Duke Mathematical Journal

Boundary regularity for fully nonlinear integro-differential equations

Xavier Ros-Oton and Joaquim Serra

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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.

Article information

Duke Math. J., Volume 165, Number 11 (2016), 2079-2154.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]

fully nonlinear integro-differential equations boundary regularity


Ros-Oton, Xavier; Serra, Joaquim. Boundary regularity for fully nonlinear integro-differential equations. Duke Math. J. 165 (2016), no. 11, 2079--2154. doi:10.1215/00127094-3476700. https://projecteuclid.org/euclid.dmj/1458133141

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