Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

TL;DR

This paper develops a mathematical framework for crystalline mean curvature flow with nonuniform driving forces, proving existence, uniqueness, and comparison principles for solutions, and addressing the effects of spatially inhomogeneous forces on curvature.

Contribution

It introduces a novel notion of solutions and establishes fundamental properties for crystalline mean curvature flow with spatially varying driving forces.

Findings

Existence and uniqueness of level set solutions are proven.

Comparison principle for continuous solutions is established.

Crystalline curvature depends on both geometry and inhomogeneous driving forces.

Abstract

A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.