Isovolumetric and isoperimetric problems for a class of capillarity functionals

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensional parametric surfaces in R3 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 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 the existence and nonexistence of volume-constrained, S2-type, minimal surfaces for the corresponding capillarity functionals. Moreover, in some cases, we show the existence of extremals for the full isoperimetric inequality.