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
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
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.