A local analysis of the radial configuration for the two-phase torsion problem in the ball (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

This is a resume of [4]. We consider the configuration given by two concentric balls, made of different materials. We get precise information on the local optimality of this radially symmetric configuration for the two-phase torsional rigidity functional under the effect of perturbations that act exclusively on the inner ball while satisfying the volume (or surface area) preserving constraint. We perform shape derivatives up to the second order and make use of spherical harmonics to aid our calculations. Depending on the difference of the two conductivities, a symmetry breaking phenomenon occurs.