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

Let Omega be a domain of R-n, n >= 3. The classical Caffarelli-Kohn-Nirenberg inequality rewrites as the following inequality: for any s is an element of [0, 2] and any gamma < (n-2)(2)/4, there exists a constant K(Omega, gamma, s) > 0 such that (integral(Omega)vertical bar u vertical bar(2)*((s))/vertical bar x vertical bar(s)dx)(2/2)*((s))<= K (Omega, gamma, s) integral(Omega)(vertical bar del u vertical bar(2)- gamma u(2)/vertical bar x vertical bar(2))dx, (HS) for all u is an element of D-1,D-2 (Omega) (the completion of C-infinity(Omega) for the relevant norm). When 0 is an element of Omega is an interior point, the range (-infinity, (n-2)(2)/4) for gamma cannot be improved: moreover, the optimal constant K(Omega, gamma, s) is independent of Omega and there is no extremal for (HS). But when 0 is an element of partial derivative Omega, the situation turns out to be drastically different since the geometry of the domain impacts: the range of gamma's for which (HS) holds. the value of the optimal constant K(Omega, gamma, s); the existence of extremals for (HS). When Omega 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 Omega. When gamma 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 gamma 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. (C) 2019 Elsevier Ltd. All rights reserved.