We examine how the use of typical techniques from non-convex vector variational problems can help in understanding optimal design problems in conductivity. After describing the main ideas of the underlying analysis and providing some standard material in an attempt to make the exposition self-contained, we show how those ideas apply to a typical optimal desing problem with two different conducting materials. Then we examine the equivalent relaxed formulation to end up with a new problem whose numerical simulation leads to approximated optimal configurations. We include several such simulations in 2d and 3d.

We study the corrector matrix $P^{\varepsilon}$  to the conductivity equations. We showthat if $P^{\varepsilon}$  converges weakly to the identity, then for any laminate $\det P^{\varepsilon}\geq 0$ at almost every point. This simple property is shown to be false forgeneric microgeometries if the dimension is greater than two in the work Briane et al. [Arch. Ration. Mech. Anal., to appear].In two dimensions it holds true for any microgeometry as a corollary of the work in Alessandrini and Nesi [Arch. Ration. Mech. Anal.158 (2001) 155-171]. We use this property of laminates to prove that, in any dimension, the classicalHashin-Shtrikman bounds are not attained by laminates, in certain regimes, when the number ofphases is greater than two. In addition we establish new bounds for the effective conductivity,which are asymptotically optimal for mixtures of three isotropic phases among a certain class ofmicrogeometries, including orthogonal laminates, which we then call quasiorthogonal.