# 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

• 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.
• 22.02.2019: The Klausureinsicht will take place on Tuesday 26 February 2019 from 11:00 to 12:00 in 03.08.011.

## Timetable

Lectures: Mondays and Fridays 12:15-14:00, MI HS 3.

## Course Material

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

Remarks
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

Lecturer:   Gero Friesecke  gf ma.tum.de   MI 03.08.054   lecture break; after lecture
Group Mo. 14:15-15:45 :   Mi-Song Dupuy   dupuy ma.tum.de   MI 03.08.021
Group Wed. 10:15-11:45 :   Mi-Song Dupuy   dupuy ma.tum.de   MI 03.08.021
Group Thu. 08:30-10:00 :   Sören Behr   behr ma.tum.de   MI 03.08.058