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Nonlinear Functional Analysis And Its Applications

Matching of asymptotic expansions of solutions of boundary value problems

In this paper, we are dealing with the mathematical modeling and homogenization of nonlinear reaction-diffusion processes in a porous medium that consists of two components separated by an interface. One of the components is connected, and the other one is disconnected and consists of periodically distributed inclusions. At the interface, the fluxes are given by nonlinear functions of the concentrations on both sides of the interface. Thus, the concentrations may be discontinuous across the interface. For the derivation of the effective (homogenized) model, we use the method of two-scale convergence. To prove the convergence of the nonlinear terms, especially those defined on the microscopic interface, we give a new approach which involves the boundary unfolding operator and a compactness result for Banach-space-valued functions. The model is motivated by metabolic and regulatory processes in cells, where biochemical species are exchanged between organelles and cytoplasm through the organellar membranes. In this context the nonlinearities are given by kinetics corresponding to multispecies enzyme catalyzed reactions, which are generalizations of the classical Michaelis--Menten kinetics to multispecies reactions.

Homogenization of Reaction--Diffusion Processes in a Two-Component Porous Medium with Nonlinear Flux Conditions at the Interface

The periodic unfolding method was introduced by D. Cioranescu, A. Damlamian and G. Griso for studying the classical periodic homogenization in fixed domains and more recently extended to periodically perforated domains by D. Cioranescu, A. Damlamian, P. Donato, G. Griso, and R. Zaki. Here, the method is adapted to two-component domains which are separated by a periodic interface. The unfolding method is then applied to an elliptic problem with a jump of the solution on the interface, which is proportional to the flux and depends on a real parameter. We prove some homogenization and corrector results, which recover and complete those previously obtained by the first author and S. Monsurr`o. Bibliography: 32 titles. Illustrations: 2 figures.

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The periodic unfolding method for a class of imperfect transmission problems

Effective boundary condition for Stokes flow over a very rough surface

Uniform resolvent convergence for strip with fast oscillating boundary

In this paper the asymptotic behaviour of a second-order linear evolution problem is studied in a domain, a part of wich has an oscillating boundary. An homogeneous Neumann condition is given on the whole boundary of the domain. Moreover the behaviour of associated optimal control problem is analyzed.

Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary

Homogenization of the wave equation in composites with imperfect interface : A memory effect

This paper concerns an equilibrium problem for a two-dimensional elastic body with a thin Timoshenko elastic inclusion and a thin rigid inclusion. It is assumed that the inclusions have a joint point and we analyze a junction problem for these inclusions. The existence of solutions is proved and the different equivalent formulations of the problem are discussed. In particular, the junction conditions at the joint point are found. A delamination of the elastic inclusion is also assumed. In this case, the inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between the crack faces. We investigate the convergence to infinity and zero of a rigidity parameter of the elastic inclusion. It is proved that in the limit, we obtain a rigid inclusion and a zero rigidity inclusion (a crack).

Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies

We present here some corrector results for the homogenization of the wave equation in a two-component composite with $\varepsilon$-periodic connected inclusions which complete the homogenization results proved in [P. Donato, L. Faella, and S. Monsurro, J. Math. Pures Appl., 87 (2007), pp. 119–143] by the authors. On the interface separating the two components we prescribe a jump of the solution proportional to the conormal derivatives via a function of order $\varepsilon^\gamma$, with $-1<\gamma\leq1$. Due to different expressions of the energies of the limit problems, the cases $-1<\gamma<1$ and $\gamma=1$ need to be treated separately. The second one, where a memory effect appears in the homogenized problem, is the most interesting. For this critical case, displaying lack of compactness, we in particulur establish the central upper semicontinuity type inequalities by splitting a related energy term into a compact part and a part vanishing in appropriate norms.

Luisa Faella

Papers

Uniform resolvent convergence for strip with fast oscillating boundary

Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary

Homogenization of the wave equation in composites with imperfect interface : A memory effect

Junction problem for rigid and Timoshenko elastic inclusions in elastic bodies

Correctors for the Homogenization of a Class of Hyperbolic Equations with Imperfect Interfaces