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On capacity and torsional rigidity
Author(s) -
van den Berg M.,
Buttazzo G.
Publication year - 2021
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12422
Subject(s) - mathematics , bounded function , lebesgue measure , rigidity (electromagnetism) , regular polygon , ball (mathematics) , measure (data warehouse) , pure mathematics , combinatorics , lebesgue integration , mathematical analysis , geometry , physics , quantum mechanics , database , computer science
We investigate extremal 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 ⊂ R dwith compact closure Ω ¯ , and prescribed Lebesgue measure. It is shown that if Ω is convex, then cap( Ω ¯ ) T q ( 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.