Stability analysis of the two-phase torsional rigidity near a radial configuration

Let Ω₀ denote the unit ball of R^N (N ≥ 2) centered at the origin. We suppose that Ω₀ contains a core, given by a smaller concentric ball D₀, made of a (possibly) different material. We discover that, depending on the relative hardness of the two materials, this radial configuration can either be a local maximizer for the torsional rigidity functional E or a saddle shape. In this paper, we consider perturbations that simultaneously act on the boundaries ∂D₀ and ∂Ω₀. This gives rise to resonance effects that are not present when ∂D₀ and ∂Ω₀ are perturbed in isolation. A detailed analysis of the sign of the second order shape derivative of E is then made possible by employing the use of spherical harmonics.