AI 中文总结
本文针对有界Lipschitz区域上的分数阶Dirichlet问题，建立了最优全局Sobolev估计，并利用Cantor集和扇形齐次解构造反例，证明所得指数范围在Dahlberg意义下不可改进。
AI 中文摘要
设$n\geq2$，$s\in(0,1)$，且$\Omega\subset\mathbb{R}^n$为有界Lipschitz区域。本文建立了分数阶Dirichlet问题\begin{equation*} \left\{\begin{aligned} (-\Delta)^su&=f & & \text{在}\\ \\ \Omega\\ \text{中},\\\\ u&=0 & & \text{在}\\ \\ \mathbb{R}^n\setminus\Omega\\ \text{中}. \end{aligned}\right. \end{equation*}的最优全局Sobolev估计。更精确地，对任意给定的$t\in[s,\min\{2s,\\,s+1/2\})$，我们证明存在一个小的正常数$\eta=\eta(n,s,t,\Omega)$使得以下结论成立。若$n=2$且$s\in(1/2,1)$，则对任意$q\in(1,\frac{4}{1+2s}+\eta)$和$p\in(1,\frac{nq}{n-(2s-t)q}]$，有\begin{equation*} \\|u\\|_{W^{t,p}(\mathbb{R}^n)}\le C\\|f\\|_{L^q(\Omega)}, \end{equation*}且若$n=2$且$s\in(0,1/2]$，或$n\ge 3$且$s\in(0,1)$，则对任意$q\in(1,\frac{n(2s+1)}{nt+(2s-t)(2s+1)}+\eta)$和$p\in(1,\frac{nq}{n-(2s-t)q}]$，有\begin{equation*} \\|u\\|_{W^{t,p}(\mathbb{R}^n)}\le C\\|f\\|_{L^q(\Omega)}. \end{equation*}这里正常数$C$仅依赖于$n$，$s$，$t$，$p$，$q$和$\Omega$。所陈述的通用基线在Dahlberg意义下是最优的：对任何更大的目标指数$p$，存在一个有界Lipschitz区域，其右端项恒为$1$的解不属于$W^{t,p}(\mathbb{R}^n)$。反例利用边界上的正则Cantor集构造，在平面例外情形中，利用扇形中的齐次解构造。
英文摘要
Let $n\geq2$, $s\in(0,1)$, and let $Ω\subset\mathbb{R}^n$ be a bounded Lipschitz domain. In this paper, we establish optimal global Sobolev estimates for the fractional Dirichlet problem \begin{equation*} \left\{\begin{aligned} (-Δ)^su&=f & & \text{in}\ \ Ω, u&=0 & & \text{in}\ \ \mathbb{R}^n\setminusΩ. \end{aligned}\right. \end{equation*} More precisely, for any given $t\in[s,\min\{2s,\,s+1/2\})$, we prove that there exists a small positive constant $η=η(n,s,t,Ω)$ such that the following holds. If $n=2$ and $s\in(1/2,1)$, then, for any $q\in(1,\frac{4}{1+2s}+η)$ and $p\in(1,\frac{nq}{n-(2s-t)q}]$, \begin{equation*} \|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(Ω)}, \end{equation*} and, if either $n=2$ and $s\in(0,1/2]$ or $n\ge 3$ and $s\in(0,1)$, then, for any $q\in(1,\frac{n(2s+1)}{nt+(2s-t)(2s+1)}+η)$ and $p\in(1,\frac{nq}{n-(2s-t)q}]$, \begin{equation*} \|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(Ω)}. \end{equation*} Here the positive constant $C$ depends only on $n$, $s$, $t$, $p$, $q$, and $Ω$. The stated universal baselines are optimal in Dahlberg's sense: for any larger target exponent $p$ there is a bounded Lipschitz domain whose solution with right-hand side identically $1$ is not in $W^{t,p}(\mathbb{R}^n)$. The counterexamples are constructed using regular Cantor sets on the boundary and, in the planar exceptional case, homogeneous solutions in sectors.
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