Sobolev Gradients and Differential Equations
A Sobolev gradient of a real-valued functional is a gradient of that functional taken relative to the underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of problems in differential
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J. W. Neuberger
Sobolev Gradients and Differential Equations
Springer
Author John William Neuberger Department of Mathematics University of North Texas Denton, TX 76205, USA e-mail: [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Neuberger, John W.: Sobolev gradient and differential equations / John W. Neuberger. Berlin ; Heidelberg; New York ; Barcelona ; Budapest ; Hong Kong ; London; Milan; Paris; Santa Clara; Singapore; Tokyo : Springer, 1997 (Lecture notes in mathematics; 1670) ISBN 3-540-63537-8
Mathematics Subject Classification (1991): 35A15, 35A40, 65N99
ISSN 0075-8434 ISBN 3-540-63537-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10553348 46/3142-543210 - Printed on acid-free paper
Preface What is it that we might expect from a theory of differential equations? Let us look first at ordinary differential equations. THEOREM 0.1. Suppose that n is a positive integer and G is an open subset of R x R n containing a point (c, x) E R x R", Suppose also that j is a continuous function on G for which there is M > 0 such that
IIj(t, x) - jet, y)1I :S Mllx - yll, (t, x), (t, y) E G.
(0.1)
Then there is an open interval (a, b) containing c for which there is a unique function y on (a, b) so that y(c) = x, y'(t) = j(t,y(t)), t E (a,b). This result can be proved in several constructive ways which yield, along the way, error estimates which give a basis for numerical computation of solutions. Now this existence and uniqueness result certainly does not solve all problems (for example, two point boundary value problems) in ordinary differential equations. Nevertheless, it provides a position of strength from which to study a wide variety of differential equations. First of all the fact of existence of a solution gives us something to study in a qualitative, numerical or algebraic setting. The constructive nature of arguments for the above result gives one a good start toward di
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