Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains

TL;DR

This paper establishes Besov regularity estimates for solutions to the fractional Laplacian Dirichlet problem in Lipschitz domains, showing no loss of regularity compared to smooth domains using elementary methods.

Contribution

It provides the first Besov regularity estimates for the fractional Laplacian in Lipschitz domains with explicit constants, demonstrating regularity preservation despite boundary irregularities.

Findings

Regularity estimates match those in smooth domains.

No regularity loss due to Lipschitz boundaries.

Elementary proof techniques used.

Abstract

We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order $s$ in bounded Lipschitz domains $\Omega$: \[ \begin{aligned} \|u\|_{\dot{B}^{s+r}_{2,\infty}(\Omega)} \le C \|f\|_{L^2(\Omega)}, & \quad r = \min\{s,1/2\}, & \quad \mbox{if } s \neq 1/2, \\ \|u\|_{\dot{B}^{1-\epsilon}_{2,\infty}(\Omega)} \le C \|f\|_{L^2(\Omega)}, & \quad \epsilon \in (0,1), & \quad \mbox{if } s = 1/2, \end{aligned} \] with explicit dependence of $C$ on $s$ and $\epsilon$. These estimates are consistent with the regularity on smooth domains and show that there is no loss of regularity due to Lipschitz boundaries. The proof uses elementary ingredients, such as the variational structure of the problem and the difference quotient technique.