Interior regularity for nonlocal fully nonlinear equations with Dini continuous terms (Q267471)

In this paper, the author studies the interior regularity of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations with Dini-continuous terms. Regularity theory of viscosity solutions for integro-differential equations has been studied by many authors under a uniform ellipticity assumption. It was initiated by a series of seminal papers by L. A. Caffarelli and L. Silvestre, where \(C^{\alpha}\) regularity, \(C^{1+\alpha}\) regularity and the Evans-Krylov theorem for nonlocal fully nonlinear elliptic equations were established. In this paper, the author obtains \(C^{\alpha}\) regularity estimates with Dini-continuous data in two steps. First, he generalizes the recursive Evans-Krylov theorem for translation invariant nonlocal fully nonlinear equations from the case of Hölder-continuous data to the Dini-continuous case. Then, the \(C^{\alpha}\) regularity estimates are obtained via perturbative methods.