Estimates of Eigenvalues and Eigenfunctions in Periodic Homogenization

TL;DR

This paper investigates the convergence rates of eigenvalues and eigenfunctions for elliptic operators with periodic coefficients, providing $O(\epsilon)$ estimates in $H^1$ norm to understand homogenization effects.

Contribution

It introduces new convergence rate estimates for eigenvalues and eigenfunctions in periodic homogenization, specifically an $O(\epsilon)$ bound in $H^1$ norm.

Findings

Convergence rates for Dirichlet eigenvalues are established.

Bounds for normal derivatives of eigenfunctions are derived.

An $O(\epsilon)$ estimate in $H^1$ norm is achieved.

Abstract

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an $O(\epsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.