Partial Differential Equations [MA3005] - WS 18/19
News Timetable Course Material People
I will give a self-contained introduction to partial differential equations, including: basic examples of elliptic, parabolic and hyperbolic problems; elementary solution methods; weak solutions;
variational methods, including Dirichlet's principle; operator approach.
Prerequisites: MA1001 Analysis 1, MA1002 Analysis 2, MA2003 Measure and Integration, MA2004 Vector Analysis.
Literature: I will closely follow the book by L.C.Evans, Partial Differential Equations, Graduate Studies in Mathematics Vol. 19, AMS, 1998. In addition I will upload the handwritten live (tablet) lecture notes after each lecture.
News
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- 12.10.2018: The first lecture will take place on Monday 15 October 2018, 12:15-14:00, MI HS3. ON FRIDAY 19 October 2018, there will be no lecture.
- 21.01.2019: The first exam will take place on Thursday 14 February 2019, 11:00-12:30, PI HS1 (2501, Rudolf-Mößbauer-Hörsaal). The exam will take 90 minutes, no auxiliary materials are allowed. Further information on the exam here.
Timetable
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Lectures: Mondays and Fridays 12:15-14:00, MI HS 3.
Course Material
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Handwritten lecture notes will be uploaded here after each lecture.
| |
Topic |
|
| Lecture 1: |
Introduction |
Lecture 1 |
| Lecture 2: |
Transport equations |
Lecture 2 Matlab |
| Lecture 3: |
Transport equations; radial harmonic functions |
Lecture 3 |
| Lecture 4: |
Poisson's equation |
Lecture 4 |
| Lecture 5: |
Reconstructing harmonic functions from their boundary values: mean value formulas, maximum principle, uniqueness |
Lecture 5 |
| Lecture 6: |
Reconstructing harmonic functions from their boundary values: Green's function |
Lecture 6 |
| Lecture 7: |
Solving Laplace's equation for given boundary values; regularity of harmonic functions |
Lecture 7 |
| Lecture 8: |
More about mollification; more about Poisson's formula; converse to mean value formulas; Gauss-Seidel method |
Lecture 8 Matlab |
| Lecture 9: |
Ten revision questions on Laplace's and Poisson's equation |
Lecture 9 Student Solutions* |
| Lecture 10: |
Two reasons to be interested in the heat or diffusion equation |
Lecture 10 |
| Lecture 11: |
Fourier transform; solution formula for the heat equation in R^n |
Lecture 11 |
| Lecture 12: |
Proof that the sol'n formula yields the given initial and boundary values; smoothing; proof of Fourier reconstruction |
Lecture 12 |
| Lecture 13: |
Heat equation in bounded domains: gallery of boundary conditions, maximum principle, uniqueness |
Lecture 13 |
| Lecture 14: |
Long-time behaviour via inequalities of Poincare and Gronwall; What do solutions look like at intermediate timescales? An example. Definition of weak solutions for transport equations. |
Lecture 14 Matlab |
| Lecture 15: |
Weak solutions for Poisson's equation. Minus Laplace of the fundamental solution is the delta function. |
Lecture 15 |
| Lecture 16: |
Introduction to conservation laws: shock formation; Rankine-Hugoniot condition |
Lecture 16 |
| Lecture 17: |
Introduction to conservation laws: nonuniqueness; traffic flow modelling |
Lecture 17 |
| Lecture 18: |
Cole-Hopf transformation; uniqueness of viscosity solutions to Burgers; Lax-Oleinik formula |
Lecture 18 |
| Lecture 19: |
Revision: L^p spaces. Weak derivatives; definition of Sobolev spaces. |
Lecture 19 |
| Lecture 20: |
Basic properties of Sobolev spaces |
Lecture 20 |
| Lecture 21: |
Approximation of Sobolev functions |
Lecture 21 |
| Lecture 22: |
Boundary values of Sobolev functions |
Lecture 22 |
| Lecture 23: |
Sobolev inequality |
Lecture 23 |
| Lecture 24: |
Morrey's inequality. General Sobolev embedding theorem |
Lecture 24 |
| Lecture 25: |
Existence and uniqueness of solutions to Poisson's equation with Dirichlet boundary conditions |
Lecture 25 |
| Lecture 26: |
Some weak formulations are non-symmetric; Lax-Milgram theorem |
Lecture 26 |
| Lecture 27: |
Neumann boundary condition; Poincare-Wirtinger inequality |
Lecture 27 |
| Lecture 28: |
Interior regularity for Poisson's equation with Dirichlet boundary conditions |
Lecture 28 |
| Lecture 29: |
Brief discussion of regularity up to the boundary; weak formulation of eigenvalue problems |
Lecture 29 |
*Solution proposals kindly provided by F. Schnack, F. Roll, V. Lachner, E. Grune, L. Mayrhofer, N. Emmermann and M. Bulté.
People
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| |
Name |
eMail |
Room |
Consultation |
| Lecturer: |
Gero Friesecke |
gf ma.tum.de |
MI 03.08.054 |
lecture break; after lecture |
| Group Mo. 14:15-15:45 : |
Mi-Song Dupuy |
dupuyma.tum.de |
MI 03.08.021 |
|
| Group Wed. 10:15-11:45 : |
Mi-Song Dupuy |
dupuyma.tum.de |
MI 03.08.021 |
|
| Group Thu. 08:30-10:00 : |
Sören Behr |
behrma.tum.de |
MI 03.08.058 |
|
--
GeroFriesecke - 12 Oct 2018
