Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

TL;DR

This paper proves the convergence of threshold dynamics schemes to anisotropic mean curvature flows with forcing, extending previous results to more general kernels and forces, and analyzes properties like convexity preservation and asymptotic shape.

Contribution

It generalizes convergence results for threshold dynamics schemes to anisotropic mean curvature flows with forcing terms, including anisotropic kernels and convexity preservation.

Findings

Convergence of schemes to anisotropic mean curvature flows with forcing.

Convexity of evolving fronts is preserved under certain conditions.

Large initial sets with constant forcing tend toward Wulff shape.

Abstract

We establish the convergence of threshold dynamics-type approximation schemes to propagating fronts evolving according to an anisotropic mean curvature motion in the presence of a forcing term depending on both time and position, thus generalizing the consistency result obtained in [Ishii-Pires-Souganidis, 1999] by extending the results obtained in [Caffarelli-Souganidis, 2010] for $\alpha \in [1,2)$ to anisotropic kernels and in the presence of a driving force. The limit geometric evolution is of a variational type and can be approximated, at a large scale, by eikonal-type equations modeling dislocations dynamics. We prove that it preserves convexity under suitable convexity assumptions on the forcing term and that convex evolutions of compact sets are unique. If the initial set is bounded and sufficiently large, and the driving force is constant, then the corresponding generalized front propagation is asymptotically similar to the Wulff shape.