Nonlinear Potential Theory and Weighted Sobolev Spaces
The book systematically develops the nonlinear potential theory connected with the weighted Sobolev spaces, where the weight usually belongs to Muckenhoupt's class of Ap weights. These spaces occur as solutions spaces for degenerate elliptic partial diffe
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B. Teissier, Paris
1736
Springer
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Bengt Ove Turesson
Nonlinear Potential Theory and Weighted Sobolev Spaces
Springer
Author Bengt Ove Turesson Matematiska istitutionen Linkopings Universitet SE-58183 Linkoping, Sweden E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Turesson, Bengt Ove: Nonlinear potential theory and weighted Sobolev spaces I Bengt Ove Turesson. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in mathematics; 1736) ISBN 3-540-67588-4
Mathematics Subject Classification (2000): 31C45,46E35 ISSN 0075-8434 5ISBN 3-540-67588-4 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 is a company in the BertelsmannSpringer publishing group. © Springer-Verlag Berlin Heidelberg 2000 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 Printed on acid-free paper SPIN: 10725034 41/3143/du
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Introduction Let w be a weight on R N , i.e., a locally integrable function on R N such that w(x) > 0 for a.e, x ERN. Let 0 C R N be open, 1 :::; p < 00, and m a nonnegative integer. The weighted Sobolev space consists of all functions u with weak derivatives DOlU, [o] :::; m, satisfying lIullw,;;"p(o) = (
L
IIDOlulPw 0
dX)
lip
< 00.
In the case w = 1, this space is denoted Wm,P(O). Sobolev spaces without weights occur as spaces of solutions for elliptic and parabolic partial differential equations. Typically, 2m is the order of the equation and the case p = 2 corresponds to linear equations. Details can be found in almost any book on partial differential equations. For degenerate partial differential equations, i.e., equations with various types of singularities in the coefficients, it is natural to look for solutions in weighted Sobolev spaces; see, e.g., Fabes, Kenig, and Serapioni [36], Fabes, Jerison, and Kenig [35], Fabes, Kenig, and Jerison [37], and Heinonen, Kilpelainen, and Martio [59]. A class of weights, which is particularly well understood, is the class of Ap weights that was introduced by B. Mucken
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