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

Syllabus:

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 L

iv) Fourier integral operators

last update: January 4, 2016