Sobolev spaces

275 Pages·1975·5.176 MB·iii-xviii, 1-268\275

by  Robert A. Adams

SOBOLEV SPACES
ROBERT A. ADAMS
Department of Mathematics
The University of British Columbia
Vancouver, British Columbia, Canada
A C A D E M I C P R E S S New York San Francisco London 1975
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT (B 1975, BY ACADEMPIRCE SS, INC.
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
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Library of Congress Cataloging in Publication Data
Adams, Robert A
Sobolev spaces.
(Pure and applied mathematics series; v. 65)
Bibliography: p.
Includes index.
1. Sobolev spaces. I. Title.
11. Series: Pure and applied mathematics; a series
of monographs and textbooks; v. 65
QA3.P8 [QA323] 51W.8~[5 15'.7] 74-17978
ISBN 0-12-044150-0
AMS (MOS)1 970 Subject Classifications: 46E30,46E35
PRINTED IN THE UNITED STATES OF AMERICA
Preface
This monograph is devoted to a study of properties of certain Banach
spaces of weakly differentiable functions of several real variables which arise
in connection with numerous problems in the theory of partial differential
equations and related areas of mathematical analysis, arid which have become
an essential tool in those disciplines. These spaces are now most often asso-
ciated with the name of the Soviet mathematician S. L. Sobolev, though their
origins predate his major contributions to their development in the late 1930s.
Sobolev spaces are very interesting mathematical structures in their own
right, but their principal significance lies in the central role they, and their
numerous generalizations, now play in partial differential equations.
Accordingly, most of this book concentrates on those aspects of the theory of
Sobolev spaces that have proven most useful in applications. Although no
specific applications to problems in partial differential equations are discussed
(these are to be found in almost any modern textbook on partial differential
equations), this monograph is nevertheless intended mainly to serve as a
textbook and reference on Sobolev spaces for graduate students and researchers
in differential equations. Some of the material in Chapters 111-VI has grown
out of lecture notes [ 181 for a graduate course and seminar given by Professor
Colin Clark at the University of British Columbia in 1967-1968.
The material is organized into eight chapters. Chapter I is a potpourri of
standard topics from real and functional analysis, included, mainly without
Xi
Xii PREFACE
proofs, because they form a necessary background for what follows. Chapter
I1 is also largely “background” but concentrates on a specific topic, the
Lebesgue spaces LP(R), of which Sobolev spaces are special subspaces. For
completeness, proofs are included here. Most of the material in these first
two chapters will be quite familiar to the reader and may be omitted, or simply
given a superficial reading to settle questions of notation and such. (Possible
exceptions are Sections 1.25-1.27, 1.31, and 2.21-2.22 which may be less
familiar.) The inclusion of these elementary chapters makes the book fairly
self-contained. Only a solid undergraduate background in mathematical
analysis is assumed of the reader.
Chapters III-VI may be described as the heart of the book. These develop
all the basic properties of Sobolev spaces of positive integral order and
culminate in the very important Sobolev imbedding theorem (Theorem 5.4)
and the corresponding compact imbedding theorem (Theorem 6.2). Sections
5.33-5.54 and 6.12-6-50 consist of refinements and generalizations of these
basic imbedding theorems, and could be omitted from a first reading.
Chapter VII is concerned with generalization of ordinary Sobolev spaces
to allow fractional orders of differentiation. Such spaces are often involved in
research into nonlinear partial differential equations, for instance the Navier-
Stokes equations of fluid mechanics. Several approaches to defining fractional-
order spaces can be taken. We concentrate in Chapter VII on the trace-
interpolation approach of J. L. Lions and E. Magenes, and discuss other
approaches more briefly at the end of the chapter (Sections 7.59-7.74). It is
necessary to develop a reasonable body of abstract functional analysis (the
trace-interpolation theory) before introducing the fractional-order spaces.
Most readers will find that a reading of this material (in Sections 7.2-7.34,
possibly omitting proofs) is essential for an understanding of the discussion
of fractional-order spaces that begins in Section 7.35.
Chapter VIII concerns Orlicz-Sobolev spaces and, for the sake of com-
pleteness, necessarily begins with a self-contained introduction to the theory
of Orlicz spaces. These spaces are finding increasingly important applications
in applied analysis. The main results of Chapter VIII are the theorem of N. S.
Trudinger (Theorem 8.25) establishing a limiting case of the Sobolev imbed-
