# Homogenization of convex functionals which are weakly coercive and not equi-bounded from above

### Marc Briane

Institut de Recherche Mathématique de Rennes, INSA de Rennes, France### Juan Casado-Díaz

Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain

## Abstract

This paper deals with the homogenization of nonlinear convex energies defined in $W_{0}(Ω)$, for a regular bounded open set $Ω$ of $R_{N}$, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists $q>N−1$ if $N>2$, and $q⩾1$ if $N=2$, such that any sequence of bounded energy is compact in $W_{0}(Ω)$. Under this assumption the $Γ$-convergence of the functionals for the strong topology of $L_{∞}(Ω)$ is proved to agree with the $Γ$-convergence for the strong topology of $L_{1}(Ω)$. This leads to an integral representation of the $Γ$-limit in $C_{0}(Ω)$ thanks to a local convex density. An example based on a thin cylinder with very low and very large energy densities, which concentrates to a line shows that the loss of the weak coercivity condition can induce nonlocal effects.

## Cite this article

Marc Briane, Juan Casado-Díaz, Homogenization of convex functionals which are weakly coercive and not equi-bounded from above. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 4, pp. 547–571

DOI 10.1016/J.ANIHPC.2012.10.005