Topics in Differential Equations: Parabolic PDE

This is an introductory course to Parabolic PDE. We start with a brief review of 2nd order elliptic PDE, then discuss basic theory about the heat equation on the Euclidean space, including fundamental solution, Schauder and Lp estimates, maximal principle, construction and estimate of the heat kernel, energy methods. We then discuss heat equation on manifolds, Harnack principle of Li-Yau. As applications, we plan to cover a derivation of the log-Sobolev inequality on the Euclidean space via parabolic estimates, and the connectionof the inequality to Perelman's W-functional.