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

ABSTRACT

Let ${\mathrm{\Omega }}_0$ denote the unit ball of ${\mathbb{R}}^N$ ($N\ge 2$) centered at the origin. We suppose that ${\mathrm{\Omega }}_0$ 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 $\mathrm{\partial }{D}_0$ and $\mathrm{\partial }{\mathrm{\Omega }}_0$. This gives rise to resonance effects that are not present when $\mathrm{\partial }{D}_0$ or $\mathrm{\partial }{\mathrm{\Omega }}_0$ 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.

KEYWORDS:

• Torsion problem
• optimization problem
• elliptic PDE
• shape derivative
• spherical harmonics

AMS SUBJECT CLASSIFICATION:

• 49Q10

Additional information

Funding

This research was partially supported by the Grants-in-Aid for Scientific Research (B) [grant number 26287020] and Challenging Exploratory Research [grant number 16K13768] of Japan Society for the Promotion of Science.