GENERALIZED MOTION BY MEAN-CURVATURE WITH NEUMANN CONDITIONS AND THE ALLEN-CAHN MODEL FOR PHASE-TRANSITIONS

We study a sharp-interface model for phase transitions which incorporates the interaction of the phase boundaries with the walls of a container Omega. In this model, the interfaces move by their mean curvature and are normal to partial derivative Omega. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with a transition-layer model. We prove that if Omega is convex, the transition-layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.