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A Geometric Approach to Free Boundary Problems

Luis Caffarelli and Sandro Salsa
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Matheamatics 68
[Reviewed by
Vicentiu Radulescu
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One of the loveliest parts of mathematics is the subject of free or moving boundary problems, which has been intensively developed in the last few decades and has notable applications to various fields, such as shape optimization, phase transitions, fluid dynamics, probability and statistics. This work is devoted to the study of some partial differential equations whose nonlinear character gives rise to a free boundary, that is, the boundary of an a priori unknown region of positive measure where the solution of the equation vanishes identically. Such a phenomenon holds under some adequate balance between the terms of the equation representing diffusion, absorption, convection, evolution, etc. These particular balances take place neither in the case of linear equations nor in every nonlinear equation. The characterization of these special balances for several classes of nonlinear partial differential equations with major role in applications, is one of the main goals of this volume.

The present book falls into three parts, corresponding roughly to the main topics: elliptic problems, evolution problems, and an outline of the main tools applied in the work. The first six chapters cover the theory of elliptic free boundary problems. A special emphasis is given to viscosity solutions and their asymptotic developments, as well as to the role of the regularity of the free boundary. The existence of a viscosity solution is also discussed in detail. This construction relies upon a variant of the Perron method, which consists in taking the infimum of admissible supersolutions.

Part 2 is devoted to the study of evolution problems and is composed of four chapters. The topics mainly include weak and strong results on Lipschitz free boundaries. Also discussed are the main difficulties encountered in this study, including the role of time or the contribution of different homogeneities. These difficulties explain why stronger hypotheses have to be made, either about the geometry of the problem or about the starting configuration. This part of the book includes celebrated examples, such as Stefan-type problems.

The main tools developed in the last thirty years for the study of free boundary problems are the object of the last three chapters. These complementary chapters include basic facts on the boundary behavior of harmonic functions, monotonicity formulas and properties of caloric functions.

I think that the book will be a great resource, especially for scientists with an application in mind who want to find out what a free boundary problem-based approach can offer them. The applications are of interest in their own right and are not included just for the sake of being examples. Some sections of the book, in particular the introductory chapters, as well as chapters 11 and 12, should be accessible to a first-year graduate student with a basic knowledge of functional analysis, the qualitative theory of elliptic and parabolic partial differential equations, and an interest in applications. Other sections would be more suited to an advanced graduate student or experienced researcher.

The book is written by two of the most renowned specialists in the study of free boundary problems, with deep contributions in this field. The bibliography contains 45 entries, many of them very recent. The authors’ original contributions are indicated by 25 of their publications since 1981 up to present.

For anyone who later will do research on free boundary problems, this is probably the best introduction ever written. But the potential audience of this volume is much wider; his approach is just right for a book at the introductory level. The result is not only a comprehensive overview of the area itself, but also a very informative and inspiring monograph. Overall, this is a fine text for a graduate or postgraduate course in free boundary problems and a valuable reference that should be on the shelves of researchers and those teaching applied partial differential equations.

Vicentiu Radulescu ([email protected]) is Professor of Mathematics at the University of Craiova, Romania. He received both his Ph.D. and his Habilitation at the Université Pierre et Marie Curie (Paris VI) under the guidance of Professor Haim Brezis. His research interests are at the interface between nonlinear functional analysis, mathematical physics, and variational calculus. He is coauthor of the monograph “Variational and non-variational methods in nonlinear analysis and boundary value problems”, published by Kluwer Academic Publishers. Recently he proposed several problems in the Monthly, Electronic Journal of Differential Equations, Elemente der Mathematik. His web page is

Elliptic problems

* An introductory problem
* Viscosity solutions and their asymptotic developments
* The regularity of the free boundary
* Lipschitz free boundaries are $C^{1,\gamma}$
* Flat free boundaries are Lipschitz
* Existence theory

Evolution problems

* Parabolic free boundary problems
* Lipschitz free boundaries: Weak results
* Lipschitz free boundaries: Strong results
* Flat free boundaries are smooth

Complementary chapters: Main tools

* Boundary behavior of harmonic functions
* Monotonicity formulas and applications
* Boundary behavior of caloric functions
* Bibliography
* Index