On Hardy inequalities with singularities on the boundary [Sur les inégalités de Hardy avec des singularités sur la frontière]

Abstract

In this Note we present some Hardy-Poincaré inequalities with one singularity localized on the boundary of a smooth domain. Then, we consider conical domains in dimension N≥3 whose vertex is on the singularity and we show upper and lower bounds for the corresponding optimal constants in the Hardy inequality. In particular, we prove the asymptotic behavior of the optimal constant when the amplitude of the cone tends to zero.