Patrizia Donato

Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions

This article is devoted to the homogenization of a quasilinear elliptic equation with oscillating coefficients in a periodically perforated domain. A nonlinear Robin condition is prescribed on the boundary of the holes, depending on a real parameter γu2009≥u20091. We suppose that the data satisfy some suitable hypotheses which ensure, as proved by the authors in Cabarrubias and Donato [B. Cabarrubias and P. Donato, Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions, Carpathian J. Math. (2) (2011) (to appear)], the existence and the uniqueness of a solution of the problem. In particular, suitable growth conditions are assumed on the nonlinear boundary term, as done in Cioranescu–Donato–Zaki [D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal. 53 (2007), pp. 209–235]. On the quasilinear term, some assumptions on the modulus of continuity introduced in Chipot [M. Chipot, Elliptic Equations: An Introductory Course, Birkhauser Verlag AG, Germany, 2009] and weaker than a Lipschitz condition are prescribed. We study the convergence to a limit problem, which is identified by using the periodic unfolding method. We also prove the well-posedness of the limit system. To do that, we show that the homogenized operator inherits the modulus of continuity of the initial problem. As a consequence, the uniqueness of a solution of the homogenized quasilinear problem follows.

Quasilinear elliptic equations with quadratic growth in unbounded domains

On donne des resultats dexistence et de regularite pour des equations elliptiques quasilineaires a croissance quadratique par rapport au gradient du type Au+f(x,u,grad u)=0 dans Ω, u=0 sur ∂Ω

Homogénéisation et contrôlabilité approchée de l'équation de la chaleur dans des domaines perforés

Resume Nous etudions lhomogeneisation et les correcteurs de lequation de la chaleur avec des coefficients oscillants dans un domaine periodiquement perfore par des trous de la meme taille que la periode. Nous etudions aussi un probleme de controlabilite approchee dans ce domaine et obtenons a la limite la controlabilite du probleme homogeneise. Dans le probleme limite, lapproximation de letat final est alteree par une constante qui depend de la proportion de materiau dans le domaine perfore et qui est egale a l sil ny a pas de trous.

Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method

We study an ε-periodic model of a medium with double porosity which consists of two components, one of them being connected. We assume that the elasticity of the medium in the inclusion is of order ε2 and also, on the interface between the two components, we consider a jump of the displacement vector condition, proportional to the stress tensor which is continuous. The aim of the paper is to prove the convergence of the homogenization process using the periodic unfolding method.

Unfolding homogenization in doubly periodic media and applications

We define a reiterated unfolding operator for a doubly periodic domain presenting two periodicity scales. Then we show how to apply it to the homogenization of both linear and nonlinear problems. The main novelty is that this method allows the use of test functions with one scale of periodicity only and it considerably simplifies the proofs of the convergence results. We illustrate this new approach on a Poisson problem with Dirichlet boundary conditions and on the flow of a power law fluid in a doubly periodic porous medium.