Homogenization of periodic non self-adjoint problems with large drift and potential

¹ Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, Paris, France; gregoire.allaire@polytechnique.fr
² Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain; rafael.orive@uam.es

Received: 11 July 2005
Revised: 29 June 2006

Abstract

We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the macroscopic scale and on the periodic microscopic scale. Denoting by ε the period, the potential or zero-order term is scaled as and the drift or first-order term is scaled as . Under 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 prove an exponential localization at this minimum point. The homogenized problem features a diffusion equation with quadratic potential in the whole space.