Holder regularity for fully nonlinear nonlocal equations

TL;DR

This paper surveys the regularity of solutions to fully nonlinear nonlocal elliptic equations, extending previous results to include extremal nonlocal operators with gradient terms.

Contribution

It extends existing regularity results to fully nonlinear extremal nonlocal operators with gradient terms, broadening the understanding of solution behavior.

Findings

Proves Hölder continuity for viscosity solutions of these equations.

Extends prior work to include fully nonlinear extremal nonlocal operators.

Provides a comprehensive survey of recent advances in the field.

Abstract

In this survey we prove H\"older regularity results for viscosity solutions of fully nonlinear nonlocal uniformly elliptic second order differential equations with local gradient terms. This extends the nonlocal counterpart of the work of G. Barles, E. Chasseigne and C. Imbert in JEMS, 2011, to fully nonlinear extremal nonlocal operators.