Torsion functions and the Cheeger problem: a fractional approach

Let $\mathrm{\Omega }$ be a Lipschitz bounded domain of ${\mathbb{R}}^N$, $N\ge 2$. The fractional Cheeger constant $h_s(\mathrm{\Omega })$, $0<s<1$, is defined by $$h_s(\mathrm{\Omega })=\underset{E\subset \mathrm{\Omega }}{\inf }\frac{P_s(E)}{\mathrm{\mid }E\mathrm{\mid }},\text{ where }{P}_s(E)={\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{\mathrm{\mid }{\chi }_E(x)-{\chi }_E(y)\mathrm{\mid }}{\mathrm{\mid }x-y{\mathrm{\mid }}^{N+s}}dxdy,$$ with ${\chi }_E$ denoting the characteristic function of the smooth subdomain $E$. The main purpose of this paper is to show that $$\underset{p\to 1^{+}}{\lim }{\mid {\varphi }_p^s\mid }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}^{1-p}=h_s(\mathrm{\Omega })=\underset{p\to 1^{+}}{\lim }{\mid {\varphi }_p^s\mid }_{L^1(\mathrm{\Omega })}^{1-p},$$ where ${\varphi }_p^s$ is the fractional $(s,p)$-torsion function of $\mathrm{\Omega }$, that is, the solution of the Dirichlet problem for the fractional $p$-Laplacian: $-(\mathrm{\Delta }{)}_p^{s}u=1$ in $\mathrm{\Omega }$, $u=0$ in ${\mathbb{R}}^N\setminus \mathrm{\Omega }$. For this, we derive suitable bounds for the first eigenvalue ${\lambda }_{1,p}^s(\mathrm{\Omega })$ of the fractional $p$-Laplacian operator in terms of ${\varphi }_p^s$. We also show that ${\varphi }_p^s$ minimizes the $(s,p)$-Gagliardo seminorm in ${\mathbb{R}}^N$, among the functions normalized by the $L^1$-norm.