Abstract:In materials science, it is popular to describe the motion of phase boundaries by anisotropic curvature flows. A crystalline mean curvature flow is a typical example. It is a kind of a gradient flow whose energy functional is singular so the meaning of a solution itself is not clear. A prototype is the total variation flow and its speed of a part of slope zero is determined by some nonlocal quantity.
For a planar motion, a level-set approach for a crystalline curvature flow was established two decades ago. The situation is easier since the expected speed of a flat part called a facet is constant depending on its length. However, for a surface motion, the speed of a facet may not be constant and facet-splitting or facet-bending may occur. If the speed is constant, the facet is often called calibrable and its value must be an anisotropic Cheeger ratio.
In this talk, we survey recent progress of a level-set approach to crystalline mean curvature flows including spatially inhomogeneous driving forces. A key point is how to establish the notion of a solution based on crystalline curvature based on the theory of viscosity solutions. One should be careful to give a “correct” notion so that the unique solution is given by the limit of solutions of approximate problems, especially when there is an inhomogeneous driving force. This talk is based on my recent joint work with N. Požár (Kanazawa University).
Conference number：948 9114 4904