Perron's method for nonlocal fully nonlinear equations

TL;DR

This paper develops Perron's method to establish the existence of viscosity solutions for nonlocal fully nonlinear equations that are not translation invariant, including proving regularity under uniform ellipticity.

Contribution

It introduces a novel application of Perron's method to nonlocal equations, constructing discontinuous solutions and proving their regularity under ellipticity conditions.

Findings

Constructed discontinuous viscosity solutions for nonlocal equations

Proved Hölder continuity of solutions under uniform ellipticity

Extended Perron's method to non-translation invariant nonlocal equations

Abstract

This paper is concerned with existence of viscosity solutions of non-translation invariant nonlocal fully nonlinear equations. We construct a discontinuous viscosity solution of such nonlocal equation by Perron's method. If the equation is uniformly elliptic in the sense of \cite{SS}, we prove the discontinuous viscosity solution is H\"older continuous and thus it is a viscosity solution.