Hardy and Rellich inequalities on the complement of convex sets

We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_p(Ω)$ where $Ω= \Ri^d\backslash K$ with $K$ a closed convex subset of $\Ri^d$. Let $Γ=\partialΩ$ denote the boundary of $Ω$ and $d_Γ$ the Euclidean distance to $Γ$. We consider weighting functions $c_Ω=c\circ d_Γ$ with $c(s)=s^δ(1+s)^{δ'-δ}$ and $δ,δ'\geq0$. Then the Hardy inequalities take the form $\int_Ωc_Ω\,|\nablaφ|^p\geq b_p\int_Ωc_Ω\,d_Γ^{\;-p}\,|φ|^p$ and the Rellich inequalities are given by $\int_Ω|Hφ|^p\geq d_p\int_Ω|c_Ω\,d_Γ^{\,-2}φ|^p$ with $H=-\divv(c_Ω\nabla)$. The constants $b_p, d_p$ depend on the weighting parameter $δ,δ'\geq0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.

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