On functionals involving the $p$-capacity and the $q$-torsional rigidity

TL;DR

This paper establishes bounds for the $p$-capacity and explores extremal properties of the product of $p$-capacity and $q$-torsional rigidity for convex sets under various geometric constraints, identifying optimal shapes.

Contribution

It provides new bounds for $p$-capacity and characterizes extremal convex sets for the product of $p$-capacity and $q$-torsional rigidity under geometric constraints.

Findings

Bounds for $p$-capacity of compact sets in $ ^d$.

Identification of the ball as extremal shape under certain conditions.

Relationships between $p$-capacity and $q$-torsional rigidity for convex sets.

Abstract

Upper bounds are obtained for the $p$-capacity of compact sets in $\R^d$, with $d \ge 2$ and $1<p<d$. Upper and lower bounds are obtained for the product of $p$-capacity and powers of the $q$-torsional rigidity over the collection of all non-empty, open, bounded and convex sets in $\R^d$ with either a perimeter constraint, or a measure constraint, or a combination of perimeter and measure constraints. For some range of parameters we identify the ball as the unique (up to homotheties) maximiser or minimiser respectively.