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The Dirichlet problem for nonlocal Lévy-type operators

Artur Rutkowski

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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.

Article information

Publ. Mat., Volume 62, Number 1 (2018), 213-251.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35S15: Boundary value problems for pseudodifferential operators 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 60G51: Processes with independent increments; Lévy processes

Dirichlet problem nonlocal operator maximum principle weak solutions extension operator


Rutkowski, Artur. The Dirichlet problem for nonlocal Lévy-type operators. Publ. Mat. 62 (2018), no. 1, 213--251. doi:10.5565/PUBLMAT6211811.

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