Transmission conditions obtained by homogenisation

TL;DR

This paper investigates the asymptotic behavior of solutions to minimization problems in domains with shrinking sets, establishing transmission conditions on the limit manifold characterized by measures supported on it.

Contribution

It introduces a novel approach to derive transmission conditions via homogenization, linking measures on the limit manifold to the sequence of compact sets.

Findings

Limit solutions satisfy transmission conditions expressed through measures.

Characterization of all possible measures obtained as limits.

Connection between the measures and the sequence of compact sets via local minimization.

Abstract

Given a bounded open set in $\mathbb{R}^n$, $n\ge 2$, and a sequence $(K_j)$ of compact sets converging to an $(n-1)$-dimensional manifold $M$, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on $\Omega\setminus K_j$, with Neumann boundary conditions on $\partial(\Omega\setminus K_j)$. We prove that the limit of these solutions is a minimiser of the same functional on $\Omega\setminus M$ subjected to a transmission condition on $M$, which can be expressed through a measure $\mu$ supported on $M$. The class of all measures that can be obtained in this way is characterised, and the link between the measure $\mu$ and the sequence $(K_j)$ is expressed by means of suitable local minimum problems.