Tempered fractional Laplacian: Boundary regularity

Abstract

We study the boundary regularity of solutions for the tempered fractional Laplacian \[ (-\Delta)^s_\lambda u=f \quad \text{in }\Omega, \qquad u=0 \quad \text{in }\mathbb{R}^n\setminus\Omega, \] where \(00\), and \(\Omega\subset\mathbb{R}^n\) is a bounded Lipschitz domain satisfying the exterior ball condition. The tempered operator is defined by \[ (-\Delta)^s_\lambda u(x) = c_{n,s}\operatorname{P.V.} \int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}} e^{-\lambda|x-y|}\,dy, \] 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 \(\mathcal Q_\lambda^s\), and combining approximation, localization, and iterative arguments with interpolation inequalities, we establish the global estimate \[ \|u\|_{C^s(\mathbb{R}^n)} \le C\|f\|_{L^\infty(\Omega)}. \] For more regular data, we derive interior and boundary-weighted Schauder estimates. In particular, under suitable regularity assumptions on \(f\), these estimates yield \[ |D^\nu u(x)| \le C\,\delta(x)^{s-|\nu|}, \quad |\nu|>0, \] where \(\nu\) is a multi-index and \(\delta(x)=\operatorname{dist}(x,\partial\Omega)\). 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.