Quasi-Harnack inequality

TL;DR

This paper extends the classical Harnack inequality to functions with two-scale behavior, broadening its applicability to various complex settings like nonlocal equations and homogenization.

Contribution

It introduces a quasi-Harnack inequality for functions not satisfying an infinitesimal equation but showing two-scale comparison and estimate properties.

Findings

Extended Harnack inequality for two-scale functions

Applications to nonlocal and discrete equations

Insights into quasi-minimal surfaces

Abstract

In this paper we obtain some extensions of the classical Krylov-Safonov Harnack inequality. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. We require that at scale larger than some $r_0>0$ (small) the functions satisfy the comparison principle with a standard family of quadratic polynomials, while at scale $r_0$ they satisfy a Weak Harnack type estimate. We also give several applications of the main result in very different settings such as discrete difference equations, nonlocal equations, homogenization and the quasi-minimal surfaces of Almgren.