Office Rm. Blocker 614A, Telephone (979)862-3257
E-mail: kuchment@math.tamu.edu, Home Page: /~kuchment
This class is devoted to the theory of linear partial differential equations. Assuming some initial familiarity of students with classical PDEs (Laplace, wave, and heat equation), the following three main topics will be addressed:
Classification of equations of 2nd order
Sobolev spaces
Second order elliptic equations (including boundary value problems theory and spectral theory)
Linear evolution equations (including initial-and-boundary-value parabolic and hyperbolic problems)
Besides numerous applications inside mathematics, the PDEs form the core part of our scientific understanding of the physical world: from physics to chemistry, to biology, to meteorology, you name it.
The class will be based on the second part "Theory for linear partial differential equations" of the well respected recent textbook by L. Evans.
MATH 611, its equivalent, or instructor's consent (if unsure, please contact the instructor).
Grading will be based on home assignments and a take-home final exam.
Week |
Topics and sections |
Home assignments |
Exams |
---|---|---|---|
01/14 - 01/21 |
Classification of 2nd order PDEs. |
TBA |
n/a |
01/21-02/04 |
Chapter 5: Sobolev spaces |
TBA |
n/a |
02/06-03/09 |
Chapter 6: Elliptic problems |
TBA |
n/a |
03/11- 03/13 |
Spring break |
Have some rest :-) |
n/a |
03/18-04/24 |
Chapter 7: Evolution equations |
na |
Take home final |
GRADING POLICY
Percentage of points |
Grade |
---|---|
90% and higher |
A |
80% and higher |
B |
70% and higher |
C |
60% and higher |
D |
Less than 60% |
F |
Fritz John, Partial Differential Equations, Springer Verlag. Although outdated and more limited that Evans' book, this is still a very good introduction to PDEs.
L. Bers, F. John, and M. Schechter, Partial Differential Equations, Interscience, New York, 1964. American Mathematical Society, 1979. A wonderful (albeit old) book introducing properties of main types of PDEs.
Richard Courant and David Hilbert, Methods of Mathematical Physics, two volume set (any edition). This is an absolutely wonderful classics. In spite of being half of a century old, it is still a very important book, in many instances not surpassed by anything else. A must reading for anyone using PDEs extensively.
I. G. Petrovsky, Lectures on Partial Differential Equations, Dover 1991. A good small textbook on basics of PDEs. Much more limited and out dated than Evans, but still valuable.
David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag (latest edition is 2001). A classics on second order elliptic equations.
W. D. Evans and D. E. Edmunds, Spectral Theory and Differential Operators (Oxford Mathematical Monographs), Oxford Univ Press 1997. A terrific book on Sobolev spaces, elliptic differential operators, and spectral theory.
L. Hormander, The Analysis of Linear Partial Differential Operators, Springer Verlag. A 4 volume set of fundamental books written by the world leading expert and covering most important techniques and results on linear PDEs (for an advanced reader).
Michael E. Taylor, Partial Differential Equations: Basic Theory (Texts in Applied Mathematics, 23), Springer Verlag, and its two consecutive volumes. A comprehensive set of books covering all major topics of PDEs from contemporary points of view. Very geometric, in most cases equations are considered on manifolds.