Convergence of the Allen–Cahn equation to the mean curvature flow with 90°-contact angle in 2D

Abstract

We consider the sharp interface limit of the Allen–Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $\Omega$, in the situation where an interface has developed and intersects $\partial \Omega$. Here a parameter $\epsilon >0$ in the equation, which is related to the thickness of the diffuse interface, is sent to zero. The limit problem is given by mean curvature flow with a 90°-contact angle condition and convergence using strong norms is shown for small times. Here we assume that a smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that it can be parametrized suitably. With the aid of asymptotic expansions we construct an approximate solution for the Allen–Cahn equation. In order to estimate the difference of the exact and approximate solution we use a spectral estimate for the linearized Allen–Cahn operator.