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

Abstract

Let $Ω$ 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 $u\in C_c^\infty(Ω)$ and $C=C(Ω,s,N,λ)>0$. We show that the best constant in $(0.1)$ is achieved if and only if $λ>λ^*(s,Ω)$, for some $λ^*(s,Ω)\in\mathbb{R}$. As a by-product, we derive in particular that the best constant in Hardy inequality $μ_{N,s}(Ω)$ is achieved if and only if $μ_{N,s}(Ω)<\mathfrak{h}_{N,s}$, with $\mathfrak{h}_{N,s}$ being the best constant for the fractional Hardy inequality in the half space. Moreover, if $Ω$ is a convex open set, we obtain a lower bound for $λ^*(s,Ω)$ in terms of the volume of $Ω$. Specifically, we prove that $λ^*(s,Ω)\geq a(N,s)|Ω|^{-\frac{2s}{N}}$ 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 $Ω$. More precisely, we establish that $μ_{N,s}(Ω)=\mathfrak{h}_{N,s}$. 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.