Course Information

Lecture Notes:

  1. de Giorgi-Nash-Moser Theory (Updated Feb 10)
  2. Viscosity Solutions (Updated May 2)

Partial differential equations (PDE) are ubiquitous in mathematics, science, engineering and physics. A brief list of applications includes diffusion of heat, propagation of waves, optimal control theory, fluid dynamics, computer vision, and image processing. This course is the second part of a rigorous graduate level introduction to PDE. We will cover the theory of initial/boundary value problems for second-order elliptic, parabolic, and hyperbolic equations, as well as the DeGiorgi-Nash-Moser regularity theory for second-order elliptic PDE. We will also cover selected topics in nonlinear PDE, including the calculus of variations, and optimal control and viscosity solutions. Provided there is time, we may discuss approximation schemes for viscosity solutions, and the theory of viscosity solutions for second order degenerate elliptic equations.

The animation above is a simulation of a nonlinear diffusion partial differential equation (the Perona-Malik equation) applied to a color image. Note: The png animation may not play in some browsers. Click here for for a lower quality gif.


Instructor Jeff Calder (Office: 1053 Evans, Email: jcalder at berkeley dot edu)
Lectures Mon, Wed, Fri, 11am-12pm in 5 Evans
Office Hours Mon 2pm-3pm, Wed 1pm-2pm, Fri 12pm-1pm in 1053 Evans
Textbook Partial Differential Equations, L.C. Evans (1st or 2nd Edition)
Recommended readings
  1. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, M. Bardi and I.C. Dolcetta (1997)
  2. Elliptic Partial Differential Equations of Second Order, D. Gilbarg and N.S. Trudinger (1998)
  3. Elliptic Partial Differential Equations, Q. Han and F. Lin (2nd edition)
Grades Your grade will be based on homework (10%), a midterm exam (30%), and the final exam (60%). The midterm exam will be distributed in class on March 7, and will be due by 5pm on March 11. The final exam dates will be announced later in the semester. You must pass the final exam in order to pass the course.
Homework Homework will be assigned on a weekly basis, and late homework will not be accepted. I will drop the lowest two homework grades when computing your final grade. You are encouraged to collaborate on homework assignments, but you must write up your solutions in your own words. If you do collaborate with another student, please indicate this on the top of your assignment.
Piazza We have a Piazza website for student discussions. To sign up for Piazza and join our class, click here. Please note the Piazza site is only for student discussions--the homework and schedule information will be posted on this website.
Academic Integrity The mathematics department expects that students in mathematics courses will not engage in cheating or plagiarism. Cheating, plagiarism, or other forms of academic dishonesty will result in a grade of zero on the homework assignment or exam in question, and, in severe cases, a failing grade in the course and a referral to the Center for Student Conduct.