Threshold Dynamics for Anisotropic Surface Energies∗

We study extensions of Merriman, Bence, and Osher’s threshold dynamics scheme to weighted mean curvature flow, which arises as gradient descent for anisotropic (normal dependent) surface energies. In particular, we investigate, in both two and three dimensions, those anisotropies for which the convolution kernel in the scheme can be chosen to be positive and / or to possess a positive Fourier transform. We provide a complete, geometric characterization of such anisotropies. This has implications for the unconditional stability and, in the two-phase setting, the monotonicity, of the scheme. We also revisit previous constructions of convolution kernels from a variational perspective, and propose a new one. The variational perspective differentiates between the normal dependent mobility and surface tension factors (both of which contribute to the normal speed) that results from a given convolution kernel. This more granular understanding is particularly useful in the multiphase setting, where junctions are present.