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On the strong maximum principle for nonlocal operators

  • Sven Jarohs
  • Tobias Weth
Article
  • 12 Downloads

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.

Keywords

Nonlocal operator Strong maximum principle Weak maximum principle 

Notes

Acknowledgements

The authors thank Moritz Kassmann for valuable discussions.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikGoethe-Universität, FrankfurtFrankfurtGermany

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