The $p$-Faber-Krahn Inequality Noted

TL;DR

This paper revisits the $p$-Faber-Krahn inequality for the principal $p$-Laplacian eigenvalue, showing it can be improved and characterized via various mathematical methods such as capacity, volume, and inequalities.

Contribution

It introduces improvements and new characterizations of the $p$-Faber-Krahn inequality using multiple analytical techniques.

Findings

The $p$-Faber-Krahn inequality can be strengthened.

Characterizations involve Maz'ya's capacity, volume, and classical inequalities.

Provides a unified view of the inequality through different mathematical frameworks.

Abstract

When revisiting the Faber-Krahn inequality for the principal $p$-Laplacian eigenvalue of a bounded open set in $\mathbb R^n$ with smooth boundary, we simply rename it as the $p$-Faber-Krahn inequality and interestingly find that this inequality may be improved but also characterized through Maz'ya's capacity method, the Euclidean volume, the Sobolev type inequality and Moser-Trudinger's inequality.