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.

Keywords: Differential Harnack inequality, monotonicity formula

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@article{COCV_2017__23_3_827_0, author = {Lee, Paul W.Y.}, title = {Generalized {Li\ensuremath{-}Yau} estimates and {Huisken{\textquoteright}s} monotonicity formula}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {827--850}, publisher = {EDP-Sciences}, volume = {23}, number = {3}, year = {2017}, doi = {10.1051/cocv/2016015}, mrnumber = {3660450}, zbl = {1369.58018}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016015/} }

TY - JOUR AU - Lee, Paul W.Y. TI - Generalized Li−Yau estimates and Huisken’s monotonicity formula JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 827 EP - 850 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016015/ DO - 10.1051/cocv/2016015 LA - en ID - COCV_2017__23_3_827_0 ER -

%0 Journal Article %A Lee, Paul W.Y. %T Generalized Li−Yau estimates and Huisken’s monotonicity formula %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 827-850 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016015/ %R 10.1051/cocv/2016015 %G en %F COCV_2017__23_3_827_0

Lee, Paul W.Y. Generalized Li−Yau estimates and Huisken’s monotonicity formula. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 3, pp. 827-850. doi : 10.1051/cocv/2016015. http://archive.numdam.org/articles/10.1051/cocv/2016015/

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