On capacity and torsional rigidity

M. van den Berg, G. Buttazzo

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

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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 languageEnglish
Pages (from-to)347
Number of pages359
JournalBulletin of the London Mathematical Society
Volume53
Issue number2
Early online date6 Oct 2020
DOIs
Publication statusPublished - 3 Apr 2021

Keywords

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

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