Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, p. 547-571

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.

DOI : https://doi.org/10.1016/j.anihpc.2012.10.005
Classification:  35B27,  35B50,  35J60
Keywords: Homogenization, Convex functionals, Nonlinear elliptic equations, Weak coercivity, Maximum principle
@article{AIHPC_2013__30_4_547_0,
author = {Briane, Marc and Casado-D\'\i az, Juan},
title = {Homogenization of convex functionals which are weakly coercive and not equi-bounded from above},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {30},
number = {4},
year = {2013},
pages = {547-571},
doi = {10.1016/j.anihpc.2012.10.005},
zbl = {1288.35039},
mrnumber = {3082476},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2013__30_4_547_0}
}

Briane, Marc; Casado-Díaz, Juan. Homogenization of convex functionals which are weakly coercive and not equi-bounded from above. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 4, pp. 547-571. doi : 10.1016/j.anihpc.2012.10.005. http://www.numdam.org/item/AIHPC_2013__30_4_547_0/

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