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Continuous solutions and approximating scheme for fractional Dirichlet problems on Lipschitz domains

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Representations of solutions

Published online by Cambridge University Press:  26 December 2018

Patricio Felmer  and
Erwin Topp
Patricio Felmer
Affiliation:
Departamento de Ingenierí a Matemática and CMM (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (pfelmer@dim.uchile.cl)
Erwin Topp
Affiliation:
Departamento de Matemática y C. C., Universidad de Santiago de ChileCasilla 307, Santiago, Chile (erwin.topp@usach.cl)
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Abstract

In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.

Keywords

Elliptic equationsnonlocal operatorDirichlet problemLipschitz domainsviscosity solutions

MSC classification

Primary: 35D40: Viscosity solutions
Secondary: 35R10: Partial functional-differential equations 35B35: Stability 35B50: Maximum principles
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 533 - 560
Copyright
Copyright © Royal Society of Edinburgh 2018 

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