Math 557 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.


Multivariable analysis (the equivalent of Math 532).


  1. First order PDE
    • Characteristics
    • Existence, blowup, shocks
    • Linear, quasilinear, fully nonlinear
    • Burger's equation, Hamilton Jacobi
  2. Conservation laws
    • Weak solutions
    • Shocks, Rankine-Hugoniot condition
    • Entropy condition
  3. Second order PDE
  4. 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
  5. 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
  6. Heat equation
    • Fundamental solution
    • Properties of solutions: maximum principle, regularity
    • Inhomogeneous equations, Duhamel principle
    • Boundary value problems
  7. Fourier transform
    • Definition and properties of Fourier transform
    • L2 theory
    • Distributions, tempered distributions
    • applications to PDE
  8. 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.