Torsion functions and the Cheeger problem: a fractional approach

Let $Ω$ be a Lipschitz bounded domain of $\mathbb{R}^N$, $N\geq2$. The fractional Cheeger constant $h_s (Ω)$, $0<s<1$, is defined by $h_s(Ω)=\inf_{E\subsetΩ}\frac{P_s(E)}{|E|},\: \text{ where } \: P_s (E)=\int_{\mathbb{R}^N }\int_{\mathbb{R}^N }\frac{|χ_{E}(x)-χ_{E}(y)|}{|x-y|^{N+s}} dx dy,$ with $χ_{E}$ denoting the characteristic function of the smooth subdomain $E$. The main purpose of this paper is to show that $\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^{\infty}(Ω)}^{1-p}=h_s (Ω)=\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^1(Ω)}^{1-p},$ where $ϕ_p^s$ is the fractional $(s,p)$-torsion function of $Ω$, that is, the solution of the Dirichlet problem for the fractional $p$-Laplacian: $-(Δ)_p^s\,u=1$ in $Ω$, $u=0$ in $\mathbb{R}^N \setminusΩ$. For this, we derive suitable bounds for the first eigenvalue $λ_{1,p}^s(Ω)$ of the fractional $p$-Laplacian operator in terms of $ϕ_p^s$. We also show that $ϕ_p^s$ minimizes the $(s,p)$-Gagliardo seminorm in $\mathbb{R}^N$, among the functions normalized by the $L^1$-norm.

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