Hardy-Sobolev inequalities with singularities on non smooth boundary: Hardy constant and extremals. Part I: Influence of local geometry
Résumé
Let $\Omega$ be a domain of $\mathbb{R}^n$, $n\geq 3$. The classical Caffarelli-Kohn-Nirenberg inequality rewrites as the following inequality: for any $s\in [0,2]$ and any $\gamma<\frac{(n-2)^2}{4}$, there exists a constant $K(\Omega,\gamma,s)>0$ such that
$$\left(\int_{\Omega}\frac{|u|^{\crit}}{|x|^s}\, dx\right)^{\frac{2}{\crit}}\leq K(\Omega,\gamma,s)\int_{\Omega}\left(|\nabla u|^2-\gamma\frac{u^2}{|x|^2}\right)\, dx,\eqno{(HS)}$$
for all $u\in D^{1,2}(\Omega)$ (the completion of $C^\infty_c(\Omega)$ for the relevant norm). When $0\in\Omega$ is an interior point, the range $(-\infty, \frac{(n-2)^2}{4})$ for $\gamma$ cannot be improved: moreover, the optimal contant $K(\Omega,\gamma,s)$ is independent of $\Omega$ and there is no extremal for $(HS)$. But when $0\in\partial\Omega$, the situation turns out to be drastically different since the geometry of the domain impacts :
\begin{itemize}
\item the range of $\gamma$'s for which $(HS)$ holds;
\item the value of the optimal constant $K(\Omega,\gamma,s)$;
\item the existence of extremals for $(HS)$.
\end{itemize}
When $\Omega$ is smooth, the problem was tackled by Ghoussoub-Robert \cite{GR} 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 \cite{HCA2}. As an intermediate result, we prove the symmetry of some solutions to singular pdes that has an interest on its own.
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