The Method of Intrinsic Scaling A Systematic Approach to Regularity
This set of lectures, which had its origin in a mini course delivered at the Summer Program of IMPA (Rio de Janeiro), is an introduction to intrinsic scaling, a powerful method in the analysis of degenerate and singular PDEs. In the first part, the theory
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José Miguel Urbano
The Method of Intrinsic Scaling A Systematic Approach to Regularity for Degenerate and Singular PDEs
1930
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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1930
José Miguel Urbano
The Method of Intrinsic Scaling A Systematic Approach to Regularity for Degenerate and Singular PDEs
123
Author José Miguel Urbano CMUC, Department of Mathematics University of Coimbra 3001-454 Coimbra Portugal [email protected]
ISBN: 978-3-540-75931-7 e-ISBN: 978-3-540-75932-4 DOI: 10.1007/978-3-540-75932-4 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008921371 Mathematics Subject Classification (2000): 35D10, 35K65 c 2008 Springer-Verlag 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, 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: WMXDesign GmbH Printed on acid-free paper 987654321 springer.com
To Martim, Xavier and Catarina.
Preface
When I started giving talks on regularity theory for degenerate and singular parabolic equations, a fixed-point in the conversation during the coffeebreak that usually followed the seminar was the apparent contrast between the beauty of the subject and its technical difficulty. I could not agree more on the beauty part but, most of the times, overwhelmingly failed to convince my audience that the technicalities were not all that hard to follow. As in many other instances, it was the fact that the results in the literature were eventually stated and proved in their most possible generality that made the whole subject seem inexpugnable. So when I had the chance of preparing a short course on the method of intrinsic scaling, I decided to present the theory from scratch for the simplest model case of the degenerate p-Laplace equation and to leave aside technical refinements needed to deal with more general situations. The first part of the notes you are about to read is the result of that effort: an introductory and self-contained approach to intrinsic scaling, aiming at bringing to light what is really essential in this powerful tool in the analysis of degenerate and singular equations. As another striking feature of the method is its pervasiveness in terms of the applic
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