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

Abstract

This paper deals with the homogenization of nonlinear convex energies defined in $W_0^{1,1}(\Omega )$, for a regular bounded open set $\Omega$ of ${\mathbb{R}}^N$, 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 $W_0^{1,q}(\Omega )$. Under this assumption the $\Gamma$-convergence of the functionals for the strong topology of $L^{\infty }(\Omega )$ is proved to agree with the $\Gamma$-convergence for the strong topology of $L^1(\Omega )$. This leads to an integral representation of the $\Gamma$-limit in $C_0^1(\Omega )$ 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.