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 R n , n ≥ 3. The classical Caffarelli–Kohn–Nirenberg inequality rewrites as the following inequality: for any s ∈ [ 0 , 2 ] and any γ < (n − 2) 2 4 , there exists a constant K (Ω , γ , s) > 0 such that (H S) ∫ Ω | u | 2 ⋆ (s) | x | s d x 2 2 ⋆ (s) ≤ K (Ω , γ , s) ∫ Ω | ∇ u | 2 − γ u 2 | x | 2 d x , for all u ∈ D 1 , 2 (Ω) (the completion of C c ∞ (Ω) for the relevant norm). When 0 ∈ Ω is an interior point, the range (− ∞ , (n − 2) 2 4) for γ cannot be improved: moreover, the optimal constant K (Ω , γ , s) is independent of Ω and there is no extremal for (H S). 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 (H S) holds. • the value of the optimal constant K (Ω , γ , s) ; • the existence of extremals for (H S). 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 (H S). 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. [ABSTRACT FROM AUTHOR]