**Instructor:** Prof. Robert (Bob) Haslhofer

**Contact Information:** roberth(at)math(dot)toronto(dot)edu, BA6208, 8-4632

**Lectures:** Tuesday 11--1 and Thursday 11--12 in BA6183

**Office Hours:** Monday 2--4 (or by appointment) in BA6208

**Website:** http://www.math.toronto.edu/roberth/pde2.html

**Course Description:** This is a sequel to MAT 1060. The focus will be on elliptic and parabolic PDEs.

**Topics to be covered:**

*1. Elliptic equations:* Calculus of Variations, Existence of Minimizers, Regularity (Hilberts 19th problem), Lagrange multipliers, Mountain Pass lemma, applications to semilinear elliptic PDEs, Pohozaev identity, Plateau's problem, Surfaces of prescribed curvature.

*2. Parabolic equations:* Existence of weak solutions for linear parabolic equations, integral estimates, maximum principle, fixed points theorems and existence for nonlinear equations, Li-Yau Harnack inequality, curve shortening flow, short time existence, derivative estimates, Huisken's monotonicity formula, Hamilton's Harnack inequality, distance comparison principle, convergence theorem.

**Assignments:** Problem Set 1, Problem Set 2, Problem Set 3, Problem Set 4

**Main References:** The main textbook is "*Partial Differential Equations*" by L.C. Evans.

I'll also provide lecture notes for selected topics: Schauder estimates, Moser iteration, Curve shortening flow

**Secondary References:**

D. Gilbarg, N.S. Trudinger: *Elliptic Partial Differential Equations of Second Order*, Springer, 2001

G.M. Lieberman: *Second Order Parabolic Differential Equations*, World Scientific Publishing, 1996

M. Struwe: *Variational Methods. Applications to nonlinear PDEs and Hamiltonian systems*, Springer, 2008

M. Taylor: *Partial Differential Equations III. Nonlinear Equations*, Springer, 1996

L. Simon: *Schauder estimates by scaling*, Calc. Var. PDE 5, no.5, 391--407, 1997

M. Struwe: *Plateau's problem and the calculus of variations*, Princeton University Press, 1988

B. Osgood, R. Phillips, P. Sarnak: *Extremals of Determinants of Laplacians*, J. Funct. Anal. 80, no.1, 148--211, 1988

P. Li, S.T. Yau: *On the parabolic kernel of the Schrödinger operator*, Acta Math. 156, no. 3--4, 153--201, 1986

G. Huisken: *A distance comparison principle for evolving curves*, Asian J. Math. 2, no.1, 127--133, 1998

**Grading Scheme:** Attendance and participation 15%, Assignments 35%, Exam 50%

**Final Exam:** Monday, April 11 from 3pm--6pm in BA6183.