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

Briane, Marc  ; Casado-Díaz, Juan 

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 4, pp. 547-571

Résumé

This paper deals with the homogenization of nonlinear convex energies defined in $W_0^{1,1}\left(\Omega \right)$, 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 $W_0^{1,q}\left(\Omega \right)$. Under this assumption the Γ-convergence of the functionals for the strong topology of $L^{\infty }\left(\Omega \right)$ is proved to agree with the Γ-convergence for the strong topology of $L^1\left(\Omega \right)$. This leads to an integral representation of the Γ-limit in $C_0^1\left(\Omega \right)$ 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.

Détail

DOI : 10.1016/j.anihpc.2012.10.005

Classification : 35B27, 35B50, 35J60
Keywords: Homogenization, Convex functionals, Nonlinear elliptic equations, Weak coercivity, Maximum principle