Lectures: MWF 9:30 - 10:20 am in Wean Hall room 7201

Professor: Robert Pego

• Office: 6127 Wean Hall.   Email: rpego AT cmu.edu    Phone: (412)268-2553.   Stop by during my office hours: MTW 11-12, or any another time by arrangement.

Assignments:

• Assignment 1, due W 9/14   pdf
• Assignment 2, due F 10/7   pdf
• Assignment 3, due W 11/9   pdf
• Assignment 4, due F 12/9   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.

Course Outline:

This semester's topics deal with modern uses of function spaces to study solutions of PDE.

1. Distributions - basic properties. Convergence, derivatives, convolutions, Fourier transform.
2. Sobolev space essentials. Sobolev inequalities, traces, compactness.
3. Elliptic PDE:
   1. Existence of weak solutions: Lax-Milgram, Fredholm Alternative.
   2. Regularity of weak solutions: interior regularity, regularity up to the boundary.
   3. Eigenvalues and eigenfunctions.
4. Parabolic PDE: Sobolev spaces involving time. Existence of weak solutions (Galerkin method). The Navier-Stokes equations.
5. Selected topics (may vary or differ)
   1. Semigroups and operator-theoretic methods
   2. Direct method of the calculus of variations
   3. Concentration compactness
   4. Homogenization - basic examples

Prerequisites:   Very helpful: measure theory, functional analysis, basic complex variables

Grading: Based on approximately 6 homework sets and a project with presentation.

Homework will be posted online at http://www.math.cmu.edu/~bobpego/21832/ .

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.