Capillary Surfaces in Non-Cylindrical Domains

Abstract

This paper is concerned with the capillary problem in a class of non-cylindrical domains in $K\subset {\mathbb{R}}^{n+1}$ obtained by scaling a bounded cross-section ­ $\Omega \subset {\mathbb{R}}^n$ along the vertical axis. The capillary surfaces are described in two different ways. In the first model, they are described as the boundary of a Caccioppoli set and in a second model, after transforming $K$ to a cylinder, they are described as graphs of functions on $\Omega$­. The volume of the fluid is prescribed. For both models, the energy functional is derived and declared on the appropriate function space consisting of $BV$-functions. Main results are existence and a priori bounds of minimizers, using the direct methods in the calculus of variations. For the special case of a cone over the domain $\Omega$­, a criterion is given to assure that the tip is not filled with liquid. Another point of examination concerns modelling the volume restriction by means of a Lagrange multiplier.