Div-Curl Young Measures and Optimal Design in Any Dimension

We explicitly introduce and exploit div-curl Young measures to examine optimal design problems governed by a linear state law in divergence form. The cost is allowed to depend explicitly on the gradient of the state. By means of this family of measures, we can formulate a suitable relaxed version of the problem, and, in a subsequent step, put it in a similar form as the original optimal design problem with an appropriate set of designs and generalized state law. Many of the issues involved has been analyzed elsewhere. The emphasis here is placed on the fact that, by using div-curl Young measures, we make the treatment dimension-independent.