This book gives a unified presentation of different mathematical tools used to solve classical problems like Plateau's problem, Bernstein's problem, Dirichlet's problem for the Minimal Surface Equation and the Capillary problem.
The fundamental idea is a quite elementary geometrical definition of codimension one surfaces. The isoperimetric property of the Euclidean balls, together with the modern theory of partial differential equations are used to solve the 19th Hilbert problem. Also included is a modern mathematical treatment of capillary problems.
Table of Contents
Introduction. 1. Differential Properties of Surfaces. 2. Sets of Finite Perimeter and Minimal Boundaries. 3. The Dirichlet Problem for the Minimal Surface Equation. 4. Unbounded Solutions.