Curvature flow in heterogeneous media

TL;DR

This paper studies curvature-driven flows of planar curves in heterogeneous media with space-dependent forcing, providing homogenization estimates and analyzing limit solutions, especially for special cases like graph initial curves.

Contribution

It introduces homogenization estimates for curvature flows with space-dependent forcing and rigorously analyzes limit solutions in specific scenarios, including non-continuous forcing terms.

Findings

Homogenization estimates depend only on the uniform norm of the forcing.

Limit solutions can be rigorously characterized for graph initial curves.

A framework for defining solutions with bounded, non-continuous forcing is developed.

Abstract

In recent years, there has been a growing interest in geometric evolution in heterogeneous media. Here we consider curvature driven fows of planar curves, with an additional space-dependent forcing term. Motivated by a homogenization problem, we look for estimates which depend only on the uniform norm of the forcing term. By means of an asymptotic analysis, we discuss the properties of the limit solutions of the homogenization problem, which we can rigorously solve in some special cases: that is, when the initial curve is a graph, and the forcing term does not depend on the vertical direction. As a by-product, in such cases we are able to define a soluton of the geometric evolution when the forcing term is just a bounded, not necessarily continuous, function.