Course overview: This course aims to teach the basics of Partial differential equations (PDEs),
a subject that touches on many branches of pure mathematics, applied mathematics, as well as physics and applied science.
It is addressed to first and second-year graduate students, or anyone with an interest in the topic.
Partial differential equations are a very rich subject; so much so that at a research level most workers in the field
specialize in one of the many sub-fields.
The aim of this one-semester course is both to give an overview of the subject as much as possible, and introduce some tools
that are used throughout. This should prepare students adequately for the many more advanced courses in PDEs that
are (and will be) offered in the department.
Pre-requisites: The main pre-requisite is some basic real analysis, primarily Lebesgue measure on Rn, the basics of the Fourier transform on Rn
and some functional analysis,
along with multivariable calculus (Stokes' theorem).
Familiarity with the general theory of Ordinary Differential equations
is desirable, but not a prerequisite. Exposure to an undergraduate-level PDE course will be helpful, but not required.
Focus: The focus on this course is on the general theory of PDEs. There will be very little discussion of finding
explicit solutions to specific equations. There will be no discussion of numerical analysis related to PDEs.
(The latter being a very interesting topic, on which many good courses are offered, although not this year).
The main textbook that we will use is Partial Differential Equations by L. C. Evans,
(American Math Society, 2010). This will be complemented by further materials: Either notes from the instructor,
or referrals to other textbooks. Whenever such material is used, I will be distributing hand-outs
or referring to specific textbooks, for the students' convenience.
Course outline:
The course will start with a quick overview of basic facts on
the local existence theory for systems of ordinary differential equations: Existence, Uniqueness,
continuous dependence on initial data. Gronwall's inequality.
We will continue with a detailed study of the basic examples of the most basic linear PDEs:
The transport equation, Laplace and Poisson's equations,
the Heat equation and the wave equation. (Chapter 2 in Evans). Chapters 3
and 4 (Non-linear first-order PDEs and Representation methods for solutions)
will be covered briefly.
The bulk of the course will be devoted to the general theory of existence, uniqueness
and regularity for general linear and non-linear second order elliptic, parabolic and hyperbolic equations.
To reach this topic, we
will cover Sobolev spaces (Chapter 5 in Evans) and then second order linear elliptic equations (Chapter 6),
and second order parabolic and hyperbolic equations (Chapter 7).
The existence and regularity theory for nonlinear parabolic and hyperbolic equations will be covered
using my notes.
Time permitting, we may discuss a little about the calculus of variations.
Marking:
The final grade will be calculated based on Homework and the final exam.
There will be Homework assignments every two weeks.
The relative weights here will be HW 50% and Final exam 50%.