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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.

Reviewer: P.P.Zabrejko

MSC:
49Q05Minimal surfaces (calculus of variations)
49-02Research monographs (calculus of variations)
26B30Absolutely continuous functions, functions of bounded variation (several real variables)
35J20Second order elliptic equations, variational methods
35B65Smoothness and regularity of solutions of PDE
49Q15Geometric measure and integration theory, integral and normal currents (optimization)
49Q20Variational problems in a geometric measure-theoretic setting
53A10Minimal surfaces, surfaces with prescribed mean curvature