Optimal Design Problems for Two-Phase Conducting Composites with Weakly Discontinuous Objective Functionals

We consider the relatively simple but non-trivial example of an optimal design problem with a weakly discontinuous objective functional. The objective functional is quadratic and was suggested by Tartar in his “Remarks on Optimal Design” paper. We analyze a problem of finding a layout of a conducting composite such that the fields in both phases provide least squares fit to a given field. The main result of the paper is an explicit formula for the relaxed optimal design problem, suitable for numerical solution. Further analysis of our explicit formula shows that the optimal layout is a rank one laminate locally, lending some support for Tartar's conjecture that the minimizing sequences always converge strongly.