Hardy-type inequalities and principle frequency of the $p-$Laplacian

TL;DR

This paper establishes sharp weighted Hardy inequalities involving boundary distance for domains in Euclidean space and explores their implications for the principal frequency of the p-Laplacian, with improvements under specific subharmonic conditions.

Contribution

It introduces new sharp weighted Hardy inequalities involving boundary distance and demonstrates their applications to the principal frequency of the p-Laplacian, including improvements under subharmonic assumptions.

Findings

Established sharp $L^p$ weighted Hardy inequalities involving boundary distance.

Improved inequalities when $- ext{log} \, ext{distance}$ is subharmonic.

Applied inequalities to analyze the principal frequency of the p-Laplacian.

Abstract

We prove a sharp $L^p$ weighted Hardy inequality involving boundary distance $\delta$ for any domain $\Omega\subsetneq \mathbb R^n$. The inequality may be improved substantially under the additional assumption that $-\log \delta$ is subharmonic. Applications of these inequalities to the principle frequency of the $p-$Laplacian are given.