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 ${\mathcal{L}}_{\ast }\subset {\mathcal{L}}_0$, 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 ${\mathcal{L}}_{\ast }$, we prove that solutions to $\mathrm{I}u=f$ in $\Omega$, $u=0$ in ${\mathbb{R}}^n\setminus \Omega$, satisfy $u/d^s\in C^{s+\gamma }\left(\overline{\Omega }\right)$, where $d$ is the distance to $\partial \Omega$ and $f\in C^{\gamma }$. We expect the class ${\mathcal{L}}_{\ast }$ 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 $s\uparrow 1$, we recover the celebrated boundary regularity result due to Krylov for fully nonlinear elliptic equations.