Torsion functions and the Cheeger problem: a fractional approach

Abstract

Let $\Omega$ be a Lipschitz bounded domain of ${\mathbb{R}}^N$, $N\ge 2$. The fractional Cheeger constant $h_s(\Omega )$, $0<s<1$, 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 ${\chi }_E$ denoting the characteristic function of the smooth subdomain $E$. The main purpose of this paper is to show that \[\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^{\infty}(Ω)}^{1-p}=h_s (Ω)=\lim_{p\rightarrow1^+}\left|ϕ_p^s\right|_{L^1(Ω)}^{1-p},\] where ${\varphi }_p^s$ 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 ${\varphi }_p^s$. 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.

References in corpus (1)

• Solutions of the Cheeger problem via torsion functions