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DISTRIBUTIONS,PARTIAL DIFFERENTIAL EQUATIONS
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Partial Differential Equations
 Classical linear equations: transport, Laplace, heat, and wave equations (basic properties, fundamental solutions, mean value properties, maximum principles,energy methods, Fourier transform method)
 First order nonlinear PDE's (characteristics, conservation laws, shocks)
 Special solutions (similarity solutions, traveling waves, power series methods)
 Sobolev spaces (distributions, weak derivatives, weak convergence)
 More on Sobolev spaces (traces, Poincaré and Sobolev inequalities, embeddings)
 Second order elliptic equations (fixed point theorems, weak solutions, regularity,maximum principles, eigenvalues, applications)
 Evolution equations (weak solutions, Galerkin method, regularity,maximum principles and propagation of disturbance) and applications
 Semigroup theory (infinitesimal generators, HilleYoshida theorem, applications)
 Calculus of variations and its applications (direct method, Mountain pass theorem)
 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 M5134. AfterM5134, PSU offers two topics courses per semester in the broad areaof PDE, Numerical Analysis, and Applied Mathematics. While youare studying M5134 in the first year, you can attend lower level courses or upperlevel topics courses depending on your preparedness.