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of A.Grigor'yan
Partial Differential Equations
SS 2023 03.04.2023- 14.07.2023
Lectures
Mo 14:15-15:45 H10
Do 12:15-13:45 V2-205
Exercises
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
Literature
- Courant R., Hilbert D., Methods of mathematical physics, Vol. 2. (Methoden der mathematischen Physik, Band
2)
- DiBenedetto E., Partial differential equations, Birkhäuser, 2010
- Burg K., Haf H., Wille F., Meister A., Partielle Differentialgleichungen
und funktionalanalytische Grundlagen, Vieweg+Teubner, 2010
- Drabek P., Holubova G., Elements of partial differential equations, De Gruyter, 2014
- Evans L.C., Partial differential equations, Graduate Studies in Mathematics
19, AMS, 1997, 2008, 2010
- Farlow S.J., Partial differential equations for scientists and engineers,
John Wiley & Sons, 1982.
- Hattori H., Partial differential equations: methods, applications and theories,
World Scientific Publ., 2013
- Jost J., Partial differential equations, Graduate Texts in Mathematics 214,
Spinger, 2013.
- Komech A., Komech A., Principles of partial differential equations, Springer, 2009
- Leis R., Vorlesungen über partielle Differentialgleichungen zweiter Ordnung,
Mannheim : Bibliograph. Inst., 1967
- Olver P.J., Introduction to partial differential equations, Undergraduate
Texts in Mathematics, Springer, 2014
- Petrowskij I.G., Vorlesungen über partielle Differentialgleichungen , Leipzig : Teubner, 1955
- Pinchover Y., Rubinstein, J., An introduction to partial differential equations, Cambridge Univ. Press, 2007
- Schweizer B., Partielle Differentialgleichungen, Springer 2013
- Shearer M., Levy R., Partial differential equations : an introduction to theory and applications Princeton Univ. Press, 2015
and many more.