# Propagation of a Mean Curvature Flow in a Cone

## TL;DR

This paper studies mean curvature flow within a cone, establishing global existence of solutions, and analyzes the homogenization limit as the contact angle oscillates periodically, using self-similar solutions for characterization.

## Contribution

It introduces a priori estimates for radially symmetric solutions and characterizes the homogenization limit using the slowest self-similar solution.

## Key findings

- Global existence of radially symmetric solutions
- Homogenization limit characterized by self-similar solutions
- Error estimate of order $O(1)\varepsilon^{1/6}$ in the homogenization process

## Abstract

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being $\varepsilon$-periodic in its position. First, by constructing a family of self-similar solutions, we give a priori estimates for the radially symmetric solutions and prove the global existence. Then we consider the homogenization limit as $\ve\to 0$, and use {\it the slowest self-similar solution} to characterize the solution, with error $O(1)\ve^{1/6}$, in some finite time interval.