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

**Contact Information:** roberth(at)math(dot)toronto(dot)edu

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

**Lectures:** Monday 10am--12noon and Friday 2pm--3pm (
zoom link for lectures on quercus )

**Virtual office hours (changed!):** Monday 1pm--2pm (
zoom link for office hours on quercus )

**Grading Scheme:** Attendance and participation 30%, Assignments 30%, Final exam 40%

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

**Final Exam:** Mo, Apr 19th, 10am--1pm Final Exam

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

I'll also provide lecture notes for selected topics.

**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 Schroedinger 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

**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.

**Weekly schedule:**

**Week 1**

Jan 11: Plateau's problem, First and second variation (Evans 8.1) notes

Jan 15: Existence of minimizers (Evans 8.2) notes

**Week 2**

Jan 18: Weak solutions, energy estimates (Evans 8.2 and 8.3) notes

Jan 22: Reminder about embedding theorems, regularity continued (PDE1 and Evans 8.3) notes

**Week 3**

Jan 25: Schauder estimates (Notes on Schauder estimates) notes

Jan 29: DeGiorgi-Nash-Moser estimates (Notes on epsilon-regularity) notes

**Week 4**

Feb 1: Lagrange multipliers (Evans 8.4) notes

Feb 5: Min-max theory (Evans 8.5) notes

**Week 5**

Feb 8: Min-max theory and applications (Evans 8.5) notes

Feb 12: Pohozaev identity (Evans 9.4) notes

**Week 6**

Feb 22: Plateau's problem (Struwe: Plateau's problem...) notes

Feb 26: Surfaces with prescribed curvature (Osgood-Phillips-Sarnak) notes

**Week 7**

Mar 1: Intro to parabolic PDEs, time-dependent function spaces (Evans 5.9) notes

Mar 5: Weak solutions for linear parabolic PDEs (Evans 7.1) notes

**Week 8**

Mar 8: Existence and regularity (Evans 7.1) notes

Mar 12: Maximum principle (Evans 7.1) notes

**Week 9**

Mar 15: Harnack inequality and strong maximum principle (Evans 7.1, Li-Yau) notes

Mar 19: Existence for nonlinear parabolic PDEs (Evans 9.2) notes

**Week 10**

Mar 22: Curve shortening flow basics, Hamilton's Harnack (Notes on CSF) notes

Mar 26: Existence and uniqueness (Notes on CSF) notes

**Week 11**

Mar 29: Huisken's monotonicity formula and applications (Notes on CSF) notes

**Week 12**

Apr 5: Huisken's distance comparison principle (Notes on CSF) notes

Apr 9: Grayson's convergence theorem (Notes on CSF) notes