Some estimates of the torsional rigidity of heterogeneous rods

A well-known problem in elasticity consists in placing two linearly elastic materials (of different shear moduli) in a given plane domain $\Omega$, so as to maximize the torsional rigidity of the resulting rod; moreover, the proportion of these materials is prescribed. Such a problem may not have a classical solution as the optimal design may contain homogenization regions, where the two materials are mixed in a microscopic scale. Then, the optimal torsional rigidity becomes difficult to compute. In this paper we give some different theoretical upper and lower bounds for the optimal torsional rigidity, and we compare them on some significant domains.