ON LIPSCHITZ REGULARITY FOR LEVEL-SET FORCED MEAN CURVATURE FLOW UNDER THE NEUMANN BOUNDARY CONDITION (Innovation of the theory for evolution equations: developments via cross-disciplinary studies)

Abstract

Here, we study a level-set forced mean curvature flow with the homo-geneous Neumann boundary condition. We 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 continuous, and we obtain the large time behavior of the solution in this setting. Also, we give an example to demonstrate that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz continuous.