L∞-ESTIMATES FOR CAPILLARY SURFACES
We study the geometric properties of liquid drops in tt^ sitting on a plane or being trapped between two parallel planes. Bounds on the diameter depending on the included volume are proved. Cases with and without gravity are treated. A M S Classification code: 49F20, 53A10, 76B45 0 INTRODUCTION In this article we study solutions of minimum problems associated to capillary surfaces in Liquid drops are models for minimizers of energy functionals determined by surface tension, Wetting energy, and possibly gravitational forces. For treating problems of this type it is helpfui to use mecisure-theoretic methods introduced by E. De Giorgi in the fifties. In this framework a hypersurface in J?" is the boundary of a measurable set E. The characteristic function of E has derivatives in a distributional sense which are Radon meMures. The total Variation of DXE gives the (n — l)-dimensional area of the boundary dE. Exact definitions are: 0.1 DEFINITIONS (1) Let n C JR" be open and let / e £ ' ( 0 ) . Define the total Variation of the measure Df I \Df\ = snp{Jfdivgdx ; g e |ff(x)| < 1 Vx e n } , n n where Df is given hy jg Df = Jdivg f dx Vjf 6 R"). n n (2) The Space BV{Ü) is the space of all functions in with bounded total Variation.