Harnack inequality for degenerate and singular operators of $p$-Laplacian type on Riemannian manifolds

TL;DR

This paper establishes Harnack inequalities for degenerate and singular p-Laplacian type operators on Riemannian manifolds, extending classical results to more general geometric and nonlinear settings.

Contribution

It proves Harnack inequalities for p-Laplacian operators on manifolds with Ricci curvature bounds, including nonlinear perturbations, using ABP estimates.

Findings

Harnack inequality holds for p-Laplacian on manifolds with Ricci curvature bounded below

Extension of inequalities to nonlinear perturbations of Ricci curvature

Use of ABP estimates in the geometric PDE context

Abstract

We study viscosity solutions to degenerate and singular elliptic equations of $p$-Laplacian type on Riemannian manifolds. The Krylov-Safonov type Harnack inequality for the $p$-Laplacian operators with $1<p<\infty$ is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear $p$-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.