ABP maximum principles for fully nonlinear integro-differential equations with unbounded inhomogeneous terms

TL;DR

This paper extends Aleksandrov-Bakelman-Pucci maximum principles to a class of fully nonlinear integro-differential equations with order close to 2, improving existing estimates to depend only on the inhomogeneous term's L^n norm.

Contribution

It provides new maximum principles for nonlinear integro-differential equations that remove dependence on the L^ finity norm of the inhomogeneous term.

Findings

Established maximum principles for equations of order near 2.

Improved estimates to depend solely on the L^n norm of the inhomogeneous term.

Extended applicability of ABP maximum principles to broader class of equations.

Abstract

Aleksandrov-Bakelman-Pucci maximum principles are studied for a class of fully nonlinear integro-differential equations of order $\sigma\in [2-\varepsilon_0,2)$, where $\varepsilon_0$ is a small constant depending only on given parameters. The goal of this paper is to improve an estimate of Guillen and Schwab (Arch. Ration. Mech. Anal., 206, 2012) in order to avoid the dependence on $L^\infty$ norm of %the to the estimate depending only on $L^n$ norm the inhomogeneous term.