We study the singular limit problem of the Allen-Cahn equation with Neumann boundary conditions. It is well-known that sharp interfaces can be obtained from the singular limit of the Allen-Cahn equation and a family of these interfaces is the mean curvature flow. For the problem of Neumann boundary conditions, we can heuristically expect that the sharp interfaces intersect the boundary of the domain at a 90-degree angle. Katsoulakis, Kossioris, and Reitich studied the relationship between this singular limit of the Neumann problem of the Allen-Cahn equation and the level set equation of the mean curvature flow with right angle boundary condition. On the other hand, there was no result about the mean curvature flow with right angle boundary condition from geometric measure theory.
In this talk, we heuristically observe that the Allen-Cahn equation with Neumann boundary conditions converge to the mean curvature flow with 90-degree boundary conditions. We introduce the notion of weak solutions of mean curvature flow with 90-degree boundary conditions and show that the energy measure of the Allen-Cahn equation converge to the weak solution of the mean curvature flow with 90-degree boundary conditions.
This is a joint work with Professor Yoshihiro Tonegawa (Tokyo Institute of Technology).