were arbitrary we have the desired density result. That H1 .C / is a core for C.m/ follows directly from the density of H1 .C / in HmC1 .C /. Frequently, it is more convenient to consider a general densely defined, closed linear operator C W D.C / H0 ! H1 , H0 ; H1 Hilbert spaces. Then Sobolev chains can still be associated with jC jCi and jC jCi. Correspondingly, jC j1 and jC j1 denote the extension of jC j and jC j to H1 .jC j C i/ and H1 .jC j C i/, respectively. In such a case, however, the interaction of C and C needs closer investigation. Lemma 2.1.16. Let C W D.C / H0 ! H1 be a closed and densely defined linear operator between Hilbert spaces H0 and H1 . Then6 , C1 jC jk1 D jC jk1 C1 jC jk k on H1 .jC j i / and C1 1 D jC j1 C1 on H1 .jC j i /, k 2 N. Here C1 and C1 denote the continuous extensions C1 W H1 .jC j C i / ! H1 .jC j C i / and W H1 .jC j C i / ! H1 .jC j C i / C1
of C jH1 .jC jCi/ and C jH1 .jC jCi/ , respectively. Proof. By polar decomposition, see e.g. [25, VI §2.7], there is a unitary mapping U W jC jŒH0 ! C ŒH0 ; such that C D U jC j D jC jU on D.C / \ C ŒH1 (recall that jC jŒH0 D N.jC j/? D N.C /? D C ŒH1 ). We e of U shall consider for convenience the extension U e W H0 ! H1 U U e ŒN.jC j/ D U e ŒN.C / D ¹0º. With this we have with U e jC j U e jC j: U 6
A Relation of the form CW DV C
for suitable operators W and V is called transmutation relation or intertwining property for C .
Section 2.1 Sobolev Chains
49
Consequently, e jC jU e C D jC j C C jC j D U on H2 .jC jCi/. By induction and continuous extension to H1 .jC jCi/ we obtain the first result. The second follows by interchanging the rôle of C and C . Continuous extendability follows from the two continuity statements. These in turn, however, follow from the following calculation 2 jC jH D j.jC j C i/s C j20 D jC .jC j C i/s j20 s .jC jCi/ 2 D jjC j .jC j C i/s j20 j jH sC1 .jC jCi/
for 2 H1 .jC j C i/, s 2 Z, and its analogue for C interchanged with C . As a consequence we have that C.k/ W HkC1 .jC j C i/ Hk .jC j C i/ ! Hk .jC j C i/; x 7! C1 x; C.k/
W HkC1 .jC j C i/ Hk .jC j C i/ ! Hk .jC j C i/; y 7! C1 y
are – as the notation suggests – adjoints of each other. Indeed, for we have for some f 2 Hk .jC j C i/ that
2 D..C.k/ / /
h jf iHk .jC jCi/ D hC.k/ j iHk .jC jCi/ D h.jC j1 C i/k C.k/ j.jC j1 C i/k i0 D h.jC j1 C i/k C1 j.jC j1 C i/k i0 D hC.jC j1 C i/k j.jC j1 C i/k i0 D h jf iHk .jC jCi/ D h.jC j1 C i/k j.jC j1 C i/k f i0 for all 2 HjkjC1 .jC j C i/. Thus, we read off that .jC j1 C i/k H1 .jC j C i/ and C .jC j1 C i/k
D .jC j1 C i/k C.k/
2 D.C / D
D .jC j1 C i/k f:
This confirms that .C.k/ / C.k/ :
Since the converse inclusion can also be shown by reversing the above reasoning, it follows indeed that C.k/ D .C.k/ / :
50
Chapter 2 Sobolev Lattices
As an immediate consequence we have that an operator matrix of the form7 0 C C 0 extends continuously not only to a skew-selfadjoint8 operator9 C0 C in the 0 .k/ 0 C spaces Hk C 0 C 1 but also in Hk .jC j C i/ ˚ Hk .jC j C i/, k 2 Z. Indeed, we have 0 C Hk C 1 D Hk .jC j C i/ ˚ Hk .jC j C i/ C 0 Hk .jC j C i/ D ; Hk .jC j C i/ for k 2 Z. The domain of the skew-selfadjoint (extended) operator C0 C is 0 .k/ clearly given by HkC1 .jC j C i/ ˚ HkC1 .jC j C i/. We shall make use of these observations in dealing with block operator matrix systems. Example 2.1. Consider the multiplication-by-the-argument operator m W D.m/ L2 .R/ ! L2 .R/ in L2 .R/ with D.m/ WD ¹ 2 L2 .R/ j m 2 L2 .R/º. The operator m is clearly ı
Hermitian and indeed densely defined since C 1 .R/ D.m/. Since .m˙i/1 j ı ı
ı
C 1 .R/
is a bijection in C 1 .R/, we also have that m is selfadjoint with C 1 .R/ dense in D.m/. An associated Sobolev chain .Hk .i m ˙ 1//k2Z is easily characterized since all elements of any of the spaces Hk .i m ˙ 1/, k 2 Z, are in L2;loc .R/, i.e. every f 2 H1 .i m ˙ 1/ satisfies I f 2 L2 .R/ for all bounded intervals I R. More precisely, we have Hk .i m ˙ 1/ D ¹ 2 L2;loc .R/ j .i m ˙ 1/k 2 L2 .R/º or Hk .i m ˙ 1/ D .i m ˙ 1/k ŒL2 .R/: The simple structure of this chain will be utilized in connection with spectral representations for the operator @ considered later in Example 2.3. 0 jC j 7 The typical block operator 0 C is unitarily equivalent on its closed range to D C 0 jC j 0 0 1 1 0 jC j, which is why one could refer to this as a Hamiltonian structure. The associated Hamiltonian H D .p; q/ 7! hpjjC jpi C hqjjC jqi yields jC j p q D rH.p; q/. Thus, pP 0 1 D rH.p; q/ qP 1 0 is indeed formally a Hamiltonian type system of equations, which support the terminology. 8 An operator A is called skew-selfadjoint if A D A . This is equivalent to saying that 1 A is i selfadjoint. 9 Recall the earlier definition of D densely defined linear operator .k/ constructed from a closed, C 1. D H ˚ H , H Hilbert space, with 0 2 %.D/, here applied to D D C0 C 0
Section 2.1 Sobolev Chains
51
Example 2.2. The Sobolev chain associated with jDj in R. Consider the formal operators D D p1 .m @/; D D p1 .m C @/. The set ¹k j k 2 Nº with k WD 2
1
jD k jL2 .R/
2
D k , where .x/ D exp. 12 x 2 /, x 2 R, denotes the Gauss distribution
function, is known to be a complete orthonormal set (Hermite functions) in L2 .R/. Since D D DD D 1 and D D 0, it is straightforward to show by induction that D D j C1 D .j C 1/ D j (2.1.11) for all j 2 N. As a first consequence we note that hD j C1 j D j C1 iL2 .R/ D hD j j D D j C1 iL2 .R/ D .j C 1/hD j j D j iL2 .R/
(2.1.12)
D .j C 1/Š h j iL2 .R/ p D .j C 1/Š for j 2 N. Now, defining the mappings DjLinC ¹k j k2Nº W LinC ¹k j k 2 Nº L2 .R/ ! L2 .R/; 1 ' 7! p .m' @'/ 2 and D jLinC ¹k j k2Nº W LinC ¹k j k 2 Nº L2 .R/ ! L2 .R/; 1 ' 7! p .m' C @'/ 2 we see that DjLinC ¹k j k2Nº and D jLinC ¹k j k2Nº are densely defined, formally adjoint operators. Indeed, by integration by parts we find hD kC1 j D j iL2 .R/ D hDD k j D j iL2 .R/ D hD k j D D j iL2 .R/ ; for all j; k 2 N. Therefore both operators are closable. Denoting the closure of DjLinC ¹k j k2Nº again by D, we now want to determine its adjoint D . An element
is in the domain of D if and only if there is an f 2 L2 .R/ such that hD kC1 j iL2 .R/ D hD k j f iL2 .R/
52
Chapter 2 Sobolev Lattices
for all k 2 N. Utilizing the completeness of ¹k j k 2 Nº we have
D
D
X
hD k j iL2 .R/
k2N
hD k j D k iL2 .R/
h j iL2 .R/ h j iL2 .R/
C
Dk
X
hD k j f iL2 .R/
k2N
hD kC1 j D kC1 iL2 .R/
D kC1 :
This shows that is not only in the domain of D but also in the domain of the closure of D jLinC ¹k j k2Nº . Indeed, we have utilizing (2.1.12) D D
X hD k j D iL2 .R/ hD k j D k iL2 .R/
k2N
D
X hD kC1 j iL2 .R/ hD k j D k iL2 .R/
k2N
D
Dk Dk
X
hD kC1 j iL2 .R/
k2N
.k C 1/hD k j D k iL2 .R/
D D
D D kC1
X
hD kC1 j iL2 .R/
k2N
hD kC1 j D kC1 iL2 .R/
D kC1 :
Therefore, we have in particular D D D jLinC ¹k j k2Nº . Further, D D is selfadjoint and from (2.1.11) we have that D k is an eigenfunction of D D associated with the eigenvalue .k C 1/, k 2 N. The resolvent can be given explicitly by . D D/1 D
X
hD k j iL2 .R/
k2N
. .k C 1// hD k j D k iL2 .R/
Dk
for all 2 L2 .R/ and … N>0 . In particular, 0 2 %.D D/. The (unique, nonnegative, selfadjoint) root of D D is given by p X k C 1 hD k j iL2 .R/ p D D D jDj WD D k : hD k j D k iL2 .R/ k2N
Clearly, we have jDj 1: Thus in particular 0 2 %.jDj/ and the corresponding Sobolev chain .Hk .jDj/k2Z can be constructed. It can be shown that H1 .jDj/ D S 0 the space of so-called tempered distributions and H1 .jDj/ D S the Schwartz space of smooth, rapidly
Section 2.1 Sobolev Chains
53
decaying functions, see e.g. [38, 37, 4]. Note that LinC ¹k j k 2 Nº is a core for every power of jDj. Indeed, by the Riesz–Fischer theorem, is in the domain of jDjj if and only if X .k C 1/j jhD k j iL2 .R/ j2 hD k j D k iL2 .R/
k2N
< 1:
Now, to simplify calculations, we establish the density of ı
C 1 .R/ WD ¹u 2 C1 .R/j u D 0 outside of a bounded setº ı
in Hk .jDj/; k 2 Z. First, we show that C 1 .R/ is contained in H1 .jDj/. It suffices to consider k D 2t; t 2 N. Now .D D/t is selfadjoint since it is a power of the selfadjoint operator D D and it has core LinC ¹s j s 2 Nº D LinC ¹D s j s 2 Nº. Therefore, f 2 L2 .R/ is in Hk .jDj/ D D.jDjk / D D..D D/t / if and only if there is a g 2 L2 .R/ such that h.D D/t D s j f iL2 .R/ D hD s j giL2 .R/ ı
for all s 2 N. For f 2 C 1 .R/ L2 .R/, this can be confirmed by repeated integration by parts. Now, let f 2 Hs .jDj/ be given (without loss of generality s D 2t , t 2 N). We ı
want to approximate f by elements ' 2 C 1 .R/. Since LinC ¹D k j k 2 Nº is a core for .D D/t , it suffices to consider f D D k , k 2 N. Such an f is already ı
smooth, and so we only need to ‘cut-off’ the support. Let 2 C 1 .R/, 0 1, with D 1 on Œ1; C1 and for R 2 R>0 define R WD .=R/. Then R ! 1 and @j R ! 0 point-wise as R ! 1 for every j 2 Z>0 . Observing that, with .R .m/ f /.x/ WD R .x/ f .x/; x 2 R, 1 D D R .m/ R .m/ D D D .@2 R /.m/ .@R /.m/ @; 2 we see that .D D/t .R .m/ f / D .D D/t f C o.1/ as R ! 1. This shows R .m/ f ! f
in Hs .jDj/
as R ! 1: ı
However, since f D D k , k 2 N, we have R .m/ f 2 C 1 .R/. ı
By the density of C 1 .R/ in Hk .jDj/ we may regard Hk .jDj/ as the completion ı
of C 1 .R/ with respect to the norm ´ j.D D/k=2 jL2 .R/ j jHk .jDj/ D jD .D D/.k1/=2 jL2 .R/
for k even; for k odd:
54
Chapter 2 Sobolev Lattices
Example 2.3. The Sobolev chain associated with @ ˙ 1, 2 R, in R. Consider the Hilbert space L2 .R; exp.2x/ dx/ WD ¹f 2 L2;loc .R/ j exp.m/ f 2 L2 .R/º; 2 R, equipped with the norm given by sZ
7! j j;0 WD
sZ
R
j exp.x/ .x/j2 dx D
R
j .x/j2 exp.2x/ dx:
Then exp.m/ W L2 .R; exp.2x/ dx/ ! L2 .R/; ' 7! exp.m/ ' where .exp.m/ '/.x/ D exp.x/ '.x/; x 2 R, is unitary. We define @ WD exp.m/ @0 exp.m/; where @0 denotes the closure of “differentiation” considered as an operator in L2 .R/ ı
defined on C 1 .R/. Here exp.m/ W L2 .R/ ! L2 .R; exp.2x/ dx/; ' 7! exp.m/ ' is the inverse of exp.m/. By construction, @0 and @ are unitarily equivalent and so @ W exp.m/ D.@/ L2 .R; exp.2x/ dx/ ! L2 .R; exp.2x/ dx/ is ı
skew-selfadjoint if 1i @0 defined on C 1 .R/ is essentially selfadjoint in the Hilbert space L2 .R/, i.e. has a selfadjoint closure. The latter is, however, easily shown ı
and this is left as an exercise. Moreover, since C 1 .R/ is indeed a core for @0 and ı
ı
ı
exp.m/ W C 1 .R/ ! C 1 .R/ is a bijection, we also have that C 1 .R/ is a core for @ ; 2 R. Therefore, we may construct the Sobolev chain .Hk .@ ˙ 1//k2Z based on the normal operators .@ ˙ 1/. Their inverses .@ ˙ 1/1 can explicitly be given by Z ..@ 1/1 /.x/ WD exp.. C 1/x/
1 x
exp.. C 1/y/ .y/ dy
and Z 1
..@ C 1/
/.x/ WD exp.. 1/x/
x
1
exp.. 1/y/ .y/ dy
Section 2.1 Sobolev Chains
55
ı
for 2 C 1 .R/. By an elementary induction, we see that for k 2 N ı
.@ ˙ 1/k ŒC 1 .R/ C1 .R/ \ Hk .@ ˙ 1/: ı
Since C 1 .R/ is dense in H0 .@ ˙ 1/ D L2 .R; exp.2x/ dx/, we get C1 .R/ \ Hk .@ ˙ 1/ dense in Hk .@ ˙ 1/ for all k 2 Z. Using a cut-off technique we obtain ı
C 1 .R/ dense in Hk .@ ˙ 1/ ı
for all k 2 Z; 2 R. Indeed, let u 2 C1 .R/ \ Hk .@ ˙ 1/ and ' 2 C 1 .R/. Then, applying the multiplication operator '.m/ to u we obtain ı
'.m/ u D 'u 2 C 1 .R/: ı
Now, as in the previous example, let 2 C 1 .R/ be such that 0 1, D 1 on j Œ1; C1 and for R 2 R>0 define R WD .=R/. Then R ! 1 and @0 R ! 0 pointwise as R ! 1 for every positive integer j . Observing that with .R .m/ f /.x/ WD j j R .x/ f .x/; x 2 R, we see that, by Leibniz’ rule, @0 .R .m/ f / D R .m/@0 f Co.1/ as R ! 1. This, however, yields R .m/ f ! f
in Hk .@ ˙ 1/
as R ! 1 ı
for every k 2 N. We have, however, already observed that R .m/ f 2 C 1 .R/. This ı
shows denseness of C 1 .R/ in Hk .@0 ˙ 1/ for k 2 N. By the denseness of the emı
bedding of Hk .@0 ˙ 1/ in Hk .@0 ˙ 1/ for k 2 N we get the denseness of C 1 .R/ in Hk .@0 ˙ 1/ for k 2 Z. Example 2.4. The Sobolev chain associated with jD j, 2 R, in R. Starting from the same base space L2 .R; exp.2x/ dx/ as in the previous example, we define D WD exp.m/ D exp.m/ and D WD exp.m/ D exp.m/; where D has been defined in Example 2.2. This yields D D WD exp.m/ D D exp.m/; i.e. D D , D and D are unitarily equivalent to D D, D and D, respectively. The complete orthogonal set of eigenfunctions for D D is given by ¹exp.m/ D k j k 2 Nº:
56
Chapter 2 Sobolev Lattices ı
As a consequence of the reasoning in Example 2.3 and since C 1 .R/ is dense in Hk .jDj/ according to Example 2.2, we can conclude that ı
C 1 .R/ dense in Hk .jD j/ for all k 2 Z; 2 R. Note that jDj D jD0 j.
2.2
Sobolev Lattices
Subject to some minor adjustments, the considerations of the previous subsection carry over to a more general situation associated with a finite family C D .Ci /iD0;:::;n of pair-wise commuting10 , densely defined, closed linear operators Ci W D.Ci / H ! H with 0 2 %.Ci /; i D 0; : : : ; n. Definition 2.2.1. Let C WD .Ci /iD0;:::;n , n 2 N, be a family of commuting, densely defined, closed linear operators Ci W D.Ci / H ! H , H a Hilbert space, with 0 2 %.Ci /; i D 0; : : : ; n. The family .H˛ .C //˛2ZnC1 ; where H˛ .C /, ˛ D .˛0 ; : : : ; ˛n / 2 ZnC1 , denotes11 the completion of D.C ˛C / with respect to the norm jyj˛ WD jC ˛ yj0 for y 2 D.C ˛ /; where jj0 denotes the norm of H , will be called the Sobolev lattice associated with C . 10
By definition, for C0 , C1 to be commuting we must have C01 C11 D C11 C01 ;
(2.2.13)
i.e. the commutativity of the resolvents. From this we obtain, however, equivalently that C1 C01 C11 D C01 and since 0 2 %.C1 / implies C1 ŒD.C1 / D H , we have again equivalently C01 C1 C1 C01 : For y 2 D.C1 C0 / we have C0 y 2 D.C1 / and so C01 C1 C0 y D C1 y: Thus, we obtain C1 C0 C0 C1 : For symmetry reasons we have C1 C0 D C0 C1 as a requirement equivalent to (2.2.13). Thus, in this case we recover the naive idea of commutativity even if C0 and C1 are unbounded operators. 11 As a matter of convenience, we introduce here the so-called multi-index notation. In particular, ˛
C ˛ WD C0 0 Cn˛n
Section 2.2 Sobolev Lattices
57
That the spaces H˛ .C / are Hilbert spaces is easily seen, since the parallelogram equality of H implies that the norm j j˛ is a pre-Hilbert space norm. As a first result we note the following fact. Lemma 2.2.2. Let C WD .Ci /iD0;:::;n , n 2 N, be a family of commuting, densely defined, closed linear operators Ci W D.Ci / H ! H , H a Hilbert space, with 0 2 %.Ci /; i D 0; : : : ; n, and let the family .H˛ .C //˛2ZnC1 , be the Sobolev lattice associated with C . Then, for ˛ 2 N nC1 , the domain D.C ˛ / is complete with respect to the norm j j˛ and hence H˛ .C / D D.C ˛ /. Proof. Let .'k /k be a Cauchy sequence with respect to j j˛ in D.C ˛ / for ˛ 2 N nC1 . Then, .C ˛ 'k /k is a Cauchy sequence in H . Let w 2 H be its limit. By the continuity of C for 2 N nC1 we have C 'k ! C ˛ w
in H
as k ! 1 for all ˛, 2 N nC1 . Letting v WD C ˛ w 2 D.C ˛ / H˛ .C / we see that C ˛ 'k ! C ˛ v as k ! 1. Thus, v 2 D.C ˛ / is the limit of the Cauchy sequence .'k /k in H˛ .C /. Lemma 2.2.3. Let C WD .Ci /iD0;:::;n be a family of commuting, densely defined, closed linear operators Ci W D.Ci / H ! H , H a Hilbert space, with 0 2 %.Ci /; i D 0; : : : ; n, n 2 N, and let the family .H˛ .C //˛2ZnC1 ; be the Sobolev lattice associated with C . Then we have the canonical dense and continuous embeddings H˛ .C / ,! Hˇ .C /
for ˛ ˇ; ˛; ˇ 2 ZnC1 ;
where ˛ ˇ is to be understood in the component-wise sense, i.e. with ˛ D .˛0 ; : : : ; ˛n /, ˇ D .ˇ0 ; : : : ; ˇn / we define ^ ˛ ˇ W, ˛i ˇi : iD0;::: n
for ˛ D .˛0 ; : : : ; ˛n / 2
ZnC1
and we define
sup¹˛; ˇº WD .max¹˛0 ; ˇ0 º; : : : ; max¹˛n ; ˇn º/; inf¹˛; ˇº WD .min¹˛0 ; ˇ0 º; : : : ; min¹˛n ; ˇn º/ for ˛ D .˛0 ; : : : ; ˛n /; ˇ D .ˇ0 ; : : : ; ˇn / 2 ZnC1 . In particular, ˛C WD sup¹˛; 0º;
˛ WD inf¹˛; 0º:
58
Chapter 2 Sobolev Lattices
Proof. By a simple induction argument it suffices to show that H˛Cej .C / ,! H˛ .C /; where ej D .¹j º .i //iD0;:::;n D ¹j º 2 ZnC1 for j D 0; : : : ; n, is a continuous embedding with dense range. Recalling the multiindex notation, we see that, since D.C ˛C Cej / D.C ˛C / and jyj˛Cej D jC ˛Cej yj0 D jCj C ˛ yj0 jC ˛ yj0 jCj1 j1 D jCj1 j1 jyj˛ for all y 2 D.C ˛C Cej /, the continuity of the embedding of the completed spaces follows. To see that the continuous extension is indeed an embedding, we have to show that a Cauchy sequence in H˛Cej .C /, which corresponds to a null sequence in H˛ .C /, also converges to zero in H˛Cej .C /. So, let . k /k be a Cauchy sequence in D.C ˛C / Cauchy convergent in H˛Cej .C /, such that k ! 0 in H˛ .C / as k ! 1. Then Cj C ˛ k D C ˛Cej k ! w in H as k ! 1 for some w 2 H and so C ˛ k ! Cj1 w
as k ! 1:
in H
Since on the other hand C ˛ k ! 0 in H
as k ! 1;
we have, by the uniqueness of limits, that Cj1 w D 0 and so w D 0. The density of the embedding, which is just the continuous extension of the inclusion D.C ˛C Cej / ! D.C ˛C /;
7! (and therefore rightfully called “canonical embedding”), remains to be shown. For this it suffices to prove that D.C ˛C Cej / H˛Cej .C / is dense in D.C ˛C /, (which in turn is dense in H˛ .C / by construction). Since D.Cj / D Hej .C / is dense in H D H0 .C / by assumption, given w 2 H we have
k ! w
in H
as k ! 1
Section 2.2 Sobolev Lattices
59
for some sequence . k /k in D.Cj /. Consequently, we have for z 2 D.C ˛C / and w D C ˛C z C ˛C C ˛C k D k ! C ˛C z D w in H0 .C / as k ! 1. Consequently, C ˛ C ˛C k D C ˛ C ˛C C ˛C k ! C ˛ C ˛C z D C ˛ z as k ! 1, or
C ˛C k ! z
in H0 .C /
in H˛ .C /
as k ! 1. Thus, we have that C ˛C ŒD.Cj / is dense in D.C ˛C / as a subspace of H˛ .C /. We finally note that C ˛C ŒD.Cj / D C ˛C ŒCj1 ŒH D C ˛C ej ŒH D D.C ˛C Cej / D.C ˛Cej / and thus the desired density result follows. Remark 2.2.4. A Sobolev lattice is indeed a lattice with respect to lattice operations induced by the canonical dense and continuous embedding as order relation. The lattice structure becomes obvious if, assuming for simplicity that all operators of the family are unbounded, we define for ˛; ˇ 2 ZnC1 sup¹H˛ .C /; Hˇ .C /º WD Hinf¹˛;ˇ º .C /; inf¹H˛ .C /; Hˇ .C /º WD Hsup¹˛;ˇ º .C /: As in the one operator case, we define \ H1 .C / WD H˛ .C /;
H1 .C / WD
˛2ZnC1
[
H˛ .C /
˛2ZnC1
and we obtain analogously the following density result (the proof of which may be left as an exercise). Lemma 2.2.5. Let .H˛ .C //˛2ZnC1 be the Sobolev lattice associated with the family C D .Ck /kD0;:::;n of linear operators. Then H1 .C / WD
\
H˛ .C /
˛2ZnC1
is dense in Hˇ .C / for every ˇ 2 ZnC1 . Now we define the operators C˛;ˇ W H˛ .C / ! Hˇ .C /;
˛ D .˛0 ; : : : ; ˛n /;
ˇ D .ˇ0 ; : : : ; ˇn / 2 ZnC1 ;
60
Chapter 2 Sobolev Lattices
as the continuous extensions of the isometries
7! C ˛ˇ from H˛ .C / to Hˇ .C / densely defined on D.C ˛C ˇ / D H˛C ˇ .C /. That this mapping is an isometry is clear since jC ˛ˇ yjˇ D jC ˇ C ˛ˇ yj0 D jC ˛ yj0 D jyj˛ for all y 2 D.C ˛C / such that C ˛ y 2 D.C ˇ /, i.e. y 2 D.C ˛C ˇ /. Noting that C ˛ˇ ŒD.C ˛C ˇ / D C ˛ˇ ˛C Cˇ ŒH D C ˛ ˇC ŒH D D.C ˛ CˇC / D.C ˇC / we see that these isometries are not only densely defined (noting ˛C ˛ D ˛C C ˛ ˛ and so H˛C ˇ .C / ,! H˛C .C / ,! H˛ .C /), but also have dense range (since likewise ˇC ˇ D ˇC C ˇ ˇ and so H˛ CˇC .C / ,! HˇC .C / ,! Hˇ .C / ). Their closure must therefore be a surjective isometry with domain H˛ .C /. In other words, the mappings C˛;ˇ are unitary. Analogous to the one operator case we are interested specifically in Cj;˛ W H˛Cej .C / H˛ .C / ! H˛ .C /; x 7! C˛Cej ;˛ which is a densely defined, closed linear operator for all ˛ 2 ZnC1 ; j D 0; : : : n. As in the one operator case, we find [ Cj;˛ Cj;ˇ Cj; DW Cj;1 2ZnC1
for ˛; ˇ 2 ZnC1 ; ˛ ˇ; j D 0; : : : ; n. We also find that the operators Cj;˛ , j D 0; 1; : : : ; n, retain commutativity, ˛ 2 ZnC1 , and so ˛ ˇ
˛n ˇn ˛ˇ 0 0 Cn;1 DW C1 C˛;ˇ C0;1
where ˛; ˇ 2 ZnC1 . We also have the characterization ˛ H˛ .C / D C1 ŒH0
for ˛ 2 ZnC1 ;
where H0 WD H0 .C / D H0 .C / D H0 .C / D H0 .C / D H , and C WD .Ci /iD0;:::;n . Moreover, e
j Cj;1 D C1
for j D 0; : : : ; n:
We shall call the family C1 WD .C0;1 ; : : : ; Cn;1 / the (continuous) extension of C D .C0 ; : : : ; Cn / to H1 .C /. In summary we have the following result.
Section 2.2 Sobolev Lattices
61
Lemma 2.2.6. Let C WD .Ci /iD0;:::;n be a family of commuting, densely defined, closed linear operators Ci W D.Ci / H ! H , H a Hilbert space, with 0 2 %.Ci /; i D 0; : : : ; n, n 2 N, and let the family .H˛ .C //˛2ZnC1 ; be the Sobolev lattice associated with C . Then the extension C1 of C to H1 .C / has the following properties:
.1/ C1 \ .H˛ .C / ˚ H˛ .C // is unitary for all ; ˛ 2 ZnC1 , ˛Cıˇ
ı j D jC .2/ jC1 ˛ 1
jˇ for all 2 H˛Cı .C /, ˛; ˇ; ı 2 ZnC1 , ˛CıCˇ
ı .3/ jC1
j˛ kC 1 k jC1
jQ ˇ for all 2 H˛CıC .C /, ˛; ˇ; ı 2 nC1 Z , 2 N nC1 , where kC 1 k WD iD0;:::;n kCi1 ki .
Proof. The first statement follows immediately since C˛;˛ D C1 \ .H˛ .C / ˚ H˛ .C //
is unitary. For the second statement, we first observe that ˛
j0 j j˛ D jC1
for all 2 H˛ .C /:
ı , assuming 2 H Replacing by C1 ˛Cı .C /, we obtain as desired ı ˛Cı jC1
j˛ D jC1
j0 ; ˇ ˛Cıˇ D jC1 C1
j0 ; ˛Cıˇ D jC1
jˇ
for all 2 H˛Cı .C /:
Finally, replacing ˇ by ˇ yields ı ˛CıCˇ
j˛ D jC1
jˇ jC1
for all 2 H˛Cı .C /: ˛CıC
For 2 H˛CCı .C / with 2 N nC1 , we find C1
2 H0 .C / and so
ı ˛CıC jC1
j˛ D jC1 C1
j0 ˛CıC D jC C1
j0 Y ˛CıC kCi1 ki jC1
j0 iD0;:::;n
from which the desired inequality follows since ˛CıC ˇ ˛CıCˇ ˛CıCˇ jC1
j0 D jC1 C1
j0 D jC1
jˇ
for all 2 H˛CCı .C /.
62
Chapter 2 Sobolev Lattices ˛ , ˛ 2 ZnC1 , are continuous in the following sense. The extended operators C1
Definition 2.2.7. Let .H˛ .C //˛2ZnC1 , .Hˇ .D//ˇ 2ZmC1 be Sobolev chain associated with operator families C D .C0 ; : : : ; Cn / and D D .D0 ; : : : ; Dm /, m; n 2 N, respectively. A mapping A W H1 .C / ! H1 .D/ will be called continuous if for every ˛ 2 ZnC1 there is an ˇ 2 ZmC1 such that H˛ .C / ! Hˇ .D/; x 7! Ax is continuous. In the case C D D a mapping A W H1 .C / ! H1 .C / will be called continuous of finite order if there is a 2 ZnC1 such that for all ˛ 2 ZnC1 we have that H˛ .C / ! H˛C .C /; x 7! Ax
(2.2.14)
is continuous. If can be arbitrary then we say A is of order 1. We shall say that A is of order !, if ! 2 .Z [ ¹1º/nC1 is such that (2.2.14) is continuous for all 2 ZnC1 with ! in the sense of component-wise order relation for all indices k D 0; : : : ; n with !k 2 Z. According to our last lemma, for C1 constructed from a family C D .C0 ; : : : ; Cn / of commuting, unbounded, densely defined and closed linear operators, we have that ˛ is of order ˛, ˛ 2 ZN C1 . C1 Theorem 2.2.8. Let C D .C0 ; : : : ; Cn /, n 2 N, be a commutingTfamily of closed, densely defined, linear operators in Hilbert space H with 0 2 kD0;:::;n %.Ck /. Then C D .C0 ; : : : ; Cn / is also a commuting densely defined, T family of closed, linear operators in Hilbert space H with 0 2 kD0;:::;n %.Ck / and we have for all ˛ 2 ZnC1 . (2.2.15) H˛ .C / D H˛ .C / in the sense of a canonical unitary correspondence. Moreover, the unitary equivalence is given by H˛ .C / ! H˛ .C / ;
7! .
˛ 7! h.C1 /˛ jC1 i0 /
where ˛ 2 ZnC1 . Proof. Since the family of linear operators C consists of adjoints, which are always closed and which are densely defined, since C D C is an operator, it remains to
Section 2.2 Sobolev Lattices
63
showTthat commutativity T is maintained. But since %.Ck / D %.Ck /, k D 0; : : : ; n, 0 2 kD0;:::;n %.Ck / D kD0;:::;n %.Ck / , and we find .Cj /1 .Ci /1 D .Ci1 Cj1 / D .Cj1 Ci1 / D .Ci /1 .Cj /1 for all i; j D 0; : : : ; n. We claim that the inverse of the unitary correspondence12 expressed in the above equality (2.2.15) is given by the mapping C0;˛ C˛;0 R˛ W H˛ .C / ! H˛ .C /; ˛ ˛
7! C1 C1 R˛
where R˛ W H˛ .C / ! H˛ .C / denotes the Riesz map between the noted spaces. Moreover, as we shall see this unitary correspondence is canonical in the sense that it is a continuous extension of the “identity”. That C0;˛ C˛;0 R˛ W H˛ .C / ! H˛ .C /
is unitary follows, since it is a composition of unitary mappings. It remains to show that C0;˛ C˛;0 R˛ is the inverse of the natural embedding of H˛ .C / in H˛ .C / given in the statement of the lemma. Indeed, for 2 H˛ .C / H the mapping f 7! hC˛;0
j C˛;0 f i0
defines a linear functional ‰ on H˛ .C /. We need to show that
D C0;˛ C˛;0 R˛ ‰ :
We read off that hC˛;0 C0;˛ C˛;0 R˛ ‰ jC˛;0 f i0 D hC˛;0 R˛ ‰ jC˛;0 f i0
D hR˛ ‰ jf i˛ D ‰ .f / D hC˛;0
j C˛;0 f i0
for all f 2 H˛ .C /. Thus, we have C˛;0
D C˛;0 R˛ ‰ 12
Therefore we claim here that the unitary mapping of the lemma is given by / D R˛ C0;˛ C˛;0 ; R˛ .C˛;0 / .C0;˛
˛ 2 ZnC1 :
64
Chapter 2 Sobolev Lattices
and so indeed
D C0;˛ C˛;0 R˛ ‰ :
We note that in particular for 2 H1 .C / H0 .C /, f 2 H1 .C /, ‰ .f / D hC .˛/ jC ˛ f i0 D h jf i0 and so ‰ D (by our identification of H0 .C / with H0 .C / ). Thus, we have C0;˛ C˛;0 R˛ D C˛;0 R˛ is indeed an extension of the for 2 H1 .C /, which shows that C0;˛ identity and thus canonical. Note that since C0;˛ C˛;0 R˛ is unitary and H1 .C / is dense in H˛ .C /, we also have H1 .C / dense in H˛ .C / .
According to our findings, there is a corresponding Sobolev lattice .H˛ .C //˛2ZnC1 associated with the commuting family of closed, linear operators C D .C0 ; : : : ; Cn / in Hilbert space H having 02
\
\
%.Ck / D
kD0;:::;n
%.Ck / :
kD0;:::;n
Note that H D H0 .C / D H0 .C /. To distinguish norm and inner product for H˛ .C / from the ones for H˛ .C / we shall add an index , i.e. j j;˛ WD j.C /˛ j0 and h j i;˛ WD h.C /˛ j .C /˛ i0 , for all ; 2 H˛ .C /, ˛ 2 ZnC1 . Note in particular hj i;0 D hj i0 . The canonical unitary equivalence expressed in equation (2.2.15) motivates the identification of H˛ .C / with H˛ .C / as we shall do henceforth. This identifi cation yields C0;˛ C˛;0 R˛ D 1H˛ .C / D 1H˛ .C / and so the operator equality ˛ R˛ D C0;˛ C˛;0 D C1 .C1 /˛
on H˛ .C / D H˛ .C / :
With this we have in particular for all 2 H1 .C / and f 2 H1 .C /, ˛ 2 ZnC1 , h j f i0 D .f / D hR˛ j f i˛ D h.C /˛ C ˛ C ˛ R˛ j C ˛ f i0 D h.C /˛ j C ˛ f i0 ; and so jh j f i0 j j.C /˛ j0 jC ˛ f j0 D j j;˛ jf j˛ :
Section 2.3 Sobolev Lattices from Tensor Products of Sobolev Chains
65
By the density of H1 .C / in H˛ .C / and H1 .C / in H˛ .C /, and noting the possibility of continuous extension of the inner product to H˛ .C / H˛ .C /, we are led to re-utilize the inner product notation h j f i0 WD .f / for all 2 H˛ .C / and f 2 H˛ .C /, ˛ 2 ZnC1 . Thus, by continuous extension we also have f .g/ D hf jgi0 D hgjf i0 D g.f / for all f 2 H˛ .C / D H˛ .C / and g 2 H˛ .C / D H˛ .C / . Henceforth, we shall also (with little danger of confusion) drop the index 1 from C1 to avoid notational clutter. In what sense the operators C should be understood shall be determined by the context.
2.3
Sobolev Lattices from Tensor Products of Sobolev Chains
A particular instance of the Sobolev lattice construction of the previous subsection is based on tensor products of Sobolev chains. The starting point is a finite family .C.i/ /iD0;:::;n , n 2 N, of densely defined, closed linear operators C.i/ W D.C.i/ / H .i/ ! H .i/ , H .i/ Hilbert spaces, with 0 2 %.C.i/ /; i D 0; : : : ; n. Theorem 2.3.1. Let .Hk .C.i/ //k2Z be the Sobolev chain associated with C.i/ W D.C.i/ / H .i/ ! H .i/ ; H .i/ Hilbert spaces, i D 0; : : : ; n. Then .Hk0 .C.0/ / ˝ ˝ Hkn .C.n/ //kD.k0 ;:::;kn /2ZnC1 ; is the Sobolev lattice associated with C WD .Ci /0D1;:::;n , where Ci W D.Ci / H ! H with Ci WD 1H .0/ ˝ ˝ 1H .i1/ ˝ C.i/ ˝ 1H .iC1/ ˝ ˝ 1H .n/ , i D 0; : : : ; n, and H WD H .0/ ˝ ˝ H .n/ . Proof. That C is a family of closed, densely, defined, linear operators with 0 2 %.Ci / is clear from the construction of Ci as a tensor product, i D 0; : : : ; n. Moreover, 1 Ci1 D 1H .0/ ˝ ˝ 1H .i1/ ˝ C.i/ ˝ 1H .iC1/ ˝ ˝ 1H .n/ ;
66
Chapter 2 Sobolev Lattices
from which it follows that C is also a commuting family of operators. Moreover, since the inclusion Hk0 .C.0/ / ˝ ˝ Hkn .C.n/ / H.k0 ;:::;kn / .C / is dense, we must have Hk0 .C.0/ / ˝ ˝ Hkn .C.n/ / D H.k0 ;:::;kn / .C /: The following examples are intended to illustrate the construction and to provide insight into the particular spaces utilized later. Example 2.5. The Sobolev lattice associated with @ ˙ e, 2 RnC1 , n 2 N, e D .1; : : : ; 1/, where @ ˙ e D .@0;0 ˙ 1; : : : ; @n;n ˙ 1/, D .0 ; : : : ; n /, and @k;k D
k1 O
n O
1L2 .R;exp.2s x/ dx/ ˝ @k ˝
sD0
1L2 .R;exp.2s x/ dx/
sDkC1
represents a weighted partial derivative with respect to the variable at position k, k D 0; : : : ; n. The Sobolev lattice .H˛ .@ ˙ e//˛2Zn will play a crucial role in our discussion of partial differential equations (and systems) with constant coefficients in RnC1 . The abstract construction is here more easily accessible due to the density of ı
C 1 .RnC1 / in H˛ .@ ˙ e/ for every ˛ 2 Zn . Due to the lattice structure of the dense embeddings it suffices to show this for large multi-indices ˛ 2 N nC1 . We know ı
that C 1 .RnC1 / is dense in H0N .@ ˙ e/, which is just the weighted L2 -type space n 2 nC1 2 ˛ W L .R ; exp.2x/ dx/ WD kD0 L .R; exp.2k x/ dx/. Since .@ ˙ e/ H0 .@ ˙ e/ ! H˛ .@ ˙ e/ is unitary we deduce ı
.@ ˙ e/˛ ŒC 1 .RnC1 / dense in H˛ .@ ˙ e/: ı
By induction, as in Example 2.4, we find that .@ ˙ e/˛ ŒC 1 .RnC1 / C1 .RnC1 /. Using the cut-off technique utilized in Example 2.2 with .n/
R WD
n O
ı
R 2 C 1 .RnC1 /;
R 2 R>0 ;
kD0
as .n C 1/-dimensional cut-off function, we obtain, as in the 1-dimensional case, the ı
density of C 1 .RnC1 / in H˛ .@ ˙ e/. Indeed, we can even show in this way that a O
ı
C 1 .R/ is dense in H˛ .@ ˙ e/:
kD0;:::n
On
a O kD0;:::n
ı
C 1 .R/
Section 2.3 Sobolev Lattices from Tensor Products of Sobolev Chains
67
it is now clear that @ D .@0 ; : : : ; @n / .0 ; : : : ; n /; where @k is just the classical partial derivative, k D 0; : : : ; n. Example 2.6. The Sobolev lattice associated with .m0 ; : : : ; mn /, n 2 N. In RnC1 we have .n C 1/ “variables” to account for and consequently .n C 1/ multiplication-byargument operators, which can be understood as resulting from the canonical tensor product construction for the 1-dimensional multiplication-by-argument operator. We have n O H˛ .i .m0 ; : : : ; mn / ˙ e/ WD H˛k .i m ˙ 1/: kD0
We shall again write m for .m0 ; : : : ; mn / to simplify notation; it will always be clear from the context which dimension is implied. It is not hard to see that m D .m0 ; : : : ; mn / constitutes a family of commuting, selfadjoint operators and that indeed mk WD
k1 O
1L2 .R/ ˝ m ˝
sD0
n O
1L2 .R/ :
sDkC1
This Sobolev lattice will later serve as a reference lattice for the Sobolev lattice introduced in Example 2.5. To avoid confusion, we shall denote the norm and inner product in H˛ .i m ˙ 1/ by j j^;˛ and h j i^;˛ , respectively. In the spirit of the ı
previous examples for Sobolev lattices we note that the density of C 1 .RnC1 / is obı
vious in this case, since multiplication by .i m ˙ e/˛ is a bijection of C 1 .RnC1 / for every ˛ 2 ZnC1 . The Fourier–Laplace transform L , 2 R, which we are about to introduce, is a prominent example of a so-called spectral representation. It is a spectral representation for the differential operator 1i @ , which just means that an equality of the form (2.3.16) below holds. Moreover, it provides a unitary mapping between H˛ .@ ˙ e/ and the spaces H˛ .i m ˙ e/, ˛ 2 ZnC1 , currently under discussion. To consider this in more detail, let us first consider the 1-dimensional case. Recallı ing that the Fourier transform given on, say, C 1 .R/ by Z 1 .F '/.y/ .L0 '/.y/ D p exp.i y x/'.x/ dx; y 2 R; 2 R extends continuously to a unitary mapping F W L2 .R/ ! L2 .R/; we obtain that the Fourier–Laplace transform L WD F exp.m/ is also a unitary mapping L W H0 .@ ˙ 1/ ! L2 .R/ since it is a composition of unitary mappings.
68
Chapter 2 Sobolev Lattices
Moreover, we have the unitary equivalence of 1i @ and multiplication by the argument, i.e. 1 @ D L m L : (2.3.16) i This follows from the corresponding fact for the Fourier transform, i.e. for the case D 0. In particular, we find .@ ˙ 1/k D L .i m ˙ 1/k L for all k 2 Z. This yields for all 2 D..@ ˙ 1/k / j j;k D j.@ ˙ 1/k j;0 D j.i m ˙ 1/k L j0;0 DW jL j^;k ; where j j^;k denotes the norm of Hk .i m ˙ 1/, k 2 Z. This proves that L extends continuously to a unitary mapping L;k W Hk .@ ˙ 1/ ! Hk .i m ˙ 1/ for all k 2 Z. We shall re-use the notation L to symbolize the induced Fourier– Laplace transform L W H1 .@ ˙ 1/ ! H1 .i m ˙ 1/;
2 R:
The special case D 0 reduces to the Fourier transform case. These considerations can be lifted routinely to RnC1 by taking tensor products. With L WD
n O
Lk
kD0
for D .0 ; : : : ; n / 2 RnC1 , which is clearly a unitary mapping from H0 .@ ˙ e/ to L2 .RnC1 / D H0 .i m ˙ e/, we obtain, by continuous extension of L jH1 .@ ˙e/ , the unitary mappings L;˛ W H˛ .@ ˙ e/ ! H˛ .i m ˙ e/: nC1 , 2 RnC1 , we can consider the mapping Since S L;˛ L;ˇ for ˛ ˇ, ˛; ˇ 2 Z nC1 , to a ˛2ZnC1 L;˛ as the maximal extension of all the mappings L;ˇ , ˇ 2 Z mapping from H1 .@ ˙1/ to H1 .im˙e/. We shall, as above, denote this maximal mapping again by L , 2 RnC1 , and refer to it as the Fourier–Laplace transform.
Section 2.3 Sobolev Lattices from Tensor Products of Sobolev Chains
69
Example 2.7. We consider the Sobolev lattice associated with jD j, 2 RnC1 , n 2 N. Here jD j WD .jD0;0 j; : : : ; jDn;n j/; D .0 ; : : : ; n /, and Dk;k D
k1 O
n O
1L2 .R;exp.2s x/ dx/ ˝ Dk ˝
sD0
1L2 .R;exp.2s x/ dx/ ;
sDkC1
where the one-dimensional case has been discussed in Example 2.4. The associated Sobolev lattice .H˛ .jD j//˛2Zn will play an essential role in our discussion of partial differential equations and systems with constant coefficients in RnC1 . Also in ı
this case, for ease of calculation, we need to establish that C 1 .RnC1 / is dense in H˛ .jD j/, ˛ 2 ZnC1 . From Example 2.2 we see that by unitary equivalence with respect to the unitary mapping
exp. .m1 ; : : : ; mn // WD
n O
exp.k m/
kD1
we may restrict our attention to D .0; : : : ; 0/, (note that exp. .m0 ; : : : ; mn // is ı
a bijection of C 1 .RnC1 / onto itself). For D .0; : : : ; 0/ we know, however, that ˇ
¹D0 jˇ 2 N nC1 º is a complete orthogonal set in H˛ .jD0 j/ for every ˛ 2 ZnC1 . Here we have re-used13 the notation for the .n C 1/-dimensional function .x0 ; : : : ; xn / 7! .x0 / .xn /. We have, of course, ˇ
.D0 /.x0 ; : : : ; xn / D .D ˇ1 /.x0 / .D ˇn /.xn / for all ˇ D .ˇ0 ; : : : ˇn / 2 N nC1 , x D .x0 ; : : : ; xn / 2 RnC1 . The corresponding complete orthogonal set for H˛ .jD j/, ˛ 2 ZnC1 , for general 2 RnC1 is then given by ˇ
¹exp. m/ D0 jˇ 2 N nC1 º: ı
The density of C 1 .RnC1 / in H˛ .jD j/, ˛ 2 ZnC1 , now follows again by the cutoff technique of the Example 2.5. As in the previous example, we can extend the properties of the Laplace transform. For D .0 ; : : : ; n / 2 RnC1 , k D 0; : : : ; n, we 13 There is little harm in doing so since it is always clear from the context what the number of arguments of is.
70
Chapter 2 Sobolev Lattices
find 1 L Dk;k ' D p L exp.k mk /.mk @k / exp.k mk /' 2 1 D p F .mk @k / exp. m/' 2 i D p .@k mk /F exp. m/' 2 1 D Dk L '; i L Dk;k ' D L exp.k mk /.mk C @k / exp.k mk /' 1 D p F .mk C @k / exp. m/' 2 i D p .@k C mk /F exp. m/' 2 D i Dk L ' and consequently L Dk; Dk;k ' D Dk Dk L ' k ı
for ' 2 C 1 .RnC1 /. Hence, by continuous extension we obtain a mapping from H1 .jD j/ to H1 .jDj/ which is unitary if restricted to H˛ .jD j/ mapping onto H˛ .jDj/, ˛ 2 ZnC1 , and which we still shall denote simply by L , 2 RnC1 , L W H1 .jD j/ ! H1 .jDj/: The relations 1 Dk L ; i D i Dk L
L Dk;k D L Dk; k
for D .0 ; : : : ; n / 2 RnC1 , k D 0; : : : ; n, still hold on H1 .jD j/ with the ı
equality being valid in H1 .jDj/. This is due to the fact that, for ' 2 C 1 .RnC1 /, we have by integration by parts j.mk ˙ @k /'j20 D jmk 'j20 C j@k 'j20 ˙ 2 Rehmk 'j@k 'i0 D jmk 'j20 C j@k 'j20 j'j20 for k D 0; : : : ; n, and so also 'j20 C j'j20 D jDk;k 'j20 jDk; k
Section 2.3 Sobolev Lattices from Tensor Products of Sobolev Chains
71
for all D .0 ; : : : ; n / 2 RnC1 . By induction we obtain in particular that H˛ .jD j/ and H˛ .jD j C i e/ are Hilbert spaces based on the same linear space but equipped with equivalent norms (yielding identical topologies). Thus, Dk; W H˛ .jD j C i e/ ! H˛ek .jD j/ k
and Dk;k W H˛ .jD j/ ! H˛ek .jD j C i e/ can – according to our considerations in Lemma 2.1.16 – not only be considered as continuous mappings in the stated sense, but also as continuous mappings from H˛ .jD j/ to H˛ek .jD j/, ˛ 2 ZnC1 . We shall have occasion to consider later alternative characterizations of the associated norms for the case ˛ 2 N nC1 .
Chapter 3
Linear Partial Differential Equations with Constant Coefficients in RnC1, n 2 N
3.1
Linear Partial Differential Equations with Constant Coefficients in H1 .@ C e/
3.1.1 First Steps Towards a Solution Theory We are now ready to systematically investigate partial differential equations on the basis of the Hilbert space concepts discussed earlier. In this chapter we shall investigate differential expressions that are linear combinations of powers of partial derivatives @k , k D 0; : : : ; n: X X P .@/ D P .@0 ; : : : ; @n / D a.˛0 ;:::;˛n / @˛0 0 @˛nn : (3.1.1) ˛0
˛n
As a matter of convenience we shall here also employ notation abbreP the multi-index P viating @˛0 0 @˛nn D @.˛0 ;:::;˛n / . Then P .@/ D ˛0 ˛n a.˛0 ;:::;˛n / @.˛0 ;:::;˛n / P or more concisely P .@/ D ˛ a˛ @˛ , where the range of summation is now N nC1 , with only finitely many of the coefficients a˛ , ˛ 2 N nC1 , being non-zero. The coefficients may just be complex numbers, but since our discussion is for most parts completely analogous if the coefficients are complex matrices, we shall from the beginning include this possibility since it is also the more commonly encountered case. Thus for a˛ D .a˛;p;q /p;q 2 C .sC1/.tC1/ , s; t 2 N, we have X X a ˛ @˛ D a˛;p;q @˛ : P .@/ D ˛
˛
p;q
Our interest will be to develop a solution theory for equations of the form P .@/u D f;
(3.1.2)
where f is prescribed in a suitable function space, which is a system of .s C 1/ equations in .t C 1/ unknown generalized functions u D .u0 ; : : : ; u t / considered as a column matrix .u0 u t /> . The case of a single partial differential equation in a single unknown is included as a special case (s D t D 0). Another special case, an ordinary differential equation or system with constant coefficients occurs here when n D 0. Complex ..s C 1/ .t C 1//-matrices are considered as elements of C .sC1/.tC1/ ; s; t 2 N. To be more precise we give a formal definition.
Section 3.1 Partial Differential Equations in H1 .@ C e/
73
Definition 3.1.1. Let .a˛ /˛2N nC1 be a family of matrices in C ..sC1/.tC1// such that a˛ D 0 for all but a finite number of multi-indices ˛ 2 N nC1 , n 2 N. Then P ˛ ˛2N nC1 a˛ @ is called a linear (..s C 1/ .t C 1//-) partial differential expression (with constant coefficients) and of order #P WD sup¹j˛j1 j a˛ ¤ 0 ^ ˛ 2 N nC1 º, Pn j.˛0 ; : : : ; ˛n /j1 WD : : : ; ˛n / 2 N nC1 . The corresponding equation kD0 ˛k , .˛0 ;P of the form (3.1.2) with P .@/ WD ˛2N nC1 a˛ @˛ will be referred to as a (linear) (..s C 1/ .t C 1//-)partial differential equation (with constant coefficients) (of order #P ). If s D t D 0, we shall also speak of a scalar partial differential equation and if s > 0 or t > 0 of a system of (.s C 1/) partial differential equations (in .t C 1/ unknown functions). If s > t such a system will be called (formally) over-determined, if s < t it will be called (formally) under-determined. If s D t the system will be called square. P If we wished to give a finite range for the sum P .@/ D ˛2N nC1 a˛ @˛ , we could take P WD sup¹˛ j a˛ ¤ 0 ^ ˛ 2 N nC1 º, i.e. the supremum of the degrees occurring in P , with respect to the (partial) order relation ^
.x0 ; : : : ; xn / .y0 ; : : : ; yn / W,
xk yk
(3.1.3)
kD0;:::;n
for all .x0 ; : : : ; xn /; .y0 ; : : : ; yn / 2 RnC1 and write P .@/ D
X
a ˛ @˛ :
0˛P
It should, however, be noted that in general j P j1 ¤ #P . Moreover, we do not assume anything about the particular form of the polynomial matrix P .@/, thus including extreme cases such as P .@/ not depending on certain derivatives and even P .@/ just being a constant matrix. Thus our solution theory also includes what is known as linear (partial or ordinary) differential-algebraic systems. Example 3.1. Consider the partial differential expression P .@/ WD @20 @1 C @21 of order 3 in R2 having P D .2; 2/; thus, j P j1 D 4 ¤ #P D 3. Example 3.2. Consider the partial differential expression in R2 P .@/ WD
10 00
@0 C
10 00
@1 C
10 11
of order 1 having P D .1; 1/; note, j P j1 D 2 ¤ #P D 1. This leads to a system of two equations @0 u0 C @1 u0 C u0 D f0 ; u0 C u1 D f1 :
74
Chapter 3 Partial Differential Equations with Constant Coefficients
In the sense of the above definition this would still be called a system of partial differential equations1 , although the second equation is of degree 0. Even a purely algebraic expression such as 10 P .@/ WD 01 will be called a partial differential expression, if only of degree 0. We need to determine the generalized function spaces in which to consider (3.1.2). Our general framework, and in particular the examples presented in Chapter 2, suggest that we consider spaces built-up from the Sobolev lattice2 .H˛ .@ C e//˛2ZnC1 , 2 RnC1 . In order to discuss an ..s C 1/ .t C 1//-partial differential equation (PDE for short) we need to assemble corresponding “matrices of spaces” HA .@ C e/ WD
t M s M j D0
H˛;ij .@ C e/ .H1 .@ C e//.sC1/.tC1/
iD0
with A D .˛;ij /iD0;:::;sIj D0;:::;t , i.e. a matrix of multi-indices ˛;ij 2 ZnC1 , i D 0; : : : ; s; j D 0; : : : ; t , suggestively written as HA .@ C e/ D .H˛;ij .@ C e//iD0;:::;sIj D0;:::;t 0 1 H˛;00 .@ C e/ H˛;0t .@ C e/ B C :: :: :: D@ A: : : : H˛;s0 .@ C e/ H˛;st .@ C e/ Considering A D .˛;ij /iD0;:::;sIj D0;:::;t as an element of .ZnC1 /.sC1/. tC1/ , we recognize that .HA .@ C e//A2.ZnC1 /.sC1/.t C1/ is again a lattice of Hilbert spaces. We shall use the notation HnC1;sC1;tC1;˙1 .@ C e/ WD .H˙1 .@ C e//iD0;:::;sIj D0;:::;t or briefly again just H˙1 .@ C e/ if the parameters s; t are clear from the context. By construction, the inner product of HA .@ C e/ is given by X h ij j ij i;˛;ij (3.1.4) h j i;A WD i;j 1
In the literature such a combination of partial differential equations with purely algebraic equations is referred to as partial differential-algebraic equations (PDAEs), in the framework presented here, however, we shall continue to speak of partial differential equations (PDEs). 2 For a finer regularity analysis one could also use a finer Sobolev lattice structure by constructing analogously .HA .j@ C ej//A2.RnC1 /.sC1/.t C1/ (here j@ C ej D .j@0 ;0 C 1j; : : : ; j@n1 ;n1 C 1j/), which interpolates the Sobolev lattice .HA .@ C e//A2.ZnC1 /.sC1/.t C1/ . Note that HA .@ C e/ D HA .j@ C ej/ for A 2 .ZnC1 /.sC1/.t C1/ and HnC1;sC1;t C1;˙1 .@ C e/ D HnC1;sC1;t C1;˙1 .j@ C ej/. The inner product corresponds to (3.1.4) with A 2 .RnC1 /.sC1/.t C1/ instead of A 2 .ZnC1 /.sC1/.t C1/ . For sake of simplicity we shall, however, only make use of the integer multi-index case here.
Section 3.1 Partial Differential Equations in H1 .@ C e/
75
for A D .˛;ij /iD0;:::;sIj D0;:::;t 2 .ZnC1 /.sC1/.tC1/ and D . ij /iD0;:::;sIj D0;:::;t , D . ij /iD0;:::;sIj D0;:::;t 2 HA .@ Ce/. To simplify notation we shall write again simply Hˇ .@ C e/ for HA .@ C e/ if ˛;ij D ˇ for i D 0; : : : ; sI j D 0; : : : ; t and leave the determination of the actual matrix size to the context. The partial differential expression P .@/ will correspondingly be interpreted as the operator P .@ C / acting in the obvious (matrix-)way on elements in H1 .@ C e/. The focus of our discussion will be on the case of ..s C 1/ .s C 1//-partial differential expressions and equations and so we shall for most of the following assume that partial differential expressions are square matrices. For solutions and data, however, we allow the numbers of columns to be arbitrary. Our main task for this section is to discuss the question: Is the equation P .@/u D
s X
Pij .@/uj k
j D0
iD0;:::;sIkD0;:::;t
D .fik /iD0;:::;sIkD0;:::;t DW f
for a ..s C 1/ .s C 1//-partial differential expression P .@/ D .Pij .@//i;j D0;::: s and every given f 2 HnC1;sC1;tC1;1 .@ C e/ reasonably solvable by an element u D .uj k /j D0;:::;sIkD0;:::;t in a function space HnC1;sC1;tC1;1 .@ C e/, n; s; t 2 N? A spectral representation of 1i @ is given by the (component-wise applied) Fourier– Laplace transform L . Its application suggests that the solution u is given as the inverse Fourier–Laplace transform of P .i m C /1 L f provided a suitable parameter vector 2 RnC1 can be found so that P .i m C / can be inverted. The inverse can formally be given by Cramer’s rule P .i x C /1 D
1 cof.P .i x C //> : det.P .i x C //
Here cof.A/> denotes the transpose of the co-factor matrix cof.A/ D ..1/rCs det..aij /iD0;:::;r1;rC1;:::;sI j D0;:::;k1;kC1;:::;s //r;kD0;:::s of the matrix A, also known as the adjunct matrix3 . Note that P .i x C / is an ..s C 1/ .s C 1//-matrix with polynomials as entries. In the scalar case, i.e. s D 0, we have det.P .i xC// D P .i xC/ and cof.P .i xC// D 1. It is clear from linear algebra that cof.P .i x C //> is a matrix with polynomials as entries of the same size, so that the invertibility of P .i x C / solely rests on the invertibility of det.P .i x C //. This observation leads to the following definition. 3 The adjunct matrix cof.A/> is frequently and confusingly referred to as adjoint matrix, a habit we shall not adopt.
76
Chapter 3 Partial Differential Equations with Constant Coefficients
Definition 3.1.2. Let P .@/ be a linear ..s C1/.s C1//-partial differential expression and 2 RnC1 be given. We shall call P .@/ invertible in H1 .@ C e/ (or invertible with respect to ) if the equation det.P .i x C // D 0 has no solutions x 2 RnC1 . Remark 3.1.3. This is essentially the well-posedness of the partial differential equation in the sense of Petrovskii (see [15]). In order to have a proper inverse we would expect a continuity property for this inverse and the invertibility condition defined here does not appear to warrant this. However, it turns out that the simple algebraic condition in Definition 3.1.2 is sufficient to obtain continuity of the inverse. For this we need to explore the asymptotic behavior of polynomials in some detail.
3.1.2 The Tarski–Seidenberg Theorem and some Consequences We need some algebraic results on polynomials, in particular a consequence of the Tarski–Seidenberg theorem describing the asymptotic behavior of polynomials. With a modified notation we quote essentially from [21], where the proofs of the core results can be found. nC1
Definition 3.1.4. Let A.RnC1 / denote the smallest subset of the power set 2R of RnC1 , which is closed4 with respect to (finite) union, (finite) intersection and taking complements5 and contains all sets of the form ŒR>0 P WD ¹x 2 RnC1 j P .x/ > 0º; where P W RnC1 ! R is a polynomial function. The elements of A.RnC1 / are called semi-algebraic sets in RnC1 , n 2 N, or simply semi-algebraic if the dimension can be concluded from the context. Clearly, if P W RnC1 ! R is a polynomial function, then ¹x 2 RnC1 j P .x/ D 0º;
¹x 2 RnC1 j P .x/ 0º
(3.1.5)
are both semi-algebraic sets in RnC1 , n 2 N. Indeed, ¹x 2 RnC1 j P .x/ D 0º D .RnC1 n ¹x 2 RnC1 j P .x/ > 0º/ \ .RnC1 n ¹x 2 RnC1 j P .x/ > 0º/; ¹x 2 RnC1 j P .x/ 0º D RnC1 n ¹x 2 RnC1 j P .x/ > 0º: 4
I.e. A.RnC1 / is given as the intersection ˇ \° ^ nC1 ˇ A 2R ŒR>0 P 2 A ^ ˇ
^
nC1
u;v2A
P 2RR 5
polynomial
± u \ v; u [ v; RnC1 n u 2 A :
From elementary set theory we know that we can obtain a normal form as a union of intersection or an intersection of unions.
