Jian-Hua Hao | Searcharxiv

arXiv2026

Differential Harnack Estimates and Frequency Monotonicity for Filtration Equations on Riemannian Manifolds

In this work we derive local differential Harnack estimates for positive solutions of filtration equations on complete Riemannian manifolds with a lower Ricci curvature bound by using Nash--Moser iteration. Inspired by earlier work on parabolic frequency functions, we establish some new frequency monotonicity formulas for these equations. The results cover the porous-medium and fast-diffusion equations, as well as several non-power diffusion laws.

math.DG↗

arXiv2026

Gradient estimates and Liouville theorems for the \(Φ\)-Laplacian equations on Riemannian manifolds

This paper establishes gradient estimates and Liouville-type theorems for the \(Φ\)-Laplacian equation \(Δ_Φ(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = Λ_Φ(u)\) on compact Riemannian manifolds. Using a nonlinear \(Φ\)-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds under the lower bound assumption of Ricci curvature and suitable conditions on \(Φ\) and \(G\), which leads to Liouville theorems for global solutions. For the parabolic case, we employ the maximum principle to derive gradient estimates on compact Riemannian manifolds, and subsequently obtain Liouville-type results. Our work provides a unified framework that generalizes prior results for \(p\)-harmonic functions and other quasilinear equations.

math.DG↗