Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Abstract

In [1], for 1<p<∞{1<p<\infty}, we proved the Wloc2⁢s,p{W^{2s,p}_{\mathrm{loc}}} local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian (-Δ)s{(-\Delta)^{s}} on an arbitrary bounded open set of ℝN{\mathbb{R}^{N}}. Here we make a more precise and rigorous statement. In fact, for 1<p<2{1<p<2} and s≠12{s\neq\frac{1}{2}}, local regularity does not hold in the Sobolev space Wloc2⁢s,p{W^{2s,p}_{\mathrm{loc}}}, but rather in the larger Besov space (Bp,22⁢s)loc{(B^{2s}_{p,2})_{\mathrm{loc}}}.

Keywords

• fractional laplacian
• dirichlet boundary condition
• weak solutions
• local regularity
• 35b65
• 35r11
• 35s05