Generic level sets in mean curvature flow are BV solutions

TL;DR

This paper demonstrates that generic level sets of viscosity solutions to mean curvature flow are BV solutions that optimally dissipate energy, extending previous foundational work linking viscosity solutions and Brakke flows.

Contribution

It establishes that generic level sets in mean curvature flow are BV solutions with optimal energy dissipation, broadening the understanding of the solution structure.

Findings

Generic level sets are BV solutions.

Level sets saturate the optimal energy dissipation rate.

Extends the connection between viscosity solutions and Brakke flows.

Abstract

We show that a generic levelset of the viscosity solution to mean curvature flow is a distributional solution in the framework of sets of finite perimeter by Luckhaus and Sturzenhecker, which in addition saturates the optimal energy dissipation rate. This extends the fundamental work of Evans and Spruck (J. Geom. Anal. 1995), which draws a similar connection between the viscosity solution and Brakke flows.