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

This paper deals with the homogenization of nonlinear convex energies defined in , for a regular bounded open set Ω of RN, 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 . 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 L1(Ω). This leads to an integral representation of the Γ-limit in 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.