15 August 2016 Boundary regularity for fully nonlinear integro-differential equations
Xavier Ros-Oton, Joaquim Serra
Duke Math. J. 165(11): 2079-2154 (15 August 2016). DOI: 10.1215/00127094-3476700


We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s(0,1). We consider the class of nonlocal operators LL0, which consists of infinitesimal generators of stable Lévy processes belonging to the class L0 of Caffarelli–Silvestre. For fully nonlinear operators I elliptic with respect to L, we prove that solutions to Iu=f in Ω, u=0 in RnΩ, satisfy u/dsCs+γ(Ω¯), where d is the distance to Ω and fCγ. We expect the class L to be the largest scale-invariant subclass of L0 for which this result is true. In this direction, we show that the class L0 is too large for all solutions to behave as ds. The constants in all the estimates in this article remain bounded as the order of the equation approaches 2. Thus, in the limit s1, we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.


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Xavier Ros-Oton. Joaquim Serra. "Boundary regularity for fully nonlinear integro-differential equations." Duke Math. J. 165 (11) 2079 - 2154, 15 August 2016. https://doi.org/10.1215/00127094-3476700


Received: 2 April 2014; Revised: 23 August 2015; Published: 15 August 2016
First available in Project Euclid: 16 March 2016

zbMATH: 1351.35245
MathSciNet: MR3536990
Digital Object Identifier: 10.1215/00127094-3476700

Primary: 35J60
Secondary: 45K05

Keywords: boundary regularity , fully nonlinear integro-differential equations

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 11 • 15 August 2016
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