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

TL;DR

This paper extends the concept of relative p-capacity with Robin boundary conditions and proves that, under volume constraints, the minimal capacity configuration occurs when the sets are concentric balls, using rearrangement techniques.

Contribution

It introduces a generalized p-capacity with Robin boundary conditions and establishes a minimality result for concentric balls under volume constraints.

Findings

Minimal p-capacity achieved by concentric balls

Use of rearrangement techniques in capacity analysis

Extension of p-capacity concept to Robin boundary conditions

Abstract

In this paper, we generalize the notion of 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 and a derearrangement technique.