On Dirichlet's Boundary Value Problem An Lp-Theory Based on a Genera
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268 Christian G. Simader Mathematisches Institut der Universit~t M~Jnchen, MLinchen/Deutschland
On Dirichlet's Boundary Value Problem An LP-Theory Based on a Generalization of G&rding's Inequality
Springer-Verlag Berlin-Heidelberg
New York 1972
A M S S u b j e c t Classifications (1970): 39 A 15, 35J 05, 35J 40
I S B N 3-540-05903-2 S p r i n g e r - V e r l a g B e r l i n • H e i d e l b e r g - N e w Y o r k I S B N 0-387-05903-2 S p r i n g e r - V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payabie to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg 1972. Library of Congress Catalog Card Number 72-85089. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents Outline Chapter I
I
A priori estimates for solutions of linear elliptic functional equations with constant coefficients
~3
8 I.
Some definitions. Formulation of & basic a priori estimate
13
8 2.
Construction of certain "testing functions"° Analytic tools
21
83.
Proofs of local and global a priori estimates and regularity theorems
41
Chapter II
: A representation for continuous linear functionals on ~ ' P ( G ) (I < p < ~ ) and its &pplications: A generalization of G~rding's inequality and existence theorems
84
84.
A representation for continuous linear functionals on w~'P(G) (I < p < ~)
85
85.
Bilinear forms and a generalization of the Lax-Milgram-theorem
97
86.
Some coerciviness inequalities generalizing G~rd!ng's inequality
101
Existence theorems in the case of uniformly strongly elliptic Dirichlet billnear forms
121
Chapter III
: Regularity and existence theorems for uniformly
133
elliptic functional equations
§8.
Some properties of the spaces
§9.
Differentiability theorems
137
§ 10.
Fredholm's alternative for uniformly elliptic functional equations
163
§11.
Further regularity theorems
186
wk'P(G)
134
IV Appendix
I
200
Appendix 2
228
List of notations
230
Bibliography
234
At this point I want to thank my academic teachers Prof. E. Heinz and Prof. E. Wienholtz.
Further I want to thank the editors of the
"Lecture Notes" Series,
Prof. A. Dold and Prof. B. Eckmann,
Springer-Verlag
and
for publishing my manuscript in this series. Last but
not least I thank Mr. G. Abersfelder for typing the manuscript.
Outline
The starting point of the study of elliptic boundary value problems was Dirichlet's p r o b l e m for the Laplacian
A .
lized as follows.
be a uniformly elliptic par-
Let
L =
7-
as(x)D s
This question is genera-
tial differential
Isl 0
= 0
(L*v
for every
equivalent
tion have to be
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