Estimates of eigenvalues and eigenfunctions in elliptic homogenization with rapidly oscillating potentials

TL;DR

This paper investigates the convergence rates and eigenvalue bounds for second-order elliptic equations with rapidly oscillating periodic coefficients and potentials, providing new estimates relevant to homogenization theory.

Contribution

It introduces new $H^1$ convergence rate estimates and bounds for Dirichlet eigenvalues and eigenfunctions in elliptic homogenization with oscillating potentials.

Findings

Established $H^1$ convergence rates using Dirichlet correctors.

Derived bounds for Dirichlet eigenvalues and eigenfunctions.

Provided $O( ext{epsilon})$ estimates for solutions with Dirichlet boundary conditions.

Abstract

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.