Preprint
Inserted: 14 jan 2020
Last Updated: 4 sep 2020
Year: 2020
Abstract:
We investigate extremality properties of shape functionals which are products of Newtonian capacity ${\rm cap}(\overline{\Omega})$, and powers of the torsional rigidity $T(\Omega)$, for an open set $\Omega\subset{\bf R}^d$ with compact closure $\overline{\Omega}$, and prescribed Lebesgue measure. It is shown that if $\Omega$ is convex then ${\rm cap}(\overline{\Omega})T^q(\Omega)$ is (i) bounded from above if and only if $q\ge1$, 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 $q\le0$, then the product is minimised among all bounded sets by a ball of measure $1$.
Keywords: Torsional rigidity, Newtonian capacity, Dirichlet boundary condition
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