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

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Let Ω₀ denote the unit ball of Rⁿ (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 or ∂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.