Tempered fractional Laplacian: Boundary regularity
Abstract
We study the boundary regularity of solutions for the tempered fractional Laplacian (−Δ)sλu=f in Ω, u=0 in ℝn∖Ω, where 0<s<1, λ>0, and Ω⊂ℝn is a bounded Lipschitz domain satisfying the exterior ball condition. The tempered operator is defined by (−Δ)sλu(x) = cn,sP.V. ∫ℝn (u(x)−u(y))/(|x−y|n+2s) e−λ|x−y| dy, which was originally proposed in [Koponen, Phys. Rev. E (1995), 1197-1199]. % Exponential tempering preserves the leading singularity of % the kernel but destroys its homogeneity, preventing a direct % application of classical boundary regularity estimates. % Extending these estimates [Ros-Oton and Serra, J. Math. Pures Appl. (2014), 275--302] to the tempered setting therefore % requires overcoming the difficulties caused by the loss % of scale invariance. By decomposing the operator into the classical fractional Laplacian and the correction operator 𝒬λs, and combining approximation, localization, and iterative arguments with interpolation inequalities, we establish the global estimate |u|Cs(ℝn) ≤ C|f|L∞(Ω). For more regular data, we derive interior and boundary-weighted Schauder estimates. In particular, under suitable regularity assumptions on f, these estimates yield |Dνu(x)| ≤ C δ(x)s−|ν|, |ν| > 0, where ν is a multi-index and δ(x)=dist(x,∂Ω). These bounds quantify the possible growth of higher-order derivatives near the boundary and provide a regularity foundation for numerical error analysis of the tempered problem.
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BibTeX
@misc{zhao2026tempered,
title = {{Tempered fractional Laplacian: Boundary regularity}},
author = {Jie Zhao and Minghua Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16437},
eprint = {2610.16437},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16437},
}
RIS
TY - UNPB
TI - Tempered fractional Laplacian: Boundary regularity
AU - Jie Zhao
AU - Minghua Chen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16437
UR - https://arxiv.org/abs/2610.16437
AB - We study the boundary regularity of solutions for the tempered fractional Laplacian (−Δ)^s_λu=f in Ω, u=0 in ℝ^n∖Ω, where 0<s<1, λ>0, and Ω⊂ℝ^n is a bounded Lipschitz domain satisfying the exterior ball condition. The tempered operator is defined by (−Δ)^s_λu(x) = c_n,sP.V. ∫_ℝ^n (u(x)−u(y))/(|x−y|^n+2s) e^−λ|x−y| dy, which was originally proposed in [Koponen, Phys. Rev. E (1995), 1197-1199]. % Exponential tempering preserves the leading singularity of % the kernel but destroys its homogeneity, preventing a direct % application of classical boundary regularity estimates. % Extending these estimates [Ros-Oton and Serra, J. Math. Pures Appl. (2014), 275--302] to the tempered setting therefore % requires overcoming the difficulties caused by the loss % of scale invariance. By decomposing the operator into the classical fractional Laplacian and the correction operator 𝒬_λ^s, and combining approximation, localization, and iterative arguments with interpolation inequalities, we establish the global estimate |u|_C^s(ℝ^n) ≤ C|f|_L^∞(Ω). For more regular data, we derive interior and boundary-weighted Schauder estimates. In particular, under suitable regularity assumptions on f, these estimates yield |D^νu(x)| ≤ C δ(x)^s−|ν|, |ν| > 0, where ν is a multi-index and δ(x)=dist(x,∂Ω). These bounds quantify the possible growth of higher-order derivatives near the boundary and provide a regularity foundation for numerical error analysis of the tempered problem.
ER -