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Partial Differential Equations

  1. Classical linear equations: transport, Laplace, heat, and wave equations (basic properties, fundamental solutions, mean value properties, maximum principles,energy methods, Fourier transform method)
  2. First order nonlinear PDE's (characteristics, conservation laws, shocks)
  3. Special solutions (similarity solutions, traveling waves, power series methods)
  4. Sobolev spaces (distributions, weak derivatives, weak convergence)

  5. More on Sobolev spaces (traces, Poincaré and Sobolev inequalities, embeddings)
  6. Second order elliptic equations (fixed point theorems, weak solutions, regularity,maximum principles, eigenvalues, applications)
  7. Evolution equations (weak solutions, Galerkin method, regularity,maximum principles and propagation of disturbance) and applications
  8. Semigroup theory (infinitesimal generators, Hille-Yoshida theorem, applications)
  9. Calculus of variations and its applications (direct method, Mountain pass theorem)
The above consists of the core part of the first-year graduate study on the subject of Partial Differential Equations at PSU. Each instructor may add a few additional topics. Math 513-4 is a year long course covering the above and provide an introduction to the fundamental theories and methods in partial differential equations. The first course M513 will covertopics 1--4 listed above and the second course M514 will cover the rest. Most of the first 5 chapters of Evans' book will be covered in M513.

  • Text 1: Partial Differential Equations by L. C. Evans
  • Text 2: Introduction to Partial Differential Equations, G. B. Folland, 2nd Ed (1995)

Additional References

3. Partial Differential Equations, R. McOwen
4. Hilbert Space Methods for Partial Differential Equations, R. Showalter (available free electronically)
5. Elliptic Partial Differential Equations of Second Order, D. Gilbarg and N. S. Trudinger, 2nd Ed or later.
6. Partial Differential Equations, F. John

Qualifying exam problems of previous times: Ask Becky
PS: To pursue a Ph.D. in the area of PDE, we recommend that you keepin mind ODE is an important component. PSU has M411 (ODE/Fourier Series),M412 (PDE), and M511 (ODE) before the course level of M513-4. AfterM513-4, PSU offers two topics courses per semester in the broad areaof PDE, Numerical Analysis, and Applied Mathematics. While youare studying M513-4 in the first year, you can attend lower level courses or upperlevel topics courses depending on your preparedness.