Extremal decomposition problems for p-harmonic radius

TL;DR

This paper generalizes classical extremal problems involving conformal and harmonic radii to the setting of p-harmonic radii in n-dimensional Euclidean spaces, using advanced techniques of curve family modulii.

Contribution

It extends known results from planar and harmonic cases to p-harmonic radii in higher dimensions, providing new analogues and proof techniques.

Findings

Derived inequalities for p-harmonic radii in n-dimensional spaces.

Extended classical results to a broader mathematical setting.

Utilized modulii of curve families and dissymmetrization techniques.

Abstract

We extend classical results by Lavrent'ev and Kufarev concerning the product of the conformal radii of planar non-overlapping domains. We also extend relatively recent results for the case of domains in the $n$-dimensional Euclidean space, $n \geq 3$, with conformal radii replaced by harmonic ones. Namely, we get analogues of these results in $n$-dimensional Euclidean space in terms of $p$-harmonic radius. The proofs are based on technique of modulii of curve families and dissymmetrization of such families.