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
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 L2 continuity
iv) Fourier integral operators
last update: January 4, 2016