Multiscale analysis of diffusion processes in composite media

Abstract

a b s t r a c tThe goal of this paper is to present some homogenization results for a nonlinear problem arising in the modeling of diffusion in a periodic structure formed by two media with different properties, separated by an active interface. Our setting is relevant for modeling heat diffusion in composite materials with imperfect interfaces or electrical conduction in biological tissues. The approach we follow is based on the periodic unfolding method, which allows us to deal with general media.

On the homogenization of a two-conductivity problem with flux jump

“…In the pioneering work [8], using the asymptotic expansion method, the authors study the homogenization of a thermal problem in a two-component composite with interfacial barrier in the particular case in which the conductivities of the two components are both of order one. For this problem, the convergence results were rigorously justified later by using various mathematical methods: the energy method in [21] and [33], the two-scale convergence method in [23] and the unfolding method for periodic homogenization in [20], [39], [40] and [32], to quote just a few of them. Also, for problems involving jumps in the solution in other contexts, such as heat transfer in polycrystals with interfacial resistance, linear elasticity problems or problems modeling the electrical conduction in biological tissues, see [4], [5], [24], [25], [26], [30] and [41].…”

Homogenization results for the calcium dynamics in living cells

“…Using the above convergence results and Lebesgue's convergence theorem, we can pass to the limit in (4.7) (see, for details, [6,9,21] and [22]). Thus, we get:…”

Exponential decay for a nonlinear model for electrical conduction in biological tissues

“…Composite materials have widespread applications in science and technology and, for this reason, have been extensively studied especially using homogenization techniques (we quote, among others, [15,17,19,31,32,35,36]). In this framework the authors, and co-workers, have investigated a problem arising in electric conduction in biological tissues with the purpose of obtaining some useful results for applications in electrical tomography (see [4][5][6][7][8][9][10][11][12]).…”