Existence of weak solutions to volume-preserving mean curvature flow with obstacles

TL;DR

This paper establishes the existence of global weak solutions for volume-preserving mean curvature flow with obstacles using phase field methods, extending previous models to include obstacles and spatially dependent forcing.

Contribution

It proves the convergence of Allen-Cahn solutions with a multiplier to weak solutions of the flow in all dimensions, incorporating obstacles and spatial forcing effects.

Findings

Existence of global weak solutions for obstacle-including flow

Convergence of Allen-Cahn solutions with multiplier to flow solutions

Vanishing of the discrepancy measure despite spatial forcing

Abstract

We prove the existence of global-in-time weak solutions to volume-preserving mean curvature flow with in the presence of obstacles by the phase field method in all dimensions. Namely, we prove the convergence of solutions to the Allen-Cahn equation with a multiplier to a weak solution to the flow. The choice of the multiplier is motivated from [Mugnai-Seis-Spadaro '16], [Kim-Kwon '20], and [Takasao '23], which enables us to complete the comparison between the multiplier and the forcing that stops the intrusion into the obstacle. We also prove the vanishing of the discrepancy measure by dealing with the forcing term that is now spatially dependent due to the obstacles.