The Method of Intrinsic Scaling

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.

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)

Klappentext

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 is presented from scratch for the model case of the degenerate p-Laplace equation. This approach brings to light what is really essential in the method, leaving aside technical refinements needed to deal with more general equations, and is entirely self-contained.

The second part deals with three applications of the theory to relevant models arising from flows in porous media and phase transitions. The aim is to convince the reader of the strength of the method as a systematic approach to regularity for this important class of equations.

Inhalt
The Method of Intrinsic Scaling.- Weak Solutions and a Priori Estimates.- The Geometric Setting and an Alternative.- Towards the Hölder Continuity.- Some Applications.- Immiscible Fluids and Chemotaxis.- Flows in Porous Media: The Variable Exponent Case.- Phase Transitions: The Doubly Singular Stefan Problem.