论文标题
扭转功能和Cheeger问题：分数方法
Torsion functions and the Cheeger problem: a fractional approach
论文作者
论文摘要
令$ω$为$ \ mathbb {r}^n $，$ n \ geq2 $的Lipschitz有限域。 \ [h_s（ω）= \ inf_ {e \subsetΩ} \ frac {p_s（e）} {| e | e | e | e | e |}，\ where} {where} {where} {where} {where} {where} {p_s（e） } \ int _ {\ Mathbb {r}^n} \ frac {|χ_{e}（x） - χ_{e}（y）|} {| x-y |^{n+s}}} dx dx dx dx dy，\]，带有$ qu__ {本文的主要目的是证明\ [\ lim_ {p \ rightarrow1^+} \ left | ϕ_p^s \ right | _ {l^{\ infty}（ω）}^{1-p} = h_s = h_s （ω）= \ lim_ {p \ rightArrow1^+} \ left | ϕ_p^s \ right | _ {l^1（ω）}^{1-p}，\]，其中$ ϕ_p^s $是fractional $（s，p）$ - $ - $ prifation $ pripation $ pripation for friclet的问题 - $ - （δ）_p^s \，u = 1 $ in $ω$，$ u = 0 $ in $ \ mathbb {r}^n \setMinusΩ$。为此，我们以$ ϕ_p^s $表示的第一个特征值$λ__{1，p}^s（ω）$得出合适的界限。我们还表明，$ ϕ_p^s $最小化$（s，p）$ - gagliardo seminorm in $ \ mathbb {r}^n $，在由$ l^1 $ norm标准化的功能中。
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.
