# Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian

U. Biccari, M. Warna, E. Zuazua Addendum: Local elliptic regularity for the Dirichlet fractional Laplacian
Adv. Nonlinear Stud., Vol. 17, Nr. 4 (2017), pp. 837-839. DOI: 10.1515/ans-2017-6020

Abstract:

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