Existence of isovolumetric extremals for capillarity functionals

TL;DR

This paper investigates capillarity functionals, which combine surface area and additional terms, proving the existence of saddle-type extremals in cases where no minimal surfaces exist due to non-homogeneity.

Contribution

It establishes the existence of isovolumetric extremals for non-homogeneous capillarity functionals lacking minimal surfaces, using variational methods.

Findings

Existence of saddle-type critical points for certain capillarity functionals.

No volume-constrained minimal surfaces exist in the considered class.

Variational techniques effectively identify extremals in non-homogeneous cases.

Abstract

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 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 S2-type minimal surface. Using variational techniques, we prove existence of extremals characterized as saddle-type critical points.