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On the loss of maximum principles for higher-order fractional Laplacians


Authors: Nicola Abatangelo, Sven Jarohs and Alberto Saldaña
Journal: Proc. Amer. Math. Soc. 146 (2018), 4823-4835
MSC (2010): Primary 35B50; Secondary 35S15, 35J35
DOI: https://doi.org/10.1090/proc/14165
Published electronically: August 8, 2018
MathSciNet review: 3856149
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Abstract: We study the existence and positivity of solutions to problems involving higher-order fractional Laplacians $ (-\Delta )^s$ for any $ s>1$. In particular, using a suitable variational framework and the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for $ s\in (n,n+1)$ with $ n\in \mathbb{N}$ odd, and we mention some particular domains where positivity preserving properties do hold.


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Additional Information

Nicola Abatangelo
Affiliation: Département de Mathḿatiques, Université Libre de Bruxelles CP 214, boulevard du Triomphe, 1050 Ixelles, Belgium
Email: nicola.abatangelo@ulb.ac.be

Sven Jarohs
Affiliation: Institut für Mathematik, Goethe-Universität, Frankfurt, Robert-Mayer-Straße 10, 60054 Frankfurt am Main, Germany
Email: jarohs@math.uni-frankfurt.de

Alberto Saldaña
Affiliation: Institut für Analysis, Karlsruhe Institute for Technology, Englerstraße 2, 76131, Karlsruhe, Germany
Email: alberto.saldana@partner.kit.edu

DOI: https://doi.org/10.1090/proc/14165
Received by editor(s): March 6, 2018
Published electronically: August 8, 2018
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2018 American Mathematical Society