On the motion by singular interfacial energy

Anisotropic curvature flow equations with singular interfacial energy are important for good understanding of motion of phase-boundaries. If the energy and the interfacial surface were smooth, then the speed of the interface would be equal to the gradient of the energy. However, this is not so simple in the case of non-smooth crystalline energy. But it's well-known that a unique gradient characterization of the velocity is possible if the interface is a curve in the two-dimensional space. In this paper we propose a notion of solution in the three-dimensional space by introducing geometric subdifferentials and characterizing the speed. We also give a counterexample to a problem concerning the Cahn-Hoffman vector field on a facet, a flat portion of the interface.