Gradient Bounds and Liouville theorems for Quasi-linear equations on compact Manifolds with nonnegative Ricci curvature

TL;DR

This work develops gradient bounds and rigidity results for the quasi-linear elliptic equation $\mathrm{div}(\Phi'(|\nabla u|^2)\nabla u)=F'(u)$ on compact manifolds with nonnegative Ricci curvature. It introduces the $P$-function $P(u;x)=2\Phi'( |\nabla u|^2 )|\nabla u|^2 - \Phi(|\nabla u|^2) - 2F(u)$ and obtains a sharp pointwise gradient bound $2\Phi'( |\nabla u|^2 )|\nabla u|^2 - \Phi(|\nabla u|^2) \le 2F(u)$, under structural assumptions (A) or (B) on $\Phi$ and $F$. The paper then proves Liouville-type theorems (including a result when $F''\ge 0$) and develops a Harnack-type inequality and an ABP gradient estimate for $|\nabla u|$, with dependence on curvature and gradient bounds; a local splitting theorem is established when equality occurs, yielding a neighborhood isometric to a product with a one-dimensional solution profile. Together, these results extend classical gradient bounds and rigidity phenomena from Euclidean settings to compact Riemannian manifolds and connect stability, regularity, and geometric structure via $\Lambda(t)=2t\Phi''(t)+\Phi'(t)$.

Abstract

In this work we establish a gradient bound and Liouville-type theorems for solutions to Quasi-linear elliptic equations on compact Riemannian Manifolds with nonnegative Ricci curvature. Also, we provide a local splitting theorem when the inequality in the gradient bound becomes equality at some point. Moreover, we prove a Harnack-type inequality and an ABP estimate for the gradient of solutions in domains contained in the manifold.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Lemma 2.1
• Corollary 2.2
• proof : Proof of Lemma \ref{['PfunctionEllipticEstimateForGrBound']}