First-order expansions for eigenvalues and eigenfunctions in periodic homogenization

TL;DR

This paper develops first-order asymptotic expansions for eigenvalues and eigenfunctions of elliptic operators with periodic coefficients, advancing homogenization theory in smooth, convex, and general domains.

Contribution

It introduces a new first-order correction term for eigenfunctions and extends homogenization results to boundary layer problems in various domains.

Findings

Derived explicit first-order eigenvalue expansions.

Established new correction terms for eigenfunctions.

Extended homogenization techniques to boundary layers.

Abstract

For a family of elliptic operators with periodically oscillating coefficients, $-\text{div}( A(\cdot/\varepsilon) \nabla) $ with tiny $\varepsilon>0$, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions (eigenspaces) for both Dirichlet and Neumann problems in bounded, smooth and strictly convex domains (or more general domains of finite type). A new first-order correction term is introduced to derive the expansion of eigenfunctions in $L^2$ or $H^1_{\text{loc}}$. Our results rely on the recent progress on the homogenization of boundary layer problems.