Phase field method for mean curvature flow with boundary constraints

This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result and then show some numerical comparisons of these two different models