This book offers a comprehensive treatment of the subject, rich in methods and results. It provides a presentation of elliptic and parabolic free boundary problems and techniques used for treating basic results. Part 1 concerns elliptic problems. After a formulation of the problem and its description in an introductory chapter, viscosity solutions are introduced and studied in chapter 2. Chapters 3 - 5 are dedicated to the regularity of a free boundary. In chapter 6, the existence theory for viscosity solutions of free boundary problems in a smooth (Lipschitz) domain is constructed. Part 2 is devoted to evolution problems, including the chapters: 7) Parabolic free boundary problems; 8) Lipschitz free boundary: weak results; 9) Lipschitz free boundary: strong results; 10) Flat free boundary are smooth. In Part 3 (Complementary chapters: main tools), the boundary behaviour of harmonic functions (chapter 11) and the boundary behaviour of caloric functions (chapter 13) in Lipschitz domains are studied. Monotonicity formulas and their applications are considered in chapter 12.

As the explained material becomes quickly rather complicated, it might be useful to add a list of symbols to the next edition; it would be highly appreciated by any reader who needs to study some separate “internal” parts of this important and valuable monograph.

Reviewer:

oj