Uniform resolvent convergence for strip with fast oscillating boundary

TL;DR

This paper studies how elliptic operators behave on a strip with rapidly oscillating boundaries, establishing uniform resolvent convergence and describing how boundary conditions can change during homogenization.

Contribution

It provides a comprehensive analysis of homogenization for elliptic operators on oscillating boundaries, including convergence rates and boundary condition transformations.

Findings

Uniform resolvent convergence established

Rates of convergence depend on oscillation amplitude and period

Boundary condition types can change after homogenization

Abstract

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change.