Local Gradient Estimate for $p$-harmonic functions on Riemannian Manifolds

TL;DR

This paper establishes gradient estimates and Harnack inequalities for positive p-harmonic functions on Riemannian manifolds, with constants depending only on curvature bounds, dimension, p, and the domain radius.

Contribution

It introduces a new approach using Moser iteration to derive gradient estimates that depend solely on Ricci curvature lower bounds, differing from previous sectional curvature methods.

Findings

Gradient estimates depend only on Ricci curvature lower bounds.

Harnack inequalities are established with explicit constants.

Method applies to positive p-harmonic functions on Riemannian manifolds.

Abstract

For positive $p$-harmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension $n$, $p$ and the radius of the ball on which the function is defined. Our approach is based on a careful application of the Moser iteration technique and is different from Cheng-Yau's method employed by Kostchwar and Ni, in which a gradient estimate for positive $p$-harmonic functions is derived under the assumption that the sectional curvature is bounded from below.