2012/11/23 by Paul W. Y. Lee, Lee, Paul W. Y.
Mathematics · #58J35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:58J35
paper · pdf · doi:10.48550/arxiv.1211.5559
31 pages
arxiv created 2013/09/03 · arxiv updated 2013/09/04
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.