A curl-free improvement of the Rellich-Hardy inequality with weight

TL;DR

This paper derives an explicit best constant for a weighted Rellich-Hardy inequality under curl-free conditions, extending previous results and showing the constant cannot be attained, thus advancing understanding of vector field inequalities.

Contribution

It provides a new explicit best constant for the curl-free Rellich-Hardy inequality with weights, and proves this constant is not attainable, improving prior bounds and understanding.

Findings

Explicit best constant computed for curl-free Rellich-Hardy inequality

Proved the best constant is not attainable

Extended previous results to weighted, curl-free vector fields

Abstract

We consider the best constant in the Rellich-Hardy inequality (with a radial power weight) for curl-free vector fields on $\mathbb{R}^N$, originally found by Tertikas-Zographopoulos \cite{Tertikas-Z} for unconstrained fields. This inequality is considered as an intermediate between Hardy-Leray and Rellich-Leray inequalities. Under the curl-free condition, we compute the new explicit best constant in the inequality and prove the non-attainability of the constant. This paper is a sequel to \cite{CF_MAAN,CF_Re}.