Convergence of the Scalar- and Vector-Valued Allen-Cahn Equation to Mean Curvature Flow with $90${\deg}-Contact Angle in Higher Dimensions

TL;DR

This paper proves the convergence of scalar- and vector-valued Allen-Cahn equations to mean curvature flow with a 90-degree contact angle in higher dimensions, using asymptotic expansions and spectral estimates.

Contribution

It establishes the convergence of Allen-Cahn equations to mean curvature flow with contact angle conditions in arbitrary dimensions, extending previous results.

Findings

Convergence in strong norms for well-prepared initial data.

Construction of curvilinear coordinates for asymptotic analysis.

Spectral estimates for the linearized Allen-Cahn operator.

Abstract

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 $N\geq 2$ in the situation when a two-phase diffuse interface has developed and intersects the boundary $\partial\Omega$. The limit problem is mean curvature flow with $90${\deg}-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 time $T>0$. 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. Finally, we prove a spectral estimate for the linearized Allen-Cahn operator in both cases in order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument.