Corrector results for a parabolic problem with a memory effect

Abstract

Abstract. The aim of this paper is to provide the correctors associated to the homogenization of a parabolic problem describing the heat transfer. The results here complete the earlier study in [Jose, Rev. Roumaine Math. Pures Appl. 54 (2009) 189-222] on the asymptotic behaviour of a problem in a domain with two components separated by an ε-periodic interface. The physical model established in [Carslaw and Jaeger, The Clarendon Press, Oxford (1947)] prescribes on the interface the condition that the flux of the temperature is proportional to the jump of the temperature field, by a factor of order ε γ . We suppose that −1 < γ ≤ 1. As far as the energies of the homogenized problems are concerned, we consider the cases −1 < γ < 1 and γ = 1 separately. To obtain the convergence of the energies, it is necessary to impose stronger assumptions on the data. Mathematics Subject Classification. 35B27, 35K20, 82B24.

Homogenization of diffusion problems with a nonlinear interfacial resistance

“…For similar homogenization problems of elliptic type we refer the reader to [9,27,28,30,33,34,39] and for related parabolic problems [22][23][24]29,31], together with the references therein. For the homogenization of linear and quasilinear elliptic problems with a nonlinear Robin condition containing a nonlinear term with the same growth as that in the boundary condition considered in the present paper, we refer to [8,16], respectively, where the periodic unfolding method was used.…”

Correctors for the Nonlinear Wave Equations in Perforated Domains

“…For the semilinear case, Brahim-Otsman et al [1] gave the homogenization and corrector results for a fixed domain. Our study is also related to that for the parabolic case in [14,15,17,18,20,27]. This paper is organized as follows.…”

Homogenization and exact controllability for problems with imperfect interface

“…In Section 2, we describe in details the two component domain Ω. In Section 3, at first, we recall the definitions and the properties of specific functional spaces, suitable for the solutions of these kinds of interface problems, introduced in [21,23,28,49]. Then, we remind the definitions and the main properties of two unfolding operators for the two component domain Ω, defined for the first time in [7,26].…”

“…For previous homogenization results in the case of weakly converging data, we quote Tartar (see [9], Proposition 8.17, Remark 8.18 and Theorem 8.19) and [13,52]. As regards evolution problems in domains with imperfect interface, we refer to [21,22,23,54,55].…”

“…We refer the reader to [38,39] for the optimal control of hyperbolic problems in composites with imperfect interface and to [42] for the optimal control of rigidity parameters of thin inclusions in composite materials. We quote [23]÷ [25] and [34] for the correctors and the approximate control for a class of parabolic equations with interfacial contact resistance. In [30], the approximate controllability of linear parabolic equations in perforated domains is considered.…”