On mean curvature flow with forcing

TL;DR

This paper studies a volume-dependent mean curvature flow, proving preservation of star-shapedness, well-posedness, instant smoothness, and exponential convergence to equilibrium, using viscosity solutions and variational methods.

Contribution

It introduces a new analysis of mean curvature flow with volume-dependent forcing, establishing preservation of geometric properties and solution regularity.

Findings

Preservation of $ ho$-reflection property over time.

Solutions become instantly smooth from $ ho$-reflection initial data.

Flow converges exponentially to a unique equilibrium.

Abstract

This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the $\rho$-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with $\rho$-reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow's exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed in [Kim-Feldman, 2014] to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.