Section 3.1 Partial Differential Equations in H1 .@ C e/
77
In particular, ¹x 2 RnC1 j P .x/ ¤ 0º D ¹x 2 RnC1 j P .x/ > 0º [ ¹x 2 RnC1 j P .x/ > 0º is semi-algebraic. Our next result shows that the Cartesian product of semi-algebraic sets is again semi-algebraic. Lemma. Let A 2 A.RnC1 / and B 2 A.RmC1 /, n; m 2 N, then A B 2 A.RnCmC2 /: Proof. Since AB D .ARmC1 /\.RnC1 B/ and RnC1 B is related to B RnC1 by an exchange of Cartesian factors, which does not affect the property of being semialgebraic, it suffices to show the case A D RnC1 and, since RnC1 is, for positive n, a Cartesian product of the semi-algebraic set R, it suffices to consider n D 0. As stated above, a semi-algebraic set B can be expressed as a union of intersections of finitely many sets of the form ŒR>0 Pk ;
ŒR0 Ps ;
k; s 2 ¹0; : : : ; N º
for some N 2 N, where Pk W RnC1 ! R are polynomial functions for all k 2 e k W RnC2 ! R given by ¹0; : : : ; N º. By replacing Pk by the polynomial function P nC1 e k .t; x/ D Pk .x/ for t 2 R; x 2 R P , we get e k D R ŒR>0 Pk ; ŒR>0 P
e s D R ŒR0 Ps ; ŒR0 P
k; s 2 ¹0; : : : ; N º:
e k ; ŒR0 P e s , respectively, for k; s 2 By replacing ŒR>0 Pk ; ŒR0 Ps by ŒR>0 P ¹0; : : : ; N º in the finite union of intersections representing B we obtain R B, which therefore is seen to be a semi-algebraic set in RnC2 . The observation in the above proof, that interchanging factors in a Cartesian product of two semi-algebraic sets yields again a semi-algebraic set, can clearly be generalized to more than two factors. Let m; n 2 N with m n. We define mappings ˛n;m W RnC1 ! RmC1 ; .x0 ; : : : ; xn / 7! .x˛0 ; : : : ; x˛m /; where ˛ D .˛0 ; : : : ; ˛m / W ¹0; : : : ; mº ! ¹0; : : : ; nº is an injection, i.e. a permutation of a collection of .m C 1/ of the numbers in ¹0; : : : ; nº. In particular, ˛n;n is simply a permutation of the factors in the Cartesian product RnC1 . The above observation could be stated as ˛n;n ŒŒA WD ¹˛n;n ŒAI A 2 A.RnC1 /º D A.RnC1 / for n 2 N and ˛ a permutation of ¹0; : : : ; nº. The Tarski–Seidenberg theorem, compare [21, appendix], states a rather non-trivial extension of this observation.
78
Chapter 3 Partial Differential Equations with Constant Coefficients
Theorem (Tarski–Seidenberg). If A 2 A.RnC1 / then ˛n;m ŒA 2 A.RmC1 / for all permutations ˛ W ¹0; : : : ; mº ! ¹0; : : : ; nº, n; m 2 N, n m. In particular, for A 2 A.RnC1 / we have RsC1 n ˛m;s ŒRmC1 n ˇn;m ŒA 2 A.RsC1 / for all permutations ˛ W ¹0; : : : ; sº ! ¹0; : : : ; mº, ˇ W ¹0; : : : ; mº ! ¹0; : : : ; nº, n; m; s 2 N, n m s. Since ˇn;m ŒA ˇ ° ˇ D .aˇ0 ; : : : ; aˇm / 2 RmC1 ˇ
± .aˇ0 ; : : : ; aˇm ; b0 ; : : : ; bnm / 2 ˇn;n ŒA
_ b2Rnm
and RsC1 n ˛m;s ŒB ˇ ° ˇ D RsC1 n .b˛0 ; : : : ; b˛s / 2 RsC1 ˇ ˇ ° ˇ D .b˛0 ; : : : ; b˛s / 2 RsC1 ˇ ˇ ° ˇ D .b˛0 ; : : : ; b˛s / 2 RsC1 ˇ
^
_
± .b˛0 ; : : : ; b˛s ; c0 ; : : : ; cms / 2 ˛m;m ŒB
c2Rms
± .b˛0 ; : : : ; b˛s ; c0 ; : : : ; cms / 2 RmC1 n ˛m;m ŒB
c2Rms
^
± .b˛0 ; : : : ; b˛s ; c0 ; : : : ; cms / 2 ˛m;m ŒRmC1 nB ;
c2Rms
we get, with e ˛ as a permutation of ¹0; : : : ; nº with ˛ e ˛, RsC1 n ˛m;s ŒRmC1 n ˇn;m ŒA ˇ ^ ± ° ˇ .b˛0 ; : : : ; b˛s ; c0 ; : : : ; cms / 2 ˛m;m Œˇn;m ŒA D .b˛0 ; : : : ; b˛s / 2 RsC1 ˇ ˇ ° ˇ D .b˛0 ; : : : ; b˛s / 2 RsC1 ˇ
c2Rms
^
c2Rms
± .b˛0 ; : : : ; b˛s ; c0 ; : : : ; cms / 2 ˇn;m ŒA ıe ˛
ˇ ° ˇ D .b˛0 ; : : : ; b˛s / 2 RsC1 ˇ ± _ ^ .b˛0 ; : : : ; b˛s ; c0 ; : : : ; cms ; b0 ; : : : ; bnm / 2 ˇn;n ŒA : ˛ ıe c2Rms b2Rnm
With this general observation, we see for example that ˇ^ _ _ ± ° ˇ ." 0 _ .u y " ^ .x; u; z/ 2 E// .x; y/ 2 R2 ˇ "2R u2R z2RnC1
Section 3.1 Partial Differential Equations in H1 .@ C e/
79
is semi-algebraic if E 2 A.RnC3 /. Indeed, since ¹."; x; u; z; y/ 2 RnC5 j" 0 _ .u y C " 0 ^ .x; u; z/ 2 E/º D ¹."; x; u; z; y/ 2 RnC5 jP0 ."; x; u; z; y/ WD " 0º [ .¹."; x; u; z; y/ 2 RnC5 jP1 ."; x; u; z; y/ WD y u " 0º \ .R E R// is clearly semi-algebraic, the result is obvious from the above. Defining fE W R ! R [ ¹C1; 1º; x 7! fE .x/ where 8 .nC2;0/ ˆ sup .1;0;2;:::;nC2/ ŒE \ ¹xº R RnC1 ˆ ˆ ˆ ˆ .nC2;0/ ˆ ˆ if .1;0;2;:::;nC2/ ŒE \ .¹xº R RnC1 / ¤ ; bounded; < fE .x/ D
.nC2;0/ ˆ ˆ 1 if .1;0;2;:::;nC2/ ŒE \ .¹xº R RnC1 / D ;; ˆ ˆ ˆ ˆ ˆ :C1 if .nC2;0/ ŒE \ .¹xº R RnC1 / unbounded, .1;0;2;:::;nC2/
we see that the set ¹.x; y/ 2 R2 j y fE .x/º – the so-called subgraph of fE – is also semi-algebraic. Indeed, we have ¹.x; y/ 2 R2 j y fE .x/º ˇ^ _ ° ˇ D .x; y/ 2 R2 ˇ
_
± " 0 _ .u y " ^ .x; u; z/ 2 E/ :
"2R u2R z2RnC1
Functions f having a semi-algebraic subgraph ¹.x; y/ 2 R2 j y f .x/º have a relatively simple asymptotic structure, see [21, appendix]. Theorem 3.1.5. Let f W R ! R[¹C1; 1º be such that ¹.x; y/ 2 R2 j y f .x/º is semi-algebraic. Then R can be decomposed into a finite number of intervals where f .x/ D C1, f .x/ D 1 or f is a continuous algebraic function. If f .x/ is finite for all sufficiently large positive x and not identically zero, then f .x/ D C x a .1 C o.1//
as x ! C1;
where C 2 R n ¹0º and a 2 Q, i.e. a is a rational number.
80
Chapter 3 Partial Differential Equations with Constant Coefficients For our purposes we record the following consequence:
Corollary 3.1.6. Let E be a semi-algebraic subset of R3Cn , n 2 N, and fE W R ! R [ ¹C1; 1º be finite for large positive arguments. Then fE is either identically zero for such arguments or is a continuous algebraic function such that fE .x/ D C x a .1 C o.1//
as x ! C1;
where C 2 R n ¹0º and a 2 Q. As a first implication for the asymptotic behavior of polynomials we note: Corollary 3.1.7. Let P be a real (scalar) polynomial in RnC1 , n 2 N, with P ./ > 0 for all 2 RnC1 . Then there is a positive constant C and a rational number c 2 Q such that6 P ./ C ji ˙ ejce for all 2 RnC1 : Proof. Noting that the set E defined as E WD ¹.x; y; / 2 R3Cn j P ./ C y D 0 ^ x ji ˙ ej2e D 0º is semi-algebraic in R3Cn , we also see that subgraph fE of this set is well-defined for all sufficiently large positive arguments. Indeed, fE is a negative function since P is positive. Hence by the previous result we have fE .x/ D C1 x a .1 C o.1// as x ! C1, where C1 2 R<0 and a 2 Q. On the other hand, we see fE .x/ D sup¹y 2 R j .x; y; / 2 Eº D sup¹P ./ j x D ji ˙ ej2e º for x 1, and so P ./ inf¹P ./ j ji ˙ ej2e D ji ˙ ej2e º D .A/ .ji ˙ ej2e /a .1 C o.1// as x ! C1. Since P is locally bounded below by a positive constant, we have indeed a constant C 2 R>0 such that (with c D 2a) P ./ C ji ˙ ejce 6
for all 2 RnC1 :
Here and in the following an expression of the form jzj˛ , z 2 C nC1 , ˛ 2 RnC1 , abbreviates jz0 j˛0 jzn j˛n :
The symbol e here is our abbreviation for .1; : : : ; 1/.
Section 3.1 Partial Differential Equations in H1 .@ C e/
81
Corollary 3.1.8. Let P be a polynomial in RnC1 , n 2 N, with P .i / ¤ 0 for all 2 RnC1 , then there a positive constant C and a rational number c 2 Q such that jP .i /j C ji ˙ ejce
for all 2 RnC1 :
Proof. Applying the previous corollary to the polynomial 7! jP ./j2 yields the result. For later use we note the following result. Corollary 3.1.9. Let P be a polynomial in RnC1 , n 2 N, with P .i C % / ¤ 0 for a fixed 2 RnC1 ; jj D 1, and all 2 RnC1 ; % 2 Œ%0 ; 1Œ; %0 2 R. Then there is a positive constant C and a rational number c 2 Q such that7 jP .i C % /j C .ji C % j2 C e/ce
for all % 2 Œ%0 ; 1Œ; 2 RnC1 :
Proof. Noting that E WD ¹.x; y; ; %/ 2 R1C1C.nC1/C1 j x .j i C % j2 C e/e D 0 ^ jP .i C % /j2 C y D 0 ^ % %0 º is also semi-algebraic and applying the arguments of the proof of Corollary 3.1.7 to the real polynomial .; %/ 7! jP .i C % /j2 the result follows. Finally, Corollary 3.1.8 also answers our current question about the meaning of invertibility for partial differential equations. Theorem 3.1.10. Let P .@/ be a linear ..s C 1/ .s C 1//-partial differential expression in RnC1 , n; s 2 N. If P .@/ is invertible in H1 .@ C e/ then P .@/1 W HnC1;sC1;tC1;1 .@ C e/ ! HnC1;sC1;tC1;1 .@ C e/ exists as a continuous8 Here and in the following we use as a convenient abbreviation x 2 C e WD .x02 C 1; : : : ; xn2 C 1/, nC1 . With some care, there is little danger confusing this with the quite different x D .x0 ; : : : ; xP n/ 2 C n 2 notation x D kD0 xk2 , x D .x0 ; : : : ; xn / 2 C nC1 , which we also shall use. The first interpretation is only used if x 2 occurs in conjunction with another vector. 8 By this we mean that for every A 2 .ZnC1 /.sC1/.t C1/ there is a B 2 .ZnC1 /.sC1/.t C1/ such that 7
HA .@ C e/ ! HB .@ C e/; ˆ 7! P .@/1 ˆ is a continuous mapping. Indeed, closer inspection of the proof yields a stronger result, namely that there is a D 2 .ZnC1 /.sC1/.t C1/ such that for every A 2 .ZnC1 /.sC1/.t C1/ HA .@ C e/ ! HAD .@ C e/; ˆ 7! P .@/1 ˆ is continuous.
82
Chapter 3 Partial Differential Equations with Constant Coefficients
operator for every t 2 N. The inverse is given by Cramer’s rule as P .@/1 D P .@ C /1 D det.P .@ C //1 cof.P .@ C //> : Proof. Of course we would expect the solution operator to be given as P .@ C /1 D det.P .@ C //1 cof.P .@ C //> : Since cof.P .@ C//> as a partial differential operator is continuous in H1 .@ Ce/, it remains to be seen that det.P .@ C //1 W H1 .@ C e/ ! H1 .@ C e/ is continuous. For this it suffices to interpret this as the inverse of a scalar partial differential operator. The latter is equivalent to showing that det.P .i m C //1 W H1 .i m C e/ ! H1 .i m C e/ is continuous. Let ' 2 .CV 1 .RnC1 //.sC1/.tC1/ and consider the function x 7! det.P .i x C %0 //1 '.x/ with 0 2 RnC1 , j0 j D 1, i.e. a normalized vector, and % 2 Œ0; 1Œ, which is a well-defined element of .CV 1 .RnC1 //.sC1/.tC1/ since P .@/ was assumed to be invertible in H1 .@ C e/, D % 0 . By Corollary 3.1.8, there is a rational exponent c such that jdet.P .i C //j C .ji C j2 C e/ce
for all 2 RnC1 :
(Recall that in ji Cj2 Ce the term ji Cj2 stands for .ji 0 C0 j2 ; : : : ; ji n Cn j2 /, whereas in the expression ji Cj2 C1 the term ji Cj2 would have to be interpreted as ji 0 C 0 j2 C C ji n C n j2 in order to make the expressions meaningful.) Therefore .ji C /j2 C e/ce jdet.P .i C //j1 C 1 for all 2 RnC1 . Moreover, we estimate jP .i C /1b ' .i C /j D jdet.P .i C //1 cof.P .i C //>b ' .i C /j C 1 j.ji C j2 C e/ce cof.P .i C //>b ' .i C /j for all 2 RnC1 . Since all entries in cof.P .i C //> can be uniformly estimated by a term of the form C 0 .ji C /j C e/˛ for some suitably large ˛ 2 N nC1 , we find e .jj2 C e/ˇ jb ' .i C /j C ' .i C /j jP .i C /1b
(3.1.6)
e 2 R>0 , a multi-index ˇ 2 ZnC1 and all 2 RnC1 . Applying for some constant C the inverse Fourier–Laplace transform to the resulting norm inequality we get e j'j;ACˇE jP .@ C /1 'j;A C
(3.1.7)
for every A 2 .ZnC1 /.sC1/.tC1/ where we have used the notation ˇE WD .ˇ/j D0;:::;sIkD0;:::;t : Since .CV 1 .RnC1 //.sC1/.tC1/ is dense in HACˇE .im C e/, we conclude by continuous extension that (3.1.7) holds for all ' 2 HACˇE .@ C e/.
Section 3.1 Partial Differential Equations in H1 .@ C e/
83
e do not Remark 3.1.11. We note that the multi-index ˇ and the continuity constant C depend on t 2 N. As a matter of convenience we shall henceforth again write simply CV 1 .RnC1 / for the linear space of matrix-valued functions .CV 1 .RnC1 //.sC1/.tC1/ and, as in the case of the notation H˙1 .@ C /, leave it to the context to determine the appropriate matrix size. Theorem 3.1.10 exhibits the solution theory for partial differential equations with constant coefficients which we are pursuing. There are, however, several finer points which still need to be discussed. The continuity of the inverse of P .@/ is a particular case of the general concept of continuity for operators in a Sobolev lattice. We recall that an operator W W HnC1;sC1;tC1;1 .@ C e/ ! HnC1;sC1;tC1;1 .@ C e/ is said to be continuous in HnC1;sC1;tC1;1 .@ C e/ if ^
_
A2.ZnC1 /.sC1/.t C1/
B2.ZnC1 /.sC1/.t C1/
W jHA .@ Ce/ W
HA .@ C e/ ! HB .@ C e/ continuous: In the particular case of interest to us, we have the stronger property of continuity: ^
_
^
2N B2.ZnC1 /.sC1/.t C1/ A2.ZnC1 /.sC1/.t C1/ ;sup¹Ak jkD0;:::;sºinf¹Aj jj D0;:::;sºD
W jHA .@ Ce/ W HA .@ C e/ ! HAB .@ C e/ continuous: Indeed, if W is a function of @ , then continuity is equivalent to this stronger continuity property. Proposition 3.1.12. Let F .i / W RnC1 ! C .sC1/.sC1/ be a matrix of Borel functions Fij .i / W RnC1 ! C, i; j D 0; : : : ; s, and let F .@ /jHA .@ Ce/ W HA .@ C e/ ! HB .@ C e/, 7! F .@ / WD .Fij .@ / /i;j be continuous for some A 2 .ZnC1 /.sC1/.uC1/ , B 2 .ZnC1 /.sC1/.uC1/ , s; u 2 N. Then, for every w 2 N, 2 N, we have that for all C 2 .ZnC1 /.sC1/.wC1/ such that sup¹Ckj jk D 0; : : : ; sI j D 0; : : : ; wº inf¹Ckj jk D 0; : : : ; sI j D 0; : : : ; wº D there is a D 2 .RnC1 /.sC1/.wC1/ such that F .@ /jHC .@ Ce/ W HC .@ C e/ ! HC D .@ C e/ continuous:
84
Chapter 3 Partial Differential Equations with Constant Coefficients
Proof. By assumption jF .@ / j;B C0 j j;A for some A 2 .ZnC1 /.sC1/.uC1/ ; B 2 .Z nC1 /.sC1/.uC1/ , some positive constant C0 and for all 2 HA .@ C e/. Since the concept of continuity does not depend on the number of columns and therefore works for all u if it is only assumed to hold for u D 0, it is sufficient to consider the case u D 0. Observing that by the function calculus of spectral theory Fij .@ / j@ C ej ' D j@ C ej Fij .@ /' for all ' in a dense subset of HBi C .@ C e/, 2 ZnC1 , we find that s s ˇX ˇX ˇ2 ˇ2 ˇ ˇ ˇ ˇ Fik .@ / 'k ˇ Dˇ .@ C e/ Fik .@ / 'k ˇ ˇ kD0
;Bi C
;Bi
kD0
s ˇX ˇ2 ˇ ˇ Dˇ Fik .@ / .@ C e/ 'k ˇ
;Bi
kD0
C02
s X
j.@ C e/ 'k j2;Ak
kD0
D C02
s X
j'k j2;Ak C
kD0
follows by continuous extension for all 'k 2 HAk C .@ C e/; k D 0; : : : ; s, with 2 ZnC1 arbitrary. Thus, jF .@ / 'j2;BCG C02
s X s X
j'j2;Ak CGj
j D0 kD0
holds for all ' 2 HACG .@ C e/, where G 2 .ZnC1 /sC1 is arbitrary. For Ak C Gj Ck we can estimate jF .@ / 'j2;BCG C02
s X
j'j2;ACGk C02 .s C 1/j'j2;C :
kD0
This is the case for Gj D inf¹.Ck Ak /jk D 0; : : : ; sº, j D 0; : : : ; s. With this B C G D .Bj C inf¹.Ck Ak /jk D 0; : : : ; sº/j D0;:::;s and B C G C D holds for D WD .sup¹Ck jk D 0; : : : ; sº inf¹Bj jj D 0; : : : ; sº inf¹Ck jk D 0; : : : ; sº C sup¹Ak jk D 0; : : : ; sº/j D0;:::;s .Cj Bj inf¹.Ck Ak /jk D 0; : : : ; sº/j D0;:::;s D C B G:
Section 3.1 Partial Differential Equations in H1 .@ C e/
85
We are led to the following definition. Definition 3.1.13. Let F .i / W RnC1 ! C .sC1/.sC1/ be a matrix of Borel functions Fij .i / W RnC1 ! C, i; j D 0; : : : ; s, and let F .@ / be continuous in H1 .@ C e/, 2 R, s 2 N. If A WD .Ak /kD0;:::;s 2 .N nC1 /.sC1/ is such that9 _ ^ ^ jFkj .i /j C0 .ji C j2 C e/Ak =2 a:e:
C0 2R>0 2RnC1 k;j 2¹0;:::;sº
then we shall say that F .@ / has a regularity loss of at most A . Correspondingly, if ^ ^ _ jFkj .i /j C0 .ji C j2 C e/Ak =2 a:e:
C0 2R>0 2RnC1 k;j 2¹0;:::;sº
we shall speak of a regularity gain of at least A for F .@ /. If there is a possible regularity loss A which is minimal with respect to the order relation (3.1.3), then we shall call A a minimal regularity loss of F .@ / in H1 .@ Ce/, (i.e. there is no regularity loss B with component-wise B A and B ¤ A). A regularity gain A will be called maximal regularity gain of F .@ / in H1 .@ C e/, if A is maximal with respect to the same order relation, (i.e. there is no regularity gain B A with B ¤ A). As a matter of convenience we shall allow for an entry 1, if the F .@ / if the regularity gain or loss is unbounded and still speak of maximal regularity gain and minimal regularity loss. Of course our main interest here is to use these concepts for partial differential operators P .@ C / and their inverses. Let us consider some examples. P Example 3.3. Consider the (negative) Laplacian WD @2 WD nkD0 @2k as a .1 1/-partial differential expression. We find 2 D 02 C C n2
n Y
.j2 C 1/
j D0
showing a regularity loss of at most .2; : : : ; 2/ in H1 .@ C e/. In this Sobolev chain, there is no possibility of improving this estimate. Indeed, assuming 2 D 02 C C n2 C
n Y
.j2 C 1/˛j =2
j D0
with one of the ˛j , say for j D 0, satisfying 0 ˛0 < 2, and setting all but the 0-th component of equal to zero, gives 02 C .1 C 02 /˛0 =2 ; 9 Here the symbol is saying that needs only hold almost everywhere in the sense of the Lebesgue measure in RnC1 . a:e:
86
Chapter 3 Partial Differential Equations with Constant Coefficients
which leads to a contradiction for large arguments 0 2 R. Therefore, the only minimal regularity loss in H1 .@ C e/ is indeed equal to .2; : : : ; 2/. The differential operator @2 C 1 is invertible in H1 .@ C e/ and, in contrast to the minimal regularity loss statement for @2 C 1 above, .@2 C 1/1 has regularity gain of (at least) ˛ for all ˛ 2 N nC1 with j˛j1 D 2. Indeed, for ˛ 2 N nC1 such that j˛j1 D 2 and all x 2 RnC1 we have with y D .x02 ; : : : ; xn2 / n Y
.y C e/˛ D
.xj2 C 1/˛j
j D0 n X n Y
j D0
˛j xk2 C 1
kD0
2
where x 2 WD
.x C 1/2 ;
Pn
2 kD0 xk
is used as an abbreviation. For any ˛ 2 N nC1 with k D
j˛j1 > 2 we cannot have such an estimate, since
.t 2 C1/k=2 .nC1/ t 2 C1
is unbounded10 for t 2
R>0 . Thus, every ˛ 2 N nC1 with j˛j1 D 2 yields a maximal regularity gain for .@2 C 1/1 . Example 3.4. Consider P .@/ WD .@0 @1 C1/2 @21 in R2 . P .@/ is obviously invertible in H1 .@ C e/. P .@/1 features a regularity loss. In fact, we find .x02 C 1/ P .i x0 ; i x1 / 1 showing a regularity loss of at most .2; 0/. Moreover, this estimate implies .x02 C 1/s=2 P .i x0 ; i x1 / .x02 C 1/s=21 ! 0 for all s 2 1; 2Œ. That the lower estimate for such s cannot be improved can be seen by observing that .x02 C 1/s=2 P .i x0 ; i x1 / D .x02 C 1/s=21 for all s 2 1; 2Œ when x1 D
x0 . x02 C1
Furthermore,
.x12 C 1/s=2 P .i x0 ; i x1 / D for all s 2 R when x1 D
x0 x02 C1
1C
s=2 x02 .x02 C1/2 x02 C 1
!0
as x0 ! 1. As a consequence we also find
.x02 C 1/s=2 .x12 C 1/t=2 P .i x0 ; i x1 / ! 0 for all s 2 1; 2Œ; t 2 R, x1 D
x0 x02 C1
and x0 ! 1. Therefore, ˛ D .2; 0/ is the
(only) minimal (multi-index) regularity loss for P .@/1 . 10
This quotient refers to the case x D t e.
Section 3.1 Partial Differential Equations in H1 .@ C e/
87
Example 3.5. Consider the 2 2-partial differential expression11 @2 @0 b P .@0 ; b @/ WD 1 @0 in R1Cn . Then det.P .@0 ; b @// D @20 b @2 is invertible with respect to D .%; 0; : : : ; 0/, % 2 R n ¹0º. We have for every x0 2 R, b x D .x1 ; : : : ; xn / 2 Rn , det.P .i x0 C %; ib x // D .i x0 C %/2 C b x 2 D x02 C %2 C b x 2 C 2 i % x0 and so jdet.P .i x0 C %; ib x //j2 D .x02 %2 b x 2 /2 C 4 %2 x02 D .x02 b x 2 /2 C %4 C 2 x02 %2 C 2 %2 b x2 %4 C 2 x02 %2 C 2 %2 b x2
(3.1.8)
%2 .%2 C 2 x02 C 2b x2/ x2/ %2 .%2 C x02 C b %2 .%2 C xk2 /
for k D 0; : : : ; n. This proves invertibility with respect to all such as well as a @2 /1 of at least ˛ 2 N nC1 with j˛j1 D 1. Moreover, regularity gain for .@20 b x 2 /2 C %4 C 2 x02 %2 C 2 %2 b x 2 / D %2 .%2 C 2 x02 C 2b x2/ ..x02 b D %2 .%2 C 4 x02 / for x02 D b x 2 . Let k; s 2 ¹1; : : : ; nº, k ¤ s, and xk D xs D
p1 2
x0 . Then
.xs2 C 1/1 .1 C xk2 /1 %2 .%2 C 4 x02 / D 4 .2 C x02 /2 %2 .%2 C 4 x02 / ! 0 as x0 ! 1. If one of the indices, say s, is equal to zero then choosing xk D x0 also yields .x02 C 1/1 .1 C xk2 /1 %2 .%2 C 4 x02 / D .1 C x02 /2 %2 .%2 C 4 x02 / ! 0 as x0 ! 1. This shows that the regularity gains of ˛ 2 N nC1 with j˛j1 D 1 for .@20 b @2 /1 are maximal. However, @0 b @2 P .@0 ; b @/1 D .@20 b @2 /1 1 @0 11
Since here b @ abbreviates .@1 ; : : : ; @n /, we have b @2 D
P
2 kD1;:::;n @k .
88
Chapter 3 Partial Differential Equations with Constant Coefficients
features a regularity loss. We consider the absolute value squared corresponding to the matrix entry with the most detrimental regularity property, i.e. .b x 2 /2 .x02 b x 2 /2 C %2 .%2 C 2 .x02 C b x 2 //
.b x 2 /2 1 !1 2 max.1; %4 / .x02 b x 2 /2 C 1 C x02 C b x2
for x02 D b x 2 and x0 ! 1 showing that there can be no regularity gain. On the other hand, by elementary calculus we estimate .b x 2 /2 1 .b x 2 /2 min.1; %4 / .x02 b .x02 b x 2 /2 C %2 .%2 C 2 .x02 C b x 2 // x 2 /2 C 1 C x02 C b x2
1 .1 C x02 /; min.1; %4 /
suggesting a regularity loss of at most .1; 0; : : : ; 0/ for det.P .@0 ; b @//1 b @2 and so, for 1 P .@0 ; b @/ , a (maximal) regularity loss ..1; 0; : : : ; 0/; .0; 0; : : : ; 0//. Example 3.6. Consider the 2 2-partial differential expression @0 @1 P .@0 ; @1 / WD @1 @0 in R1C1 . Then, by the previous example with n D 1, det.P .@0 ; @1 // D @20 @21 is invertible with respect to D .%; 0/, % 2 R n ¹0º, and so we have that P .@0 ; @1 / is also invertible. However, considering i x0 C % i x1 P .i x0 C %; i x1 /1 D .x02 %2 x12 2 i % x0 /1 ; i x1 i x0 C % yields a regularity loss ..0; 0/; .0; 0//. Indeed, we estimate x12 1 2: 2 2 2 2 2 2 2 2% .x0 x1 / C % .% C 2 .x0 C x1 // There is also a maximal regularity gain equal to ..0; 0/; .0; 0//. Example 3.7. Consider P .@0 ; b @/ WD @0 b @2 in R1C.nC1/ , n 2 N. Then P .@0 ; b @/ is invertible with respect to D .%; 0; : : : ; 0/ for % > 0, since jP .i x0 C %; ib x /j2 D x 2 C %/2 %2 > 0. Moreover, x02 C .b x02 C .b x 2 C %/2 min.1; %2 / .x02 C .b x 2 C 1/2 /
Section 3.1 Partial Differential Equations in H1 .@ C e/
89
and x 2 C %/2 x02 C .b x 2 /2 C %2 x02 C .b min.1; %2 / .x02 C 1/u ..b x 2 /2 C 1/v min.1; %
2
/ .x02
u
C 1/
n Y
.xj4 C 1/ˇj v
j D1
min.1; %2 / .x02 C 1/u
n 1 Y 2 .xj C 1/2ˇj v ; 2 j D1
for every ˇ 2 N nC1 ; jˇj1 D 1, u; v 2 ¹0; 1º, u C v D 1. Therefore, all multi-indices of the form ˛ D .˛0 ; 2 / with 2 N n ; j j1 D 1 ˛0 , and ˛0 2 ¹0; 1º are maximal regularity gains for P .@0 ; b @/1 in H1 .@ C e/.
3.1.3 Regularity Loss .0; : : : ; 0/ Of particular interest are ..s C1/.s C1//-partial differential operators P .@ C/ for which the solution operator P .@ C /1 features no regularity loss, because these can be considered as normal operators with zero in the resolvent set in each space HA .@ C e/ of the Sobolev lattice .HA .@ C e//A2.ZnC1 /.sC1/.t C1/ . In this case the inverse P .@ C /1 jHA .@ Ce/ W HA .@ C e/ ! HA .@ C e/ is the inverse of the normal operator P .@ C/jHA .@ Ce/ W DA .P .@ C// HA .@ Ce/ ! HA .@ Ce/. Here DA .P .@ C // WD P .@ C /1 HA .@ C e/ is the domain of the normal operator P .@ C /jHA .@ Ce/ . From the general theory of Sobolev lattices and earlier observations on the independence on the number of columns t of the data, it is clear that it suffices to consider the case of multi-index A D 0 2 .ZnC1 /sC1 , i.e. there is only one column. To have zero in the resolvent set, we would need an estimate of the form jP .@ C /uj;0 c0 juj;0 for some c0 2 R>0 and all u 2 D0 .P .@ C //. Applying the Fourier–Laplace transform, we see that this is equivalent to v uX Z s ˇX ˇ2 u s ˇ ˇ t Pkj .i x C / uj .x/ˇ dx D jP .i m C /uj0 ˇ kD0
RnC1 j D0
c0 juj0 v uX Z u s t D c0 j D0
RnC1
juj .x/j2 dx
90
Chapter 3 Partial Differential Equations with Constant Coefficients
for all u 2 D0 .P .i m C //. We observe that s ˇX s ˇ2 X ˇ ˇ Pkj .i x C / uj .x/ˇ ˇ kD0 j D0
D
s s X X
Pkj .i x C / uj .x/
D
uj .x/
D
j D0
s s X X
Pkj .i x C / Pkm .i x C / um .x/
mD0 kD0
j D0 s X
Pkm .i x C / um .x/
mD0
kD0 j D0 s X
s X
uj .x/
s s X X
Pkj .i x C / Pkm .i x C / um .x/:
mD0 kD0
Thus, we are led to the point-wise condition P .i x C / P .i x C / c02 ;
(3.1.9)
where P .i x C / D .Pj i .i x C / /i;j D1;:::;s . We notice that P .i m C / D P .i m C /, so that (3.1.9) is the (strict) positive definiteness of the partial differential expression P .@ C / P .@ C /. Linear algebra tells us that (3.1.9) can be stated as the uniform positivity of the smallest eigenvalue of P .i x C / P .i x C /, in other words the uniform positivity of the smallest characteristic value of P .i x C/.
3.1.4 Classification of Partial Differential Equations The main prerequisite for our solution theory for a partial differential expression is the requirement to ensure that there are no real zeros of the corresponding polynomial obtained as the determinant of the Fourier–Laplace transformed partial differential expression. We shall say that an ..s C 1/ .s C 1//-partial differential expression P .@/ is well-composed, if there is a 2 RnC1 such that P .@/ is invertible with respect to ; (we shall then say that P .@/ is well-composed with respect to ). Clearly, P .@/ is well-composed with respect to a parameter vector 2 RnC1 , if and only if the problem of finding u 2 H1 .@ C e/ solving P .@/u D f for every given f 2 H1 .@ C e/ is a well-posed problem, i.e. P .@ C /1 exists as a continuous operator on H1 .@ C e/. A problem involving such a P .@/ is also called well-posed in the sense of Petrovskii (with respect to ). As indicated in Remark 3.1.11 there are further distinctions between various types of partial differential operators. They hinge eventually on the issue of causality. Postponing this deeper discussion until later, we would here like to give a preparatory, purely formal classification of well-composed partial differential expressions, the importance of which will only become clear later, when the issue of causality is actually
Section 3.1 Partial Differential Equations in H1 .@ C e/
91
discussed. We mention this background here to motivate the terminology used in the next definition. Definition 3.1.14. Let P .@/ be an ..s C 1/ .s C 1//-partial differential expression 1 in RnC1 and 2 RnC1 n ¹0º. Define .0/ WD jj , the direction of the vector . (a) If there is a %0 2 R such that
det.P .i x C % .0/ // has no real zero for all % 2 Œ%0 ; 1Œ; P .@/ is called evolutionary in direction .0/ . (b) If P .@/ is evolutionary in directions .0/ and .0/ , then P .@/ is called reversibly evolutionary in direction .0/ . (c) If P .@/ is evolutionary in direction .0/ and non-evolutionary in direction .0/ , then P .@/ will be called irreversibly evolutionary in direction .0/ . (d) If P .@/ is for a direction .0/ 2 RnC1 not evolutionary, then P .@/ will be called non-evolutionary in direction .0/ . (e) If P .@/ is non-evolutionary in direction .0/ for all directions .0/ 2 RnC1 , then P .@/ will be called non-evolutionary. We shall illustrate this classification by examples. The classification of systems follows the classification of their determinant. Therefore it suffices to consider scalar partial differential expressions. P Example 3.8. The negative Laplacian @2 D nkD0 @2k is non-evolutionary in RnC1 ; n 2 N>0 . Indeed, .i C % .0/ /2 D 2 C %2 C 2 i % .0/ D 0 for % 2 R>0 is equivalent to ? .0/ and jj D %, a system which always has a real solution in RnC1 for n 2 N>0 . Example 3.9. Consider an ..sC1/.sC1//-partial differential expression P .@/ in R, which simply is an ..s C 1/ .s C 1//-matrix of ordinary differential expressions with constant coefficients. Then, det.P .@// is a scalar ordinary differential expression and Q WD det.P .// a 1-dimensional polynomial. In R the only possibilities for directions are .0/ D ˙1. Let %0 > max¹jRe j j Q. / D 0; 2 Cº. Then Q.i x ˙ %/ D 0 has no real solutions for % 2 Œ%0 ; 1Œ: This proves that in the 1-dimensional case, P .@/ is always reversibly evolutionary. Example 3.10. Q.@0 ; b @/ D @20 b @2 in R1C.nC1/ , n 2 N, where b @ D .@1 ; : : : ; @nC1 / Choosing .0/ D .˙1; 0; : : : ; 0/, we see from estimate (3.1.8) that Q.@0 ; b @/ is reversibly evolutionary in direction .0/ . Let us now consider a general direction .0/ WD ..0/;0 ; .1/ / WD ..0/;0 ; .0/;1 ; : : : ; .0/;nC1 / 2 R1C.nC1/
92
Chapter 3 Partial Differential Equations with Constant Coefficients
with .1/ ¤ 0. Then for % 2 R>0 Q.i x0 C % .0/;0 ; i b x C % .1/ / x C % .1/ /2 D .i x0 C % .0/;0 /2 .ib 2 D x02 C 2 i % x0 .0/;0 C b x 2 2 i %b x .1/ %2 .j.1/ j2 .0/;0 / 2 D x02 C b x 2 %2 .j.1/ j2 .0/;0 / C 2 i % .x0 .0/;0 b x .1/ /:
This is zero if and only if x0 2 R and b x 2 RnC1 satisfy x0 .0/;0 D b x .1/ ; 2 b x 2 D x02 C %2 .j.1/ j2 .0/;0 /:
Since b x D .b x .1/ / j.1/ j2 .1/ C with ? .1/ , for this to hold we require 2 .b x .1/ /2 j.1/ j2 C 2 D x02 .0/;0 j.1/ j2 C 2
and 2 2 0 D x02 x02 .0/;0 j.1/ j2 2 C %2 .j.1/ j2 .0/;0 / x02 2 D .j.1/ j2 .0/;0 / %2 C 2: j.1/ j2 2 2 This is possible if and only if j.1/ j2 .0/;0 0. Since j.1/ j2 C .0/;0 D 1, the 1 b latter is equivalent to j.0/;0 j p . In other words, Q.@0 ; @/ is non-evolutionary for
all directions v.0/ with j.0/;0 j j.0/;0 j > sets
p1 2
2 p1 . 2
In all other directions .0/ , i.e. directions with (and therefore j.1/ j < p1 ), Q.@0 ; b @/ is reversibly evolutionary. The 2
¹.t; x/ 2 R1C.nC1/ jt 2 D x 2 ^ ˙t 0º D ¹.t; x/ 2 R1C.nC1/ jt D ˙jxj º describe a double cone12 hypersurface. This double cone separates the regions of the parameter vector 2 R1C.nC1/ , where Q.@0 ; b @/ D Q..@0 ; b @/ C / is invertible (reversibly evolutionary in direction jj ) from those where it is non-invertible (non@/ C / evolutionary in direction ). For on the double cone hypersurface Q..@0 ; b jj
is also non-invertible (and if ¤ 0, also non-evolutionary in direction
). jj
12 The two cones are closely connected to the so-called forward or backward light-cone, respectively, although their discussion here occurs after Fourier–Laplace transformation.
Section 3.1 Partial Differential Equations in H1 .@ C e/
93
Example 3.11. Consider P .@0 ; b @/ WD @0 b @2 in R1C.nC1/ , n 2 N, where b @ D .@1 ; : : : ; @nC1 /. Then from Example 3.7 we see that P .@0 ; b @/ is evolutionary with respect to the direction .1; 0; : : : ; 0/. Let us consider an arbitrary direction .0/ WD ..0/;0 ; .1/ / WD ..0/;0 ; .0/;1 ; : : : ; .0/;nC1 / 2 R1C.nC1/ with .1/ ¤ 0. Then x C % .1/ / D i x0 C % .0/;0 .ib x C % .1/ /2 P .i x0 C % .0/;0 ; ib D i x0 C % .0/;0 C b x 2 2 i %b x .1/ %2 j.1/ j2 D i .x0 2 %b x .1/ / .%2 j.1/ j2 b x 2 % .0/;0 /: As in the Example 3.10, for this to have a real zero we need 1 b x 2 D x02 %2 j.1/ j2 C 2 4 and hence
1 %2 j.1/ j2 % .0/;0 D x02 %2 j.1/ j2 C 2 ; 4 which is only possible for 1 %2 j.1/ j2 % .0/;0 x02 %2 j.1/ j2 0: 4 @/ can be evolutionary only in directions for which %j.1/ j2 This shows that P .@0 ; b .0/;0 < 0 for all sufficiently large % 2 R>0 . Therefore, .1; 0; : : : ; 0/ is the only direction in which P .@0 ; b @/ is evolutionary and so P .@0 ; b @/ is irreversibly evolutionary in this direction. Example 3.12. The system of Example 3.6 belongs to a particular class of partial differential expressions, the so-called symmetric hyperbolic systems. They are defined as ..s C 1/ .s C 1//-partial differential expressions in R1C.nC1/ of the form E0 @0 C A.b @/ C B
P where A.i / D i nC1 kD1 Ak k with Ak , k D 1; : : : ; n C 1, real symmetric, which yields a skew-selfadjoint matrix A.i / for every 2 RnC1 , where B 2 R.sC1/.sC1/ is arbitrary and E0 is assumed to be a positive definite matrix. Now consider the matrix .i C %/ E0 C A.i / C B where ; % 2 R and 2 RnC1 . We find Re..i C %/ x E0 x C x A.i / x C x B x/ D % x E0 x C x Re.B/ x
(3.1.10)
94
Chapter 3 Partial Differential Equations with Constant Coefficients
for all x 2 C sC1 . Using the Cauchy–Schwarz inequality we obtain .j%j "0 ˇ0 /jxj j..i C %/ E0 C A.i / C B/xj for all 2 R; 2 RnC1 ; x 2 C sC1 , where "0 is the smallest eigenvalue of E0 and ˇ0 the largest absolute value of the eigenvalues of Re.B/ D 12 .B C B /. As a consequence, .i C%/ E0 CA.i /CB must be non-singular for all 2 R; 2 RnC1 and therefore invertible for all D .%; 0; : : : ; 0/; % 2 R; j%j > ˇ"00 . Thus, symmetric hyperbolic systems are in particular reversibly evolutionary in direction .1; 0; : : : ; 0/ and their solution operator features no regularity loss. Remark 3.1.15. It is worth noting that the above classification scheme also contains possibly surprising limit cases. For example, any P .@/ that is invertible with respect to D .0; : : : ; 0/, where P is a polynomial in .n C 1/ variables, can be considered as reversibly evolutionary with respect to direction .0; : : : ; 0; 1/ if P is considered as a polynomial in .n C 2/ variables. Indeed, considering P as a polynomial function in .n C 2/ variables, i.e. considering Q W C nC1 C ! C; .x0 ; : : : ; xn ; xnC1 / 7! P .x0 ; : : : ; xn / we find that .x0 ; : : : ; xn ; xnC1 / 7! Q.i x0 ; : : : ; i xn ; i xn C %/ does not have any real zeros for arbitrary % 2 R. Thus, Q.@/ is evolutionary in direction .0; : : : ; 0; ˙1/.
3.1.5 The Classical Classification of Partial Differential Equations Although in our approach we shall not need to use the classical classification into elliptic, parabolic and hyperbolic differential expressions, it is instructive to establish the relationship to the above classification. At the same time it is convenient to expand our ‘jargon’ and to link up with other sources of reference. In order to make these connections we first have to gain more insight into the concept of ‘evolutionarity’. The question of finding real zeros can be reduced to the discussion of real zeros of 1-dimensional polynomials considering the other variables merely as parameters. Let P be a polynomial in .n C 1/ variables, n 2 N. Then using x D .x .0/ / .0/ C
with WD .x .x .0/ / .0/ / ? .0/ we may write P .i x C % .0/ / D P .i . .0/ C //; with WD x .0/ i %. Noting that, conversely, if for a given ? .0/ there is a root of 7! P .i . .0/ C // with I m D %, then x WD .Re / .0/ C is a real root of x 7! P .i x C % .0/ /, we obtain the following characterization.
Section 3.1 Partial Differential Equations in H1 .@ C e/
95
Proposition 3.1.16. P .@/ is evolutionary in direction .0/ if and only if all roots of 7! P .i . .0/ C // have imaginary parts which are bounded below uniformly with respect to all ? .0/ ; 2 RnC1 . In other words, defining ° ˇ _ ± ˇ P .i . .0/ C // D 0 ; N.P .i /; .0/ / WD ˇ ?.0/
we must have I mŒN.P .i /; .0/ / bounded below. P .@/ is reversibly evolutionary in direction .0/ if and only if I mŒN.P .i /; .0/ / is also bounded above, i.e. I mŒN.P .i /; .0/ / is a bounded set, otherwise P .@/ is irreversibly evolutionary. P .@/ is non-evolutionary in direction .0/ if and only if I mŒN.P .i /; .0/ / is neither bounded below nor above. Proof. The result is clear from the transformation: RnC1 ! R ˚ ¹.0/ º? ; x 7! .x .0/ ; x .x .0/ / .0/ / with R ˚ ¹.0/ º? ! RnC1 ; .s; / 7! s .0/ C
as inverse. Remark 3.1.17. We note that the requirements on can be relaxed to admitting all
2 RnC1 . Indeed, if is a zero of 7! P .i . .0/ C // for an arbitrary 2 RnC1 , then C . .0/ / is a root of 7! P .i . .0/ C . . .0/ / .0/ ///, where now . . .0/ / .0/ / ? .0/ . Moreover, I m D I m. C . .0/ //, so that the characterization given in the proposition does not change. P If P .x/ WD ˛ a˛ x ˛ is a polynomial of degree #P , then P .i . .0/ C // D
X ˛
a˛ .i . .0/ C //˛ D
#P X
Qk . / k
kD0
where Qk is a polynomial of degree at most equal to #P k for k D 0; : : : ; #P . Here kŠ Qk . / equals the k-th derivative of the polynomial 7! P 1-dimensional ˛ . The polynomial P .i . .0/ C //, in particular Q#P . / D i#p a ˛ j˛j1 D#P .0/ part of P of highest order is called the principal part Pmain . With this notion we have that Q#P . / is independent of and equal to Pmain .i .0/ / D i#P Pmain ..0/ /. It may happen that Pmain ..0/ / D 0 in which case the direction .0/ is called a characteristic direction, otherwise it is called a non-characteristic direction of P .@/. The same terminology is used for square systems P .@/, if det.P .@//main has such directions.
96
Chapter 3 Partial Differential Equations with Constant Coefficients
Proposition 3.1.18. Let P .@/ be a scalar partial differential expression evolutionary in a non-characteristic direction .0/ . Then Pmain .@/ is reversibly evolutionary in direction .0/ . Moreover, N.Pmain .i /; .0/ / R. P ˛ Proof. Let P .@/ D ˛ aa @ . Since .0/ is assumed to be non-characteristic, the degree #P .i . .0/ C// of 7! P .i . .0/ C // is equal to #P and the leading coefficient of 7! P .i . .0/ C // is just i#P Pmain ..0/ / ¤ 0. The proposition follows if we can show that all roots of 7! Pmain . .0/ C / are real. For D 0 we have Pmain . .0/ C / D Pmain . .0/ / D Pmain ..0/ / #P , i.e. D 0 is the only root and has multiplicity #P . So, let ¤ 0 and define p.; / WD P .i . .0/ C //: Then, with % WD j j, WD %1 , WD %1 , we get p.; / D
#P X
pk .; / D
kD0
#P X
%k pk . ; /;
kD0
P
where pk .; / WD Pk .i. .0/ C // WD i k j˛jDk a˛ . .0/ C /˛ are homogeneous polynomials of degree k. In particular, we have p#P D i#P Pmain and our task is to show that p#P .; / D 0 has only real roots. For this we consider q% . ; / WD %#P p.; / D
#P X
%k#P pk . ; /
kD0
D p#P . ; / C
(3.1.11)
#P 1 #P k1 1 1 X pk . ; / % % kD0
and observe that q% . ; / ! p#P . ; / as % ! 1. Moreover, using the fact that the roots of a polynomial may be arranged to be continuous functions of the coefficients, we may consider the family of roots . k .1=%; //kD1;:::;#P of q% . ; / (with possible repetitions according to their multiplicity) as a family of continuous functions .s; / 7! k .s; / in .R n ¹0º/ RnC1 . Now, we achieve the result by an indirect argument. Assume there is a real 0 ? .0/ , j0 j D 1, such that we have a root 0 of p#P . ; 0 / D 0 with I m 0 ¤ 0. Since p#P . ; 0 / D .1/#P p#P . ; 0 / we may assume that I m 0 < 0. With the continuous dependence of the roots on the coefficients, see e.g. [21, appendix], we must therefore have a k 2 ¹1; : : : ; #P º such that k .s; 0 / ! 0 as s ! 0. A-fortiori we have I m k .s; 0 / ! I m 0 as s ! 0. Observing that % WD % k .1=%; 0 / is a root of p.; %0 / D P .i . .0/ C %0 // we have I m % % I m 0 D o.%/ as % ! 1
Section 3.1 Partial Differential Equations in H1 .@ C e/
97
and so, recalling I m 0 < 0 we see that I mŒN.P . i /; .0/ / cannot be bounded below, contradicting the assumption of P .@/ being evolutionary. Remark 3.1.19. A scalar partial differential expression satisfying the assumption of this proposition is called hyperbolic in the sense of Gårding or Gårding hyperbolic or – briefly – hyperbolic in direction .0/ . Correspondingly, an ..sC1/.sC1//-partial differential expression is called Gårding hyperbolic in direction .0/ if its determinant is Gårding hyperbolic in direction .0/ . As a by-product of the proof we also have the following consequence. Corollary 3.1.20. Let P .@/ D Pmain .@/ be evolutionary in a non-characteristic direction .0/ then P .@/ is reversibly evolutionary in direction .0/ . Remark 3.1.21. The proposition and the corollary state that every partial differential expression which is Gårding hyperbolic in direction .0/ has a Gårding hyperbolic principal part and that the principal part is evolutionary in a non-characteristic direction .0/ , i.e. Gårding hyperbolic in direction .0/ , if and only if it is reversibly evolutionary in this direction. The converse holds under an additional assumption. Proposition 3.1.22. Let P .@/ be a scalar partial differential expression such that .0/ is a non-characteristic direction for P .@/ and 7! Pmain . .0/ C / has only simple zeros for all ? .0/ ; 2 RnC1 n ¹0º. Then P .@/ is reversibly evolutionary in direction .0/ . Proof. The assumed simplicity of the roots implies (by the inverse function theorem) the smooth dependence of the roots on the coefficients of the polynomial, compare e.g. [21, appendix]. In particular, we have Lipschitz continuous dependence, a fact we will have to utilize in proving this result. We shall use the notations of the proof of the previous proposition. So let a normalized 0 ? .0/ be given and let 0 be a solution of p#P . ; 0 / D 0. By assumption we have I m 0 D 0. Therefore, we must have a k 2 ¹1; : : : ; #P º such that I m k .s; 0 / ! I m 0 D 0 as s ! 0. By the Lipschitz continuity of .s; 0 / 7! k .s; 0 / we get jI m k .t; 0 / I m k .s; 0 /j C jt sj for some C 2 R>0 uniformly with respect to 0 2 RnC1 , j0 j D 1, and letting s ! 0 jI m k .t; 0 /j D jI m k .t; 0 / I m 0 j C jt j:
(3.1.12)
Recalling that % WD % k .1=%; 0 / is a root of p.; %0 / D P .i . .0/ C %0 // D 0 we have, using (3.1.12), with some generic constant C 2 R>0 jI m % j D j% I m k .1=%; 0 /j C
for all % 2 R>0 :
98
Chapter 3 Partial Differential Equations with Constant Coefficients
Observing that any root of p.; / for 2 RnC1 n BRC1 .0; %0 / yields such 0 and 0 , the desired boundedness of I mŒN.P .i /; .0/ / follows. Remark 3.1.23. The assumptions of this proposition give rise to the concept of strict hyperbolicity. An ..s C 1/ .s C 1//-partial differential expression is called strictly hyperbolic in direction .0/ if .0/ is a non-characteristic direction of det.P .@// and 7! det.P /main . .0/ C / has only simple zeros for all ? .0/ ; 2 RnC1 n ¹0º. That the additional assumption is not merely technical is shown in the following example, where a Gårding hyperbolic principal part does not yield an evolutionary partial differential expression due to multiple roots of the principal part. Considering only the principal part is therefore not sufficient to establish well-(com)posedness. Example 3.13. Let P .@0 ; @1 / WD .@0 @1 /2 @1 . We see that Pmain .1; 0/ D 1, i.e. .1; 0/ is a non-characteristic direction of P .@0 ; @1 /. Pmain .@0 ; @1 / D .@0 @1 /2 is reversibly evolutionary (as the square of the reversibly evolutionary (and strictly hyperbolic) operator @0 @1 ) and so also hyperbolic (but not strictly hyperbolic) in this direction. Investigating the zeros of .ix0 C%i x1 /2 i x1 yields .x0 x1 /2 D %2 and 2 % .x0 x1 / D x1 D ˙2 %2 . Thus, x1 D ˙2 %2 and x0 D ˙% C x1 provides 4 real zeros for every % 2 R n ¹0º showing that P .@0 ; @1 / is not invertible with respect to arbitrary 2 R ¹0º. That there are reversibly evolutionary problems which are not Gårding hyperbolic can be seen by the following example. Example 3.14. Consider P .@0 ; b @/ WD @0 C i b @2 , b @ D .@1 ; : : : ; @n /, the so-called Schrödinger operator. Clearly, Pmain .1; 0; : : : ; 0/ D 0 and so .1; 0; : : : ; 0/ is a characteristic direction of @0 C i b @2 . However, i x0 C % i x 2 has no real zeros for % ¤ 0. 2 b Therefore, @0 C i @ is reversibly evolutionary in direction .1; 0; : : : ; 0/. We have restricted our attention to the scalar case, since the classification is expressed in terms of determinants. We will, however, summarize our findings for general ..s C 1/ .s C 1//-partial differential expressions. Theorem 3.1.24. Let P .@/ be an ..s C 1/ .s C 1//-partial differential expression. P .@/ is evolutionary in a non-characteristic direction .0/ if and only if P .@/ is hyperbolic in this direction. If det.P .@//main is strictly hyperbolic in direction .0/ then P .@/ is hyperbolic in this direction. Proof. The result follows immediately from the definitions of the terms (reducing the issues to the scalar case) and the previous propositions. Proposition 3.1.16 shows how trying to characterize being reversibly evolutionary in terms of terms of imaginary parts of roots can fail. Clearly, I mŒN.P .i /; .0/ /
Section 3.1 Partial Differential Equations in H1 .@ C e/
99
has to be unbounded for P .@/ not to be reversibly evolutionary in direction .0/ . We define N.P .i . .0/ C /// WD ¹ jP .i . .0/ C // D 0º
(3.1.13)
and [
N.P; .0/ ; r/ WD
N.P .i . .0/ C ///
?.0/ ;jjr
° ˇ _ ± ˇ D ˇ det.P /.i . .0/ C // D 0 ^ j j r :
(3.1.14)
?.0/
Then N.P .i . .0/ C /// D
[ r2R0
D
[
N.P; .0/ ; r/ [
N.P .i . .0/ C ///:
r2R0 ?.0/ ;jjr
Definition 3.1.25. Let P .@/ be irreversibly evolutionary in a characteristic direction .0/ . If additionally inf I mŒN.det.P /; .0/ ; r/ ! 1
as r ! 1
(3.1.15)
then we shall call P .@/ purely irreversibly evolutionary in direction .0/ . If the polynomial 7! det.P .i . .0/ C /// is non-constant and has a highest order term with a coefficient, which is independent of , then P .@/ is called parabolic in direction .0/ . Remark 3.1.26. The additional condition (3.1.15) serves to avoid contamination with reversibly evolutionary parts, i.e. to avoid the possibility that some roots may have bounded imaginary parts. Generally, the highest order term of the 1-dimensional polynomial 7! det.P /.i . .0/ C // will depend on , in fact even the degree #det.P /.i . .0/ C// of 7! det.P /.i . .0/ C // may depend on . We have that the leading coefficient is given by the polynomial Q#P .i . .0/ C// . / D
1 #det.P /.i . .0/ C// Š
.#det.P /.i . .0/ C// /
. 7! det.P /.i . .0/ C ///
For the parabolic case it is assumed that this is a constant.
:
100
Chapter 3 Partial Differential Equations with Constant Coefficients
A final case of failing to be evolutionary is the following. Definition 3.1.27. Let P .@/ be a ..s C 1/ .s C 1//-partial differential expression that is non-evolutionary in all direction .0/ and let .I mŒN.det.P /; .0/ ; r//˙ WD I mŒN.det.P /; .0/ ; r/ \ ˙ŒR>0 : If additionally inf .I mŒN.det.P /; .0/ ; r//C ! 1 and
sup .I mŒN.det.P /; .0/ ; r// ! 1
(3.1.16)
as r ! 1, then we shall call P .@/ purely non-evolutionary. If, additionally, all directions .0/ are non-characteristic, then P .@/ is called elliptic. Note that, since P .@/ is non-evolutionary, the sets .I mŒN.det.P /; .0/ ; r//˙ must be non-empty and are obviously bounded below or above, respectively, by zero. Example 3.15. P .@0 ; @1 / WD @20 @21 is purely non-evolutionary and also elliptic (note P ..0/ / D Pmain ..0/ / D 1 for all directions .0/ ). Example 3.16. P .@0 ; @1 / WD @20 C @41 is purely non-evolutionary, but not elliptic (note Pmain .1; 0/ D 0). Elliptic operators are particularly easy to characterize. Proposition 3.1.28. Let P .@/ be an ..s C 1/ .s C 1//-partial differential expression in .n C 2/ variables, n 2 N. Then P .@/ has no characteristic directions if and only if P .@/ is elliptic. Proof. If P .@/ is elliptic, then by definition there are no characteristic directions. We need to show that it suffices to require all directions to be non-characteristic in order to have an elliptic partial differential expression, (in other words, we need to demonstrate that the conditions on the imaginary parts of the roots are superfluous). Without loss of generality we may again restrict our attention to the scalar case. So, let Pmain ..0/ / ¤ 0 for all directions .0/ , i.e. for all .0/ 2 RnC2 with j.0/ j D 1. Let ˇ _ ° ± ˇ 0 WD inf jI m j ˇ .jj D 1 ^ Pmain . .0/ C / D 0/ : ?.0/
We claim that 0 > 0. Assume the contrary, i.e. let there be a sequence of zeros . k /k and normalized parameter vectors .k /k such that I m k ! 0 as k ! 1. Without loss of generality we may assume that this sequence .k /k is convergent to some 1 (otherwise pick a convergent subsequence). Since by continuous dependence
Section 3.1 Partial Differential Equations in H1 .@ C e/
101
of the corresponding roots we must also have boundedness of the sequence of corresponding zeros . k /k , we may likewise assume without loss of generality that . k /k is convergent to some 1 . Going to the limit we have Pmain . 1 .0/ C 1 / D 0
C
1 .0/ and therefore j 1 as a characteristic direction, contradicting the assumption. 1 .0/ C 1 j This confirms that 0 > 0. Defining analogously ˇ _ ° ± ˇ 1=% WD inf jI m j ˇ .jj D 1 ^ q% . ; / D 0/
?.0/
we have by the continuous dependence of the roots, which carries over to the infimum of the mapping s 7! s that 1=% ! 0 as % ! 1. In particular, we have 1=% 0 =2 for all sufficiently large % 2 R. As in the proof of Proposition 3.1.18, we conclude that the roots of P .i . .0/ C // satisfy 1 jI m j % 1=% % 0 2 and therefore jI mŒN.P; .0/ ; r/j ! 1 as r ! 1:
(3.1.17)
Ellipticity in the sense of our definition follows if we can show that .I mŒN.det.P /; .0/ ; r//˙ are both non-empty. Noting that by homogeneity Pmain .i . ..0/ / // D .1/#P Pmain .i . .0/ C // we see that both signs must occur for the imaginary parts of the roots. By continuous dependence of the roots on the coefficients the same must hold true for all sufficiently large j j and so .I mŒN.det.P /; .0/ ; r//˙ are non-empty sets and, using (3.1.17) we get inf .I mŒN.det.P /; .0/ ; r//C ! 1; sup .I mŒN.det.P /; .0/ ; r// ! 1; as r ! 1.
102
Chapter 3 Partial Differential Equations with Constant Coefficients
Remark 3.1.29. This last result could motivate the definition of ellipticity as P .@/ having no characteristic directions, which is actually the usual way. For systematic reasons the above definition appears to be more appropriate. Example 3.17. The partial differential expression @1 C i @2 in R2 is called the Cauchy–Riemann operator13 . Since there are no non-vanishing real roots .x1 ; x2 / of i x1 x2 , we see that @1 C i @2 is elliptic. Likewise .@1 C i @2 /2 is elliptic. Both differential expressions are non-invertible with respect to all 2 R2 . So this example gives an elliptic operator for which our solution theory utterly fails. This is not too surprising since the spectrum of @1 C i @2 as a normal operator in L2 .R2 / is all of C, so that a reasonable solution theory cannot be expected in this framework. We shall later develop an extension of our theory which in some instances gives a possibility of solutions in the spectrum, see Subsection 3.2. How bad such operators really can be, even ignoring our special approach, is seen from the following related example by Bitsadze (1953). Let f be analytic in a neighborhood U of the closed unit disc in R2 , i.e. f is smooth and satisfies .@1 C i @2 /f D 0 (classically) in U . Then u defined as the mapping .x; y/ 7! .1 x 2 y 2 / f .x; y/ solves the equation14 .@1 C i @2 /2 u D 0 and u satisfies the homogeneous Dirichlet boundary condition u D 0 on the boundary of the unit disc. Thus, since f is arbitrary, this seemingly simple boundary value problem has already an infinite number of solutions. In order to avoid pathologies like Bitsadze’s example other concepts of ellipticity have been devised. Definition 3.1.30. A scalar elliptic partial differential expression P .@/ is called strictly elliptic if P .@/ is elliptic and the number of zeros of P . .0/ C / (counted according to their multiplicities) with positive imaginary parts is equal to the number with negative imaginary parts for all directions .0/ and all ? .0/ with j j sufficiently large. Equivalently, we could have said that the number of zeros of Pmain . .0/ C / (counted according to their multiplicities) with positive imaginary parts is equal to the number with negative imaginary parts for all directions .0/ and all ? .0/ ; ¤ 0. P .@/ strictly elliptic implies that P must be of even order. Therefore, @1 C i @2 cannot be strictly elliptic, but also .@1 C i @2 /2 cannot be strictly elliptic because it inherits this failure from @1 C i @2 . 13 Recall that functions u satisfying .@ C i @ /u D 0 in the classical sense in a domain of R2 are 1 2 known as analytic functions in . Using the complex notation .x; y/ 7! x C i y the Cauchy–Riemann operator is often written as @z@ WD .@1 C i @2 / with z D x C i y and so z D x i y. 14 Equivalently, in complex notation this is . @ /2 u D 0. Note that @ .1 jzj2 / D z is analytic. @z @z
Section 3.1 Partial Differential Equations in H1 .@ C e/
103
Remark 3.1.31. It is remarkable that strict ellipticity is really only needed for the two-dimensional case. Lopatinski has shown, see e.g. [24], that for n 3 every elliptic differential expression is strictly elliptic. The proof is based on a homotopy argument, which we briefly describe here. Let P .@/ be elliptic and assume that Pmain ..0/ C / has different numbers of zeros with positive and negative imaginary parts for a particular direction .0/ and some ? .0/ ; jj D 1. Then, by homogeneity, Pmain ..0/ / must have the reverse discrepancy. If we can find a continuous connection between C and , then we must have a vanishing imaginary part of a zero of the polynomial for some other value of contradicting the ellipticity. Indeed, we may take a continuous family of rotations15 .A t / t2Œ0;1 leaving the direction .0/ invariant and such that A0 D and A1 D . Let . k .A t //kD1;:::;#P be the continuous family of roots of Pmain ..0/ C A t /. Then at least one of them must change sign to account for the reversal of the discrepancy between roots with positive and negative imaginary parts and by continuity this root must have a vanishing imaginary part for some intermediate t 2 0; 1Œ. This would yield a real root contradicting the assumed ellipticity. The reasoning of the last remark also yields the following statement. Proposition 3.1.32. Let P .@/ be a scalar elliptic partial differential expression. Then P .@/ is strictly elliptic if and only if the number of zeros of Pmain . .0/ C / (counted according to their multiplicities) with positive imaginary parts is equal to the number with negative imaginary parts for some direction .0/ and some ? .0/ , ¤ 0. Proof. Let the number of roots of Pmain . .0/ C / with positive imaginary parts equal the number of those with negative imaginary parts. We may assume without loss of generality that j j D 1. Now, let .A t / t2Œ0;1 be a continuous family of rotations turning .0/ into e .0/ and into e
. Then the continuity argument16 of Remark 3.1.31 yields that the equality of the number of roots with positive and negative imaginary parts must be maintained. We found that the principal part of a scalar partial differential expression P .@/ has no characteristic direction if and only if P .@/ is elliptic. This non-vanishing of Pmain on the unit sphere can be enforced by the following stronger condition. Definition 3.1.33. A scalar elliptic partial differential expression P .@/ in RnC1 is called strongly elliptic if P .@/ is elliptic and there is a 2 C such that Re. Pmain .x// > 0 15
for all x 2 RnC1 n ¹0º:
This is where the dimension n 3 comes into play. Note that this rotation can be done in dimensions greater or equal to 2. Moreover, recall that all ordinary differential expressions with constant coefficients are reversibly evolutionary. Thus, there are no 1-dimensional elliptic operators. This contradicts some statements found in the literature where ellipticity is characterized by regularity properties. 16
104
Chapter 3 Partial Differential Equations with Constant Coefficients
Remark 3.1.34. Clearly, we may normalize such a number to have absolute value 1. By the homogeneity of the polynomial Pmain it would also be sufficient to impose this condition on the unit sphere in RnC1 . It can be shown that strong ellipticity implies strict ellipticity. In two dimensions the two concepts actually coincide. That this is not the case in higher dimensions can be seen from the example @41 C @42 @43 C i .@21 C @22 /@23 which turns out to be strictly elliptic but not strongly elliptic, see [24].
