Gradient estimates via two-point functions for elliptic equations on manifolds

TL;DR

This paper develops new gradient estimates for solutions to elliptic equations on manifolds, relating solution values at two points to their distance, with implications for various geometric settings and boundary conditions.

Contribution

It introduces a novel two-point function method to derive sharp gradient bounds for elliptic equations on manifolds, extending to Finsler and bounded geometry cases.

Findings

Derived pointwise solution estimates depending only on dimension and Ricci curvature bounds.

Established sharp gradient bounds relating solution gradients to symmetric model solutions.

Extended results to Finsler manifolds and manifolds with boundary conditions.

Abstract

We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the Ricci curvature. These estimates imply sharp gradient bounds relating the gradient of an arbitrary solution at given height to that of a symmetric solution on a warped product model space. We also discuss the problem on Finsler manifolds with nonnegative weighted Ricci curvature, and on complete manifolds with bounded geometry, including solutions on manifolds with boundary with Dirichlet boundary condition. Particular cases of our results include gradient estimates of Modica type.