On the strong maximum principle for a fractional Laplacian

Abstract

In this paper, we obtain a version of the strong maximum principle for the spectral Dirichlet Laplacian. Specifically, let $d \in \{1,2,3,\ldots \}$, $s \in (\frac{1}{2},1)$, and $\Omega \subset \mathbb {R}^d$ be open, bounded, connected with Lipschitz boundary. Suppose $u \in L^1(\Omega )$ satisfies $u \ge 0$ a.e. in $\Omega$ and $(-\Delta )^s u$ is a Radon measure on $\Omega$. Then u has a quasi-continuous representative ${\tilde{u}}$. Let $a \in L^1(\Omega )$ be such that $a \ge 0$ a.e. in $\Omega$. Then if

$$\begin{aligned} (-\Delta )^s u + au \ge 0 \quad \text {a.e.} \text { in } \Omega \end{aligned}$$

and ${\tilde{u}} = 0$ on a subset of positive $H^s$-capacity of $\Omega$, then $u = 0$ a.e. in $\Omega$.