Fractional Hardy inequalities on $C^{1,1}$ open sets

Abstract

Let $\Omega$ be a bounded open set of class $C^{1,1}$ in ${\mathbb{R}}^N$ and $s\in (\frac{1}{2},1)$. We study a family of fractional Hardy-type inequalities \begin{equation} \frac{c_{N,s}}{2}\displaystyle\iint_{Ω\timesΩ}\frac{(u(x)-u(y))^2}{|x-y|^{N+2s}}\ dxdy-\displaystyleλ\int_Ωu^2\ dx\geq C\displaystyle\int_Ω\frac{u^2}{δ^{2s}}\ dx,~~~\quad\forallλ\in\mathbb{R},~~~~~~~(0.1) \end{equation} with $u\in C_c^{\infty }(\Omega )$ and $C=C(\Omega ,s,N,\lambda )>0$. We show that the best constant in $(0.1)$ is achieved if and only if $\lambda >{\lambda }^{\ast }(s,\Omega )$, for some ${\lambda }^{\ast }(s,\Omega )\in \mathbb{R}$. As a by-product, we derive in particular that the best constant in Hardy inequality ${\mu }_{N,s}(\Omega )$ is achieved if and only if ${\mu }_{N,s}(\Omega )<{\mathfrak{h}}_{N,s}$, with ${\mathfrak{h}}_{N,s}$ being the best constant for the fractional Hardy inequality in the half space. Moreover, if $\Omega$ is a convex open set, we obtain a lower bound for ${\lambda }^{\ast }(s,\Omega )$ in terms of the volume of $\Omega$. Specifically, we prove that ${\lambda }^{\ast }(s,\Omega )\ge a(N,s)|\Omega {|}^{-\frac{2s}{N}}$ with an explicit constant $a(N,s)>0$. Finally, for bounded $C^{1,1}$ domains, we prove that, for $s$ sufficiently close to $\frac{1}{2}$, the optimal Hardy constant is independent of both the geometry and the topology of $\Omega$. More precisely, we establish that ${\mu }_{N,s}(\Omega )={\mathfrak{h}}_{N,s}$. This behavior is in sharp contrast with the local case, where the topology/geometry of the domain strongly influences the value of the optimal constant, and reveals a new rigidity phenomenon in the nonlocal setting.