Open Access
2016 Nonlocal Equations in Bounded Domains: A Survey
Xavier Ros-Oton
Publ. Mat. 60(1): 3-26 (2016).


In this paper we survey some results on the Dirichlet problem

\[\begin{cases} L u =f&\text{in }\Omega \\ u=g &\text{in }\mathbb{R}^n\backslash\Omega \end{cases} \]

for nonlocal operators of the form

\[ Lu(x)=\operatorname{PV}\int_{\mathbb{R}^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)\,dy. \]

We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain.

In order to include some natural operators~$L$ in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations.

We hope that this survey will be useful for both novel and more experienced researchers in the field.


Download Citation

Xavier Ros-Oton. "Nonlocal Equations in Bounded Domains: A Survey." Publ. Mat. 60 (1) 3 - 26, 2016.


Published: 2016
First available in Project Euclid: 22 December 2015

zbMATH: 1337.47112
MathSciNet: MR3447732

Primary: 35B65 , 47G20 , 60G52

Keywords: bounded domains , Integro-differential equations , regularity

Rights: Copyright © 2016 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.60 • No. 1 • 2016
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