期刊
PUBLICACIONS MATEMATIQUES
卷 60, 期 1, 页码 3-26出版社
UNIV AUTONOMA BARCELONA
DOI: 10.5565/PUBLMAT_60116_01
关键词
Integro-differential equations; bounded domains; regularity
类别
资金
- [MTM2011-27739-C04-01]
- [2009SGR345]
In this paper we survey some results on the Dirichlet problem {Lu = f in Omega u = g in R-n\Omega for nonlocal operators of the form Lu(x) = PV integral(Rn){u(x) - u(x + y)}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.
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