# Local elliptic regularity for the Dirichlet fractional Laplacian

## TL;DR

This paper investigates the local elliptic regularity of solutions to the Dirichlet problem for the fractional Laplacian, establishing regularity in Besov and Sobolev spaces depending on the integrability parameter p.

## Contribution

It provides new regularity results for weak solutions of the fractional Laplacian Dirichlet problem, using two different analytical methods to analyze local elliptic equations.

## Key findings

- Solutions are in $B^{2s}_{p,2,loc}( ext{Omega})$ for $1<p<2$.
- Solutions are in $W^{2s,p}_{loc}( ext{Omega})$ for $2 extless p< ext{infinity}$.
- Two methods: variational formulation and heat kernel representation.

## Abstract

We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-\Delta)^s$ on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\textrm{loc}}(\Omega)$, while for $2\leq p<\infty$ we show that the solutions belong to $W^{2s,p}_{\textrm{loc}}(\Omega)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.