Partial Differential Equations

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.

L.C. Evans: Partial differential equations, GSM 19, American mathematical society and

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer 2010.