##
**A geometric approach to free boundary problems.**
*(English)*
Zbl 1083.35001

Graduate Studies in Mathematics 68. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3784-2/hbk). ix, 270 p. (2005).

Free or moving boundary problems appear in many areas of analysis, geometry, and applied mathematics. There are two possible ways of investigating free boundary problems. The first is connected with the classical explicit description of free boundary conditions. This approach is convenient in hte one-dimensional situation or in multi-dimensional situations under special geometric assumptions. The second way is based on a weak statement of the problem with free boundary conditions, “hidden” by introducing some integral identity or by minimizing a variational integral. The authors intend to build a bridge between these directions of investigation of free boundary problems. The presentation of the main ideas and tools is restricted for simplicity to one elliptic and one parabolic problem. The book consists of three parts.

The first part deals with an elliptic problem of the following form \[ \Delta u \text{ in } \Omega^+(u):=\{ u>0 \} \text{ and in } \Omega^-(u):=\{ u\leq 0\} ^0, \]

\[ G(u_\nu^+,u_\nu^-)=0 \text{ on the free boundary } F(u):=\partial \Omega^+(u), \] where \(u_\nu^+\) and \(u_\nu^-\) denote the normal derivatives in inward direction w.r.t. \(\Omega^+(u)\) and \(\Omega^-(u)\), respectively. By using descriptive heuristic considerations the authors introduce the reader to the theory of viscous solutions for free boundary problems with application to a detailed description of the behaviour of the solution near the free boundary and regularity properties of the free boundary itself.

The second part is devoted to the Stefan-like problem \[ \Delta u^{\pm}-a^{\pm}u_t^{\pm}=0 \text{ in } \Omega^{\pm}(u), \]

\[ V_\nu-G(u_\nu^+,u_\nu^-)=0 \text{ on the free boundary } F(u):=\partial \Omega^+(u), \] where \(V_\nu(\cdot,\tau)\) is the speed of the surface \(F(u)\cup \{ t=\tau \}\) in the direction \(\nu=\nabla u^+/ | \nabla u ^+ |\). The authors discuss the difficulties arising in transferring the method of the preceding part to the non-stationary case and carry out a comprehensive investigation of regularity of the viscous solution of the evolution problem.

The third part of this book reproduces the general framework of applied methods and includes important, deep results of the theory of harmonic and caloric measures in Lipschitz domains in connection with growth properties of the solutions, i.e. boundary Harnack inequality and monotonicity formulas.

The tools and ideas presented in this book will serve as a basis for the study of more complex phenomena and problems. The book is well written and the style is clear. It is suitable for graduate students and researchers interested in partial differential equations.

The first part deals with an elliptic problem of the following form \[ \Delta u \text{ in } \Omega^+(u):=\{ u>0 \} \text{ and in } \Omega^-(u):=\{ u\leq 0\} ^0, \]

\[ G(u_\nu^+,u_\nu^-)=0 \text{ on the free boundary } F(u):=\partial \Omega^+(u), \] where \(u_\nu^+\) and \(u_\nu^-\) denote the normal derivatives in inward direction w.r.t. \(\Omega^+(u)\) and \(\Omega^-(u)\), respectively. By using descriptive heuristic considerations the authors introduce the reader to the theory of viscous solutions for free boundary problems with application to a detailed description of the behaviour of the solution near the free boundary and regularity properties of the free boundary itself.

The second part is devoted to the Stefan-like problem \[ \Delta u^{\pm}-a^{\pm}u_t^{\pm}=0 \text{ in } \Omega^{\pm}(u), \]

\[ V_\nu-G(u_\nu^+,u_\nu^-)=0 \text{ on the free boundary } F(u):=\partial \Omega^+(u), \] where \(V_\nu(\cdot,\tau)\) is the speed of the surface \(F(u)\cup \{ t=\tau \}\) in the direction \(\nu=\nabla u^+/ | \nabla u ^+ |\). The authors discuss the difficulties arising in transferring the method of the preceding part to the non-stationary case and carry out a comprehensive investigation of regularity of the viscous solution of the evolution problem.

The third part of this book reproduces the general framework of applied methods and includes important, deep results of the theory of harmonic and caloric measures in Lipschitz domains in connection with growth properties of the solutions, i.e. boundary Harnack inequality and monotonicity formulas.

The tools and ideas presented in this book will serve as a basis for the study of more complex phenomena and problems. The book is well written and the style is clear. It is suitable for graduate students and researchers interested in partial differential equations.

Reviewer: Nikolai V. Krasnoschok (Donetsk)

### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35R35 | Free boundary problems for PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35K20 | Initial-boundary value problems for second-order parabolic equations |