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

  • Marc Briane

    Institut de Recherche Mathématique de Rennes, INSA de Rennes, France
  • Juan Casado-Díaz

    Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain

Abstract

This paper deals with the homogenization of nonlinear convex energies defined in , for a regular bounded open set of , the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists if , and if , such that any sequence of bounded energy is compact in . Under this assumption the -convergence of the functionals for the strong topology of is proved to agree with the -convergence for the strong topology of . This leads to an integral representation of the -limit in 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.

Cite this article

Marc Briane, Juan Casado-Díaz, Homogenization of convex functionals which are weakly coercive and not equi-bounded from above. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 4, pp. 547–571

DOI 10.1016/J.ANIHPC.2012.10.005