Non-Homogeneous Boundary Value Problems and Applications Volume II
I. In this second volume, we continue at first the study of non homogeneous boundary value problems for particular classes of evolu tion equations. 1 In Chapter 4 , we study parabolic operators by the method of Agranovitch-Vishik [lJ; this is step (i) (
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Herausgegeben von
J. 1. Doob . 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
Geschaftsfuhrende 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 by P.Kenneth
Volume II
Springer-Verlag Berlin Heidelberg New York 1972
J. L. Lions
E. Magenes
University of Paris
University of Pavia
Title of the French Original Edition: Problemes aux limites non homogenes et applications (tome II) Publisher: S. A. Dunod, Paris 1968
Translator:
P. Kenneth Paris
Geschiiftsfiihrende Herausgeber:
B.Eckmann Eidgenossische Technische Hochschule Zurich
B. L. van der Waerden Mathematisches Institut der Universitiit Zurich
Primary
AMS Subject Classifications (1970) 35J20, 35J25, 35J30, 35J35, 35J40, 35K20, 35K35, 35L20, Secondary 46E35
e-ISBN-13: 978-3-642-65217-2 ISBN-13: 978-3-642-65219-6 001: 10.1007/978-3-642-65217-2 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 payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag, Berlin . Heidelberg 1972. Softcover reprint of the hardcover 1st edition 1972
Library of Congress Catalog Card Number 71-151407
Introduction I. In this second volume, we continue at first the study of nonhomogeneous boundary value problems for particular classes of evolution equations. In Chapter 4 1 , we study parabolic operators by the method of Agranovitch-Vishik [lJ; this is step (i) (Introduction to Volume I, Section 4), i.e. the study of regularity. The next steps: (ii) transposition, (iii) interpolation, are similar in principle to those of Chapter 2, but involve rather considerable additional technical difficulties. In Chapter 5, we study hyperbolic operators or operators welldefined in thesense of Petrowski or Schroedinger. Our regularity results (step (i)) seem to be new. Steps (ii) and (iii) are all3.logous to those of the parabolic case, except for certain technical differences. In Chapter 6, the results of Chapter'> 4 and 5 are applied to the study of optimal control problems for systems governed by evolution equations, when the control appears in the boundary conditions (so that non-homogeneous boundary value problems are the basic tool of this theory). Another type of application, to the characterization of "all" well-posed problems for the operators in question, is given in the Appendix. Still other applications, for example to numerical analysis, will be given in Volume 3. 2. For the same ,>ystems {P, Qj}, Volume 3 will proceed in an analogous way, but starting from regularity results
