Duke Mathematical Journal

Boundary regularity for fully nonlinear integro-differential equations

Xavier Ros-Oton and Joaquim Serra

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1458133141

Digital Object Identifier
doi:10.1215/00127094-3476700

Mathematical Reviews number (MathSciNet)
MR3536990

Zentralblatt MATH identifier
1351.35245

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

Keywords
fully nonlinear integro-differential equations boundary regularity

Citation

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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