Nonlocal elliptic equations in bounded domains: a survey

Understand

In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form \[Lu(x)=\textrm{PV}\int_{\mathbb R^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)dy.\] 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 have no analogue for local equations.