Second domain variation for the $p$ - capacity and the $q$ - torsional rigidity

TL;DR

This work analyzes second-order domain variations for two functionals in $\mathbb{R}^d$, $1<p<d$, $q>1$: the $p$-capacity ${\mathcal C}_p(\overline{\Omega})$ and the $q$-torsion ${\cal T}_q(\Omega)$. It develops Hadamard-type volume-preserving perturbations, derives first- and second-variation formulas, and identifies critical domains via constant boundary gradients $|\nabla u|^p$ (or $|\nabla\psi|^q$). For the ball, the analysis reduces to exterior Steklov spectral data: $\mu_k=(d-2+k)/R$ and $\hat{\mu}_k=k/R$, allowing explicit expressions for $\ddot{\mathcal C}_p(0)$ and $\ddot{\mathcal T}_q(0)$ and sharp sign conclusions. The results show the ball is a local minimizer of ${\mathcal C}_p$ when $p>p^*=1+\frac{d-1}{d}$ and a local maximizer of ${\cal T}_q$ for $q>1$, with the expected reverse behavior possible when $1<p<p^*$, offering insight into isoperimetric-type optimality for capacity and torsion under shape perturbations.

Abstract

The second domain variation of the $p$-capacity and the $q$ - torsional rigidity for compact sets in $R^d, d\geq3$ with $1<p<d$ is computed. Conditions on $p$ and $q>1$ are given such that the ball is a local minimzer or maximizer of the product.

Theorems & Definitions (9)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Proposition 5
• Proposition 6
• Example 7
• Proposition 8
• Proposition 9