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

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 $θ$, 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.