An isoperimetric result for an energy related to the $p$-capacity

Abstract

In this paper, we generalize the notion of the relative $p$-capacity of $K$ with respect to $\Omega$, by replacing the Dirichlet boundary condition with a Robin one. We show that, under volume constraints, our notion of $p$-capacity is minimal when $K$ and $\Omega$ are concentric balls. We use the $H$-function (see Bossel (1986) and Daners (2006)) and a derearrangement technique.