Non-Homogeneous Boundary Value Problems and Applications Vol. 1
1. We describe, at first in a very formaI manner, our essential aim. n Let m be an op en subset of R , with boundary am. In m and on am we introduce, respectively, linear differential operators P and Qj' 0 ~ i ~ 'V. By "non-homogeneous boundary value prob
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Herausgegeben von
J. 1. Doab . A. Grothendieck .E. Heinz. F. Hirzebruch E. Hopf . W. Maak . S. MacLane. W. Magnus. J. K. Moser M. M. Postnikov . F. K. Schmidt. D. S. Scott· K. Stein
Geschäftsführende Herausgeber B. Eckmann und B. 1. van der Waerden
J. L. Lions . E. Magenes
Non-Homogeneous Boundary Value Problems and Applications Translated from the French P.Kenneth
py
Volume I
t· .
.
"
Springer-Verlag Berlin Heidelberg New York 1972
]. L. Lions
E. Magenes
University of Paris
University of Pavia
Tille of the Freneh Original Edition: Problemes aux limites non homog~nes et applications (tarne Ij Publisher: S. A. Dunod, Paris 1968
Translator:
P. Kenneth Paris
Geschäftsführende Herausgeber:
B.Eckmann Eidgenössische Technische Hochschule Zürich
B. L. van der Waerden Mathematisehes Institut der Universität Zürich
Primary
AMS Subject Classifications (1970) 35J20, 35J25, 35J30, 35J35, 35J40, 35K20, 35K35, 35L20 Secondary 46E35
ISBN -13 :978-3-642-65163-2
e- ISBN -13:978-3-642-65161-8
DOI: 10.1007/978-3-642-65161-8 This work is subject to copyright. All rights are rcserved, whether the whole or part of the material is concerned,specifically those of translation, reprinting, rc-tise 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 payable to the publisher, the amount of the fee to be determined by agreement with the publisher. @) by Springer-Verlag, Berlin' Heidelberg 1972. Softeover reprint of the hardeover 1st edition 1972
Library of Congrcss Catalog Card Number 71-151407
Preface 1. We describe, at first in a very formaI manner, our essential aim. Let mbe an op en subset of R n, with boundary am. In mand on am we introduce, respectively, linear differential operators P and Qj'
0 ~
i
~
'V.
By "non-homogeneous boundary value problem" we mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be given in function space s F and Gj , F being a space" on m" and the G/ s spaces" on am" ; we seek u in a function space u/t "on m" satisfying (1)
(2)
Pu =
Qju = gj on
f in m,
am,
0 ~
i
~ 'v«])).
Qj may be identically zero on part of am, so that the number of boundary conditions may depend on the part of am considered 2. We take as "working hypothesis" that, for fEF and gjEGj , the problem (1), (2) admits a unique solution u E U/t, which depends continuously on the data 3 . But for alllinear probIems, there is a large number of choiees for the space s u/t and {F; Gj } (naturally linke d together). Generally speaking, our aim is to determine families of spaces 'ft and {F; Gj }, associated in a "natural" way with problem (1), (2) and convenient for applications, and also all possible choiees for u/t and {F; Gj } in these families. Let us make this explicit by means of two examples, chosen as the simplest possible ones, but which already demonstrate th" utility of non-homogeneous probIems. «1» The Q/s will be called "boundary ope
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