ding theorem, and the imbedding theorems of Trudinger and T. K. Donaldson
for Orlicz-Sobolev spaces given in Sections 8.29-8.40.
The existing mathematical literature on Sobolev spaces and their general-
izations is vast, and it would be neither easy nor particularly desirable to
include everything that was known about such spaces between the covers of
one book. An attempt has been made in this monograph to present all the core
material in sufficient generality to cover most applications, to give the reader
an overview of the subject that is difficult to obtain by reading research
papers, and finally, as mentioned above, to provide a ready reference for
PREFACE xiii
someone requiring a result about Sobolev spaces for use in some application.
Complete proofs are given for most theorems, but some assertions are left for
the interested reader to verify as exercises. Literature references are given in
square brackets, equation numbers in parentheses, and sections are numbered
in the form m.n with m denoting the chapter.
Acknowledgments
We acknowledge with deep gratitude the considerable assistance we have
received from Professor John Fournier in the preparation of this monograph.
Also much appreciated are the helpful comments received from Professor
Bui An Ton and the encouragement of Professor Colin Clark who originally
suggested that this book be written. Thanks are also due to Mrs. Yit-Sin Choo
for a superb job of typing a difficult manuscript. Finally, of course, we accept
all responsibility for error or obscurity and welcome comments, or
corrections, from readers.
xv
List of Spaces and Norms
The numbers at the right indicate the sections in which the symbols are
introduced. In some cases the notations are not those used in other areas
of analysis.
II * Ile 7.2
II1I * ;B 1 +B2II 7.11
;B SPP(R")II 7.67
*
II *;Bs*p(a)ll 7.72
1.1 1.1
1.25
1.25
1.25
1.25
1.25
1.26
1.27
5.2
8.37
1.51
1.52
7.7, 7.9
xviii LlST OF SPACES AND NORMS
7.29
8.14
3.1
3.12
7.33
8.7
1-40, 1.46
1.53
2.1
3.1
2.5
5.22
7.3
7.3
7.5
7.62
7.66
8.9
1.1
7.14
7.24, 7.26, 7.32
7.32
7.35
7.43
3.1
3.1
3.11
6.54
7.11
7.30
7.36,7.39
7.39
7.48
7.49
8.27
8.27
8.27
1.4, 1.6, 6.51
1.5,1.10
1.22
I
Introductory Topics
Notation
1.1 Throughout this monograph the term domain and the symbol R shall be
reserved for an open set in n-dimensional, real Euclidean space R". We shall
be concerned with differentiability and integrability of functions defined on
Q-these functions are allowed to be complex valued unless the contrary is
stated explicitly. The complex field is denoted by @. For c E @ and two
+
functions u and u the scalar multiple cu, the sum u 11, and the product uu are
always taken to be defined pointwise as
(4
( 4 = cu(x),
+
(u+u)(x) = u(x) u(x),
(uu) (x) = u(x),
at all points x where the right sides make sense.
(Az ty=pic,al p oint in R" is denoted by x = (xl, ..., x,,); its norm 1x1 =
x~')~''T. he inner product of x and y is x.y = xjyj.
If 01 = (a1, ..., E,) is an n-tuple of nonnegative integers aj, we call a a
multi-index and denote by xu the monomial x";' ..ex:, which has degree
la1 = C'j= aj. Similarly, if Dj = i?/axj for 1 sj -< n, then
D' = D"1 ...D:
denotes a differential operator of order Jal. D(o*.-*=o )u.u
1
2 I INTRODUCTORY TOPICS
If a and fl are two multi-indices, we say PI c1 provided /3, I aj for
1 ~j s n. In this case a-B is also a multi-index and la-/3I + 1/31 = IQI. We
also denote
...
a! = a,! a"!
and if fl I; a,
The reader may wish to verify the Leibniz formula
valid for functions u and u that are la1 times continuously differentiable near x.
1.2 If G c R", we denote by G the closure of G in R". We shall write G c c R
provided G c R and G is a compact (Lea,c losed and bounded) subset of R".
If u is a function defined on G, we define the support of u as
SUPPU = {xEG:u(x)#O}.
We say that u has compact support in R if supp u c c R. We shall denote by
"bdry G" the boundary of G in R", that is, the set G n @where G' = R" G =
N
{x E R" :x 4 G} is the complement of G.
If x E R" and G c R", we denote by "dist(x, G)" the distance from x to G,
that is, the number inf,,,lx-yl. Similarly, if F, G c R",
dist(F,G) = infdist(y,G) = inflx-yl.
Y ~ F XEG
YEF
Topological Vector Spaces
1.3 We assume that the reader is familiar with the concept of a vector space
over the real or complex scalar field, and with the related notions of dimension,
subspace, linear transformation, and convex set. We also assume familiarity
with the basic concepts of general topology, Hausdorff topological spaces,
weaker and stronger topologies, continuous functions, convergent sequences,
topological product spaces, subspaces, and relative topology.
Let it be assumed throughout this monograph that all vector spaces referred
to are taken over the complex field unless the contrary is explicitly stated.
1.4 A topological vector space, hereafter abbreviated TVS, is a Hausdorff
topological space that is also a vector space for which the vector space oper-