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

TL;DR

This paper computes the second domain variation for the p-capacity and q-torsional rigidity in higher dimensions, identifying conditions under which the ball is a local extremum of their product.

Contribution

It provides the first computation of second domain variations for these functionals and characterizes when the ball is a local extremum based on p and q.

Findings

Derived conditions for local minimality or maximality of the ball.

Computed second domain variations for p-capacity and q-torsional rigidity.

Identified parameter ranges where the ball is optimal.

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.