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In this paper we consider phase separations on (generalized) hypersurfaces inEuclidian space. We consider a diffuse surface area (line tension) energy ofModica-Mortola type and prove a compactness and lower bound estimate in thesharp interface limit. We use the concept of generalized BV functions overcurrents as introduced by Anzellotti et. al. [Annali di Matematica Pura edApplicata, 170, 1996] to give a suitable formulation in the limit and achievethe necessary compactness property. We also consider an application to phaseseparated biomembranes where a Willmore energy for the membranes is combinedwith a generalized line tension energy. For a diffuse description of suchenergies we give a lower bound estimate in the sharp interface limit.

Phase separation on varying surfaces and convergence of diffuse interface approximations