Stable solutions of elliptic partial differential equations. (English) Zbl 1228.35004

Dupaigne, Louis

Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 143. Boca Raton, FL: CRC Press (ISBN 978-1-4200-6654-8/hbk). xiv, 321 p. (2011).

This volume is concerned with a modern research subject and offers a self-contained presentation of the notion of stability in elliptic partial differential equations. The basic questions of regularity and classification of stable solutions are mainly considered in this book. The author is an authorized expert in this field and he offers a summary of the most recent developments of the theory, such as nonlocal and higher-order equations, refined versions of the maximum principle, the standard regularity theory for linear elliptic equations, as well as some of the fundamental functional inequalities commonly used in this field. There are also included two additional topics, namely the inverse-square potential and some background material on submanifolds of the Euclidean space.
The content of the book is divided into eight chapters, as follows: I. Defining stability; II. The Gelfand problem; III. Extremal solutions; IV. Regularity theory of stable solutions; V. Singular stable solutions; VI. Liouville theorems for stable solutions; VII. A conjecture of E. De Giorgi; VIII. Further readings. Three appendices (Maximum principles; Regularity theory for elliptic operators; Geometric tools) complement the content of the book and provide useful notions and properties.
This is a very well-written and well-structured volume with clearly explained proofs and a good supply of interesting comments. The author clearly loves the subject he has developed in this book. The proofs are clear and the author makes great efforts to advise readers how to navigate through these and how to develop their understanding of the techniques. This book is suitable for graduate students and researchers who have a good background in functional analysis and partial differential equations and have mastered such key techniques as weak convergence, the maximum principle, the implicit function theorem, monotonicity methods in PDEs, etc. This reviewer strongly appreciates that the book under review deserves to be widely read.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

MSC:

35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations
35B35	Stability in context of PDEs
35A16	Topological and monotonicity methods applied to PDEs
35Jxx	Elliptic equations and elliptic systems
35B50	Maximum principles in context of PDEs
35B53	Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

nonlinear elliptic equation; Gelfand problem; conjecture of E. De Giorgi; implicit function theorem; monotonicity methods