Recent progress in the theory of homogenization with oscillating Dirichlet data

TL;DR

This paper reviews recent advances in the homogenization of elliptic systems with oscillating coefficients and boundary data, highlighting convergence rates, the homogenized limit system, and boundary layer effects.

Contribution

It summarizes the authors' recent work that solves a longstanding open problem by analyzing the homogenization process with oscillating boundary conditions.

Findings

Solutions converge in L2 with a power rate

Identified the homogenized limit system and boundary data

Revealed the influence of boundary layers on the homogenized system

Abstract

This note is a summary of the recent paper [9]. Here, we study the homogenization of elliptic systems with Dirichlet boundary condition, when both the coefficients and the boundary datum are oscillating. In particular, in the paper [9], we showed that, the solutions converge in L2 with a power rate, and we identified the homogenized limit system and the homogenized boundary data. Due to a boundary layer phenomenon, this homogenized system depends in a non trivial way on the boundary. The analysis in [9] answers a longstanding open problem, raised for instance in [4]