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

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