Gradient estimates for mean curvature flow with Neumann boundary conditions

TL;DR

This paper establishes boundary gradient estimates for mean curvature flow of graphs with Neumann boundary conditions, proving existence and regularity results using a Huisken-type monotonicity formula and boundary analysis.

Contribution

It introduces new boundary gradient estimates and regularity conditions for mean curvature flow with Neumann boundary conditions, expanding understanding of flow behavior near boundaries.

Findings

Derived boundary gradient estimates for mean curvature flow

Proved existence of flow of graphs under Neumann conditions

Established regularity criteria for transport terms

Abstract

We study the mean curvature flow of graphs both with Neumann boundary conditions and transport terms. We derive boundary gradient estimates for the mean curvature flow. As an application, the existence of the mean curvature flow of graphs is presented. A key argument is a boundary monotonicity formula of a Huisken type derived using reflected backward heat kernels. Furthermore, we provide regularity conditions for the transport terms.