Elliptic Partial Differential Equations Volume 1: Fredholm Theory of

The theory of elliptic partial differential equations has undergone an important development over the last two centuries. Together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications, it has become one of the m

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Managing Editors: H. Amann Universität Zürich, Switzerland J.-P. Bourguignon IHES, Bures-sur-Yvette, France K. Grove University of Maryland, College Park, USA P.-L. Lions Université de Paris-Dauphine, France Associate Editors: H. Araki, Kyoto University F. Brezzi, Università di Pavia K.C. Chang, Peking University N. Hitchin, University of Warwick H. Hofer, Courant Institute, New York H. Knörrer, ETH Zürich K. Masuda, University of Tokyo D. Zagier, Max-Planck-Institut Bonn

Vitaly Volpert

Elliptic Partial Differential Equations Volume 1: Fredholm Theory of Elliptic Problems in Unbounded Domains

Vitaly Volpert Institut Camille Jordan, CNRS Université Claude Bernard Lyon 1 43 boulevard du 11 novembre 1918 69622 Villeurbanne cedex France [email protected]

2010 Mathematics Subject Classification: 35J, 47F, 47A53, 47J05 ISBN 978-3-0346-0536-6 e-ISBN 978-3-0346-0537-3 DOI 10.1007/978-3-0346-0537-3 Library of Congress Control Number: 2011923524 © Springer Basel AG 2011 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 other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

To my mathematical family

Tous les g´eom`etres et physiciens semblent aujourd’hui d’accord que le domaine d’application des math´ematiques n’a d’autres limites que les limites de nos connaissances mˆemes. Il serait pourtant bien t´em´eraire d’affirmer que nous nous trouvons d´ej`a en possession des symboles les mieux appropri´es pour interpr´eter simplement les ph´enom`enes de la nature. I1 est au contraire beaucoup plus probable que bien des th´eories math´ematiques aujourd’hui en estime ne seront admir´es plus tard que comme des chefs d’oeuvre historiques, et d’autres th´eories aussi belles et aussi parfaites, mais d’une application plus large viendront les remplacer. Serge Bernstein, 1904 ([63])

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

1 Introduction 1 Elliptic problems in applications . . . . . . . . . . 1.1 Heat and mass transfer . . . . . . . . . . . . . 1.2 Electrostatic and gravitational fields . . . . . 1.3 Hydrodynamics . . . . . . . . . . . . . . . . . 1.4 Biological applications . . . . . . . . . . . . . 2 Classical theory of linear elliptic problems . . . . . 2.1 Function spaces . . . . . . . . . . . . . . . . . 2.2 Elliptic problems . . . . . . . . . . . . . . . . 2.3 A priori estimates . . . . . . . . . . . . . . . . 2.4 Fredholm operators . . . . . . . . . . . . . . . 3 Elliptic problems in unbounded domains . . . . . . 3.1 Function spaces . . . . . . . . . . . . . . . . . 3.2 Limiting problems . . . . . . . . . . . . . . . 3.3 Fredholm property and solvability conditions 3.4 Inde

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