Professor Klaus Ecker, Free University of Berlin
«Entropies, Logarithmic Sobolev inequalities,
Differential Harnack inequalities and
starting from Wednesday 12-6-2013
at 11:15-12:45 B214 (Mathematics building)
In this lecture course we present an entropy functional for evolving domains which is essentially the one used by Perelman for Ricci flow as part of his solution of the Poincare conjecture, but we augment it by a boundary integral which incorporates the speed function for the boundary of the evolving domain. This entropy more or less arises as the time-derivative of the more standard entropy quantity also featuring in thermodynamics where we integrate the function u log u over a (potentially evolving) domain. Here, u is a solution of the heat equation or the backward heat equation on this domain and satisfies suitable Neumann boundary conditions related to the velocity of the boundary of the domain. We prove basic properties of this functional, in particular show its relation to logarithmic Sobolev inequalities. We also explain logarithmic Sobolev inequalities and show how they can be derived from ordinary Sobolev inequalities. Furthermore, we show a connection between the time-derivative of the entropy and differential Harnack inequalities, the latter in particular for the standard heat equation. We will also mention potential applications of this to geometric evolution equations such as the mean curvature ow. Moreover, we shall explain Perelman's original arguments pertaining to this entropy in the context of Ricci flow, which is both interesting from the nonlinear PDE and the differential geometry point of view. We assume basic familiarity with linear PDE but background on differential geometry will be provided in lectures whenever required.
Lecture Notes: Part1