# On capacity and torsional rigidity

M. van den Berg, G. Buttazzo

Research output: Contribution to journalArticle (Academic Journal)peer-review

## Abstract

We investigate extremal properties of shape functionals which are products of Newtonian capacity $\cp(\overline{\Om})$, and powers of the torsional rigidity $T(\Om)$, for an open set $\Om\subset \R^d$ with compact closure $\overline{\Om}$, and prescribed Lebesgue measure. It is shown that if $\Om$ is convex then $\cp(\overline{\Om})T^q(\Om)$ is (i) bounded from above if and only if $q\ge 1$, and (ii) bounded from below and away from $0$ if and only if $q\le \frac{d-2}{2(d-1)}$. Moreover a convex maximiser for the product exists if either $q>1$, or $d=3$ and $q=1$. A convex minimiser exists for $q< \frac{d-2}{2(d-1)}$. If $q\le 0$, then the product is minimised among all bounded sets by a ball of measure $1$.
Original language English 347 359 Bulletin of the London Mathematical Society 53 2 6 Oct 2020 https://doi.org/10.1112/blms.12422 Published - 3 Apr 2021

## Keywords

• 49Q10 (primary)
• 49J40
• 49J45
• 35J99 (secondary)