On the Harnack inequality for degenerate and singular elliptic equations with unbounded lower order terms via sliding paraboloids

TL;DR

This paper develops a method using sliding paraboloids to prove a Harnack inequality for a broad class of degenerate and singular elliptic equations with unbounded lower order terms, including Monge-Ampère equations.

Contribution

It introduces a novel approach with sliding paraboloids to establish Harnack inequalities for complex elliptic equations with unbounded coefficients.

Findings

Proves a Harnack inequality for degenerate and singular elliptic equations.

Establishes a doubling estimate for solutions with large gradients.

Applies to equations with unbounded lower order terms, including Monge-Ampère.

Abstract

We use the method of sliding paraboloids to establish a Harnack inequality for linear, degenerate and singular elliptic equation with unbounded lower order terms. The equations we consider include uniformly elliptic equations and linearized Monge-Amp\`ere equations. Our argument allows us to prove the doubling estimate for functions which, at points of large gradient, are solutions of (degenerate and singular) elliptic equations with unbounded drift.