Diffuse interfaces with sharp corners and facets: Phase field models with strongly anisotropic surfaces

We provide the general outline of an analysis of the motion of diffuse interfaces in the order-parameter (phase field) formulation which includes nondifferentiable and nonconvex gradient energy terms. Nondifferentiability leads to equilibrium and motion equations that are apparently undefined and nonconvexity leads to equations that are apparently ill-posed. The problem of nondifferentiability is resolved by using nonlocal variations to move entire facets or line segments with orientations having such facets or line segments in the underlying Wulff shape. The problem of ill-posedness is resolved by using varifolds (infinitesimally corrugated diffuse interfaces) constructed from the edges and corners in the underlying Wulff shape, or equivalently by using the convexification of the gradient energy term and then reinterpreting the solutions as varifolds. This is justified on a variational basis. We conclude that after an initial transient, level sets move by weighted mean curvature, in agreement with the sharp interface limit. We provide equations for tracking the shocks that develop as edges and corners in level sets.