Hypersurfaces of prescribed
weighted mean curvature, or F-mean curvature, are introduced as critical immersions
of anisotropic surface energies, thus generalizing minimal surfaces and surfaces of
prescribed mean curvature. We first prove enclosure theorems in ℝn+1 for such
surfaces in cylindrical boundary configurations. Then we derive a general
second variation formula for the anisotropic surface energies generalizing
corresponding formulas of do Carmo for minimal surfaces, and Sauvigny for
prescribed mean curvature surfaces. Finally we prove that stable surfaces of
prescribed F-mean curvature in ℝ3 can be represented as graphs over a planar
strictly convex domain Ω, if the given boundary contour in ℝ3 is a graph over
∂Ω.
