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 $Ω= \Ri^d\backslash K$ 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 $c(s)=s^{\delta }(1+s{)}^{{\delta }^{\prime }-\delta }$ and $\delta ,{\delta }^{\prime }\ge 0$. 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 $\delta ,{\delta }^{\prime }\ge 0$ and the Hausdorff dimension of the boundary. We compute the optimal constants in a broad range of situations.

References in corpus (2)

• In between the inequalities of Sobolev and Hardy
• Uniqueness of diffusion on domains with rough boundaries

Cited by in corpus (1)

• The weighted Hardy constant