Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation☆

By Junfang Li, Xiangjin Xu. 2009

125 citations recorded by the source corpus.

Abstract In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci ( M ) ⩾ − k , k ∈ R . As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci ( M ) ⩾ − k , which generalizes a result of L. Ni (2004, 2006) [20] , [21] . As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/df256958c096d843751888aad0c13112860a1c10

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Gradient Estimates for the Porous Medium Equations on Riemannian Manifolds (2011, background): “Recently, Li and Xu [8] obtained new Li–Yau type gradient estimates for positive solutions of the heat equation (1.”
The Li-Yau inequality and applications under a curvature-dimension condition (2014, background): “…curvature case, let us mention the various bounds (1.2) for any α > 1, derived by B. Davies in [5, Chapter 5.3], and derived by S.-T. Yau [22], itself improved by the ﬁrst author and Z. Qian [3], as Meanwhile, R. Hamilton [8] had proved the inequality (1.4) and, most recently, J. Li and X. Xu [10]”