Level set mean curvature flow with Neumann boundary conditions

TL;DR

This paper explores the connection between level set and varifold methods for mean curvature flow with Neumann boundary conditions, proving that the level sets satisfy key inequalities and boundary conditions under certain initial data.

Contribution

It establishes a rigorous link between the level set and varifold approaches for mean curvature flow with Neumann boundary conditions, including proof of Brakke's inequality and boundary conditions.

Findings

Level sets satisfy Brakke's inequality.

Almost all level sets meet generalized Neumann boundary conditions.

Results hold for appropriate initial data.

Abstract

We investigate the relation between the level set approach and the varifold approach for the mean curvature flow with Neumann boundary conditions. With an appropriate initial data, we prove that the almost all level sets of the unique viscosity level set solution satisfy Brakke's inequality and a generalized Neumann boundary condition.