A brush problem. Homogenization involving thin domains and PDEs in graphs

TL;DR

This paper studies the homogenization of elliptic PDEs in complex comb-like domains with thin teeth, revealing how the limit problem can be represented as a differential equation on a graph, even without periodicity.

Contribution

It introduces a non-periodic homogenization framework for PDEs in comb domains with thin structures, extending the unfolding operator method to this setting.

Findings

Limit problem can be interpreted as a differential equation on a graph.

Homogenization results hold even when the teeth distribution is non-periodic.

The approach accommodates variable asymptotic density of teeth.

Abstract

This work analyses the homogenization of a linear elliptic equation with Neumann boundary conditions in a comb/brush domain, composed of a fixed base and a family of thin teeth. The teeth are defined as rescalings of order less than or equal to $\varepsilon$ of a model tooth of arbitrary shape. Periodicity in their distribution is not assumed; instead, the existence of an asymptotic limit density $\theta$, which may vanish in certain regions, is assumed. The convergence analysis is performed using an adaptation of the unfolding operator method to a non-periodic framework. Finally, it is shown that, under certain conditions on the geometry of the teeth, the resulting limit problem can be interpreted as a differential equation on a graph.