Convergence of the optimality criteria method for multiple state optimal design problems

Abstract We consider multiple state optimal design problems with two isotropic materials from the conductivity point of view. Since the classical solutions of these problems usually do not exist, a proper relaxation of the original problem is obtained, using the homogenization method. In [1] we derive necessary conditions of optimality of the relaxed problem, which enables us to implement a new variant of the optimality criteria method. It appears that this variant gives converging sequence of designs for the energy minimization problems. In this work we prove convergence of the method for energy minimization problems in the spherically symmetric case and in a case when the number of states is less than the space dimension.