MAT 1061 Partial Differential Equations II (Winter 2021)

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

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