Interacting phase fields yielding phase separation on surfaces

In the present article we study diffuse interface models for two-phase biomembranes. We will do so by starting off with a diffuse interface model on Rⁿ defined by two coupled phase fields $u,v$. The first phase field u is the diffuse approximation of the interior of the membrane; the second phase field v is the diffuse approximation of the two phases of the membrane. We prove a compactness result and a lower bound in the sense of Γ-convergence for pairs of phase functions (u_ε,v_ε). As an application of this first result, we consider a diffuse approximation of a two-phase Willmore functional plus line tension energy.