Oscillatory integrals and periodic homogenization of Robin boundary value problems

TL;DR

This paper studies the homogenization of elliptic systems with oscillating Robin boundary conditions, providing qualitative results on Lipschitz domains and quantitative convergence rates on smooth, convex domains.

Contribution

It establishes homogenization results for Robin boundary problems on Lipschitz domains and derives explicit convergence rates for smooth, convex domains.

Findings

Qualitative homogenization on Lipschitz domains under non-resonance.

Quantitative convergence rates in $L^2$ for smooth, convex domains.

Use of oscillatory integral estimates to obtain dimension-dependent rates.

Abstract

In this paper, we consider a family of second-order elliptic systems subject to a periodically oscillating Robin boundary condition. We establish the qualitative homogenization theorem on any Lipschitz domains satisfying a non-resonance condition. We also use the quantitative estimates of oscillatory integrals to obtain the dimension-dependent convergence rates in $L^2$, assuming that the domain is smooth and strictly convex.