Instructor: Bob Haslhofer
Contact Information: roberth(at)math(dot)toronto(dot)edu
Website: http://www.math.toronto.edu/roberth/pde2.html
Lectures: Monday and Wednesday 12:30--2:00 in BA6183
Office hours: Wednesday 2:30--3:30 in BA6208
Grading Scheme: attendance and participation 20%, assignments 30%, final exam 50%
Assignments: via crowdmark
Final Exam: Monday, April 13, 1pm--4pm in BA 6183
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. Taylor: Partial Differential Equations III. Nonlinear Equations, Springer, 1996
M. Struwe: Plateau's problem and the calculus of variations, Princeton University Press, 1988
L. Simon: Schauder estimates by scaling, Calc. Var. PDE 5, no.5, 391--407, 1997
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, weak and strong maximum principle, fixed points
theorems and existence for nonlinear equations, classical and differential Harnack
inequality,
curve shortening flow, short time existence, derivative
estimates, Huisken's monotonicity formula, Hamilton's Harnack
inequality, distance comparison principle, Grayson's convergence theorem.
Weekly schedule:
Week 1
Plateau's problem, first and second variation, existence of minimizers (Evans 8.1 and 8.2) notes
Week 2
review of Sobolev spaces (Notes on Sobolev spaces), weak solutions, energy estimates, regularity (Evans 8.2 and 8.3) notes
Week 3
Schauder estimates (Notes on Schauder estimates), DeGiorgi-Nash-Moser estimates (Notes on epsilon-regularity) notes
Week 4
Lagrange multipliers, min-max theory (Evans 8.4 and 8.5) notes
Week 5
applications of min-max theory, Pohozaev identity (Evans 8.5 and 9.4) notes
Week 6
Plateau's problem, surfaces with prescribed curvature (Struwe's book and Osgood-Phillips-Sarnak paper) notes
Week 7
intro to parabolic PDEs, time-dependent function spaces, weak solutions (Evans 5.9 and 7.1) notes
Week 8
existence and regularity, weak and strong maximum principle (Evans 7.1) notes
Week 9
differential and classical Harnack inequality, existence for nonlinear parabolic PDEs (Li-Yau paper, Evans 9.2) notes
Week 10
curve shortening flow basics, existence and uniqueness (Notes on CSF) notes
Week 11
Huisken's monotonicity formula, Hamilton's Harnack inequality (Notes on CSF) notes
Week 12
distance comparison principle, Grayson's convergence theorem (Notes on CSF) notes