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), p. 547-571 / Harvested from Numdam

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Résumé

This paper deals with the homogenization of nonlinear convex energies defined in $W_{0}^{1, 1} (Ω)$ , 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 $W_{0}^{1, q} (Ω)$ . Under this assumption the Γ-convergence of the functionals for the strong topology of $L^{\infty} (Ω)$ is proved to agree with the Γ-convergence for the strong topology of $L^{1} (Ω)$ . This leads to an integral representation of the Γ-limit in $C_{0}^{1} (Ω)$ 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.

Publié le : 2013-01-01

MR 3082476 | Zbl 1288.35039

DOI : https://doi.org/10.1016/j.anihpc.2012.10.005
Classification: 35B27, 35B50, 35J60

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     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},
     volume = {30},
     year = {2013},
     pages = {547-571},
     doi = {10.1016/j.anihpc.2012.10.005},
     mrnumber = {3082476},
     zbl = {1288.35039},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 30 (2013) pp. 547-571. doi : 10.1016/j.anihpc.2012.10.005. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_4_547_0/

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