Singular perturbations and homogenization in stratified media

Abstract

We consider a heterogeneous structure which is stratified in some direction (say x1). The strips are assumed to have very high conductivity which means that the conductivity coefficient a{x1) ranges over |0.+ ∞| oscillating without the usual boundedness assumption (with respect to the L∞-norm) of a and a^-1 This problem has been already considered by C.Picard J. Mossino and B.Heron who studied in particular the asymptotic behaviour as ∊-0 of the diffusion equation:

boundary condition under the assumption that the sequences (a∊) and (a∊)^-1 have limits for the weak topology of L¹.

In this paper (see also preprint [19]). we propose a new method (convex functionals on measures ranging into L² which allow us to pass to the limit in (*) when (a∊) and (a∊)¹converges ∗-weakly to general Radon measures, say m and tr. Due to the loss of coercivity, the limit of the sequence of solutions (u∊) may have discontinuities whose energy (associated to the singular part of ∊) is concentrated on hyperplanes orthogonal to the On the same way , diffusions surface energies will appear associated with the singular part of m. Extensions to nonlinear problems are also considered.

Keywords:

• heterogeneous material
• weak *convergence
• Radon measures
• convex functionals

^∗UFR Sciencc,. Université de Toulon et du Var. BP 132. 83957 La Gardc Crdm (France)

^∗∗Lab.Analyse NurriCrique.Bat. 325 . Universlté de Paris-Sud. 91405 Orasay et Faculté Mahtématiques et Informatique 33 rue Samt Leu,80039 Amiens

^∗UFR Sciencc,. Université de Toulon et du Var. BP 132. 83957 La Gardc Crdm (France)

^∗∗Lab.Analyse NurriCrique.Bat. 325 . Universlté de Paris-Sud. 91405 Orasay et Faculté Mahtématiques et Informatique 33 rue Samt Leu,80039 Amiens

Notes

^∗UFR Sciencc,. Université de Toulon et du Var. BP 132. 83957 La Gardc Crdm (France)

^∗∗Lab.Analyse NurriCrique.Bat. 325 . Universlté de Paris-Sud. 91405 Orasay et Faculté Mahtématiques et Informatique 33 rue Samt Leu,80039 Amiens