3.1.6 Elliptic, Parabolic, Hyperbolic? The terms ‘elliptic’, ‘parabolic’, ‘hyperbolic’ have their historical origin in the discussion of scalar, second order, partial differential expressions in R2 . In this subsection we want to indicate, the origin of this choice of terminology. The partial differential expression with real scalar coefficients P .@1 ; @2 / WD a11 @21 2 a12 @1 @2 a22 @22 is Fourier transformed into the quadratic form q.x1 ; x2 / WD a11 x12 C 2 a12 x1 x2 C a22 x22 a11 a12 x1 D . x1 ; x2 / : a12 a22 x2
a12 If A WD aa11 is positive (or negative) definite, i.e. the matrix has two eigenvalues 12 a22 of equal sign, then this quadratic form is the highest order term in the equation of an ellipse and P .@1 ; @2 / is called elliptic. If A has two eigenvalues of opposite sign, then q is the highest order term of the equation of a hyperbola. In this case there is a characteristic direction, since q.x/ D x > A x assumes both signs (e.g. for the two normalized eigenvectors) and must therefore vanish for a direction .0/ (compare the argument in the proof of Proposition 3.1.32). Since it is a homogeneous, quadratic form we have q.˙.0/ / D 0. These are the only direction where the quadratic form is zero and any other direction .0/ is therefore non-characteristic. To show this, we calculate q. .0/ C / D a11 . .0/;1 C 1 /2 C 2 a12 . .0/;1 C 1 /. .0/;2 C 2 / C a22 . .0/;2 C 2 /2 > D q..0/ / 2 C 2 .0/ A C q. / > and so if q. / q..0/ / ..0/ A /2 < 0 we would have two different real roots which would make P .@1 ; @2 / strictly hyperbolic. In the present two dimensional case we have 2 ŒR..0/;2 ; .0/;1 / for ? .0/ .
Section 3.1 Partial Differential Equations in H1 .@ C e/
105
Then, with j.0/ j D 1, > q. / q..0/ / ..0/ A /2 2 2 D .a11 .0/;1 C 2 a12 .0/;1 .0/;2 C a22 .0/;2 / 2 2 .a11 .0/;2 2 a12 .0/;1 .0/;2 C a22 .0/;1 / 2 2 a12 .0/;2 C a22 .0/;2 .0/;1 /2 .a11 .0/;1 .0/;2 C a12 .0/;1
D det.A/ < 0; for all ¤ 0. Finally, if A has one non-vanishing eigenvalue and one eigenvalue equal to zero, then q is as we shall see the leading term of the equation of a parabola. Let x.0/ 2 R2 > x/ C be the normalized eigenvector for the eigenvalue 0. Then, with x D x.0/ .x.0/ b x .0/ .b x> x/ where b x .0/ WD .x.0/;2 ; x.0/;1 /> , we get .0/ 2 q.x/ D x > A x D b x> x .0/ .b x> .0/ Ab .0/ x/ D q.x i s x.0/ /
(3.1.18)
for all s 2 R. Since eigensolutions for different eigenvalues are orthogonal, b x .0/ is the other eigenvector and 1 D b x> Ab x ¤ 0 the corresponding eigenvalue. With re.0/ .0/ gards to P .@1 ; @2 / the direction x.0/ is characteristic. Therefore with D .1 ; 2 / D % x.0/ ; % ¤ 0, recalling Ax.0/ D 0, i.e. a11 x.0/;1 C a12 x.0/;2 D a12 x.0/;1 C a22 x.0/;2 D 0, we get P .@/ D P .@ C % x.0/ / D a11 .@1 ;1 C 1 /2 2a12 .@1 ;1 C 1 /.@2 ;2 C 2 / a22 .@2 ;2 C 2 /2 D P .@ / C %2 P .x.0/ / 2% .a11 x.0/;1 @1 ;1 C a12 x.0/;1 @2 ;2 C a12 x.0/;2 @1 ;1 C a22 x.0/;2 @2 ;2 / D P .@ / 2% ..a11 x.0/;1 C a12 x.0/;2 /@1 ;1 C .a12 x.0/;1 C a22 x.0/;2 /@2 ;2 / D P .@ /;
(3.1.19)
> and using column vector notation we can decompose .@1 ;1 @2 ;2 /> D x.0/ .x.0/ @ /C > b x .0/ .b x .0/ @ / to find that
P .@ / D .@1 ;1 @2 ;2 / A .@1 ;1 @2 ;2 /> D .@1 ;1 @2 ;2 / A .x.0/;2 x.0/;1 /> ..x.0/;2 x.0/;1 /@ / D 1 ..x.0/;2 x.0/;1 /@ /2 :
(3.1.20)
106
Chapter 3 Partial Differential Equations with Constant Coefficients
This shows that the principal part is just a multiple of a second order directional derivative. Consider now the more general second order partial differential expression Q.@/ D P .@ / C a> @ C b, a 2 C 2 (written as a column vector) and b 2 C associated with P .@/. From (3.1.19) and (3.1.20) we see Q.@ C / D P .@ / C a> .@ C / C b D 1 ..x.0/;2 ; x.0/;1 /@ /2 C a> .@ C / C b: After Fourier–Laplace transformation we are led to consider the polynomial y 7! 1 ..x.0/;2 ; x.0/;1 / y/2 C a> .i y C / C b: We see that we would have real zeros if I m a> y C Re a> C Re b C 1 ..x.0/;2 ; x.0/;1 / y/2 D 0
(3.1.21)
and Re a> y C I m a> C I m b D 0:
(3.1.22)
Representing a as a linear combination of x.0/ and b x .0/ WD .x.0/;2 ; x.0/;1 / we obtain > a D a0 C b a0 with a0 WD x.0/ .x.0/ a/; b a0 WD b x> .b x .0/ a/ and so from (3.1.21) we .0/ obtain 0 D I mb a> a> x .0/ y/2 C Re a0> I m a0> y 0 y C Reb 0 C Re b C 1 .b D I m.b x .0/ a/b x> x .0/ y/2 I m.x.0/ a/ x0> y C % Re.x.0/ a/; 0 y C Re b C 1 .b and from (3.1.22) we get > > x> 0 D Re.b x .0/ a/b 0 y C Re.x.0/ a/ x0 y C I m.x.0/ a/ x0 C I m b > D Re.b x .0/ a/b x> 0 y C Re.x.0/ a/ x0 y C % I m.x.0/ a/ C I m b:
If Re.x.0/ a/ ¤ 0, then we can eliminate x0> y D and obtain
I m.b x .0/ a/ I m.x.0/ a/
Re.b x .0/ a/b x> 0 yC% I m.x.0/ a/CI m b Re.x.0/ a/
Re.b x .0/ a/ x .0/ y/2 b x> 0 y C Re b C 1 .b Re.x.0/ a/
I m.x.0/ a/
% I m.x.0/ a/ C I m b % Re.x.0/ a/ D 0: Re.x.0/ a/
To avoid real zeros for all sufficiently large % 2 R>0 we must have 1 Re.x.0/ a/ < 0:
Section 3.1 Partial Differential Equations in H1 .@ C e/
107
In this case Q.@/ would be irreversibly evolutionary in direction x.0/ . Moreover, the leading coefficient a> x.0/ of the polynomial 7! Q.i . x.0/ C s b x .0/ // is constant and therefore Q.@/ is parabolic. We find that the principal part of Q.@/ is given by P .@/ D Qmain .@/ D Qmain .@ C / D P .@ / D 1 .b x .0/ @ /2 D 1 .b x .0/ @/2 : Comparison with (3.1.18) shows Qmain .i x C / D q.i x C /. This coincidence of the highest order parts of Q.i x C /, ¤ 0 and A D 0, with the highest order term q.x/ of a parabola finally motivates the use of the term ‘parabolic’17 for this type of partial differential expression. Note, however, that with regards to the full polynomial Q.i C / this correspondence is rather superficial since, in the parabolic case, Q.i C / has non-real coefficients.
3.1.7 Evolutionary Partial Differential Expressions in Canonical Form It is common and convenient to single out the particular direction in which a specific partial differential expression is evolutionary as one of the coordinate directions. We shall choose the coordinate labelled by 0 to make its particular role apparent. In many applications this direction is identified as the forward time direction and hence @0 would be the time-derivative. The other directions (orthogonal to time) are then referred to as spatial directions. To emphasize the particular role that the time direction plays we have chosen to refer to the underlying domain as R1Cn , where 1 refers to the time dimension and n 2 N is the spatial18 dimension. In order to standardize our approach, we need to determine how the structure we are working in is affected by a change of coordinates. Lemma 3.1.35. Let A 2 R.nC1/.nC1/ be a non-singular matrix, n 2 N. Then the induced rescaling operator A j V
C1 .RnC1 /
W CV 1 .RnC1 / L2 .RnC1 / ! CV 1 .RnC1 / L2 .RnC1 /
defined by A j V
C1 .RnC1 /
WD jdet.A/j1=2 .A /
for 2 CV 1 .RnC1 / extends to a unitary mapping A in L2 .RnC1 /. 17
It should be observed that this terminology is only formally motivated. The Fourier transform of e.g. the parabolic partial differential expression @0 @21 yields the polynomial .x0 ; x1 / 7! i x0 C x12 which has complex coefficients and so does not represent a parabola in R2 . The correspondence to a parabola can, however, be established in a formal way, if we think of @ as real (ignoring that @ is actually purely imaginary, i.e. skew-selfadjoint), 2 ŒR>0 .1; 0/. 18 In the case n D 0 we are dealing with evolutionary ordinary differential expressions. (To ensure that the spatial dimension is at least 1 we sometimes write R1C.nC1/ for n 2 N).
108
Chapter 3 Partial Differential Equations with Constant Coefficients
Proof. The result follows by an elementary calculation. Following the standard convention of identifying x D .x0 ; : : : ; xn / 2 RnC1 with the column matrix 0 1 x0 B :: C .nC1/1 @ : A2R xn we have
Z
Z 2
RnC1
j.A /.x/j dx D Z D
j .Ax/j2 jdet.A/j dx
RnC1
j .y/j2 dy
RnC1
for all 2 CV 1 .RnC1 /, from which the result follows by taking limits. Lemma 3.1.36. Let A 2 R.nC1/.nC1/ be a non-singular matrix. Then for every P s 2 RnC1 we have (with @ s WD nkD0 sk @k ) that A .@ s/ D .@ A1 s/ A holds for all 2 CV 1 .RnC1 / and all s 2 RnC1 . Proof. Let t 2 RnC1 . Then by the chain rule ..@ t / A /.x/ D jdet.A/j1=2
n X
ti @j .Ax/ Aj i
i;j D0
D .A .@ .At // /.x/: With s WD A t we obtain
.@ A1 s/ A D A .@ s/
on CV 1 .RnC1 /. Formally @ s can and will also be interpreted as the matrix product s > @ D @> s, where @ D .@0 ; : : : ; @n / is analogously identified with the formal ..n C 1/ 1/-matrix 0 1 @0 B :: C @ : A; @n i.e. the vector analytical nabla operator. In this spirit we write .A /1 @ and obtain according to the last lemma ..A /1 @ s/A D A .@ s/ for any s 2 RnC1 .
(3.1.23)
Section 3.1 Partial Differential Equations in H1 .@ C e/
109
Lemma 3.1.37. Let A 2 R.nC1/.nC1/ be a non-singular matrix. Then A j V
C1 .RnC1 /
W CV 1 .RnC1 / H˛ .@ C e/ ! H˛ ..A /1 @ C e/
extends to a unitary mapping for every ˛ 2 ZnC1 . Thus, the Sobolev lattices .Hˇ ..A /1 @ C e//ˇ 2ZnC1 ;
.Hˇ .@ C e//ˇ 2ZnC1 ;
are (unitarily) equivalent. Proof. With the previous lemma we find ..A /1 @ C e/ A D A .@ C e/ and so A .@ C e/˛ D ..A /1 @ C e/˛ A for all ˛ 2 ZnC1 . Moreover, 2 jA jH 1 Ce/ ˛
[email protected] /
D
Z RnC1
j...A /1 @ C e/˛ A /.x/j2 dx
Z D
RnC1
j...A /1 @ C e/˛ .A //.x/j2 jdet.A/j dx
Z D
RnC1
j..@ C e/˛ /.Ax/j2 jdet.A/j dx
Z D D
j..@ C e/˛ /.y/j2 dy
RnC1 2 j jH ˛ .@Ce/
for all 2 CV 1 .RnC1 /. Again by taking limits the result follows for arbitrary 2 H˛ .@ C e/. The required density of CV 1 .RnC1 / in H˛ ..A /1 @ C e/, however, needs to be shown. First we recall the density of CV 1 .RnC1 / in H˛ .@ C e/ for all ˛ 2 ZnC1 . We also have the estimate . 2 C e/kei D .i2 C 1/k .jj2 C 1/k . 2 C e/ke for all 2 RnC1 , k 2 N. Replacing by .A /1 we get ...A /1 /2 C e/˛ .j.A /1 j2 C 1/j˛j1 max¹1; k.A /1 k2 ºj˛j1 .jj2 C 1/j˛j1 max¹1; k.A /1 k2 ºj˛j1 . 2 C e/j˛j1 e
110
Chapter 3 Partial Differential Equations with Constant Coefficients
for all 2 RnC1 ; ˛ 2 N nC1 . Similarly, we estimate . 2 C e/˛ .jj2 C 1/j˛j1 max¹1; kA k2 ºj˛j1 .j.A /1 j2 C 1/j˛j1 max¹1; kA k2 ºj˛j1 ...A /1 /2 C e/j˛j1 e for all 2 RnC1 ; ˛ 2 N nC1 . This shows, via an application of the Fourier transform L0 , that H˙1 ..A /1 @ C e/ D H˙1 .@ C e/ in the sense that ^
_
Hˇ ..A /1 @ C e/ ,! H˛ .@ C e/
(3.1.24)
H˛ .@ C e/ ,! Hˇ ..A /1 @ C e/:
(3.1.25)
ˇ 2ZnC1 ˛2ZnC1
and
^
_
˛2ZnC1 ˇ 2ZnC1
The estimate also yields the inverse continuity statements ^
_
Hˇ ..A /1 @ C e/ ,! H˛ .@ C e/
(3.1.26)
H˛ .@ C e/ ,! Hˇ ..A /1 @ C e/:
(3.1.27)
˛2ZnC1 ˇ 2ZnC1
and
^
_
ˇ 2ZnC1 ˛2ZnC1
This shows in particular the denseness of CV 1 .RnC1 / in Hˇ ..A /1 @ C e/ for every ˇ 2 ZnC1 . Let Hˇ ..A /1 @ C e/ be given with ˇ 2 N nC1 large. Then there is a ˛ˇ ˇ; ˛ˇ 2 N nC1 , such that H˛ˇ .@ C e/ ,! Hˇ ..A /1 @ C e/. Then there is also a ˇ ˛ˇ ; ˇ 2 N nC1 , such that Hˇ ..A /1 @ C e/ ,! H˛ˇ .@ C e/. Since Hˇ ..A /1 @ C e/ is dense in Hˇ ..A /1 @ C e/, we also have the density of the continuous embedding H˛ˇ .@Ce/ ,! Hˇ ..A /1 @Ce/. Now the density of CV 1 .RnC1 / in H˛ˇ .@ C e/ yields the desired density of CV 1 .RnC1 / in Hˇ .@ .A /1 C e/. The continuity statements (3.1.24), (3.1.25), (3.1.26), (3.1.27) also show that the lattices .Hˇ .@ .A /1 Ce//ˇ 2ZnC1 and .Hˇ .@Ce//ˇ 2ZnC1 are equivalent. Finally, from our initial calculation for functions in CV 1 .RnC1 / we conclude that the correspondence is indeed unitary. The last result can now be refined further.
Section 3.1 Partial Differential Equations in H1 .@ C e/
111
Theorem 3.1.38. Let A 2 R.nC1/.nC1/ be a non-singular matrix. Then A j V
C1 .RnC1 /
W CV 1 .RnC1 / H˛ .@ C e/ ! H˛ ..A /1 @A C e/
extends to a unitary mapping for every ˛ 2 .ZnC1 /.sC1/.tC1/ ; s; t 2 N; 2 RnC1 . Moreover, the Sobolev lattices .Hˇ ..A /1 @A C e//ˇ 2.ZnC1 /.sC1/.t C1/ and .Hˇ .@ C e//ˇ 2.ZnC1 /.sC1/.t C1/ are equivalent. Proof. We may restrict our attention to the case s D t D 0, (recall the convention not to note the matrix size for .CV 1 .RnC1 //.sC1/.tC1/ ). The result for D 0 is the previous lemma. The case of arbitrary 2 RnC1 follows similarly: 2 jA jH 1 @ Ce/ ˛ ..A / A Z D j..A /1 @A C e/˛ A .x/j2 exp.2 A x/ dx RnC1 Z D j...A /1 @ C e/˛ . .A ///.x/j2 jdet.A/j exp.2A x/ dx RnC1 Z D j..@ C e/˛ /.Ax/j2 jdet.A/j exp.2 Ax/ dx nC1 R Z D j.@ C e/˛ .y/j2 exp.2 y/ dy
D
RnC1 2 j jH ; ˛ .@ Ce/
for all 2 CV 1 .RnC1 /. The remaining results follow in the same way as in the proof of the previous lemma by using LA as the appropriate spectral representation (rather than the (n-dimensional) Fourier transform F D L0 /. Finally, to arrive at the corresponding spaces with @ A replacing @ we note the following result. Theorem 3.1.39. Let A 2 R.nC1/.nC1/ be a non-singular matrix. Then the Sobolev lattices .Hˇ .@A C e//ˇ 2.ZnC1 /.sC1/.t C1/ and .Hˇ .@ C e//ˇ 2.ZnC1 /.sC1/.t C1/ are equivalent. Proof. We may again restrict our attention to the case s D t D 0. According to the previous theorem we only need to show that the Sobolev lattices .H˛ ..A /1 @A C e//˛2ZnC1 ;
.H˛ .@ A C e//˛2ZnC1
are equivalent. Using the Laplace transform LA , this amounts to comparing the operators i .A /1 x C e and i x C e. Let B D .Bij /i;j D0;:::;n , so that the adjoint is
112
Chapter 3 Partial Differential Equations with Constant Coefficients
B D .Bji /i;j D0;:::;n . Obviously, we can estimate X
1C
xi2 j.i x C e/2e j D .1 C x02 / .1 C xn2 /:
iD0;:::;n
Therefore we obtain Y
j.i B x C e/˛ j2 D
j D0;:::;n
Y
X 2 ˛j 1C xi Bji iD0;:::;n
1C
j D0;:::;n
X iD0;:::;n
°
max 1 C max 1 C
jBji j2
˛j
iD0;:::;n
ˇ ±j˛j1 ˇ jBij j2 ˇj D 0; : : : ; n 1C
X iD0;:::;n
°
X
xi2
X
X
xi2
j˛j1
iD0;:::;n
ˇ ±j˛j1 ˇ jBij j2 ˇj D 0; : : : ; n j.i x C e/2e jj˛j1
iD0;:::;n
and so ²s j.i B x C e/ j max 1C
X
˛
iD0;:::;n
jBij
ˇ ³j˛j1 ˇ j.i x C e/j˛j1 e j ˇ D 0; : : : ; n
j2 ˇj
for all x 2 RnC1 . Choosing B D A1 we obtain in particular j.i .A /1 x C e/˛ j ˇ ²s ³j˛j1 X ˇ max 1C j.A1 /ij j2 ˇˇj D 0; : : : ; n j.i x C e/j˛j1 e j
(3.1.28)
iD0;:::;n
for all x 2 RnC1 . On the other hand, substituting y WD B x we also get ²s j.i y C e/˛ j max
1C
X iD0;:::;n
ˇ ³j˛j1 ˇ jBij j2 ˇˇj D 0; : : : ; n j.i .B /1 y C e/j˛j1 e j
for all y 2 RnC1 and choosing B D A we obtain ²s j.i y C e/˛ j max
1C
X iD0;:::;n
ˇ ³j˛j1 ˇ jAij j2 ˇˇj D 0; : : : ; n j.i .A /1 y C e/j˛j1 e j
(3.1.29) for all y 2 RnC1 . Equations (3.1.28) and (3.1.29) show the equivalence of the Sobolev lattices, since ˛ 2 ZnC1 is arbitrary.
Section 3.1 Partial Differential Equations in H1 .@ C e/
113
This theorem leads to the following consideration. Let P .@/ be an arbitrary partial differential operator evolutionary in direction 0 2 RnC1 . Then with D % 0 , % 2 R>0 , n X P .@/ D P .@ C / D P .0 @ C %/ 0 C .i @/ i iD1
where ¹i j i D 1; : : : nº is an orthonormal set chosen such that ¹0 º [ ¹i j i D 1; : : : nº is a complete orthonormal set in RnC1 . Choosing a unitary transformation A such that this complete orthonormal set is transformed into the canonical basis ¹ei j i D 0; : : : ; nº by A1 such that Ae0 D 0 ;
Aei D i
for i D 1; : : : ; n;
i.e. in block matrix notation A D .0 1 n / written as a row of columns, leads to A P .@/ D P .A @/ A ; (3.1.30) Pn where we have abbreviated A@ D @0 0 C iD1 @i i . Note that since A is unitary by construction, we have .A /1 D A: So, rather than solving the equation P .@/u D f in H1 .@ C e/, we may, according to the last two theorems, solve the equation Q.@0 ; : : : ; @n / w D g WD A f in H1 .@e0 C e/, where Q.@0 ; : : : ; @n / D P .A@/ is evolutionary in direction e0 . Following the convention introduced in Example 3.5, we shall for sake of emphasis write Q.@0 ; b @/ WD Q.@0 ; @1 ; : : : ; @n /; where it is now understood that Q.@0 ; b @/ is evolutionary in direction e0 , which is the time direction and all other directions e1 ; : : : ; en are now spatial directions. Instead of referring to RnC1 as the underlying domain we write R1Cn to underscore that the ‘zeroth’ variable is singled out as the time-direction. An ..s C 1/ .s C 1//-partial differential operator Q.@0 ; b @/ in R1Cn , where e0 is understood as the direction in which Q.@0 ; b @/ is evolutionary, is said to be in standard form. This partial differential expression can now be discussed in an (equivalent) associated Sobolev lattice with D e0 .
114
Chapter 3 Partial Differential Equations with Constant Coefficients
Example 3.18. The partial differential expressions of the wave equation @20 b @2 , @2 and of the heat equation @0 b @2 are typical of the Schrödinger equation @0 i b examples of evolutionary partial differential expressions in standard form. In many circumstances it will be convenient to assume that a problem is already formulated in standard form and we shall make use of this option whenever it appears to be advantageous.
3.1.8 Functions of @ and Convolutions So far we have encountered polynomials and their inverses as particular functions of the Abelian system, i.e. the family of commuting selfadjoint operators, 1 1 1 @ WD @0;0 ; : : : ; @n;n : i i i Of course, more general functions may be of interest also. As a generalization of so called convolution integrals given by Z
.x y/ .y/ dy . /.x/ WD RnC1
for suitably well-behaved functions, e.g. ; 2 CV 1 .RnC1 /, convolutions in the framework of Sobolev lattices turn out to be closely related to the concept of functions of @ . We give the following formal definition extending the idea of convolutions to matrix operators. Definition 3.1.40. Let f 2 HnC1;sC1;tC1;1 .@ C e/, n; s; t 2 N. Then, with19 b. i / WD L f; f
(3.1.31)
This notation is motivated by the custom to denote the Fourier transform of an f as b f . Indeed, for
2 CV 1 .RnC1 / we recall Z b
.x/ D .2/.nC1/=2 exp.ixy/ .y/ dy 19
RnC1
for x 2 RnC1 , which – by complex extension – yields Z b
. i / WD .2/.nC1/=2 exp.i. i/y/ .y/ dy Z D .2/.nC1/=2
RnC1 RnC1
exp.i y/ exp.y/ .y/ dy
D .L /. / for 2 RnC1 , 2 RnC1 . This observation justifies the notation b f . i / for L f .
Section 3.1 Partial Differential Equations in H1 .@ C e/
115
the operator b f WD .2/.nC1/=2 f
1 .@ C / i
b.m i / L D .2/.nC1/=2 L f is called the convolution operator associated with f . For g 2 HnC1;tC1;rC1;1 .@ C e/, r 2 N, we call f g 2 HnC1;sC1;rC1;1 .@ C e/ the convolution of f with b.m i /b g in H1 .@ C e/, whenever it is well-defined, i.e. if f g . i / 2 HnC1;sC1;rC1;1 .i m C e/. Remark 3.1.41. According to the definition we see for example that, if we have an b.x element f 2 HnC1;sC1;tC1;1 .@ C e/, n; s; t 2 N, such that x 7! .i x C e/˛ f nC1 i / is a bounded measurable matrix-valued function for some ˛ 2 Z , then f g is the well-defined convolution of f with g in H1 .@ Ce/ for arbitrary g 2 H1 .@ C e/. Note that for a convolution to make sense we must of course have that g is in the b. 1 @ i / and this means in particular that the matrix sizes must fit domain of f i b. 1 @ i / does not depend on r 2 N in any significant together. As noted earlier, f i way. Recalling that, for every 2 RnC1 , the multiplication operator exp. m/ is a unitary mapping from L2 .RnC1 ; exp.2 x/ dx/ to L2 .RnC1 /, we see that this mapping extends by continuity to the Sobolev lattice .H˛ .@ C e//˛2ZnC1 . We have, for 2 CV 1 .RnC1 /, j j;˛ D j.@ C e/˛ j;0 D j exp. m/ .@ C e/˛ j0;0 D j.@0 C e/˛ exp. m/ j0;0 D j exp. m/ j0;˛ ; and this extends by density to all 2 H˛ .@ C e/, ˛ 2 ZnC1 . Recalling that exp. m/ maps CV 1 .RnC1 / onto itself, this shows that the mapping exp. m/ W H˛ .@ C e/ ! H˛ .@0 C e/ is unitary for every ˛ 2 ZnC1 . Since @ D exp. m/ @0 exp. m/ and therefore @ D @.0 ;:::;n / is unitarily equivalent to @0 D @.0;:::;0/ , we also have that, for every Borel function f W RnC1 ! C, 1 1 f @ D exp. m/ f @0 exp. m/: i i
116
Chapter 3 Partial Differential Equations with Constant Coefficients
In particular we have that exp. m/ W H˛ .@ C e/ ! H˛ .@0 C e/ is unitary for ˛ 2 ZnC1 ; 2 RnC1 . Moreover, noting that exp. m/ is the inverse operator of exp. m/, we also find that .@ C e/˛ exp.2 m/ .@ C e/˛ W H˛ .@ C e/ ! H˛ .@ C e/ and .@ C e/˛ exp.2 m/ .@ C e/˛ W H˛ .@ C e/ ! H˛ .@ C e/ are unitary for all ˛ 2 ZnC1 ; 2 RnC1 . Thus, we see that H˛ .@ C e/ ,! H0 .@0 C e/ ,! H˛ .@ C e/; where the unitary mappings exp. m/ .@ C e/˛ and .@ C e/˛ exp. m/ are interpreted as embeddings. Recall that H0 .@0 C e/ is simply L2 .RnC1 /. We want to show that this constitutes a Gelfand chain. For this we need to show that H˛ .@ C e/ D H˛ .@ C e/; where now, however, the duality pairing is obtained by continuously extending the L2 .RnC1 /-inner product rather than the H0 .@ C e/-inner product. So, let f 2 H˛ .@ C e/ . Then for all 2 H˛ .@ C e/ f . / D hR˛; f j i;˛ D h.@ C e/˛ R˛; f j .@ C e/˛ i;0 D hexp. m/ .@ C e/˛ R˛; f j exp. m/ .@ C e/˛ i0;0 D h.@0 C e/˛ exp. m/ R˛; f j .@0 C e/˛ exp. m/ i0;0 D h.@ C e/˛ exp.2 m/ .@ C e/˛ exp.2 m/ R˛; f j exp.2 m/ i0;0 : We have .@ C e/˛ exp.2 m/ .@ C e/˛ exp.2 m/ R˛; f 2 H˛ .@ C e/ and exp.2 m/ 2 H˛ .@ C e/. Thus, .@ Ce/˛ exp.2m/ .@ Ce/˛ exp.2m/ R˛; W H˛ .@ Ce/ ! H˛ .@ Ce/ is unitary and so indeed H˛ .@ C e/ D H˛ .@ C e/ in the sense of unitary equivalence. Since the Riesz map R˛; can be factorized in the form R˛; D .@ C e/˛ .@ C e/˛ ;
Section 3.1 Partial Differential Equations in H1 .@ C e/
117
we find .@ C e/˛ exp.2 m/ .@ C e/˛ exp.2 m/ R˛; D .@ C e/˛ exp.2 m/ .@ C e/˛ exp.2 m/ .@ C e/˛ .@ C e/˛ D .@ C e/˛ exp. m/ .@ C e/˛ exp. m/ D .@ C e/˛ .@ C e/˛ D 1: In other words, the stated unitary equivalence is canonical, as it is just the identity. We therefore have for f 2 H˛ .@ C e/ D H˛ .@ C e/, 2 H˛ .@ C e/, that f . / D hf j exp.2 m/ i0;0 : In particular, the sesqui-linear form h j i0;0 , as the continuous extension of the inner product of L2 .RnC1 /, gives the duality pairing between H˛ .@ C e/ and H˛ .@ C e/ for every ˛ 2 ZnC1 ; 2 RnC1 . In this framework we shall see that the convolution, as the notation suggests, is largely independent of 2 RnC1 . We have, for ; g component-wise in CV 1 .RnC1 / and f 2 H1 .@ C e/, hf g j i;0 b.m i /b D .2/.nC1/=2 hf g . i / j b
. i /i0;0 Z X bij .m i / b
ik .m i / dx D .2/.nC1/=2 f gj k .m i / b i;j;k
D .2/.nC1/=2
XZ i;j
.nC1/=2
D .2/
RnC1
RnC1
bij .m i / f
X
ik .m i / dx b gj k .m i / b
k
b.m i / j b hf
.m i /b g . i / i0;0 :
We note that, with20 .1 g/.x/ WD g.x/, x 2 RnC1 , Z 1 .L g/ .x/ D exp.ix y/ exp. y/ g.y/ dy .2/.nC1/=2 RnC1 Z 1 exp.ix y/ exp. y/ g.y/ dy D .2/.nC1/=2 RnC1 Z 1 D exp.ix y/ exp. y/ exp.2 y/ g.y/ dy .2/.nC1/=2 RnC1 D .L .1 exp.2 m/ g //.x/ 20
In general A is the rescaling operator given by p .A '/.x/ WD jdet.A/j '.Ax/
for a regular linear mapping A W RnC1 ! RnC1 , ' a complex-valued function on RnC1 , n 2 N. Here A is just multiplication by 1.
118
Chapter 3 Partial Differential Equations with Constant Coefficients
and so .2/.nC1/=2 L .m/ .L g/ D L . 1 exp.2 m/ g /, where XZ
1 exp.2 m/ g D
ik . y/ exp.2 y/ gj k .y/ dy: k
RnC1
Thus, hf g j i;0 D hf g j exp.2 m/ i0;0 D hf j 1 exp.2 m/ g i;0 D hf j exp.2 m/ 1 exp.2 m/ g i0;0 D hf j .exp.2 m/ / 1 g i0;0 ; and in particular hf g j i0;0 D hf j
1 g i0;0
(3.1.32)
for all f 2H˛ .@ C e/ and ; g 2 CV 1 .RnC1 /. Equality (3.1.32) shows that the b. 1 @ i/ is dependent on 2 RnC1 convolution operator f D .2/.nC1/=2 f i only as far as the domain is concerned. This appears plausible taking into account that formally @ C D @ C D @. The solution theory specifies that the solution of the ..s C 1/ .s C 1//-partial differential equation P .@ C / u D f can be found in the form u D L P .i m C /1 L f assuming that P .@/ is invertible with respect to 2 RnC1 . Noting that P .i C/1 2 HnC1;sC1;sC1;1 . i m C e/ and defining G WD .2/.nC1/=2 L P .i C/1 2 HnC1;sC1;sC1;1 .@ C e/ we have that L G WD .2/.nC1/=2 P .i C/1 and so u D L .2/.nC1/=2 .L G/.m/ L f D G f: The element G is known as the fundamental solution or Green’s function (s D 0) or Green’s tensor (for s 1) of P .@/. Clearly, the fundamental solution satisfies the equation P .@ C / G D .2/.nC1/=2 L 1..sC1/.sC1// ;
Section 3.1 Partial Differential Equations in H1 .@ C e/
119
where 1..sC1/.sC1// stands for the ..s C 1/ .s C 1//-identity matrix, which is considered here as an element of HnC1;sC1;sC1;1 .i m C e/. More precisely we have 1 0 1 0 0 :: C : B : C B 0 :: C 2 HA .i m C e/ B :: : : @ : : 0A 0 0 1 with
1 e C1 C1 :: C : B : C B C1 : : ADB : C: :: @ :: : C1 A C1 C1 e 0
Since 0 2 H1 .i m C e/, we only need to see that the constant function 1 is in He .i m C e/. This follows since j.i m C e/e 1jL2 .RnC1 / D j.i C e/e jL2 .RnC1 / nC1 D j.i C 1/1 jL 2 .R/
Z D
R
1 dx 1 C x2
.nC1/=2 D .nC1/=2 < 1:
Consequently, the right-hand side of the equation for the fundamental solution satisfies L 1..sC1/.sC1// 2 HA .@ C e/: The element 0 B B ı 1..sC1/.sC1// WD B @
ı 0 0 :: : : 0 :: :: :: : 0 : 0 0 ı
1 C C C WD .2/.nC1/=2 L 1..sC1/.sC1// A
where the so-called Dirac ı-distribution is given by ı D .2/.nC1/=2 L 1:
120
Chapter 3 Partial Differential Equations with Constant Coefficients
To understand its action we calculate hı j i;0 D .2/.nC1/=2 hL 1 j i;0 D .2/.nC1/=2 h1 j L i0;0 Z .nC1/=2 b
.x i / dx D .2/ RnC1
D .2/.nC1/=2 exp. 0/
Z RnC1
exp.i 0 x/ b
.x i / dx
D .L b
. i //.0/ D .0/ for all 2 CV 1 .RnC1 /. By density it follows that for any 2 He .@ C e/ we have that .0/ is well-defined as limk!1 k .0/ for any sequence . k /k2N converging to in He .@ C e/. The number .0/ is referred to as the evaluation or trace of at 0. The Dirac ı-distribution can be distinguished from 0 only by elements 2 He .@ C e/ with .0/ ¤ 0. It is therefore used to model strongly localized objects or phenomena such as elementary particles, mass points, light spots, short impulses etc. in varied applications. The particular relevance in the current context is that the fundamental solution G of an invertible ..s C 1/ .s C 1//-partial differential expression P .@/ D P .@ C /, 2 RnC1 is given as the solution 21 of P .@/ G D ı 1..sC1/.sC1// : As indicated above, the solution for an arbitrary right-hand side f is given by convolution with the fundamental solution. It is interesting to note that finding the fundamental solution is essentially a scalar problem. We found that by Cramer’s rule P .i C /1 D
1 cof.P .i C //> ; det.P .i C //
D cof.P .i m C //>
1 1..sC1/.sC1// : det.P .i C //
Recalling that cof.P .@ C //> is also an ..s C 1/ .s C 1//-partial differential expression, we have that the fundamental solution G of P .@ C / is given by G D cof.P .@ C //> g 1..sC1/.sC1// ; 21 Using the latter imagery for the Dirac ı-distribution as a ‘short impulse’ the fundamental solution G is also referred to as the impulse response. In the current framework the fundamental solution may depend on the choice of 2 RnC1 .
Section 3.1 Partial Differential Equations in H1 .@ C e/
121
where g is the fundamental solution of the scalar partial differential expression given by det.P .@ C //, i.e. g D .2/.nC1/=2 L
1 : det.P .i C //
We shall later have the opportunity to explicitly find fundamental solutions for particular differential expressions. The apparently rather intimate relationship between general partial differential expressions and scalar partial differential expressions will now be investigated further.
3.1.9 Systems and Scalar Equations The use of the Fourier–Laplace transform as a spectral representation in conjunction with the Sobolev lattice structure makes dealing with partial differential expressions almost a purely algebraic issue. We shall exploit this fact to reveal the equivalence of systems of partial differential equations with scalar partial differential equations, so that it will appear merely as a matter of convenience which approach to use. Let us assume we have P .@ C / u D f 2 H1 .@ C / with P .@ C / being an ..s C 1/ .s C 1//-partial differential expression, 2 RnC1 . We can arrive at s scalar partial differential equations with identical scalar partial differential expression satisfied by u component-wise. The main tool here is the Cayley–Hamilton theorem which we shall recall now. Theorem 3.1.42 (Cayley–Hamilton theorem). Let A 2 C ..sC1/.sC1// be a matrix PsC1 with Q.t / WD det.t 1..sC1/.sC1// A/ DW kD0 ak t k as characteristic polynomial, s 2 N. Then sC1 X ak Ak D 0: kD0
In other words, a matrix A is a ‘root’ of its characteristic polynomial. Let k , k D 0; : : : ; s, denote the roots of Q. Then Q.t / D
s Y
.t k /:
kD0
In case of multiple roots it can happen that some of the corresponding multiple linear factors .t k / can be omitted without destroying the property that A is a ‘root’ of the resulting polynomial of lower degree. Such a non-trivial polynomial of lowest degree is called the minimal polynomial22 of A. Note that the leading coefficient in any case is 1. As an obvious consequence we have our next result. 22 For normal matrices the minimal polynomial consists of the product of the linear factors associated with the distinct roots (ignoring multiplicity). This is the situation we encounter in many applications.
122
Chapter 3 Partial Differential Equations with Constant Coefficients
P# Theorem 3.1.43. For the minimal polynomial .t / WD kD0 ck t k of degree # s C 1, associated with an ..s C 1/ .s C 1//-matrix A we have # X
ck Ak D 0:
kD0
Note that if A is invertible then c0 ¤ 0 as of course is a0 D .1/s det.A/ ¤ 0. In this case, multiplication by A1 yields # 1 1
A
D
X ckC1 Ak c0
(3.1.33)
s X akC1 k A : a0
(3.1.34)
kD0
and A1 D
kD0
The coefficients are polynomials in the entries of A. Applying these findings to P .@/ in place of A yields an interesting way to construct P .@/1 in terms of powers of P .@/. We also find det.P .@// D .1/sC1 a0 .@/ D .1/s
sC1 X
ak .@/ P .@/k
(3.1.35)
kD1
or, using the minimal polynomial, c0 .@/ D
# X
ck .@/ P .@/k ;
kD1
where ak .@/; cj .@/ are scalar partial differential expressions (to be applied component-wise), k D 0; : : : ; s C1I j D 0; : : : ; # . Applying these formulas to the solution u of P .@/u D f we get det.P .@// u D .1/sC1 a0 .@/ u D .1/s
sC1 X
ak .@/ P .@/k u
kD1
D .1/s
sC1 X
(3.1.36)
ak .@/ P .@/k1 f;
kD1
or using the minimal
polynomial23
c0 .@/ u D
# X kD1
23
instead
ck .@/ P .@/k u D
# X
ck .@/ P .@/k1 f:
kD1
We make here the implicit assumption that the coefficients are of polynomial type, i.e. we take the minimal polynomial to be the polynomial of lowest degree with polynomial coefficients.
Section 3.1 Partial Differential Equations in H1 .@ C e/
123
Summarizing, we have found the following correspondence. Theorem 3.1.44. Let P .@/ be an ..s C 1/ .s C 1//-partial differential expression, invertible with respect to 2 RnC1 and let u; f 2 H1 .@ C e/ be such that P .@/ u D f:
(3.1.37)
PsC1 Furthermore, let t 7! a .i x C / t k be the characteristic polynomial of PrC1 kD0 k P .i xC/ and t 7! kD0 ck .i xC/ t k its minimal polynomial (asC1 D crC1 D 1). Then u also solves the equations det.P .@// u D .1/s
s X
akC1 .@/ P .@/k f
(3.1.38)
kD0
and c0 .@/ u D
r X
ckC1 .@/ P .@/k f
(3.1.39)
kD0
where ak .@/; cj .@/ are scalar partial differential expressions, k D 0; : : : ; s C 1I j D 0; : : : ; r C 1. Conversely, every solution of (3.1.38) or (3.1.39) also solves (3.1.37). Proof. The first part of the result is clear from our considerations above. It remains to see the converse. For sake of brevity, we only consider the case of equation (3.1.38); the other case follows analogously. From (3.1.35) we have det.P .@// D .1/s
sC1 X
ak .@/ P .@/k
kD1
and using this in the right-hand side of (3.1.38) leads to det.P .@// .u P .@/1 f / D 0: Since det.P .@// is invertible (since P .@/ was assumed to be invertible) we get u P .@/1 f D 0 and so u solves (3.1.37). This result shows that the components of u do indeed solve a scalar partial differential equation of the form det.P .@// w D h or c0 .@/ w D h;
124
Chapter 3 Partial Differential Equations with Constant Coefficients
respectively. Thus a system of partial differential equations is always equivalent to a diagonal system of scalar partial differential equations involving the same scalar partial differential expression. Sometimes we wish to convert a scalar equation into a system while reducing the order of differentiation with respect to a particular choice of coordinate direction. Although, we do not need the partial differential expression to be evolutionary, we shall write P .@0 ; b @/ for the scalar partial differential expression, which is to be converted to a system that is first order with respect to @0 . More specifically let @/ WD P .@0 ; b
sC1 X
ak .b @/ @k0
kD0
with asC1 .b @/ D 1. Then the well-known procedure from the theory of ordinary differential equations translates this into the ..s C 1/ .s C 1//-partial differential equation 0 1 1 0 0 @0 1 0 1 0 u 0 B C : : : :: :: :: B 0 CB B : C @0 B C B @0 u C C B :: C B : CB :: :: :: C D B C: B :: C : : : : B C 0 :: C B CB A @ 0 A @ B C 0 @0 1 @ 0 A @s0 u f a0 .b @/ a1 .b @/ as1 .b @/ @0 C as .b @/ With 0 B B B B b A.@/ WD B B B @
this simplifies to
0
1
0
0 :: :
0 :: :
1 :: :
0 0 b b a0 .@/ a1 .@/
1 0 :: C : C C C ; 0 C C C 0 1 A b as1 .@/ as .b @/ :: : :: :
1 u C B B @0 u C B V WD B : C C; @ :: A
1 0 B :: C B C @// V D F WD B : C: .@0 C A.b @0A f
0
@s0 u
0
(3.1.40)
@// D P .@0 ; b @/ and so invertibility is preserved. We note that det.@0 C A.b Let V be a solution of (3.1.40). Then with u WD V0 we recover from the first s rows Vk D @0 Vk1 D @k0 u for k D 1; : : : ; s
Section 3.1 Partial Differential Equations in H1 .@ C e/
125
as well as P .@0 ; b @/u D f from the last row. Of course there are many systems equivalent to a given scalar partial differential equation and the above way may not be the smartest way to get an equivalent system. Examples 3.5 and 3.6 provide two systems equivalent to the wave equation of which only the second features regularity loss 0. The first system – displaying a non-vanishing regularity loss – is, however, obtained in the way described above. Thus, since every evolutionary higher order scalar equation with constant leading coefficient with respect to powers of @0 can be brought into the form of a system of first order with respect to @0 , we may consider an ..s C1/.s C1//-partial differential expression evolutionary in direction .1; 0; : : : ; 0/ of the form P .@0 ; b @/ D @0 C A.b @/; where A.b @/ is an ..s C 1/ .s C 1//-partial differential expression in the spatial derivatives only, as the standard first order form of an ..s C 1/ .s C 1//-partial differential expression evolutionary in direction .1; 0; : : : ; 0/ 2 R1Cn .
3.1.10 Causality Intrinsic in the term ‘evolutionary’ is the expectation that solutions associated with evolutionary operators are ‘causal’. With the direction of evolution as the time direction we know what ‘before’ and ‘after’ means and causality will be understood as the property that the solution can be non-zero only ‘after’ the right-hand side is non-zero. To make this more precise we need to define the support of an element in 1 H1 .@ C e/ in direction 0 WD jj , 2 RnC1 n ¹0º. Definition 3.1.45. Let W RnC1 ! C be a continuous function. We define the support supp of as [ supp WD RnC1 n ¹ j open in RnC1 ^ D 0 in º: V
nC1
Two linear functionals f; g 2 C C1 .R / are called equal on if f g D 0 on , i.e. f . / D g. / for all 2 CV 1 .RnC1 / with supp . Moreover, we define the support supp f of a linear functional f on CV 1 .RnC1 / as [ supp f WD RnC1 n ¹ j open in RnC1 ^ f D 0 in º: Let .0/ be a particular direction in RnC1 . Then [ supp.0/ f WD R n ¹I j I open in R ^ f D 0 in ŒI .0/ C ¹.0/ º? º is called the support of f in direction .0/ . By definition we have that the orthogonal projection of supp f on LinR ¹.0/ º D ŒR .0/ is given by Œsupp.0/ f .0/ :
126
Chapter 3 Partial Differential Equations with Constant Coefficients
If .0/ D .1; 0; : : : ; 0/ D e0 is the evolutionary direction of a partial differential expression in R1Cn under consideration, then we shall briefly speak of the time support of f and write supp0 f WD suppe0 f: Here .0/ D .1; 0; : : : ; 0/ is identified as the positive time direction e0 . Remark 3.1.46. A straight-forward calculation shows [ supp.0/ f D R n ¹I j I open in R ^ RnC1 n supp f ŒI .0/ C ¹.0/ º? º \ D ¹R n I j I open in R ^ supp f ŒR n I .0/ C ¹.0/ º? º \ D ¹C j C closed in R ^ supp f ŒC .0/ C ¹.0/ º? º: As a matter of convenience, we may also rotate .0/ into a suitable direction. Indeed, let A W RnC1 ! RnC1 be unitary, so that A is its inverse. We note that 7! ı A is a bijection on CV 1 .RnC1 / and define V
nC1 /
A W C C1 .R
V
nC1 /
! C C1 .R
;
u 7! Au nC1
V
with u. ı A / DW Au. /, where C C1 .R / is the linear space of functionals on CV 1 .RnC1 /. We find ^ f D 0 in , f . / D 0 2CV 1 ./
,
^
f . / D 0
ıA2CV 1 .A Œ /
,
^
f . ı A ı A / D 0
ıA2CV 1 .A Œ /
,
^
Af . ı A/ D 0
ıA2CV 1 .A Œ /
,
^
Af . / D 0
2CV 1 .A Π/
,
Af D 0 in A Œ
and so supp Af D A Œsupp f :
Section 3.1 Partial Differential Equations in H1 .@ C e/
127
As a consequence we have supp f ŒC .0/ C ¹.0/ º?
,
A Œsupp f A ŒŒC .0/ C ¹.0/ º?
,
supp Af ŒC A .0/ C ¹A .0/ º?
and therefore also supp.0/ f D suppA .0/ Af: This observation allows for the convenient assumption that .0/ D .1; 0; : : : ; 0/ is the standard direction of concern in the discussion of causality (compare Lemma 3.1.48 below). We are now ready to formulate our concept of causality. Definition 3.1.47. Let W be a mapping from a subset of linear functional defined on CV 1 .RnC1 / with values in the set of such functionals. If inf supp.0/ .f g/ inf supp.0/ .W .f / W .g//
(3.1.41)
for all f; g in the domain of W then we call W (forward) causal in direction .0/ . If W is (forward) causal in direction .0/ , we shall also say that W is backward causal in direction .0/ or just backward causal, if the direction is understood from the context (in particular if .0/ is the positive time direction). Here we interpret inf supp.0/ f D 1 if supp.0/ f is not bounded below and inf supp.0/ f D C1 if supp.0/ f is empty, so that (3.1.41) is only restrictive if we take f with supp.0/ f bounded below. If W is linear then W .0/ D 0 and we may assume g D 0 for an equivalent description of causality. As a word of warning, we point out that, without topology in the background, the concepts of support and causality as introduced here can be quite meaningless. It is, however, convenient to introduce these concepts as a “jargon” without explicit reference to a particular topology, which in fact will vary. The gap can, in these cases, which we consider, be completed via a density argument and continuous extension with respect to the particular topology used. For example, any element in H1 .@ C e/ can and will be identified with a linear functional on CV 1 .RnC1 / via CV 1 .RnC1 / ! C;
(3.1.42)
' 7! h j exp.2m/'i;0 : In the above terminology we obviously have, for any 2 RnC1 , that P .@/ is causal in any direction .0/ . Indeed, hP .@ C / u j i;0 D hu j P .@ C i / i;0
128
Chapter 3 Partial Differential Equations with Constant Coefficients
for all 2 CV 1 .RnC1 /. Since D 0 in implies P .@ C i / D 0 in , we have supp P .@ C i / supp and so u D 0 in implies P .@ C / u D 0 in . Thus supp P .@ C / u supp u: Recalling the above remark, this implies in particular that supp.0/ P .@ C / u supp.0/ u and so inf supp.0/ u inf supp.0/ P .@ C / u; where the direction .0/ is arbitrary and need not be linked to . The interesting part is whether the inverse, i.e. the solution operator, is also causal. In the light of our earlier consideration of transforming coordinates and in consequence Sobolev lattices, it is worth noting the following general result. V
nC1
V
nC1
Lemma 3.1.48. Let W W D.W / C C1 .R / ! C C1 .R / be causal in direction .0/ and let A W RnC1 ! RnC1 be a rotation. Then the mapping A ı W ı A defined by u 7! A .W .A u// is causal in direction A .0/ . Proof. The condition inf supp.0/ .f g/ inf supp.0/ .W .f / W .g// is equivalent to inf suppA .0/ .Af Ag/ inf suppA .0/ .A.W .f // A.W .g/// inf suppA .0/ .A.W .A Af // A.W .A Ag///: Replacing Af and Ag for f and g, we obtain the desired result. Example 3.19. As an important example, consider the 1-dimensional situation @ C . C ˇ/ W H1 .@ C 1/ ! H1 .@ C 1/ for 2 R. According to the above, this is a causal operator in direction ˙1. Since ji x C . C ˇ/j2 D x 2 C . C ˇ/2 . C ˇ/2
Section 3.1 Partial Differential Equations in H1 .@ C e/
129
we see that @ C . C ˇ/ is reversibly evolutionary24 . Note, however, that if ˇ D we have @ C . C ˇ/ D @ , which is not even invertible with respect to . Indeed, 0 2 .@ / and we shall see later that invertibility in H1 .@ C / also fails. Let us now consider .@ C . C ˇ//1 for ˇ ¤ . To study this operator we first solve the ordinary differential equation @u C ˇ u D f for, say, f 2 CV 1 .R/ classically. By the variation of constant method, the general solution is given by u.x/ D C exp. ˇ x/ Z C exp. ˇx/
exp.ˇs/ f .s/ ds: 1;x
To make sure that u 2 H1 .@ C i/, we have to adjust the constants and find, for ˇ > , Z u.x/ D exp.ˇx/ exp.ˇs/ f .s/ ds: 1;x
Similarly, for ˇ < we get Z u.x/ D exp.ˇx/ With
´ gˇ; .x/ WD sgn.ˇ C /
exp.ˇs/ f .s/ ds: Œx;1Œ
exp.ˇx/ 0
for sgn.ˇ C / x 0; for sgn.ˇ C / x < 0
we simply obtain u D gˇ; f in the sense of a convolution integral. Note in particular that gˇ; 2 H0 .@ C e/: 24 This is of course also clear from our general observation that ordinary differential expressions are always reversibly evolutionary.
130
Chapter 3 Partial Differential Equations with Constant Coefficients
Indeed, we calculate jgˇ; j2;0
Z D
R
jgˇ; .x/j2 exp.2x/ dx
Z D
sgn.ˇ C/ R>0
exp.2ˇx/ exp.2x/ dx
Z D
sgn.ˇ C/ R>0
exp.2.ˇ C /x/ dx
Z D D
R>0
exp.2 jˇ C j x/ dx
1 : 2 jˇ C j
Moreover, by uniqueness and continuous extension we have that gˇ; D .@ C .ˇ C //1 for 2 R; ˇ ¤ . Now, observe that hgˇ; f j i;0 D h.@ C .ˇ C //1 f j i;0 D hf j.@ C .ˇ C //1 i;0 D hf j.@ C .ˇ //1 i;0 D hf j.@ C . C .ˇ 2///1 i;0 D hf jgˇ 2; i;0 ; for all 2 CV 1 .R/, C ˇ ¤ 0. Let a WD inf supp f 2 R so that f D 0 on 1; aŒ. With supp 1; aŒ we therefore have hf j i;0 D 0: We also read off from the convolution integral that gˇ 2; 2 C1 .R/ and sup supp sup supp gˇ 2; , if ˇC > 0. Moreover, since the multiplication operator .i m C .ˇ C //1 maps H1 .i m C e/ into itself, we also have gˇ 2;
2 H1 .@ C e/. Since a simple cut-off with25 j1 ŒN;N .m/ yields that j1 Here j" .x/ WD "1 j.x="/, x 2 R, with j 2 CV 1 .R/ and supp.j / D Œ1; 1 may be chosen arbitrarily. A specific well-known choice is given by 1 j.x/ D C exp 1 x2 25
for x 2 1; 1Œ, and j.x/ D 0 otherwise, where C WD R 1;1 Œ
1 : 1 exp 1x 2 dt
Section 3.1 Partial Differential Equations in H1 .@ C e/
131
ŒN;N .m/ gˇ 2; ! gˇ 2; in H1 .@ C e/ as N ! 1, we also see that 0 D lim hf jj1 ŒN;N .m/ gˇ 2; i;0 N !1
D hf j gˇ 2; i;0 D h.@ C .ˇ C //1 f j i;0 for all 2 CV 1 .R/ with supp 1; aŒ . This proves that, in the case ˇ C > 0, .@ C .ˇ C //1 f D 0
on 1; aŒ
or inf supp.@ i .ˇ C //1 f a. Since a D inf supp f , the desired inequality follows for ˇ C > 0 proving that .@ C .ˇ C //1 is causal in direction C1. In the case ˇ C < 0 causality in direction 1 can be shown analogously. Since sgn.ˇ C / D sgn./ for jj sufficiently large, indeed for jj > jˇj, we may reformulate our findings in the following way: .@ C .ˇ C //1 is causal in direction sgn./ for all sufficiently large jj (in fact for jj > jˇj). Example 3.20. The last causality result carries over to the higher dimensional case following the same line of reasoning, since the cut-off procedure and the integration to obtain the solution to the ordinary differential equations can be carried out for directional derivatives in arbitrary dimensions. Thus, for any direction .0/ 2 RnC1 and D % .0/ 2 RnC1 ; % 2 R>0 , and ˇ 2 RnC1 such that k C ˇk ¤ 0, k D 0; : : : ; n, .@ C .ˇ C //˛ W H1 .@ C e/ ! H1 .@ C e/ is causal in direction .0/ for all ˛ 2 ZnC1 . This follows by induction from the causality of .@;k C .ˇk C k //1 W H1 .@ C e/ ! H1 .@ C e/ in direction .0/ . This is clear if .0/k ¤ 0. For .0/k D 0 we have .@;k C .ˇk C k //1 D .@k C ˇk /1 : This operator, however, cannot have any influence on the support in direction .0/ , since in this case .0/ ? ek . Considering the translation operator h W H1 .@ C e/ ! H1 .@ C e/, h 2 RnC1 , defined as the continuous extension of the mapping .h /.x/ WD .x C h/
for all 2 CV 1 .RnC1 /:
132
Chapter 3 Partial Differential Equations with Constant Coefficients
We can confirm that causality is, for (Borel) functions of @ , neutral with respect to shifts, i.e. translation of the argument. We calculate first Z jh j2;0 D j .x C h/j2 exp.2 x/ dx nC1 R Z D j .y/j2 exp.2 .y h// dy RnC1
D exp.2 h/ j j2;0 for all 2 CV 1 .RnC1 /. Therefore exp. h/ h W H0 .@ C e/ ! H0 .@ C e/ is unitary since h is clearly onto with inverse h W H0 .@ C e/ ! H0 .@ C e/. Moreover, we have h .@ C e/ D .@ C e/ h (3.1.43) on CV 1 .RnC1 / and so by induction h .@ C e/˛ D .@ C e/˛ h for all ˛ 2 ZnC1 . Consequently, we also get by continuous extension that exp. h/ h W H˛ .@ C e/ ! H˛ .@ C e/ is unitary for every ˛ 2 ZnC1 . In addition, due to the above commutativity property, we have that h commutes with P .@/ D P .@ C/ D P ..@ Ce/C.e// and so also with P .@/1 if P .@/ is invertible with respect to . We shall exploit this fact in the proof of our next theorem. Alternatively, these commutativity results can of course be confirmed (and extended to Borel functions of @ ) by applying the Fourier–Laplace transform. In particular, we note for future consideration that for functions of @ it suffices to test causality with such elements f satisfying inf supp.0/ f D 0: Remark 3.1.49. That h commutes with differentiation is not much of a surprise if we note that h is actually a function of @ . Indeed, we have26 h D exp.h / exp.h @ / D exp.h .@ C // and so exp. h/ h D exp.h @ / which confirms again the above unitarity statement. 26
It may be interesting to note that this formally corresponds to Taylor’s formula, since exp.hz/ D hk k k2N kŠ z .
P
Section 3.1 Partial Differential Equations in H1 .@ C e/
133
Theorem 3.1.50. Let P .@/ be an ..s C 1/ .s C 1//-partial differential expression evolutionary in direction .0/ 2 RnC1 , s; n 2 N. Then P .@ C /1 W H1 .@ C e/ ! H1 .@ C e/ is well-defined and causal in direction .0/ with D % .0/ for all sufficiently large % 2 R>0 . Proof. That P .@ C /1 W H1 .@ C e/ ! H1 .@ C e/ exists as a continuous operator for all D % .0/ with % 2 R>0 sufficiently large follows from our solution theory. Moreover, we have the existence of an ˛ D .˛0 ; b ˛ / 2 N nC1 , b ˛ 2 N n , such ˛ 1 nC1 that .i x C e/ P .i x C / is uniformly bounded in x 2 R , 2 ŒR>0 .0/ , with jj %0 > 0 for some %0 2 R>0 . According to our earlier considerations and in particular due to Lemma 3.1.48, we may assume without loss of generality that .0/ D .1; 0; : : : ; 0/. Therefore, we are concerned with an expression of the form ˛ .i x0 C 1/˛0 .ib x C e/b P .i x0 C %; ib x /1
bounded uniformly for x0 2 R, b x 2 Rn , % 2 R%0 . Clearly ji x0 C 1j2 D x02 C 1 x02 C .1 C %/2 D ji x0 C % C 1j2 and so there exists a constant C 2 R>0 such that ˛ x C e/b P .i x0 C %; ib x /1 j C j..i x0 C %/ C 1/˛0 .ib
(3.1.44)
for all x0 2 R, b x 2 Rn , % 2 R%0 . Since .@ C C e/˛ is causal in every direction and the composition of causal operators is again causal, it suffices to show that .@ C C e/˛ P .@ C /1 is causal in direction .0/ in order to show that P .@ C /1 is causal in this direction. For this we first note that (3.1.44) can be written in the form j.i C C %0 C 1; i C e/˛ P .i C C %0 ; i /1 j C for all 2 R, 2 R>0 , 2 Rn . Letting z D i we obtain that j.i z C %0 C 1; i C e/˛ P .i z C %0 ; i /1 j C
(3.1.45)
for all 2 Rn , z 2 R i R>0 . Now let f 2 Hˇ .@ C e/, ˇ 2 .ZnC1 /s1 D .ZnC1 /s be such that a D inf supp.0/ f 2 R. Then, we have to show that inf supp.0/ ..@ C . C e//˛ P .@ C /1 f / a:
134
Chapter 3 Partial Differential Equations with Constant Coefficients
Without loss of generality we may assume that f 2 H0 .@ C e/ since, from Example 3.20, we have that the bounded bijections .@ C e C / W H .@ C e/ ! H0 .@ C e/, 2 ZnC1 are already causal in direction .0/ . Due to the above noted interaction with translation, we may focus our attention on the case a D 0. Under these simplifying assumptions, we first observe that .L .@ C C e/˛ P .@ C / f /.x/ D .i z C %0 C 1; i x1 C 1; : : : ; i xn C 1/˛ b.z i %0 ; x1 ; : : : ; xn / P .i z C %0 ; i x1 ; : : : ; i xn /1 f for almost all x D .x1 ; : : : ; xn / 2 Rn . The mapping .z; x1 ; : : : ; xn / 7! .i z C %0 C 1; i x1 C 1; : : : ; i xn C 1/˛ b.z i %0 ; x1 ; : : : ; xn / P .i z C %0 ; x1 ; : : : ; xn /1 f
(3.1.46)
can be considered as a mapping from R i R>0 into L2 .Rn / and so, given any 2 CV 1 .Rn /, we may define a mapping S W R i R>0 ! C; b.z i %0 ; /i0 : z 7! hFn j .i z C%0 C1; i C e/˛ P .i z C%0 ; i /1 f By the converse of the Paley–Wiener theorem (see e.g. [52]), we know that the mapb.z i %0 ; ; : : : ; / is analytic27 and ping F W R i R>0 ! L2 .Rn / given by z 7! f so the mapping S is also analytic in R i R>0 . Indeed, t 7! exp.%0 t /h j f .t; /i0 2 L2 .R>0 / and so its Fourier transform z 7! h j .L%0 ˝ 1/f .z; /i0 D z 7! hFn j .L%0 ˝ Fn /f .z; /i0 b.z i %0 ; /i0 D z 7! hFn j f extends to an analytic function in R i R>0 . It can be seen that b.z i %0 ; /i0 S 0 .z/ D hFn j Q.z; / f b 0 .z i %0 ; /i0 C hFn j .i z C %0 C 1; i C e/˛ P .i z C %0 ; i /1 f where Q denotes the derivative of z 7! .i z C %0 C 1; i C e/˛ P .i z C %0 ; i /1 : 27
Recall that weak analyticity is equivalent to analyticity, see e.g. [46].
