Convergence of the scalar- and vector-valued Allen-Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result

We consider the sharp interface limit for the scalar-valued and vector-valued Allen-Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Omega of arbitrary dimension N >= 2 in the situation when a two-phase diffuse interface has developed and intersects the boundary partial derivative Omega. The limit problem is mean curvature flow with 90 degrees-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on [0, T] for some timeT > 0. Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen-Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen-Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)).