**The lecture this week Wednesday (June 30th) is canceled and shifted to Tuesday, July 6th, 9:45--11:15 in 1C-01.**

# Current evaluation of the course

# Topics

The course consists of topics on linear elliptic partial differential equations, where the differential equations are coupled with boundary conditions. Boundary value problems of such kind appear e.g. in modeling of reaction-, convection- and diffusion processes. Special cases of boundary value problems are so called eigenvalue problems, which appear e.g. in quantum mechanics or in vibrations of elastic materials. If time permits we will also investigate basic properties of time-dependent problems (parabolic initial-boundary value problems) like the heat equation, which models heat conduction.

In the course I will cover results on existence of weak solutions in Sobolev spaces, estimates of such solutions together with qualitative and regularity properties.

1. Motivation & examples

2. Explicit solution of the Poisson boundary value problem on balls

3. Weak derivatives and Sobolev spaces

- Poincaré and Sobolev inequalities, imbedding theorems

4. Elliptic boundary value problems

- Maximum and comparison principles, existence results, Fredholm alternative
- Regularity properties of solutions

5. Elliptic eigenvalue problem

Eigenvalues, eigenfunctions, completeness, variational characterization

6. Parabolic initial-boundary value problems

Audience: Mathematicians, physicists, engineers;

Prerequisites: Analysis I--III (or similar lectures), basics in functional analysis