Preservation of concavity properties by the Dirichlet heat flow and applications

I will present some recent results, obtained in collaboration with K. Ishige, Q. Liu and A. Takatsu, about the preservation of concavity properties of the initial datum by the heat flow and porous medium equation. Time permitting, I will show how these results can be used to prove some important functional inequalities, the so-called Borell-Brascam-Lieb inequalities.