Homogenization of systems with equi-integrable coefficients

Marc Briane (1) , Juan Casado-Diaz (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 EDAN US - Departamento de Ecuaciones Diferenciales y Analisis Numerico

In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for scalar equations does not hold for vector-valued systems, we present an alternative approach based on an approximation result by Lipschitz functions due to Acerbi and Fusco combined with a Meyers $L^p$-estimate adapted to the functional ellipticity condition. The present framework includes in particular the elasticity case and the reinforcement by stiff thin fibers.

Mots clés: equi-integrable coefficients; H-convergence; homogenization; vector-valued systems; not equi-bounded coefficients

Domaines: Equations aux dérivées partielles [math.AP]

Marc Briane, Juan Casado-Diaz. Homogenization of systems with equi-integrable coefficients. ESAIM: Control, Optimisation and Calculus of Variations, 2014, 20 (4), pp.1214-1223. ⟨10.1051/cocv/2014013⟩. ⟨hal-01102183⟩