# The Dirichlet problem for nonlocal L\'evy-type operators

## TL;DR

This paper develops a comprehensive theory for the Dirichlet problem involving nonlocal Le9vy-type operators, establishing fundamental properties, solution existence, and bounds, even for measures not absolutely continuous.

## Contribution

It introduces a new framework for analyzing nonlocal Le9vy operators with general measures, including Sobolev space theory and maximum principles.

## Key findings

- Proved existence and uniqueness of weak solutions.
- Established maximum principles and L^a0 bounds.
- Extended the theory to C^{1,1} domains.

## Abstract

We present the theory of the Dirichlet problem for nonlocal operators which are the generators of general pure-jump symmetric L\'evy processes whose L\'evy 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.