Generalized Li-Yau estimates and Huisken's monotonicity formula

TL;DR

This paper extends classical estimates like Li-Yau and Huisken's monotonicity formula to a broader class of parabolic equations, providing new inequalities and sharpness results for these generalizations.

Contribution

It introduces generalized Li-Yau and Huisken's estimates for second order linear parabolic equations, including matrix versions and sharpness analysis.

Findings

New Cheeger-Yau inequality derived

Generalized Huisken's monotonicity formula established

Sharpness of inequalities demonstrated for fundamental solutions

Abstract

We prove a generalization of the Li-Yau estimate for a board class of second order linear parabolic equations. As a consequence, we obtain a new Cheeger-Yau inequality and a new Harnack inequality for these equations. We also prove a Hamilton-Li-Yau estimate, which is a matrix version of the Li-Yau estimate, for these equations. This results in a generalization of Huisken's monotonicity formula for a family of evolving hypersurfaces. Finally, we also show that all these generalizations are sharp in the sense that the inequalities become equalities for a family of fundamental solutions, which however different from the Gaussian heat kernels on which the equality was achieved in the classical case.