# Hardy and Rellich inequalities on the complement of convex sets

## TL;DR

This paper proves the existence of weighted Hardy and Rellich inequalities on the complement of convex sets in Euclidean space, providing explicit formulas for optimal constants depending on geometric and weight parameters.

## Contribution

It establishes new weighted Hardy and Rellich inequalities on convex set complements with explicit optimal constants, extending classical inequalities to more general weighted and geometric contexts.

## Key findings

- Derived explicit forms of Hardy and Rellich inequalities on convex set complements.
- Computed optimal constants for these inequalities in various geometric scenarios.
- Extended classical inequalities to weighted versions with boundary distance functions.

## Abstract

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_p(\Omega)$ where $\Omega= \Ri^d\backslash K$ with $K$ a closed convex subset of $\Ri^d$. Let $\Gamma=\partial\Omega$ denote the boundary of $\Omega$ and $d_\Gamma$ the Euclidean distance to $\Gamma$. We consider weighting functions $c_\Omega=c\circ d_\Gamma$ with $c(s)=s^\delta(1+s)^{\delta'-\delta}$ and $\delta,\delta'\geq0$. Then the Hardy inequalities take the form \[ \int_\Omega c_\Omega\,|\nabla\varphi|^p\geq b_p\int_\Omega c_\Omega\,d_\Gamma^{\;-p}\,|\varphi|^p \] and the Rellich inequalities are given by \[ \int_\Omega|H\varphi|^p\geq d_p\int_\Omega |c_\Omega\,d_\Gamma^{\,-2}\varphi|^p \] with $H=-\divv(c_\Omega\nabla)$. The constants $b_p, d_p$ depend on the weighting parameter $\delta,\delta'\geq0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.