A lower bound for the weighted-Hardy constant for domains satisfying a uniform exterior cone condition

TL;DR

This paper establishes a lower bound for the weighted Hardy constant in domains with a uniform exterior cone condition, linking geometric boundary properties to functional inequalities.

Contribution

It provides a new lower bound for the weighted Hardy constant based on the domain's geometric cone aperture, advancing understanding of Hardy inequalities in geometric domains.

Findings

Lower bound depends on cone aperture

Applicable to domains with uniform exterior cone condition

Enhances understanding of Hardy inequalities in geometric settings

Abstract

We consider weighted Hardy inequalities involving the distance function to the boundary of a domain in the $N$-dimensional Euclidean space with nonempty boundary. We give a lower bound for the corresponding best Hardy constant for a domain satisfying a uniform exterior cone condition. This lower bound depends on the aperture of the corresponding infinite circular cone.