Section 3.1 Partial Differential Equations in H1 .@ C e/
135
The family of matrix-valued functions z 7! Q.z;b x /, b x 2 Rn , is by the rules of differentiation still entry-by-entry absolutely and locally uniformly bounded above by a term of the form C .ib x C e/ for some 2 Zn , C 2 R>0 . Thus, S is seen to be analytic. Moreover, by (3.1.45) we have jS. i /j0;0 C j j0 jF . i /j0;0 D C j j0 jf j.%0 C ;0;:::;0/;0 C j j0 jf j.%0 ;0;:::;0/;0 for all 2 R>0 . Thus, we also get that S is in the so-called Hardy–Lebesgue space ˇ ° ˇ (3.1.47) HL WD f W ŒR iŒR>0 ! C ˇ f analytic ; ± ^ f . i/ 2 L2 .R/; sup¹jf . i/jL2 .R/ j 2 R>0 º < 1 : 2R>0
The Paley–Wiener theorem (see e.g. [52]), now yields that S can be continuously extended to the closed halfplane ŒR iŒR0 yielding on R an element s of L2 .R/ such that F s 2 L2 .R/ has support in R0 , i.e. h j F si0;0 D h exp.% m1 / j L% si%;0 D 0 for all
2 CV 1 .R/ with supp
R<0 and so also h j L% si%;0 D 0
for all
2 CV 1 .R/ with supp
R<0 . By taking limits we may even allow 2 L2 .R; exp.2 x/ dx/
with supp
R<0 . Thus, we obtain 0 D h .i m C C e/˛ P .i m C /1 L f jL%
˝ Fn i0;0
D h .i m C C e/˛ P .i m C /1 L f jL .
˝ /i0;0
D h L .i m C C e/˛ P .i m C /1 L f j
˝ i;0
D h .@ C C e/˛ P .@ C /1 f j for all 2 L2 .R; exp.2 x/ dx/ with supp CV 1 .Rn /, we get
˝ i;0
R<0 . Since was arbitrary in
.@ C C e/˛ P .@ C /1 f D 0 on ŒR<0 .0/ C ¹.0/ º? as desired, which finally confirms the claimed causality of P .@ C /1 .
136
Chapter 3 Partial Differential Equations with Constant Coefficients
The last causality result implies in particular that the fundamental solution G of an evolutionary ..s C 1/ .s C 1//-partial differential expression in H1 .@ C e/, D % .0/ , % 2 R>0 sufficiently large, must satisfy supp G ŒR0 .0/ C ¹.0/ º? since supp.0/ ı 1..sC1/.sC1// D ¹0º: This observation also implies a certain independence of G from the size % of D % .0/ as long as % %0 for some sufficiently large %0 2 R>0 . Lemma 3.1.51. Let P .@/ be an ..s C 1/ .s C 1//-partial differential expression evolutionary in direction .0/ . Then there is a %0 2 R>0 such that the fundamental solutions G with D % .0/ satisfy G% .0/ D G%0 .0/
for all % %0
as functionals on CV 1 .RnC1 /. Proof. Let %0 2 R>0 be such that for all % %0 P .@ C / is invertible with D % .0/ . Then, if 0 2 C1 .RnC1 / with supp.0/ 0 bounded below and 0 D 1 in a neighborhood of ŒR0 .0/ C ¹.0/ º? , we get hG j i;0 D hG j
0 i;0
D hG j
0 i;0
for all 2 CV 1 .RnC1 /, since .1 R<0 .0/ C ¹.0/ º? . Now, hG%0 .0/ j
0/
C hG j .1
0 / i;0
2 CV 1 .RnC1 / with supp.1
0/
0 i%0 .0/ ;0
D hG%0 .0/ j exp.2%0 .0/ m/
0 i0;0
D hG%0 .0/ j exp.2%.0/ m/ exp.2.% %0 /.0/ m/
0 i0;0
and so in particular jhG%0 .0/ j exp.2%.0/ m/ exp.2.% %0 /.0/ m/ i0;0 j Cj Letting
0 j%0 .0/ ;˛
D C j exp..% %0 /.0/ m/
0 j% .0/ ;˛ :
D exp.2.% %0 /.0/ m/ , this yields with % %0
jhG%0 .0/ j exp.2%.0/ m/ i0;0 j C j exp..% %0 /.0/ m/ e j j% .0/ ;˛ C
0
j% .0/ ;˛
Section 3.1 Partial Differential Equations in H1 .@ C e/
137
2 CV 1 .RnC1 /. This can be rewritten as
for all
e j j% .0/ ;˛ jhG%0 .0/ j i% .0/ ;0 j C for all 2 CV 1 .RnC1 / showing that, by continuous extension, G%0 .0/ can be identified with an element in H˛ .@% .0/ C e/ for which we shall use the same notation G%0 .0/ . Then, with WD % .0/ we have hP .@ C / G%0 .0/ j i;0 D hG%0 .0/ j P .@ C / i;0 D hG%0 .0/ j exp.2%.0/ m/ P .@ C / i0;0 D hG%0 .0/ j exp.2%.0/ m/ P .@2 / i0;0 D hG%0 .0/ j P .@/ exp.2%.0/ m/ i0;0 D hG%0 .0/ j exp.2%0 .0/ m/ P .@/ exp.2%.0/ m/ i%0 .0/ ;0 D hG%0 .0/ j P .@%0 .0/ C %0 .0/ / exp.2.% %0 /.0/ m/ i%0 .0/ ;0 D hP .@%0 .0/ C %0 .0/ / G%0 .0/ j exp.2.% %0 /.0/ m/ i%0 .0/ ;0 D hı 1..sC1/.sC1// j exp.2.% %0 /.0/ m/ i%0 .0/ ;0 D hı 1..sC1/.sC1// j i%0 .0/ ;0 : Thus, G%0 .0/ satisfies not only P .@%0 .0/ C %0 .0/ / G%0 .0/ D ı 1..sC1/.sC1// but also P .@% .0/ C % .0/ / G%0 .0/ D ı 1..sC1/.sC1// : By uniqueness we may conclude G% .0/ D G%0 .0/ . Remark 3.1.52. It should be noted that the calculation applied to the fundamental solution in the last proof extends to more general element f 2 H1 .@%0 .0/ C e/. We have, quite generally, the embedding ¹f 2 H1 .@%0 .0/ C e/ j supp f bounded belowº ,! H1 .@% .0/ C e/ for all % %0 . Moreover, for f 2 H1 .@%1 .0/ C e/ \ H1 .@%2 .0/ C e/ with %1 ; %2 %0 , the solutions u%j of P .@%j .0/ C %j .0/ / u%j D f;
j D 0; 1; 2;
coincide. A remarkable consequence of causality is the semi-local character of our solution theory in the evolutionary case. Indeed, due to linearity, the property of causality
138
Chapter 3 Partial Differential Equations with Constant Coefficients
implies that solutions coincide ‘as long as’ the right-hand sides coincides. More precisely if f; g 2 H1 .@% .0/ C e/ are such that supp.0/ .f g/ is bounded below by a 2 R, then, if P .@/ is evolutionary in direction .0/ , uf WD P .@% .0/ C % .0/ /1 f , ug WD P .@% .0/ C % .0/ /1 g also satisfy supp.0/ .uf ug / a whenever % is sufficiently large. This observation leads to an even more dramatic generalization. C .@ Definition 3.1.53. A linear functional f on CV 1 .RnC1 / is said to be in H1 % .0/ C nC1 e/ or in H1 .@% .0/ C e/ if, for every 2 C1 .R / which is, together with all its derivatives, bounded and has supp.0/ bounded above or below, respectively, we have .m/ f 2 H1 .@% .0/ C e/;
where
.m/ f is defined as the linear functional satisfying . .m/ f /. / WD f .
.m/ /
(3.1.48)
;C for all 2 CV 1 .RnC1 /. The set of such will be denoted by CV 1.0/ .RnC1 / or ; C .@ CV 1.0/ .RnC1 /, respectively. If the functional f is in H1 %.0/Ce/\H1 .@%.0/C loc ˙ e/, then we say f is in H1 .@% .0/ Ce/. For a sequence .fk /k in H1 .@% .0/ Ce/ we ˙ .@ ˙ shall write limk!1 fk 2 H1 % .0/ Ce/ and fk ! limk!1 fk in H1 .@% .0/ Ce/ as k ! 1, if .m/ fk ! .m/ lim fk k!1
in H1 .@% .0/ C e/ for all
;˙
2 CV 1.0/ .RnC1 /, respectively.
Using the suggestive inner product notation for general functionals on CV 1 .RnC1 /, the defining property (3.1.48) reads h .m/ f j i0;0 WD hf j for all 2 CV 1 .RnC1 / and all
.m/ i0;0
;C ; in CV 1.0/ .RnC1 / or CV 1.0/ .RnC1 /, respectively.
Lemma 3.1.54. Let P .@/ be a partial differential expression evolutionary in direction .0/ and let %0 2 R>0 be such that P .@% .0/ C% .0/ / is invertible in H1 .@% .0/ Ce/ ;C for all % %0 . Moreover, given R 2 R>0 , let .0/ ;R 2 CV 1.0/ .RnC1 / be defined by nC1 , with28 .0/ ;R .x/ WD 0 ..0/ x R/; x 2 R Z 0 .t / WD
j.s/ ds Œt;1Œ
Here j is the well-known element of CV 1 .R/ constructed earlier. Of course, any element ' 2 R CV 1 .RnC1 / with R '.x/ dx D 1 could be used. 28
Section 3.1 Partial Differential Equations in H1 .@ C e/
139
C .@ for all t 2 R. Then, for every % %0 and f 2 H1 % .0/ C e/, the limit
U WD lim P .@% .0/ C % .0/ /1 R!1
.0/ ;R
f
C .@ exists in H1 % .0/ C e/ and satisfies
hU j P .@/ i0;0 D hf j i0;0
(3.1.49)
for all 2 CV 1 .RnC1 /.
Proof. First we note that
.0/ ;R
;C
2 CV 1.0/ .RnC1 / so that
.0/ ;R .m/ f
2 H1 .@% .0/ C e/:
The solution uR 2 H1 .@% .0/ C e/ of P .@% .0/ C % .0/ / uR D
.0/ ;R .m/ f
is, for % %0 , well-defined. From the definition of inf supp.0/ .
.0/ ;R1 .m/
.0/ ;R2 .m// f
.0/ ;R , we get, for all R1 ; R2
inf supp.0/ .
.0/ ;R1
D inf supp. . R1 /
2 R,
.0/ ;R2 /
. R2 //
min.R1 ; R2 / 1: Therefore, by causality in direction .0/ , the corresponding solutions must also satisfy inf supp.0/ .uR1 uR2 / min.R1 ; R2 / 1: In other words, we have uR1 D uR2
in ŒR<min¹R1 ;R2 º1 .0/ C ¹.0/ º? :
Thus R 7! huR j i%0 .0/ ;0 is eventually constant (i.e. for sufficiently large argument); we can define a linear functional U by U. / WD hU j i%0 .0/ ;0 WD lim huR j i%0 .0/ ;0 R!1
for all 2 CV 1 .RnC1 /. Moreover, by the same argument it follows that for every ;C 2 CV 1.0/ .RnC1 / we have .m/ U D .m/ uR if R is sufficiently large. Thus, in ;C particular, .m/ U 2 H1 .@% .0/ C e/ and, since 2 CV 1.0/ .RnC1 / was arbitrary, C .@% .0/ C e/: U 2 H1
140
Chapter 3 Partial Differential Equations with Constant Coefficients
Moreover, we have hU j P .@/ i0;0 D lim huR j P .@/ i0;0 R!1
D lim huR j exp.2%.0/ m/ P .@/ i%0 .0/ ;0 R!1
D lim huR j P .@% .0/ C % .0/ / exp.2%.0/ m/ i%0 .0/ ;0 R!1
D lim hP .@% .0/ C % .0/ / uR j exp.2%.0/ m/ i%0 .0/ ;0 R!1
D lim h
.0/ ;R .m/ f
j exp.2%.0/ m/ i%0 .0/ ;0
D lim h
.0/ ;R .m/ f
j i0;0
R!1 R!1
D lim hf j R!1
.0/ ;R .m/ i0;0
D hf j i0;0 D f . / for all 2 CV 1 .RnC1 /. Remark 3.1.55. The particular form of .0/ ;R is used here only for the sake of definiteness. Any sequence . k /k in C1 .RnC1 / with supp.0/ k bounded above as well as supp.0/ .1 k / bounded below and inf supp.0/ .1 k / ! 1 as k ! 1 could be used to serve the same purpose, producing the same U . Given any linear functional U on CV 1 .RnC1 /, and defining hU j i;0 WD hU j exp.2 m/ i0;0 WD U.exp.2 m/ / for 2 CV 1 .RnC1 /, 2 RnC1 , we see that (3.1.49) can equivalently be written (with D % .0/ ) as hU j i;0 D hf j P .@ C / i;0 : This notation makes (3.1.49), in a more suggestive way, a generalization of the solution theory in H1 .@% .0/ C e/. We have chosen to write (3.1.49) in order to be closer to the usual concept of a weak solution. C .@ The element U 2 H1 % .0/ C e/ will – because of the relation (3.1.49) – be C .@ called a weak solution in H1 % .0/ C e/ of the equation C P .@/ U D f 2 H1 .@% .0/ C e/:
This solution U will also be referred to as a solution in the weak sense. The last lemma shows its existence; the next shows uniqueness.
Section 3.1 Partial Differential Equations in H1 .@ C e/
141
Lemma 3.1.56. Let P .@/ be a partial differential expression evolutionary in direction .0/ and let %0 2 R>0 be such that P .@% .0/ C% .0/ / is invertible in H1 .@% .0/ Ce/ C .@ for all % %0 . A weak solution of the equation P .@/ U D f in H1 % .0/ C e/ is unique. C .@ Proof. Let U 2 H1 % .0/ C e/ be a weak solution of
P .@/ U D 0: Then
.0/ ;R .m/ U
2 H1 .@% .0/ C e/ solves the equation
P .@/
.0/ ;R .m/ U
D .P .@/
.0/ ;R .m/
.0/ ;R .m/ P .@// U
2 H1 .@% .0/ C e/:
Since .P .@/ .0/ ;R .m/ .0/ ;R .m/ P .@// contains terms with at least one differentiation acting on .0/ ;R we have inf supp.0/ .P .@/
.0/ ;R .m/
.0/ ;R .m/ P .@// U
R 1:
Therefore, by causality we must also have inf supp.0/
.0/ ;R .m/ U
R 1:
Letting R ! 1 we see that U. / D 0 for all 2 CV 1 .RnC1 /, i.e. U D 0. This lemma establishes the existence of the solution operator C C P .@/1 W H1 .@% .0/ C e/ ! H1 .@% .0/ C e/:
Theorem 3.1.57. Let P .@/ be a partial differential expression evolutionary in direction .0/ and let %0 2 R>0 be such that P .@% .0/ C % .0/ / is invertible in C .@ H1 .@% .0/ C e/ for all % %0 . Then, for every f 2 H1 % .0/ C e/, we have a C unique solution U 2 H1 .@% .0/ C e/ solving P .@/ U D f in the weak sense. Moreover, U depends continuously on f in the sense that the mapping 1 .0/ ;R .m/ f 7! .0/ ;R2 .m/ P .@/ f C .@ is continuous in H1 % .0/ C e/ for every R 2 R.
Proof. The unique existence has been established in Lemma 3.1.54 (existence) and Lemma 3.1.56 (uniqueness). The continuity statement needs to be shown. We find P .@/
.0/ ;R .m/ U
D
.0/ ;R .m/ f
C .P .@/
.0/ ;R .m/
2 H1 .@% .0/ C e/
.0/ ;R .m/ P .@// U
142
Chapter 3 Partial Differential Equations with Constant Coefficients
and so .0/ ;R .m/ U
D uR C P .@% .0/ C % .0/ /1 .P .@/
.0/ ;R .m/
.0/ ;R .m/ P .@// U:
Recalling from the proof of Lemma 3.1.56 that, due to causality, inf supp.0/ P .@% .0/ C % .0/ /1 .P .@/
.0/ ;R .m/
.0/ ;R .m/ P .@// U
R1
we find .0/ ;R2 .m/ U
D
.0/ ;R2 .m/
.0/ ;R .m/ U
D
.0/ ;R2 .m/ uR :
This yields the desired continuity result. Thus, we see that we really do not lose much generality by restricting our attention to the globally constrained Sobolev lattices H1 .@ C e/. For this reason we shall also henceforth not bother to point out the corresponding semi-local in time result, which can always easily be obtained by the above ideas.
3.1.11 Initial Value Problems As we have shown earlier, we may for an evolutionary problem always assume that the direction of evolution is the first coordinate direction. To emphasize this, it is usual to renumber the coordinate to refer to the direction of evolution as the direction of the ‘0-th’ variable. We recall that the corresponding direction .0/ D e0 D .1; 0; : : : ; 0/ 2 R1Cn , n 2 N, is frequently referred to as ‘time’. We have adopted this terminology already to discuss various applications we shall turn to in detail later. In this spirit we shall, henceforth, always consider evolutionary problems in the form P .@0 ; b @/ u D f , where it is understood that P .@0 ; b @/ is evolutionary in direction e0 making @0 the timederivative and b @ D .@1 ; : : : ; @n / indicates the spatial derivatives. We shall consider solving this equation in H1 .@% e0 C .1; e// D H1 .@0;% C 1; b @ C e/, for % 2 R>0 b sufficiently large (to assure that P .@0;% C%; @/ is invertible in H1 .@0;% C1; b @Ce/), e WD .1; : : : ; 1/ 2 Rn . Recalling the tensor product construction used to obtain H1 .@0;% C 1; b @ C e/, we note that @ C e/ D H˛0 .@0;% C 1/ ˝ Hb .b @ C e/ H˛ .@0;% C 1; b ˛ where ˛ D .˛0 ; ˛1 ; : : : ; ˛n / 2 Z1Cn , b ˛ D .˛1 ; : : : ; ˛n / 2 Zn . When ˛ is a multi.ij / index ..s C 1/ .t C 1//-matrix .˛ /iD0;:::;sIj D0;:::;t with .1 C n/-tuples as entries .ij / .ij / .ij / .ij / ˛D ˛ .ij / D .˛0 ; ˛1 ; : : : ; ˛n / in Z1Cn , then ˛0 D .˛0 /iD0;:::;sIj D0;:::;t and b .ij / .ij / .ij / .ij / n ˛ D .˛1 ; : : : ; ˛n / in Z . It is .b ˛ /iD0;:::;sIj D0;:::;t with multi-index entries b
Section 3.1 Partial Differential Equations in H1 .@ C e/
143
to single out the time variable that we shall also write @ C e/ H1 .@0;% C 1/ ˝ H1 .b and H˛0 .@0;% C 1/ ˝ H1 .b @ C e/
for
for H1 .@0;% C 1; b @ C e/ [ ˛ b
H˛0 .@0;% C 1/ ˝ Hb .b @ C e/: ˛
Thus we shall recover the intuition of an initial value problem from an abstract problem of the form P .@0 ; b @/ u D f where in this context f has a specific form. To analyze this issue more closely we note that the polynomial matrix P can be written in the form P .i ; i / D
p X
aj .i / .i /j
j D0
where aj is a polynomial matrix for j D 0; : : : ; p, p 2 N. We assume that ap is not the zero polynomial matrix, so that p is indeed the maximal degree of the single variable polynomial matrix 7! P .i ; i / for 2 RnC1 . For the type of problem we wish to discuss here, it is customary to assume that ap is a constant invertible matrix and without loss of generality we assume ap 1.ss/ :
(3.1.50)
Thus, we have P .@0 ; b @/ D
p X
j p aj .b @/ @0 D @0 C
j D0
p1 X
j aj .b @/ @0 :
j D0
Now let u be smooth on R0 Rn , e.g. u D R>0 .m0 / ; 2 CV 1 .RnC1 /, and define f WD P .@0 ; b @/ u. Then f is smooth on R>0 and by definition u solves classically the equation
P .@0 ; b @/ u D f
on R>0 . The regular part of f , i.e. the part in H0 .@0;% C 1; b @ C e/, is just f0 WD R>0 .m0 / P .@0 ; b @/ . Moreover, u D f D 0 on the time interval R<0 , i.e. on R<0 Rn . So f differs from its regular part f0 by an element with time support in ¹0º, i.e. supp0 .f f0 / ¹0º: @/ R>0 .m0 / R>0 .m0 / P .@0 ; b @// and letting Now, f f0 D .P .@0 ; b e WD exp.2% m0 /
144
Chapter 3 Partial Differential Equations with Constant Coefficients
we get @/ R>0 .m0 / R>0 .m0 / P .@0 ; b @// j i% e0 ;0 h.P .@0 ; b D h.P .@0 ; b @/ R>0 .m0 / R>0 .m0 / P .@0 ; b @// j exp.2% m0 / i0;0 D h j R>0 .m0 / P .@0 ; b @/ ei0;0 hR>0 .m0 / P .@0 ; b @/ j ei0;0 Z D .h j P .@0 ; b @/ ei0 .t / hP .@0 ; b @/ j ei0 .t // dt Œ0;1Œ
Z D
p X
Œ0;1Œ j D1
Z D
j .haj .b @/ j .@0 /j ei0 .t / h.@0 aj .b @/ /.t; / j e.t; /i0 / dt
p X
j .h@0 aj .b @/ j .@0 /j 1 ei0 .t / h@0 aj .b @/ j ei0 .t // dt
Œ0;1Œ j D1
Z
p X
.@0 haj .b @/ j .@0 /j 1 ei0 .t // dt:
(3.1.51)
Œ0;1Œ j D1
Continuing in this fashion we obtain eventually @/ R>0 .m0 / R>0 .m0 / P .@0 ; b @// j i% e0 ;0 h.P .@0 ; b Z
D
D
jX 1 p X h@k0 aj .b @/ j .@0 /j k1 ei0 .t / dt @0
Œ0;1Œ j D1
1 p jX X
kD0
haj .b @/ .@k0 /.0; / j ..@0 /j k1 e/.0; /i0
j D1 kD0
D
1 p jX X
hı ˝ .aj .b @/ .@k0 /.0; // j .@0 /j k1 ei0;0
j D1 kD0
D
1 p jX X
j k1
ı ˝ .aj .b @/ .@k0 /.0; // j ei0;0
j k1
ı ˝ .aj .b @/ .@k0 /.0; // j i;0 :
h@0
j D1 kD0
D
1 p jX X
h@0
j D0 kD0
Thus we have found f f0 D
1 p jX X j D0 kD0
j k1
@0
ı ˝ .aj .b @/ .@k0 /.0; //:
(3.1.52)
Section 3.1 Partial Differential Equations in H1 .@ C e/
145
In other words, u solves the equation @/ u D f0 C P .@0 ; b
1 p jX X
j k1
@0
ı ˝ .aj .b @/ .@k0 /.0; //:
j D0 kD0
Obviously, we have .@k0 u/.0C; / D .@k0 /.0; / for k D 0; : : : ; p 1, which is what in classical terms would be called initial conditions at time zero. This suggests the following definition of an initial value problem in the framework of our Sobolev lattice structure. Definition 3.1.58. Let P .@0;% C %; b @/ be an evolutionary partial differential expression and %0 2 R>0 be such that P .@0;% C %; b @/ is invertible for all % %0 . Then the problem of finding the solution u 2 H1 .@0;% C 1; b @ C e/ of P .@0 ; b @/ u D f0 C
1 p jX X
j k1
@0
ı ˝ .aj .b @/ u0;k /;
(3.1.53)
j D0 kD0
where f0 2 H0 .@0;% C 1/ ˝ H1 .b @ C e/ with supp0 f0 R0 and u0;k 2 b H1 .@ C e/, k D 0; : : : ; p 1, % %0 , is called the abstract initial value problem for P .@0 ; b @/ with initial data .u0;k /kD0;:::;p1 and source term f0 . If specifically f0 D 0, we shall speak of the abstract initial value problem for P .@0 ; b @/ with initial data .u0;k /kD0;:::;p1 . That the abstract initial value problem has a unique solution u 2 H1 .@0;% C 1; b @ C e/ which depends continuously on the data f0 and .u0;k /kD0;:::;p1 is clear from the general theory. The question arises, however, in what sense does this solution assume its initial data, i.e. do we and in which sense do we have right-continuity (with respect to time) of the solution u and its first .p 1/ time derivatives. Pp Pj 1 j k1 Before we explore this question let us first consider the term j D0 kD0 @0 ı ˝ .aj .b @/ u0;k / on the right-hand side of (3.1.53). Recalling that in the 1-dimensional case we found @0 R>0 D ı;
(3.1.54)
it is not hard to see that
j @0
8 kj m0 ˆ ˆ < .kj /Š R>0
mk0 D R >0 ˆ kŠ R>0 ˆ : j k1 @0 ı
for j < k; for j D k; for j > k
146
Chapter 3 Partial Differential Equations with Constant Coefficients
for all j; k 2 N. This yields, for j > k, j k1
@0
j k
ı D @0
j
R>0 D @0
mk0 kŠ R>0
and so 1 p jX X
j k1
@0
j D0 kD0
D P .@0 ; b @/
p1 X
p1 X kD0
D P .@0 ; b @/
jX 1
aj .b @/ @0 j
j D0
kD0
D P .@0 ; b @/
p X
ı ˝ .aj .b @/ u0;k / D
p1 X kD0
mk0 ˝ u0;k kŠ R>0
p X
kD0 j aj .b @/ @0
j D0
mk0 ˝ u0;k kŠ R>0
p1 X kDj
mk0 ˝ u0;k kŠ R>0
p p1 X X mkj mk0 0 R>0 ˝ u0;k aj .b @/ ˝ u0;k kŠ .k j /Š R>0 j D0
kDj
pj p X X1 mk mk0 0 R>0 ˝ u0;k aj .b @/ ˝ u0;kCj : kŠ kŠ R>0 j D0
kD0
This shows that solving the abstract initial value problem for data f0 , .u0;k /kD0;:::;p1 is equivalent to solving the abstract initial value problem with source term f0
p X
aj .b @/
j D0
pj X1 kD0
mk0 ˝ u0;kCj kŠ R>0
and vanishing initial data .0/kD0;:::;p1 . Let w denote the solution of the latter problem. Then the solution u of the first problem is given by uDwC
p1 X kD0
mk0 ˝ u0;k : kŠ R>0
(3.1.55)
As a matter of simplification we may therefore assume that the initial data are all vanishing. To connect with point-wise concepts we shall use the following simple version of a so-called Sobolev embedding lemma. Lemma 3.1.59. Let D .%; 0; : : : ; 0/, where % 2 R>0 and ˛ 2 Zn , n 2 N. Then the mapping CV 1 .R1Cn / H1 .@0;% C 1/ ˝ H˛ .b @ C e/ ! ¹ 2 C0 .R; H˛ .b @ C e//j j j;˛;1;1=2 < 1º; u 7! .t 7! u.t; //
Section 3.1 Partial Differential Equations in H1 .@ C e/
147
has a continuous extension to all of H1 .@0;% C 1/ ˝ H˛ .b @ C e/ (sometimes called the “trace operator” or the “operator of point-wise evaluation in time”) with respect to the norm of H1 .@0;% C 1/ ˝ H˛ .b @ C e/ and the image norm29 j j;˛;1;1=2 WD sup¹j exp.%t / .t /j˛ j t 2 Rº ² j exp.%t / .t / exp.%s/ .s/j˛ C sup jt sj1=2
ˇ ³ ˇ ˇ t; s 2 R ^ t ¤ s ˇ
@ C e/ for 2 CV 1 .R1Cn /. Moreover, for all u 2 H1 .@0;% C 1/ ˝ H˛ .b 1 sup¹j exp.%t /. u/.t /j˛ j t 2 Rº p j@0 uj;.0;˛/ 2% and
ˇ ³ j exp.%t /. u/.t / exp.%s/. u/.s/j˛ ˇˇ sup t; s 2 R ^ t ¤ s j@ uj;.0;˛/ : ˇ jt sj1=2 ²
@ C e/ and . k /k a sequence in CV 1 .RnC1 / In particular, if f 2 H1 .@0;% C 1/ ˝ H˛ .b converging to f in H1 .@0;% C1/˝H˛ .b @Ce/, then limk!1 k .t; / exists in H˛ .b @Ce/ for all t 2 R, is independent of the particular approximating sequence and equals f . Furthermore, the mapping is injective. Proof. For 2 CV 1 .R1Cn / we have ˇZ t ˇ ˇ ˇ ˇ @0 .u/ duˇˇ j .t / .s/j˛ D ˇ s
˛
sˇ Z ˇsˇ Z t ˇ ˇ ˇ ˇ ˇ t 2 ˇˇ exp.2%u/ duˇˇ ˇˇ j@0 .u/j˛ exp.2%u/ duˇˇ s s s j exp.2%t / exp.2%s/j j@0 j;.0;˛/ 2% s p j exp.2%t / exp.2%s/j j@0 j;.0;˛/ jt sj 2%jt sj p jt sj sup¹exp.%x/j x 2 ¹t; sºº j@0 j;.0;˛/
from which we can read off the desired continuity. With s ! 1, we also see from the second inequality that s 1 j .t /j exp.%t / j j;.1;˛/ : 2% 29
Note that the first term on the right is the so-called Morgenstern norm, [32].
148
Chapter 3 Partial Differential Equations with Constant Coefficients
Moreover, we also calculate j exp.%t / .t / exp.%s/ .s/j˛ ˇZ t ˇ ˇ ˇ D ˇˇ .@0 .exp.%m0 / //.u/ duˇˇ s ˛ sˇ Z ˇ p ˇ ˇ t jt sj ˇˇ j.@0 .exp.%m0 / //.u/j2˛ duˇˇ s sˇ Z ˇ p ˇ ˇ t 2 ˇ j.@0;% /.u/j˛ exp.2%u/ duˇˇ jt sj ˇ s p jt sj j@ j;.0;˛/ ; which shows that the mapping under consideration is a well-defined continuous linear mapping. This mapping can now be extended by the obvious uniform continuity to all of H1 .@0;% C 1/ ˝ H˛ .b @ C e/ due to the density of CV 1 .R1Cn / in H1 .@0;% C 1/ ˝ @ C e/. H˛ .b By substituting k j for in the first or second estimate, we see that . k .t; //k is a Cauchy sequence in H˛ .b @ C e/, if . k /k is a Cauchy sequence in H1 .@0;% C 1/ ˝ b H˛ .@ C e/. Moreover, let . k /k be another such sequence. Then, with k k in place of , we see that k .t; / k .t; / ! 0 in H˛ .@ C e/ as k ! 1, since k k ! 0 in H1 .@0;% C 1/ ˝ H˛ .@ C e/ as k ! 1. This shows that the extension is indeed well-defined by f .t / D lim k .t; /; t 2 R; k!1
where f WD limk!1 k . Finally, to see that W H1 .@0;% C 1/ ˝ H˛ .b @ C e/ ! ¹ 2 C0 .R; H˛ .b @ C e//j j j;˛;1;1=2 < 1º is injective, assume . k /k 2 CV 1 .RnC1 /N with j k j;˛;1;1=2 ! 0 as k ! 1 is @ C e/ with limit f 2 H1 .@0;% C 1/ ˝ a Cauchy sequence in H1 .@0;% C 1/ ˝ H˛ .b b H˛ .@ C e/. We need to show that f D 0. Letting k ! 1 in the equality h@0 k j i.%;0/;.0;˛/ D h k j .@0;% C %/ i.%;0/;.0;˛/ ; where
2 CV 1 .RnC1 / is arbitrary, we obtain h@0 f j i.%;0/;.0;˛/ D 0
Section 3.1 Partial Differential Equations in H1 .@ C e/ for every
149
2 CV 1 .RnC1 / from which we conclude @0 f D 0
@ C e/ and so in H1 .@0;% C 1/ ˝ H˛ .b f D0
in H1 .@0;% C 1/ ˝ H˛ .b @ C e/
follows. Remark 3.1.60. From the estimate p j .t / .s/j˛ jt sj sup¹exp.%x/j x 2 ¹t; sºº j@0 j;.0;˛/ we see that, even for % D 0, we obtain the continuity estimate p j .t / .s/j˛ jt sj j@0 j0;.0;˛/ for all 2 H1 .@0 C 1/ ˝ H˛ .b @ C e/. It is common practice to identify u 2 H1 .@0;% C1/˝H˛ .b @Ce/ with the continuous mapping t 7! u.t / and so to simply write u.t / for . u/.t /, t 2 R. In the evolutionary case, we know that u D 0 on R<0 for a solution u of an initial value problem and, in particular, for u 2 H1 .@0;% C 1/ ˝ H˛ .b @ C e/, we have u.0/ D 0: With the last lemma we see that we can define for any f 2 H1 .@0;% C1/˝H˛ .@Ce/ the point-wise evaluation in time (or “time trace”) as the mapping t 7! f .t /, where f .t / is given as the element of the singleton ˇ ° ± ˇ lim k .t; /ˇ. k /k 2 CV 1 .RnC1 /N ^ lim k D f : k!1
k!1
Moreover, according to the second estimate of Lemma 3.1.59, we have that t 7! f .t / as a mapping into H˛ .b @ C e/ is (locally) Hölder continuous on R with Hölder 1 exponent 2 . Having made this connection to point-wise continuity of the point-wise evaluation in time we need the next result to obtain sufficient regularity. Note that this is the point where the assumption ap D 1 is used. Pp1 Lemma 3.1.61. Let P .@0;% C %; b @/ D .@0;% C %/p C j D0 aj .b @/ .@0;% C %/j be an evolutionary partial differential expression and let %0 2 R>0 be such that P .@0;% C %; b @/ is invertible for all % %0 . Then @/1 ŒHk .@0;% C 1/ ˝ H1 .b @ C e/ HpCk .@0;% C 1/ ˝ H1 .b @ C e/ P .@0;% C %; b
150
Chapter 3 Partial Differential Equations with Constant Coefficients
for all k 2 Z. Moreover, for every k 2 Z there is a ˇ 2 Zn such that jP .@0;% C %; b @/1 f j;.pCk;˛/ C jf j;.k;˛Cˇ / for some constant C 2 R>0 and all ˛ 2 Zn , f 2 Hk .@0;% C 1/ ˝ Hˇ .b @ C e/. @/u D f . Then Proof. Let P .@0 ; b p p P .@0 ; b @/ u D @0 @0 f
and so p @0
.u
p @0
f/C
p1 X
j aj .b @/ @0 u D 0
j D0
or p
p
@0 .u @0 f / C
p1 X
j p aj .b @/ @0 .u @0 f / D
j D0
p1 X
j p aj .b @/ @0 @0 f
j D0
D
p X
aps .b @/ @s 0 f
sD1
D
@1 0
p1 X
j apj 1 .b @/ @0 f:
j D0
Defining recursively f0 WD f
and
fkC1 WD
p1 X
j apj 1 .b @/ @0 fk
j D0
for k 2 N, we first get f1 2 Hk .@0;% C 1/ ˝ H1 .b @ C e/ and p @/ .u @0 f0 / D @1 P .@0 ; b 0 f1 :
By repeating this construction we confirm by a simple induction that fj 2 Hk .@0;% C @ C e/ for all j 2 N and 1/ ˝ H1 .b N 1 X j p P .@0 ; b @/ u @0 @0 fj D @N fN 0 j D0 p
for all N 2 N. Indeed, the case N D 0 is clear and with uN WD u@0 assuming P .@0 ; b @/ uN D @N fN ; 0
PN 1 j D0
j
@0 fj ,
Section 3.1 Partial Differential Equations in H1 .@ C e/
151
we get p1 X j p N 1 b P .@0 ; @/ .uN @0 @0 fN / D @0 apj 1 .b @/ @0 @N fN 0 j D0 .N C1/
D @0
p1 X
j apj 1 .b @/ @0 fN
j D0 .N C1/
D @0
fN C1
and p
fN uN C1 D uN @0 @N 0 Du
p @0
N 1 X
j
p
@0 fj @0 @N fN 0
j D0 p
D u @0
N X
j
@0 fj :
j D0
@ C e/. Then, since P .@0 ; b @/1 has a finite Now, let f 2 Hk .@0;% C 1/ ˝ H1 .b regularity loss, for N 2 N sufficiently large we get that u
p @0
N X
j @0 fj 2 HpCk .@0;% C 1/ ˝ H1 .b @ C e/
j D0
and so, since
PN
j j D0 @0
fj 2 Hk .@0;% C 1/ ˝ H1 .b @ C e/, also
u 2 HpCk .@0;% C 1/ ˝ H1 .b @ C e/: We also read off that by construction 1 @/1 f D P .@0;% C %; b @/1 @N fN C1 C @0 u D P .@0;% C %; b 0
p
N X
j
@0 fj
j D0
from which finally the desired continuity estimate follows (for N 2 N sufficiently large). In other words, we have shown that the time-regularity of a solution is always equal to time-degree p of the polynomial plus the time-regularity of the right-hand side f .
152
Chapter 3 Partial Differential Equations with Constant Coefficients
Pp1 Theorem 3.1.62. Let P .@0;% C %; b @/ D .@0;% C %/p C j D0 aj .b @/ .@0;% C %/j be an evolutionary partial differential expression and let %0 2 R>0 be such that @/ is invertible for all % %0 . Furthermore, let u 2 H1 .@0;% C1; b @Ce/ P .@0;% C%; b be the solution of the corresponding abstract initial value problem for a source term f and initial data .u0;k /kD0;:::;p1 , % %0 . Then, .@k0 u/.0C/ D u0;k
for k D 0; : : : ; p 1:
Proof. We have 1 p jX X
@/ u D f C P .@0 ; b
j k1
@0
ı ˝ .aj .b @/ u0;k /:
j D0 kD0
According to (3.1.55) we also have pj p1 p X mk X X1 mk b R>0 ˝ u0;k D f aj .b @/ ˝ u0;kCj : P .@0 ; @/ u kŠ kŠ R>0 j D0
kD0
kD0
Since the right-hand side is in H0 .@0;% C 1/ ˝ H1 .b @ C e/, we find from Lemma 3.1.61 that u
p1 X kD0
mk ˝ u0;k 2 Hp .@0;% C 1/ ˝ H1 .b @ C e/: kŠ R>0
This yields, in particular, p1 X
j
j
j
p X
Uj WD @0 u @0
kD0
D @0 u
kDj
jX 1
mkj ˝ u0;k .k j /Š R>0
j k1
@0
mk ˝ u0;k kŠ R>0 (3.1.56)
ı ˝ u0;k 2 H1 .@0;% C 1/ ˝ H1 .b @ C e/
kD0
for j D 0; : : : ; p 1. We have Uj .0/ D 0
for j D 0; : : : ; p 1;
(3.1.57)
since causality yields that Uj D 0 on R<0 RnC1 and therefore also its point-wise evaluation Uj .t / D 0 for t 2 R<0 . From (3.1.56), (3.1.57) we read off that j
j
@0 u.0C/ u0;j D @0 u.0/ D 0
for j D 0; : : : ; p 1:
Section 3.1 Partial Differential Equations in H1 .@ C e/
153
This theorem confirms that a solution of an abstract initial value problem satisfies its initial conditions classically in the sense of equality in H1 .b @ C e/. Indeed, here we have the actual justification for calling problem (3.1.53) an initial value problem. Corollary 3.1.63. Under the assumptions of Theorem 3:1:62, the solution u of the abstract initial value problem is .p 1/-times continuously differentiable on R>0 . Proof. We first recall that
w WD u
p1 X kD0
mk ˝ u0;k 2 Hp .@0;% C 1/ ˝ H1 .b @ C e/: kŠ R>0
Since by assumption p 2 N, this shows that t 7! w.t / must be continuous (indeed Hölder continuous with exponent 12 ). Observing that h1 . .t C h; x/ .t; x// .@0 /.t; x/ Z 1 h D ..@0 /.t C s; x/ .@0 /.t; x// ds h 0 Z Z 1 h tCs 2 D .@0 /.r; x/ dr ds; h 0 t for all 2 CV 1 .RnC1 /, h 2 R, we get, similar to the proof of Lemma 3.1.59, jh1 . .t C h; x/ .t; x// .@0 /.t; /j˛ ˇZ Z ˇ ˇ 1 ˇˇ h tCs 2 ˇ j.@
/.r; /j dr ds ˛ 0 ˇ ˇ jhj 0 t ˇZ Z ˇ1=2 ˇ Z h Z tCs ˇ1=2 ˇ ˇ ˇ 1 ˇˇ h tCs 2 2 ˇ ˇ ˇ exp.2% r/ dr ds j.@
/.r; /j exp.2% r/ dr ds 0 ˛ ˇ ˇ ˇ jhj ˇ 0 t 0 t ˇ1=2 ˇZ h ˇ exp.2% s/ 1 ˇˇ 1 p ds ˇ j@20 j.%;0/;.0;˛/ exp.%t / ˇˇ 2% jhj 0 ˇ ˇ p ˇ exp.2% h/ 1 2%h ˇ1=2 2 ˇ ˇ j@ j.%;0/;.0;˛/ jhj exp.%t / ˇ 0 ˇ .2%h/2 r jhj exp.%t / exp.%jhj/ j@20 j.%;0/;.0;˛/ 2 r jhj exp.%t / exp.%jhj/ .1 C %/2 j j.%;0/;.2;˛/ : 2
154
Chapter 3 Partial Differential Equations with Constant Coefficients j 1
Replacing by @0
and taking limits we obtain j 1
j 1
j
jh1 .@0 .t C h; / @0 .t; // .@0 /.t; /j˛ r jhj j 1 exp.%t / exp.%jhj/ .1 C %/2 j@0 j.%;0/;.2;˛/ 2 r jhj exp.%t / exp.%jhj/ .1 C %/j C1 j j.%;0/;.j C1;˛/ 2 for all 2 Hp .@0;% C 1/ ˝ H˛ .b @ C e/, j D 1; : : : ; p 1, ˛ 2 ZnC1 . This proves that every 2 Hp .@0;% C 1/ ˝ H˛ .b @ C e/ is .p 1/-times classically differentiable and the classical derivatives at a point t coincide with the point-wise in time evaluation of the generalized derivative at t . We already know that these derivatives are continuous (indeed Hölder continuous with exponent 12 ). Having seen that w is .p 1/-times continuously differentiable and since
wjR>0 D ujR>0
p1 X kD0
.mjR>0 /k ˝ u0;k ; kŠ
we get the same differentiability property for u on R>0 . The regularity mechanism shown in the last result is actually a consequence of a more general situation described in the following lemma. @ C e/. Then Lemma 3.1.64. Let u 2 R>0 .m0 / Hp .@0;% C 1/ ˝ H1 .b
u
p1 X kD0
mk ˝ .@k0 u/.0C/ 2 Hp .@0;% C 1/ ˝ H1 .b @ C e/: kŠ R>0
Proof. Let w D u H1 .b @ C e/ and
Pp
mk kD0 kŠ
R>0 ˝ .@k0 u/.0C/. Then w 2 H0 .@0;% C 1/ ˝
p
h@0 wj i.%;0/;0 D h wj.@0;% C %/p i.%;0/;0 D h uj.@0;% C %/p i.%;0/;0
p1 XZ kD0
R>0
tk h u0;k j..@0;% C %/p /.t /i0 exp.2%t / dt: kŠ
Section 3.1 Partial Differential Equations in H1 .@ C e/
155
Now, by assumption u D R>0 .m0 / U with U 2 Hp .@0;% C 1/ ˝ H1 .b @ C e/ and so p
h@0 wj i.%;0/;0 D h R>0 .m0 / U j.@0;% C %/p i.%;0/;0 Z D hU.t /j..@0;% C %/p /.t /i0 exp.2%t / dt R>0
Z D
R>0
hU.t /j..@0 /p exp.2%m0 / /.t /i0 dt
Z D
R>0
h@0 U.t /j..@0 /p1 exp.2%m0 / /.t /i0 dt
C hU.0/j..@0 /p1 exp.2%m0 / /.0/i0 : Repeating this integration by parts argument we obtain by induction that p
p
h @0 uj i.%;0/;0 D hR>0 .m0 / @0 U j i.%;0/;0 C
p1 X
h.@k0 U /.0/j..@0 /pk1 exp.2%m0 / /.0/i0
kD0 p
D hR>0 .m0 / @0 U j i.%;0/;0 C
p1 X
hı ˝ .@k0 U /.0/j.@0 /pk1 exp.2%m0 / i0;0
kD0 p
D hR>0 .m0 / @0 U j i.%;0/;0 C
D p1 X
pk1
@0
ˇ E ˇ ı ˝ .@k0 U /.0/ˇ
kD0
.%;0/;0
:
Comparison with the above yields p
p
h@0 wj i.%;0/;0 D hR>0 .m0 / @0 U j i.%;0/;0 : This can be estimated by p
p
jh@0 wj i.%;0/;0 j j@0 U j.%;0/;˛ j j.%;0/;˛ p for some sufficiently large ˛ 2 N nC1 , proving that @0 w 2 H0 .@0;% C1/˝H˛ .b @Ce/. b Since @0 D @0;% C % is a bijection between HkC1 .@0;% C 1/ ˝ Hˇ .@ C e/ and @ C e/ for all k 2 Z; ˇ 2 ZnC1 , we get Hk .@0;% C 1/ ˝ Hˇ .b
w 2 Hp .@0;% C 1/ ˝ H˛ .b @ C e/:
156
Chapter 3 Partial Differential Equations with Constant Coefficients
These findings suggest that we formulate the classical initial value problem in the Sobolev lattice space H1 .@.%;0/ C e/, with % 2 R>0 sufficiently large, and seek to find u 2 R>0 .m0 / ŒHp .@0;% C 1/ ˝ H1 .b @ C e/ such that P .@0 ; b @/ u D f 2 R>0 .m0 / ŒH0 .@0;% C 1/ ˝ H˛ .b @ C e/ and
j .@0 u/.0C/ D u0;j 2 H1 .b @ C e/
on R>0
for j D 0; : : : ; p 1:
In the above we have just proved existence of a solution to this problem.
3.1.12 Some Applications to Linear Partial Differential Equations of Mathematical Physics To illustrate the framework developed so far, we shall now consider a number of examples and explicitly calculate fundamental solutions as elements of the Sobolev lattice associated with @ C e. The technical ideas are of course classical and well known, see e.g. [48]. 3.1.12.1
Transport Equation
Possibly the simplest evolutionary partial differential expression is featured by the transport equation .@0 C s b @/ u D f; 1 s with where s 2 Rn n ¹0º, which describes transport of a substance in direction jsj scalar velocity jsj. The term s b @ is frequently referred to as a drift or convection
term. Considering the polynomial expression ji C % C i s j2 D . C s /2 C %2 , % 2 Rn¹0º, we see that .@0 Cs b @/ is reversibly evolutionary (in the forward, % > 0, or backward, % < 0, time direction). As our earlier considerations suggest the (forward) fundamental solution s;C can be found by calculating the inverse Fourier–Laplace transform 1 1 s;C D L .%;0/ i m0 C % C i s m .2/.nC1/=2 which can be done by applying Cauchy’s integral formula. By definition we have (with D .%; 0; : : : ; 0/, % 2 R>0 ,) for 2 CV 1 .RnC1 / ˇ ˇ 1 1 ˇ b h s;C j i;0 D
. i / .2/.nC1/=2 i m0 C % C i s m ˇ 0;0 Z Z 1 1 b
. i %; / d d (3.1.58) D .2/.nC1/=2 R Rn i C % i s Z R Z 1 1 b D
. i %; / d d : lim .2/.nC1/=2 R!1 R Rn i C % i s
Section 3.1 Partial Differential Equations in H1 .@ C e/ Assuming as a matter of convenience that D 0 ˝ Z
R
R
Z
D Z D and then Z
Z Rn
Z
Rn
Rn
Z
157
, we evaluate further
1 b
. i %; / d d i C % i s R R R R
1 b
0 . i %/ b./ d d i C % i s 1 b
0 . i %/ d b./ d ; i C % i s
R
1 b
0 . i %/ d R i C % i s Z Z R 1 1 exp.i t / exp.%t / 0 .t / dt d Dp 2 R i C % i s R Z Z R 1 1 Dp exp.i t / exp.%t / 0 .t / d dt: 2 R R i C % i s
Now, Cauchy’s integral formula yields Z 1 exp.i t / d D 2 exp. t %/ exp.i t s / R; i C % i s where R; is the closed curve connecting consecutively the points R, R i R, R i R, R; R in the complex plane by straight lines. Taking R;C as the closed curve connecting consecutively the points R; R; R C i R; R C i R; R in the complex plane by straight lines, since the pole i % s is outside of the contour R;C , we get Z 1 exp.i t / d D 0: R;C i C % i s These observations yield Z
R
1 exp.i t / d i C % is R Z R 1 exp.i.R C i u/t / du Ci R C i u C i% C s 0 Z R 1 exp.i .u C i R/ t / du i u C i R C i% C s R Z R 1 exp.i.R C i u/t / du D 0 i R C i u C i% C s 0
(3.1.59)
158
Chapter 3 Partial Differential Equations with Constant Coefficients
and Z i
R
1 exp.i t / d C i% C s
R
Z
Ci Z Ci Z i
R
1 exp.i.R i u/t / du R i u C i % C s
R
1 exp.i .u i R/ t / du u iR C i% C s
0
R R 0
(3.1.60)
1 exp.i.R i u/t / du R iu C i% C s
D 2 exp. t %/ exp.i t s /: For t > 0 we see from (3.1.60) that Z
R
1 exp.i t / d ! 2 exp. t %/ exp.i t s / C i% C s
i R
as R ! 1. For t < 0 we likewise see from (3.1.59) that Z
R
1 exp.i t / d ! 0 C i% C s
R
as R ! 1. The latter confirms our knowledge that s;C D 0 on R<0 . By an exchange of limits we have Z
R
1 b
0 . i %/ d C i % Cs R Z R>0 .t / exp.2 t %/ exp.i t s / 0 .t / dt ! .2/1=2
i
R
as R ! 1. Substituting this into (3.1.58) we get Z Z n=2 h s;C j i;0 D .2/ exp.2 t %/ exp.i t s / 0 .t / dt b./ d Z D .2/n=2 Z D .2/n=2 and noting that 1 .2/n=2
Rn
Rn
R>0
Z Rn
R>0
Z
R>0
Z
Rn
exp.i t s / 0 .t / b./ exp.2 t %/ dt d exp.i t s / b./ d 0 .t / exp.2 t %/ dt;
exp.i t s / b./ d D
.t s/
Section 3.1 Partial Differential Equations in H1 .@ C e/
159
we get Z h s;C j i;0 D
R>0
0 .t / .t s/ exp.2 t %/ dt:
(3.1.61)
Let M be a d -dimensional submanifold in RmC1 and define the linear functional Z ıM . / WD
.x/ d Vx (3.1.62) x2M
for all 2 CV 1 .RmC1 /, where d V denotes the d -dimensional volume element30 on M . We shall call ıM a ı-layer31 on M . Using the ı-layer on the line t 7! .t; t s/, i.e. on ŒR>0 .1; s/ as a 1-dimensional submanifold in Rn , we see that Z 1
.t; x/ exp.2 t %/ d V.t;x/ h s;C j i;0 D p 1 C s 2 .t;x/2ŒR>0 .1;s/ 1 D p ıŒR>0 .1;s/ .exp.2 m0 %/ /: 1 C s2 Writing the latter as usual in the form ˇ ˇ 1 1 ıŒR>0 .1;s/ .exp.2 m0 %/ / D p ıŒR>0 .1;s/ ˇˇ ; p 2 2 1Cs 1Cs ;0 we get s;C D p
1 1 C s2
ıŒR>0 .1;s/ :
As a by-product we have that @ C e/: ıŒR>0 .1;s/ 2 H1 .@0;% C 1; b From the general theory we know that p
1 1 C s2
ıŒR>0 .1;s/
30 In the case d D 1, we shall also write ds for d V and speak of a line element. If d D 2 or x x d D m then M is a surface or hypersurface, respectively, and we also write dSx for d Vx and call dSx the surface element at x. 31 Even the case d D 0 can be incorporated by noting that a 0-dimensional manifold is composed of isolated points and for an isolated point set ¹aº we let
ı¹aº WD a ı: In this sense we have ı¹0º D ı:
160
Chapter 3 Partial Differential Equations with Constant Coefficients
satisfies the initial value problem with initial data ı. In particular, we have then 1 ıŒR>0 .1;s/ .0C/ D ı p 1 C s2
in H1 .b @ C e/:
An alternative description of s;C can be given by observing that (3.1.61) can be understood as Z h s;C j i;0 D
.t; u C t s/ exp.2 t %/ d V.t;u/ Z
.t;u/2R>0 ¹0º
Z
t2R>0
D D
t2R>0
hıj t s .t; /i0 exp.2 t %/ dt ht s ıj .t; /i0 exp.2 t %/ dt:
In other words, s;C .t C/ D R0 .t / t s ı D R0 .t / ı¹t sº D R0 .t / ı. t s/; where the last expression is a common abuse of notation motivated by formally applying the rules of substitution for integrals. 3.1.12.2
Acoustics
The propagation of acoustic waves in a homogeneous isotropic medium is, with suitable scaling, modelled approximately by the linearized equations of acoustics given in block matrix form by a .4 4/-partial differential equation !! v 0 b @ @0 C D f; > b p @ 0 where
0
1 @1 b @ WD @ @2 A D grad @3
and b @> D .@1 @2 @3 / D div
and v denotes the velocity field and p the pressure distribution in R1C3 . For 0 1 @ 0 0 0 @1 ! ! B 0 @ 0 0 @2 C @ @0 b 0 b @ C B D P .@0 ; b @/ D @0 C D @ 0 0 @0 @3 A b b @> 0 @ > @0 @1 @2 @3 @0
Section 3.1 Partial Differential Equations in H1 .@ C e/ we find
161
det P .@0 ; b @/ D .@20 b @2 / @20 :
Recall that P .@0 ; b @/ is reversibly evolutionary in the (forward or backward) time direction. In the general a case of a homogeneous medium, there is an additional (con0 stant) block matrix 0 c describing the acoustic material properties. Here a is a positive definite, symmetric .3 3/-matrix and c a positive constant. The acoustic field is then governed by the equation ! 1 a0 v b D f; (3.1.63) A0 .@/ @0 0c p @ where A0 .b @/ WD 0> b . b @ 0 These equations are closely related to the symmetric hyperbolic system
a0 0c
@0 C
3 X 0 ei @i ei> 0 iD1
where ek , k D 1; 2; 3, are the canonical basis vectors of R3 , which is therefore also reversibly evolutionary. The system decomposes into Newton’s law @0 v C a1 b @p D @> v D f2 . Applying the f1 and the conservation of mass equation @0 p C c 1 b substitution R given by .t; x/ 7! .c 1=2 t; a1=2 x/ p we get with the associated isometric re-scaling operation R (R u WD det.R/uıR), R @0 D @0 c 1=2 R ;
R b @> D b @> a1=2 R
and so @0 C
R
a0 0c
1
D @0 c D
1=2
C
a1=2 0 0 c 1=2
a0 0c
1
0 b @ > b @ 0 1
and
R b @ D a1=2 b @ R
!!
