Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions

Abstract: "We study a sharp-interface model for phase transitions that 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 [delta 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 non-convex domains."