Relaxation of a control problem in the coefficients with a functional of quadratic growth in the gradient

We study an optimal design problem consisting in mixing two anisotropic (electric or thermal) materials in order to minimize a functional depending on the gradient of the state. It is known that this type of problem has no solution in general, and then it is necessary to introduce a relaxed formulation. Here we prove that this relaxation is obtained by using composite materials, is constructed by homogenization, and takes a particular extension of the cost functional to these new materials. We obtain an integral representation of this relaxed cost functional. Besides, we show that our results contain some previous results obtained by other authors for isotropic materials.Ministerio de Educación y CienciaJunta de Andalucí