Homogenization of periodic systems with large potentials

Abstract

We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by ε the period, the potential is scaled as ε⁻². Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be approximately factorized as the product of a fast oscillating cell eigenfunction and of a slowly varying solution of a scalar second-order equation. This result applies to various types of equations such as parabolic, hyperbolic or eigenvalue problems, as well as fourth-order plate equation. We also prove that, for well-prepared initial data concentrating at the bottom of a Bloch band, the resulting homogenized tensor depends on the chosen Bloch band. Our method is based on a combination of classical homogenization techniques (two-scale convergence and suitable oscillating test functions) and of Bloch waves decomposition.