Hardy–Sobolev inequalities with singularities on non smooth boundary: Hardy constant and extremals. Part I: Influence of local geometry

Abstract

Let Ω be a domain of Rn, n≥3. The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s∈[0,2] and any γ<(n−2)24, there exists a constant K(Ω,γ,s)>0 such that (HS)∫Ω|u|2⋆(s)|x|sdx22⋆(s)≤K(Ω,γ,s)∫Ω|∇u|2−γu2|x|2dx,for all u∈D1,2(Ω) (the completion of Cc∞(Ω) for the relevant norm). When 0∈Ω is an interior point, the range (−∞,(n−2)24) for γ cannot be improved: moreover, the optimal constant K(Ω,γ,s) is independent of Ω and there is no extremal for (HS). But when 0∈∂Ω, the situation turns out to be drastically different since the geometry of the domain impacts : •the range of γ’s for which (HS) holds.•the value of the optimal constant K(Ω,γ,s);•the existence of extremals for (HS). When Ω is smooth, the problem was tackled by Ghoussoub–Robert (2017) where the role of the mean curvature was central. In the present paper, we consider nonsmooth domain with a singularity at 0 modeled on a cone. We show how the local geometry induced by the cone around the singularity influences the value of the Hardy constant on Ω. When γ is small, we introduce a new geometric object at the conical singularity that generalizes the ”mean curvature”: this allows to get extremals for (HS). The case of larger values for γ will be dealt in the forthcoming paper (Cheikh-Ali, 2018). As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own.