!! 0 a1=2 b @ R b @> a1=2 0 ! 1=2 1=2 0 b @ a 0 c c 1=2 a1=2 @0 C > b 0 1 0 @ 0
Thus, from the equation (3.1.63) above, we obtain !! 1=2 a 0 b @ 0 c 1=2 a1=2 R v D R f: @0 C b R p 0 c 1=2 @> 0
0 1
! R :
162
Chapter 3 Partial Differential Equations with Constant Coefficients
This shows that without loss of generality we may restrict our attention to the isotropic case, i.e. a D 1.33/ , c D 1. The fundamental solution is given by !! ! 0 b @ 0 b @ GC WD @0 @1 0 g1;C 1.44/ C R>0 ˝ ı 1.44/ ; b b @> 0 @> 0 where g1;C is the fundamental solution for the scalar partial differential expression @20 b @2 of the so-called wave equation. This becomes apparent from considering the minimal polynomial 7! .@0 / ..@0 /2 b @2 /: @ , @/ WD 0> b It follows that, with A0 .b b @ 0 @//1 .@0 A0 .b 2 b2 1 D .3 @20 3 .@0 A0 .b @// @0 C .@0 A0 .b @//2 b @2 / @1 0 .@0 @ / 2 b2 1 D .@20 C 3 A0 .b @/ @0 2 A0 .b @/ @0 C A0 .b @/2 b @2 / @1 0 .@0 @ / 2 b2 1 D .@20 C A0 .b @/ @0 C A0 .b @/2 b @2 / @1 0 .@0 @ / :
We can re-write this as 2 b2 1 @//1 D .@20 b @2 C A0 .b @/.@0 C A0 .b @/// @1 .@0 A0 .b 0 .@0 @ / 2 b2 1 D .@0 C A0 .b @// A0 .b @/ @1 C @1 0 .@0 @ / 0 :
(3.1.64)
In order to determine the fundamental solution of the partial differential equations of acoustics it therefore remains to calculate the fundamental solution g1;C of the wave equation, which can again be done routinely by applying residue calculus of complex analysis. As in the previous case we find .2/2 hg1;C j i;0 ˇ ˇ 1 ˇb D
. i / .i m0 C %/2 C m2 ˇ 0;0 Z Z 1 b D
. i %; / d d 2 2 (3.1.65) R R3 .i C %/ C Z Z 1 b D lim lim
. i %; / d d "!0C R!1 2ŒR;R 2R3 nB.0;"/ 2 i jj .i C%Ci jj/ Z Z 1 b
. i %; / d d : C lim lim "!0C R!1 2ŒR;R 2R3 nB.0;"/ 2 i jj .i C%i jj/
Section 3.1 Partial Differential Equations in H1 .@ C e/ With D 0 ˝ we evaluate further Z Z
2ŒR;R
Z
D
2R3 nB.0;"/
Z
2R3 nB.0;"/
Z D
Z
2R3 nB.0;"/
1 b
. i %; / d d 2 jj . C i % ˙ jj/
2ŒR;R
1 b
0 . i %/ b./ d d 2 jj . C i % ˙ jj/
2ŒR;R
1 b
0 . i %/ d b./ d ; 2 jj . C i % ˙ jj/
where Z
1 b
0 . i %/ d
2ŒR;R C i % ˙ jj Z Z 1 1 exp.i t / exp.%t / 0 .t / dt d Dp 2 2ŒR;R C i % ˙ jj R Z Z 1 1 exp.i t / exp.%t / 0 .t / d dt: Dp C i % ˙ jj 2 R 2ŒR;R
Now, residue calculus yields Z 1 exp.i t / d D 2i exp. t %/ exp.˙i t jj/ R; C i % ˙ jj Z
and
R;C
1 exp.i t / d D 0; C i % ˙ jj
where R;˙ are the contours in the previous example. For t > 0 we get Z
2ŒR;R
1 exp.i t / d ! 2i exp. t %/ exp.˙i t jj/ C i % ˙ jj
and for t < 0 we have Z
2ŒR;R
1 exp.i t / d ! 0 C i % ˙ jj
as R ! 1. Thus, by an exchange of limits we have now Z 1 b
. i %/ d 2 jj2 0 . C i %/
2ŒR;R p Z 2 R>0 .t / exp.2 t %/ sin.t jj/ 0 .t / dt ! jj R
163
164
Chapter 3 Partial Differential Equations with Constant Coefficients
as R ! 1. Substituting this into (3.1.65) yields hg1;C j i;0 1 D .2/3=2 D
Z lim
"!0C
1 .2/3=2 lim
2R3 nB.0;"/
"!0C R!1
R>0
2B.0;R/nB.0;"/
Z lim
exp.2 t %/
sin. t jj/
0 .t / dt b./ d jj
Z
lim
1 .2/3=2 lim
R>0
Z
"!0C R!1
D
Z
Z R>0
2B.0;R/nB.0;"/
exp.2 t %/
sin. t jj/
0 .t / dt b./ d jj
sin. t jj/ b ./ d exp.2 t %/ 0 .t / dt: jj
Using Z 2B.0;R/nB.0;"/
Z
D 4
r2Œ";R
sin. t jj/ exp.i x/ d jj
sin.t r/ sin.r jxj/ dr; jxj
(3.1.66)
we get with an interchange of integration and introducing polar coordinates hg1;C j i;0 1 D2 lim lim .2/2 "!0C R!1
1 D2 lim lim .2/2 "!0C R!1
Z
Z R>0
Z
Z
R3
Z R>0
R>0
sin.t r/ sin.r jxj/ dr .x/ dx jxj r2Œ";R exp.2 t %/ 0 .t / dt Z sin.t r/ sin.r u/ dr .u/ u du r2Œ";R
exp.2 t %/ 0 .t / dt; R where .u/ WD 2S 2 .u / dS is the integral of 7! .u / over the 2-dimensional unit sphere S 2 , u 2 R>0 . Using
.u/ D .u/
Section 3.1 Partial Differential Equations in H1 .@ C e/ for u 2 R we obtain Z Z 2 R>0
Z
D
R>0
165
sin.t r/ sin.r u/ dr .u/ u du
r2Œ";R
Z
r2Œ";R
.cos..t u/ r/ cos..t C u/ r// dr .u/ u du
sin..t u/ R/ sin..t C u/ R/
.u/ u du D .t u/ .t C u/ R>0 Z sin..t u/ "/ sin..t C u/ "/
.u/ u du .t u/ .t C u/ R>0 Z Z sin..t u/ R/ sin..t u/ "/ D
.u/ u du
.u/ u du: .t u/ .t u/ R R Z
Employing the fact that
Z lim
R!1
R
Z
and lim
R!0C
R
sin.u R/ '.u/ du D '.0/ u
sin. u R/ '.u/ du D 0 u
for all ' 2 CV 1 .R/, we get (recall D .%; 0/ 2 R>0 R3 ) Z 1
.t / t exp.2 t %/ 0 .t / dt hg1;C j i;0 D .2/2 R>0 Z Z 1 D .t / dS t exp.2 t %/ 0 .t / dt 4 R>0 2S 2 Z Z 1 1 .x/ dSx exp.2 t %/ 0 .t / dt D 4 R>0 t x2t ŒS 2 Z 1 1 exp.2 jxj %/ .jxj; x/ dx D 4 x2R3 jxj 1 1 D
p ıŒR>0 .¹1ºS 2 / exp.2 m0 %/ m0 4 2 ˇ ˇ 1 1 D ; p ıŒR>0 .¹1ºS 2 / ˇˇ m0 ;0 4 2 for 0 2 CV 1 .R n ¹0º/. Consequently, we have g1;C .t / D
1 1 p ıŒR>0 .¹1ºS 2 / .t; / for t 2 R n ¹0º: 4 2 t
(3.1.67)
166
Chapter 3 Partial Differential Equations with Constant Coefficients
To complete the proof we need to show (3.1.66) and (3.1.67). To see (3.1.66) we first realize that the substitution 7! A, where A is a rotation, does not change the value of the integral Z sin. t jj/ exp.i x/ d jj 2B.0;R/nB.0;"/ Z sin. t jA j/ D exp.i A Ax/ d jA j 2B.0;R/nB.0;"/ Z sin. t jj/ exp.i Ax/ d : D jj 2B.0;R/nB.0;"/ Therefore, we may choose without loss of generality x D .0; 0; jxj/ and we get Z sin. t jj/ exp.i x/ d jj 2B.0;R/nB.0;"/ Z Z D 2 sin. t r/ r exp.i r cos # jxj/ sin # dr d # #2Œ0;
r2Œ";R
Z D 2
s2Œ1;C1
Z D 2
r2Œ";R
Z D 4
Z
r2Œ";R
Z
r2Œ";R
s2Œ1;C1
sin. t r/ exp.i r s jxj/ r dr ds sin. t r/ exp.i r s jxj/ ds r dr
sin. t r/ sin. r jxj/ dr: jxj
It remains to show (3.1.67). But since ' has compact support, we have, for some W 2 R>0 sufficiently large, Z Z sin. u R/ sin. u R/ '.u/ du D '.u/ du u u R ŒW;W Z '.u/ '.0/ D sin. u R/ du u ŒW;W Z sin. u R/ du: C '.0/ u ŒW;W Since u 7! ŒW;W .u/ '.u/'.0/ is in L1 .R/, i.e. Lebesgue integrable over R, the u Riemann–Lebesgue lemma shows that Z '.u/ '.0/ du ! 0 as R ! 1: sin. u R/ u ŒW;W The desired result now follows since Z Z sin.u R/ sin.v/ du D dv u v ŒW;W ŒW R;W R
Section 3.1 Partial Differential Equations in H1 .@ C e/
167
converges to the well-known improper integral Z sin.u/ du D u R as R ! 1. This concludes the construction of the fundamental solution of the wave equation. The cone ŒR>0 .¹1º S 2 / D ¹.t; x/ 2 R1C3 j jxj D t ^ t 2 R>0 º is called the (forward) light cone and its closure is the support of the fundamental solution. Comparing with the abstract initial value problem .@20 b @2 /u D @0 ı ˝ u0;0 C ı ˝ u0;1 with initial data .u0;0 ; u0;1 / D .0; ı/, the formula for the fundamental solution yields g1;C .0C/ D 0; so that t 7! g1;C .t C/ is continuous as a mapping into H1 .b @ C e/ and @0 g1;C .0C/ D ı:
(3.1.68)
To consider this more closely we have to calculate @0 g1;C : h@0 g1;C j i;0 D hg1;C j .@0 C 2 %/ i;0 Z Z 1 D .t / dS t .@0 .exp.2 m0 %/ 0 /.t / dt 4 R>0 2S 2 Z Z 1 .t / dS exp.2 t %/ 0 .t / dt D 4 R>0 2S 2 Z Z 1 .t @/ .t / dS exp.2 t %/ 0 .t / dt; C 4 R>0 2S 2 for D 0 ˝
. Thus, we obtain ˇ ˇ 1 h@0 g1;C j i;0 D ıŒR>0 .¹1ºS 2 / ˇˇ p 2 4 2 t ;0 ˇ ˇ 1 b ˇ C ıŒR>0 .¹1ºS 2 / ˇ.b m @/ p 4 2 t 2 ;0
for 0 2 CV 1 .R n ¹0º/. Consequently, we have @0 g1;C .t / D
4
1 p
2 t2
ıR>0 ¹1ºS 2 .t; /
1 p
4 2 t 2
.b @m b/ ıR>0 ¹1ºS 2 .t; /
168
Chapter 3 Partial Differential Equations with Constant Coefficients
for t 2 R n ¹0º. In particular, we have with (3.1.68) 1 1 ıR>0 ¹1ºS 2 .t; / p .b @m b/ ıR>0 ¹1ºS 2 .t; / ! ı p 2 4 2 t 4 2 t 2 Since we know that t 7!
1p 1 ı 2 .t; / 4 2 t ŒR>0 .¹1ºS /
as t ! 0 C :
is continuous on R n ¹0º in the
sense of continuity in H1 .b @ C e/ and reading off that ˇ Z ˇ 1 1 1 1 ˇ D .t / .x/ dSx p ıŒR>0 .¹1ºS 2 / .t; /ˇ 4 R>0 t x2t ŒS 2 4 2 t 0 we also have R>0 .t / 1 1 ıŒR>0 .¹1ºS 2 / .t; / D ı t ŒS 2 p 4 t 4 2 t for t 2 Rn¹0º. For the fundamental solution, we also have an alternative formulation. We first observe that, by the rules of substitution for integrals, we have for bounded, measurable w, ' 2 CV 1 .R/ and 2 CV 1 .R1C3 / 2hw.m0 / '.m20 m2 /j i;0 Z Z Z D2 '.t 2 r 2 / .t; r x0 / r 2 dx0 dr w.t / exp.2% t / dt R
Z D
Z
Z
1;t 2
R
Z D
S2
Œ0;1Œ
Z
S2
Z
'.u/ R>0
Z
C
R
Z
'.u/ R<0
R
p p t 2 u x0 / t 2 u dx0 du w.t / exp.2% t / dt
'.u/ .t;
S2
.t;
p
Z
S2
.t;
t 2 u x0 /
p
p
t 2 u x0 /
t 2 u dx0 w.t / exp.2% t / dt du
p
t 2 u dx0 w.t / exp.2% t / dt du:
This suggests the definition (by letting ' ! ı) Z Z 1 2 2 hw.m0 / ı.m0 m /j i;0 WD
.t; jt j x0 / jt j dx0 w.t / exp.2% t / dt: 2 R S2 For w D R>0 this is Z Z 1 2 2 hR>0 .m0 / ı.m0 m /j i;0 D
.t; t x0 / t dx0 exp.2% t / dt 2 Œ0;1Œ S 2 (3.1.69) and so we obtain 1 g1;C .t; / D .t / ı.t 2 . /2 / 2 R>0 1 .t / ı t ŒS 2 D 4 t R>0 1 D p ıR ¹1ºS 2 .t; / for t 2 R; t ¤ 0: 4 t 2 >0
Section 3.1 Partial Differential Equations in H1 .@ C e/
169
Remark 3.1.65. In this expression for g1;C .t /, restricting our attention to t ¤ 0 is not important since the behavior at t D 0 is known explicitly. In general one may define (in accordance with the rules of substitution for functions32 ) the functional representation ı ı ‰, at least on an open set U , by hı ı ‰j i0;0 Z WD
x2N.‰/
.x/ dSx p jdet.d0 ‰.x//j det.1 C d1 ‰.x/ .d0 ‰.x/ d0 ‰.x/ /1 d1 ‰.x//
for all 2 CV 1 .U /, where ‰ W U RnC1 ! RmC1 , n; m 2 N, m < n, is continuously differentiable on an open set U with det.d0 ‰.x0 ; x1 // ¤ 0, x D .x0 ; x1 / 2 U with x0 2 RmC1 and x1 2 Rnm , in a neighborhood of N.‰/ D ¹x 2 U RnC1 j ‰.x/ D 0º. For m D 0 this simplifies to a line integral Z hı ı ‰j i0;0 WD
.x/ x2N.S/
1 dsx : jgrad ‰.x/j
p With ‰.t; x/ D t 2 x 2 we get jgrad ‰.t; x/j D 2 t 2 C x 2 (n D 3; m D 0) and so the representation (3.1.69) is recovered as a particular case on N.‰/ D ŒR>0 .¹1º S 2 / with U D R4 n ¹0º. The particular shape of the standard reversibly evolutionary partial differential equations of mathematical physics shared by the system of linear acoustics also suggests an alternative approach reducing the system size to just a scalar partial differential equation. We note that @20 A0 .b @/2 D .@0 C A0 .b @// .@0 A0 .b @// has block diagonal form and therefore decouples v and p. Thus we may focus our attention on just one part, the other (block) unknown being easily reconstructed from the original first order equation. More specifically, we may focus on the scalar equation .@20 b @/ p D .@20 b @> b @ 2 / p D @ 0 f2 C b @ f1 which is just the wave equation. The velocity field can then easily be reconstructed since 1 b v D @1 0 @p C @0 f2 : 32
Here d0 and d1 denote differentiation with respect to the first and second block variable, i.e. d0 ‰.x0 ; x1 / WD d.y 7! ‰.y; x1 //.x0 /; d1 ‰.x0 ; x1 / WD d.y 7! ‰.x0 ; y//.x1 /:
170
Chapter 3 Partial Differential Equations with Constant Coefficients
3.1.12.3
Thermodynamics
According to Fourier, heat conduction obeys the linear partial differential equation P .@0 ; b @/ # D f with
P .@0 ; b @/ D @0 b @2 ;
the so-called heat equation33 . Here # denotes the scalar heat distribution function in R1C3 and f models heat sources (or sinks). Inserting an additional drift term s b @ for increased generality yields @0 C s b @ b @2 : The corresponding fundamental solution gs is given by gs D
1 1 : L.%;0/ 2 4 i m0 C i s m bCm b2 C %
Once again we give the explicit evaluation. As in the case of the transport equation, we find using residue calculus hgs j 0 ˝
i;0 Z
Z 1 exp. t jj2 / exp.i t s / 0 .t / b./ d exp.2%t / dt .2/3=2 R>0 R3 Z Z Z 1 D exp. t jj2 / 0 .t / exp.i .x t s/ / .x/ dx d .2/3 R>0 R3 R3 exp.2%t / dt: D
33
The equation results from considering the determinant of a system associated with ! 0 1 b q @ : D Q # b @> @0
where q denotes the so-called heat flux, Q models the influence of given heat sources. Other thermal parameters are assumed to be constants and eliminated by suitable scaling. The first row models Fourier’s law. Our standard solution procedure can be started as usual, noting that 7! . 1/. 2 .1 C @0 / C .@0 b @2 // is the minimal polynomial. Alternatives to Fourier’s law have been proposed leading to quite different systems. E.g. the Catta b neo model of heat conduction associated with 1C @0 @ or the Jeffreys type heat conduction model b @> @0 b b associated with 1C @0 @C @0 @ , ; 2 R>0 . The determinants are @20 . @20 C @0 b @2 / and
b @>
@0
@2 /, respectively, showing that the Cattaneo model is reversibly evolution@20 . @20 C @0 .1 C @0 /b ary, whereas the Jeffreys model is irreversibly evolutionary.
Section 3.1 Partial Differential Equations in H1 .@ C e/
171
We observe that for t 2 R>0 Z 1 .2 t /3=4 exp. t jj2 / exp.i .x t s/ / d D .F .p2t //.x t s/: .2/3=2 R3 Since the Gauss distribution function is invariant under the spatial Fourier transform F D L0 , its interaction with rescaling yields hgs j 0 ˝ i;0 Z Z D .4 t /3=2 exp. jx t sj2 =.4t // 0 .t / .x/ dx exp.2%t / dt R>0
R3
for 0 2 CV 1 .R n ¹0º/. This proves that gs .t; x/ D R>0 .t / .4 t /3=2 exp.jx t sj2 =.4t //
for t 2 R; t ¤ 0; x 2 R3 :
In particular, we have here that gs is indeed given by a function. From our general results on initial value problems we also know the behavior at t D 0. We therefore have in particular .4 t /3=2 exp.j t sj2 =.4t // ! ı
as t ! 0C;
where the convergence is in H1 .b @ C e/. 3.1.12.4
Electrodynamics
The Maxwell System.
With 0
1 0 @3 @2 b @ WD @ @3 0 @1 A @2 @1 0 representing the operator curl, the differential operator of Maxwell’s equations J E b (3.1.70) M.@0 ; @/ D K H governing the propagation of electromagnetic fields .E; H / in a homogeneous medium occupying R3 can be written in (block) matrix form as a .66/-partial differential expression ! 1 b " 0 0 @ M.@0 ; b @/ WD @0 0 b @ 0 ! @0 "1b @ D ; 1 b @ @0
172
Chapter 3 Partial Differential Equations with Constant Coefficients
where "; are positive definite, symmetric .3 3/-matrices. Here the first block row is called Ampère’s law and the second block row is Faraday’s law. With A D det. /1=2 1=2 one finds formally with the associated isometric re-scaling34 operation A @ D 1=2 b @ 1=2 A : A b Transformation with A therefore leads to: ! ! @0 "1b @0 @ "1 1=2 b @ 1=2 A D A @ @0 @ 1=2 1=2b @0 1 b and so @ @0 1=2 "1 1=2 b b @ @0 34
!
1=2 A E 1=2 A H
D
1=2 0 0 1=2
A F:
Generally, for any symmetric real matrix A we have by a transformation of variables Ab @ D .A1b @/ A ;
where .A1b @/ is to be understood as a formal cross product operation. Moreover, one finds Ab @ D .A1b @/ A D
1 Ab @ A A : det.A/
Indeed, following the algebraic argument in [28] we first observe that for x; y; z 2 R3 we have by definition of the cross product x .y z/ D det..x y z// where .x y z/ is the block matrix notation with x; y; z represented as column vectors. Thus, we have x ..Ay/ .Az// D det..x Ay Az// D det..A A1 x Ay Az// D det.A .A1 x y z// D det.A/ det..A1 x y z// D det.A/ A1 x .y z/ D det.A/ x A1 .y z/ for all x; y; z 2 R3 and so, letting u D Ay, we obtain the matrix equality 1 A .u/ A D .A1 u/ det.A/ for every u 2 R3 . The latter implies, by a Fourier transform argument, the stated transformation formula.
Section 3.1 Partial Differential Equations in H1 .@ C e/
173
Since 1=2 "1 1=2 D j"1=2 1=2 j2 is also symmetric and positive definite, without loss of generality we may assume that D 1.33/ . In the so-called vacuum case we have – up to scaling – that " D D 1.33/ . In this case we find simply ! @0 b @ det D @20 .@20 b @2 /2 ; b @ @0
@0 b @
@2 are is reversibly evolutionary, since @0 and @20 b b @ @0 reversibly evolutionary. In the general case we have that ! !! 1 " 0 0 b @ @0 "1b @ " 0 D @0 0 0 @ @0 b @ 0 1 b
which re-confirms that
and, with ei , i D 1; 2; 3, the canonical basis vectors of R3 , ! 3 X " 0 " 0 0 ei 0 b @ @0 @0 D @i 0 0 ei 0 b @ 0 iD1
is still a symmetric hyperbolic system, which shows that M.@0 ; b @/ is reversibly evolutionary according to our general result for such systems. For the vacuum case, the fundamental solution can be determined in the following way. We first observe ! !2 ! 0 b @ 0 b @ 2 b D 0; 1.66/ @ b @ 0 b @ 0 which indicates that the minimal polynomial of M.@0 ; b @/ WD
@0 b @
b @
@0
is
@2 /: .@0 / ..@0 /2 b Noting that this is the same as in the acoustic case, replacing A.b @/ with M0 .b @/ WD M.0; @/, we find immediately that 2 b2 1 @/1 D .@20 b @2 C M0 .b @/.@0 C M0 .b @/// @1 M.@0 ; b 0 .@0 @ / 2 b2 1 D .@0 C M0 .b @// M0 .b @/ @1 C @1 0 .@0 @ / 0 1.66/ :
(3.1.71)
Thus, the fundamental solution is given by @/1 ı 1.66/ GC WD M.@0 ; b D .@0 C M0 .b @// M0 .b @/ @1 0 g1;C 1.66/ C R0 ˝ ı 1.66/ D M0 .b @/ g1;C 1.66/ C M0 .b @/2 @1 0 g1;C 1.66/ C R0 ˝ ı 1.66/ ;
174
Chapter 3 Partial Differential Equations with Constant Coefficients
where g1;C denotes the fundamental solution of the wave equation. Using M0 .b @/2 D .M0 .b @/2 b @2 / .@20 b @2 / C @20 , we also obtain GC D .@0 C M0 .b @// g1;C 1.66/ C .M0 .b @/2 b @2 / @1 0 g1;C 1.66/ : From the vector identity jj2 x D . x/ x, we have by inverse Fourier transform with respect to that b @2 1.33/ D b @b @> C b @ b @ and it follows that ! b grad div 0 @b @> 0 2 b2 b .M0 .@/ @ 1.66/ / D D ; 0 grad div 0 b @b @> where grad and div denote the generalized vector analytic operations J given by ap> b b plying @ and @ , respectively. Therefore, if the right-hand side K of (3.1.70) is E is simply divergence-free, i.e. satisfies div J D div K D 0, then the solution H given by E J g1;C J b D GC D .@0 C M0 .@// ; H K g1;C K where the convolution operator g1;C is to be understood component-wise. Remark 3.1.66. Given that the electromagnetic field .E; H / is real-valued, an alternative approach could have been taken. We may write F WD E C i H as a complexvalued representation of the electromagnetic field. Then Maxwell’s equations reduce to .@0 C i b @/ F D .@0 C i curl/ F D S WD J C i K; where curl stands for b @ and is just a generalization of the vector analytic operation with the same name. Here 7! .@0 / ..@0 /2 b @2 / is the characteristic polynomial and consequently, with M0 .b @/ replaced by i b @ in (3.1.71), we obtain 2 b2 1 .@0 C i b @/1 D .@0 i b @/ i b @ @1 C @1 0 .@0 @ / 0 2 b2 1 b b D i b @ .@20 b @2 /1 @1 C @1 0 @ @ .@0 @ / 0 :
Using b @2 1.33/ D b @b @> C b @ b @, we get bb> 2 b2 1 @/1 D .@0 i b @/ .@20 b @2 /1 @1 .@0 C i b 0 @ @ .@0 @ / : This leads to the corresponding (forward) fundamental solution bb> C WD .@0 i b @/ gC 1.33/ @1 0 @ @ gC 1.33/ :
Section 3.1 Partial Differential Equations in H1 .@ C e/
175
The relation b @2 D b @b @> C b @ b @ can be also considered in a different way. Consider Pcurl WD .b @2 /1b @2 /1 b @ b @; Pgrad WD .b @b @> ; @ C e/ for every ˛ 2 which are well-defined as bounded operators in H˛ .@0;% C 1; b ZnC1 . Indeed they are orthogonal projectors with Pcurl C Pgrad D 1.33/ ;
(3.1.72)
which is a form of the so-called Helmholtz decomposition35 . With Pcurl 0 Pgrad 0 Pcurl ˚ Pcurl D D ; Pgrad ˚ Pgrad ; 0 Pcurl 0 Pgrad we have for the fundamental solution of Maxwell’s equations @// gC 1.66/ b @2 Pgrad ˚ Pgrad @1 GC D .@0 C M0 .b 0 gC 1.66/ D .@0 C M0 .b @// Pcurl ˚ Pcurl gC 1.66/ C @0 Pgrad ˚ Pgrad gC 1.66/ b @2 Pgrad ˚ Pgrad @1 0 gC 1.66/ D .@0 C M0 .b @// Pcurl ˚ Pcurl gC 1.66/ C Pgrad ˚ Pgrad R>0 ˝ ı 1.66/ ; where the term Pgrad ˚ Pgrad R>0 ˝ ı 1.66/ is the part of the fundamental solution in the null space of M0 .b @/. If the material is conducting, then the current density j in the right-hand side J D "1 j contains a contribution j0 produced by the electromagnetic field itself. The dependence on the field is commonly given by Ohm’s law: j0 D E; where the non-negative symmetric matrix denotes the conductivity (and so J0 WD "1 j0 D "1 E). In this case, Maxwell’s equations assume the form ! @0 C "1 "1b @ E J ; D 1 b K H @0 @ where now J WD J J0 ; K are given external forcing terms. Considering again the vacuum case " D D 1 and as multiplication by a positive constant, we get ! @ @0 C b D @0 .@0 C / .@20 C @0 b @2 /2 ; det b @ @0 35 The classical Helmholtz decomposition states – roughly speaking – that every sufficiently smooth field in RnC1 can be written as a sum of a curl of a certain field and a gradient of a certain potential.
176
Chapter 3 Partial Differential Equations with Constant Coefficients
so that the minimal polynomial is 7! .@0 / .@0 C / ..@0 /2 C .@0 / b @2 /: Without elaborating the details, we note that in this case we need to find the fundamental solution for the scalar expression @2 ; @20 C @0 b which for > 0 is the partial differential expression of a damped wave equation, also known as the telegraph equation. Investigating the real zeros of the polynomial 7! .i C %/2 C .i C %/ C 2 for 2 Rn , n 2 N, % 2 R n ¹0º, from 2 C 2i % C %2 C i C % C 2 D 0 we obtain the system of real equations 2 C I m D %2 C 2 C Re %; 2% C Re C I m % D 0: Here we have assumed 2 C, which is more general than currently needed. For Im % % ¤ 12 Re , the second equation implies D 2%CRe , and so
Im % 2% C Re
2
2 Im .%2 %.2% C Re // 2% C Re 2 Im D .%2 C Re %/ 2% C Re
.I m /2 % D 2% C Re
D %2 C 2 C Re % from the first equation. Thus, the constraint %2 C Re % > 0 prevents real zeros. From this we see that for j%j > jRe j there are no real roots for these choices of % 2 R. Thus, reversible evolutionarity is maintained for any 2 C. Recall that, for the telegraph equation, we have specifically 2 R>0 . Similar to earlier considerations, the partial differential expression of the telegraph equation re-appears by eliminating the magnetic field from Maxwell’s equations. Differentiating @0 E C E b @ H D J with respect to time and substituting @0 H D b @ E C K yields @20 E C @0 E C b @ b @ E D @0 J C b @ K:
Section 3.1 Partial Differential Equations in H1 .@ C e/
177
@ D 0 we get b @> E D .@0 C /1 b @> J (in H1 .@0;% C 1; b Since b @> b @ C e/ for > 2 b b b b b % 2 R>0 ), and using we @ @ E @ E D C@ @ E get the .3 3/-telegraph equation @20 E C @0 E b @2 E D @0 J C b @ K .@0 C /1 b @b @> J : The magnetic field can now be recovered from the electric field 1 b H WD @1 0 @ E C @0 K:
The telegraph equation can be modified by the following observation. It is easy to see that exp.s m0 / W H˛ .@0;% C 1; b @ C e/ ! H˛ .@0;%Cs C 1; b @ C e/ is unitary for every ˛ 2 Z1C3 , %; s 2 R. With % 2 R>0 and s D 2 2 R>0 , we get that ! @0 C 12 b @ exp. m0 =2/ J exp. m0 =2/ E D : b exp. m0 =2/ H exp. m0 =2/ K @ @0 12 @ C 1 b @ is now The corresponding minimal polynomial of 0 2 b @ @0 12 1 1 2 2 b2 7! @0 @0 C .@0 / @ : 2 2 4
The relevant scalar partial differential expression 2 @2 @20 b 4 is known as the partial differential expression of the modified telegraph equation. Again investigating the associated polynomial 7! .i C %/2 C 2
2 4
for 2 Rn , n 2 N, % 2 R n ¹0º, in the more general case 2 C, from 2 C 2i % C %2 C 2
2 D0 4
we obtain the system of real equations 2 C
.Re /2 .I m /2 D %2 C 2 ; 4 Re I m 2% C D 0: 2
178
Chapter 3 Partial Differential Equations with Constant Coefficients
For % ¤ 0 the second equation implies D Re 4%I m and so .Re /2 .I m /2 .Re /2 .I m /2 D %2 C 2 : C 16%2 4 This leads to the non-existence of real roots if .Re /2 .I m /2 .Re /2 .I m /2 < %2 ; C 2 16% 4 which is the same as .4%2 .Re /2 /.4%2 C .I m /2 / > 0: Thus, there are no real roots and the system is reversibly evolutionary if j%j >
1 jRe j: 2
Recall that for the telegraph equation we have assumed 2 R>0 . In the case Re D 0 and I m ¤ 0 we obtain the so-called Klein–Gordon equation. The Extended Maxwell System. It is frequently convenient to consider Maxwell’s 36 equations in the following J form. Emodified b @> b Noting that M.@0 ; @/ H D K implies (since b @ D 0) @0 b @>
E H
WD
@> E @0 b @> H @0 b
! D
! b @> J ; b @> K
we may write Maxwell’s equations in the redundant form 0 0 11 0 1 0 b> 0 b @> 0 0 0 @1 0 @ J Bb B C C B @ 0 CC B J EC B@ 0 b B CDB B@0 B CC B B @ A b b K @ 0 @ 0 @ AA H @ @ 1b> > 0 b @ 0 0 @ 0 0 @ K 36
1 C C C A
Variants of this idea to reformulate Maxwell’s equations have been discovered and re-discovered by a number of authors in various contexts. Possibly the earliest reference is [34] and consequently one finds the term “Ohmura system”. It was discovered in the context of a quaternionic formulation of Maxwell’s equations. It was later re-discovered independently in various forms e.g. in [19], [36], [11]. In [36], however, the full functional analytical context has been described in the general anisotropic, inhomogeneous case, showing that Maxwell’s equations are indeed equivalent to an evolution equation involving two commuting selfadjoint operators. The extended Maxwell system can be utilized for standard elliptic regularity for the spatial part of Maxwell’s equations and shows advantages in applications involving the integral equations method, see e.g. [45, 47].
Section 3.1 Partial Differential Equations in H1 .@ C e/ Remark 3.1.67. It may be preferable to write 0 11 0 0 0 0b @ b @ H B CC B @ > 0 CC B 0 B 0 0b B CC B B@0 B b b @ @ @ 0 0 AA @ E @ 0 b @> 0 0 0
1
0
K 1b> C B @0 @ J CDB A B J @ 1b> @0 @ K
179
1 C C C: A
In this way we see that also the extended Maxwell system is of the typical block form of reversibly evolutionary expressions of mathematical physics @0 W ; W @0 @ b @ . This transformation is a particular instance of a general here with W WD b @> 0 b mechanism for tri-diagonal skew-selfadjoint block operator matrix A D .Aj;k /j;k with AkC1;k D Ak;kC1 for k D 1; : : : ; 2n 1 (all other entries being zero). Indeed, consider the permutation ' given by ¹1; : : : ; 2nº ! ¹1; : : : ; 2nº; k 7! 2.k b.k 1/=ncn/ 1 C b.k 1/=ncn which induces a row operation %' W B D .Bi;j /i;j D1;:::;2n 7! .B'.i/;j /i;j D1;:::;2n . Then the image of the block identity matrix corresponding to the block operator matrix structure of A yields a mapping P' , such that %' .A/ D P' A: With P' we now obtain that P' AP' is of the desired superblock structure 0 W : W 0 The .8 8/-partial differential expression corresponding to the extended Maxwell system is 0 1 @> 0 0 @0 b B b C @ 0 C B @ @0 b MM.@0 ; b @/ WD B (3.1.73) C @ 0 b @ A @ @0 b 0 0 b @> @0 and now has @2 /4 .@20 b as determinant and 7! .@0 /2 b @2
180
Chapter 3 Partial Differential Equations with Constant Coefficients
as minimal polynomial. Indeed, we find that 11 0 0 0 0 0 b @> 0 0 0 b @> C B Bb B C B @ 0 CC B @ 0 B@ 0 b Bb B @0 C B C B @0 B C B @ 0 b @ 0 b @ @ 0 b @ AA @ @ > b 0 0 @ 0 0 0
0 b @ 0 b @>
11 0 CC 0 CC @2 ; CC D @20 b b @ AA 0
and so the fundamental solution of this form of Maxwell’s equation has the simple form 11 0 0 0 b @> 0 0 CC B Bb @ 0 CC B B@ 0 b CC g1;C 1.88/ ; B@0 C B @ @ 0 b @ 0 b @ AA 0 0 b @> 0 where g1;C is the fundamental solution of the wave equation in R1C3 . That the solution is indeed of the form 0 1 0 BEC B C @H A 0 is merely a structural condition on the data, which must have the form 1 0 1 0 1b> F1 @0 @ F2 C B F2 C B F2 C B CDB B C: @ F3 A @ F3 A 1 > b F4 @0 @ F3 Conversely, if (3.1.74) holds then the solution 0 1 u1 B u2 C B C @ u3 A u4 must satisfy @> u2 D b @> F2 ; @0 b @0 u1 b @> u2 D F1 ;
@0 b @> u3 D b @> F3 ; @0 u4 C b @> u3 D F4 ;
and so b> @0 u1 D F1 C b @> u2 D F1 C @1 0 @ F2 D 0; b> @0 u4 D F4 b @> u3 D F4 @1 0 @ F3 D 0; implying u1 D u4 D 0.
(3.1.74)
Section 3.1 Partial Differential Equations in H1 .@ C e/ 3.1.12.5
181
Elastodynamics
Let the displacement of an elastic solid with mass density matrix M from its equilibrium state be given by a function u D .us /sD1;2;3 of space and time. Then the motion of the body is governed by the equation M @20 u D div T C F; where T D .Tij /i;j D1;2;3 denotes the stress tensor and F an external forcing term. The elastic properties of a medium can be described in terms of the rate of change of the displacement vector field u, i.e. the derivative b @u WD .@k us /s;kD1;2;3 . The stress tensor T D .Tij /i;j D1;2;3 can formally be considered as a matrix-valued function depending on this rate of change. In the simplest case this dependence is linear and homogeneous, i.e. we have the so-called Hooke’s law Tij D
3 X
Cij ks @k us
k;sD1
for i; j D 1; 2; 3, or
T D Cb @u
for short, where C D .Cij ks /i;j;k;sD1;2;3 with real constants Cij ks known as elasticity moduli, i; j; k; s D 1; 2; 3, should be symmetric in the sense that Cij ks D Cksij
(3.1.75)
for i; j; k; s D 1; 2; 3. The symmetry requirement reduces the original 81 constants to P 45 elasticity moduli. Denoting the trace 3iD1 Xi i of a matrix X D .Xij /i;j D1;2;3 2 C 33 by trace.T /, we see that .X; Y / 7! trace.X Y / provides an inner product in C 33 . The induced norm p X 7! trace.X X / is known as the Frobenius norm. In order to obtain an evolutionary equation, C should be non-negative with respect to this inner product. In general there may be non-trivial elements in the null space Œ¹0ºC of C W C 33 ! 33 C . The projection theorem yields the orthogonal decomposition C 33 D Œ¹0ºC ˚ C ŒC 33 and in particular the subspace C ŒC 33 reduces C so that C ŒC 33 ! C ŒC 33 ; U 7! C U is a well-defined, positive definite mapping.
182
Chapter 3 Partial Differential Equations with Constant Coefficients
More specifically, it is commonly assumed that Œ¹0ºC D ¹U 2 C 33 j U C U > D 0º; i.e. that C vanishes on antisymmetric matrices. This amounts to the additional symmetry constraint Cij ks D Cijsk (3.1.76) for i; j; k; s D 1; 2; 3 effectively reducing the number of elasticity moduli to 21. This assumption makes the stress tensor T (real) symmetric, i.e. T > D T . Indeed, Tij D
3 X
Cij ks @k us D
k;sD1
3 X
Cksj i @k us D
k;sD1
3 X
Cj iks @k us D Tj i
k;sD1
for i; j D 1; 2; 3. Moreover, we see from (3.1.75) and (3.1.76) that Cij kl D Clkj i D Cklj i D Cij lk D Cj ilk for i; j; k; l D 1; 2; 3 holds. Consequently, 1 T D .T C T > / 2 1 D .C @u C .C @u/> / 2 3 3 X 1 X D Cij kl @k ul C Cj ikl @k ul i;j 2 k;lD1
D
1 2
3 X
k;lD1
Cij kl @k ul C
k;lD1
Cj ilk @l uk
k;lD1
i;j
1 .@k ul C @l uk / i;j 2 k;lD1 1 .@k ul C @l uk / DC : 2 k;l D
3 X
3 X
Cij kl
Thus, we see that only the real symmetric part of @u is needed in Hooke’s law. We recall that, under these assumptions, we have merely 21 real constants left as elasticity moduli – rather than 45 in the general symmetric case. Noting that 0
X11 @ X21 X31
PD W C 33 ! C 33 ; 1 0 1 X11 0 X12 X13 0 X22 X23 A 7! @ 0 X22 0 A X32 X33 0 0 X33
Section 3.1 Partial Differential Equations in H1 .@ C e/
183
defines an orthogonal projector with respect to the Frobenius inner product, it is preferable (under the additional symmetry assumption (3.1.76)) to use the modified Frobenius inner product (inducing an equivalent norm) h j i WD .X; Y / 7! trace..PD X / PD Y / C 3 X
D
1 trace...1 PD /X / .1 PD /Y / 2
Xks Yks
k;sD1;ks
for all complex .3 3/-matrices X D .Xij /i;j D1;2;3 and Y D .Yij /i;j D1;2;3 with X> D X, Y > D Y . The tensor C is still positive definite with respect to the modified inner product in the sense that, for some positive constant c0 , we have 3 X
hX jC X i D
3 X
Xij Cij ks Xks
i;j D1; ij k;sD1
D
3 3 X X iD1 k;sD1
D
3 3 X X
3 X
Xii Ci iks Xks C
C2
iD1 kD1
C
Xij Cij ks Xks
i;j D1; i<j k;sD1
Xii Ci ikk Xkk
3 X
3 X
3 X
3 X
Xij Cij ks Xks
iD1 k;sD1;k<s 3 X
Xij Cij kk Xkk
i;j D1; i<j kD1
C2
3 X
3 X
Xij Cij ks Xks
i;j D1; i<j k;sD1;k<s 3 X
D hC X jX i c0
Xks Xks
k;sD1;ks
for all complex .3 3/-matrices X D .Xij /i;j D1;2;3 with X > D X . Following the so-called Voigt convention [49] we may re-write the above Cartesian tensors in an easier accessible matrix form. For this we first observe that the space of symmetric .3 3/-matrices can be identified with 6-vectors due to the relation hKjKi D
3 X i;j D1;ij
jKij j2 ;
184
Chapter 3 Partial Differential Equations with Constant Coefficients
for symmetric matrices K D .Kij /i;j D1;2;3 , which shows the isometry of the mapping V given by 1 0 K11 B K22 C 0 1 C B K11 K12 K13 V B K33 C C: B @ K12 K22 K23 A 7 !B K23 C B C K13 K23 K33 @ K31 A K12 We also see that V
0
1 .@k ul C @l uk / 2 k;l
1 @1 u1 B C @2 u2 B C C B @3 u3 C B1 D B .@2 u3 C @3 u2 / C D W B2 C B1 C @ 2 .@3 u1 C @1 u3 / A 1 2 .@1 u2
with
0
1 B0 B B0 W DB B0 B @0 0
0 1 0 0 0 0
C @2 u1 / 0 0 1 0 0 0
0
@1 B 0 B B 0 B B 0 B @ @3 @2
0 @2 0 @3 0 @1
1 0 0 1 0 C C u1 C @3 C @ A u2 @2 C C u3 @1 A 0
1 0 0 0 0 0 0C C 0 0 0C C: 1 C 2 0 0C 1 0 0A 2
0 0
1 2
Hooke’s law can now be written in the matrix form as 1 .@k ul C @l uk / W V T DW V C 2 k;l 0 1 1 0 T11 @1 0 0 B 0 @2 0 C 0 1 B T22 C B C C B B T33 C B 0 0 @3 C u1 1 B C C B @ A DW V C V W B C u2 D B T23 C 0 @ @ 3 2 B B C C @ T31 A @ @3 0 @1 A u3 @2 @1 0 T12 and so
1 T11 B T22 C C B B T33 C 1 C B B T23 C D W V C V W C B @ T31 A T12 0
0
@1 B 0 B B 0 B B 0 B @ @3 @2
0 @2 0 @3 0 @1
1 0 0 1 0 C C u1 @3 C C @ u2 A: @2 C C u3 @1 A 0
Section 3.1 Partial Differential Equations in H1 .@ C e/
185
Thus, C is transformed into the matrix 0
C1111 C1122 C1133 2 C1123 B C1122 C2222 C2233 2 C2223 B B C1133 C2233 C3333 2 C2333 W B B 2 C1123 2 C2223 2 C2333 4 C2323 B @ 2 C1113 2 C1322 2 C1333 4 C1323 2 C1112 2 C1222 2 C1233 4 C1223 0 C1111 C1122 C1133 C1123 C1113 B C1122 C2222 C2233 C2223 C1322 B B C1133 C2233 C3333 C2333 C1333 DB B C1123 C2223 C2333 C2323 C1323 B @ C1113 C1322 C1333 C1323 C1313 C1112 C1222 C1233 C1223 C1213
1 2 C1113 2 C1112 2 C1322 2 C1222 C C 2 C1333 2 C1233 C CW 4 C1323 4 C1223 C C 4 C1313 4 C1213 A 4 C1213 4 C1212 1 C1112 C1222 C C C1233 C C: C1223 C C C1213 A C1212
Hooke’s law takes on the form 1 0 T11 C1111 B T22 C B C1122 C B B B T33 C B C1133 C B B B T23 C D B C1123 B C B @ T31 A @ C1113 T12 C1112 0
C1122 C2222 C2233 C2223 C1322 C1222
C1133 C2233 C3333 C2333 C1333 C1233
C1123 C2223 C2333 C2323 C1323 C1223
C1113 C1322 C1333 C1323 C1313 C1213
10 C1112 @1 B 0 C1222 C CB B C1233 C CB 0 C C1223 C B B 0 C1213 A @ @3 C1212 @2
0 @2 0 @3 0 @1
1 0 0 1 0 C C u1 @3 C C @ u2 A: @2 C C u3 @1 A 0
Now, with little danger of confusion, again using T to denote the column matrix 0
1 T11 B T22 C B C B T33 C B C B T23 C; B C @ T31 A T12 C for the matrix 0
C1111 B C1122 B B C1133 B B C1123 B @ C1113 C1112
C1122 C2222 C2233 C2223 C1322 C1222
C1133 C2233 C3333 C2333 C1333 C1233
C1123 C2223 C2333 C2323 C1323 C1223
C1113 C1322 C1333 C1323 C1313 C1213
1 C1112 C1222 C C C1233 C C C1223 C C C1213 A C1212
186
Chapter 3 Partial Differential Equations with Constant Coefficients
and Grad for
0
@1 B 0 B B 0 B B 0 B @ @3 @2
0 @2 0 @3 0 @1
1 0 0 C C @3 C C; @2 C C @1 A 0
we get Hooke’s law as T D C Grad u:
(3.1.77)
The Newton law governing the elastic displacement is now M @20 u Div T D F;
(3.1.78)
where 1 T11 1 B T22 C 0 C @ 1 0 0 0 @3 @ 2 B B T33 C B C A @ D 0 @2 0 @ 3 @ 1 0 B T23 C B C 0 0 @3 @ 2 @ 1 0 @ T31 A T12 0
Div T WD
3 X kD1
@k Tj k
j D1;2;3
and the .3 3/-matrix M 2 R33 is assumed to be positive definite and symmetric. Substituting Hooke’s law into (3.1.78), we obtain the system .@20 M 1 Div C Grad/ u D M 1 F
(3.1.79)
for the displacement. Before we analyze this system, let us take a look at an alternative option which is to formulate this system as first order in time revealing that the equations of elastodynamics can also be formulated as a symmetric hyperbolic system. For this we let v WD @0 u to obtain M @0 v Div T D F and, by differentiating Hooke’s law, we get @0 T D C Grad v: Thus we are led to the block matrix system 1 1 v M 0 0 Div v M F @0 D : T 0 C Grad 0 T 0
(3.1.80)
Section 3.1 Partial Differential Equations in H1 .@ C e/
187
We see that (3.1.80) is a symmetric hyperbolic system. The corresponding partial differential expression is 1.33/ @0 M 1 Div b E.@0 ; @/ WD @/; D @0 E0 .b C Grad 1.66/ @0 where E0 .b @/ WD E.0; b @/. Studying a general 9 9-matrix system more explicitly would require us to find roots of a 9-th order polynomial. This can only be successfully done in very special cases. To reduce the number of parameters let us consider simpler materials. First we assume that M D c 1.33/ for c 2 R>0 . Without loss of generality we may then also assume c D 1 (otherwise we re-scale). A case which can be handled in further detail, is the so-called isotropic case where 0 1 2 C 0 0 0 B 2 C 0 0 0C B C B 2 C 0 0 0 C B C: C DB 0 0 0 0C B 0 C @ 0 0 0 0 0A 0
0
0
0
0
The eigenvalues of C are ; 2 C 3k; 2 with multiplicities 3; 2 and 1, respectively. The quantities ; are known as Lamé constants and for the medium to be isotropic (C positive definite) we must have > 0;
2 C 3 > 0
or
2 0 < C : 3 The characteristic polynomial in this case is found to be > 0;
(3.1.81)
.@0 /3 ..@0 /2 b @2 /2 ..@0 /2 .2 C / b @2 /: Noting that (3.1.81) implies 2 C > 0 we see that there are two wave equations involved with different time-scales (velocities). A closer look reveals that the minimal polynomial is of smaller degree .@0 / ..@0 /2 b @2 / ..@0 /2 .2 C / b @2 /: Indeed, in this particular case we find Div C Grad 0 2 b E0 .@/ D 0 C Grad Div and calculate E0 .b @/.E0 .b @/2 b @2 / .E0 .b @/2 .2 C / b @2 / D 0:
188
Chapter 3 Partial Differential Equations with Constant Coefficients
The solution operator evaluates to E.@0 ; b @/1 b 2 b2 1 C E0 .b D @1 @/ .@20 .2 C / b @2 /1 0 C E0 .@/ .@0 @ / 2 2 b2 1 C E0 .b b2 1 C E0 .b @/2 @1 @/2 @1 0 .@0 @ / 0 .@0 .2 C / @ /
C .E0 .b @/3 .@0 C E0 .b @// @0 E0 .b @// .@20 b @2 /1 .@20 .2 C / b @2 /1 2 b2 1 2 b2 1 C E0 .b @/4 @1 0 .@0 @ / .@0 .2 C / @ / :
The (forward) fundamental solution is then given by GE;C D R>0 ˝ ı C E0 .b @/ g ;C C E0 .b @/ g2 C;C C E0 .b @/2 .R>0 ˝ ı/ g ;C C E0 .b @/2 .R>0 ˝ ı/ g2 C;C C .E0 .b @/3 .@0 C E0 .b @/// @0 E0 .b @/ g ;C g2 C;C C E0 .b @/4 .R>0 ˝ ı/ g ;C g2 C;C ; where g;C denotes the (forward) fundamental solution of .@20 b @2 /, 2 R>0 , which can be obtained from the fundamental solution of the wave equation above by a simple rescaling of space g;C D 3=4 1 ˝ 1=p g1;C D 3=2 g1;C .; 1=2 /: Indeed, @2 / 1 ˝ 1=p g1;C D 1 ˝ 1=p .@20 b @2 / g1;C D 1 ˝ 1=p ı .@20 b and h1 ˝ 1=p ıj i;0 D hıj1 ˝ p i;0 D 3=4 .0/: Note that to recover the displacement from the velocity vector an additional integration, i.e. application of @1 0 or convolution with .R>0 ˝ ı/, is needed. As convenient as the formulation as a symmetric hyperbolic system sometimes is, it suffers from the considerable size of the resulting system. In the present case of elastodynamics we could obtain a smaller system by the observation made for the acoustics equations. We find, however, that the original equation (3.1.78) already has the desired form 2 1 @20 u M 1 Div C Grad u D @1 Div C Grad/ v 0 .@0 M
D M 1 F
(3.1.82)
and the stress tensor T can be found from Hooke’s law (3.1.77) as soon as we have solved (3.1.82). The .3 3/-partial differential expression @/ WD @20 M 1 Div C Grad DW @20 E0 .b @/ E.@0 ; b
Section 3.1 Partial Differential Equations in H1 .@ C e/
189
can be analyzed more comfortably due to its smaller size. We only consider here the case M D 1.33/ and briefly discuss several special classes of media: (a) (monoclinic media) In this case we have generically 0
C1111 B C1122 B B C1133 C DB B 0 B @ 0 C1112
C1122 C2222 C2233 0 0 C1222
C1133 0 0 C2233 0 0 C3333 0 0 0 C2323 C1323 0 C1323 C1313 C1233 0 0
1 C1112 C1222 C C C1233 C C 0 C C 0 A C1212
with 13 material constants. This case is characterized by leaving the equations invariant under reflexion .x; y; z/ 7! .x; y; z/: The 13 constants have to satisfy certain constraints to make sure that C is positive definite. We certainly must require that 1 C1111 C1122 C1133 @ C1122 C2222 C2233 A and C2323 C1323 are positive definite C1323 C1313 C1133 C2233 C3333 0
(3.1.83)
and C1212 > 0;
(3.1.84)
as well as det.C / > 0. Due to the particular block structure C D we find
1 W B 1 0 1
A W W B
A W W B
1 B 1 W
0 1
D
A W B 1 W 0 0 B
and so we see that C is positive definite if and only if in addition to (3.1.83), (3.1.84) we have that 1 0 1 0 C1112 C1111 C1122 C1133 @ C1122 C2222 C2233 A .C1212 /1 @ C1222 A C1112 C1222 C1233 C1133 C2233 C3333 C1233 is positive definite.
190
Chapter 3 Partial Differential Equations with Constant Coefficients
(b) (rhombic media) This case is characterized by mirror symmetries with respect to all coordinate planes .x; y; z/ 7! .x; y; z/; .x; y; z/ 7! .x; y; z/; .x; y; z/ 7! .x; y; z/: These symmetries reduce the number of free parameters even further. We find in this case that C reduced to 0 1 0 0 C1111 C1122 C1133 0 B C1122 C2222 C2233 0 0 0 C B C B C1133 C2233 C3333 0 0 0 C B C: B 0 0 0 C2323 0 0 C B C @ 0 0 0 0 C1313 0 A 0 0 0 0 0 C1212 For C to be positive definite we must require that the .3 3/-matrix 1 0 C1111 C1122 C1133 @ C1122 C2222 C2233 A C1133 C2233 C3333 is positive definite and C2323 ; C1313 ; C1212 > 0: (c) (hexagonal media) In media with a hexagonal symmetry we have 0 1 0 0 C1111 C1122 C1133 0 B C1122 C1111 C1133 0 C 0 0 B C B C1133 C1133 C3333 0 C 0 0 B C C DB C 0 0 0 C 0 0 2323 C B A @ 0 0 0 0 C2323 0 1 0 0 0 0 0 2 .C1111 C1122 / with 5 material constants. Positive definiteness of C requires the constraints C1111 C1122 > 0;
C2323 > 0;
1 2 .C1111 C C1112 /C3333 > C1133 : 2
(d) (cubic media) If there are three orthogonal symmetry axes then 0 1 0 0 C1111 C1122 C1122 0 B C1122 C1111 C1122 0 0 0 C B C B C1122 C1122 C1111 0 C 0 0 C C DB B 0 0 0 C2323 0 0 C B C @ 0 0 0 0 C2323 0 A 0 0 0 0 0 C2323
Section 3.1 Partial Differential Equations in H1 .@ C e/
191
with only 3 material constants. Indeed, this case is characterized by leaving the equations invariant under spatial transformations .x; y; z/ 7! .y; x; z/; .x; y; z/ 7! .x; z; y/; .x; y; z/ 7! .z; y; x/; as well as under reflexions .x; y; z/ 7! .x; y; z/; .x; y; z/ 7! .x; y; z/; .x; y; z/ 7! .x; y; z/: Geometrically this means axial symmetry with respect to all coordinate axes and mirror symmetry with respect to all coordinate planes. Indeed, the mirror symmetries reduce C to 1 0 0 0 C1111 C1122 C1133 0 C B B C1122 C2222 C2233 0 0 0 C B C C BC C C 0 0 0 C B 1133 2233 3333 C: B B 0 C 0 0 C 0 0 2323 B C B 0 0 0 0 C1313 0 C @ A 0 0 0 0 0 C1212 This is a rhombic medium. The additional axial symmetries now lead to 1 0 0 C1111 C1122 C1122 0 C B B C1122 C1111 C1122 0 0 0 C C B BC 0 0 C C B 1122 C1122 C1111 0 C: B C B 0 0 0 C 0 0 2323 B C B 0 0 0 0 C2323 0 C @ A 0 0 0 0 0 C2323 0
The required conditions for C to be positive definite can be easily found as C2323 > 0;
C1111 C1122 > 0;
C1111 C 2 C1122 > 0
or C2323 > 0;
1 C1111 > C1122 > C1111 : 2
192
Chapter 3 Partial Differential Equations with Constant Coefficients
The isotropic case mentioned above is a particular instance of such a cubic system. Utilizing the fact that the only linearly independent Cartesian 4-tensors invariant under rotation, i.e. isotropic tensors, are37 .i; j; k; s/ 7! ¹0º .i j / ¹0º .k s/; .i; j; k; s/ 7! ¹0º .i k/ ¹0º .j s/; .i; j; k; s/ 7! ¹0º .i s/ ¹0º .k j / we find that in the isotropic case we must have Cij ks D ¹0º .i j / ¹0º .k s/ C ¹0º .i k/ ¹0º .j s/ C ¹0º .i s/ ¹0º .k j / for i; j; k; s D 1; 2; 3 with parameters ; ; 2 R. Together with the symmetry constraint (3.1.76) on C we find that D and so Cij ks D ¹0º .i j / ¹0º .k s/ C ¹0º .i k/ ¹0º .j s/ C ¹0º .i s/ ¹0º .k j / for i; j; k; s D 1; 2; 3. This yields in the cubic case an additional reduction C1111 D 2 C ; C1122 D ; C1212 DD C1221 D ; leaving just 2 elasticity moduli known as Lamé constants. Note that C D 1.66/ and the rather special case 1 0 0 0 C1111 C1122 C1122 0 B C1122 C1111 C1122 0 0 0 C C B B C1122 C1122 C1111 0 0 0 C B C C DB 0 0 0 C1122 0 0 C B C @ 0 0 0 0 C1122 0 A 0 0 0 0 0 C1122 are examples of cubic media. The latter has been singled out since in this case we obtain a decoupled system, i.e. the system is diagonal. Indeed, for cubic systems, we @/ the matrix obtain for E0 .b 0 2 1 @1 0 0 @ b @ @/ D .C1111 C1122 2 C2323 / @ 0 @22 0 A C2323 b E0 .b 0 0 @23 C .C1122 C 2 C2323 / b @b @> 37 The characteristic functions used here are nothing but the components of the so-called Kroneckerı-tensor.
Section 3.1 Partial Differential Equations in H1 .@ C e/
193
reducing to E0 .b @/ D b @ b @ C.2 C / b @b @> in the isotropic case. The fundamental solution in the case of a cubic medium can now in principle be calculated according to our usual procedure in terms of the fundamental solution of a sixth order scalar partial differential equation. Depending on the type of cubic medium, the corresponding polynomial may be reducible thus further simplifying the construction of a fundamental solution. Here we consider only the isotropic case, where the characteristic polynomial is given by det.E.@0 ; b @/ 1.33/ / D .@20 .2 C / b @2 /.@20 b @2 /2 : The minimal polynomial in the isotropic case is .@20 .2 C / b @2 / .@20 b @2 / as can be seen from .E0 .b @/ .2 C / b @2 / .E0 .b @/ b @2 / D 0; which is implied by @/ .2 C / b @2 / D . C / b @ b @; .E0 .b .E0 .b @/ b @2 / D . C / b @b @> :
(3.1.85)
Thus, .@20 .2 C / b @2 / .@20 b @2 / 133 D E.@0 ; b @/2 C .2 @20 .3 C / b @2 / E.@0 ; b @/ and so .@20 .2 C /b @2 /.@20 b @2 / E.@0 ; b @/1 D E.@0 ; b @/ C 2 @20 .3 C /b @2 D @20 C E0 .b @/ .3 C / b @2 or E.@0 ; b @/1 D .@20 .3 C / b @2 / .@20 .2 C / b @2 /1 .@20 b @2 /1 C E0 .b @/ .@20 .2 C / b @2 /1 .@20 b @2 /1 :
(3.1.86)
194
Chapter 3 Partial Differential Equations with Constant Coefficients
This can be further evaluated as E.@0 ; b @/1 1 2 1 1 .@ .2 C / b @2 D @2 / C .@20 b @2 / b 2 0 2 2 1 2 b .2 C / @ .@20 .2 C / b @2 /1 .@20 b @2 /1 2 C E0 .b @/ .@20 .2 C / b @2 /1 .@20 b @2 /1 1 1 D .@20 .2 C / b @2 /1 C .@20 b @2 /1 2 2 1 C .E0 .b @/ .2 C / b @2 / .@20 .2 C / b @2 /1 .@20 b @2 /1 2 1 C .E0 .b @/ b @2 / .@20 .2 C / b @2 /1 .@20 b @2 /1 2 1 1 D .@20 .2 C / b @2 /1 C .@20 b @2 /1 2 2 1 @ b @ .@20 .2 C / b C . C / b @2 /1 .@20 b @2 /1 2 1 @b @> .@20 .2 C / b C . C / b @2 /1 .@20 b @2 /1 : 2 A corresponding formula for the fundamental solution in terms of the fundamental solutions g ;C and g2 C;C follows. Moreover, we find with (3.1.85) b @ b @ .@20 E0 .b @// D .@20 b @2 / b @ b @ and
b @b @> .@20 E0 .b @// D .@20 .2 C / b @2 / b @b @> :
@ b @ Cb @b @> , we get by subtracting Since b @2 D b b @2 .@20 E0 .b @// D .@20 b @2 / b @ b @ .@20 .2 C / b @2 / b @b @> : In terms of the fundamental solutions g ;C and g2 C;C , we have .@20 b @2 /b @ b @ g ;C 1.33/ D b @ b @ .@20 E0 .b @//g ;C 1.33/ Db @ b @ ı 1.33/ and @2 /b @b @> g2 C;C 1.33/ D b @b @> .@20 E0 .b @//g2 C;C 1.33/ .@20 .2 C /b D b @b @> ı 1.33/ :
Section 3.1 Partial Differential Equations in H1 .@ C e/
195
By adding we get b @2 .@20 E0 .b @// GE;C D .@20 E0 .b @// .b @ b @ g ;C 1.33/ b @b @> g2 C;C 1.33/ / and so
b @2 GE;C D b @ b @ g ;C C b @b @> g2 C;C :
With this we get GE;C D Pcurl g ;C C Pgrad g2 C;C : Here Pcurl g ;C is associated with the so-called shear waves and Pgrad g2 C;C with the so-called pressure waves. b grad .i / b Remark 3.1.68. It is interesting to note that P
./ WD .F Pgrad /./ and 3 b V b P curl .i / ./ WD .F Pcurl /./, 2 R n ¹0º, 2 C1 .R3 /, define projections on the eigenspaces of E0 .i / for all 2 R3 n ¹0º, (here F and b denote the spatial Fourier transform). Indeed, a spectral representation of a general matrix of polynomials A.i / of the form X A.i / D k .i / Pk .i /; k
where the eigenvalues 7! k .i / and the entries of the projections 7! Pk .i / onto corresponding eigenspaces are Borel functions, can be used to obtain the solution operator in the form X @//1 D .@0 k .b @//1 Pk .b @/: .@0 A.b k
Analogously, we have @//1 D .@20 A.b
X
.@20 k .b @//1 Pk .b @/:
k
3.1.12.6
Fluid Dynamics
The dynamics of viscous fluids is commonly described by the Navier–Stokes system. If v denotes the velocity distribution and p the pressure density of the fluid, Newton’s law, assuming constant mass density, takes the particular form @0 v b @/v D f @2 v C b @p C .v >b with the typical non-linearity appearing as a directional derivative of the velocity field in its own direction, 2 R>0 . The system needs to be amended by suitable conditions
196
Chapter 3 Partial Differential Equations with Constant Coefficients
on the pressure density. In the so-called incompressible case it is assumed that the velocity field is divergence-free, i.e. b @> v D 0: Dropping the non-linear term we end up with the so-called Stokes system in the incompressible case which can be discussed within our framework. I Stokes equation (constant pressure, incompressible case). According to the above introductory remarks, the Stokes equation in the incompressible case is given by the system 0 10 1 0 1 @2 0 0 @1 @0 b f1 v1 B CB C B C 2 b B 0 @0 @ 0 @2 C B v2 C B f2 C D ; (3.1.87) B C @ 0 0 @0 b @2 @3 A @ v3 A @ f3 A p 0 @1 @2 @3 0 which is associated with a non-invertible operator. Assuming, however, that a solution p of 0 1 f1 2 >@ b b f2 A ; @ p D @ f3 is known, determining v reduces to discussing the .3 3/-partial differential expression S0 .@0 ; b @/ WD @0 C b @ b @: Since, from a mathematical point of view, the effect of this is equivalent to assuming b @p D 0, we refer to this case as the constant pressure case. The characteristic polynomial of S0 .@0 ; b @/ is .@0 / .@0 b @2 /2 and the minimal polynomial is @2 / D 2 .2 @0 b @2 / C @0 .@0 b @2 /: .@0 / .@0 b This shows that S0 .@0 ; b @/ is irreversibly evolutionary. Moreover, we find b2 1 .@0 C b @ b @/1 D .@0 b @2 /1 .b @2 C b @ b @/@1 0 .@0 @ / b2 1 D .@0 b @2 /1 b @b @> @1 0 .@0 @ / ; and the fundamental solution GS is therefore given by @b @> @1 GS D g0 1.33/ b 0 g0 1.33/ ;
Section 3.1 Partial Differential Equations in H1 .@ C e/
197
where g0 is the fundamental solution of the heat equation. Using the projections Pcurl ; Pgrad we obtain @2 @1 GS D g0 1.33/ Pgradb 0 g0 1.33/ D Pcurl g0 1.33/ C Pgrad R>0 ˝ ı 1.33/ : The condition that the velocity field u satisfying @/ u D f S0 .@0 ; b
(3.1.88)
is divergence-free appears as a constraint on the data rather than an additional equation (which would make this system formally overdetermined). Indeed, applying div D b @> to (3.1.88) yields @0 b @> u D b @> f and so u is divergence-free whenever the data f are divergence-free. I Linearized Navier–Stokes equation (constant pressure, incompressible case), Oseen system. Adding a drift term yields the linearized38 Navier–Stokes operator in the constant pressure case N0;s .@0 ; b @/ WD @0 C @ @ Cs > @: The determinant is given by @/ .@0 C s >b @ b @2 /2 @/ D .@0 C s >b det N0;s .@0 ; b and so the characteristic polynomial is now @/ .@0 C s >b @ b @2 /2 : 7! .@0 C s >b The minimal polynomial is @/ .@0 C s >b @ b @2 / 7! .@0 C s >b and so @ Cb @ b @/1 D .@0 C s >b @ b @2 /1 .@0 C s >b @/1 .@0 C s >b @ b @2 /1 : b @b @> .@0 C s >b 38
In the non-linear case the direction of drift s=jsj is proportional to the velocity field itself. The non-linear drift term is then v 7! .v >b @/ v; with 2 R>0 . Using a more concise block matrix notation, the Navier–Stokes equations are of the form ! ! v f @0 Cb @ b @ b @ v .v >b @/ 0 : D C 0 p p b 0 0 @> 0
198
Chapter 3 Partial Differential Equations with Constant Coefficients
The fundamental solution is now given by @b @> s;C gs 1.33/ ; GN WD gs 1.33/ b where s;C is the (forward) fundamental solution of the transport equation and gs is the fundamental solution of the heat equation with drift term. Using Pgrad we can separate the pure shift contribution and obtain @2 s;C gs 1.33/ GN D gs 1.33/ Pgrad b @/gs ı/ 1.33/ D gs 1.33/ Pgrad s;C ..@0 C s >b D gs 1.33/ C Pgrad s;C 1.33/ Pgrad gs 1.33/ D Pcurl gs 1.33/ C Pgrad s;C 1.33/ : As in the previous case, the condition that the velocity field u satisfying @/ u D f N0;s .@0 ; b
(3.1.89)
is divergence-free appears as a constraint on the data rather than an additional equation (which would make this system formally overdetermined). Applying @> to (3.1.89) leads to .@0 C s >b @/ b @> u D b @> f; which is a transport equation for div u. Again we have that u is divergence-free if the data f are. 3.1.12.7
Quantum Mechanics
The last examples to illustrate the utility of our Sobolev lattice framework are taken from quantum mechanics. I Schrödinger equation in vacuum. The scalar partial differential expression associated with this equation is of the form @0 i b @2 ; which is reversibly evolutionary (as a curiosity we mention that the Schrödinger equation is – in a different context – also known as the parabolic wave equation). Here the (forward) fundamental solution can be found by the same techniques as before and is given by the function39 gS;C .t; x/ D R>0 .t / .4 t /3=2 exp.i jxj2 =.4t // exp.3i=4/ 39 It may be interesting to note that this fundamental solution appears also by the formal substitution t 7! i t into the fundamental solution of the heat equation.
