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

In this paper, we generalize the notion of relative $p$-capacity of $K$ with respect to $\mathrm{\Omega }$, 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 $\mathrm{\Omega }$ are concentric balls. We use the $H$-function and a derearrangement technique.