Some smoothness results for the optimal design of a two-composite material which minimizes the energy

摘要

We study the smoothness properties of the optimal solutions corresponding to the minimization of the potential energy for the mixture of two isotropic materials (electric, thermic, elastic membrane,). As it is well known this problem has not solution in general and therefore our results refer to a relaxed formulation. We show that the state function is twice derivable and that the optimal proportion is derivable in the direction of . We also get some uniqueness results and applications to the non-existence of classical (unrelaxed) solutions.