Hardy–Sobolev inequalities with singularities on non smooth boundary: Hardy constant and extremals. Part I: Influence of local geometry. (May 2019) - Cheikh Ali, Hussein
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.