Quasiconvexification in 3-D for a variational reformulation of an optimal design problem in conductivity

Abstract

We analyze a typical 3-D conductivity problem which consists in seeking the optimal layout of two materials in a given design domain Ω ⊂ IR3 by minimizing the L2-norm of the electric field under a constraint on the amount on each material that we can use. We utilize a characterization of the three-dimensional divergence-free vector fields which is especially appropriate for a variational reformulation. By using gradient Young measures as a main tool, we can give an explicit form of the ”constrained quasiconvexification” of the cost density. This result is similar to the one in the 2-D situation. However, the characterization of the divergence-free vector fields introduces a certain nonlinearity in the problem that needs to be addressed properly.