Boundary regularity for fully nonlinear integro-differential equations (Q320242)

\textit{L. Caffarelli} and \textit{L. Silvestre} [Commun. Pure Appl. Math. 62, No. 5, 597--638 (2009; Zbl 1170.45006)] studied the interior regularity for fully nonlinear nonlocal problems. In this important paper, the Authors analyse the case of the boundary regularity. They consider the class of nonlocal operators \({\mathcal L}_* \subset {\mathcal L}_o\) which consists of infinitesimal generators of stable Lévy processes belonging to the class \( {\mathcal L}_o\) introduced by Caffarelli and Silvestre. Consider the operator \(I\) elliptic with respect to \({\mathcal L}_* \). Let \(u\) be a the Dirichlet solution of \(Iu=f\) in \(\Omega\) with \(f\in C^\gamma\), then \(\frac{u}{d^s}\) is \(s+\gamma\) Hölder continuous in \(\bar{\Omega}\). The constants are stables when \(s \rightarrow 1\) and therefore the celebrated Krylov result is recovered by this approach.