On obstacle problem for Brakke's mean curvature flow

TL;DR

This paper establishes the global existence of weak solutions to the obstacle problem for Brakke's mean curvature flow in 2D and 3D, using Allen-Cahn equations with forcing terms.

Contribution

It introduces a method to prove global existence of weak solutions for obstacle problems in Brakke's mean curvature flow with $C^{1,1}$ boundaries.

Findings

Proves global existence of weak solutions in 2D and 3D.

Utilizes Allen-Cahn equation with forcing to construct solutions.

Addresses obstacle problems with smooth boundary conditions.

Abstract

We consider the obstacle problem of the weak solution for the mean curvature flow, in the sense of Brakke's mean curvature flow. We prove the global existence of the weak solution with obstacles which have $C^{1,1}$ boundaries, in two and three space dimensions. To obtain the weak solution, we use the Allen-Cahn equation with forcing term.