First-order expansions for eigenvalues and eigenfunctions in periodic homogenization

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, s...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 150; no. 5; pp. 2189 - 2215
Main Author: Zhuge, Jinping
Format: Journal Article
Language:English
Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.10.2020
Cambridge University Press
Subjects:
Boundary layers
Dirichlet problem
Domains
Eigenvalues
Eigenvectors
Homogenization
eigenfunctions
Homogenization
eigenvalues
35B27
boundary layers
35P20
ISSN:0308-2105, 1473-7124
Online Access:Get full text
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Summary: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.
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content type line 14
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2019.8