PARTIAL DIFFERENTIAL EQUATIONS I


MATH 393C (Unique # 55795) and CAM 393C (Unique # 65080)

FALL SEMESTER 2010





INSTRUCTOR: Thomas Chen
Office: RLM 12.138, Phone: 471 7180
E-Mail: t c A_T m a t h . u t e x a s . e d u
Office Hours: Mo, Fr 1 - 2 PM.

TIME AND LOCATION: Classes meet on MWF 12:00PM - 1:00PM in RLM 11.176.


COURSE DESCRIPTION

This is the first part of a two semester graduate course on PDE's.
Recent topics have included quantum mechanics, statistical physics, ergodic theory, group representations,
statistical mechanics, quantum field theory, introductory partial differential equations, monotone operators and partial
differential equations, Hilbert space methods for partial differential equations, Hamiltonian dynamics, nonlinear functional
analysis, Euler and Navier-Stokes equations, microlocal calculus and spectral asymptotics, calculus of variations.
Prerequisite: Graduate standing and consent of instructor.

TEXTBOOK: Lawrence C. Evans, Partial Differential Equations, AMS.
Additional lecture notes are available on Blackboard.


SCHEDULE OF TOPICS

The material covered in this course includes:
  1. 1st and 2nd order constant coefficient linear PDE's.
    1. Laplace equation: Harmonic functions, mean value property, maximum principle, Harnack inequality, regularity, Poisson equation, uniqueness.
    2. Heat equation: Heat balls, mean value property, maximum principle on bounded and unbounded domains, uniqueness of solutions, regularity.
    3. Wave equation: Spherical means, Kirchhoff solution, method of descent, uniqueness.
    4. Transport equation: Characteristics.
  2. Nonlinear 1st order PDE's.
    1. Method of characteristics, applications.
    2. Analysis of conservation laws via characteristics, Burgers equation, Rankine-Hugoniot condition, shocks, rarefaction waves, entropy solutions.
  3. Power series, majorant method, Cauchy-Kovalevskaya theorem.
  4. Fourier transform methods.
    1. Uncertainty principle, Riesz-Thorin interpolation, Hausdorff-Young inequality, Lp Fourier transform.
    2. Weak Lp, Marcinkiewicz interpolation, Hardy-Littlewood maximal function, Lebesgue differentiation, Hardy-Littlewood-Sobolev inequality.
    3. Calderon-Zygmund operators, Mikhlin-Hormander theorem, Paley-Littlewood theory, square function estimates, Sobolev inequalities.
    4. Stationary phase estimates. Stein-Tomas restriction theorem.
  5. Schrodinger equation
    1. Linear Schrodinger equation; stationary phase analysis; Kato smoothing.
    2. Strichartz estimates, T-T* argument, relation to Stein-Tomas theorem.
    3. Spectral theory (see here, chapter 3.1), Birman-Schwinger principle, Agmon bound, Mourre estimate, scattering.
    4. Local well-posedness of the Cauchy problem for some nonlinear Schrodinger equations.
  6. Linear evolution equations; semigroup theory; method of Galerkin approximations for weak solutions.

HOMEWORK ASSIGNMENTS

The homework assignments will be posted here.
  1. HW 0: Please read chapter 5 in Evans (Sobolev spaces).
  2. HW 1, please hand in at the beginning of class on Friday, September 3.
    Note: The HW sheet has been updated on 8/26/10, 2:50 PM. The solution has been posted on Sep 3.
  3. HW 2, please hand in at the beginning of class on Monday, September 13.
    Note: The HW sheet has been updated on 9/3/10, 10 PM.
    Hint: For problem 1, use Cauchy's derivative formula for complex analytic functions (justify why it can be applied) to estimate |h_n|.
    Note: The solution has been posted on Sep 13; last update on 9/13, 9:50 PM (some absolute value signs were missing).
  4. HW 3, please hand in at the beginning of class on Monday, September 20.
    Note: The HW sheet has been updated on 9/19/10, 9:55 PM.
  5. HW 4, please hand in at the beginning of class on Monday, September 27.
    Note: The HW sheet has been updated on 9/21/10, 10:30 PM.
    Note: The solution has been posted on Oct 20.
  6. HW 5, please hand in at the beginning of class on Monday, October 4.
    Note: The solution has been posted on Oct 20.
  7. HW 6, please hand in at the beginning of class on Monday, October 11.
    Note: The HW sheet has been updated on 10/5/10, 11:30 AM.
    Note: The solution has been posted on Oct 20.
  8. HW 7, please hand in at the beginning of class on Monday, October 25.
    Note: The solution has been posted on Nov 9.
  9. MIDTERM on Friday, October 22. Please check Blackboard.
    Note: The midterm solution has been posted on Oct 28.
  10. HW 8, please hand in at the beginning of class on Monday, November 1.
    Note: The solution has been posted on Nov 9.
  11. HW 9, please hand in at the beginning of class on Monday, November 8.
    Note: The HW sheet has been updated on 11/7/10 at 9:35PM (typos in problem 1 corrected).
  12. HW 10, please hand in at the beginning of class on Monday, November 15.
    Note: The solution is part of the discussion in class.
    Note: The solution has been posted on Dec 8.
  13. HW 11, please hand in at the beginning of class on Monday, November 22.
    Note: The HW sheet has been updated on 11/21/10 at 5:35PM.
    Note: The solution is contained in the lecture notes available on blackboard.
  14. HW 12, please hand in at the beginning of class on Friday, December 3.
    Note: The HW sheet has been updated on 12/1/10 at 2:45PM.
    Note: The solution has been posted on Dec 7.
  15. FINAL EXAM on Friday, December 10. Please check Blackboard.
    Note: The solution has been posted on Dec 14 (updated on Dec 15, 1:50 AM).
For solutions to HW assignments and additional course material, see Blackboard.

GRADING

Homework will be assigned regularly. Students are encouraged to work in groups; however, each student must write up his or her own work.
A midterm exam, and a final exam will be given. The course grade will be based on the homework grade and the two exam grades.


MIDTERM on Friday, October 22. PLEASE NOTE: All exam information is posted on Blackboard.


FINAL EXAM on Friday, December 10. PLEASE NOTE: All exam information will be posted on Blackboard.


It is implicit in your registration for this class that, barring some unforeseen calamity, you affirm to be present to take the final examination at this time.

The class grade will be determined as follows:

Homework: 15 percent
Midterm: 35 percent
Final Exam: 50 percent

BIBLIOGRAPHY

  1. E. Lieb, M. Loss, Analysis, AMS.
  2. F. John, Partial Differential Equations, Springer.
  3. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer.
  4. T. Cazenave, Semilinear Schrodinger Equations (Courant Lecture Notes), AMS.
  5. T. Tao, Local And Global Analysis of Nonlinear Dispersive And Wave Equations (Cbms Regional Conference Series in Mathematics), AMS.
  6. M. Reed, B. Simon, Methods of Modern Mathematical Physics 1 - 4, Academic Press.
  7. T. Kato, Perturbation Theory for Linear Operators, Springer.
  8. G. Teschl, Mathematical Methods in Quantum Mechanics, AMS.

The University of Texas at Austin provides upon request appropriate academic accommodations for qualified students with disabilities.
For more information, contact the Office of the Dean of Students at 471-6259, 471-4641 TTY.