21-832: Partial Differential Equations II
Lectures: MWF 9:30 - 10:20 am in Wean Hall room 7218
Professor: Robert Pego
- Office: 6130 Wean Hall.
Email: rpego AT cmu.edu
Stop by during my office hours: MTW 11-12, or another time by arrangement.
Partial Differential Equations, by L. C. Evans,
published by the American Mathematical Society.
(Graduate Studies in Mathematics, 2nd ed. 2010. )
- Homework 1, due W 2/1: pdf
- Homework 2, due F 2/17: pdf
- Homework 3, due W 3/7: pdf
- Homework 4, due F 4/13: pdf
Potentially useful reference books:
- An Introduction to Partial Differential Equations
by M. Renardy and R.C. Rogers (Texts in Applied Mathetmatics 13), Springer 1993.
- Elliptic Partial Differential Equations of Second Order
by D. Gilbarg and N.S. Trudinger, Springer, 2nd ed. 1983.
This semester's topics deal with modern uses of function spaces
to study solutions of PDE.
- Distributions - basic properties. Convergence, derivatives, convolutions,
Sobolev space essentials. Sobolev inequalities, traces, compactness.
- Existence of weak solutions: Lax-Milgram, Fredholm Alternative.
- Regularity of weak solutions:
interior regularity, regularity up to the boundary.
- Eigenvalues and eigenfunctions.
Sobolev spaces involving time.
Existence of weak solutions (Galerkin method).
The Navier-Stokes equations.
Selected topics (may vary or differ)
- Semigroups and operator-theoretic methods
- Direct method of the calculus of variations
- Concentration compactness
- Homogenization - basic examples
Very helpful: PDE I, measure theory, functional analysis, basic complex
Grading: Based on approximately 6 homework sets
and a project with presentation.
Homework will be posted online at
Project and presentation
You will be asked to do a project on a topic that goes beyond
the material discussed in class. Most of the projects are based
on reading articles in the research literature and/or
chapters in research monographs. At the end of the semester
everyone will give a presentation. Details on the project
as well as a list of suggested topics and articles will be
provided in class.
Academic integrity requires that your tests and homework solutions
are your independent work and not copied from other sources. On homework you
are encouraged, however, to discuss with others and consult other resources to
improve your understanding.