Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

TL;DR

This paper extends the theory of mean curvature flows by proving convergence of a minimizing movements scheme to solutions for anisotropic and inhomogeneous cases, removing translation invariance assumptions.

Contribution

It generalizes recent results on mean curvature evolution by establishing convergence without translation invariance, including in low dimensions and under energy convergence conditions.

Findings

Proves convergence of minimizing movements scheme to level set solutions

Establishes convergence to distributional solutions in low dimensions

Generalizes mean curvature flow theory to anisotropic and inhomogeneous cases

Abstract

In this paper we address anisotropic and inhomogeneous mean curvature flows with forcing and mobility, and show that the minimizing movements scheme converges to level set/viscosity solutions and to distributional solutions \textit{\`a la} Luckhaus-Sturzenhecker to such flows, the latter holding in low dimension and conditionally to a convergence of the energies. By doing so we generalize recent works concerning the evolution by mean curvature by removing the hypothesis of translation invariance, which in the classical theory allows to simplify many arguments.