Dr. Alessandro Goffi “Global geometric estimates for the heat equation“

In this talk I will discuss some first and second order geometric estimates for the heat equation introduced by [P. Li – S.T. Yau, Acta Math. 1986], [R. Hamilton, Comm. Anal. Geom. 1993], and then developed in [B. Chow- R. Hamilton, Invent. Math.1997], [P. Souplet – Q.S. Zhang, Bull. LMS 2006], [D. Bakry, I. Gentil, M.Ledoux, Springer 2014], along with log-concavity preserving properties for the heat flow studied by [H. Brascamp – E. Lieb, J. Func. Anal. 1976]. I will focus on the heat flow on unbounded (convex) domains with Neumann boundary conditions. The analysis is based on a duality method à la Evans and exploits stability properties of Kolmogorov-Fokker-Planck equations in Lebesgue spaces and an adjoint version of the Bernstein method. During the talk I will outline the connection among these geometric estimates and semiconcavity estimates for first and second order Hamilton-Jacobi equations studied by S. Kruzhkov and W. Fleming, Aronson-
Bénilan bounds for the porous medium flow and Oleinik estimates for conservation laws.

This is a joint work with G. Tralli (Ferrara).