Relaxation of an optimal design problem with an integral-type constraint

We study a new relaxation for a two-dimensional optimal design problem in conductivity consisting of determining how to mix two given conducting materials in order to minimize the amount of one of them, subject to a constraint on the efficiency of the conducting properties of the mixture. Our approach here is different from that obtained in [10], and based on a local reformulation of the optimal design problem by means of the introduction of new potentials. The concept of constrained quasiconvexification is used in an important way.