Sobolev Gradients and Differential Equations
A Sobolev gradient of a real-valued functional on a Hilbert space is a gradient of that functional taken relative to an underlying Sobolev norm. This book shows how descent methods using such gradients allow a unified treatment of a wide variety of proble
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J.W. Neuberger
Sobolev Gradients and Differential Equations Second Edition
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J.W. Neuberger Department of Mathematics University of North Texas 1155 Union Circle #311430 Denton, TX 76105-5017 USA [email protected]
ISBN: 978-3-642-04040-5 e-ISBN: 978-3-642-04041-2 DOI: 10.1007/978-3-642-04041-2 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009938945 Mathematics Subject Classification (2000): 35A05, 35A15, 35A35, 65D10, 65D25, 46E35, 46T99, 34B15, 34B27, 31C25, 81V99 c Springer-Verlag Berlin Heidelberg 1997, 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. 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. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
What is expected from a theory of differential equations? Look first at the fundamental theorem for ordinary differential equations: Theorem 0.1. Suppose that n is a positive integer and G is an open subset of R × Rn which contains a point (c, w). Suppose also that f : G → Rn is a continuous function for which there is M > 0 such that f (t, x) − f (t, y) ≤ M x − y for all (t, x), (t, y) ∈ G.
(0.1)
Then there is an open interval (a, b) containing c for which there is a unique function u on (a, b) so that u(c) = w, u (t) = f (t, u(t)), t ∈ (a, b). This result can be proved in several constructive ways which yield, along the way, error estimates giving a basis for numerical computation of solutions. Now this existence and uniqueness result certainly does not solve all problems in ordinary differential equations. For one thing, the result is only local. For just one other instance, it doesn’t tell about two point boundary value problems, even though it has relevance there. Nevertheless, it provides a position of strength from which to study a wide variety of ordinary differential equations. The fact of existence and uniqueness 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 discerning properties of solutions. Many agree that it would be good to have a similar position of strength for partial differential equations but su
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