## Publicacions Matemàtiques

### The Dirichlet problem for nonlocal Lévy-type operators

Artur Rutkowski

#### 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
Revised: 15 May 2017
First available in Project Euclid: 16 December 2017

https://projecteuclid.org/euclid.pm/1513393236

Digital Object Identifier
doi:10.5565/PUBLMAT6211811

Mathematical Reviews number (MathSciNet)
MR3738190

Zentralblatt MATH identifier
06848693

#### 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