UNIFORM SPECTRAL ASYMPTOTICS FOR SINGULARLY PERTURBED LOCALLY PERIODIC OPERATORS

ABSTRACT

We consider the homogenization of the spectral problem for a singularly perturbed diffusion equation in a periodic medium. Denoting by ε the period, the diffusion coefficients are scaled as ε² and vary both on the macroscopic scale and on the periodic microscopic scale. We make a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain with non-degenerate quadratic behavior. We then prove an exponential localization phenomena at this minimum point. Namely, the k-th original eigenfunction is shown to be asymptotically given by the product of the first cell eigenfunction (at the ϵ scale) times the k-th eigenfunction of an homogenized problem (at the scale). The homogenized problem is a diffusion equation with quadratic potential in the whole space. We first perform asymptotic expansions, and then prove convergence by using a factorization strategy.

ACKNOWLEDGMENTS

This work was partly done when both authors enjoyed the hospitality of the Laboratoire d'Analyse Numérique at the Université Pierre et Marie Curie in Paris. The final version of the paper was achieved during the stay of A. Piatnitski at Ecole Polytechnique in January 2001.