The first PDEs encountered in this course are - linear - first order
equations. We discuss a classification of linear PDEs of any order -
mainly constant coefficients. We start with an introduction to PDEs via
Fourier series methods. These methods are motivated via the classical
examples of hyperbolic (wave), parabolic (diffusion) and elliptic
(Laplace) equations. A brief introduction to Sobolev spaces and Hilbert
space methods accumulates in the proof of the existence of weak
solutions of linear elliptic equations by the Lax-Milgram Lemma, as
well as some basic regularity theory.
Semi-group theory (Hille-Yosida theorem) is introduced to treat evolution equations (parabolic and hyperbolic equations) from a Hilbert space point of view. Furthermore, eigenvalue problems and expansions of eigenfunctions are studied through these methods.