Abstract
This paper is concerned with the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Omega subset of R-2. We assume that a diffuse interface already has developed and that it is in contact with the boundary partial derivative Omega. The boundary condition is designed in such a way that the limit problem is given by the mean ...
Abstract
This paper is concerned with the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Omega subset of R-2. We assume that a diffuse interface already has developed and that it is in contact with the boundary partial derivative Omega. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant alpha-contact angle. For alpha close to 90 degrees we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen-Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen-Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.
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