Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains

Abstract

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.