Theory of weak solutions of linear elliptic equations of second order (Evans, Ch. 6):
- Existence of weak solutions (Evans 6.1 und 6.2)
- Regularity of weak solutions (Evans 6.3)
- Maximum Principles (Evans 6.4)
- Eigenvalues and eigenfunctions (Evans 6.5)
Separation of variables with applications to initial value problems for the heat - and the wave equation
The above topics use fundamentals from linear functional analysis (Evans, Appendix D and Fourier series theorem). Some of the results from functional analysis will be proved in lectures. No prerequisites are required for this except for Analysis 1 - 3 and Linear Algebra 1 and 2.