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

Abstract

In this work, we analyze a 3-d dynamic optimal design problem inconductivity governed by the two-dimensional wave equation. Underthis dynamic perspective, the optimal design problem consists inseeking the time-dependent optimal layout of two isotropic materialson a 2-d domain ($\Omega\subsetR^2$); this is done by minimizinga cost functional depending on the square of the gradient of thestate function involving coefficients which can depend on time,space and design. The lack of classical solutions of this type ofproblem is well-known, so that a relaxation must be sought. Weutilize a specially appropriate characterization of 3-d($(t,x)\inR\timesR^2$) divergence free vector fields throughClebsh potentials; this lets us transform the optimal design probleminto a typical non-convex vector variational problem, to which Youngmeasure 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 inthis work: the 2-d hyperbolic state law, and the vector character ofthe problem. Though these two ingredients have been previouslyconsidered separately, we put them together in this work.