Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

TL;DR

This paper extends fundamental estimates and inequalities, specifically the Alexandroff-Bakelman-Pucci estimate and Harnack inequality, to a class of degenerate or singular fully non-linear elliptic equations, including those from stochastic control.

Contribution

It introduces new methods to establish ABP estimates and Harnack inequalities for degenerate elliptic equations involving the m-Laplace operator and Bellman-Isaacs equations.

Findings

Established ABP estimate for degenerate elliptic equations

Proved Harnack inequality for viscosity solutions

Applicable to equations with degeneracy when the gradient is small

Abstract

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate elliptic equations.