H-convergence and regular limits for stratified media with low and high conductivities

Abstract

This paper considers a sequence of linear elliptic problems (PC) defined on structures which are stratified in one direction (say x₁). It gives the corresponding homogenized problem in cases where neither the conductivity matrices A^∊(x₁) nor their inverses are uniformly elliptic with respect to ∊ (for fixed ∊ however, these matrices have bounded coefficients and they are symmetric, uniformly elliptic with respect to xl). Conditions of H-convergence type on the matrices A^∊ are given which ensure that the solution u^∊ of (P^∊) converges in L² to the solution u of a problem of the same form (it is known that this is not general). They are : some relaxed ellipticity condition, convergence in the sense of measures of the matrice B∊classically associated to A' and regularity assumptions on the limit. Moreover, if an additional condition of equiintegrability with respect to intervals is fulfilled, the convergence of 11' to u is uniform with respect to x₁.