Homogenization of eigenvalue problem for Laplace–Beltrami operator on Riemannian manifold with complicated ‘bubble‐like’ microstructure | Zendy

Khrabustovskyi Andrii | Zendy

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Homogenization of eigenvalue problem for Laplace–Beltrami operator on Riemannian manifold with complicated ‘bubble‐like’ microstructure

Author(s) -

Khrabustovskyi Andrii

Publication year - 2009

Publication title -

mathematical methods in the applied sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.719

H-Index - 65

eISSN - 1099-1476

pISSN - 0170-4214

DOI - 10.1002/mma.1128

Subject(s) - laplace–beltrami operator , homogenization (climate) , mathematics , eigenfunction , riemannian manifold , eigenvalues and eigenvectors , laplace operator , mathematical analysis , laplace's equation , bubble , manifold (fluid mechanics) , heat equation , operator (biology) , p laplacian , boundary value problem , physics , mechanical engineering , biodiversity , ecology , biochemistry , chemistry , repressor , quantum mechanics , gene , mechanics , transcription factor , engineering , biology

We study the asymptotic behavior of the eigenvalues and the eigenfunctions of the Laplace–Beltrami operator on a Riemannian manifold M ε depending on a small parameter ε>0 and whose structure becomes complicated as ε→0. Under a few assumptions on scales of M ε we obtain the homogenized eigenvalue problem. In addition we study the behavior of the heat equation on M ε and investigate the large time behavior of the homogenized equation. Copyright © 2009 John Wiley & Sons, Ltd.

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