  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.
