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 $Iu=c(x)u$ in $\Omega$, where $\Omega \subset {\mathbb{R}}^N$ is a domain, $c\in L^{\infty }(\Omega )$ and $I$ is an operator of the form $Iu(x)=P.V.{\int }_{{\mathbb{R}}^N}(u(x)-u(y))j(x-y)dy$ 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.

To appear in Math. Z

References in corpus (1)

• Strong maximum principles for fractional Laplacians

Cited by in corpus (1)

• Strong comparison principle for the fractional $p$-Laplacian and applications to starshaped rings