Level-set forced mean curvature flow with the Neumann boundary condition

TL;DR

This paper investigates the properties and long-term behavior of solutions to a level-set forced mean curvature flow with Neumann boundary conditions, establishing Lipschitz regularity and conditions for global Lipschitz continuity.

Contribution

It provides new results on Lipschitz regularity, large time behavior, and the sharpness of conditions on the forcing term for this geometric flow.

Findings

Solution is Lipschitz in time and locally Lipschitz in space.

Under an additional condition, the solution is globally Lipschitz.

Examples show the necessity of the additional condition for global Lipschitz continuity.

Abstract

Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.