On the strong maximum principle for a fractional Laplacian,Archiv der Mathematik

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On the strong maximum principle for a fractional Laplacian
Archiv der Mathematik ( IF 0.5 ) Pub Date : 2021-06-04 , DOI: 10.1007/s00013-021-01624-x
Nguyen Ngoc Trong , Do Duc Tan , Bui Le Trong Thanh

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 $.

中文翻译：

关于分数拉普拉斯算子的强最大值原理

在本文中，我们获得了谱 Dirichlet Laplacian 的强最大值原理的一个版本。具体来说，让$d \in \{1,2,3,\ldots \}$、$s \in (\frac{1}{2},1)$和$\Omega \subset \mathbb {R}^d$是开的，有界的，与 Lipschitz 边界相连。假设$u \in L^1(\Omega )$满足$u \ge 0$ ae in $\Omega $和$(-\Delta )^su$是\ (\欧米茄\)。那么你有一个准连续代表${\tilde{u}}$。令$a \in L^1(\Omega )$使得$a \ge 0$ ae in$\Omega $。那么如果