Existence of isovolumetric ${\mathbb{S}}^2$-type stationary surfaces for capillarity functionals

Abstract

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in ${\mathbb{R}}^3$ as the sum of the area integral and a non homogeneous term of suitable form. Here we consider the case of a class of non homogenous terms vanishing at infinity for which the corresponding capillarity functional has no volume-constrained ${\mathbb{S}}^2$-type minimal surface. Using variational techniques, we prove existence of extremals characterized as saddle-type critical points.