Generalized Li−Yau estimates and Huisken’s monotonicity formula
Lee, Paul W.Y.1
1 Room 216, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 827-850.

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.

DOI : 10.1051/cocv/2016015
Classification : 58J35
Mots-clés : Differential Harnack inequality, monotonicity formula
Lee, Paul W.Y. 1

1 Room 216, Lady Shaw Building, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China. 
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Lee, Paul W.Y. Generalized Li−Yau estimates and Huisken’s monotonicity formula. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 827-850. doi : 10.1051/cocv/2016015. http://www.numdam.org/articles/10.1051/cocv/2016015/

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