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We consider the sharp interface limit for the scalar-valued and vector-valuedAllen-Cahn equation with homogeneous Neumann boundary condition in a boundedsmooth domain $\Omega$ of arbitrary dimension $N\geq 2$ in the situation when atwo-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 forwell-prepared initial data as long as a smooth solution to the limit problemexists. 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 suitablecurvilinear coordinates and set up an asymptotic expansion for thescalar-valued and the vector-valued Allen-Cahn equation. Finally, we prove aspectral estimate for the linearized Allen-Cahn operator in both cases in orderto estimate the difference of the exact and approximate solutions with aGronwall-type argument.

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