Weak and viscosity solutions of the fractional Laplace equation

Abstract

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation $$ \begin{cases} (-\Delta)^s u=f &\text{in }\Omega\\ u=g &\text{in }\mathbb R^n\setminus\Omega \end{cases} $$ are also continuous solutions (up to the boundary) of this problem in the viscosity sense.

Here $s\in(0,1)$ is a fixed parameter, $\Omega$ is a bounded, open subset of $\mathbb R^n$ ($n\geqslant1$) with $C^2$-boundary, and $(-\Delta)^s$ is the fractional Laplacian operator, that may be defined as $$(-\Delta)^su(x):=c(n,s)\int\limits_{\mathbb R^n}\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy,$$ for a suitable positive normalizing constant $c(n,s)$, depending only on $n$ and $s$.

In order to get our regularity result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem.

As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of $(-\Delta)^s$ is strictly positive in $\Omega$.