Section 3.1 Partial Differential Equations in H1 .@ C e/
199
for t 2 R n ¹0º; x 2 R3 . The behavior at t D 0 is again determined by our general results on initial value problems. In particular, R>0 .t / .4 t /3=2 exp.i j j2 =.4t // exp.3i=4/ ! ı
as t ! 0 C :
@/ is usually given as the .44/-partial I Dirac equation. The Dirac operator Q0 .@0 ; b differential expression with the block matrix form ! b C i C. @/ @ 0 Q0 .@0 ; b @/ WD : C.b @/ @0 i Here C.b @/ WD
@3 @1 i @2 @1 Ci @2 @3
…1 WD
0 C1 ; C1 0
D
P3 kD1
…k @k ; where
…2 WD
0 i ; Ci 0
…3 WD
C1 0 0 1
are known as Pauli matrices. Applying the unitary transformation given by the block C1 b matrix p1 Ci i C1 to Q0 .@0 ; @/ we obtain 2
! b i C i C. @/ @ 0 @/ WD Q1 .@0 ; b i i C.b @/ @0 ! 1 Ci C1 @/ i @0 1 C C.b @/ i @0 C 1 C C.b D 2 i C1 i C.b @/ C @0 i i C.b @/ C @0 i ! 1 Ci C1 @0 C i C.b @/ i Ci D : C1 C1 2 i C1 C.b @/ @0 i @/ has the typical form of reversibly The latter may be a preferable form since Q1 .@0 ; b evolutionary expressions of mathematical physics @0 W ; W @0 where W WD i i C.b @/. It is worth noting that 0 i C.b @/ b i C.@/ 0 so that b @2 .
0
C.b @/
C.b @/ 0
!2 D
C.b @/2 0 0 C.b @/2
! Db @2
is seen to be a (selfadjoint) square root of the negative Laplacian
200
Chapter 3 Partial Differential Equations with Constant Coefficients
The characteristic polynomial of course remains unchanged by this transformation and is given by det
@0 C i C.b @/ b C.@/ @0 i
! D det
@/ @0 i C i C.b b i i C.@/ @0
!
@2 C 1/2 : D ..@0 /2 b We obtain .@0 /2 b @2 C 1 D @20 b @ 2 C 1 2 @0 C 2 as minimal polynomial of the two variants of Dirac operators Q0 .@0 ; b @/ and Q1 .@0 ; b @/ as the calculation 0 i C i C.b @/ b i i C.@/ 0
!2 D
1.22/ .b @2 1/ 0 0 1.22/ .b @2 1/
!
@2 1/; D 1.44/ .b confirms. Correspondingly, we have, for i D 0; 1, Qi .@0 ; b @/1 D .2 @0 Qi .@0 ; b @// .@20 b @2 C 1/1 D Qi .@0 ; b @/ .@20 b @2 C 1/1 : Therefore, the (forward) fundamental solution Fi;C associated with Qi .@0 ; b @/ is given by Fi;C WD Qi .@0 ; b @/ gKG;C where gKG;C is the (forward) fundamental solution of the (unit mass) Klein–Gordon operator @2 b @2 C 1: 0
By comparison we see that this is formally the partial differential expression of the modified telegraph equation with e D i. We note j..i 0 C %/2 C 2 /1 j j..i 0 C %/ C i jj/1 ..i 0 C %/ i jj/1 j %2 and therefore j.@20 b @2 /1 jH
b
b
˛ .@0;% C1;@Ce/!H˛ .@0;% C1;@Ce/
%2
Section 3.1 Partial Differential Equations in H1 .@ C e/
201
for all ˛ 2 ZnC1 . Thus, we obtain, for % > 1, the Neumann series representation of .@20 b @2 C 1/1 .@20 b @2 C 1/1 D .@20 b @2 /1 .1 C .@20 b @2 /1 /1 D .@20 b @2 /1
1 X
.b @2 @20 /k
kD0
D
1 X
.1/k .@20 b @2 /k1 ;
kD0
in every H˛ .@0;% C 1; b @ C e/, ˛ 2 ZnC1 . Applying this to the Dirac-ı-distribution yields 1 X k gKG;C D g1;C .1/k g1;C ; kD2 k k where g1;C denotes the k-fold convolution g1;C D g1;C g1;C of the fundamental solution g1;C of the wave equation with itself. P k k The term 1 kD2 .1/ g1;C evaluates to the function p J1 . t 2 j j2 / 1 .t j j/ p ; .t; x/ 7! 4 R>0 t 2 j j2
where J1 is the Bessel function with index 1, see e.g. [30], therefore we have p 1 1 J1 . t 2 j j2 / gKG;C .t / D .t / ı t ŒS 2 .t j j/ p 4 t R>0 4 R>0 t 2 j j2 p 1 1 J1 . t 2 j j2 / 2 2 D .t / ı.t j j / .t j j/ p ; 2 R>0 4 R>0 t 2 j j2 for t 2 R n ¹0º. The fundamental solution gKG;C can be calculated as a variant of the fundamental solution of the wave equation. The behavior at t D 0 is again known from our general results. 40 Remark 3.1.69. As a curiosity unit i as we observe that – interpreting @3 the@complex qi E3 i @2 0 1 the real 2 2-matrix C1 0 – we may re-write the term @1 Ci @2 1@ E2 i E1 3 40
It should be emphasized that complex numbers can be (and are perhaps best) defined as real 2 2matrices of the form x y ; x; y 2 R: y x The usual matrix multiplication coincides with complex multiplication. It may be interesting to note that the block matrix structure x y ; x; y 2 C; y x generalizes the complex numbers to quaternions.
202
Chapter 3 Partial Differential Equations with Constant Coefficients
as
@3 q C .@1 E2 @2 E1 / @1 q C .@2 E3 @3 E2 /
As a consequence we see
Ci
@3 E3 @2 E2 @1 E1 : @2 q C .@3 E1 @1 E3 /
1 q iE3 C @0 i C i C.b @/ B B E2 iE1 C @ b p iH3 A i i C.@/ @0 H2 iH1 1 0 q iE3 p iH3 p iH3 b B @0 E2 iE1 C i C.@/ H2 iH1 i H2 iH1 C C DB @ A p iH3 q iE3 q iE3 b @0 i C.@/ i H2 iH1 E2 iE1 E2 iE1 0 !
0
1 @0 q @3 H3 @2 H2 @1 H1 C H3 @3 p C .@1 H2 @2 H1 / @0 E3 p @ @1 p C .@2 H3 @3 H2 / @0 E1 H2 @2 p C .@3 H1 @1 H3 / @0 E2 C H1 A: @3 q C .@1 E2 @2 E1 / C @0 H3 C q @0 p @3 E3 @2 E2 @1 E1 E3 i C @1 q C .@2 E3 @3 E2 / C @0 H1 C E2 @2 q C .@3 E1 @1 E3 / C @0 H2 E1 i
D
Assuming that the functions p; q and the vector fields E D .E1 ; E2 ; E3 / and H D .H1 ; H2 ; H3 / – as the applications require – are real-valued, we get 1 0 @0 p div E E3 B @0 E1 @1 p .@2 H3 @3 H2 / C H2 C C B B @0 E2 @2 p .@3 H1 @1 H3 / H1 C C B B @0 E3 @3 p .@1 H2 @2 H1 / C p C C B B @0 H1 C @1 q C .@2 E3 @3 E2 / C E2 C C B B @0 H2 C @2 q C .@3 E1 @1 E3 / E1 C C B @ @0 H3 C @3 q C .@1 E2 @2 E1 / C q A @0 q C div H H3 10 1 0 p @0 div 0 B grad @0 rot 0 C B E C CB C DB @ 0 rot @0 grad A @ H A 0 0 div @0 q 0 .0 1 0/ 1 0 .0 0 0/ 1 0.0/1 0 B 0 0 10 0 0 0 B B @ 1 A @ 0 0 0 A @ 1 0 0 A B B 0 0 00 000 CB B001 0 0 1 01 00 0 01 B B @ 0 A @ 1 0 0 A @ 0 0 0 A B @ 0 000 0 00 .0/ .0 0 0/ .0 0 1/ 0
1 0.0/1 0 C C0 @0AC p C CB E 0 B 0 1C @ 0 C C H C @0AC q 1 A .0/
showing a correspondence to the extended Maxwell system.
1 C C A
Section 3.1 Partial Differential Equations in H1 .@ C e/
203
Indeed, transformation with the block matrix 0 1 0 010 B 1 0 0 0 C B C @ 0 1 0 0A 0 001 yields 0
0 B 1 B @ 0 0
0 0 1 0
1 0 0 0
10 0 p BE 0C CB 0A@H 1 q
1
0
1 H C B p C CDB C A @ E A q
and so 10 1> 10 0 0 010 0 @0 div 0 CB C B 0C C B grad @0 rot 0 C B 1 0 0 0 C A @ A @ 0 rot @0 grad 0 1 0 0A 0 0 0 div @0 0 001 1 10 10 1 0 0 010 @0 div 0 0 1 0 0 B 1 0 0 0 C B grad @0 rot 0 C B 0 0 1 0 C CB CB C DB @ 0 1 0 0A@ 0 rot @0 grad A @ 1 0 0 0 A 0 001 0 0 div @0 0 0 01 1 0 10 0 010 0 @0 div 0 B 1 0 0 0 C B rot grad @0 0 C CB C DB @ 0 1 0 0 A @ @0 0 rot grad A 0 001 0 div 0 @0 0 1 0 rot grad @0 B 0 @ 0 C 0 div C DB @ rot grad @0 0 A div 0 0 @0
0
0 B 1 B @ 0 0 0
0 0 1 0
1 0 0 0
as well as 0
0
0 B 1 B @ 0 0
0 0 1 0
1 0 0 0
0.0/1 B 0 1B @ A 0 B B 1 B 0 0C CB0 1 0AB B 0 1 B B@0A @ 0 .0/
.0 0/ 0 1 1 000 @0 0 0A 0 000 1 0 10 @ 1 0 0 A 0 00 .0 0 0/
0 .0 0 0/ 1 0 10 @ 1 0 0 A 0 0 0 01 000 @0 0 0A 000 .0 0 1/
1 0.0/1 0 C C0 @0AC 0 C C B 1 0 B 0 1C @ 0 C C 0 @0AC C 0 1 A .0/
0 0 1 0
1 0 0 0
1> 0 0C C 0A 1
204
Chapter 3 Partial Differential Equations with Constant Coefficients 0
0
0 B B 1 DB @ 0 0
0 0 1 0
1 0 0 0
.0/ B001 1B 0 B B@1A CB 0CB 0 CB0 1 0AB 0 B @ A 1 B B 0 @ 0 .0/
.0 1 0/ 0 1 000 @0 0 0A 000 0 1 0 10 @ 1 0 0 A 0 00 .0 0 0/
0
0
0 B B 1 DB @ 0 0
0 0 1 0
1 0 0 0
.0 0 0/ B0 0 1 01 1B 0 B B @ 1 0 0 A CB 0CB 0 00 1 CB 0 0AB 0 0 0 B @ A 1 B B 000 @ 000
1 000 B @0 0 0A B B 000 B B .0 0 0/ B 1 D B0 B 0 10 B B @ 1 0 0 A B @ 0 00 .0
0 1/
.0 1 0/ 0 1 000 @0 0 0A 000 0 1 0 10 @ 1 0 0 A
0 .0/
0 00 .0 0 0/
0 10 1 0 0 10 @ 0 A @ 1 0 0 A 0 0 00 .0/ .0 0 1/ 0 1 0 1 0 000 @0A @0 0 0A 1 .0/
1 .0/ 0 1C 0 C0 @0AC C 0 B 0 C CB0 0 1CB 0 C@1 C @0AC 0 C 1 A
.0/ 1 0 @ 1 A 0 0 1 0 @0A 0
.0 0 1/ 0 0
.0 0 0/ 1 0 10 @ 1 0 0 A 0 00 0 1 000 @0 0 0A 000 .0 0 1/ 0
000 .0 0 0/
1 0 0 0
0 1 0 0
1 0 C 0C C 0A 1
.0/ 1 .0/ 0 1C 0 C @0AC C C 0 C 0 1C 0 C C @0AC C 1 A .0/
0 11 0 @0AC C 1 C C C .0/ C 0 1 C: 0 C C @0AC C 0 A .0/
Thus we obtain again the block structure of reversibly evolutionary systems of @0typical W mathematical physics W @0 , where
rot div
W WD
0
! grad 0
00
1 0 11 0 10 0 B@ C A @ 1 0 0 0AC B CB C 0 00 1 A @ .0 0 1/ .0/
1 0 1 0 B rot C @ 1 0 0 A B DB 0 0 0 @ div C.0 0 1/ 0
0 11 0 @ grad C 0 A C C C: 1 A 0
Section 3.2 Partial Differential Equations in H1 .jD j/
3.2
205
Linear Partial Differential Equations with Constant Coefficients in H1 .jD j/
3.2.1 Extension of the Solution Theory to H1 .jD j/. The solution theory developed and exemplified in the previous sections clearly is wholly satisfactory for evolutionary problems as well as for non-evolutionary but invertible differential expressions. It is a remarkable fact, however, that for every partial differential expression with constant coefficients a fundamental solution can be constructed in H1 .jDj/ D S , the space of tempered distributions. This result is originally due to Hörmander, [20]. It has later found a beautiful expression as an application of algebraic methods in the study of rings of differential operators, [12]. Since exp.m/ P .@/ D P .@C/ exp.m/ and exp.m/ W H1 .jD j/ ! H1 .jDj/ are continuous, we have from this result the existence of a fundamental solution in H1 .jD j/ for every 2 RnC1 . In the evolutionary case it may happen, however, that the causal fundamental solution, which is of foremost interest, is not, due to exponential growth, in H1 .jDj/ as examples from ordinary differential equations readily show. Thus, the fundamental solution in H1 .jDj/ may not be the one we are looking for. Moreover, in the non-invertible case we shall see non-uniqueness results. In many such cases of interest, however, additional constraints can be found to enforce uniqueness. Despite these added difficulties the prospect of discussing fundamental solutions even in non-invertible cases is a strong incentive for an extension of the solution theory. These remarks may suffice to motivate the expansion of our solution theory to the Sobolev lattice .H˛ .jD j//˛2ZnC1 , 2 RnC1 , which in contrast to the previously considered exponentially weighted Sobolev lattice will be referred to as the tempered, exponentially weighted Sobolev lattice. Rather than re-doing the whole approach, however, we shall perform this expansion more simply by continuous extension of the solution theory already established. We first recall that the lattice .Hˇ .jD j//ˇ 2ZnC1 satisfies H˛ .jD j/ ,! H˛ .@ C e/ for all ˛ 2 N nC1 . By duality this implies the continuity of the dense embedding H˛ .@ C e/ ,! H˛ .jD j/ for all ˛ 2 N nC1 . This shows that in particular H1 .@ C e/ ,! H1 .jD j/ densely and continuously. We shall use the notation HnC1;sC1;tC1;1 .jD j/, s; t 2 N, 2 RnC1 , for H1 .jD j/ in the analogous sense to HnC1;sC1;tC1;1 .@ C e/ for our new solution spaces, if the added information is needed for sake of clarity.
206
Chapter 3 Partial Differential Equations with Constant Coefficients
As a preparatory step to simplify further calculations, we need the following lemmata. Lemma 3.2.1. For all ˇ 2 N nC1 , 2 H1 .jD j/ jjD jˇ j;0 D jjD jˇ 2bˇ=2c .D D /bˇ=2c j;0 D jDˇ 2bˇ=2c .D D /bˇ=2c j;0 : Moreover, C
X
(3.2.90)
jm˛ @ j2;0
˛Cˇ;˛;2N nC1
jDˇ 2bˇ=2c .D D /bˇ=2c j2;0 X C 1 jm˛ @ j2;0
(3.2.91)
˛Cˇ;˛;2N nC1
and C
X
j@ m˛ j2;0
˛Cˇ;˛;2N nC1
jDˇ 2bˇ=2c .D D /bˇ=2c j2;0 X C 1 j@ m˛ j2;0
(3.2.92)
˛Cˇ;˛;2N nC1
for some fixed C 2 R>0 and for all ˇ 2 N nC1 , 2 CV 1 .RnC1 /. Proof. Equality (3.2.90) follows from abstract functional analytical arguments. It remains to consider (3.2.91) and (3.2.92). Again we observe that it suffices to deal with the case D 0. We first recall that formally D D p1 .m @/; D D p1 .m C @/ 2 2 so that 2jDk j20;0 D j@k j20;0 C jmk j20;0 2 Reh@k j mk i0;0 D j@k j20;0 C jmk j20;0 C j j20;0 and 2jDk j20;0 D j@k j20;0 C jmk j20;0 j j20;0 : Combining these equalities we get 4jDk Dk j20;0 D 2j@k Dk j20;0 C 2jmk Dk j20;0 2jDk j20;0 D j @2k C mk @k C j20;0 C j mk @k C m2k j20;0 (3.2.93) j@k j20;0 jmk j20;0 j j20;0 :
Section 3.2 Partial Differential Equations in H1 .jD j/
207
The first term on the right-hand side can be further evaluated to yield j@2k @k mk j20;0 D j@2k mk @k j20;0 D j@2k j20;0 C jmk @k j20;0 2 Reh.@2k 1/ j mk @k i0;0 D j@2k j20;0 C j j20;0 C jmk @k j20;0 2 Reh@2k j i0;0 2 Reh@2k j mk @k i0;0 C 2 Reh j mk @k i0;0 D j@2k j20;0 C jmk @k j20;0 C 3j@k j20;0 : We get similarly jmk @k m2k j20;0 D jmk @k j20;0 C jm2k j20;0 C 3jmk j20;0 and so for the original term in (3.2.93) 4jDk Dk j20;0 D j@2k j20;0 C 2jmk @k j20;0 C 2j@k j20;0 C 2jmk j20;0 C jm2k j20;0 j j20;0 : The equivalence of norms (3.2.91) now follows by induction. We have 2Dk Dk D .mk C @k /.mk @k / D @2k C m2k C 1 and .Dk Dk /sC1 D .Dk Dk /s Dk Dk D 12 .Dk Dk /s .@2k C m2k C 1/ and so also 1 j.Dk Dk /sC1 j20;0 D j.Dk Dk /s .@2k C m2k C 1/ j20;0 2 X j C jmik @k .@2k C m2k C 1/ j20;0 iCj 2s; i;j 2N
X
C
j C2
jmik @k
j
C miC2 @k j20;0 k
iCj 2s; i;j 2N
C 0 jjDk j2sC1 j20;0 ; where we have estimated the lower order terms by the last term according to the induction assumption. Evaluating j C2
jmtk @k
j
C mtC2 @k j20;0 k
j C2
D jmtk @k
j
j20;0 C jmtC2 @k j20;0 k j
j C2
2 RehmtC2 @k j mtk @k k j C2
D jmtk @k
i0;0
j
j20;0 C jmtC2 @k j20;0 k j C1
C 2jmtC1 @k k
j
j C1
j20;0 C 4 .t C 1/ RehmtC1 @k j mtk @k k
i0;0 ;
208
Chapter 3 Partial Differential Equations with Constant Coefficients
we obtain X
j.Dk Dk /sC1 j20;0 C
j
jmtk @k j20;0
tCj 2sC2; t;j 2N
C 00 jjDk j2sC1 j20;0 : Together with j.Dk Dk /sC1 j0;0 D jjDk j2sC2 j0;0 D jDk jDk j2sC1 j0;0 jjDk j2sC1 j0;0 ; we obtain the desired lower estimate X
jjDk j2sC2 j20;0 C
j
jmik @k j20;0
iCj 2sC2; i;j 2N
for some generic constant C 2 R>0 provided we have X
jjDk j2sC1 j20;0 C
j
jmtk @k j20;0
tCj 2sC1; t;j 2N
for some generic constant C 2 R>0 . The odd case, however, follows in a similar fashion if we assume it holds for the even case. Indeed, jjDk j2sC1 j20;0 D jDk jDk j2s j20;0 jjDk j2s Dk j20;0 C 0 jjDk j2s j20;0 X j C1 j C jmtk @k mtC1 @k j20;0 k tCj 2s; t;j 2N
C 0 jjDk j2s j20;0 X j C jmtk @k j20;0 C 0 jjDk j2s j20;0 ; tCj 2sC1; t;j 2N
for generic constants C; C 0 2 R>0 . Together with jjDk j2sC1 j0;0 D jDk jDk j2s j0;0 jjDk j2s j0;0 the desired estimate follows also in this case. Thus, we have by induction jjDk js j0;0 C
X tCj s; t;j 2N
j
jmtk @k j20;0
Section 3.2 Partial Differential Equations in H1 .jD j/
209
for all 2 CV 1 .RnC1 / and s 2 N, where the constant C 2 R>0 only depends on s. The corresponding upper estimate follows by a much simpler induction argument using the triangle inequality, the details of which may be left as an exercise. The case 2 RnC1 now follows by unitary equivalence. The general result (3.2.91) is now clear since the occurring operations commute for different variables. Since mk formally commutes with @k up to terms of lower order the remaining estimate (3.2.92) now follows. In the proof of the previous lemma we have made use of a rough estimate of ‘lower order terms’ due to differentiating products. The interaction of differentiation and multiplication can of course be made precise in the form of a ‘Leibniz rule’. Lemma 3.2.2. A Leibniz rule holds for D and D in the sense that D˛ .
X
/D
2jˇ j1 =2
ˇ CD˛; ˇ;2N nC1
and D˛ .
X
/D
ˇ CD˛; ˇ;2N nC1
˛ ˇ
2jˇ j1 =2
.@/ˇ D
˛ ˇ
@ˇ D
2 for all 2 C1 .RnC1 / such that Dˇ isQbounded ˛i for all ˇ ˛ and all ˛ n nC1 H1 .D /, ˛; ˇ 2 N . Here ˇ D iD0 ˇi for ˛ D .˛0 ; : : : ; ˛n /, ˇ D .ˇ0 ; : : : ; ˇn / 2 N nC1 . Proof. Let Then Dk; .
2 CV 1 .RnC1 /. 1 / D p .@k C mk C / . / 2 1 1 1 D p .@k / p @k C p .mk C / 2 2 2 1 D p .@k / C Dk; : 2
(3.2.94)
The Leibniz rule follows in this case by the usual induction argument. The case with D instead of D follows in the same way. It is clear that the role of and can be formally interchanged. Based on this observation, we shall now derive somewhat of a converse of the Leibniz rule. We
210
Chapter 3 Partial Differential Equations with Constant Coefficients
claim
D˛
X
D
jˇ j1 =2
2
ˇ CD˛; ˇ;2N nC1
˛ ˇ
D ..@ˇ / /:
(3.2.95)
This follows also by induction. Indeed,
D˛Cek
D
D˛
Dk;
X
D
jˇ j1 =2
2
ˇ CD˛; ˇ;2N nC1
˛ ˇ
D .@ˇ Dk;
/:
According to (3.2.94) we have @ˇ Dk;
1 / C p @ˇ Cek 2
D Dk; .@ˇ
and so
D˛Cek
X
D
jˇ j1 =2
2
ˇ CD˛; ˇ;2N nC1
X
D
jˇ j1 =2
2
ˇ CD˛; ˇ;2N nC1
X
C
ˇ CD˛; ˇ;2N nC1
ˇ CD˛Cek ; k
ˇ CD˛; ˇk
D .Dk; .@ˇ
/ C @ˇ Cek
/
DCek .@ˇ
/
2jˇ Cek j1 =2 D .@ˇ Cek 2jˇ j1 =2
1;ˇ;2N nC1
2jˇ j1 =2
1; ˇ;2N nC1
˛ ˇ
/
D .@ˇ
˛ ˇ ek
/
D .@ˇ
/:
m mC1 (as a simple consequence of m C k1 D k ˛ ˛ ˛Cekk ˛Ce k D 0 D ˛ D ˛Cek , we get the desired for m; k 2 N, k 1) and 0 equality. The analogous result holds for D replaced by D , i.e. we also have Since
ˇ
C
D˛
˛ ˇ ek
D
˛ ˇ
X
C
˛ ˇ
X
D
˛
˛ ˇ
D
˛Cek ˇ
X ˇ CD˛; ˇ;2N nC1
2jˇ j1 =2
˛ ˇ
D ..@/ˇ
/:
(3.2.96)
Section 3.2 Partial Differential Equations in H1 .jD j/
211
Now let f 2 H1 .D /. Then we have using (3.2.96) hD˛ . f /j i;0 D h f jD˛ i;0 D hf j D˛ i;0 ˇ X ˇ D f ˇˇ ˇ ˇ D f ˇˇ D
2jˇ j1 =2
ˇ CD˛Iˇ;2N nC1
X
2jˇ j1 =2
ˇ CD˛Iˇ;2N nC1
X
2jˇ j1 =2
ˇ CD˛Iˇ;2N nC1
˛ ˇ
˛ ˇ ˛ ˇ
D ..@/ˇ / ;0
D ...@/ˇ / / ;0
h..@/ˇ /D f j i;0 :
Since 2 CV 1 .RnC1 / was arbitrary, the stated Leibniz rule also follows in this case. With these results we can now extend our earlier solution theory which was summarized in Theorem 3.1.10. Let P .@/ be invertible with respect to 2 RnC1 . Then x 7! .i x C e/˛ P .i x C /1 is bounded for some ˛ 2 N nC1 and using the product rule for matrix valued functions we obtain @k .i m C e/˛ P .i m C /1 D i .˛/ .i m C e/˛ek P .i m C /1 C i.im C e/˛ P .im C /1 .@k P /.im C /P .i m C /1 C .im C e/˛ P .im C /1 @k D i .˛/ .i m C e/ek ..i m C e/˛ P .i m C /1 / C i.i m C e/˛ ..i m C e/˛ P .i m C /1 /.@k P /.i m C / ..i m C e/˛ P .i m C /1 / C .i m C e/˛ P .i m C /1 @k : This yields by a lengthy but in principle elementary induction, making heavy use of Lemma 3.2.1, that _ ^ .i mCe/˛ P .i mC/1 jH .jDj/ W H .jDj/ ! H .jDj/ is continuous: 2N nC1 2N nC1
212
Chapter 3 Partial Differential Equations with Constant Coefficients
Thus, with ˇ 2 N nC1 given and noting that P enjoys the same continuity property as P , we find a possibly rather large 2 N nC1 such that jhP .i m C /1 j i0;0 j D jh jP .i m C /1 i0;0 j j j0;ˇ jP .i m C /1 j0;ˇ D j j0;ˇ j.i m C e/˛ .i m C e/˛ P .i m C /1 j0;ˇ C1 j j0;ˇ j.i m C e/˛ P .i m C /1 j0;˛Cˇ D C1 j j0;ˇ jjDj˛Cˇ .i m C e/˛ P .i m C /1 j0;0 C2 j j0;ˇ jjDj
j0;0
D C2 j j0;ˇ j j0; for all ; 2 CV 1 .RnC1 /. This, together with the lattice structure of the Sobolev lattice .H˛ .jDj//˛2ZnC1 and the density of CV 1 .RnC1 / in H1 .jDj/, shows that P .i m C /1 W H1 .jDj/ ! H1 .jDj/
(3.2.97)
is continuous. Thus, we find that the solution theory carries over to the larger space H1 .jDj/ simply by continuous extension. To confirm that P .i m C / indeed has an inverse we need to see that P .i m C / u D 0 has only the trivial solution. This follows from 0 D hP .i m C / uj i0;0 D hujP .i m C / i0;0 for all
2 CV 1 .RnC1 /, and by noting that P .i m C /j V
C1 .RnC1 /
W CV 1 .RnC1 / !
CV 1 .RnC1 / is a bijection (due to the assumed invertibility). Indeed, this yields huj i0;0 D 0 for all 2 CV 1 .RnC1 /, i.e. u D 0. Theorem 3.2.3. Let P .@/ be a ..s C 1/ .s C 1//-partial differential expression in RnC1 . If P .@/ is invertible with respect to 2 RnC1 then P .@/1 WD P .@ C /1 W HnC1;sC1;tC1;1 .jD j/ ! HnC1;sC1;tC1;1 .jD j/ exists as a continuous operator for every t 2 N, via continuous extension of P .@ C /1 W HnC1;sC1;tC1;1 .@ C e/ HnC1;sC1;tC1;1 .jD j/ ! HnC1;sC1;tC1;1 .jD j/:
Section 3.2 Partial Differential Equations in H1 .jD j/
213
Proof. It should be noted that by applying the inverse Fourier–Laplace transform we get from (3.2.97) the existence of the continuous operator S WD L P .i m C /1 L . The unitary equivalence of S with P .i m C /1 yields that S is indeed the inverse of P .@ C /: S D P .@ C /1 : Remark 3.2.4. In contrast to the theory in H1 .@ C e/ we do not get a fixed loss of regularity ˇ ˛ for P .@/1 jH˛ .jD j/ W H˛ .jD j/ ! Hˇ .jD j/, where ˇ 2 ZnC1 ˇ ˛, is assumed to be maximal. In other words, we do have continuity of P .@/1 W H1 .jD j/ ! H1 .jD j/, but not of finite order. This is the price we have to pay for enlarging the class of admissible right-hand sides. It is also important that the concept of causality of the solution operator carries over to H1 .jD j/. For this we identify elements in H1 .jD j/ as linear functionals on CV 1 .RnC1 / in the same way as in (3.1.42) by considering ' 7! h j exp.2m/'i;0 on CV 1 .RnC1 /. Indeed, let P .@/ be evolutionary in direction .0/ and P .@/1 f D 0 in an open set Ga0 ;.0/ WD .a0 C R>0 / .0/ C ¹.0/ º? , whenever f D 0 in Ga0 ;.0/ . Then, if additionally f 2 H1 .@ C e/, D % .0/ , with % 2 R>0 sufficiently large, we have by definition hP .@/1 f j i;0 D 0 for all 2 CV 1 .RnC1 / with supp Ga0 ;.0/ . This property carries over from f 2 H1 .@ C e/ to f 2 H1 .jD j/ by approximation (using a standard cut-off technique). Thus, we obtain the following causality result. Theorem 3.2.5. Let P .@/ be an ..s C 1/ .s C 1//-partial differential expression in RnC1 evolutionary in direction .0/ . Then, with D % .0/ and % 2 R>0 sufficiently large, P .@ C /1 W H1 .jD j/ ! H1 .jD j/ is causal in direction .0/ . Proof. We know that P .@ C /1 jH1 .@ Ce/ W H1 .@ C e/ ! H1 .@ C e/ is causal in direction .0/ . Therefore, if g 2 Hˇ .@ C e/, with ˇ a matrix of multiindices in .N nC1 /s1 D .N nC1 /s , is such that a D inf supp.0/ g 2 R, then we have that inf supp.0/ P .@ C /1 g a, or hP .@ C /1 gj i;0 D 0
214
Chapter 3 Partial Differential Equations with Constant Coefficients
for all 2 CV 1 .RnC1 / with sup supp.0/ < a. Now, let ! 2 C1 .R/ be such that supp ! is bounded above and supp.1 !/ is bounded below. Then !R 2 C1 .RnC1 / given by !R .x0 ; : : : ; xn / WD !.x0 R/ !.xn R/ for x D .x0 ; : : : ; xn / 2 RnC1 , R 2 R>0 , satisfies !R .x/ ! 1 as R ! 1 for all x 2 RnC1 . Moreover, @˛ !R .x/ ! 0 as R ! 1 for all x 2 RnC1 , ˛ 2 N nC1 n ¹0º. As a consequence of the uniform boundedness of these terms, we find by an application of the Leibniz rule !R .m/ g 2 Hˇ .@ C e/
and
!R .m/ g ! g
in Hˇ .jD j/
as R ! 1
for all g 2 Hˇ .jD j/. Thus, we get in particular, with a D inf supp.0/ g 2 R, hP .@ C /1 !R .m/ gj i;0 D 0 for all R 2 R, 2 CV 1 .RnC1 / with sup supp.0/ < a. Letting R ! 1 and using the continuity of P .@ C /1 , we get hP .@ C /1 gj i;0 D 0 for all 2 CV 1 .RnC1 / with sup supp.0/ < a. This proves that causality in direction .0/ is preserved. Thus we have extended our previous solution theory to the new Sobolev lattice. The added benefit is that even if P .@/ is not invertible, there is a fundamental solution in H1 .jD j/, which can always be constructed. In our specific cases of interest the construction is indeed fairly elementary. It is interesting to note that the concepts of the spectrum and the resolvent set of an operator in H1 .C /, where C D @ C e or C D jD j, 2 RnC1 , are of particular interest, carry over from the Hilbert space case in the obvious way. Definition 3.2.6. Let .H˛ .C //˛2ZnC1 be a Sobolev lattice associated with an operator family C and let W W H1 .C / ! H1 .C / be a continuous linear operator in H1 .C /. Then the (Sobolev lattice) resolvent set of W is given by %1 .W / WD ¹ 2 C j.W / injective ^ .W /ŒH1 .C / D H1 .C / ^ .W /1 continuousº: and correspondingly the spectrum of W in H1 .C / is 1 .W / WD C n %1 .W /:
Section 3.2 Partial Differential Equations in H1 .jD j/
215
The spectral parts – (Sobolev lattice) point spectrum, (Sobolev lattice) residual spectrum and (Sobolev lattice) continuous spectrum, respectively – are defined (as in the Hilbert space case): P 1 .W / WD ¹ 2 C j.W / not injectiveº; R1 .W / WD ¹ 2 Cj.W / injective ^ .W /ŒH1 .C / ¤ H1 .C /º; C 1 .W / WD ¹ 2 Cj.W / injective ^ .W /H1 .C / D H1 .C / ^ .W /1 not continuousº: Returning to our current special situation, we distinguish the exponentially weighted Sobolev lattice from the tempered, exponentially weighted Sobolev lattice in the following by writing a lower index ‘ ’ to denote the spectral parts of the case C D jD j, 2 RnC1 . Example 3.21. Consider the special situation of the differential expression @0 p b @ in R1Cn , n 2 N>0 , p 2 Rn ; jpj D 1, in H1 .@0; C 1; b @ C e/, 2 R. Since @ D @0; p b @ . / @0 p b is invertible if Re ¤ , we see that @/: C n .i ŒR C / %1 .@0 p b Let now D 1i . / 2 R. Application of the Fourier–Laplace transform leads to b i W H1 .i m0 C 1; i m b C e/ ! H1 .i m0 C 1; i m b C e/: i m0 i p m Since the spectral information is preserved, i.e. @/ D P 1 .i m0 i p m b C / D P 1 .i m0 i p m b / C ; P 1 .@0 p b R1 .@0 p D/ D R1 .i m0 i p m b C / D R1 .i m0 i p m b/ C ; C 1 .@0 p D/ D C 1 .i m0 i p m b C / D C 1 .i m0 i p m b/ C ; b. From the above we already have it suffices to consider i m0 i p m b/ R 1 .m0 p m b C e/ can We would like to show equality. Since – as we know – H1 .i m0 C 1; i m 1Cn / such that be characterized as those elements of Lloc .R 2 b C e/˛ .i m0 C 1/˛0 .i m
2 L2 .R1Cn /
216
Chapter 3 Partial Differential Equations with Constant Coefficients
for some ˛ D .˛0 ; b ˛ / 2 Z1Cn and since S WD ¹.x0 ; y/j x0 2 R; y 2 Rn ; x0 p y D º b / is injective. Moreover, since is a Lebesgue null set, we see that .m0 p m CV 1 .R1Cn n S / is dense in L2 .R1Cn / and b C e/˛ .m0 p m b /j V .i m0 C 1/˛0 .i m
C1 .R1Cn nS /
is a bijection in CV 1 .R1Cn nS /, we have that .m0 pb m /ŒH1 .im0 C1; ib m Ce/ is dense in H1 .im0 C 1; i m b C e/. Let u" WD .m0 p m b i "/1 " , with " WD ".nC1/=2 .. . ; 0//="/, " 2 R>0 , and 2 CV 1 .RnC1 / such that supp BRnC1 .0; 1/ and Z j .y/j2 dy D 1: RnC1
Then we have b / u" D .m0 p m b / .m0 p m b i "/1 " .m0 p m D " C i " .m0 p m b i "/1 " : b C e/, ˇ 2 Z1Cn , but The right-hand side is bounded in every Hˇ .i m0 C 1; i m Z Z 2 .x02 C 1/˛0 .b x 2 C e/˛ ju" .x0 ;b x /j2 dx0 dx juj˛ D Z D
RnC1
RnC1
inf¹.x02
!1
Z
R
R
.x02 C 1/˛0 .b x 2 C e/˛ j " .x0 ;b x /j2 dx0 dx .x0 p b x /2 C "2
C 1/˛0 .b x 2 C e/˛ j .x0 ;b x / 2 BRnC1 .. ; 0/; "/º .3 C jpj2 /"2
as " ! 0:
b /1 is unbounded and so Consequently, we find that .m0 p m 1 .m0 p m b/ D C 1 .m0 p m b/ D R: Thus we have 1 .@0; C p b @/ D C 1 .@0; C p b @/ D i ŒR C : b @ acting in H1 .jD0; j; jDj/. Alternatively, we could have considered @0 p b b Then we would have found C n .i ŒR C / %1; .@0; C p @/. For D
Section 3.2 Partial Differential Equations in H1 .jD j/ 1 i .
217
/ 2 R we see that for any f 2 CV 1 .R/ we have that 2 C1 .R1Cn / given
by
.x0 ; y/ WD exp. x0 / exp.i x0 / f .x0 C p y/ b for x0 2 R and y 2 Rn can be considered as an element of H1 .jD0; j; jDj/. Indeed, jh j i;0 j2 D jh j exp.2m0 / i0;0 j2 ˇZ ˇ2 ˇ ˇ D ˇˇ exp.i x0 / f .x0 C p b x / exp.x0 / .x0 ;b x / d.x0 ;b x /ˇˇ R1Cn
Z
R1Cn
2 sup¹jf .t /jjt 2 Rº exp.x0 / j .x0 ;b x /j d.x0 ;b x/
sup¹jf .t /j2 jt 2 Rº .nC1/ Z .x02 C 1/.b x 2 C e/e exp.2x0 /j .x0 ;b x /j2 d.x0 ;b x/ R1Cn
b Similarly we for all 2 CV 1 .R1Cn /. Thus, we deduce 2 H.1;e/ .jD0; j; jDj/. b Moreover, can show that classical derivatives of are also in H.1;e/ .jD0; j; jDj/. we find by construction of that .@0 p b @ .i C // D 0 since point-wise we have @0 .x0 ; y/ D .i C / exp. x0 / exp.i x0 / f .x0 C p y/ C exp. x0 / exp.i x0 / .@f /.x0 C p y/ and
b @ .x0 ; y/ D exp. x0 / exp.i x0 / .@f /.x0 C p y/ p
(where @f denotes the 1-dimensional derivative of f ) for all x0 2 R and y 2 Rn (recall jpj D 1). Thus, we find in this case – in contrast to the above – 1; .@0; C p b @/ D P 1; .@0; C p b @/ D i R C : Example 3.22. Consider the differential expression @2 in RnC1 , n 2 N>0 , as a continuous operator in H1 .@ C e/ for some fixed 2 RnC1 . By the Fourier– Laplace transform we are led to discuss .i m C /2 D m2 2 2 i m
218
Chapter 3 Partial Differential Equations with Constant Coefficients
in H1 .i m C e/. For ¤ 0 and with WD x jj2 .x / (and noting that ? ) we calculate j .i x C /2 j2 D .x 2 2 Re /2 C .2 x C I m /2 D ..jj2 .x / C /2 2 Re /2 C .2 x C I m /2 D .jj4 .x /2 C 2 2 Re /2 C .2 x C I m /2 : For this to be zero for an x 2 R we must have x D 12 I m and jj4 .x /2 C 2 2 Re D 14 jj4 .I m /2 C 2 2 Re D 0 so that 1 x D I m jj2 C 2 with ? in RnC1 such that 1 2 D 2 C Re jj4 .I m /2 : 4 For n 2 N1 this is possible if only if 1 2 C Re jj4 .I m /2 0; 4 i.e. for in the ‘interior’ ¹ 2 C j 2 C Re 14 jj4 .I m /2 0º enclosed by the parabola ¹ 2 C j 2 C Re 14 jj4 .I m /2 D 0º. In the ‘exterior’ of this parabola given by the open set ¹ 2 C j 2 C Re 14 jj4 .I m /2 < 0º the differential expression @2 is invertible in H1 .@ C e/. Consequently, we have ˇ ³ ² ˇ 2 1 4 2 ˇ 2 C ˇ C Re jj .I m / < 0 %1 .@2 /: 4 For 2 ¹ 2 C j 2 C Re 14 jj4 .I m /2 0º we see that the real zeros of the polynomial x 7! .i x C /2 constitute a Lebesgue null set (indeed if n > 1 an .n 2/-dimensional sphere in a hyperplane orthogonal to ). By the argument of the previous example we find again ˇ ² ³ ˇ 2 1 4 2 ˇ 2 C ˇ C Re jj .I m / 0 C 1 .@2 / [ %1 .@2 /: 4 To show that actually ˇ ³ ² ˇ 2 1 4 2 ˇ 2 C ˇ C Re jj .I m / 0 D C 1 .@2 / 4
(3.2.98)
Section 3.2 Partial Differential Equations in H1 .jD j/
219
we need to see the failure of continuity of .@2 /1 for all values in the set ¹ 2 C j 2 C Re 14 jj4 .I m /2 0º. A similar but slightly refined argument to the one used in the previous example yields the result. Choose a real zero 1 x D I m jj2 C 2 with ? in RnC1 such that 1 2 D 2 C Re jj4 .I m /2 4 and consider u" WD
1 j.i m C
/2
C j C i "
"
with " WD ".nC1/=2 .. x /="/, where 2 CV 1 .RnC1 / is such that supp
BRnC1 .0; 1/ and Z RnC1
j .y/j2 dy D 1:
Then we have ..i m C /2 / u" D D
..i m C /2 /
" j.i m C /2 C j C i " ..i m C /2 / C i " 1
" i "
" : 2 2 j.i m C / C j C i " j.i m C / C j C i "
The right-hand side is bounded in every Hˇ .i m C e/, ˇ 2 ZnC1 , but for u" we find Z ju" j2˛ D .x 2 C e/˛ ju" .x/j2 dx Z D
RnC1
RnC1
.x 2 C e/˛ j " .x/j2 dx j.i x C /2 C j2 C "2
inf¹.x 2 C e/˛ j x 2 BRnC1 .x ; "/º ! 1 as " ! 0C: sup¹j.x i /2 j2 j x 2 BRnC1 .x ; "/º C "2
This yields the desired result (3.2.98). Considering @2 in H1 .jD j/, we see that for 2 C 12 22 jj4 .1 2 /2 0, D .1 C i 2 /2 , 2 RnC1 n ¹0º, we have w WD ı¹x º is an element in H1 .jDj/ that solves the equation ..i m C /2 / w D 0: Consequently, u WD L w 2 H1 .jD j/ solves .@2 / u D 0:
220
Chapter 3 Partial Differential Equations with Constant Coefficients
In this case we get ˇ ² ³ ˇ 2 1 4 2 ˇ 2 C ˇ C Re jj .I m / 0 D P 1; .@2 / D 1; .@2 /: 4 In the remaining case D 0, we find by the same reasoning that R0 D C 1 .@2 / D P 1; .@2 /: As the reasoning – in particular for the last example – indicates, there is a general spectral result here. Theorem 3.2.7. Let P .@/ be an .L L/-differential expression in RnC1 , L 2 N>0 . If the set of all real zeros N WD ¹xjx 2 RnC1 ; det.P .i x C / / D 0º of x 7! det.P .i xC/ / is a Lebesgue null set for every 2 C, i.e. x 7! det.P .i xC/ / is not the zero polynomial, then we have 1 .P .@ C // D C 1 .det P .@ C // D P 1; .det P .@ C // D 1; .P .@ C // D ¹ 2 Cj N non-emptyº and %1 .P .@ C // D %1; .P .@ C //: Proof. Let 2 C be such that N is empty. Then we have, according to our solution theory, 2 %1 .det P .@ C // D %1 .P .@ C //. Since N is in any case a Lebesgue null set (by assumption), the injectivity of det.P .i mC/ / and therefore .P .i m C / / in H1 .i m C e/ is clear. Likewise, we have the density of the range of det.P .i m C / / in H1 .i m C e/, since CV 1 .RnC1 n N / is dense in H1 .i m C e/ and det.P .i m C / / W CV 1 .RnC1 n N / ! CV 1 .RnC1 n N / is bijective. 1 Let u" WD jdet.P .i mC//jCi
, " WD ".nC1/=2 .. x /="/, x 2 N , 2 " " CV 1 .RnC1 / such that supp BRnC1 .0; 1/ and Z j .y/j2 dy D 1: RnC1
Then we have det.P .i m C / /
" jdet.P .i m C / /j C i " det.P .i m C / / C i "
" D jdet.P .i m C / /j C i " 1
" : i" jdet.P .i m C / /j C i "
det.P .i m C / / u" D
Section 3.2 Partial Differential Equations in H1 .jD j/
221
The right-hand side is bounded in every Hˇ .i m C e/, ˇ 2 ZnC1 , but for u" we find Z ju" j2˛ D Z D
RnC1
.x 2 C e/˛ ju" .x/j2 dx .x 2 C e/˛ j " .x/j2 dx jdet.P .i x C / /j2 C "2
RnC1
inf¹.x 2 C e/˛ j x 2 BRnC1 .x ; "/º sup¹jdet.P .i ." x C x / C / /j2 j x 2 BRnC1 .x ; "/º C "2
! 1 as " ! 0: This confirms that 2 C 1 .det P .@ C //: Thus, we have shown ¹ j N emptyº %1 .det P .@ C // D %1 .P .@ C // and ¹ j N non-emptyº C 1 .det P .@ C //: Consequently, we have 1 .det P .@ C // D 1 .P .@ C // ¹ j N non-emptyº C 1 .det P .@ C // and so the desired equality follows. Considering the case of H1 .jD j/, we note that %1 .P .@ C // %1; .P .@ C //: Now ı¹x º 2 H1 .jDj/ satisfies det.P .i m C / / ı¹x º D 0 and so u WD L ı¹x º D exp. m/ exp.i x m/ 2 H1 .jD j/ satisfies det.P .@ C / / u D 0: Thus, we find ¹ j N non-emptyº P 1; .det P .@ C // and the remaining equalities follow.
222
Chapter 3 Partial Differential Equations with Constant Coefficients
Finally, the definition of a convolution carries over to the present situation in a slightly modified form. Via Fourier–Laplace transform, the concept of convolution is reduced to multiplication. Since for F 2 H1 .jDj/ pointwise evaluation may rarely make sense, we define F .m/ W H1 .jDj/ H1 .jDj/ ! H1 .jDj/;
7! .m/ F with h .m/ F j for ;
i0;0 D hF j .m/
i0;0
2 H1 .jDj/. Note that
.m/
2 H1 .jDj/
follows from a Leibniz type formula, see Subsection 3.2.3. Definition 3.2.8. Let f 2 HnC1;sC1;tC1;1 .jD j/, n; s; t 2 N. Then the mapping b 1 .@ C / D .2/.nC1/=2 L f b.m i / L f WD .2/.nC1/=2 f i is continuous in HnC1;sC1;tC1;1 .jD j/ and will be called the convolution operator associated with f . If f W HnC1;sC1;tC1;1 .jD j/ HnC1;sC1;tC1;˛ .jD j/ ! HnC1;sC1;tC1;ˇ .jD j/ is continuous for some ˛; ˇ 2 ZnC1 , then the continuous extension to the space b. 1 .@ C // HnC1;sC1;tC1;˛ .jD j/ will again be denoted by f or .2/.nC1/=2 f i and will also be referred to as a convolution operator. In this case, for g 2 HnC1;tC1;rC1;˛ .jD j/, r 2 N, we call f g the convolution of f with g in H1 .jD j/. The proof that convolutions of this form are well-defined is postponed to Subsection 3.2.3. Indeed, we shall not develop the idea of convolution in full generality, since we are mainly interested in very particular types of convolution, e.g. convolution with fundamental solutions and convolution with elements having compact support. The details of the issues involved in considering such special cases will be presented in Subsection 3.2.3. For our present purposes it suffices to consider the invertible case and convolution with the fundamental solution. According to our extension of the solution theory, this particular convolution carries over to H1 .jD j/. Let P .@/ be invertible with respect to 2 RnC1 and let G be the corresponding fundamental solution. Then by definition P .@/1 D G
Section 3.2 Partial Differential Equations in H1 .jD j/
223
on H1 .jD j/ H1 .@ Ce/ but also in H1 .@ Ce/ H1 .jD j/. By continuous extension we obtain that P .@/1 D G as continuous mappings in H1 .jD j/.
3.2.2 Some Applications to Linear Partial Differential Equations of Mathematical Physics 3.2.2.1
Helmholtz Equation in R3
We recall that @2 . ˙ i /2 D @2 C .i /2 D @2 C .˙i /2 is invertible for every 2 R>0 , e.g. in H1 .@ C e/. The fundamental solution can easily be determined as the inverse Fourier transform of x 7! .2/3=2 .x 21.˙i /2 / . By applying residue calculus we find that the fundamental solution g˙i is given by the function g˙i .x/ D Indeed, with
exp.˙i jxj/ exp. jxj/ 4 jxj
for x ¤ 0:
2 CV 1 .R3 /,
hg i j i0;0
Z 1 1 b./ d D lim 2 3=2 "!0C 2R3 nB.0;"/ . ˙ i /2 .2/ Z 1 1 b./ d lim D .2/3=2 "!0C 2R3 nB.0;"/ 2 jj .jj C ˙ i / Z 1 1 b./ d : lim C 3=2 "!0C 2R3 nB.0;"/ 2 jj .jj i / .2/
Introducing polar coordinates, Z 1 1 b./ d 2 2 3=2 .2/ 2R3 nB.0;"/ . ˙ i / Z Z 1 r2 b.r0 / dr dS D 0 .2/3=2 r2Œ";1Œ 0 2S 2 r 2 . ˙ i /2 Z Z Z r2 1 D exp.ir0 x/ .x/ dx dr dS 0 .2/3 r2Œ";1Œ 0 2S 2 x2R3 r 2 . ˙ i /2 Z Z Z 1 r2 D exp.ir0 x/ dS 0 .x/ dr dx .2/3 x2R3 r2Œ";1Œ r 2 . ˙ i /2 0 2S 2 Z Z 1 r sin.r jxj/ dr .x/ dx: D 2 2 2 2 jxj x2R3 r2Œ";1Œ r . ˙ i /
224
Chapter 3 Partial Differential Equations with Constant Coefficients
Here we have made use of the rotational symmetry of the domain and integrand to turn x 2 R3 into 0
1 0 @ 0 A jxj in
R 0 2S 2
exp.i r0 x/ dS 0 . In standard polar coordinates we obtain 0
1 cos ' cos B C 0 D @ sin ' cos A sin and dS 0 D cos d' d and so for x 2 R3 n ¹0º Z
Z 0 2S 2
exp.i r0 x/ dS 0 D 2
exp.i rjxj sin / cos d Z
D 2 D 4
2Œ=2;C=2
exp.i rjxj s/ ds s2Œ1;C1
sin.rjxj/ : r jxj
R r Now, r2R>0 r 2 .˙i sin. r jxj/ dr can again be evaluated using residue calculus. /2 First we observe Z Z 1 r r sin. r jxj/ dr D sin. r jxj/ dr 2 2 2 2 r2R r . ˙ i /2 r2R>0 r . ˙ i / Z r 1 D exp.i r jxj/ dr 2 2i r2R r . ˙ i /2 Z 1 1 D exp.i r jxj/ dr 4i r2R r C . ˙ i / Z 1 1 exp.i r jxj/ dr: C 4i r2R r . ˙ i / Then, for 2 R n ¹0º, we find Z r2R
´ 1 2 i exp. i jxj/ exp. jxj/ for > 0; exp. i r jxj/ dr D r . C i / 0 for < 0
Section 3.2 Partial Differential Equations in H1 .jD j/
225
and Z r2R
´ 1 2 i exp. i jxj/ exp. jxj/ for < 0; exp. i r jxj/ dr D r C . C i / 0 for > 0:
Consequently, for > 0, we get Z r sin.r jxj/ dr D exp.˙ i jxj/ exp. jxj/ 2 . ˙ i /2 r 2 r2R>0 and so hg i j i0;0
Z Z r sin. r jxj/ 1 dr D lim 2 2 2 2 .2/ "!0C jxj x2R3 r2Œ";1Œ r . ˙ i / Z exp.˙ i jxj/ exp. jxj/ D .x/ dx; 3 4 jxj x2R
.x/ dx
from which the result follows. Since g˙i 2 L1;loc .R3 /, i.e. locally Lebesgue integrable (even in L2;loc .R3 / and even uniformly with respect to 2 R, 2 R>0 ), we have that g˙i is a regular distribution. Note that we must have g i D .L˙ ˝ 1/ g1;˙ . ; / where L˙ ˝ 1 is the Fourier–Laplace transform with respect to time only and g1;˙ is the forward or backward fundamental solution for @20 @2 , respectively. Obviously we also have that g˙i 2 H1 .b @ C e/ as long as 2 R>0 . Since g˙i ! g˙i 0
in H1 .jDj/
where g˙i 0 .x/ D
as ! 0C;
exp.˙ i jxj/ ; 4 jxj
we can use g˙i 0 as fundamental solutions for @2 2 , 2 R. Usually, we assumes 2 Œ0; 1Œ and call the wave number. For ¤ 0 the non-uniqueness is obvious (there are infinitely many solutions u 2 H1 .jDj/ of the equation .@2 2 /u D 0). But for D 0, in which case gL WD g0˙i 0 is given by gL .x/ D
1 4 jxj
for x 2 R3 n ¹0º;
we also have non-uniqueness. The regular distribution given by gL is a fundamental solution for the so-called Laplace operator @2 .
226 3.2.2.2
Chapter 3 Partial Differential Equations with Constant Coefficients Helmholtz Equation in R2
In R2 the situation is slightly more complicated. We consider again (for > 0) hg i j i0;0
Z
1 b./ d "!0C R!1 2B.0;R/nB.0;"/ . ˙ i /2 Z Z r 1 b.r 0 / ds dr lim lim D 0 2 "!0C R!1 r2Œ";R r 2 . ˙ i /2 0 2S 1 Z r 1 lim lim D .2/2 "!0C R!1 r2Œ";R r 2 . ˙ i /2 Z Z exp.ir0 x/ .x/ dx ds 0 dr
1 D 2
lim
lim
0 2S 1
x2R2
2
Z Z r 1 lim lim D 2 2 .2/ "!0C R!1 x2R2 r2Œ";R r . ˙ i /2 Z exp.i r0 x/ ds 0 dr .x/ dx: 0 2S 1
Now the usual rotational symmetry argument yields Z Z 1 1 exp.i r0 x/ ds 0 D exp.i r jxj cos '/ d' DW J0 . r jxj/: 2 0 2S 1 2 '2Œ0;2 Thus we have hg i j i0;0 1 D 2
lim
R!1
Z x2R2
Z r2Œ0;R
r2
r J0 . r jxj/ dr . ˙ i /2
.x/ dx:
Fortunately, for x ¤ 0 the integral Z Z r s J . r jxj/ dr D J .s/ ds 2 . ˙ i /2 0 2 .. ˙ i / jxj/2 0 r s R>0 R>0 has known values, see [16, formula 6.532, 4]: Z s i .1/ H .. i / jxj/ J0 .s/ ds D 2 2 2 0 R>0 s .. i / jxj/ and
Z R>0
s i .2/ J0 .s/ ds D H0 .. C i / jxj/ s 2 .. C i / jxj/2 2
Section 3.2 Partial Differential Equations in H1 .jD j/ .1/
227
.1/
where H0 ; H0 are the Hankel functions for index 0 of first and second kind re.1/ .2/ spectively. Noting that H0 .z/ D H0 .z /, z 2 C n ¹0º, we have i .2/ gCi .x/ D H0 .. C i / jxj/ and 4
gi .x/ D
i .1/ H .. i / jxj/: 4 0
For ¤ 0 we can let ! 0C and obtain a limit in H1 .jDj/ i .2/ gCi 0 .x/ D H0 . jxj/; 4
i .1/ gi 0 .x/ D C H0 . jxj/: 4
These are both fundamental solutions of the Helmholtz equation in R2 .@2 2 / g˙i 0 D ı;
2 R n ¹0º:
The case D 0 is, however, not achieved by taking the limit ! 0. Indeed, the .1/ .2/ asymptotic behaviors of the Hankel functions H0 ; H0 are given by .1/
H0 .z/ D
2i ln.z/ C C.1/ C o.1/;
.2/
H0 .z/ D
2i ln.z/ C C.2/ C o.1/
as z ! 0 for some constants C.i/ ; i D 1; 2, and so we have no convergence of g˙i 0 in H1 .jDj/ as ! 0 in R. We have, however, convergence of w WD g˙i 0 ˙
1 1 . ln.j j/ C˙;sgn./ / ! ln.j j/ 2 2
in H1 .jDj/ as ! 0C or ! 0. Here C˙;sgn./ denotes a constant depending on the sign of 2 R. Clearly we have @2 w D ı C 2 w ! ı
in H1 .jDj/ as ! 0
and therefore w0 WD
1 ln.j j/ 2
may serve as a fundamental solution for @2 in the case n D 2. Another way of looking at this fundamental solution is to consider it as a limit of the solution of @2 0 D ı ˝ ı ˝ ŒR1 ;R2 in R3 (compare ‘method of descent’ below) as R1 ; R2 ! C1. For this we study the limit of g0 ı ˝ ı ˝ ŒR1 ;R2
228
Chapter 3 Partial Differential Equations with Constant Coefficients
as R1 ; R2 ! 1. To discuss this limit in a suitable sense, we first evaluate for 2 CV 1 .R3 / (and with x D .x1 ; x2 ; x3 /) hg0 ı ˝ ı ˝ ŒR1 ;R2 j i0 D hı ˝ ı ˝ ŒR1 ;R2 j g0 i0 Z Z 1 D .y1 ; y2 ; x3 y3 / dy dx3 ŒR1 ;R2 R3 4j.y1 ; y2 ; y3 /j Z Z 1 .y1 ; y2 ; y3 / dy dx3 D ŒR1 ;R2 R3 4j.y1 ; y2 ; x3 y3 /j Z Z 1 dx3 .y1 ; y2 ; y3 / dy D 3 4j.y ; y 1 2 ; x3 y3 /j R ŒR1 ;R2 p Z x3 C R1 C j.x1 ; x2 /j2 C .x3 C R1 /2 1 D ln .x/ dx: p 4 R3 x3 R2 C j.x1 ; x2 /j2 C .x3 R2 /2 Now
p j.x1 ; x2 /j2 C .x3 C R1 /2 ln p x3 R2 C j.x1 ; x2 /j2 C .x3 R2 /2 p x3 R2 C j.x1 ; x2 /j2 C .x3 R2 /2 D ln p x3 C R1 C j.x1 ; x2 /j2 C .x3 C R1 /2 j.x1 ; x2 /j2 D ln p p .x3 C R1 C j.x1 ; x2 /j2 C .x3 C R1 /2 /.R2 x3 C j.x1 ; x2 /j2 C .x3 R2 /2 /
x 3 C R1 C
D 2 ln.j.x1 ; x2 /j/ R 1 R2 C ln p p .x3 C R1 C j.x1 ; x2 /j2 C .x3 C R1 /2 /.R2 x3 C j.x1 ; x2 /j2 C .x3 R2 /2 / ln.R1 R2 /;
for sufficiently large R1 ; R2 2 R>0 . Noting that the integral only extends over the compact set supp and R 1 R2 p p 2 .x3 C R1 C j.x1 ; x2 /j C .x3 C R1 /2 /.R2 x3 C j.x1 ; x2 /j2 C .x3 R2 /2 / !1 as R1 ; R2 ! 1, we get 1 hln.R1 R2 / j i0 lim hg0 ı ˝ ı ˝ ŒR1 ;R2 j i0 C R1 ; R2 !1 4 1 D hln.j.m1 ; m2 /j/ j i0 2
Section 3.2 Partial Differential Equations in H1 .jD j/
229
for all 2 CV 1 .R3 /. Thus, we have established 21 ln.j.m1 ; m2 /j as a somewhat peculiar limit case of a three-dimensional potential. 3.2.2.3
Cauchy–Riemann Operator
Obviously, in the non-invertible case, fundamental solutions are more difficult to find. This difficulty is akin of the problems one encounters when trying to solve Au D f when 0 2 C .A/. In this respect the Cauchy–Riemann operator @C WD @1 C i @2 in R2 is a particularly notable example since C .@C / D C. Fortunately, noting that @C D @1 C i @2 D .@1 i @2 /, we have @C @C D @C @C D @2 and so the previous example yields that gCR WD @C w0 D 21 @C ln.j j/ is a fundamental solution for the Cauchy–Riemann operator. We find gCR .x1 ; x2 / D
1 2 .x1 C i x2 /
as a regular distribution which, in the usual complex notation (identifying R2 and C via the isometry .x1 ; x2 / 7! z WD x1 C i x2 ), can be re-written as gCR .z/ D 3.2.2.4
1 : 2z
Wave Equation in R2 (Method of Descent)
One may suspect that the solution of the wave equation in R2 may be found as the solution of the three-dimensional wave equation with the right-hand side independent of one of the variables. Noting that the constant function 1 … H1 .@ C 1/ but 1 2 H1 .jDj/, we see that in order to pursue this idea we require the extension of the solution theory to H1 .jD j/; D .%; 0; 0; 0/. Consider the uniquely determined solution u 2 H1 .jD0;% j/ ˝ H1 .jDj/; % 2 R>0 , of the wave equation @2 / u D ı ˝ ı ˝ ı ˝ 1 .@20 b in R1C3 . We expect u to be independent of the last variable and then u interpreted in R1C2 would solve the equation .@20 b @2 / u D ı ˝ ı ˝ ı and so u would be the fundamental solution of the wave equation in R1C2 . This approach to fundamental solutions of lower dimension is known as the method of descent. We know that u D g1;C ı ˝ ı ˝ ı ˝ 1
230
Chapter 3 Partial Differential Equations with Constant Coefficients
and calculate41 huj i;0 D huj exp.2% m0 / i0;0 D hg1;C ı ˝ ı ˝ ı ˝ 1j exp.2% m0 / i0;0 D hı ˝ ı ˝ ı ˝ 1 j .1 g1;C / exp.2% m0 / i0;0
D hı ˝ ı ˝ ı ˝ 1 j i0;0 Z D .0; 0; 0; s/ ds; R
where we have used as an ad-hoc abbreviation .t; x/ WD hg1;C j .t;x/ exp.2% m0 / i0;0 ; and .t;x/ is the translation operation given by ..t;x/ '/.s; y/ D '.s C t; y C x/ for complex-valued functions ' defined on R1C3 . We find Z .0; 0; 0; s/ ds R Z D hg1;C j .0;0;0;s/ exp.2% m0 / i0;0 ds R Z D hg1;C j .0;0;0;s/ i;0 ds R Z Z 1 exp.2 jxj %/ .jxj; x1 ; x2 ; x3 C s/ dx ds D 3 4jxj R x2R Z Z Z 1 D t exp.2 t %/
.t; t x1 ; t x2 ; s/ dSx ds dt: 4 t2Œ0;1Œ R x2S 2 Noting that Z
Z R
D2 41
x2S 2
.t; t x1 ; t x2 ; s/ dSx ds
Z
Z
R
.x1 ;x2 /2B.0;1/
.t; t x1 ; t x2 ; s/ d.x1 ; x2 / ds; q 1 x12 x22
Recall that A denotes the rescaling operator given by p .A '/.x/ D jdet.A/j '.Ax/
for a regular linear mapping A W RnC1 ! RnC1 . Here n D 2 and A is just multiplication by 1.
