## Duke Mathematical Journal

### Boundary regularity for fully nonlinear integro-differential equations

#### Abstract

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order $2s$, with $s\in(0,1)$. We consider the class of nonlocal operators $\mathcal{L}_{*}\subset\mathcal{L}_{0}$, which consists of infinitesimal generators of stable Lévy processes belonging to the class $\mathcal{L}_{0}$ of Caffarelli–Silvestre. For fully nonlinear operators $\mathrm{I}$ elliptic with respect to $\mathcal{L}_{*}$, we prove that solutions to $\mathrm{I}u=f$ in $\Omega$, $u=0$ in $\mathbb{R}^{n}\setminus\Omega$, satisfy $u/d^{s}\in C^{s+\gamma}(\bar{\Omega})$, where $d$ is the distance to $\partial\Omega$ and $f\in C^{\gamma}$. We expect the class $\mathcal{L}_{*}$ to be the largest scale-invariant subclass of $\mathcal{L}_{0}$ for which this result is true. In this direction, we show that the class $\mathcal{L}_{0}$ is too large for all solutions to behave as $d^{s}$. The constants in all the estimates in this article remain bounded as the order of the equation approaches $2$. Thus, in the limit $s\uparrow1$, 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
Revised: 23 August 2015
First available in Project Euclid: 16 March 2016

https://projecteuclid.org/euclid.dmj/1458133141

Digital Object Identifier
doi:10.1215/00127094-3476700

Mathematical Reviews number (MathSciNet)
MR3536990

Zentralblatt MATH identifier
1351.35245

#### 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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