MA6000A Theory of PDEs

Semester 2, 2011/12
Lecturer: Roger Moser
Contact details
E-mail: r.moser@bath.ac.uk
Office: 4W 5.6 (University of Bath)
Phone: 01225 386699
Lectures: Mondays from 16 January to 19 March, 10:00 – 12:00 in 3W 4.13 (for students from Bath) or via TCC.
This course is available to students from Bristol, Imperial, Oxford, and Warwick (and possibly other institutions) via the Taught Course Centre. If you would like to receive e-mail announcements, please let me have your e-mail address (just send an e-mail to the above address), regardless of your institution.
Prerequisites: I will assume that the students are familiar with the basics of functional analysis and Sobolev spaces. However, it may be possible to study the relevant topics concurrently. Please contact me if you are concerned about your background.
Assessment is in the form of a viva for students who require it. If you wish to be assessed, please let me know. You can find more information here.
Notes: The updated lecture notes are now available here.
Starboard files: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
Homework
Download problem sheets: [1] [2] [3] [4] [5] [6] [7] [8]
Download solutions: [1] [2] [3] [4] [5] [6] [7] [8]
Course content
  1. Linear elliptic equations of second order
    • Weak solutions
    • Existence
    • Regularity
    • Harnack inequality
    • maximum principle
  2. Calculus of Variations
    • First and second variation
    • Direct method
    • Caccioppoli inequality and regularity
  3. Viscosity solutions
    • Vanishing viscosity method
    • Comparison principle
    • Perron’s method
Books

Introductory texts

L. C. Evans, Partial Differential Equations
Q. Han and F.-H. Lin, Elliptic Partial Differential Equations
M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems

Further reading

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order
E. Giusti, Direct Methods in the Calculus of Variations
M. Giaquinta: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems
L. A. Caffarelli and X. Cabré: Fully Nonlinear Elliptic Equations
M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N. S.) 27 (1992), no. 1, 1–67.

Background material

E. DiBenedetto, Real Analysis
P. D. Lax, Functional Analysis
R. A. Adams and J. J. F. Fournier, Sobolev Spaces