Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms

Abstract

We prove optimal regularity estimates for viscosity solutions to a class of fully nonlinear nonlocal equations with unbounded source terms. More precisely, depending on the integrability of the source term $f\in L^p(B_1)$, we establish that solutions belong to classes ranging from $C^{\ddot{I}ƒ-d/p}$ to $C^{\ddot{I}}ƒ$, at critical thresholds. We use approximation techniques and Liouville-type arguments. These results represent a novel contribution, providing the first such estimates in the context of not necessarily concave nonlocal equations.