# Locally optimal configurations for the two-phase torsion problem in the ball

## TL;DR

This paper investigates optimal configurations for two-material torsion problems in a ball, analyzing shape derivatives to identify when symmetric or symmetry-breaking configurations maximize torsional rigidity.

## Contribution

It provides second-order shape derivative analysis for the torsional rigidity functional in two-phase materials, revealing conditions for symmetry breaking.

## Key findings

- Explicit second-order shape derivatives computed using spherical harmonics.
- Identification of parameter regimes where symmetry breaking occurs.
- Analysis of the impact of conductivity ratio and smaller ball radius on optimal configurations.

## Abstract

We consider the unit ball $\Omega\subset \mathbb{R}^N$ ($N\ge2$) filled with two materials with different conductivities. We perform shape derivatives up to the second order to find out precise information about locally optimal configurations with respect to the torsional rigidity functional. In particular we analyse the role played by the configuration obtained by putting a smaller concentric ball inside $\Omega$. In this case the stress function admits an explicit form which is radially symmetric: this allows us to compute the sign of the second order shape derivative of the torsional rigidity functional with the aid of spherical harmonics. Depending on the ratio of the conductivities and on the radius of the smaller ball, a symmetry breaking phenomenon occurs.