Weighted Hardy inequality with higher dimensional singularity on the boundary

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