Numerical results for an optimal design problem with a nonlinear cost depending on the gradient of the state.

Numerical results for an optimal design problem with a nonlinear cost depending on the gradient of the state.

We study an optimal design problem consisting in mixing two anisotropic (electric or thermal) materials in order to minimize a nonlinear functional which depends on the gradient of the state. Mathematically this problem isformulated as a control problem for a linear elliptic partial differential equation where the control variable is the diffusion matrix. It is well known that it has not a solution in general and thus it is necessary to introduce a relaxed formulation. A great difficulty to deal with this problem is that the relaxed functional is not known explicitly. We show as the solutions of the relaxed problem can be numerically approximated replacing the relaxed functional by an upper or lower one and we present some numerical experiences.