Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain

Abstract

This paper is devoted to the study of the asymptotic behavior of the solutions of singularly perturbed transmission problems in a periodically perforated domain.The domain is obtained by making in R n a periodic set of holes, each of them of size proportional to a positive parameter . We first consider an ideal transmission problem and investigate the behavior of the solution as tends to 0. In particular, we deduce a representation formula in terms of real analytic maps of and of some additional parameters. Then we apply such result to a nonideal nonlinear transmission problem. KEYWORDSideal contact conditions, periodic dilute composite, singularly perturbed domain, transmission problem 3392

Dependence of effective properties upon regular perturbations

“…In connection with doubly periodic problems for composite materials, we mention the monograph of Grigolyuk and Fil'shtinskij [27], where the authors have proposed a method of integral equations for planar periodic problems in the frame of elasticity (see also Fil'shtinskij [22] and the more recent work Filshtinsky and Mityushev [23]). Finally, in [55] the fourth named author has explicitly computed the effective conductivity of a periodic dilute composite with perfect contact as a power series in the size of the inclusions (see also [54]).…”

Perturbation analysis of the effective conductivity of a periodic composite

“…then (ũ + j ,ũ − j ) = (0, 0). The case (λ + , λ − ) ∈ ]0, +∞[ 2 was proved in Pukhtaievych [47,Prop. 1].…”