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 (Ω), for a regular bounded open set Ω of 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 q1 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 (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.

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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