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[1] U. Clarenz and H. von der Mosel. On surfaces of prescribed f-mean curvature. Pacific Journal of Mathematics, 213(1):15-36, 2004.
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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 n+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 can be represented as graphs over a planar strictly convex domain Ω, if the given boundary contour in is a graph over Ω.