Homogenization of Dirichlet parabolic problems for coefficients and open sets simultaneously variable and applications to optimal design

In a previous paper, we studied the homogenization of a sequence of parabolic linear Dirichlet problems, when the coefficients and the domains vary arbitrarily. Here, we improve the convergence result given in this paper by showing the strong convergence in L2 every time. This is applied to obtain an existence result for control problems in optimal design written in a relaxed form. The control variables are the material and the shape.