Isovolumetric and isoperimetric problems for a class of capillarity functionals

TL;DR

This paper investigates capillarity functionals, exploring the existence and nonexistence of volume-constrained minimal surfaces and extremals related to isoperimetric inequalities in R3.

Contribution

It establishes conditions for existence and nonexistence of extremals for a class of anisotropic capillarity functionals with fixed volume.

Findings

Existence of volume-constrained minimal surfaces under certain anisotropic conditions.

Nonexistence results for some classes of anisotropies vanishing at infinity.

Existence of extremals for the full isoperimetric inequality in specific cases.

Abstract

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensionals parametric surfaces in R3 as the sum of the area integral with an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of S2 and with fixed volume, extremals of capillarity functionals are surfaces whose mean curvature is prescribed up to a constant. For a certain class of anisotropies vanishing at infinity, we prove existence and nonexistence of volume- constrained, S2-type, minimal surfaces for the corresponding capillarity functionals. Moreover, in some cases, we show existence of extremals for the full isoperimetric inequality.