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
Office hours: Monday 1pm--2pm
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