Regularity of free boundaries in obstacle-type problems.

*(English)*Zbl 1254.35001
Graduate Studies in Mathematics 136. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8794-3/hbk). x, 221 p. (2012).

Obstacle problems are important examples of free-boundary formulations not only because they often appear in applications (i.e. in the modeling of real-world situations) but also because of the subtle analysis required to understand basic properties like solvability of the associated boundary-value problems and the expected regularity of the solutions.

This book contains nine chapters and focuses on the study of the regularity of solutions to a few classes of (academic) examples of partial differential equations involving free boundaries. To give an example: Find \(u\) in a suitable functional framework such that it satisfies \[ -\Delta u=f(x,u,\nabla u) \;\text{in} \;\Omega\subset \mathbb{R}^n, \] where the right-hand side \(f\) possibly admits discontinuities in the second and third variables. Typically, the discontinuity set is a priori unknown and is referred to as free.

The authors pay efforts to offer readable information, written very much in a textbook-like fashion. A selection of topics is covered here: Model problems (chapter 1), Optimal regularity of solutions (chapter 2), Preliminary analysis of the free boundary (chapter 3), Regularity of the free boundary (chapter 4), Global solutions (chapter 5), Regularity of the free boundary: uniform results (chapter 6), The singular set (chapter 7), Touch with the fixed boundary (chapter 8), The thin obstacle problem (chapter 9). Furthermore, each chapter contains a Notes and comments section as well as exercises. The book includes more than 25 years of regularity theory interest for free and moving boundary problems, probably very much driven by early works by L. A. Caffarelli and his PDEs school.

This book contains nine chapters and focuses on the study of the regularity of solutions to a few classes of (academic) examples of partial differential equations involving free boundaries. To give an example: Find \(u\) in a suitable functional framework such that it satisfies \[ -\Delta u=f(x,u,\nabla u) \;\text{in} \;\Omega\subset \mathbb{R}^n, \] where the right-hand side \(f\) possibly admits discontinuities in the second and third variables. Typically, the discontinuity set is a priori unknown and is referred to as free.

The authors pay efforts to offer readable information, written very much in a textbook-like fashion. A selection of topics is covered here: Model problems (chapter 1), Optimal regularity of solutions (chapter 2), Preliminary analysis of the free boundary (chapter 3), Regularity of the free boundary (chapter 4), Global solutions (chapter 5), Regularity of the free boundary: uniform results (chapter 6), The singular set (chapter 7), Touch with the fixed boundary (chapter 8), The thin obstacle problem (chapter 9). Furthermore, each chapter contains a Notes and comments section as well as exercises. The book includes more than 25 years of regularity theory interest for free and moving boundary problems, probably very much driven by early works by L. A. Caffarelli and his PDEs school.

Reviewer: Adrian Muntean (Eindhoven)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35R35 | Free boundary problems for PDEs |

35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |

35B65 | Smoothness and regularity of solutions to PDEs |