The Dirichlet Problem with L2-Boundary Data for Elliptic Linear Equations
The Dirichlet problem has a very long history in mathematics and its importance in partial differential equations, harmonic analysis, potential theory and the applied sciences is well-known. In the last decade the Dirichlet problem with L2-boundary data h
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen Subseries: Mathematisches Institut der Universitat Erlangen-Nurnberg Adviser: H. Bauer
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Jan Chabrowski
The Dirichlet Problem with L2_Boundary Data for Elliptic Linear Equations
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Author Jan Chabrowski Department of Mathematics The University of Queensland St. Lucia QLD 4072, Australia
Mathematics Subject Classification (1991): 35B, 35D, 35J
ISBN 3-540-54486-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54486-0 Springer-Verlag New York Berlin Heidelberg 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. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 5.1 5.2 5.3 6.1 6.2 6.3
Contents INTRODUCTIO N CHAPTER 1 Weighted Sobolev space W I ,2 Sobolev spaces The Dirichlet problem Domains with G 2-boundary Some properties of weighted Sobolev spaces Problem of traces in weighted Sobolev spaces CHAPTER 2 The Dirichlet problem in a half-space Preliminaries Assumptions and properties of solutions in Behaviour of W,;,2-solutions for small X n Traces of solutions in on X n = 0 The Dirichlet problem in The Dirichlet problem in the weighted Sobolev space CHAPTER 3 The Dirichlet problem in a bounded domain Assumptions and preliminaries Weak convergence of solutions at the boundary Traces in L 2(8Q) Energy estimate Solvability of the Dirichlet problem CHAPTER 4 Estimates of derivatives Estimates of the second order derivatives Analogue of Theorem 3.1 for Du Reverse Holder inequality Higher integrability property of Du CHAPTER 5 Harmonic measure The Dirichlet problem with continuous boundary data Harmonic measure and the Lebesgue surface measure Counter-example of Modica and Mortola CHAPTER 6 Exceptional sets on the boundary Formulation of the problem and preliminaries Uniqueness criterion for solutions in n LP(Q) Counter-examples CHAPTER 7 Applications of the L 2-method
7.1 7.2 7.3 7.4 7.5 7.6
Energy estimate for degenerate elliptic equations Traces in W 2,2(Q) Existence of solutions of the problem (7.1),(7.2) Non-local problem Energy estimate for non-local problem Weak solutions of non local problem CHAPTER 8 Domains with GI,a-boundary 8.1 Regularized distance 8.2 Analogue of Lemma 2.
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