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On capacity and torsional rigidity

Research output: Contribution to journalArticle

Original languageEnglish
Number of pages13
JournalBulletin of the London Mathematical Society
DateSubmitted - 2020

Abstract

We investigate extremality properties of shape functionals which are products of Newtonian capacity cap (Ω), and powers of the torsional rigidity T(Ω), for an open set Ω ⊂ Rd with compact closure Ω, and prescribed Lebesgue measure. It is shown that if Ω is convex then cap (Ω)Tq (Ω) is (i) bounded from above if and only if q ≥ 1, and (ii) bounded from below and away from 0 if and only if q ≤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 < d−2/2(d−1) . If q ≤ 0, then the product is minimised among all bounded sets by a ball of measure 1.

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