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

TL;DR

This paper investigates the stability of a radial configuration with a core of different material inside a unit ball under torsional rigidity, revealing conditions under which it is a local maximizer or saddle, using shape derivatives and spherical harmonics.

Contribution

It provides a detailed analysis of the second order shape derivative for the torsional rigidity functional considering simultaneous boundary perturbations, highlighting resonance effects.

Findings

Radial configuration can be a local maximizer or saddle depending on material hardness.

Resonance effects occur when perturbing both boundaries simultaneously.

Spherical harmonics facilitate the analysis of shape derivative signs.

Abstract

Let $\Omega_0$ denote the unit ball of $\mathbb{R}^N$ ($N\ge 2$) centered at the origin. We suppose that $\Omega_0$ contains a core, given by a smaller concentric ball $D_0$, 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 $\partial D_0$ and $\partial\Omega_0$. This gives rise to resonance effects that are not present when $\partial D_0$ or $\partial\Omega_0$ 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.