Phase separation on varying surfaces and convergence of diffuse interface approximations

TL;DR

This paper investigates phase separation on complex surfaces using diffuse interface models, establishing compactness and convergence results, and applies these to biomembranes with combined energy functionals.

Contribution

It introduces a generalized BV function framework over currents for phase separation on hypersurfaces and proves convergence of diffuse interface approximations.

Findings

Established compactness and lower bound estimates in the sharp interface limit.

Applied the theory to phase separated biomembranes with combined energies.

Provided a diffuse description and convergence results for complex surface energies.

Abstract

In this paper we consider phase separations on (generalized) hypersurfaces in Euclidian space. We consider a diffuse surface area (line tension) energy of Modica-Mortola type and prove a compactness and lower bound estimate in the sharp interface limit. We use the concept of generalized BV functions over currents as introduced by Anzellotti et. al. [Annali di Matematica Pura ed Applicata, 170, 1996] to give a suitable formulation in the limit and achieve the necessary compactness property. We also consider an application to phase separated biomembranes where a Willmore energy for the membranes is combined with a generalized line tension energy. For a diffuse description of such energies we give a lower bound estimate in the sharp interface limit.