Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains
Abstract
Let n≥2, s∈(0,1), and let Ω⊂ℝn be a bounded Lipschitz domain. In this paper, we establish optimal global Sobolev estimates for the fractional Dirichlet problem { (−Δ)su&=f & & in Ω, u&=0 & & in ℝn∖Ω. . More precisely, for any given t∈[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∈(1/2,1), then, for any q∈(1,4/(1+2s)+η) and p∈(1,nq/(n−(2s−t)q)], |u|Wt,p(ℝn)≤ C|f|Lq(Ω), and, if either n=2 and s∈(0,1/2] or n≥ 3 and s∈(0,1), then, for any q∈(1,(n(2s+1))/(nt+(2s−t)(2s+1))+η) and p∈(1,nq/(n−(2s−t)q)], |u|Wt,p(ℝn)≤ C|f|Lq(Ω). 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 Wt,p(ℝn). 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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BibTeX
@misc{ma2026optimal,
title = {{Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains}},
author = {Wenxian Ma and Sibei Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05753},
eprint = {2610.05753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05753},
}
RIS
TY - UNPB
TI - Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains
AU - Wenxian Ma
AU - Sibei Yang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05753
UR - https://arxiv.org/abs/2610.05753
AB - Let n≥2, s∈(0,1), and let Ω⊂ℝ^n be a bounded Lipschitz domain. In this paper, we establish optimal global Sobolev estimates for the fractional Dirichlet problem { (−Δ)^su&=f & & in Ω, u&=0 & & in ℝ^n∖Ω. . More precisely, for any given t∈[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∈(1/2,1), then, for any q∈(1,4/(1+2s)+η) and p∈(1,nq/(n−(2s−t)q)], |u|_W^t,p(ℝ^n)≤ C|f|_L^q(Ω), and, if either n=2 and s∈(0,1/2] or n≥ 3 and s∈(0,1), then, for any q∈(1,(n(2s+1))/(nt+(2s−t)(2s+1))+η) and p∈(1,nq/(n−(2s−t)q)], |u|_W^t,p(ℝ^n)≤ C|f|_L^q(Ω). 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(ℝ^n). The counterexamples are constructed using regular Cantor sets on the boundary and, in the planar exceptional case, homogeneous solutions in sectors.
ER -