Homogenization of imperfect transmission problems: in the case of weakly converging data

We study the homogenization of a second order linear evolution problem in a domain of Rn with imperfect interface. Namely, we assume that = 1" [ 2" with 1" connected and 2", union of "- periodic connected inclusions of size ", and we prescribe on " = @ 2", the interface separating 1" and 2", the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivatives through a function of order " . More precisely, we describe, for different values of the parameter 2 R, the asymptotic behavior, as " ! 0, of the solution of the following problem: 8>>>>>>< >>>>>>: u00 " − div(A"ru") = f" in ( 1" [ 2")×]0, T[, [A"ru"] · n1" = 0 on "×]0, T[, A"ru1" · n1" = −" h"[u"] on "×]0, T[, u" = 0 on @ ×]0, T[, u"(0) = U0 " in , u0 "(0) = U1 " in . where A"(x) := A(x/"), A being a periodic, symmetric, bounded and positive definite matrix field, h"(x) := h(x/"), with h positive, bounded and periodic and u" = (u1", u2"), ui" being defined in "i , i = 1, 2. We denoted by ni" the unitary outward normal to i" and by [ ] the jump trough ". This problem models the wave propagation in a medium made by two components with very different coefficients of propagation. This leads to the jump boundary condition on the interface. According to , different limit behaviors are obtained. In particular for = 1 a linear memory effect appears. Some non-standard corrector results are also given. References [1] P.Donato, L.Faella, S.Monsurr`o, Homogenization of the wave equation in composites with imperfect interface: A memory effect, J. Math. Pures Appl. (9) 87 (2007), 119-143. [2] P.Donato, L.Faella, S.Monsurr`o, Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces, SIAM J. Math. Anal., Vol. 40 (2009), No. 5, pp. 1952-1978.