Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
Marc Briane, Juan Casado‐Díaz · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2012
This paper deals with the homogenization of nonlinear convex energies defined in W_{0}^{1,1}(\Omega ) , for a regular bounded open set Ω of \mathbb{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}(\Omega ) . Under this assumption the Γ -convergence of the functionals for the strong topology of L^{\infty }(\Omega ) is proved to agree with the Γ -convergence for the strong topology of L^{1}(\Omega ) . This leads to an integral representation of the Γ -limit in C_{0}^{1}(\Omega ) 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.