Torsion functions and the Cheeger problem: a fractional approach

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