This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
PrerequisitesMultivariable analysis (the equivalent of Math 532).
- First order PDE -characteristics -existence, blowup, shocks -linear, quasilinear, fully nonlinear -burger's equation, Hamilton Jacobi
- Conservation laws -weak solutions -shocks, Rankine-Hugoniot condition -entropy condition
- Second order PDE
- Wave equation -D'Alembert formula -special computations for dimension d=1,3,2 -finite speed of propagation -energy conservation -boundary value problems and separation of variables
- Laplace and Poisson equations -harmonic functions and their properties -maximum principle, mean-value property, regularity -fundamental solution -Poisson formula for a ball -boundary value problems
- Heat equation -fundamental solution -properties of solutions: maximum principle, regularity -inhomogeneous equations, Duhamel principle -boundary value problems
- Fourier transform -Definition and properties of Fourier transform -L2 theory -Distributions, tempered distributions -applications to PDE
- Weak derivatives and Sobolev spaces -definition of weak derivative and Sobolev spaces -Sobolev inequalities -compactness -trace -weak formulation of elliptic boundary value problems in H^1
- L.C. Evans, Partial Differential Equations.