Publicacions Matemàtiques

The Dirichlet problem for nonlocal Lévy-type operators

Artur Rutkowski

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.pm/1513393236

Digital Object Identifier
doi:10.5565/PUBLMAT6211811

Mathematical Reviews number (MathSciNet)
MR3738190

Zentralblatt MATH identifier
06848693

Subjects
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

Keywords
Dirichlet problem nonlocal operator maximum principle weak solutions extension operator

Citation

Rutkowski, Artur. The Dirichlet problem for nonlocal Lévy-type operators. Publ. Mat. 62 (2018), no. 1, 213--251. doi:10.5565/PUBLMAT6211811. https://projecteuclid.org/euclid.pm/1513393236


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