First-order expansions for eigenvalues and eigenfunctions in periodic homogenization

Abstract

For a family of elliptic operators with periodically oscillating coefficients, $-\text{div}(A(\cdot /\epsilon )\nabla )$ with tiny $\epsilon >0$, 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 $H_{\text{loc}}^1$. Our results rely on the recent progress on the homogenization of boundary layer problems.

arXiv comment27 pages; comments are welcome