在本文中，我们综述了关于非局部算子形式为\[Lu(x)=\textrm{PV}\int_{\mathbb R^n}\bigl\{u(x)-u(x+y)\bigr\}K(y)dy.\]的狄利克雷问题\[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }Ω\\ u&=&g&\textrm{in }\mathbb R^n\backslashΩ\end{array}\right.\]的一些结果。我们从最基础的内容开始，证明解的存在性、最大值原理，并构造一些有用的障碍。然后，我们集中在正则性的解决方案，在内部和边界上的区域。为了在正则性理论中包括一些自然算子$L$，我们不假设核上有任何正则性。这导致了一些有趣的功能，是纯粹的非本地，在这个意义上，没有类似的本地方程。我们希望，这项调查将是有用的新的和更有经验的研究人员在该领域。
英文摘要：
In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }Ω\\ u&=&g&\textrm{in }\mathbb R^n\backslashΩ\end{array}\right.\] 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.
We hope that this survey will be useful for both novel and more experienced researchers in the field.
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