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First-order expansions for eigenvalues and eigenfunctions in periodic homogenization

Part of: Spectral theory and eigenvalue problems Partial differential equations

Published online by Cambridge University Press:  20 March 2019

Jinping Zhuge
Jinping Zhuge*
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY40506, USA ([email protected])
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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.

Keywords

Homogenizationeigenvalueseigenfunctionsboundary layers

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure
Secondary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2189 - 2215
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Copyright
Copyright © Royal Society of Edinburgh 2019

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