On the strong maximum principle for nonlocal operators

Abstract

In this paper we derive a strong maximum principle for weak supersolutions of nonlocal equations of the form $$\begin{aligned} Iu=c(x) u \qquad \text { in }\,\Omega , \end{aligned}$$ where $$\Omega \subset \mathbb {R}^N$$ is a domain, $$c\in L^{\infty }(\Omega )$$ and I is an operator of the form $$\begin{aligned} Iu(x)=P.V.\int \limits _{\mathbb {R}^N}(u(x)-u(y))j(x-y) dy \end{aligned}$$ with a nonnegative kernel function j. We formulate minimal positivity assumptions on j corresponding to a class of operators, which includes highly anisotropic variants of the fractional Laplacian. Somewhat surprisingly, this problem leads to the study of general lattices in $$\mathbb {R}^N$$ . Our results extend to the regional variant of the operator I and, under weak additional assumptions, also to the case of x-dependent kernel functions.