Minimal surfaces and functions of bounded variation.

*(English)*Zbl 0545.49018
Monographs in Mathematics, Vol. 80. Boston-Basel-Stuttgart: Birkhäuser. XII, 240 p. DM 96.00 (1984).

This book presents a theory of parametric and non-parametric minimal hypersurfaces in euclidean spaces of arbitrary dimension. The first part of the book is devoted to parametric minimal surfaces and contains 11 chapters: Functions of bounded variation and Caccioppoli Sets; Traces of BV functions; The Reduced Boundary; Regularity of the Reduced Boundary; Some Inequalities; Approximation of Minimal Sets, I and II; Regularity of Minimal Surfaces; Minimal Cones; The First and Second Variation of the Area; The Dimension of the Singular Set. Its informal contents is existence and regularity almost everywhere of solutions to the Plateau problem and the problem on the dimension of the singular set. This part presents theorems of E. De Giorgi, L. Simons, H. Federer.

The second part of the book is devoted to non-parametric minimal surfaces and contains 6 chapters: Classical Solutions of the Minimal Surface Equation; The a priori Estimate of the Gradient; Direct Methods; Boundary Regularity; A Further Extension of the Notion of Non-Parametric Minimal Surfaces; The Bernstein Problem.

At the end of the book there are three small addenda devoted to some variants of the De la Vallée Poussin theorem, the distance function and elliptic equations of the second order. On the whole this book is a research level monograph in which the latest results are presented; it will be a useful reference book for mathematicians working in minimal surfaces, elliptic differential equations, geometric measure theory, capillarity and plasticity.

The second part of the book is devoted to non-parametric minimal surfaces and contains 6 chapters: Classical Solutions of the Minimal Surface Equation; The a priori Estimate of the Gradient; Direct Methods; Boundary Regularity; A Further Extension of the Notion of Non-Parametric Minimal Surfaces; The Bernstein Problem.

At the end of the book there are three small addenda devoted to some variants of the De la Vallée Poussin theorem, the distance function and elliptic equations of the second order. On the whole this book is a research level monograph in which the latest results are presented; it will be a useful reference book for mathematicians working in minimal surfaces, elliptic differential equations, geometric measure theory, capillarity and plasticity.

Reviewer: P.P.Zabrejko

##### MSC:

49Q05 | Minimal surfaces and optimization |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

26B30 | Absolutely continuous real functions of several variables, functions of bounded variation |

35J20 | Variational methods for second-order elliptic equations |

35B65 | Smoothness and regularity of solutions to PDEs |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

49Q20 | Variational problems in a geometric measure-theoretic setting |

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |