This book presents the theory of functions spaces, now known as Sobolev spaces, which are widely used in the theory of p
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S.L. Sobolev
Applications of Functional Analysis in Mathematical Physics
T ra n sla tio n s o f M a t h e m a t ic a l M o n o g r a p h s
V o lu m e
Applications of
FUNCTIONAL ANALYSIS in Mathematical Physics by
S. L. Sobolev
Am erican M athem atical Society Providence, Rhode Is la n d 1963
7
HEKOTOPLIE FIPHMEHEHH.fi OyHKUHOHAJILHOrO AHAJIH3A B MATEMATHHECKOH OH3HKE C. JI. COBOJIEB H3flaTejibCTBO ilenmirpajiCKoro r ocyAapcTBeHHoro y HHBepcHTeTa
JleiiHHrpafl 1950 T r a n s la te d from the Russian by F. E. Browder
Publication aided by gra nt N S F - G 12381 from the NATIONAL SCIE N CE FOUNDATION
Library of Congress Card Number 63-15658 Standard Book Number 821-81557-1
Co py rig ht © 1963 by the American M a th e m a ti c a l Society All rights reserved. No portion of this book may be reproduced without the written permission of the publisher.
Second Printing, I9&1 Third Printing, 1969 Printed in the United States of America
Contents
A uthor’s P kkface C hapter I. S pecial
vii pkohlkms of functional analysis
§1. I n t r o d u c t i o n
1
1. Summahle functions (1). 2. The Holder and Minkowski inequalities (d). .'3. The reverse of the Holder and Minkowski inequalities (7). §2. B asi c p r o p e r t i e s o f t h k s lacks Lp 1. Norms. Definitions (9). 2. rI’he Riesz-Fischer Theorem (11). d. Con tinuity in the large of functions in Lr, (11). 4. Countable dense nets (Id).
9
§d. L i n k a r f u n c t i o n a l s on L,, 1. Definitions. Boundedness of linear functionals (l(i). 2. Clarkson’s in equalities (17). d. Theorem on the general form of linear functionals (22). 4. Convergence of functionals (2.7).
lb
§4. C o m p a c t n e s s o f s p a c e s 1. Definition of compactness (28). 2. A theorem on weak compactness (29). d. A theorem on strong compactness (dO). 4. Proof of the theorem on strong compactness (d l).
28
§7. G e n e r a l i z e d d e r i v a t i v e s 1. Basic definitions (dd). 2. Derivatives of averaged functions (d.7). d. Rules for differentiation (d7). 4. Independence of the domain (d9).
dd
§6. P r o p e r t i e s o f i n t e g r a l s o f p o t e n t i a l t y p e 1. Integrals of potential type. Continuity (42). 2. Membership in Lu- (4d).
42
§7. The
s p a c e s Lp] a n d Wjj' 1. Definitions (47). 2. The norms in Lp' (40). d. Decompositions of Wp' and its norming (48). 4. Special decompositions of W{p (70).
47
§8. I m b e d d i n g T h e o r e m s 1. The imbedding of Wjl1in C (70). 2. Imbedding of Wjj1 in Ltl- (77). d. e x amples (78).
70
iii
Contents
IV §9. G e n e r a l
methods of norming
V/'p
and c o r o l l a r i e s of t h e
I mbedding
60
THEOREM
1. A theorem on equivalent norms (60). 2. The general form of norms equivalent to a given one (62). 3. Norms equivalent to the special norm (64). 4. Spherical projection operators (64). 5. Nonstar-like domains (66). 6. Examples (67). §10. S ome
c o n s e q u e n c e s of t h e
I mbedding T he or em
68
1. Completeness of the space (68). 2..The imbedding of Wp] in (69). 3. Invariant norming of Wjj* (72). §11. T h e
c o mp l et e c on t in u i t y of t h e i mb e d d i n g o p e r a t o r
( K o n d r a s e v ’s
74
theorem)
1. Formulation of the problem (74). 2. A lemma on the compactness of the special integrals in C (75). 3. A lemma on the compactness of integrals in Lq• (77). 4. Complete continuity of the imbedding operator in C (82). 5. Complete continuity of the operator of imbedding in Lq■ (84).
Chapter II. Variational methods
in mathematical physics
§12. T h e D i r i c h l e t p r o b l e m 1. Introduction (87). 2. Solution of the variational problem (88). 3. Solu tion of the Dirichlet problem (91). 4. Uniqueness of the solution of the Dirichlet problem (94). 5. Hadamard’s example (97).
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§13. T he N eumann p r ob l em 1. Formulation of the problem (99). 2. Solution of the variational problem (100). 3. Solution of the Neumann problem (101).
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§14. P o l y h a r m o n i c e q u a t i o n s 103 1. The behaviour of functions from on boundary manifolds of various dimensions (103). 2. Formulation of the basic boundary value problem (105). 3. Solution ofthe variational problem (106). 4. Solution of the basic boundary value problem (108). §15. U n i q u e n e s s
of t h e s o l u t i o n of t h e basi c b o u n d a r y v a l u e p r o b l e m
FOR TH E POLYHARMONIC EQUA TIO N
112
1. Formulation of the problem (112). 2. Lemma (112). 3. The structure of the domains SI/, —SLy, (115). 4. Proof of the lemma for /c and using (1.10), we obtain
f x (Q) y (Q) P ( Q ) r f i > < 1.
(1.14)
Let X(Q) and Y(Q) now be two a r b it r a r y functions on Q inte g r a t e respectively to the powers p and p ' . T h e n for the functions x (Q)
|X($)I
.
[J \Xf Pdv^p
l n$)i
y{Q) =
[ [ 1Y \ P ' P d v y
the inequality (1.13) is valid, a n d consequently, we have the inequality (1.14), which after simplification takes the form
J I X (Q ) | • | F ( Q ) | P
(Q) dv < [ J \ X \ p P d v ^ >
*[J| Y\p ' P d v Y
,
from which follows the Holder inequality | / x « J ) Y(Q )P(Q )dv
[ J \x\p p d v^p y. X [ / | Y\p' P d v ] p \
(L15)
L 2
It is obvious t h a t the inequality sign can only hold in (1.14) if for almost all Q w e h a v e t h e e q u a l t i y xp= yp . Consequently, in the inequality (1.15) the equality sign holds only in the case in which -7--------------- =
J \ X\ PPdv
J |Y\P‘ P d v
2
2
> sign X Y = const.
almost everywhere, i.e., if the functions | X | P and | Y | p differ almost everywhere merely by a c ons ta nt factor and X and Y have almost everywhere the same sign. Fr om (1.15) follows the generalized Holder inequality for several func tions. Let Xj-f X2+ • • • + X*= 1, X*>0 a nd let the functions (j= 1,2,- ■ ■,k) be integrable in their absolute values to the powers 1/X;, i.e..
_i_ j* | Oj 2
P dv < co .
Special
0
p r ob lem s of f u n c t io n a l analysis
T h e n the p ro duc t \9r ■ -