Generalized motion by mean curvature with Neumann conditions and the Allen-Cahn model for phase transitions (Q1897238)

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 \(\partial \Omega\). 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.