Minimal surfaces of codimension one. (English) Zbl 0565.49030

Massari, Umberto; Miranda, Mario

North-Holland Mathematics Studies, 91. Notas de Matemática, 95. Amsterdam - New York - Oxford: North-Holland. XIII, 243 p. $ 34.50; Dfl. 90.00 (1984).

This book provides a thorough treatment of minimal surfaces of codimension one in a Euclidean space of any dimension. Both the non- parametric and parametric cases are covered. The first chapter provides a nice introduction to the surface differentiation operators \(\delta_ i\) used so often by the Italian school. Also in that chapter, the authors present the gradient estimate for minimal graphs (in 1.6.2), the internal estimate for curvatures of minimal surfaces (in 1.7.1), and the Bernstein theorem for five dimensional surfaces (in 1.7.2). The second chapter deals with minimal boundaries. The basic definitions and results are there, which should be appreciated by those not familiar with the theory. The regularity theory is presented in the minimal boundary context. The third chapter treats the Dirichlet problem for the minimal surface equation. The general existence theory for strictly convex integrands is presented and applied to the minimal surface problem. The De Giorgi regularity theory is included and applied. The fourth and final chapter deals with generalized solutions to the minimal surface equation: By a generalized solution is meant, for an open set \(\Omega \subset \mathbb R^ n\), a Lebesgue measurable function \(f\) such that \(E=\{(x,z)\mid x\in \Omega,\, z\in \mathbb R,\, z<f(x)\}\) has minimal boundary in \(\Omega\times {\mathbb R}\). Generalized solutions are applied to the study of the Dirichlet problem with infinite data and on unbounded domains. The final topic is a proof of Miranda’s improvement of the De Giorgi– Stampacchia removable singularities theorem.
The English is rough, but understandable. In the latter part of the book the authors have inadvertently omitted the reference numbers in some places.

Reviewer: H. Parks

MSC:

49Q05	Minimal surfaces and optimization
49-02	Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49Q20	Variational problems in a geometric measure-theoretic setting
53A10	Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35B65	Smoothness and regularity of solutions to PDEs
35Dxx	Generalized solutions to partial differential equations
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

minimal surfaces of codimension one; Bernstein theorem; minimal boundaries; regularity; Dirichlet problem