Math 554 Fall 2008

Linear Analysis and Partial Differential Equations

Room: 343 Altgeld

Time: TTh 9:00 - 10:20 am

Office: 219 Altgeld Hall
E-mail: ekirr@math.uiuc.edu
Phone: 265-5418
Office hours:
- Tuesdays 4:00 - 5:00 pm
- Wednesdays 4:00 - 5:00 pm
- By appointment

Syllabus: We will cover chapters 5-7 from the required textbook below. We will start by relating certain partial differential equations with physical phenomena. Next we will introduce Sobolev spaces and review certain properties of Banach, Hilbert spaces, as well as of linear operators defined on them. We will then use this framework to study both weak and classical solutions of the following second order equations on bounded domains:

elliptic equations: Lax-Milgram Theorem, regularity, maximum principles, eigenvalues and eigenfunction;
parabolic equations: energy estimates, regularity, maximum principle;
hyperbolic equations: energy estimates, propagation of singularities.

We will also develop the semigroup theory and show how it applies to linear evolution equations (of parabolic and hyperbolic type). The theories and the applications encountered in this course will create a strong foundation for studying nonlinear equations and nonlinear science in general.

Prerequisites: Real analysis (Math 447 or equivalent) and multivariable calculus. Measure theory (Math 540) and familiarity with partial differential equations (Math 553) would be very helpful.

Texts: Lawrence C. Evans, Partial Differential Equations (required); V. S. Vladimirov, Equations of Mathematical Physics (recommended); R. A. Adams, Sobolev Spaces (recommended).

Homework: Assignments will be given approximately every two weeks. Your grade will be based mostly on them. I encourage you to work together on the homework but solutions should be written up independently and turned in by the deadline for each assignment. Check back this page for actual assignments and deadlines. Solutions will be posted on this page, if necessary, after the due date. Late homework accepted only if you get an extension from me.

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