Section 3.2 Partial Differential Equations in H1 .jD j/
231
we get 2huj i;0 Z Z D t exp.2 t %/
R
t2Œ0;1Œ
Z D
Z .x1 ;x2 /2B.0;1/
R
Z
R
t exp.2 t %/ t2Œ0;1Œ
.x1 ;x2 /2B.0;1/
Z
Z
R
.t; t x1 ; t x2 ; s/ d.x1 ; x2 / ds dt q 1 x12 x22
.t; t x1 ; t x2 ; s/ ds d.x1 ; x2 / dt q 1 x12 x22
.t; x1 ; x2 ; s/ ds d.x1 ; x2 / dt q t2Œ0;1Œ .x1 ;x2 /2B.0;t/ t 2 x12 x22 Z Z Z exp.2 t %/
.t; x1 ; x2 ; s/ ds d.x1 ; x2 / dt D q 2 2 2 t2Œ0;1Œ .x1 ;x2 /2B.0;t/ R t x1 x2 Z ¹.t;x/2R1C2 jt >jxjº .t; x/ Z D
.t; x; s/ ds dx exp.2 t %/ dt: p t 2 jxj2 .t;x/2R1C2 R D
exp.2 t %/
R
Consequently, we have found the solution u given by the function q R>0 .t x12 C x22 / u.t; x1 ; x2 ; x3 / D for t ¤ 0; x1 ; x2 ; x3 2 R: q 2 t 2 x12 x22 In particular, we see that u is independent of the last variable in the sense that uDw˝1 for some w 2 H1 .jD0;% j/ ˝ H1 .j.D1 ; D2 /j/. Indeed, let .@20 @21 @22 /w D ı ˝ ı ˝ ı in R1C2 . Then by letting w j i;0 he w j ˝ !i;0 WD he
Z !.s/ ds R
we have a continuous embedding w 7! w e of H1 .jD0;% j/ ˝ H1 .j.D1 ; D2 /j/ into H1 .jD0;% j/ ˝ H1 .j.D1 ; D2 ; D3 /j/. Due to uniqueness of solution we must have w.t; x1 ; x2 / D
¹.t;x/2R1C2 jt >jxjº .t; x1 ; x2 / q 2 t 2 x12 x22
for t ¤ 0; x1 ; x2 2 R:
232
Chapter 3 Partial Differential Equations with Constant Coefficients
Since w coincides with the fundamental solution of the wave equation in R1C2 , we also know its behavior at t D 0. Continuing with the method of descent for one more step yields that the fundamental solution g of the wave equation in R1C1 is given by g.t; x/ D
1 .t jxj/; 2 R>0
t; x 2 R:
The details for this step are left as an exercise. 3.2.2.5
Plane Waves
The construction in the previous subsection points in a particular direction of application. The extension to the tempered, exponentially weighted Sobolev lattice allows for particular elements of interest not previously included. Another instance of this type of generalization is associated with the idea of a plane wave. Let us consider again the wave equation in R1Cn with initial data f , g: .@20 b @2 / u D ı ˝ g C @0 ı ˝ f: If the initial data satisfy
x b @ f D 0 D x b @g
for all x 2 RnC1 ; x ? p, for some fixed direction p 2 RnC1 , jpj D 1, then, due to uniqueness, we also have x b @u D 0 for all x 2 RnC1 ; x ? p. A solution with this property is called a plane wave. In this case, the wave equation reduces to .@20 .p b @/2 / u D .@0 C p b @/ .@0 p b @/ u D ı ˝ g C @0 ı ˝ f; and so decomposes into two transport equations. Indeed, we have .@20 .p b @/2 /1 D .@0 C p b @/1 .@0 p b @/1 1 D ..@0 C p b @/1 C .@0 p b @/1 / @1 0 2 and so 1 u D ..@0 C p b @/1 C .@0 p b @/1 / @1 0 .ı ˝ g C @0 ı ˝ f / 2 1 D ..@0 C p b @/1 C .@0 p b @/1 / .R0 ˝ g C ı ˝ f /: 2 Introducing the solution operator for the initial value problem for the transport equation (in direction p) W˙ f WD .@0 ˙ p b @/1 ı ˝ f
Section 3.2 Partial Differential Equations in H1 .jD j/
233
for f 2 H1 .j.D1 ; : : : ; Dn /j/ satisfying x b @u D 0 for all x 2 RnC1 ; x ? p, we obtain 1 1 1 u D .WC f C @1 0 WC g/ C .W f C @0 W g/; 2 2 showing that u is composed of a part propagating in direction p and another part propagating in the opposite direction p. If for example f D ı¹pº? and g D 0, then uD
1 .m0 / .ı¹.s;y/ j sypD0º C ı¹.s;y/ j sCypD0º / 2 R>0
or u.t / D
1 .t / .ı¹y j typD0º C ı¹y j tCypD0º /: 2 R>0
If we wanted the solution just to be R>0 .m0 / ı¹.s;y/ j sCypD0º instead, we would have to choose f D ı ? and g D p b @ ı ?. ¹pº
3.2.2.6
¹pº
Linearized Navier–Stokes Equations (Incompressible Case)
Since we are now quite familiar with the construction of fundamental solutions, we may be brief in our discussion of the variants of the linearized equations of fluid dynamics. I Stokes equation with pressure (incompressible case). The .4 4/-partial differential expression of the Stokes equation including pressure (in R1C3 ) is given by (see (3.1.87)) ! b b b C @ @ @ @ 0 S1 .@0 ; b : @/ WD b @> 0 An elementary calculation shows that the characteristic polynomial is . 2 @0 b @2 / .@0 b @2 /2 and the minimal polynomial is . 2 @0 b @2 / .@0 b @2 /: @/ is non-invertible (and so non-evolutionary), since its determiWe see that S1 .@0 ; b nant is @2 /2 : b @2 .@0 b
234
Chapter 3 Partial Differential Equations with Constant Coefficients
Nevertheless, a fundamental solution with time-support R0 can be found in the usual way expressed in terms of a fundamental solution for the Laplace operator and the fundamental solution of the heat equation. We shall not give the details in this case. The problem can be reduced to an evolutionary problem by eliminating the second component in the same way as discussed earlier, see Subsection 3.1.12.6. Note, however, that even for well-behaved right-hand sides we cannot expect the solution to be in H1 .@0;% C 1; b @ C e/ but only in H1 .jD0;% j/ ˝ H1 .j.D1 ; D2 ; D3 /j/. The non-uniqueness issue of the solution p of the equation b @2 p D g in H1 .jD0;% j/ ˝ H1 .j.D1 ; D2 ; D3 /j/ must also be addressed (see our later discussion of harmonic polynomials). I Linearized Navier–Stokes equations (incompressible case). Here we have an additional drift term: ! b b b b C @ @ s @ @ @ 0 N1;s .@0 ; b @/ WD : b @> 0 @, we find Comparing with the previous case, simply by replacing @0 with @0 s b . 2 .@0 s b @/ b @2 / .@0 s b @ b @2 /2 as characteristic polynomial and . 2 .@0 s b @/ b @2 / .@0 s b @ b @2 / @/ remains non-evolutionary since its as the minimal polynomial. Of course, N1;s .@0 ; b determinant is b @2 .@0 s b @ b @2 /2 : A fundamental solution can be found in the usual way expressed in terms of a fundamental solution for the Laplace operator and the fundamental solution of the heat equation with drift term. We omit the details. I Static Stokes equations. The static Stokes equations result from the dynamic Stokes equations, if we assume that the solution does not vary with time. The partial differential expression associated with them is given by ! b b b @ @ @ S1 .0; b @/ D b 0 @> which will be considered in R3 . The determinant is now .b @2 /3 and the characteristic polynomial is given by 7! . 2 C b @2 / . b @2 /2 :
Section 3.2 Partial Differential Equations in H1 .jD j/
235
The minimal polynomial is @2 / . b @2 / D 3 . 2 /b @2 .b @2 /2 7! . 2 C b D . 2 . 1/b @2 / .b @2 /2 : Noting that our earlier reasoning based on the Cayley–Hamilton theorem still applies, we see that a fundamental solution of S1 .0; @/is given by GsS D .S1 .0; b @/2 S1 .0; b @/ b @2 C b @2 / g 0
.2/
.2/
where g0 WD g0 g0 with g0 denoting a fundamental solution of Laplace’s equa.2/ tion. This can be re-written (using @2 g0 D g0 / as .2/ GsS D S1 .0; b @/2 g0 C S1 .0; b @/ g0 g0 :
3.2.2.7
Electro- and Magnetostatics
Maxwell’s equations in the expanded form involve the partial differential expression MM.@0 ; b @/, see (3.1.73). The corresponding static case results if we assume the solution is time-independent, which leads to 0 1 0 b @> 0 0 C B b @ 0 C B @ 0 b MM.0; b @/ D B C @ 0 b @ 0 b @A 0 0 b @> 0 in R3 . In this case we get
MM.0; b @/2 D b @2 188
and so, analogously to the Cauchy–Riemann operator, we have a fundamental solution @/ g0 188 G0;MM WD MM.0; b as a fundamental solution, where g0 is a fundamental solution of the Laplace operator b @2 . Similarly, in R2 we find 0 1 0 @1 @2 0 B @1 0 0 @2 C C MM.0; b @/ D B @ @2 0 0 @1 A 0 @2 @1 0 and again
@2 144 : MM.0; b @/2 D b
236
Chapter 3 Partial Differential Equations with Constant Coefficients
Thus, we obtain
e 0;MM WD MM.0; b G @/ g0 144
as the fundamental solution in this case42 . 3.2.2.8
Force-free Magnetic Fields
Applying the (forward) Fourier–Laplace transform to Maxwell’s equations in the form discussed in Remark 3.1.66 yields the expression .i m0 C 1 % C i b @/
(3.2.99)
where % 2 R>0 . With 2 R replacing the operator m0 , a pointwise interpretation leads to the partial differential expression . C i % b @/ D curl C C i %;
2 R:
The case % ! 0C leads in turn to considering an equation of the form b @ H H D F: If F D 0, a solution H can be regarded as a magnetic field with vanishing Lorentz force, since then b @ H and H must be parallel. In this case, H would be called force-free in the region where F D 0. The determinant det.b @ / D .b @2 2 / reveals b @ as non-invertible and so non-evolutionary. Moreover, the characteristic polynomial is found as 7! . C / .b @2 . C /2 / D 3 3 2 C .b @2 3 2 / C .b @2 2 /: Correspondingly, with g˙i 0 denoting our earlier fundamental solution for the differential expression .b @2 2 /, we find for ¤ 0 @ /2 3 .b @ / . 2 b @2 / C 2 2 / 1 g˙i 0 G0;F F WD ..b D .b @ /2 1 g˙i 0 3 .b @ / g˙i 0 2 g˙i 0 1 ı D 1 @ @ g˙i 0 @ g˙i 0 1 ı; as a fundamental solution for force-free (FF) fields. For 2 R n ¹0º, magnetic fields in H1 .j.D1 ; D2 ; D3 /j/ which are force-free in all of R3 can be found from the characteristic polynomial. We have .b @ /..b @ /2 3 .b @ / C .3 2 C b @2 // D .b @2 2 /: 42 Note that, although, the fundamental solution g of b @2 does not decay in R2 , its derivatives do de0 cay. Thus, unique representations (see later) of decaying solutions can be given in terms of convolutions with e G 0;MM .
Section 3.2 Partial Differential Equations in H1 .jD j/
237
Therefore every (scalar) solution u of the homogeneous Helmholtz equation .b @2 2 / u D 0 yields a force-free solution Ha via Ha WD ..b @ /2 3 .b @ / C .3 2 C b @2 // u a for every a 2 C 3 . Such u can be found as the inverse Fourier transform of an element in H1 .j.D1 ; D2 ; D3 /j/ with support on @B.0; jj/. Again, we could consider the expanded system ! b @ b @ 1.44/ b @> 0 instead. Here we find ! ! b @ b @ 1.44/ b @> 0
! ! b @ b @ C 1.44/ D b @> 0
b @ b @ b @> 0
!2 2 1.44/
and b @ b @ > b @ 0
!2 D D
! b @ b @ b @b @> 0 @ 0 b @>b ! b @2 1.33/ 0 D b @2 1.44/ : 0 b @2
Thus, a fundamental solution is found for all 2 C from a fundamental solution g of the Helmholtz equation: ! ! b @ b @ C 1.44/ g 1.44/ ; G WD b @> 0 (recall that g is uniquely determined for 2 C n R). In order for the equation ! ! b @ b @ H F 1.44/ D > b ' f @ 0 to reduce to the above system we must again require the compatibility condition b @> F C f D 0: Then, for 2 C n R we find .b @2 2 / ' D b @> F C f D 0
(3.2.100)
238
Chapter 3 Partial Differential Equations with Constant Coefficients
and so ' D 0. For 2 R, we cannot exclude the possibility of a non-trivial component '. However, under assumption (3.2.100), a solution given in the form F G f is always of the desired form: F G D g f
! ! b F @ b @ C 1.44/ > b f @ 0 b @ F b @f C F D g : 0
From
! ! b @ b @ 1.44/ b @> 0
! ! b @ b @ @2 2 C 1.44/ D b b @> 0
it is also clear how to obtain force-free magnetic fields in R3 in this framework. Let U D 0 and b @> U C u D 0: .b @2 2 / u Then the first component of ! ! b b U @ b @ @ U b @u C U D C 1.44/ b u 0 @> 0 is a force-free magnetic field. 3.2.2.9
Beltrami Field Expansions
Force-free magnetic fields in R3 are also known as Beltrami fields. We shall now consider an associated generalized eigensolution expansion. According to our general functional-analytical considerations, we need to find a suitable diagonalization of i m. Note that such an investigation takes places in H3;3;3;1 .@ C e/. It is only later that we shall refer to the extended space H3;3;3;1 .jDj/. With43 1 0 m1 m3 m1 m3 C qi m2 jmj q i m2 jmj pm1 p p 2 jmj B 2 jmj m21 Cm22 2 jmj m21 Cm22 C C B C B m2 m3 C i m1 jmj m2 m3 q qi m1 jmj C pm2 p p V .m/ WD B 2 2 2 2 C 2 jmj B 2 jmj m1 Cm2 2 jmj m1 Cm2 B C q q @ A 2 2 m Cm m2 Cm2 2 p1 2 jmj
pm3 2 jmj
2 p1 2 jmj
43 The matrix V .x/ is uniquely determined by the diagonal matrix only almost everywhere and up to right-multiplication by a diagonal matrix with diagonal entries of absolute value 1 almost everywhere.
Section 3.2 Partial Differential Equations in H1 .jD j/ 1 Cjmj 0 0 i m D V .m/ @ 0 0 0 A V .m/ : 0 0 jmj 0
we have
Thus, we obtain the following generalized eigensolution expansion 1 0 100 FC1 WD F V .m/ @ 0 0 0 A; 000 0 1 000 F0 WD F V .m/ @ 0 1 0 A; 000 1 0 000 WD F V .m/ @ 0 0 0 A; F1 001 and so F h WD F V .m/ h D FC1 hC1 C F0 h0 C F1 h1 ;
with
X
h WD
hj ;
hk WD Fk f;
j D1;0;C1
E.g. V .x/ could be replaced by 0 B B V .x/ B @
The structure of V .x/ is in both cases 0
x1 Ci x2 q x12 Cx22
0
0
0 0
1 0
0
v1 .x/ @ v2 .x/ v3 .x/
x1 =jxj x2 =jxj x3 =jxj
x1 i x2 q x12 Cx22
1 C C C: A
1 v1 .x/ v2 .x/ A; v3 .x/
where in the latter case 1 v1 .x/ D p .1 C .x1 =jxj C i x2 =jxj/ x1 =jxj .1 C x3 =jxj/1 /; 2 1 v2 .x/ D p .i C .x1 =jxj C i x2 =jxj/ x2 =jxj .1 C x3 =jxj/1 /; 2 1 v3 .x/ D p .x1 =jxj C i x2 =jxj/ 2 for x … ¹0º ¹0º R0 , compare [33].
239
240
Chapter 3 Partial Differential Equations with Constant Coefficients
where Fk denotes the adjoint of Fk , k D 1; 0; C1. Applying the generalized eigensolution transform, e.g. to Maxwell’s operator .@0 C i b @/, yields44 the unitary equivalence 1 0 @0 C i jmj 0 0 A F D @0 C i b 0 0 @0 F @ @ 0 0 @0 i jmj X Fj .@0 C i j jmj/ Fj : D j D1;0;C1
The solution of
@0 U C i b @U DF
is therefore given by X U D
Fj .@0 C i j jmj/1 Fj F
j D1;0;C1
D
X
Fj R0 .m0 / exp.i j m0 jmj/ 0 Fj F
j D1;0;C1
D .R0 .m0 / exp.i m0 j@j// 0 P1 F C R0 .m0 / 0 P0 F C .R0 .m0 / exp.i m0 j@j// 0 PC1 F; where Pj WD Fj Fj are orthogonal projectors for j D 1; 0; C1, and 0 denotes convolution with respect to time. As pointed out before, this derivation makes sense in H3;3;3;1 .@ C e/. Clearly, for example, a point source of the form F ı¹p0 º a0 with a0 2 C 3 n ¹0º does not make sense if p0 2 ¹0º R ¹0º. For other p0 , however, we can define F ı¹p0 º a0 WD lim F j" .m p0 / a0 "!0C
and we find F ı¹p0 º a0 D exp.i m p0 / V .p0 / a0 : 44 One might prefer to consider the unitary mapping F F instead, yielding a unitary equivalence between 1 0 @0 C jb @j 0 0 C B 0 0 @0 A @ 0 0 @0 jb @j
and @0 C ib @.
Section 3.2 Partial Differential Equations in H1 .jD j/ Moreover, we have
241
1 jmj 0 0 b @ F ı¹p0 º a0 D F @ 0 0 0 A ı¹p0 º a0 0 0 jmj 1 0 jp0 j 0 0 D F ı¹p0 º @ 0 0 0 A a0 0 0 jp0 j X D j jp0 j Fj ı¹p0 º a0 ; 0
j D1;0;C1
that is b @ Fj ı¹p0 º a0 D j jp0 j Fj ı¹p0 º a0 ;
j D 1; 0; C1:
The latter justifies the term ‘generalized eigensolution’, since every non-trivial Fj ı¹p0 º a0 2 H3;3;3;1 .jDj/ is indeed an eigenfunction corresponding to the eigenvalue j jp0 j. The question of generalized eigenfunction expansions could have been attacked with the extended Maxwell operator ! ib @ b @ @0 C : b @> 0 In this case we would need to diagonalize the block matrix i m m : m 0 The discussion proceeds in the framework of H3;4;4;1 .@Ce/. We first find a unitary matrix 0 1 m1 m3 m2 m3 i m1 jmj qi m2 jmj q pm1 pm2 p i i p 2 jmj B 2 jmj 2 jmj m21 Cm22 2 jmj m21 Cm22 C B C B m2 m2 m3 C i m1 jmj m1 m3 C i m2 jmj C m1 q q p C Bp p i ip 2 2 2 2 C B 2 jmj 2 jmj 2 jmj m Cm 2 jmj m Cm 1 2 1 2 C V .m/ WD B q C B m21 Cm22 C B m3 1 p p C Bp 0 B 2 jmj C 2 jmj 2 q @ A m21 Cm22 m 1 3 p p p 0 2
such that
0 jmj B 0 i m m D V .m/ B @ 0 m 0 0
2 jmj
2 jmj
1 0 0 0 jmj 0 0 C C V .m/ : 0 jmj 0 A 0 0 jmj
242
Chapter 3 Partial Differential Equations with Constant Coefficients
Since we have multiple diagonal elements, we are led to combine the resulting generalized eigensolution expansion in the following way: 0 1 1000 B0 1 0 0C C FC WD F V .m/ B @ 0 0 0 0 A; 0000 1 0 0000 B0 0 0 0C C F WD F V .m/ B @ 0 0 1 0 A; 0001 and so F h WD F V .m/ h D FC hC C F h
where h WD hC C h ;
h˙ WD F˙ f:
Applying generalized eigensolution transform to the extended Maxwell operator this @ b @ @0 C ib yields the unitary equivalence b @> 0 0 1 0 0 0 @0 C i jmj B C 0 @0 C i jmj 0 0 CF F B A @ 0 0 @0 i jmj 0 0 0 0 @0 i jmj ! ib @ b @ D @0 C > b @ 0 .@0 C i jmj/ FC C F .@0 i jmj/ FC : D FC
The solution of @0 U C
! ib @ b @ U DF b @> 0
is therefore given by U D FC .@0 C i jmj/1 FC F C CF .@0 i jmj/1 F F D FC .R0 .m0 / exp.i m0 jmj// 0 FC F
C F .R0 .m0 / exp.i m0 jmj// 0 F F D .R0 .m0 / exp.i m0 j@j// 0 P F C .R0 .m0 / exp.i m0 j@j// 0 PC F;
Section 3.2 Partial Differential Equations in H1 .jD j/
243
where P˙ WD F˙ F˙
are orthogonal projections. We now find F ı¹p0 º a0 D exp.i m p0 / V .p0 / a0 for a0 2 C 4 and p0 2 R3 n ¹0º. Moreover, we have ! ib @ b @ F˙ ı¹p0 º a0 D ˙ jp0 j F˙ ı¹p0 º a0 2 H3;4;4;1 .jDj/ b @> 0 showing that F˙ ı¹p0 º a0 is a generalized eigensolutions. Here we conclude our initial discussion of examples illustrating the utility of our extension of the solution theory to H1 .jD j/ for suitable choices of 2 RnC1 . The convolution concept introduced above has enabled us to represent solutions in terms of convolutions of a fundamental solution with suitable right-hand sides. Whereas in the invertible case we have seen that the unique solution can be given in terms of a convolution for arbitrary right-hand sides in the appropriate Sobolev lattice, the situation is – not surprisingly – more complicated in the non-invertible case. Although there is still a fundamental solution in H1 .jD j/, the right-hand sides cannot be chosen arbitrarily. This is reflected in difficulties to ensure the existence of convolutions in H1 .jDj/.
3.2.3 Convolutions with Particular Elements of H1 .jD j/; 2 RnC1 The existence of convolutions is often a difficult issue and we shall not attempt to consider this issue in full generality. Generally speaking we can say that, whenever the product of the Fourier–Laplace transforms is well-defined, then so is the convolution of the original distributions. For our purposes it is sufficient to describe appropriate special cases where the convolution is well-defined. We want to do this in slightly more generality than in the above examples, where only convolution with fundamental solutions in the invertible case was considered. A first special case will be the consideration of convolutions with g 2 H1 .jD j/. In this case we have formally hg
j i;0 D .2/.nC1/=2 hb g .m i / b. i /jb
. i /i0;0
. i/i0;0 : D .2/.nC1/=2 hb. i/jb g .m i / b
It suffices to show that b g .m i / jH˛ .jD0 j/ W H˛ .jD0 j/ ! H˛ .jD0 j/ for all sufficiently large ˛ 2 N nC1 , in order to confirm the existence of g in g .m i / jH˛ .jD0 j/ H1 .jD j/ for all 2 H1 .jD j/. The stated continuity of b
244
Chapter 3 Partial Differential Equations with Constant Coefficients
follows from the Leibniz rule using the fact that for g 2 H1 .jD j/ all derivatives of b g . i / 2 H1 .jDj/ are bounded. This follows from the Sobolev type estimate derived below. As by-products of this observation we have that the convolution defined in Definition 3.2.8 is indeed well-defined and that H1 .jDj/ is a multiplicative algebra, i.e.
WD .m/
D
.m/ 2 H1 .jDj/
for ; 2 H1 .jDj/. It is not always feasible to impose such strong assumptions on one of the convolution factors. For many purposes it will, however, be sufficiently general to consider convolutions with elements in H1 .jD j/ having compact support. The following theorem describes this particular class of convolutions. Theorem 3.2.9. Let f 2 H1 .jD j/ have compact support. Then g f exists in H1 .jD j/ for any g 2 H1 .jD j/. Proof. For simplicity let D 0. Then, since f 2 H1 .jDj/, we know that f is well-defined on H1 .jDj/. Moreover, since f has compact support, we may take a real-valued cut-off function 0 2 CV 1 .RnC1 / with 0 D 1 on a ball around the origin containing the support of f to obtain f D 0 .m/ f: From this we see that f 2 H1 .@ C e/: Indeed, let f 2 H2˛ .jDj/ for some ˛ 2 N nC1 . Then, from Lemma 3.2.1 and the Leibniz rule, we obtain hf j i0;0 D h 0 .m/f j i0;0 D hf j 0 .m/ i0;0 C j.D D/˛ . 0 .m/ /j0;0 X C jm @ˇ . 0 .m/ /j0;0 Cˇ 2˛
C
X
j@ˇ j0;0
Cˇ 2˛
C j.@ C e/2˛ j0;0 for some generic constant C 2 R>0 and all 2 CV 1 .RnC1 /. Thus, we see that f 2 H2˛ .@ C e/. As before, reversing the argument with the isometric re-scaling operator 1 , we find h
0
gj i0;0 D hg j .1
0/
i0;0
Section 3.2 Partial Differential Equations in H1 .jD j/
245
for all ; 0 2 CV 1 .RnC1 /. Clearly, .1 0 / 2 CV 1 .RnC1 / H1 .jDj/. We shall see that the convolution .1 0 / can be extended simply by continuous extension to 0 D f 2 H1 .jDj/ with supp. 0 / compact and indeed we still have .1 0 / 2 H1 .jDj/. We may assume that the support of 0 is contained in the set where the cut-off function 0 is equal to 1. The Fourier transform b of such a WD 1 0 D 1 0 .m/ 0 2 H1 .@ C e/ is then differentiable to any order. Moreover, since @ b 2 H1 .i m C e/ for every 2 N nC1 , we have that for every 2 N nC1 there exists ˛ 2 N nC1 such that Z j .i x C e/˛ @ b.x/j2 dx < 1: RnC1
To continue our proof and for later purposes we need a simple .n C 1/-dimensional version of Sobolev’s lemma. Lemma 3.2.10. The mapping CV 1 .RnC1 / H2e .jDj/ ! ¹ 2 C0 .RnC1 /j j j1;1=2 < 1º; u 7! .t 7! u.t // has a continuous extension to all of H2e .jDj/ (“trace operator”) with respect to the norm of H2e .jDj/ and the image norm ˇ ² ³ j .t / .s/j ˇˇ nC1 nC1 º C sup ^t ¤s : j j1;1=2 WD sup¹j .t /j j t 2 R t; s 2 R jt sj1=2 ˇ In particular, if f 2 H2e .jDj/ and . k /k is a sequence in CV 1 .RnC1 / converging to f in H2e .jDj/, then limk!1 k exists in C0 .RnC1 /, is independent of the particular approximating sequence and equals f . Furthermore, the mapping is injective. Proof. By the fundamental theorem of calculus we obtain
.x0 ; : : : ; xn / Z Z D D
@0 @n .t0 ; : : : ; tn / dtn dt0 1;x0 1;xn Z x0 Z xn j.it0 C 1/ .itn C 1/j@0 @n .t0 ; : : : ; tn / 1
j.it0 C 1/ .itn C 1/j
1
dtn dt0
and so, applying the Cauchy–Schwarz inequality, Z nC1 2 2 j .x0 ; : : : ; xn /j j.it C 1/j dt j.i m C 1/e @e j20;0
R nC1
j.i m C 1/e @e j20;0 :
246
Chapter 3 Partial Differential Equations with Constant Coefficients
Similarly, we get
.x0 ; : : : ; xn / .y0 ; : : : ; yn / n X D . .x0 ; : : : ; xk ; ykC1 ; : : : ; yn / .x0 ; : : : ; xk1 ; yk ; : : : ; yn // D
kD0 n X
Z
Z 1;x0
kD0
Z
1;yn
Z
xk
1;xk1 yk
Z 1;ykC1
@0 @n .t0 ; : : : ; tn /dtn dt0
and consequently j .x0 ; : : : ; xn / .y0 ; : : : ; yn /j2 n X n jarctan.xk / arctan.yk /j j.i m C 1/e @e j20;0 n
p
kD0 n X
jxk yk j j.i m C 1/e @e j20;0
kD0
n C 1 n jx yj j.i m C 1/e @e j20;0 ;
showing the required continuity estimate. Note that the term j.i m C 1/e @e j0;0 can be further estimated by jjDj2e j20;0 D j.D D/e j20;0 . The desired injectivity follows since convergence in H2e .jDj/ implies convergence in L2 .RnC1 /. Indeed, if f D 0, then by construction we have a sequence . k /k converging to f in H2e .jDj/, thus in L2 .RnC1 /, and to zero point-wise. Therefore, f must vanish. Returning to the proof of the last theorem, we note that, according to this lemma, we also have that for every 2 N nC1 there is an ˛ 2 N nC1 such that the function x 7! .i x C e/˛ @ b.x/ is bounded. Therefore, for any ˇ 2 N nC1 , we can find a sufficiently large ˛ 2 N nC1 , ˛ e, such that jjDjˇ b.m/ b
j20;0 X C
jm @ b.m/ b
/j20;0
Cˇ; ;2N nC1 ˇ
ˇ2 X ˇ ˇ
b bˇ ˇ C .@ /.m/ @ ˇ ˇm 0;0 Cˇ; ;2N nC1 ˇ
CD; ;2N nC1 ˇ2 X X ˇ ˇ C˛ ˛ b bˇ ˇ C .i m C e/ .@ /.m/@ ˇ ˇ.i m C e/ X
Cˇ; ;2N nC1
C 0
X
Cˇ; ;2N nC1
j20;0 C 0 jjDj˛Cˇ b
CD; ;2N nC1
j20;0 jm .i m C e/˛ @b
0;0
Section 3.2 Partial Differential Equations in H1 .jD j/
247
for some generic constant C 2 R>0 and 0 D sup¹j.i x C e/˛ @ b.x/j2 j x 2 RnC1 ; 2 N nC1 ; ˇº: We obtain an estimate of the form 0 C j.i m C e/e˛ @ bj20;0 D C j.@ C e/e˛ m 0 .m/ j20;0 C j.@ C e/e˛ j20;0 for some generic constant C 2 R>0 (independent of and ). Thus, we obtain by applying inverse Fourier transformation for every ˇ 2 N nC1 a sufficiently large 2 N nC1 such that jjDjˇ ..1
0/
/j0;0 C j.@ C e/ j0;0 jjDj Cˇ j0;0
(3.2.101)
for some generic constant C 2 R>0 (independent of and ). Since ˇ 2 N nC1 was arbitrary this shows that .1 0 / 2 H1 .jDj/. Estimate (3.2.101) finally also shows that convolution .1 0 / can be extended by continuity to .1 f / . In particular, this confirms that f g 2 H1 .jDj/ is well-defined. The result carries over to the general case 2 RnC1 . For this, note in particular that, since f has compact support, we have that f 2 H1 .jD j/ for any 2 RnC1 . For later purposes we record the following slight refinement of the above version of Sobolev’s lemma. Corollary 3.2.11. For f 2 H2e .jDj/ we have that f can be considered as a continuous function, i.e. f 2 C0 .RnC1 /. If f 2 H4e .jDj/ then f is in C1 .RnC1 /. If f 2 H1 .jDj/ then f is in C1 .RnC1 /. Proof. By the fundamental theorem of calculus we obtain @k .x0 ; : : : ; xn / D
1 h
D
1 h
Z Z
.x0 ; : : : ; xk1 ; xk C h; xkC1 ; : : : ; xn / .x0 ; : : : ; xn / h
xk Ch xk
.@k .x0 ; : : : ; xn / @k .x0 ; : : : ; xk1 ; tk ; xkC1 ; : : : ; xn // dtk
xk Ch Z xk
xk
tk
@2k .x0 ; : : : ; xk1 ; s; xkC1 ; : : : ; xn / ds dtk
and so, applying the above Sobolev lemma, ˇ2 ˇ ˇ ˇ ˇ@k .x0 ; : : : ; xn / .x0 ; : : : ; xk1 ; xk C h; xkC1 ; : : : ; xn / .x0 ; : : : ; xn / ˇ ˇ ˇ h jhj nC1 j.i m C 1/e @eC2ek j20;0 holds for all 2 H4e .jDj/. Since k D 0; : : : ; n is arbitrary, we read off that has continuous partial derivatives. The last statement follows by induction.
248
Chapter 3 Partial Differential Equations with Constant Coefficients
In special non-invertible cases it is customary to establish particular solution theories by continuous extension to suitable spaces. Uniqueness is enforced by additional constraints. It is not surprising that these adjustments have to be made, since we are trying to solve in the non-invertible case (i.e. ‘in the spectrum’), where by definition the standard solution theory fails. The observation that convolutions can sometimes be written as convolution type integral operators is frequently utilized in this context. The next subsection provides a discussion of some of the issues involved.
3.2.4 Integral Representations of Convolutions with Fundamental Solutions It may be comforting to know that convolutions as abstract objects are not always far away from convolution integrals. In order to obtain integral representations one has to further restrict the class of elements the convolution is applied to. We shall frequently utilize H1 .jD j/ for appropriate 2 RnC1 or even CV 1 .RnC1 / for this purpose. 3.2.4.1
An Integral Representation of the Solution of the Transport Equation
We recall that, for the forward fundamental solution s;C , we found, with the particular parameter choice D .%; 0; : : : ; 0/, Z h s;C j i;0 D h s;C j exp.2 t %/ i0;0 D
.t; t s/ exp.2 t %/ dt R>0
for all 2 CV 1 .R1Cn /. Thus, we find for the convolution s;C , 2 CV 1 .R1Cn /, that Z Z .1 / .t; x/ D ..t r/; .x y// .r; y/ dr dy; Rn
h s;C
R
j i0;0 D h s;C j.1 Z Z D R>0
Z D
R1Cn
/ i0;0
R1Cn
.t C r; t s C y/ .r; y/ dr dy dt
Z
R>0
Consequently,
.t C r; t s C y/ dt .r; y/ dr dy:
Z . s;C
/.t; x/ D Z D
for
2 CV 1 .R1Cn /.
R>0
1;tŒ
.t r; x r s/ dr .u; x .t u/ s/ du
Section 3.2 Partial Differential Equations in H1 .jD j/ 3.2.4.2
249
Potentials, Single and Double Layers in R3
Recalling that @2 in R3 has g0 given by x 7! find for ; 2 CV 1 .R3 / hg0
1 4 jxj
as a fundamental solution, we
j i0;0 D hg0 j.1 / i0;0 Z Z 1 D .x C y/ .y/ dy dx R3 4 jxj R3 Z Z 1 D .x C y/ .y/ dx dy R3 R3 4 jxj Z Z 1 D .z/ .y/ dz dy: 3 3 4 jy zj R R
The latter shows that g0 Z x 7!
R3
is given by the function 1 1 .z/ dz D 4 jx zj 4 r
where r.x/ D jxj and 1 WD 4 r.x/
´
1 4 jxj
for x ¤ 0;
0
for x D 0;
.x/
x 2 R3 :
Thus, for sufficiently well-behaved right-hand sides f , a particular solution u of @2 u D f
(3.2.102)
can be given as a convolution integral Z 1 1 f .x/ D f .z/ dz: u.x/ D 4 r R3 4 jx zj We already noted that we cannot expect uniqueness of solution to hold. In fact, every solution of (3.2.102) is given by uD where u0 is a solution of
1 f C u0 4 r
@2 u0 D 0:
(3.2.103)
(3.2.104)
Remark 3.2.12. For g0 , the term ‘potential’ is frequently used. This term is physically motivated since, in suitable physical contexts, grad g0 can be interpreted as a force. Conversely, a force K generated by a function V as K D grad V
250
Chapter 3 Partial Differential Equations with Constant Coefficients
is called a potential force and V is called a potential of K. If div K D 0 in an open set , then the force is called divergence-free in and the region in which this holds is called source-free. In this terminology, the above g0 is a potential for K WD grad g0 and K is divergence-free in R3 n ¹0º. This is why a solution theory for the equation @2 u D f is also called a ‘potential theory’. From (3.2.103) and (3.2.104) we see that in order to establish a solution theory for (3.2.102) we need to gain additional insight into the null space of @2 . Although we are specifically concerned here with the three-dimensional case, we shall consider the general n-dimensional case to analyze the null space. Lemma 3.2.13. If u 2 Hn;1;1;1 .jDj/ satisfies @2 u D 0, then u is a polynomial. u satisfies the Proof. Let u be a solution of @2 u D 0. Then, its Fourier transform b equation m2 b u D 0: As a consequence we have that suppb u ¹0º: We shall now show that b u 2 LinC ¹@˛ ı j˛ 2 N nC1 º so that the claim of the lemma then follows by applying the inverse Fourier transformation. We have that b u 2 H2˛ .jDj/ for some ˛ 2 N nC1 . Then, there must be a constant C 2 R>0 such that jhb u j i0;0 j C j.D D/˛ j0;0
(3.2.105)
for all 2 CV 1 .RnC1 /. Consider h 2 CV 1 .R/ with supp h Œ1; 1, and supp.1 h/ Rn 1=2; C1=2 Œ: Q Then g" .x/ WD nkD0 h.xk ="/ for x D .x0 ; : : : ; xn / 2 RnC1 and " 2 R>0 defines a family .g" /"2R>0 of functions in CV 1 .RnC1 /. First we observe that u j .1 g" / i0;0 C hb u j g" i0;0 D hb u j g" i0;0 : hb u j i0;0 D hb
Section 3.2 Partial Differential Equations in H1 .jD j/
251
For " D 2 we obtain from (3.2.105) with a new constant C 0 jhb u j i0;0 j D jhb u j g2 i0;0 j (3.2.106)
C j.D D/˛ .g2 /j0;0 X C0 j@ j0;0 02˛
for all 2 CV 1 .RnC1 /. For WD . .m/ we have 2 CV 1 .RnC1 / and
P
1 2N nC1 ;jj1 j2˛j1 C1 Š
@ .0/ m /g1
@ .0/ D 0 for multi-indices 2 N nC1 ; j j1 j2˛j1 C 1. Consequently, we have a generic constant C such that j@ .x/j C jxjj2˛j1 C1jj1 for all x 2 RnC1 and so noting that @ g" D @ g1 .="/ "jj1 we get u j g" jhb u j i0;0 j D jhb
X
i0;0 j C 0
j@ .g"
/j0;0 ! 0 as " ! 0C:
02˛
Thus, we have shown that u j g1 .m/ i0;0 hb u j i0;0 D hb ˇ X ˇ D b u ˇˇ
2N nC1 ;jj1 j2˛j1 C1
X
D
2N nC1 ;jj1 j2˛j1 C1
D
1 @ .0/ m g1 Š
0;0
1 @ .0/ hb u jm g1 i0;0 Š
X 2N nC1 ;jj1 j2˛j1 C1
hb u jm g1 i0;0
ˇ 1 ˇˇ @ ıˇ Š 0;0
for all 2 CV 1 .RnC1 /, showing that b u 2 LinC ¹@˛ ı j˛ 2 N nC1 º. The homogeneous solutions of the Laplace equation are therefore polynomials called harmonic polynomials. The last lemma implies uniqueness of solution of
252
Chapter 3 Partial Differential Equations with Constant Coefficients
(3.2.102) up to arbitrary harmonic polynomials. To find harmonic polynomials45 is a purely algebraic question about relations between coefficients. For suitable right-hand sides, existence of solutions can be established via convolution. For example, if the right-hand side f of the .n C 1/-dimensional Laplace equation @2 u D f 45 Harmonic polynomials are obtainable in a systematic way by analytic means involving the Kelvin transform. The Kelvin transform is induced by the following reflexion at the boundary of the unit ball
s W RnC1 n ¹0º ! RnC1 n ¹0º; x 7!
1 x jxj2
with s W RnC1 n ¹0º ! RnC1 n ¹0º; x 7!
1 x jxj2
as its inverse. The Kelvin transform is now given as K W C1 .RnC1 n ¹0º/ ! C1 .RnC1 n ¹0º/; u 7! jmj1n u ı s: The main feature of the Kelvin transform is that it transforms harmonic functions in an open set RnC1 n ¹0º to harmonic functions in the open set sŒ RnC1 n ¹0º. Harmonic functions in RnC1 n ¹0º are given e.g. by the fundamental solution g0 and their derivatives. Here g0 depends on the dimension in the following way: 8 1 for n D 0; ˆ < 2 jxj 1 g0 .x/ D
[email protected];1/j1 ln.jxj/ for n D 1; ˆ : 1 1 for n > 1
[email protected];1/j jxj n
for x 2
RnC1
n ¹0º. Thus, p˛ WD
K.@˛ g0 /
are all analytic in RnC1 n ¹0º, ˛ 2 N nC1 . In particular,
8 1 ˆ < 2 K.g0 /.x/ D
1 1 ˆ
[email protected];1/j : 1
[email protected];1/jn
for n D 0; ln.jxj/ for n D 1; for n > 1
for x 2 RnC1 n ¹.0; 0/º. Due to the decay at infinity of @˛ g0 for ˛ 2 N nC1 , n 2 N>1 , or ˛ 2 N 2 n¹.0; 0/º or ˛ 2 Nn¹0; 1º, the singularity of p˛ at 0 is removable and so p˛ must be a (homogeneous) harmonic polynomial in these cases. For n D 0 we have 1 p0 .x/ D ; 2
1 p1 .x/ D x 2
for x 2 R describing a basis of all harmonic polynomials in R. For n D 1 we have that .K.@˛ g0 //˛2N 2 n¹.0;0/º consists of non-constant, homogeneous, harmonic polynomials spanning all harmonic polynomials in R2 with p.0/ D 0, [44]. For n > 1 we have that .K.@˛ g0 //˛2N nC1 spans all harmonic polynomials in RnC1 , for more details see [44].
Section 3.2 Partial Differential Equations in H1 .jD j/
253
is assumed to be in CV 1 .RnC1 /, n 2 N>1 , then uniqueness of solution can be established by imposing additional decay conditions at 1 in order to exclude harmonic polynomials as contributors to a solution. Let us return to the case n D 3. (The case n D 2 will be treated separately). We first note the following regularity result. Lemma 3.2.14. Let u 2 H1 .jD1 j; jD2 j; jD3 j/ satisfy @2 u 2 CV 1 .R3 /: Then u 2 C1 .R3 /: Proof. Since (3-dimensional) polynomials are certainly in C1 .R3 /, we only need to consider the regularity of 1 uD f 4 r with f WD @2 u 2 CV 1 .R3 /. We see that Z 1 f .x z/ dz; u.x/ D R3 4 jzj
x 2 R3 ;
from which we can read off the desired regularity by noting that we may differentiate u by differentiating f ‘under the integral’. A local variant of this regularity result also holds. It will be derived for arbitrary dimensions. Lemma 3.2.15. Let u 2 H1 .jDj/ and @2 u D f C g; with g 2 C1 .RnC1 / and f; g 2 H1 .jDj/ with supp f compact, n 2 N. Then u 2 C1 .RnC1 n supp f /: Proof. Consider j" .x0 ; : : : ; xn / D "n1 j.x0 ="/ j.xn ="/, where j 2 CV 1 .R/ is chosen as in Example 3.19. Since regularity is a local property, we may just consider an arbitrary open cube Q.x; %0 / WD x0 %0 ; x0 C %0 Œ xn %0 ; xn C %0 Œ with x 2 RnC1 n supp f and %0 2 R>0 so small that Q.x; 2 %0 / RnC1 n supp f . Since u 2 H1 .jDj/ we have u 2 Hk .jDj/ for some k 2 N nC1 and with 0 WD j"0 Q.x;%0 / , 0 < "0 < %0 , we see that 0 has compact support and 0 .m/ u
2 Hk .@ C e/:
254 Letting
Chapter 3 Partial Differential Equations with Constant Coefficients 1
WD j"1 Q.x;%0 "0 / , 0 < "1 < %0 "0 , we have 1 .m/ u
D
1 .m/
0 .m/u
and @2
1 .m/ u
D
1 .m/ .f
D
1 .m/ g
C .@2
C g/ 2 .@
1 /.m/
2 .@
1 /.m/
.@
1 /.m/
0 .m/u:
@u C .@2
1 /.m/ u
0 /.m/ u
Since 1 and 0 have compact support the right-hand side is a distribution with compact support. Therefore @2
1 .m/ u
C
1u
D
1 .m/ g
C .@2
2 .@
1 /.m/
.@
1 /.m/
0 .m/u
C
0 /.m/u/ 1 .m/
0 .m/ u
which yields 1 .m/ u
D .@2 C 1/1 . C .@
2
1 /.m/
1 .m/ g 0 .m/u
2 @.@
C
1 .m/
1 .m/
0 .m/u/
0 .m/ u/
in H1 .@ C e/. We note that 1 .m/ g 2 H1 .@ C e/ and obtain that .@2 C 1/1 . 1 .m/ g C .@2 1 /.m/ 0 .m/u C 1 .m/ 0 .m/ u/ is an element of intersection T 2 2 1 is bounded for j D 0; : : : ; n. j D0;:::;n HkC2ej .@Ce/, since .@j ˙1/ .@ C1/ 2 Similarly, we find that the contributing part .@ C 1/1 @.@ 1 .m/ 0 .m/u/ is T in j D0;:::;n HkCej .@ C e/, since .@k ˙ 1/ .@2 C 1/1 @j is bounded for j; k D 0; : : : ; n. Thus, we have \ HkCej .@ C e/: 1 .m/ u 2 j D0;:::;n
Defining recursively
mC1
we obtain, assuming "j WD
WD j"mC1 Q.x;% 2j C1 m
, j 2 N, that 1
Pm 0 j D0 "j /
, 0 < "m < % 0
Pm
j D0 "j ,
on Q.x; %0 /
for 0 < < %0 and all m 2 N. With D %20 and 0 WD j%0 =8 Q.x;%0 =4/ , we have
0 .m/ u D 0 .m/ s .m/ u for all s 2 N and by induction
0 .m/ u 2 H1 .@ C e/ and so also
0 .m/ u 2 H1 .jDj/: By Lemma 3.2.10 and Corollary 3.2.11 we obtain 0 .m/ u 2 C1 .RnC1 / or u 2 C1 .Q.x; %0 =8//:
Section 3.2 Partial Differential Equations in H1 .jD j/
255
In the same way a more general regularity result follows. It is frequently referred to as Weyl’s lemma. Corollary 3.2.16. Let u 2 H1 .jDj/ and @2 u C a @u C b u D f C g; with g 2 C1 .RnC1 / and f; g 2 H1 .jDj/ with supp f compact, a 2 C nC1 ; b 2 C, n 2 N. Then u 2 C1 .RnC1 n supp f /: For the extension of the solution theory for @2 , the next inequalities – known as Poincaré estimates – are of interest. Lemma 3.2.17. Let uD
1 f 4 r
with f 2 CV 1 .R3 /. Then we have ˇ ˇ ˇ 1 ˇ C ˇ ˇ u ˇ r C i ˇ 4 j.r C i/ f j0 0 R where C WD sup¹ R3 jxyj1jyj5=2 dyjx 2 R3 ^ jxj D 1º. Proof. We have46 ˇ2 Z ˇ ˇ ˇ 1 ˇ u.x/ˇˇ dx ˇ R3 jxj C i ˇZ ˇ2 Z ˇ ˇ 1 1 ˇ ˇ dx f .z/ dz D ˇ ˇ 2 R3 jxj C 1 R3 4 jx zj Z 1 2 R3 jxj C 1 ˇ2 ˇZ ˇ ˇ 1 2 1=2 1=4 ˇ .jzj C 1/ jzj jf .z/j dz ˇˇ dx ˇ 2 1=2 1=4 jzj R3 4 jx zj .jzj C 1/ Z Z 1 1 1 dz 16 2 R3 jxj2 C 1 R3 jx zj .jzj2 C 1/ jzj1=2 Z jzj2 C 1 1=2 jzj jf .z/j2 dz dx: R3 jx zj 46
We remind the reader that for sake of brevity we are using here the customary formal estimation process, which finds its justification only in the validity of the ‘last’ estimate. The existence of all previously used integrals follows by following the formal argument in reverse order. This then finally makes the formal estimates rigorous.
256
Chapter 3 Partial Differential Equations with Constant Coefficients
We find for x ¤ 0 with the substitution y D z=jxj that Z
Z
1
R3
jx zj .jzj2 C 1/ jzj1=2
1 dz jx zj jzj5=2 Z 1 1 D dy 1=2 5=2 jxj R3 jx=jxj yj jyj C ; jxj1=2
dz
R3
(3.2.107)
where ²Z C WD sup
R3
ˇ ³ ˇ 1 3 ˇ dy x 2 R ^ jxj D 1 : jx yj jyj5=2 ˇ
Inserting this into the estimate for u and using (3.2.107) once again, we obtain as desired ˇ2 ˇ ˇ ˇ 1 ˇ u.x/ˇˇ dx ˇ R3 jxj C i Z Z C 1 jzj2 C 1 1=2 jzj jf .z/j2 dz dx 16 2 R3 .jxj2 C 1/ jxj1=2 R3 jx zj Z Z 1 C dx .jzj2 C 1/ jzj1=2 jf .z/j2 dz D 2 2 1=2 jx zj 16 R3 R3 .jxj C 1/ jxj Z C2 .jzj2 C 1/ jf .z/j2 dz: 16 2 R3
Z
Considering r as a multiplication operator in L2 .R3 /, we see that we are here concerned with the spaces H˙1 .r C i/ of the associated Sobolev chain .Hk .r C i//k2Z . The set CV 1 .R3 / can be seen to be dense in H1 .r C i/ by a simple cut-off argument, (which we will not carry out in detail). The density of CV 1 .R3 / in H1 .r C i/ (H0 .r Ci/ WD L2 .R3 /) then shows that the last lemma yields that 41 r W CV 1 .R3 / H1 .r Ci/ ! H1 .r Ci/ can be extended by continuity to a continuous mapping from H1 .r C i/ to H1 .r C i/. As a consequence we obtain Theorem 3.2.18. Let f 2 H1 .r C i/. Then @2 u D f has a unique solution in H1 .r C i/, so that .@2 /1 W H1 .r C i/ ! H1 .r C i/ is a bounded, linear operator.
Section 3.2 Partial Differential Equations in H1 .jD j/
257
Proof. Let S W H1 .r C i/ ! H1 .r C i/ denote the continuous extension of the convolution operator 41 r W CV 1 .R3 / H1 .r C i/ ! H1 .r C i/. Then, since Sf 2 H1 .r C i/ ,! H.1;1;1/ .jD1 j; jD2 j; jD3 j/, we have @2 Sf D f
in H.3;3;3/ .jD1 j; jD2 j; jD3 j/;
(3.2.108)
for f 2 CV 1 .R3 /. Since convergence of f in H1 .r C i/ implies convergence in the space H0 .jD1 j; jD2 j; jD3 j/ and convergence of Sf in H1 .r C i/ implies convergence in H.1;1;1/ .jD1 j; jD2 j; jD3 j/, we have by continuity of @2 in H1 .jD1 j; jD2 j; jD3 j/ and by taking limits that equation (3.2.108) still holds for f 2 H1 .r C i/. Thus we have existence of a solution in H1 .r C i/. Uniqueness now follows since a solution of @2 u D 0 must be a (harmonic) polynomial and the only polynomial in H1 .r C i/ is the zero polynomial. The continuity of the solution operator is known explicitly by construction according to Lemma 3.2.17 and has indeed been utilized already for the above approximation. The particular asymptotic behavior of the solutions just introduced is also reflected in the so-called Poincaré estimate in R3 . We shall record this type of estimates in the following lemma (again for arbitrary dimensions). Lemma 3.2.19. For all
2 CV 1 .RnC1 /, n 2 N, we have the following estimates.
(a) If n D 0 then jr 1 j20 4 j@ j20 with
2 CV 1 .R n ¹0º/.
(b) If n D 1 then j.r ln.r//1 with47
j20 4 .j@0
j20 C j@1
j20 /
2 CV 1 .R2 n .¹0º [ @BR2 .0; 1///.
(c) If n > 1 then jr 1 j20
n X 4 j@k .n 1/2
j20 :
kD0
Proof. For 2 CV 1 .R/ we clearly have Z j˛.r/ .r/ C @ .r/j2 r n dr 0 R0
(3.2.109)
47 Recall that we denote a ball of radius R 2 R >0 centered at x in some normed space E by BE .x; R/. Its boundary, the sphere, is denoted by @BE .x; R/.
258
Chapter 3 Partial Differential Equations with Constant Coefficients
where the sufficiently regular real-valued coefficient function ˛ is to be determined later. We first obtain at least formally Z Z j@ .r/j2 r n dr j˛.r/ .r/j2 r n dr R0
R0
Z
2 Re Z D Z D
˛.r/ .r/ @ .r/ r n dr R0
Z 2 n
R0
R0
j˛.r/ .r/j r dr
˛.r/@.j .r/j2 /r n dr R0
.r n @.r n ˛.r// ˛.r/2 /j .r/j2 r n dr:
We shall attempt to determine ˛ in such a way that r n @ .r n ˛.r// ˛.r/2 c0 ˛.r/2
(3.2.110)
is satisfied for some c0 2 R>0 . Singularities of suitable ˛ may result in further constraints on to make the above calculation rigorous. Let us first consider the case n D 0. In this case, we want to satisfy @ ˛.r/ .1 C c0 / ˛.r/2 : Clearly, ˛.r/ WD =r, 2 R, satisfies this estimate if .1 C c0 / 2 > 0 or
We set WD
1 . / > 0: 1 C c0 1 1Cc0
2 R>0 and choose c0 such that c0 .1 C c0 /2
is maximal, i.e. c0 D 1. This concludes the case n D 0. Note that in this case, we have assumed additionally that 2 CV 1 .R n ¹0º/ to avoid problems associated with the singularity of ˛. For n > 1, (3.2.110) also holds for ˛.r/ D =r if .n 1/ .1 C c0 / 2 > 0 or
.n 1/ > 0: .1 C c0 /
Section 3.2 Partial Differential Equations in H1 .jD j/ With c0 D 1 we get Z R0
jr
1
.r/j20 r n dr
4 .n 1/2
259
Z R0
j@ .r/j20 r n dr:
Replacing by r 7! .r x0 /, 2 CV 1 .RnC1 /, and integrating with respect to x0 in the unit sphere we obtain (polar coordinates!) the desired estimate jr
1
j20
ˇ ˇm 4 ˇ @ .n 1/2 ˇ jmj
Finally, for n D 1, let ˛.r/ WD
. r ln.r/
ˇ2 nC1 X ˇ 4 ˇ j@k ˇ .n 1/2 0
j20 :
kD1
Then the desired inequality (3.2.110) holds if
.1 C c0 / 2 > 0: Here we must assume additionally 2 CV 1 .R n ¹0; 1º/. The same reasoning as in the case n > 1 now yields the desired Poincaré type inequality. The constraints on imply the requirement 2 CV 1 .R2 n .¹0º [ @BR2 .0; 1///. Remark 3.2.20. The connection between these estimates and the Laplacian is given by the following observation. The above estimates are valid in the closed subspace HnC1 generated by the closure of the specified functions in CV 1 .RnC1 / with respect to the Hilbert space norm 48
7! j@ j0 in H1 .jDj/. Defining49 grad W H.nC1/ ˇ.r/1 ŒL2 .RnC1 / !
M
L2 .RnC1 /
0;:::;n 48
According to the above Poincaré type estimates this norm is equivalent to q
7! jˇ.r/ j20 C j@ j20 ;
where the weight ˇ.r/ is given by ´ ˇ.r/ WD 49
1 r
1 r ln r
for n ¤ 1; for n D 1:
Here .m/1 ŒL2 .RnC1 / abbreviates the Hilbert space ¹ 2 L2;loc .RnC1 /j .m/ 2 L2 .RnC1 /º
with norm
7! j .m/ j0 ; where is a measurable function vanishing only on a Lebesgue null set.
260
Chapter 3 Partial Differential Equations with Constant Coefficients
as the closure of DnC1 CV 1 .RnC1 / ˇ.r/1 ŒL2 .RnC1 / ! L2 .RnC1 /;
7! @ and div W D.div/
M
L2 .RnC1 / ! ˇ.r/ ŒL2 .RnC1 /
0;:::;n
as its negative adjoint, i.e. ‰ 2 D.div/ if there is an f 2 ˇ.r/ ŒL2 .RnC1 / such that hgrad j‰i0 D hˇ.r/ j ˇ.r/1 f i0 for all 2 DnC1 , where 8 V ˆ
°
ˇ ˇ f 2 ˇ.r/ŒL2 .RnC1 /ˇ
^
for n D 0; for n D 1; for n 2 N>1 ;
hgrad j‰i0 D hˇ.r/ jˇ.r/1 f i0
± ;
2DnC1
we may consider the composition div grad as a particular realization of the negative Laplacian @2 . We have D. div grad/ D ¹ 2 HnC1 j grad 2 D.div/º. Now the above Poincaré type estimates yield, for 2 D. div grad/ jˇ.r/ j20 CnC1 jgrad j20 D CnC1 hgrad
j grad i0
CnC1 h j div grad i0 CnC1 hˇ.r/ j ˇ.r/1 div grad i0 CnC1 jˇ.r/ j0 jˇ.r/1 div grad j0 ; where
´ CnC1 WD
2 jn1j
for n ¤ 1;
2
for n D 1:
Consequently, we have jˇ.r/ j0 CnC1 jˇ.r/1 div grad j0
Section 3.2 Partial Differential Equations in H1 .jD j/ for all
261
2 D. div grad/. The solution theory of the equation div grad u D f
is now provided by a simple application of Riesz representation formula. Indeed, let f 2 ˇ.r/ŒL2 .RnC1 /. Then f generates a continuous linear functional 7! hˇ.r/1 f j ˇ.r/ i0 on HnC1 , since jhˇ.r/1 f j ˇ.r/ i0 j jˇ.r/1 f j0 jˇ.r/ j0 CnC1 jˇ.r/1 f j0 j@ j0 for all 2 D.grad/ D HnC1 . This functional can be represented by an element u 2 HnC1 as hgrad u j grad i0 D h@ u j @ i0 D hˇ.r/1 f j ˇ.r/ i0 for every
2 HnC1 . By definition of div, this implies that grad u 2 D.div/ and div grad u D f:
This shows existence. Since the argument can be reversed uniqueness also follows and since the Riesz map is the solution operator, continuous dependence on the data is also clear. In particular for n D 2, our earlier solution theory for the 3-dimensional case is reproduced. Returning for sake of simplicity to the case n D 3, we now want to briefly outline the possibility of considering so-called boundary value problems in the framework of our “whole-space” solution theory. Let R3 be a bounded domain with a smooth boundary @, i.e. is a bounded, smooth submanifold with boundary in R3 . In order to find the proper formulation for a boundary value problem in this setting, let us assume that u is a smooth function on R3 such that @2 u D f: Using the characteristic function of as a ‘cut-off’-function we obtain @2 w D f C 2
3 X
.@k / @k u C .@2 / u
kD1
with w WD u. We want to evaluate the distributions @k , k D 1; 2; 3, and @2 .
262
Chapter 3 Partial Differential Equations with Constant Coefficients 2 CV 1 .R3 /
We find for all
h@k j i0 D h j @k i0 Z D .x/ @k .x/ dx R3 Z D @k .x/ dx Z D nk .x/ dSx ; @
where nk , k D 1; 2; 3, denotes the k-th component of the exterior normal vector to the boundary manifold @, (according to Gauss’ theorem). The latter, however, is by definition equal to hnk ı@ j i0 and so @k D nk ı@ :
(3.2.111)
Since ı@ appears to describe Dirac-ı-sources spread over @ one speaks of a monopole layer or single layer on @, here with a weight factor nk , k D 1; 2; 3. Now, it is easy to calculate 2
@ D
3 X
@2k
kD1
3 X
D
@k .nk ı@ / D
kD1
3 X
nk @k ı@ C
kD1
3 X
.@k nk / ı@ :
kD1
Thus, @2 D n @ ı@ C .div n/ ı@ DW @ .n ı@ /: We see that 3 DX
3 ˇ E ˇ X D E ˇ ˇ @k .nk ı@ / ˇ D ı@ ˇ n k @k 0
kD1
kD1
Z D
3 X
@
Z D
nk .x/ @k .x/ dSx
kD1
lim Z
@ h!0
D lim
h!0
0
@
.x C h n/ .x h n/ dSx 2h .x C h n/ .x h n/ dSx : 2h
The latter can be interpreted as 3 X kD1
@k .nk ı@ / D lim
h!0
1 .hn ı@ hn ı@ / 2h
Section 3.2 Partial Differential Equations in H1 .jD j/
263
and is therefore referred to with physical justification50 as a dipole layer or double layer. As a consequence we find the following form of our induced equation for w: @2 w D f C .n @u/ ı@ C u @ .n ı@ /: The monopole layer ı@ is weighted by n @ u and the dipole layer @ .n ı@ / is weighted by u. Since ı@ has support on @ the weight factors are also only used on ı@ . Consequently, n @ u and u need only be known on @ in order to recover w from the right-hand side. The data uj@ is referred to as Dirichlet boundary data and n @ uj@ is known as Neumann boundary data for the current problem. The right-hand side is a distribution with compact support and thus we have a solution w0 WD
1 . f C .n @u/ ı@ C u @ .n ı@ //: 4 r
(3.2.112)
This solution is smooth outside of the support of f and away from @ according to our above regularity lemma. Moreover, we have the following asymptotic decay property. Lemma 3.2.21. Let f 2 H1 .jD1 j; jD2 j; jD3 j/ with supp f compact. Then RnŒR;R .r/
1 f 2 H1 .r C i/ 4 r
for all sufficiently large R 2 R>0 . Proof. We have @2
1 f Df 4 r
and using a real-valued cut-off function 0 2 CV 1 .R3 / with 0 D 1 on supp f we obtain, as in the proof of Lemma 3.2.15, 1 1 1 2 @ .1 0 / f D .1 0 / f C 2 @ 0 @ f C @2 0 f 4 r 4 r 4 r 1 1 f C @2 0 f: D 2 @ 0 @ 4 r 4 r 50 Indeed, ı hn @ hn ı@ is the sum of a (shifted) monopole layer hn ı@ and another monopole layer hn ı@ of opposite sign (‘charge’) (shifted in the opposite direction). Such a configuration is referred to as a dipole. Since we consider here the limit of the scaled monopole layer sum
1 . ı C .hn ı@ // 2h hn @ as h ! 0, we are actually considering something one might call an ‘infinitesimal’ dipole layer.
