Noncritical Weighted Hardy's Inequalities with compact perturbations

TL;DR

This paper investigates weighted Hardy inequalities with sharp missing terms in bounded domains with smooth boundaries, focusing on variational problems involving distance-based weights.

Contribution

It introduces new results on noncritical weighted Hardy inequalities with compact perturbations and sharp missing terms in smooth bounded domains.

Findings

Establishment of sharp Hardy inequalities with weights based on boundary distance

Identification of conditions for compact perturbations to Hardy inequalities

Development of variational framework for these inequalities

Abstract

Let $\Omega$ be a bounded domain of $\mathbb{R}^N$ whose boundary is a $\mathbb{C}^2$ compact manifolds. In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we adopted powers of the distance function $\delta(x)$ to the boundary $\partial\Omega$.