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 )W01,1(Ω), for a regular bounded open set Ω of \mathbb{R}^{N}RN, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists q > N−1q>N−1 if N > 2N>2, and q⩾1q⩾1 if N = 2N=2, such that any sequence of bounded energy is compact in W_{0}^{1,q}(\Omega )W01,q(Ω). Under this assumption the Γ-convergence of the functionals for the strong topology of L^{\infty }(\Omega )L∞(Ω) is proved to agree with the Γ-convergence for the strong topology of L^{1}(\Omega )L1(Ω). This leads to an integral representation of the Γ-limit in C_{0}^{1}(\Omega )C01(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.