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This paper is concerned with the sharp interface limit for the Allen-Cahnequation with a nonlinear Robin boundary condition in a bounded smooth domain$\Omega\subset\mathbb{R}^2$. We assume that a diffuse interface already hasdeveloped and that it is in contact with the boundary $\partial\Omega$. Theboundary condition is designed in such a way that the limit problem is given bythe mean curvature flow with constant $\alpha$-contact angle. For $\alpha$close to $90${\deg} we prove a local in time convergence result forwell-prepared initial data for times when a smooth solution to the limitproblem exists. Based on the latter we construct a suitable curvilinearcoordinate system and carry out a rigorous asymptotic expansion for theAllen-Cahn equation with the nonlinear Robin boundary condition. Moreover, weshow a spectral estimate for the corresponding linearized Allen-Cahn operatorand with its aid we derive strong norm estimates for the difference of theexact and approximate solutions using a Gronwall-type argument.

Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to $90${\deg}