Torsion Functions and the Cheeger Problem: A Fractional Approach

Abstract

Abstract Let Ω be a Lipschitz bounded domain of ℝ N ${\mathbb{R}^{N}}$ , N ≥ 2 ${N\geq 2}$ . The fractional Cheeger constant h s ⁢ ( Ω ) ${h_{s}(\Omega)}$ , 0 < s < 1 ${0<s<1}$ , is defined by h s ⁢ ( Ω ) = inf E ⊂ Ω ⁡ P s ⁢ ( E ) | E | , where P s ⁢ ( E ) = ∫ ℝ N ∫ ℝ N | χ E ⁢ ( x ) - χ E ⁢ ( y ) | | x - y | N + s ⁢ d x ⁢ d y , $h_{s}(\Omega)=\inf_{E\subset{\Omega}}\frac{P_{s}(E)}{|E|},\quad\text{where}% \quad P_{s}(E)=\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|\chi_{E}(x)-% \chi_{E}(y)|}{|x-y|^{N+s}}\,\mathrm{d}x\,\mathrm{d}y,$ with χ E ${\chi_{E}}$ denoting the characteristic function of the smooth subdomain E. The main purpose of this paper is to show that lim p → 1 + | ϕ p s | L ∞ ⁢ ( Ω ) 1 - p = h s ( Ω ) = lim p → 1 + | ϕ p s | L 1 ⁢ ( Ω ) 1 - p , $\lim_{p\rightarrow 1^{+}}\lvert\phi_{p}^{s}|_{L^{\infty}(\Omega)}^{1-p}=h_{s}(% \Omega)=\lim_{p\rightarrow 1^{+}}\lvert\phi_{p}^{s}|_{L^{1}(\Omega)}^{1-p},$ where ϕ p s ${\phi_{p}^{s}}$ is the fractional ( s , p ) ${(s,p)}$ -torsion function of Ω, that is, the solution of the Dirichlet problem for the fractional p-Laplacian: - ( Δ ) p s ⁢ u = 1 ${-(\Delta)_{p}^{s}\,u=1}$ in Ω, u = 0 ${u=0}$ in ℝ N ∖ Ω ${\mathbb{R}^{N}\setminus\Omega}$ . For this, we derive suitable bounds for the first eigenvalue λ 1 , p s ⁢ ( Ω ) ${\lambda_{1,p}^{s}(\Omega)}$ of the fractional p-Laplacian operator in terms of ϕ p s ${\phi_{p}^{s}}$ . We also show that ϕ p s ${\phi_{p}^{s}}$ minimizes the ( s , p ) ${(s,p)}$ -Gagliardo seminorm in ℝ N ${\mathbb{R}^{N}}$ , among the functions normalized by the L 1 ${L^{1}}$ -norm.