Generalized Li−Yau estimates and Huisken’s monotonicity formula

Room 216, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China.
wylee@math.cuhk.edu.hk

Received: 20 August 2015
Accepted: 25 March 2016

Abstract

We prove a generalization of the Li−Yau estimate for a broad 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 equality for a family of fundamental solutions, which however different from the Gaussian heat kernels on which the equality was achieved in the classical case.