Open Access
2018 The Dirichlet problem for nonlocal Lévy-type operators
Artur Rutkowski
Publ. Mat. 62(1): 213-251 (2018). DOI: 10.5565/PUBLMAT6211811


We present the theory of the Dirichlet problem for nonlocal operators which are the generators of general pure-jump symmetric Lévy processes whose Lévy measures need not be absolutely continuous. We establish basic facts about the Sobolev spaces for such operators, in particular we prove the existence and uniqueness of weak solutions. We present strong and weak variants of maximum principle, and $L^\infty$ bounds for solutions. We also discuss the related extension problem in $C^{1,1}$ domains.


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Artur Rutkowski. "The Dirichlet problem for nonlocal Lévy-type operators." Publ. Mat. 62 (1) 213 - 251, 2018.


Received: 7 November 2016; Revised: 15 May 2017; Published: 2018
First available in Project Euclid: 16 December 2017

zbMATH: 06848693
MathSciNet: MR3738190
Digital Object Identifier: 10.5565/PUBLMAT6211811

Primary: 35S15 , 47G20 , 60G51

Keywords: Dirichlet problem , extension operator , maximum principle , nonlocal operator , weak solutions

Rights: Copyright © 2018 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.62 • No. 1 • 2018
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