Convergence of the Allen-Cahn Equation to the Mean Curvature Flow with ${90}^{\circ }$-Contact Angle in 2D

We consider the sharp interface limit of the Allen-Cahn equation with homogeneous Neumann boundary condition in a two-dimensional domain $\mathrm{\Omega }$, in the situation where an interface has developed and intersects $\mathrm{\partial }\mathrm{\Omega }$. 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 $90$\textdegree-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 and estimate the difference of the exact and approximate solution with the aid of a spectral estimate for the linearized Allen-Cahn operator.