# Partial Differential Equations

WS 2015/16    19.10.2015 - 12.02.2016

## Klausur (written exam)

Mi  10.02.16  16:00-18:00  U2-200

## Lectures

Mi 16:15-17:45  U2-200     Fr  14:15-15:45  U2-200

## 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

## Literature

1. Courant R., Hilbert D., Methods of mathematical physics, Vol. 2. (Methoden der mathematischen Physik, Band 2)
2. DiBenedetto E., Partial differential equations, Birkhäuser, 2010
3. Burg K., Haf H., Wille F., Meister A., Partielle Differentialgleichungen und funktionalanalytische Grundlagen, Vieweg+Teubner, 2010
4. Drabek P., Holubova G., Elements of partial differential equations, De Gruyter, 2014
5. Evans L.C., Partial differential equations, Graduate Studies in Mathematics 19, AMS, 1997, 2008, 2010
6. Farlow S.J., Partial differential equations for scientists and engineers, John Wiley & Sons, 1982.
7. Hattori H., Partial differential equations: methods, applications and theories, World Scientific Publ., 2013
8. Jost J., Partial differential equations, Graduate Texts in Mathematics 214, Spinger, 2013.
9. Komech A., Komech A., Principles of partial differential equations, Springer, 2009
10. Leis R., Vorlesungen über partielle Differentialgleichungen zweiter Ordnung, Mannheim : Bibliograph. Inst., 1967
11. Olver P.J., Introduction to partial differential equations, Undergraduate Texts in Mathematics, Springer, 2014
12. Petrowskij I.G., Vorlesungen über partielle Differentialgleichungen , Leipzig : Teubner, 1955
13. Pinchover Y., Rubinstein, J., An introduction to partial differential equations, Cambridge Univ. Press, 2007
14. Schweizer B., Partielle Differentialgleichungen, Springer 2013
15. Shearer M., Levy R., Partial differential equations : an introduction to theory and applications Princeton Univ. Press, 2015

and many more.