Syllabus

1. Explicit solution methods for PDEs: separation of variables, characteristics, d'Alembert's formula.
2. Function spaces: elementary theory of abstract Banach and Hilbert spaces, C ^k- and L ^p-spaces, contraction mapping theorem.
3. Theory of distributions: test functions, calculus of distributions, tempered distributions, Sobolev spaces.
4. Fourier analysis: Fourier series and Fourier transform in classical and distributional settings, convolution.
5. Differential equations with distributions: fundamental solutions of differential operators, Green's functions for boundary value problems.
6. Linear operators: bounded and unbounded linear operators on Banach spaces, adjoint operators, closed operators, self-adjoint and symmetric operators.
7. Spectral theory: resolvent and spectrum of a linear operator, Fredholm alternative.
8. Compact operators: spectral theory for compact and compact, self-adjoint operators in a Hilbert space, Hilbert-Schmidt operators, application to integral equations, Green's function and eigenfunctions of the Laplacian.
9. Variational methods: variational characterization of eigenvalues, including Rayleigh-Ritz and Courant-Weyl principles, application to Sturm-Liouville theory, Euler-Lagrange equations in the Calculus of Variations, Dirichlet principle.
10. Weak solutions of PDEs: weak formulation of boundary value problems, variational methods, Lax-Milgram Lemma.

Examinations