## The Dirichlet problem for nonlocal elliptic operators with $C^{0,\alpha }$ exterior data

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- by Alessandro Audrito and Xavier Ros-Oton PDF
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## Abstract:

In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $Lu=0$ in $\Omega$, $u=g$ in $\mathbb R^N\setminus \Omega$, in non-smooth domains $\Omega$. When $g$ is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right-hand side for which the boundary regularity is well understood. Here, we study the case in which $g\in C^{0,\alpha }$, and establish the optimal Hölder regularity of $u$ up to the boundary. Our results extend previous results of Grubb for $C^\infty$ domains $\Omega$.## References

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## Additional Information

**Alessandro Audrito**- Affiliation: Institute of Mathematics DISMA, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy; and Department of Mathematics, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
- MR Author ID: 1210008
- Email: alessandro.audrito@polito.it, alessandro.audrito@math.uzh.ch
**Xavier Ros-Oton**- Affiliation: Institute of Mathematics, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland; and ICREA, Passeig Lluís Companys 23, 08010 Barcelona, Spain; and Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
- MR Author ID: 920237
- Email: xavier.ros-oton@math.uzh.ch
- Received by editor(s): October 21, 2019
- Received by editor(s) in revised form: March 13, 2020
- Published electronically: July 20, 2020
- Additional Notes: The first author was supported by PRIN 2015 (MIUR, Italy), and by the GNAMPA projects “Esistenza e proprietà qualitative per soluzioni di EDP non lineari ellittiche e paraboliche” and “Ottimizzazione Geometrica e Spettrale” (Italy).

The second author was supported by the European Research Council under the Grant Agreement No. 801867 “Regularity and singularities in elliptic PDE (EllipticPDE)”, by the Swiss National Science Foundation, and by MINECO grant MTM2017-84214-C2-1-P - Communicated by: Ryan Hynd
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 4455-4470 - MSC (2010): Primary 35B65; Secondary 47G20
- DOI: https://doi.org/10.1090/proc/15121
- MathSciNet review: 4135310