Isovolumetric and Isoperimetric Problems for a Class of Capillarity Functionals

摘要

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 an anisotropic term of suitable form. In the class of parametric surfaces with the topological type of ({mathbb{S}^{2}}) 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 the existence and nonexistence of volume-constrained, ({mathbb{S}^{2}})-type, minimal surfaces for the corresponding capillarity functionals. Moreover, in some cases, we show the existence of extremals for the full isoperimetric inequality.