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First-order expansions for eigenvalues and eigenfunctions in periodic homogenization

Published online by Cambridge University Press:  20 March 2019

Jinping Zhuge*
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY40506, USA (jinping.zhuge@uky.edu)

Abstract

For a family of elliptic operators with periodically oscillating coefficients, $-{\rm div}(A(\cdot /\varepsilon )\nabla )$ with tiny ε > 0, we comprehensively study the first-order expansions of eigenvalues and eigenfunctions (eigenspaces) for both the 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 L2 or $H^1_{\rm loc}$. Our results rely on the recent progress on the homogenization of boundary layer problems.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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