Hardy and Rellich inequalities on the complement of convex sets

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.