Weighted Hardy inequality with higher dimensional singularity on the boundary

TL;DR

This paper investigates a weighted Hardy inequality with boundary singularities on submanifolds, establishing conditions for the existence of minimizers in higher-dimensional settings.

Contribution

It introduces new conditions for the existence of minimizers in weighted Hardy inequalities with boundary singularities on submanifolds.

Findings

Derived necessary and sufficient conditions for minimizer existence.

Extended Hardy inequality analysis to higher-dimensional boundary singularities.

Provided a comprehensive framework for weighted inequalities with boundary singularities.

Abstract

Let $\Omega$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $\Sigma_k$ be a closed smooth submanifold of $\delta \Omega$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $\Sigma_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.