The Method of Intrinsic Scaling

1. Front Matter

   Pages i-x

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2. The Method of Intrinsic Scaling

   1. Introduction

      Pages 1-8

   2. Weak Solutions and a Priori Estimates

      Pages 11-19

   3. The Geometric Setting and an Alternative

      Pages 21-34

   4. Towards the Hölder Continuity

      Pages 35-48

3. Some Applications

   1. Immiscible Fluids and Chemotaxis

      Pages 51-86

   2. Flows in Porous Media: The Variable Exponent Case

      Pages 87-105

   3. Phase Transitions: The Doubly Singular Stefan Problem

      Pages 107-143

4. Back Matter

   Pages 145-150

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When I started giving talks on regularity theory for degenerate and sin- lar parabolic equations, a ?xed-point in the conversation during the co?- break that usually followed the seminar was the apparent contrast between the beauty of the subject and its technical di?culty. 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 re?nements needed to deal with more general situations. The ?rst part of the notes you are about to read is the result of that e?ort: an introductory and self-containedapproachtointrinsicscaling,aimingatbringingtolightwhatis 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 applications, in the second part of the book, intrinsic scaling is applied to several models arising from ?ows in porous media, chemotaxis and phase transitions.

• DEX
• Degenerate and singular PDEs
• Scratch
• class
• equation
• intrinsic scaling
• model
• p-Laplacian
• regularity
• set
• variable
• partial differential equations

From the reviews:

"This book concerns the regularity theory for degenerate and singular parabolic equations and the focus is on a particular subject – the Hölder continuity of solutions. … The aim of this book is to describe in details the method of intrinsic scaling … and to convince the reader of the strength of this approach to regularity, by giving evidence of its wide applicability in different situations. … The book will be very useful for researchers from different branches of mathematical physics." (Vladimir N. Grebenev, Zentralblatt MATH, Vol. 1158, 2009)

• CMUC, Department of Mathematics, University of Coimbra, 3001-454, Portugal

  José Miguel Urbano

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