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 2020-10, Vol.150 (5), p.2189-2215
Main Author: Zhuge, Jinping
Format: Article
Language:English
Subjects:
Boundary layers
Dirichlet problem
Domains
Eigenvalues
Eigenvectors
Homogenization
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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.
ISSN:0308-2105
1473-7124
DOI:10.1017/prm.2019.8