Weak solutions to mean curvature flow respecting obstacles I: the graphical case

TL;DR

This paper develops a method to construct global weak solutions for mean curvature flow of hypersurfaces that are constrained by obstacles, specifically focusing on the case of complete graphs, extending previous approaches.

Contribution

It introduces a new approach based on Saez and the second author's method to handle obstacles in mean curvature flow for complete graphs, providing solutions for general initial data.

Findings

Established existence of global weak solutions respecting obstacles

Extended the approach to general initial data and one-sided obstacles

Provided a framework for analyzing obstacle-influenced mean curvature flow

Abstract

We consider the problem of evolving hypersurfaces by mean curvature flow in the presence of obstacles, that is domains which the flow is not allowed to enter. In this paper, we treat the case of complete graphs and explain how the approach of M. Saez and the second author yields a global weak solution to the original problem for general initial data and onesided obstacles.