Optimal Regularity for Fully Nonlinear Nonlocal Equations with Unbounded Source Terms

TL;DR

This paper establishes optimal regularity estimates for solutions to fully nonlinear nonlocal equations with unbounded source terms, covering various integrability regimes and including non-concave cases, using approximation and Liouville techniques.

Contribution

It provides the first regularity results for nonlocal equations with unbounded sources, extending to non-concave operators and identifying critical thresholds for solution smoothness.

Findings

Solutions belong to $C^{\sigma - d/p}$ or $C^\sigma$ depending on source integrability.

Established regularity estimates for non-concave nonlocal equations.

Used novel approximation and Liouville methods for these estimates.

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 $C^{\sigma-d/p}$ to $C^\sigma$, 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.