# Partial Differential Equations I and II

## Partial Dfferential Equations I

The course Partial Differential Equations I presents students with an opportunity to apply the techniques learned in Analysis I-III to a major area of modern mathematics. Moreover, they become acquainted with current research topics. The students should develop an understanding of the central role of partial differential equations in science and engineering.

Topics

Among others, the following topics are covered: classification of partial differential equations, introduction to potential theory, Hilbert space methods: Riesz representation theorem, Lax-Milgram theorem, Sobolev spaces, Fourier transform, trace theorems, H-regularity of weak solutions, eigenvalue problems for elliptic operators

Prerequisites

Students are required to have successfully completed the courses Analysis I-III and Linear Algebra I.

Examination

The examination consists of a written or oral exam; duration and type of the exam are announced at the beginning of the semester. For admission to the exam, a sufficient number of exercises has to be solved and submitted.

Literature:

1. M. Renardy, R. Rogers: An Introduction to Partial Differential Equations, Springer Verlag 2004;
2. L.C. Evans: Partial Differential Equations, AMS 1998;
3. D. Gilbarg, N. Trudinger: Partial Differential Operations of Second Order, Springer Verlag 2001;
4. L.C. Evans, R.F. Gariepy: Measure Theory and Fine Properties of Functions, CRC Press 1992

## Partial Differential Equations II

In continuation of Partial Differential Equations I, the course Partial Differential Equations II presents students with an opportunity to apply the techniques learned in Analysis I-III to a major area of modern mathematics. Moreover, they independently explore and understand current research topics and develop a greater understanding of the central role of partial differential equations in science and engineering.

Topics

Topics covered include: evolution equations, special equations, maximum principles, weak formulations, existence theory, regularity, nonlinear equations, qualitative theory.

Prerequisites

Taking the course requires knowledge from Partial Differential Equations I.

Examination

The examination consists of a written or oral exam; duration and type of the exam are announced at the beginning of the semester. For admission to the exam, a sufficient number of exercises has to be solved and submitted.

Literature:

1. L.C. Evans: Partial Differential Equations, AMS 1998
2. M. Renardy, R. Rogers: An Introduction to Partial Differential Equations, Springer-Verlag 2004
3. E. Di Benedetto: Partial Differential Equations, Birkhäuser 1995
4. D. Henry: Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag 1981
5. J. Smoller: Stock Waves and Reaction Diffusion Equations, Springer-Verlag 1983
6. G.R. Sell, Y. You: Dynamics of Evolutionary Equations, Springer-Verlag 2002