On the Sharp Interface Limit of a Model for Phase Separation on Biological Membranes

We rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed by Garcke et al. to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn-Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed by X. Chen for the corresponding result for the Cahn-Hilliard equation.