Mathematics 742: Applied Mathematics II
Partial Differential Equations

Instructor: W. Craig 905-525-9140 ext 23422

Course meeting times: Tuesday & Thurs 10:00 - 11:30 in HH 312
Office hours: Friday 10:00 - 11:30 or by appointment, HH418

ABSTRACT: This is a one semester course that is intended to be a graduate level introduction to the theory of partial differential equations (PDEs). The course material will cover the basic properties of solutions of of first order equations, the wave equation as well as hyperbolic systems of equations, Laplace's equation as well as other elliptic equations, and the heat equation. We will be introducing appropriate methods such as energy estimates for symmetric hyperbolic systems, elliptic regularity for elliptic equations, and Brownian motion and Wiener measure for parabolic equations. We will also cover the general theory of PDE, including versions of the Cauchy Kowalevskaya theorem, and subsequent general theorems and counterexamples to existence. Additional topics such as microlocal analysis and pseudodifferential and Fourier integral operators, and methods of nonlinear functional analysis will be covered if there is time and by consensus of the class. Throughout this course, an attempt will be made to connect the theory to relevant examples of current research interest in mathematical physics and geometry.

Reference texts:
F. John, Partial Differential Equations (Springer Verlag)
L. C. Evans, Partial Differential Equations (AMS)
G. Folland, Partial Differential Equations (Princeton University Press)
J. Rauch, Partial Differential Equations (Springer Verlag)
P. Garabedian, Partial Differential Equations (Wiley Interscience)
W. Craig, lecture notes

Course Outline:  here

Problem Sets:

Problem Set #1:   here   Due Thursday January 21
Problem Set #2:   here   Due Thursday February 4
Problem Set #3:   here   Due Thursday March 3
Problem Set #4:   here   Due Thursday March 24
Problem Set #5:   here   Due Thursday April 14
Final Exam - take home:   here   Due Thursday April 28


1) Introduction: physics and geometry
   Laplace's equation, the heat equation and the wave equation

2) First order equations

    i) Burger's equation
    ii) first order quasilinear equations
    iii) shock formation and weak solutions
    iv) the Riemann problem
    v) global weak solutions and Lax - Olenik theory
    vi) systems of conservation laws
    vii) first order fully nonlinear equations

3) Hyperbolic equations

    i) d'Alembert's formula
    ii) spherical means and the method of descent
    iii) Duhamel's principle
    iv) Huygen's principles
    v) Gårding's condition of hyperbolicity
    vi) symmetric hyperbolic systems (linear)
    vii) symmetric hyperbolic systems (nonlinear)

4) Elliptic equations

    i) Green's identities
    ii) boundary value problems
    iii) maximum principles, Hopf lemmas, the Harnak inequality
    iv) Hilbert space methods and variational principles
    v) Perron's method
    vi) elliptic regularity
    vii) Pohozaev's identity

5) Parabolic equations

    i) initial and initial/boundary value problems
    ii) maximum principles, uniqueness and regularity
    iii) Brownian motion and Wiener measure
    iv) Feynman - Kac formula

6) General theory

    i) characteristic and non-characteristic manifolds
    ii) Cauchy - Kowalevskaya method of majorants
    iii) Holmgren - John uniqueness theorem
    iv) Nirenberg - Nishida abstract Cauchy - Kowalevskaya theorem
    v) Malgrange - Ehrenpreis existence theorem
    vi) nonexistence theorems - H. Lewy's example

7) Nonlinear functional analysis

    i) calculus in Banach space
    ii) implicit function theorem
    iii) bifurcation theory
    iv) Nash - Moser implicit function theorem

8) Introduction to microlocal analysis

    i) pseudodifferential operators and their symbols
    ii) symbol calculus
    iii) locality, pseudolocality, and L2 continuity
    iv) Fourier integral operators

last update: January 4, 2016