# 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

back to top Lectures: Mondays and Fridays 12:15-14:00, MI HS 3.## Course Material

back to top Handwritten lecture notes will be uploaded here after each lecture.Topic | ||
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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 |

Remarks | |||
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Class 1: | Sheet 1 | Student Solutions* | |

Class 2: | Sheet 2 | Student Solutions* | |

Class 3: | Sheet 3 | Student Solutions* | |

Class 4: | Sheet 4 | Student Solutions* | |

Class 5: | Sheet 5 | Student Solutions* | |

Class 6: | Sheet 6 | Student Solutions* | |

Class 7: | Sheet 7 | Student Solutions* | |

Class 8: | Sheet 8 | Student Solutions* | |

Class 9: | Sheet 9 | Student Solutions* | |

Class 10: | Sheet 10 | Student Solutions* | |

Class 11: | Sheet 11 | Student Solutions* | |

Class 12: | Sheet 12 | Student Solutions* | |

Class 13: | Sheet 13 | Solutions |

**Solution proposals kindly provided by F. Schnack, F. Roll, V. Lachner, E. Grune, L. Mayrhofer, N. Emmermann and M. Bulté.*

## People

back to topName | Room | Consultation | ||
---|---|---|---|---|

Lecturer: | Gero Friesecke | gfma.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 |