Affiliation: | (1) Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau, France;(2) IRMAR et Centre de Mathématiques, INSA de Rennes, 20 av. des buttes de Coesmes, 35043 Rennes Cedex, France;(3) Narvik Institute of Technology, HiN, 385, 8505 Narvik, Norway;(4) CEA Saclay, DEN/DM2S, 91191 Gif-sur-Yvette, France;(5) IISc-TIFR Mathematics Programme, TIFR Center, 1234, Bangalore, 560012, India;(6) P.N.Lebedev Physical Institute RAS, Leninski prospect 53, Moscow, 117333, Russia |
Abstract: | We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting by the period, the potential is scaled as –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. |