264
Chapter 3 Partial Differential Equations with Constant Coefficients
The right-hand side, according to Lemma 3.2.15, is an element in CV 1 .R3 /. We calculate with a second real-valued function 1 2 CV 1 .R3 / with 1 D 1 on supp f and
0 D 1 on supp 1 : ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ ˇ 1 ˇDˇ ˇ ˇ .1 0 / f ˇˇ f ˇˇ .1 0 / ˇ ˇ ˇ ˇ 4 r 4 r 0 0 ˇ ˇ ˇ ˇ ˇ ˇ 1 ˇ .1 0 / D ˇˇ f ˇˇ ˇ 4 r 0 ˇ ˇ ˇ ˇ 1 ˇ ˇ ˇ ˇ ˇ .1 0 / D ˇ 1 f ˇ ˇ 4 r 0 ˇ ˇ ˇ ˇ 1 .1 0 / ˇˇ jf j˛ ˇˇ.@ C e/˛ 1 4 r 0
for some ˛ 2 N 3 . Considering the last term, ˇ ˇ2 ˇ ˇ ˇ.@ C e/˛ 1 1 .1 0 / ˇ ˇ ˇ 4 r 0 ˇ2 Z Z ˇ ˇ ˇ 1 ˇ.@ C e/˛ 1 .1
D .y// .y/ dy .x/ˇˇ dx 0 ˇ R3 R3 4 j yj ˇ2 ˇZ Z ˇ ˇ 1 ˛ ˇ D .y/ dy ˇˇ dx ˇ 3 .@ C e/ 1 4 j yj .x/ .1 0 .y// supp 1 R j.r C i/ j20 ˇ Z Z ˇ 2 1 ˇ ˛ .jyj C 1/ ˇ .@ C e/ 1 supp 1
R3
It is not hard to see that ˇ Z ˇ .jyj2 C 1/1 ˇˇ .@ C e/˛ 1 R3
which confirms that ˇ ˇ ˇ .1 0 / 1 f ˇ 4 r
ˇ2 ˇ 1 .x/ .1 0 .y// ˇˇ dy dx: 4 j yj
ˇ2 ˇ 1 .x/ .1 0 .y// ˇˇ dy dx < 1; 4 j yj
ˇ ˇ ˇ ˇ ˇ ˇ ˇ jf j˛ ˇ.@ C e/˛ 1 1 .1 0 / ˇ ˇ ˇ ˇ 4 r 0
jf j˛ C0 j.r C i/ j0 ; for all
2 CV 1 .R3 / and some C0 2 R>0 . This proves that .1 0 /
1 f 2 H1 .r C i/ 4 r
ˇ ˇ ˇ ˇ
0
Section 3.2 Partial Differential Equations in H1 .jD j/
265
and so in particular RnŒR;R .r/
1 1 f D RnŒR;R .r/ .1 0 / f 2 H1 .r C i/ 4 r 4 r
for all sufficiently large R 2 R>0 . From (3.2.112) and Lemma 3.2.21 we find that RnŒR;R .r/ .w0 w/ 2 H1 .r C i/
(3.2.113)
for sufficiently large R 2 R>0 . Since w0 w must be a harmonic polynomial and the only polynomial satisfying (3.2.113) is the zero polynomial we have w D w0 D
1 . f C .n @u/ ı@ C u @ .n ı@ //: 4 r
Thus we have found what is known as the representation theorem, which states how a solution can be expressed in terms of the source term and Dirichlet and Neumann boundary data. Clearly, the same reasoning can be applied to the case where is unbounded if @ and supp f remain bounded. Theorem 3.2.22. Let w 2 H1 .jD1 j; jD2 j; jD3 j/ be such that RnŒR;R .r/ w 2 H1 .r C i/
(3.2.114)
for a sufficiently large R 2 R>0 , wjN continuously differentiable in a neighborhood of the smooth, compact boundary @ of a domain R3 and @2 w D f C .n @w/j@ ı@ C wj@ @ .n ı@ /
(3.2.115)
with f 2 H0 .jD1 j; jD2 j; jD3 j/. Then wD
1 1 f C ..n @w/j@ ı@ C wj@ @ .n ı@ //: 4 r 4 r
Proof. The result is just a slight refinement of the above derivation by being more specific about the ‘smoothness’ of w. The above uniqueness argument yields the given representation. However, we need to convince ourselves that the right-hand side of (3.2.115) defines a distribution with compact support. Let us first consider terms of the form ' ı@ where ' is a continuous function on @. We have sZ ˇZ ˇ sZ ˇ ˇ '.x/ .x/ dSx ˇˇ j'.x/j2 dSx jh' ı@ j i0 j D ˇˇ @
@
@
j .x/j2 dSx
266
Chapter 3 Partial Differential Equations with Constant Coefficients
and with n as a continuously differentiable extension of the exterior unit normal to all of Z Z 2 j .x/j dSx D jnj2 j .x/j2 dSx @
@
D
3 Z X kD1
Z D
@k .nk j ./j2 /.x/ dx Z
.div n/.x/j .x/j2 dx C 2 Re
n @ .x/ .x/ dx
C12 j j2e for some constant C1 2 R>0 and all
2 CV 1 .R3 /. This implies51 sZ
jh' ı@ j i0 j C1
@
j'.x/j2 dSx j je
so that ' ı@ 2 H.1;1;1/ .@ C e/ H1 .@ C e/: Similarly we also see that ' @ .n ı@ / 2 H1 .@ C e/ if ' is continuously differentiable in a neighborhood of @. It is interesting to investigate the term 1 ..n @w/j@ ı@ C wj@ @ .n ı@ // 4 r more closely. It can indeed be written as a boundary integral. We calculate ˇ ˇ ˇ ˇ 1 1 ˇ ˇ .' ı@ / ˇ D ' ı@ ˇ 4 r 4 r 0 0 Z Z 1 .y/ dy dSx D '.x/ @ R3 4 jx yj Z Z 1 dSx .y/ dy: D '.x/ 3 4 jx yj R @ Thus we see that 1 .' ı@ / D 4 r
Z @
1 '.x/ dSx : 4 j xj
(3.2.116)
51 Here it becomes apparent that it would also be sufficient to require ' 2 L .@/. More subtle trace 2 theorems would yield even more general boundary weights '.
Section 3.2 Partial Differential Equations in H1 .jD j/ Slightly more involved, we find ˇ ˇ 1 .' @ .n ı@ // ˇˇ 4 r 0 ˇ ˇ 1 ˇ D ' @ .n ı@ / ˇ 4 r 0 ˇ ˇ 1 D @ .n ı@ / ˇˇ ' 4 r 0 ˇ 3 X ˇ 1 D nk ı@ ˇˇ @k ' 4 r kD1
D
3 Z X kD1
Z
D Z
@
nk .x/ @k ' @
Z R3
R3
0
1 .y/ dy .x/ dSx 4 j yj
1 .y/ dy dSx 4 jx yj 1 .x/ .y/ dy dSx ; .n @/ 4 j yj
.n @ '/.x/ '.x/
@
Z
Z
267
R3
and so 1 .' @ .n ı@ // 4 r Z 1 D n.y/ @ '.y/ dSy @ 4 j yj Z 1 .y/ '.y/ dSy : n.y/ @ z 7! 4 j zj @
(3.2.117)
Here the somewhat clumsy looking notation @ .z 7! 4 j1 zj / has been used to ensure that the differentiation with respect to correct variable is properly expressed. Using (3.2.116) and (3.2.117) we obtain 1 ..n @w/j@ ı@ C wj@ @ .n ı@ // (3.2.118) 4 r Z 1 1 n.y/ @w.y/ w.y/ n.y/ @ z 7! .y/ dSy : D 4 j yj 4 j zj @ Denoting the directional derivative .n@/ evaluated at a point y 2 R3 in the classical way by @
.y/ WD ..n @/ /.y/ @n.y/
268
Chapter 3 Partial Differential Equations with Constant Coefficients
the above representation can be written slightly more traditionally (but ambigiously) as Z 1 @ @ 1 1 f C w.y/ w.y/ wD dSy : 4 r 4 j yj @n.y/ @n.y/ 4 j yj @ Of course, noting that 1 2 L2 .R3 / 4 r for any compact set K, we may write the representation formula of potential theory fully in integral form: Z 1 wD f .y/ dy 4 j yj Z 1 @ @ 1 C w.y/ w.y/ dSy : 4 j yj @n.y/ @n.y/ 4 j yj @ K
N We can of course also In our construction we have assumed that w D 0 in R3 n . patch two non-trivial solutions together in the following way: w WD wC C w ; with w supported in and wC supported in R3 n , where is a bounded domain with smooth boundary. Thus, we have @2 w D f C .n @w /j@ ı@ C w j@ @ .n ı@ / and @2 wC D .n @wC /j@ ı@ wC j@ @ .n ı@ /; where n denotes the exterior normal to the boundary @ of . Note that n is then N Thus we have the exterior normal to the boundary @ of the domain R3 n . @2 w D f Œn @w@ ı@ Œw@ @ .n ı@ / with Œn @w@ WD .n @wC /j@ .n @w /j@ and Œw@ WD wC j@ w j@ as ‘jump terms’ across the boundary @ (from to its complement). The only solution with the asymptotic behavior of (3.2.114) can then be represented by Z 1 wD f .y/ dy 4 j yj Z 1 1 @ Œn @w@ .y/ Œw@ .y/ dSy : 4 j yj @n.y/ 4 j yj @
Section 3.2 Partial Differential Equations in H1 .jD j/
269
Under suitable regularity assumptions consider, for arbitrary data f and jump values g; h (g is vector-valued), the solution of @2 w D f g ı@ h @ .n ı@ / given by Z wD
@
By Gauss’ theorem Z Z f .x/ dx D
0D
@ 1 1 g.y/ h.y/ dSy : 4j yj @n.y/ 4j yj Z
2
and analogously
Therefore,
Z
1 f .y/ dy 4j yj
(3.2.119)
@ w.x/ dx D
Z
@
n.x/ @w .x/ dSx
Z @2 w.x/ dx D
N R3 n
Z
@
n.x/ @wC .x/ dSx :
Z
f .x/ dx D
@
Œn @w@ .x/ dSx :
(3.2.120)
This indicates that the data are not completely arbitrary but must be compatible in the sense of (3.2.120). In order for w to be a solution with jumps g; h respectively, it is necessary to have Z Z
f .x/ dx D
g.x/ dSx : @
If one is interested in a solution in a domain , then the actual choice of an extension to the complement of is irrelevant, and the degrees of freedom introduced by the arbitrariness of the extension can be utilized to adapt the solution in in such a way that it matches certain boundary conditions. For example, assuming f 0 and Œn @w@ D 0, we get Z @ 1 Œw@ .y/ dSy : (3.2.121) wD @n.y/ 4 j yj @ Under suitable assumptions the limit to the boundary can be performed yielding the so-called jump conditions: Z 1 1 @ w j@ D Œw@ C dSy on @ (3.2.122) Œw@ .y/ 2 @n.y/ 4 j yj @ and wC j@
1 D Œw@ C 2
Z Œw@ .y/ @
1 @ dSy @n.y/ 4 j yj
on @:
(3.2.123)
270
Chapter 3 Partial Differential Equations with Constant Coefficients
In other words we have wC j@ C w j@ D 2
Z Œw@ .y/ @
@ 1 dSy @n.y/ 4 j yj
on @:
(3.2.124)
If, for example, wC j@ D dC is given (Dirichlet boundary condition; exterior Dirichlet boundary value problem; for the interior Dirichlet boundary value problem w j@ would have to be prescribed) then Œw@ can be determined as a solution of Z 1 1 @ dSy on @: Œw@ .y/ dC D Œw@ C 2 @n.y/ 4 j yj @ That a solution exists is due to a Fredholm alternative resulting from the compactness of Z 1 @ ' 7! dSy on @ '.y/ @n.y/ 4 j yj @ as an operator in L2 .@/. Similar ideas can be utilized for the Neumann boundary condition, where n @w˙ is prescribed, respectively (exterior and interior Neumann boundary value problem). Example 3.23. It may be illuminating and entertaining to note that similar arguments can be used in the 1-dimensional case. We can even demonstrate the integral equation method in this rather trivial case. The fundamental solution is here given by x 7! 12 jxj. The decomposition of R into a bounded open interval and the unbounded complement now needs five parts: R D 1; a1 Œ [ ¹a1 º [ a1 ; aC1 Œ [ ¹aC1 º [ aC1 ; C1Œ ; a1 < aC1 . Accordingly, we assemble
D 1;a1 Œ .m/ 1 C a1 ;aC1 Œ .m/ 0 C aC1 ;1Œ .m/ C1 with k – say – harmonic in k (with 0 D a1 ; aC1 Œ , 1 D aC1 ; C1Œ and 1 D 1; a1 Œ ). Then the graph of k must be a straight line segment in k for k D 1; 0; C1. Thus, for y 2 R n ¹a1 ; aC1 º we expect (for volume source term f D 0 and using the “surface element” dS somewhat jokingly, since the boundary “integral” is of course nothing but the difference of the evaluations at the boundary points) Z 1
.y/ D .Œ@ .x/ jy xj Œ .x/ sgn jy xj/ dSx 2 @ a1 ;aC1 Œ 1 1 D C Œ@ .a1 / jy a1 j Œ@ .aC1 / jy aC1 j 2 2 1 1 Œ .a1 / sgn.y a1 / C Œ .aC1 / sgn.y aC1 /; 2 2
Section 3.2 Partial Differential Equations in H1 .jD j/
271
where jumps are taken always by subtracting the term stemming from the bounded part from the term stemming from the unbounded part Œ@ .a1 / D @ 1 .a1 / @ 0 .a1 C/; Œ@ .aC1 / D @ C1 .aC1 C/ @ 0 .aC1 /; Œ .a1 / D 1 .a1 / 0 .a1 C/; Œ .aC1 / D C1 .aC1 C/ 0 .aC1 /: Note that the unit normal vector – taken as pointing into the unbounded part – is at a1 just 1 and at aC1 it is C1. As an asymptotic condition52 enforcing unique solvability and the validity of the representation formula we may use here
.x/ C1 jxj C0 sgn.x/ ! 0
as x ! ˙1
(3.2.125)
for some constants C0 ; C1 2 C. In the 1-dimensional case the only possible harmonic polynomials are spanned by x 7! 1 and x 7! x and these are indeed excluded by (3.2.125). Since ˙1 .y/ D @ ˙1 .a˙1 ˙/ .y a˙1 / C ˙1 .a˙1 ˙/, the asymptotic assumption (3.2.125) yields 1 1 C1 D @ C1 .aC1 C/ D @ 1 .a1 / D C Œ@ .aC1 / Œ@ .a1 / 2 2 and C0 D @ C1 .aC1 / aC1 C C1 .aC1 / D @ 1 .a1 / a1 1 .a1 / 1 1 1 1 D C Œ@ .a1 / a1 Œ@ .aC1 / aC1 Œ .a1 / C Œ .aC1 /: 2 2 2 2 With regards to the above mentioned boundary integral equation method, we note here, considering the limits to the boundary, that we have “jump relations” 1 1 1
.a1 / D Œ@ .aC1 / .aC1 a1 / C Œ .a1 / Œ .aC1 /; 2 2 2 1 1 1
.a1 C/ D Œ@ .aC1 / .aC1 a1 / Œ .a1 / Œ .aC1 /; 2 2 2 1 1 1
.aC1 / D C Œ@ .a1 / .aC1 a1 / Œ .a1 / Œ .aC1 /; 2 2 2 1 1 1
.aC1 C/ D C Œ@ .a1 / .aC1 a1 / Œ .a1 / C Œ .aC1 /; 2 2 2 52 Of course in this case the asymptotic behavior is in fact the actual vanishing of .x/ C jxj 1 C0 sgn.x/ (for jxj > max¹ja1 j; jaC1 jº).
272
Chapter 3 Partial Differential Equations with Constant Coefficients
or
.aC1 / C .aC1 C/ 1 D C Œ@ .a1 / .aC1 a1 / 2 2
.a1 / C .a1 C/ 1 D Œ@ .aC1 / .aC1 a1 / 2 2
1 Π.a1 /; 2 1 Π.aC1 /: 2
Œ .a˙1 / to suitably reduce the degrees of freeAssuming Œ@ .a˙1 / D .aC1 1 a1 / dom, we obtain in the so called interior domain case for Dirichlet boundary conditions, i.e. the data .a˙1 / are prescribed, 1 Œ .aC1 /
.aC1 / D :
.a1 C/ 2 Œ .a1 / Œ .a˙1 / we obtain in the so called exteSimilarly choosing Œ@ .a˙1 / D .aC1˙1 a1 / rior domain case for Dirichlet boundary conditions, i.e. .a˙1 ˙/ prescribed, 1 Œ .aC1 /
.aC1 C/ D :
.a1 / 2 Œ .a1 / Thus, we have Œ .aC1 / D ˙2 .aC1 ˙/, Œ .a1 / D 2 .a1 ˙/, where the upper sign refers to the exterior, the lower sign to the interior domain case. Consequently,
.y/ D ˙
1 1
.a1 / jy a1 j ˙
.aC1 ˙/ jy aC1 j .aC1 a1 / .aC1 a1 /
.a1 / sgn.y a1 / ˙ .aC1 ˙/ sgn.y aC1 /; where the upper sign and lower sign provide the solution of the exterior and interior boundary value problem, respectively. To consider Neumann type boundary conditions, i.e. @ .a1 C/ D @ .aC1 / for the interior domain case53 and @ .a1 / D @ .aC1 C/ for the exterior domain case54 , we let Œ D 0 in the representation formula and obtain Z 1
.y/ D Œ@ .x/ jy xj dSx ; 2 @ a1 ;aC1 Œ 1 1 D Œ@ .a1 / jy a1 j C Œ@ .aC1 / jy aC1 j: 2 2 Differentiating we find Z 1 Œ@ .x/ sgn.y x/ dSx ; @ .y/ D 2 @ a1 ;aC1 Œ 1 1 D Œ@ .a1 / sgn.y a1 / C Œ@ .aC1 / sgn.y aC1 /: 2 2
R Note that the equality in this case means @ a1 ;aC1 Π@ .x/ dSx D 0, which is a necessary and as we shall see sufficient condition for solvability. 54 This equality of boundary data is induced by the asymptotic constraint. 53
Section 3.2 Partial Differential Equations in H1 .jD j/
273
Since uniqueness in this case only holds up to a constant, we need to impose further constraints. In the interior domain case we require, for example, Z
.x/ dx D 0; a1 ;aC1 Œ
and in the exterior domain case we impose as an additional requirement a boundary condition at infinity by sharpening the asymptotic constraint to
.x/ C1 jxj ˇ0 sgn.x/ ! 0
as x ! ˙1
for some C1 2 C with ˇ0 2 C prescribed as a “boundary condition at ˙1”. In the interior domain case we get 1 1 @ .a1 C/ D @ .aC1 / D Œ@ .a1 / Œ@ .aC1 / 2 2 and Z 0D
.x/ dx Z
D
a1 ;aC1 Œ
a1 ;aC1 Œ
1 D 2
1 1 Œ@ .a1 / jy a1 j C Œ@ .aC1 / jy aC1 j dx 2 2
Z
a1 ;aC1 Œ
.Œ@ .a1 / .y a1 / C Œ@ .aC1 / .y aC1 // dx
1 .aC1 a1 /2 .a1 aC1 /2 D Œ@ .a1 / Œ@ .aC1 / : 2 2 2 Thus, we have Œ@ .aC1 / Œ@ .a1 /
! D
@ .aC1 / @ .aC1 /
!
and so
.y/ D
1 @ .aC1 / .jy a1 j jy aC1 j/: 2
For the exterior domain case, we may argue in the following way. Here we have 1 1 @ .a1 / D @ .aC1 C/ D Œ@ .a1 / C Œ@ .aC1 /: 2 2
274
Chapter 3 Partial Differential Equations with Constant Coefficients
On the other hand, with Œ D 0 we have 1 1 Œ@ .a1 / C Œ@ .aC1 / 2 2 D @ .a1 C/ D @ .aC1 /
.a1 C/ .aC1 / aC1 a1
.a1 / .aC1 C/ D aC1 a1 .@ .a1 / a1 ˇ0 / .@ .aC1 C/ aC1 C ˇ0 / D aC1 a1 @ .aC1 C/ .a1 C a1 / C ˇ0 D 2 : aC1 a1 D
Solving for the jump terms yields Œ@ .aC1 / C1 C1 D C1 1 Œ@ .a1 /
2
@.aC1 C/ .a1 Ca1 /Cˇ0 aC1 a1
!
@ .aC1 C/
or @ .aC1 C/ .a1 C a1 / C ˇ0 C @ .aC1 C/ aC1 a1 @ .aC1 C/ .a1 C 3a1 / C ˇ0 ; D aC1 a1 @ .aC1 C/ .a1 C a1 / C ˇ0 @ .aC1 C/ Œ@ .a1 / D 2 aC1 a1 @ .aC1 C/ .3a1 C a1 / C ˇ0 : D aC1 a1
Œ@ .aC1 / D 2
Consequently, we have found
.y/ D
@ .aC1 C/ .3a1 C a1 / C ˇ0 jy a1 j 2.aC1 a1 / @ .aC1 C/ .a1 C 3a1 / C ˇ0 jy aC1 j 2.aC1 a1 /
as the solution in this case. Note that @ .y/ D
@ .aC1 C/ .3a1 C a1 / C ˇ0 sgn.y a1 / 2.aC1 a1 / @ .aC1 C/ .a1 C 3a1 / C ˇ0 sgn.y aC1 / 2.aC1 a1 /
Section 3.2 Partial Differential Equations in H1 .jD j/
275
and so @ .aC1 C/ .3a1 C a1 / C ˇ0 2.aC1 a1 / @ .aC1 C/ .a1 C 3a1 / C ˇ0 2.aC1 a1 / @ .aC1 C/ a1 @ .aC1 C/ a1 D˙ .aC1 a1 / .aC1 a1 /
@ .a˙1 ˙/ D ˙
D ˙ @ .aC1 C/: This should suffice as an outline of the ideas. We shall not pursue boundary integral methods further since we will later encounter an – in many respects – more general method to approach boundary value problems in the framework of Hilbert space methods. 3.2.4.3
Electro- and Magnetostatics (Biot–Savart’s Law)
In Subsection 3.2.2.7, a fundamental solution associated with the equation 10 1 0 0 b @> 0 0 0 0 CB b b BEC B @ 0 @ 0 E CB C B MM.0; b @/ B C @H A D B @ 0 b @ 0 b @A@H 0 0 0 0 b @> 0 0
1 J1 C B J2 C C CDB A @ J3 A DW F: 0 1
0
(3.2.126)
of electro- and magnetostatics in free space was found as 1 1.88/ : @/ G0;MM WD MM.0; b 4 r Therefore a solution of (3.2.126) is given by 0
1 0 b @> 0 0 B b C @ 0 C B @ 0 b G0;MM F D g0 MM.0; b @/ F D g0 B CF @ 0 b @ 0 b @A 0 0 b @> 0 1 1 0 0 0 0 0 .b @g0 /> 0 b @> 0 0 C C Bb B b 0 Cb @ g0 0 C @ 0 C B @g B @ 0 b g0 F D B 0 D B C C F: @ 0 b @ 0 b @ g0 0 b @g0 A @ 0 b @A 0 0 b @> 0 0 0 .b @g0 /> 0
276
Chapter 3 Partial Differential Equations with Constant Coefficients
Here 0 b @g0 D
1 4 r 3
1 m1 @ m2 A DW 1 m b 4 r 3 m3
and 1 0 @3 g0 @2 g0 b 0 @1 g0 A @g0 WD @ @3 g0 @2 g0 @1 g0 0 1 0 0 m3 m2 1 @ 1 m3 0 m1 A DW D m b: 3 4 r 4 r 3 m2 m1 0 0
It is worth noting that a specific form of the source term is necessary in order to have that the system (3.2.126) has a chance to be well-posed. Since it is equivalent to curl H D b @ H D J2 ; curl E D b @ E D J3 ; div E D b @> E D J1 ;
div H D b @> H D 0;
we must have that J2 ; J3 satisfy div Jk D 0;
k D 2; 3:
In this case we get 0 G0;MM F D
1 4 r 3 0
D
0 0 Bm Bb 0 @ 0 b m 0 0
1 1 B B r3 4 @
0 m b 0 0
1 0 0C C F 0A 0
1 0 m b J1 C r13 m b J3 C C: 1 A m b J 2 3 r 0
for suitable right-hand sides, e.g. for all data in H1 .jDj/ with compact support. Uniqueness can be enforced as in the case of potential theory. Indeed, a U satisfying MM.0; b @/ U D 0 also must satisfy MM.0; b @/ MM.0; b @/ U D b @2 U D 0. Thus, we obtain the following result (we omit the details of the argument).
Section 3.2 Partial Differential Equations in H1 .jD j/
277
Theorem 3.2.23. Let E; H 2 H1 .jD1 j; jD2 j; jD3 j/ be such that curl H D J2 ;
curl E D J3 ;
div E D J1 ;
div H D 0
(3.2.127)
with J1 ; J2 ; J3 2 H1 .jD1 j; jD2 j; jD3 j/ having compact supports and div J2 D div J3 D 0: Then E 2 C1 .R3 n .supp J1 [ supp J3 //, H 2 C1 .R3 n supp J2 / are given by ED
1 1 m b J1 C m b J3 ; 4 r 3 4 r 3
H D
1 m b J2 4 r 3
and satisfy RnŒR;R .r/ E; RnŒR;R .r/ H 2 H0 .jD1 j; jD2 j; jD3 j/
(3.2.128)
for a sufficiently large R 2 R>0 . A solution of (3.2.127) satisfying the asymptotic constraint (3.2.128) is uniquely determined. In electrostatics one usually assumes J3 D 0 and so the formula for the electric field simplifies to 1 1 J1 ; m b J1 D grad 4 r 3 4 r where J1 is a prescribed charge distribution. Thus, electrostatics yields nothing new compared to potential theory. The formula 1 H D m b J2 4 r 3 for the magnetic field H is a form of the so-called Biot–Savart law describing a magnetic field generated by a current density J2 . To model currents confined to a ‘wire’ the distribution ı is used, where is a curve in R3 modeling the wire. So letting J2 D I t ı , I 2 R, (t denotes the unit tangent to ), the induced magnetic field assumes the more specific form ED
1 m b t ı : 4 r 3 To write this in integral form, we calculate ˇ ˇ ˇ ˇ 1 1 ˇ ˇ m b t ı D I t ı m b I ˇ ˇ 4 r 3 4 r 3 0 0 Z Z 1 DI t.x/ .x y/ .y/ dy dsx 3 R3 4 jx yj Z Z 1 DI t.x/ .x y/ .y/ dsx dy 4 jx yj3 R3 Z Z 1 DI t.x/ .x y/ dsx .y/ dy 3 4 jx yj3 R H DI
278 for all
Chapter 3 Partial Differential Equations with Constant Coefficients 2 CV 1 .R3 n /. Here we have used that Z 1 1 m b D . y/ 3 3 4 r R3 4 j yj
.y/ dy:
Thus, we obtain in R3 n 1 m b t ı D I H DI 4 r 3
Z
1 . y/ t.y/ dsy : 4 j yj3
It is here implicitly assumed that div.t ı / D 0: This condition means that
Z
0 D hdiv.t ı / j i0 D hı j t grad i0 D
t.x/ grad .x/ dsx ;
which is indeed satisfied as follows from the Stokes theorem (dsx , dSx denoting the line element and the surface element, respectively) Z Z t.x/ grad .x/ dsx D n.x/ curl grad .x/ dSx D 0
F
which holds if we assume that is the boundary manifold of an oriented surface F , with n its suitably oriented normal. The integral representation holds therefore for ‘closed wire loops’. 3.2.4.4
Potential Theory in R2
The particular form of the Poincaré equality in the 2-dimensional case already hints at the additional complications of a potential theory in R2 . The solutions of @2 u D 0 are harmonic polynomials and the only harmonic polynomial w with the asymptotic behavior 1 .r/ w 2 H0 .jDj/ (3.2.129) r ln.r/ RnŒR;R at 1 is a constant. The shortcoming of this weight, which according to the Poincaré estimate in R2 is the natural weight to choose, is the failure to produce uniqueness via the asymptotic behavior at 1. Since for f 2 CV 1 .R2 / uD
1 ln.r/ f 2
Section 3.2 Partial Differential Equations in H1 .jD j/
279
is a solutions of @2 u D f; it is also clear that we should not impose a better asymptotic behavior at 1 than 1 .r/ w 2 H0 .jDj/; r .ln.r//2 RnŒR;R
(3.2.130)
in order to maintain existence in all of R2 . Thus, we get unique existence55 only up to addition of an arbitrary constant. We summarize our findings: Theorem 3.2.24. Let u 2 H1 .jD1 j; jD2 j/ and @2 u D f C g;
(3.2.131)
with g 2 C1 .R2 / and f; g 2 H1 .jD1 j; jD2 j/ with supp f compact . Then u 2 C1 .R2 n supp f /: If 1 .r/ u 2 H0 .jD1 j; jD2 j/ r .ln.r//2 RnŒR;R
(3.2.132)
for a sufficiently large R 2 R>0 then solutions of (3.2.131) can differ only by a constant. For g D 0 a solution satisfying (3.2.132) is given by56 uD
1 ln.r/ f C C; 2
where C 2 C is arbitrary. In order to enforce uniqueness, a more subtle asymptotic constraint needs to be found. For a solution u 2 H1 .jD1 j; jD2 j/ of @2 u D f with f having compact support we shall require the existence of a constant C 2 C such that u.x/ C ln.jxj/ ! 0 55
as jxj ! 1:
(3.2.133)
Uniqueness could be enforced by an ‘orthogonality-to-1’ condition: Z 1 u.x/ dx D 0 2 R2 nB 2 .0;R0 / jxj .ln.jxj// R
if we assume that supp f BR2 .0; R0 / for some R0 2 R>1 . The claimed regularity of u outside of supp f follows by Weyl’s Lemma 3.2.15, 3.2.16, in the case of R2 . Alternatively, we could require Z u.x/ dx D 0
for any fixed, bounded open set in the complement of supp f . Due to regularity outside of supp f even evaluation at a point x 2 R2 n supp f could be used to fix the unknown constant. 56 The convolution 1 ln.r/ f is often referred to as the logarithmic potential of f . 2
280
Chapter 3 Partial Differential Equations with Constant Coefficients
Indeed, since @2 u has compact support and u is smooth outside this support, we may choose a cut-off function R 2 CV 1 .R2 / with R D 1 on BR2 .0; R/ supp.@2 u/, such that @2 ..1
R /.m/ u/
D 2 .@
R .m// @u
C .@2
R /.m/ u
DW u 2 CV 1 .R2 /:
Therefore, .1 R /.m/ u D g u C P for some harmonic polynomial P . The asymptotic behavior of the right-hand side is completely determined by the growth behavior of polynomials and the convolution term, which can now be written as a convolution integral Z 1 .g u /.x/ D ln.jx yj/ u .y/ dy 2 R2 Z Z jx yj 1 D ln
u .y/ dy ln.jxj/
u .y/ dy: 2 R2 jxj R2 Since 1
jyj jxj
jxyj jxj
1C
jyj jxj
for jxj > jyj and ln is monotone, we see that Z
.g u /.x/ C ln.jxj/
R2
u .y/ dy ! 0
as jxj ! 1. Thus we have that uP is a solution with the desired asymptotic behavior. But we also know that two solutions uk , k D 0; 1, for the same right-hand side can only differ by a harmonic polynomial, say u0 u1 D P . Assuming that both also satisfy (3.2.133) with certain constants Ck , i.e. uk .x/ Ck ln.jxj/ ! 0
as jxj ! 1;
k D 0; 1;
implies P .C0 C1 / ln.jxj/ ! 0
as jxj ! 1:
Therefore, we must have P D 0 and C0 D C1 , so that u0 D u1 follows. Thus, we have found the following refinement of our previous theorem. Theorem 3.2.25. Let u 2 H1 .jD1 j; jD2 j/ satisfy @2 u D f with f 2 H1 .jD1 j; jD2 j/ having compact support. Then u 2 C1 .R2 n supp f /:
(3.2.134)
Section 3.2 Partial Differential Equations in H1 .jD j/
281
If in addition we require that u satisfies u.x/ C ln.jxj/ ! 0 as jxj ! 1 for some constant C 2 C, then the solution of (3.2.134) is determined uniquely and given by uD
1 ln.r/ f: 2
Remark 3.2.26. Compare the corresponding asymptotic condition in the 1-dimensional case (Example 3.23, (3.2.125)) that some constants C0 ; C1 2 C must exist such that u.x/ C1 jxj C0 sgn.x/ ! 0 as jxj ! 1: (3.2.135) 3.2.4.5
Cauchy’s Integral Formula
It is interesting to note how the theory of analytic functions fits into the setup developed here. Recall that a fundamental solution for the Cauchy–Riemann operator @C 1 is given by gCR WD @C w0 D 2 @C ln.j j/. Indeed, we found that gCR .x1 ; x2 / D
1 2 .x1 C i x2 /
as a regular distribution, which in the usual complex notation (identifying R2 and C via the isometry .x1 ; x2 / 7! z WD x1 C i x2 ) is written as gCR .z/ D
1 : 2 z
Thus, a solution u of @C u D f
(3.2.136)
for f 2 H1 .jD1 j; jD2 j/ with supp f compact is given by 1 f: 2 z Since any solution u of (3.2.136) with f D 0 must also solve uD
@2 u D 0; we have again that such u must be a harmonic polynomial p satisfying additionally @C p D 0: Such polynomials are called analytic polynomials and are the elements of LinC ¹.x1 ; x2 / 7! .x1 C i x2 /k j k 2 Nº: With this observation we see that uniqueness can again be enforced by a suitable decay condition. In summary, we obtain the following result.
282
Chapter 3 Partial Differential Equations with Constant Coefficients
Theorem 3.2.27. Let u 2 H1 .jD1 j; jD2 j/ and @C u D f C g;
(3.2.137)
with g 2 C1 .R2 / and f 2 H1 .jD1 j; jD2 j/ with supp f compact. Then u 2 C1 .R2 n supp f /: If in addition we require RnŒR;R .r/ u 2 H1 .r C i/
(3.2.138)
for a sufficiently large R 2 R>0 then u is uniquely determined. For g D 0 this solution is given by 1 uD f: 2 z It should be noted that a solution of (3.2.137) is also a solution of @2 u D @C f:
(3.2.139)
Conversely, for a solution u of (3.2.139), h WD @C u f must be a polynomial satisfying @C h D 0: (3.2.140) A polynomial h satisfying (3.2.140) is called a conjugate analytic polynomial and must have the form h.x1 ; x2 / D
N X
aj .x1 i x2 /j ;
j D0
which is usually written in complex notation as h.z / WD
N X
aj .z /j :
j D0
Thus, conjugate analytic polynomials are elements of LinC ¹.x1 ; x2 / 7! .x1 i x2 /k j k 2 Nº: We see, in particular, that potential theory in R2 can be considered as a generalization of the theory of the Cauchy–Riemann operator. The theory of analytic functions is particularly concerned with distributions f satisfying @C f D 0
Section 3.2 Partial Differential Equations in H1 .jD j/
283
in a domain , i.e. so called analytic functions in . In other words supp.@C f / \ D ;: From the regularity result of potential theory in R2 (Weyl’s Lemma 3.2.16) we know that f 2 C1 .R2 n supp.@C f //. Let f be analytic in a neighborhood of , i.e. supp.@C f / R2 n . Then f satisfies @C . f / D .@C / f C @C f D .@C / f: If we assume that is a bounded submanifold of R2 with sufficiently smooth boundary @ then we find by Gauss’ theorem h@C j i0 D h j @C i0 Z D @C .x/ dx Z D .n1 .x/ C i n2 .x// .x/ dsx ; @
n1
where n D n2 denotes the exterior normal to the boundary curve @ of , dsx is the line element at x 2 @. The unit tangent vector (in the counterclockwise direction) to this curve is then given by t1 n2 t WD D : t2 n1 Thus we find
Z h@C j i0 D Z
@
@
.n1 .x/ i n2 .x// .x/ dsx .t2 .x/ C i t1 .x// .x/ dsx
and so @C D i .t1 C i t2 / ı@ Using the identification .x1 ; x2 / 7! x1 C i x2 of R2 and C we reuse the notation57 t for t1 C i t2 and so @C D n ı@ D i t ı@ : 57
In this “complex notation” we have i n D t:
284
Chapter 3 Partial Differential Equations with Constant Coefficients
Applying the Solution Theorem 3.2.27, we obtain a representation formula f D
1 1 .t f ı@ / 2i z
for f in in terms of the boundary values of f on @. This is the well-known 1 1 Cauchy integral formula. This becomes apparent if we write the convolution 2i z .t f ı@ / in integral form. We have ˇ ˇ 1 1 f .t ı@ / ˇˇ 2i z 0 ˇ ˇ 1 1 ˇ ı@ ˇ f t 1 D 2i z 0 Z Z 1 1 D f .x/ t.x/ .y/ dy dsx 2i @ R2 x1 C i x2 C y1 i y2 Z Z 1 1 D f .x/ t.x/ dsx .y/ dy; 2i R2 @ x1 C i x2 C y1 i y2
for all
2 CV 1 .R2 n @/. Thus we have Z
1 . f /.y1 ; y2 / D 2i
@
t.x/
1 f .x/ dsx x1 i x2 C y1 C i y2
in R2 n @, which in complex notation with z D x1 C i x2 , dz D dx1 C i dx2 D t.x/ dsx , D y1 C i y2 , becomes 1 f ./ D 2i and 1 0D 2i
Z @
Z @
1 f .z/ dz z
1 f .z/ dz z
in
in C n :
Here we have again written for the corresponding set ¹x Ciy j .x; y/ 2 º in C. To reproduce Cauchy’s integral theorem we only need to make use of Gauss’s theorem: Z
Z
.@C f /.x/ dx D
.n1 .x/ C i n2 .x// f .x/ dsx Z D i .t1 .x/ C i t2 .x// f .x/ dsx @ Z D i f .z/ dz: @
@
Section 3.2 Partial Differential Equations in H1 .jD j/
285
This implies for f analytic in a neighborhood of that Z f .z/ dz D 0: @
We summarize our findings in the following theorem. Theorem 3.2.28. Let w 2 H1 .jD1 j; jD2 j/ be such that @C w D f with f 2 H1 .jD1 j; jD2 j/, supp f compact and RnŒR;R .r/ w 2 H1 .r C i/
(3.2.141)
for a sufficiently large R 2 R>0 . Then wD
1 f: 2 z
Moreover, if R2 is a bounded domain with smooth, compact boundary @ such that supp f \ D ;, then w is (by definition) analytic in a neighborhood of and for .x1 ; x2 / 2 R2 n @ 1 .t w ı@ / .x1 ; x2 / w.x1 ; x2 / D 2 i z Z 1 1 w.z/ dz: D 2 i @ z x1 i x2 For any w analytic in a neighborhood of we also have Z w.z/ dz D 0: @
3.2.4.6
Integral Representations of Solutions of the Helmholtz Equation in R3
Fundamental solutions associated with the Helmholtz operator @2 2 , 2 R, were found as regular distributions g˙i 0 .x/ D
exp.˙ i jxj/ ; 4 jxj
x ¤ 0;
and can be recognized as the Fourier transform g i 0 D .L0 ˝ I g1;˙ /. ; /
286
Chapter 3 Partial Differential Equations with Constant Coefficients
with respect to time of the forward and backward causal fundamental solutions g1;˙ of the wave operator @20 @2 , respectively. The direction of causality, i.e. forward or backward in time, is preserved in the sign occurring in g i 0 . Of course, since .L0 ˝ I g1; /. ; / D .L0 ˝ I g1; /. ; / D g˙i 0 D g i 0 D .L0 ˝ I g1;˙ /. ; / this distinction becomes void if we consider not as a variable, but as a fixed real number. The terms outgoing fundamental solution for gi 0 and incoming fundamental solution for gCi 0 are used as a reminder58 of the time-dependent background. The outgoing and incoming fundamental solution can be detected by the condition m RnŒR;R .jmj/ @ g i 0 ˙ i g i 0 2 H0 .r i/ D L2 .R3 / (3.2.142) jmj for all sufficiently large R 2 R>0 . These two conditions will re-appear below as the outgoing and incoming Sommerfeld radiation condition to enforce uniqueness of solution. Indeed, following our usual line of reasoning, we always have existence of a solution for the equation .@2 2 / u D f; (3.2.143) where f 2 H1 .jD1 j; jD2 j; jD3 j/ and supp f is compact. The convolutions g˙i 0 f provide two solutions for (3.2.143). Uniqueness, therefore, is a more intricate matter. We note that by the reasoning of Lemma 3.2.15, we have smoothness of any solution u of (3.2.143) outside of supp f . Since both fundamental solutions have the same decay behavior as the potential 41 r , we obtain as above that RnŒR;R .r/ g˙i 0 f 2 H1 .r i/
(3.2.144)
for all sufficiently large R 2 R>0 . Thus, we cannot hope to enforce uniqueness by this condition alone. In lieu of (3.2.142), we shall impose either the so-called outgoing Sommerfeld radiation condition m RnŒR;R .jmj/ @ u C i u 2 H0 .r i/ (3.2.145) jmj for all sufficiently large R 2 R>0 or alternatively the incoming Sommerfeld radiation condition m @ u i u 2 H0 .r i/ (3.2.146) RnŒR;R .jmj/ jmj for all sufficiently large R 2 R>0 on a solution u of (3.2.143). That imposing either one of these radiation conditions does not exclude both gi 0 f and gCi 0 f as solutions needs to be justified. 58 Historically, this terminology is used in the reverse order. This corresponds to having g 1;˙ as the Fourier transform of 7! g˙i 0 (rather than the inverse Fourier transform). The above choice is consistent with the present approach and therefore more appropriate for our framework.
Section 3.2 Partial Differential Equations in H1 .jD j/
287
Lemma 3.2.29. Let f 2 H1 .jD1 j; jD2 j; jD3 j/ and supp f compact. Then m RnŒR;R .jmj/ @ g i 0 f ˙ i g i 0 f 2 H0 .r i/ jmj for all sufficiently large R 2 R>0 , 2 R n ¹0º. Proof. It suffices to consider the outgoing case, the remaining case being analogous. So let u WD gi 0 f and 0 2 CV 1 .R3 / real-valued with 0 D 1 on supp f . Then we continue to argue as in the proof of Lemma 3.2.21. We have .@2 2 /gi 0 f D f and .@2 2 / ..1 0 / gi 0 f / D .1 0 / f C 2 @ 0 @gi 0 f C @2 0 gi 0 f D 2 @ 0 @gi 0 f C @2 0 gi 0 f: The right-hand side is an element in CV 1 .R3 /. We calculate with a second real-valued function 1 2 CV 1 .R3 / with 1 D 1 on supp f and 0 D 1 on supp 1 : ˇ ˇ ˇ ˇ ˇ ˇ m ˇ ˇ @ C i .1 0 /.m/ gi 0 f ˇˇ ˇ ˇ jmj 0 ˇ ˇ ˇ ˇ ˇ ˇ m ˇ ˇ ˇ i D ˇ gi 0 f ˇ.1 0 /.m/ @ ˇ jmj 0 ˇ ˇ ˇ ˇ ˇ ˇ m ˇ i D ˇˇ f ˇˇgCi 0 .1 0 /.m/ @ ˇ jmj 0 ˇ ˇ ˇ ˇ ˇ ˇ m ˇ i D ˇˇ 1 f ˇˇgCi 0 .1 0 /.m/ @ ˇ jmj 0 ˇ ˇ ˇ ˇ m ˛ ˇ ˇ i jf j˛ ˇ.@ C e/ 1 .m/gCi 0 .1 0 /.m/ @ ˇ jmj 0 for some ˛ 2 N 3 . Considering the last term ˇ ˇ2 ˇ ˇ ˇ ˇ.@ C e/˛ 1 .m/ gCi 0 .1 0 /.m/ @ m i ˇ ˇ jmj 0 Z ˇ Z ˇ exp.i j yj/ ˇ.@ C e/˛ 1 .m/ .1 0 .y// D ˇ 3 4j yj R R3 ˇ2 ˇ m @ i .y/ dy .x/ˇˇ dx: jmj
288
Chapter 3 Partial Differential Equations with Constant Coefficients
By integration by parts we find exp.i jx yj/ m .1 0 .y// @ i .y/ dy 4 jx yj jmj R3 Z exp.i jx yj/ exp.i jx yj/ .x y/ y .1 0 .y// C y @ 0 .y/ D 3 4 jx yj jyj 4 jx yjjyj R3 exp.i jx yj/ .x y/ y C i .1 0 .y// 1C .y/ dy 4 jx yj jx yjjyj
Z
and so – using that .a C b C c/2 3.a2 C b 2 C c 2 / – we can estimate ˇ ˇ2 ˇ ˇ ˇ ˇ.@ C e/˛ 1 .m/ gCi 0 .1 0 /.m/ @ m i ˇ ˇ jmj 0 Z ˇ2 Z ˇ ˇ ˇ exp.i j yj/ ˛ ˇ . y/y.1 0 .y// .y/ dy .x/ˇˇ dx 3 .@ C e/ 1 ˇ 3 R3 R3 4j yj jyj Z ˇ2 Z ˇ ˇ ˇ exp.i j yj/ ˛ ˇ.@ C e/ 1 C3 y@ 0 .y/ .y/ dy .x/ˇˇ dx ˇ 4j yjjyj R3 R3 Z Z ˇ ˇ .x y/ y exp.i j yj/ ˛ ˇ C 3j j ˇ.@ C e/ 1 3 4j yjjyj 1 C jx yjjyj .1 0 .y// R3 R ˇ2 ˇ .y/ dy .x/ˇˇ dx: Noting that ˇ ˇ ˇ .@ C e/˛ 1 .m/ exp.i j yj/ 1 C . y/ y .x/ ˇ j yj j yjjyj ˇ2 ˇ .1 0 .y//@ 0 .y/ ˇˇ dy dx < 1;
Z
Z R3
R3
ˇ ˇ2 ˇ ˇ ˇ .@ C e/˛ 1 .m/ exp.i j yj/ . y/ .x/ .1 0 .y// ˇ dy dx ˇ ˇ 4 j yj3
Z
Z R3
R3
< 1; and also Z
Z R3
R3
ˇ ˇ2 ˇ ˇ ˇ .@ C e/˛ 1 .m/ exp.i j yj/ .x/ @ 0 .y/ ˇ dy dx < 1; ˇ ˇ j yj
Section 3.2 Partial Differential Equations in H1 .jD j/ we find59
ˇ ˇ m ˇ @ C i .1 0 / gi 0 f ˇ jmj
289
ˇ ˇ ˇ ˇ ˇ jf j˛ C0 j j0 ˇ ˇ ˇ 0
2 CV 1 .R3 / and some C0 2 R>0 independent of . This proves that m @ C i .1 0 / gi 0 f 2 H0 .r C i/ jmj
for all
and so in particular m RnŒR;R .r/ @ C i gi 0 f jmj m @ C i .1 0 / gi 0 f 2 H0 .r C i/ D RnŒR;R .r/ jmj
for a (and therefore all) sufficiently large R 2 R>0 . The radiation condition now implies uniqueness in an intricate way. First we observe the following result. Lemma 3.2.30. Let u 2 H1 .jD1 j; jD2 j; jD3 j/ be a solution of .@2 k 2 / u D 0; If either
or
59
k 2 R n ¹0º:
(3.2.147)
m @ u C ik u 2 H0 .jD1 j; jD2 j; jD3 j/ jmj m @ u i k u 2 H0 .jD1 j; jD2 j; jD3 j/ jmj
For the second integral the required decay for large jyj and arbitrary x 2 supp 1 follows from .x y/ y jx yjjyj C .x y/ y 1C D jx yjjyj jx yjjyj
for all y 2 R3 n ¹0º.
D
jx yjjyj .x y/ .x y/ .x y/ x C jx yjjyj jx yjjyj
D
.jyj jx yj/ .x y/ x C jyj jx yjjyj
2
jxj jyj
2
sup¹jxj j x 2 supp 1 º jyj
290
Chapter 3 Partial Differential Equations with Constant Coefficients
then u 2 H0 .jD1 j; jD2 j; jD3 j/: Proof. We again restrict our attention to the outgoing case (the incoming case following by replacing k by k). Noting that u 2 C1 .R3 /, we can calculate classically Z u.x/ .@2 k 2 /u.x/ dx 0D B
Z
R3
.0;R/
Z
D
B
@u.x/ @u.x/ dx k
R
Z
C
.0;R/ 3
u.x/
@B
R3
.0;R/
2
u.x/ u.x/ dx B
.0;R/ 3
R
x @u.x/ dSx ; jxj
which implies, by taking imaginary parts (recall 2 R), that Z x @u.x/ dSx D 0: u.x/ Im jxj @B 3 .0;R/ R
Integration with respect to R 2 R>0 yields Z x @u.x/ dx 0 D Im u.x/ jxj B 3 .0;R/ R Z x D Im @u.x/ C i k u.x/ dx u.x/ jxj B 3 .0;R/ R Z ik u.x/ u.x// dx B
R3
.0;R/
and so, applying the Cauchy–Schwarz inequality, Z jkj ju.x/j2 dx B
R3
.0;R/
sZ
B
R3
.0;R/
v uZ u ju.x/j2 dx t
B
R3
.0;R/
ˇ ˇ2 ˇ ˇ x ˇ dx: ˇ @u.x/ C i k u.x/ ˇ ˇ jxj
Letting R ! 1 we obtain
ˇ ˇ ˇ 1 ˇ m @ u C i k uˇˇ juj0 p ˇˇ k jmj 0
and in particular u 2 H0 .jD1 j; jD2 j; jD3 j/:
Section 3.2 Partial Differential Equations in H1 .jD j/
291
This lemma still does not yield uniqueness, but only that solutions of the homogeneous problem satisfying the prescribed asymptotic behavior feature an even stronger decay. The missing link to proving uniqueness is the following result. Lemma 3.2.31. Let u 2 H1 .jD1 j; jD2 j; jD3 j/ and supp.@2 k 2 /u BR3 .0; R0 / for some R0 2 R>0 . Then either supp u BR3 .0; R0 / or60 Z ju.x/j2 dx p R B
R3
.0;R/nB
R3
.0;R0 /
for some p 2 R>0 and all sufficiently large R 2 R>0 . Proof. We follow the presentation in [29], p. 59–60. Let R0 2 R>0 such that supp.@2 k 2 / u BR3 .0; R0 /: Then using spherical harmonics .Sn;j /j D0;:::;2n; n2N , i.e. the system of eigenfunctions of the Laplace–Beltrami operator 0 , which form a complete orthonormal system in L2 .@BR3 .0; 1// with respect to the inner product Z . ; / 7!
.x/ .x/ dSx ; @B
R3
.0;1/
we find, calculating in polar coordinates, that Z vn;j .r/ WD r u.rx/ Sn;j .x/ dSx @B
R3
.0;1/
satisfies
n.n C 1/ 2 @ vn;j .r/ C k vn;j .r/ r2 2 Z @ 2 @ D .ru.rx// C k 2 r u.rx/ Sn;j .x/ dSx .r u.rx// C @r r @r @B 3 .0;1/ R Z 1 u.rx/ n.n C 1/ Sn;j .x/ dSx @B 3 .0;1/ r R 2 Z @ 2 @ 2 D .r u.rx// C .ru.rx// C k r u.rx/ Sn;j .x/ dSx @r r @r @B 3 .0;1/ R Z 1 C u.rx/ 0 Sn;j .x/ dSx r @B 3 .0;1/ 2
R
60
This type of estimate is referred to as Rellich’s estimate.
292
Chapter 3 Partial Differential Equations with Constant Coefficients
Z D
@B
R3
.0;1/
2
2 @ .ru.rx// r @r 1 C k 2 r u.rx/ C 0 u.rx/ Sn;j .x/ dSx r @ @r
.r u.rx// C
Z D r
@B
.0;1/ 3
..@2 k 2 / u.rx// Sn;j .x/ dSx D 0
R
for r > R0 . Here we have used that 2 1 @ 2 @ 2 C 2 0 C D@ D @r r @r r and that Sn;j is an eigenfunction of 0 associated with the eigenvalue n.n C 1/, j D 0; : : : ; 2n, n 2 N. Now if u does not eventually vanish identically then there must be a g WD vn0 ;j0 , j0 2 ¹0; : : : 2n0 º; n0 2 N, and a radius R1 > R0 such that g.R1 / ¤ 0:
(3.2.148)
Since g must satisfy the ordinary differential equation n0 .n0 C 1/ @2 g.r/ C k 2 g.r/ D 0; r2 we know that g is a linear combination of the associated fundamental system of modp p ified Bessel functions r Jn0 C1=2 .k r/, r Yn0 C1=2 .k r/ (compare [30], p. 139) associated with this equation. Thus we have p p g.r/ D a1 r Jn0 C1=2 .k r/ C a2 r Yn0 C1=2 .k r/ (with a1 a2 ¤ 0 because of (3.2.148)!). The asymptotic behavior of the Bessel functions for large arguments is known to be p n0 C 1 1 C O.r 1 /; r Jn0 C1=2 .k r/ D p cos kr 2 k p n0 C 1 1 C O.r 1 /; r Yn0 C1=2 .k r/ D p sin kr 2 k as r ! 1. Since sin and cos are orthogonal in L2 .I / where I is a period interval, we have Z n0 C 1 n0 C 1 sin kr dr cos kr 2 2 ŒR0 ;R Z 1 cos.u/ sin.u/ du D 0 D k kŒR0 ;R .n0 C1/=2
Section 3.2 Partial Differential Equations in H1 .jD j/ if k.R R0 / 2 2N. Therefore, we obtain Z Z 2 ju.x/j dx B
.0;R/nB 3
R
.0;R0 / 3
ŒR0 ; R
R
293
jg.r/j2 dr
1 j.a1 ; a2 /j2 N C O.ln.R// 2 2 k 2 1 k D .R R0 / C O.ln.R// j.a1 ; a2 /j2 2 2 k 2 2 1 j.a1 ; a2 /j2 R 8 3 k D
N C R0 2 for R D 2 k R 2 R>2=k
2 N k
C R0 , N 2 N, sufficiently large. Since for arbitrary
2 2 R k k
k .R R0 / C R0 R; 2
we have Z B
Z R3
.0;R/nB
R3
.0;R0 /
ju.x/j2 dx
B
R3
.0;R0 /nB
R3
.0;R0 /
ju.x/j2 dx
1 j.a1 ; a2 /j2 R0 8 3 k 1 j.a1 ; a2 /j2 R; 16 3 k
k for R > 2 , R > R0 sufficiently large and R0 WD 2 b 2 .R R0 /c C R0 . Therefore k k 1 2 the desired estimate follows with p WD 16 3 k j.a1 ; a2 /j > 0.
We can now derive a solution theorem for Helmholtz equation. Theorem 3.2.32. An element u 2 H1 .jD1 j; jD2 j; jD3 j/ solving the equation .@2 k 2 / u D f;
k 2 R n ¹0º;
with f 2 H1 .jD1 j; jD2 j; jD3 j/ such that supp f is compact and satisfying the Sommerfeld radiation condition m RnŒR;R .jmj/ @ u i k u 2 H0 .jD1 j; jD2 j; jD3 j/ (3.2.149) jmj for a sufficiently large R 2 R>0 , exists uniquely and is given by u D gkCi 0 f:
294
Chapter 3 Partial Differential Equations with Constant Coefficients
Proof. The existence of a solution u is clear, since we know by Lemma 3.2.149 that u D gkCi 0 f is such a solution. Now let w be another solution satisfying (3.2.149) with u replaced by w. Then the difference v WD u w satisfies .@2 k 2 / v D 0 and (3.2.149) with v in place of u. Since v must be a smooth function, we also have that the assumptions of Lemma 3.2.30 are satisfied and so v 2 H0 .jD1 j; jD2 j; jD3 j/: According to Lemma 3.2.31 we therefore must have that v D 0 everywhere. Similar representation results to those developed for potential theory can be given for the Helmholtz equation. Theorem 3.2.33. Let w 2 H1 .jD1 j; jD2 j; jD3 j/ be such that the radiation condition (3.2.149) holds for a sufficiently large R 2 R>0 , with wjN continuously differentiable in a neighborhood of the smooth, compact boundary @ of a domain R3 and @2 w k 2 w D f C
ˇ 1 1 ˇ .n @w/ˇ@ ı@ C 2 w ˇ@ @ .n ı@ / 2 2 2
(3.2.150)
with f 2 H0 .jD1 j; jD2 j; jD3 j/, supp f compact. Then w D gkCi 0 f C gkCi 0 ..n @w/j@ ı@ C wj@ @ .n ı@ // or Z
wD
exp.i k j yj/ f .y/ dy 4 j yj Z exp.i k j yj/ @ @ exp.i k j yj/ C w.y/ w.y/ dSy : 4 j yj @n.y/ @n.y/ 4 j yj @
The method of boundary integral equations indicated for the potential theoretical case can be utilized analogously. We shall, however, not pursue these ideas here. Example 3.24. For its entertainment value we shall, however, in continuation of our considerations in Example 3.23 have a look at the 1-dimensional case to see what a suitable radiation condition might be in this case. For > 0 we have x 7! .i / jxj/ exp.i as fundamental solution for @2 C .i C /2 . Thus, we have 2 i .i / ui D
1 exp.i . i / j j/ f 2 i . i /
Section 3.2 Partial Differential Equations in H1 .jD j/
295
as the solution of .@2 . i /2 /u D f: Here ui is the evaluation at 2 R n ¹0º of the Fourier–Laplace transform of the (forward causal!) solution w of .@20 @2 /w D ı ˝ f:
(3.2.151)
Letting ! 0C, we see that ui0C D 2 1i exp.i j j/ f is a solution of .@2 2 /u D f . In the same way we can derive uCi 0C D 2 1i exp.i j j/ f as the only other solution of .@2 2 /u D f resulting from a backward causal solution of (3.2.151). To enforce uniqueness we need to distinguish these two solutions. Taking our lead from the above 3-dimensional considerations, we shall assume that f has compact support and find that the following radiation conditions hold: @u i0C .x/ ˙ i sgn.x/ u i0C .x/ ! 0 as x ! C1 and x ! 1. This shows that ui0C is characterized by what we shall call the outgoing radiation condition @ui0C .x/ C i sgn.x/ ui0C .x/ ! 0 as x ! ˙1, whereas uCi0C is characterized by an analogous incoming radiation condition @uCi0C .x/ i sgn.x/ uCi0C .x/ ! 0 as x ! ˙1. Note that it suffices to require these conditions only at C1 (or at 1) to ensure the desired uniqueness. Since .@2 2 / D .i @/.i C @/, we are also led to consider – say – @ C i for 2 R n ¹0º. Here the (outgoing) fundamental solution is x 7! R>0 .x/ exp.i x/ D
1 .@ i /.x 7! exp.i jxj//: 2i
We read off (with w D .@ i /u) from the above that w.x/ ! 0 as x ! 1 now characterizes the outgoing solution of .@ C i /w D f , whereas the same asymptotic behavior as x ! C1 characterizes an incoming solution. Consider finally the system of two equal Helmholtz type equations
i @ 0 0 i C @
.i C @/u .i @/u
D
f f
296
Chapter 3 Partial Differential Equations with Constant Coefficients
or equivalently (by applying unitary and idempotent matrix 1 2
1 C1 1 1
f f
D
f 0
D
i @ @ i
p1 1 1 2 1 1
)
i u : @u
We read off (with u2 D @u, u1 D i u) that the outgoing radiation condition now assumes the form u2 .x/ C sgn.x/ u1 .x/ ! 0 as x ! ˙1. The incoming radiation condition is now of the form u2 .x/ sgn.x/ u1 .x/ ! 0 as x ! ˙1. Recall that it suffices to require these only for x ! C1 (or for x ! 1) to ensure uniqueness of a corresponding outgoing or incoming solution. Note also that these asymptotic conditions resemble the Silver–Müller radiation conditions, which will be discussed in connection with Maxwell’s equation, see Subsection 3.2.4.8 below. 3.2.4.7
Retarded Potentials
This example brings us back to the invertible case, indeed the reversibly evolutionary case. We return briefly to the wave equation. We mention it here, not because it particularly requires the extension of the solution theory we have been illustrating in the above, but because of its intimate relation to the previous cases of @2 (potential theory, Laplace equation, Poisson equation) and of @2 k 2 (Helmholtz equation). We found that the (forward causal) fundamental solution of the wave equation in R1C3 is given by g1;C .t; / D
1 .t / ı t ŒS 2 4 t R>0
for t 2 R; t ¤ 0:
We want to record a suitable integral representation also in this context. We find for – say – f 2 CV 1 .R1C3 / that j f i0;0 .g1;C f /.t; x/ D h1 .t;x/ g1;C D hg1;C j .t;x/ 1 f i0;0 Z 1 1 D f .t ju xj; u/ du: 4 y2R3 ju xj
Because of the similarity of the integral expression to a potential term and, observing the retardation juxj in the time-dependence of f , an integral expression of this form is referred to as a retarded potential. It gives an integral expression for the convolution with the fundamental solution of the wave equation. Of course, in the backward causal case, we obtain similarly an advanced potential type integral expression.
Section 3.2 Partial Differential Equations in H1 .jD j/ 3.2.4.8
297
Integral Representations of Solutions of the Time-
