Dynamic materials for an optimal design problem under the two-dimensional wave equation

In this work, we analyze a 3-d dynamic optimal design problem in conductivity governed by the two-dimensional wave equation. Under this dynamic perspective, the optimal design problem consists in seeking the time-dependent optimal layout of two isotropic materials on a 2-d domain ($\Omega\subsetR^2$); this is done by minimizing a cost functional depending on the square of the gradient of the state function involving coefficients which can depend on time, space and design. The lack of classical solutions of this type of problem is well-known, so that a relaxation must be sought. We utilize a specially appropriate characterization of 3-d ($(t,x)\inR\timesR^2$) divergence free vector fields through Clebsh potentials; this lets us transform the optimal design problem into a typical non-convex vector variational problem, to which Young measure theory can be applied to compute explicitly the "constrained quasiconvexification" of the cost density. Moreover this relaxation is recovered by dynamic (time-space) first- or second-order laminates. There are two main concerns in this work: the 2-d hyperbolic state law, and the vector character of the problem. Though these two ingredients have been previously considered separately, we put them together in this work.