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Partial Differential Equations 

SS 2023    03.04.2023- 14.07.2023


Mo 14:15-15:45    H10     
Do  12:15-13:45   V2-205   


Blatt 0 - keine Abgabe
Blatt 1 - Abgabe 14.04.23

Contents of the course

0. Introduction
Examples and origin of PDEs: Laplace equation, heat equation, wave equation, Schrödinger equation. 
Quasi-linear PDEs of second order and change of coordinates.
Classification of PDEs: elliptic, parabolic, hyperbolic.

1. Laplace equation and harmonic functions
Maximum principle and uniqueness in the Dirichlet problem.
The Green function in a ball
Solvability of the Dirichlet problem and Poisson formula
Harnack inequality and other properties of harmonic functions
Sequences of harmonic functions (Harnack theorems)
Separation of variables in the Dirichlet problem
Variational problem and Dirichlet principle

2. Heat equation
The heat kernel
Solution of the Cauchy problem
Maximum principle and uniqueness in the Cauchy problem
Mixed problem and separation of variables

3. Wave equation
Cauchy problem in dimension 1
Energy and uniqueness
Mixed problem for the wave equation
Cauchy problem in dimensions 2,3

4. The eigenvalue problem
Distributions and Sobolev spaces
Weak Dirichlet problem and Green operator
Compact embedding theorem
Eigenvalues and eigenfunctions of the weak Dirichlet problem
Higher order weak derivatives of weak solutions and eigenfunctions
Sobolev embedding theorem and smoothness of weak solutions and eigenvalues


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