Tempered fractional Laplacian: Boundary regularity

Abstract

We study the boundary regularity of solutions for the tempered fractional Laplacian \[ (-Δ)^s_λu=f \quad \text{in }Ω, \qquad u=0 \quad \text{in }\mathbb{R}^n\setminusΩ, \] where \(0<s<1\), \(λ>0\), and \(Ω\subset\mathbb{R}^n\) is a bounded Lipschitz domain satisfying the exterior ball condition. The tempered operator is defined by \[ (-Δ)^s_λu(x) = c_{n,s}\operatorname{P.V.} \int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}} e^{-λ|x-y|}\,dy, \] 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 \(\mathcal Q_λ^s\), and combining approximation, localization, and iterative arguments with interpolation inequalities, we establish the global estimate \[ \|u\|_{C^s(\mathbb{R}^n)} \le C\|f\|_{L^\infty(Ω)}. \] For more regular data, we derive interior and boundary-weighted Schauder estimates. In particular, under suitable regularity assumptions on \(f\), these estimates yield \[ |D^νu(x)| \le C\,δ(x)^{s-|ν|}, \quad |ν| > 0, \] where \(ν\) is a multi-index and \(δ(x)=\operatorname{dist}(x,\partialΩ)\). 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.