Li-Yau Harnack inequalities for geometric PDEs

In their seminal work in 1986, Li and Yau proved a differential Harnack inequality for heat equation on Riemannian manifolds. Their result not only recovers and sharpens the classical pointwise Harnack inequality, but uses a much more robust approach. As a consequence, deriving such inequalities has become an important question in many contexts. Similar inequalities were discovered by Hamilton for certain geometric flows by considering cleverly chosen quantities with many surprising computations. There is, however, no known analogue for broader classes of quasi-linear parabolic PDEs. In this talk, I will focus on a specific class of equations studied by Cafarelli and Garofalo as a good candidate for generalizing the results above (with prototypical example involving normalized (or so called "game theoretic") p-Laplacian), and discuss our work in progress with N. Garofalo and A. Banerjee.