Website for the lecture Partial Differential Equations
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News
| 18/10/2016 | The registration is activated. |
| 18/10/2016 | The lecture will always start at 8:30 and end at 10:00. |
| 20/10/2016 | A corrected version of the Homework Sheet 1 is online. We are sorry for the inconveniences this might have caused. |
| 25/10/2016 | Due to a holiday, there is no lecture at Tuesday the 1st of November. |
| 03/11/2016 | A corrected version of the Homework Sheet 3 is online. We are sorry for the inconveniences this might have caused. |
| 21/11/2016 | The notation on Homework Sheet 6 exercise 2 has been clarified. We are sorry for the inconveniences this might have caused. |
| 28/11/2016 | There was another error on Homework Sheet 6 exercise 3, which has been corrected. We are sorry for the inconveniences this might have caused. |
| 28/11/2016 | An error on Homework Sheet 7 exercise 2 has been corrected. We are sorry for the inconveniences this might have caused. |
| 06/12/2016 | An error on Homework Sheet 8 exercise 3 has been corrected. We are sorry for the inconveniences this might have caused. |
| 30/01/2017 | In this week, we don't post a homework sheet. Instead, there will be an exam preparation session on Monday, the 6th of February. Besides some general information on the exam, this class is intended to focus on some of you questions concerning the lecture. Please sent us an email in advance in case that you want a specific topic to be addressed. |
| 01/02/2017 | There will be no lecture on Thursday, 2nd of February. The course continues next week, Tuesday 7th of February. |
| 05/02/2017 | Today is the last day on which you can register for the exams. If you haven't registered yet, please do so before tomorrow. |
| 05/02/2017 | A preliminary version of the formulary for the exams is online. |
| 11/02/2017 | The exam will take place on Monday at 9:00 in the rooms B 005/B 006. Please be there in time. |
| 14/02/2017 | The results of the exam are published now. They list can be found at the notice-board next to the office B 410. The post-exam review takes place on next Monday at 10 o'clock am in our office (B 402). Please send us an e-mail in advance, if you need a Schein. |
| 31/03/2017 | There will be a preparation session on the April the 12th, at 11 o'clock(!!), for the makeup exam. This will take place in our office B 402. The makeup exam takes place at April the 20th, 2017, at 9am at room B005/B006. For further information see exams. |
| 11/04/2017 | Because of the late announcement of the exact time of the makeup exam preparation/question session (11 o'clock, see above): In case you cannot attend the preparation/question session but have urgent questions concerning lecture or homework-/tutorialsheets send us an email and we‘ll find a new appointment to discuss your questions! |
| 21/04/2017 | >The results of the makeup exam are published now. They list can be found at the notice-board next to the office B 410. The post-exam review takes place on next Monday at 10 o'clock am in our office (B 402). Please send us an e-mail in advance, if you need a Schein. |
Registration
All participating students have to register online via the Lecture Assistant. Those students who do not want to participate in one of the two tutorials may register for Tutorium X. It is important, that the data are entered correctly. Otherwise the results of the exam cannot be transmitted to the examination office.
Termine
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Lecture B 006 |
Lecture B 006 |
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Tutorial A B 040 |
Tutorial B B 041 |
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to 18:00 |
Homework Session B 006 |
Homework
Homework Sheets. Every Monday a new homework sheet is published here. On the following Monday, this sheet
is discussed in the homework session. If you want a feedback on you solutions, you may hand in you homework for corrections.
Drop your solutions into the associated letter box on the first floor (next the library) before 16:00 of the following Monday.
You will be able to retrieve your corrected solutions a week thereafter at the same place.
If you solved more than 50% of the homework exercises, your grade for this course will be improved by -0.3 in the case that you pass the exam.
Tutorial Sheets. In addition to the homework sheets we weekly provide a tutorial sheet with exercises, which are ought to be solved in the tutorial.
Exam
The exam will take place from 9:00 to 12:00 o'clock at February the 13th, 2017 at room B005/B006. The makeup exam takes place from 09:00 to 12:00 o'clock at April the 20th, 2017 at room B005/B006. To participate in the exam, you have to register via the Lecture Assistant. The deadline for your registration for either exam is February 6th, 2017. Your presence at the exam is not necessary in order to take the makeup. In the exam you will be supplied with a formulary. You can have a look on the preliminary version here.
General
Summary of the lecture
| 18/10/2016 | Introduction; PDEs as a law of nature; Notations (multiindices); The concept of a solution; Examples; Linear vs non-linear equations. |
| 20/10/2016 | Linear transport equation; Method of characteristics; Examples. |
| 25/10/2016 | The linear transport equation: further examples and the inhomogeneous case; Introduction to the Laplace equation; Harmonic functions; The mean-value properties. |
| 27/10/2016 | Harmonic functions are infinitely differentiable; Integration over spheres and balls; Harnack’s first theorem; Liouville’s theorem; Harnack’s bound |
| 03/11/2016 | The minimum principle (various versions); Smoothness of harmonic functions: a priori estimates and analyticity; Weak minimum principle for superharmonic functions; The Dirichlet Problem: uniqueness |
| 08/11/2016 | Stability with respect to the boundary condition; Poisson kernel; Solution of the Dirichlet Problem on a ball; generalization of superharmonic functions; Minimum Principle |
| 10/11/2016 | Construction of harmonic functions: the theorem of Perron |
| 15/11/2016 | Existence of a solution of the Dirichlet problem: The barrier condition; The exterior ball condition and other sufficient conditions; Lebesgue’s spine; Introduction to Poisson’s equation |
| 17/11/2016 | The fundamental solution; Elementary properties; Newton’s potential solves Poisson’s equation; The Green’s function: motivation and definition; Existence and uniqueness |
| 22/11/2016 | Estimates on the Green’s function; Solution of the Dirichlet problem with vanishing B.C.; Symmetry of the Green’s function; Representation of the solution of the Dirichlet problem for general densities and boundary values |
| 24/11/2016 | The Green’s function for the ball; Introduction to the calculus of variations; The energy functional for the Poisson equation; Uniqueness again; The set of minimizers of the energy is equal to the set of solutions of the PDE |
| 29/11/2016 | Introduction to the heat equation; Invariance under scaling; the heat kernel; Solution of the initial value problem (IVP) in R^n; Infinite propagation speed |
| 01/12/2016 | Duhamel’s Principle; Solution of the IVP with a source term; Return to equilibrium |
| 06/12/2016 | Heat balls and parabolic cylinders; The mean value property for solutions of the heat equation; |
| 08/12/2016 | The minimum and maximum principle; Uniqueness and stability of the boundary value problem in the cylinder; Uniqueness for the initial value problem on R^n; Return to equilibrium |
| 13/12/2016 | Smoothing properties of the heat equation; Analyticity; Existence for Hölder data; Energy methods: a priori estimates, forwards and backwards uniqueness |
| 15/12/2016 | The wave equation: introduction; Solution of the 1D homogeneous initial value problem, d’Alembert’s formula; Duhamel’s principle and the inhomogeneous problem |
| 20/12/2016 | The n=1 wave equation: properties of the solution; The problem of the attached string on the half-line; Kirchhoff’s formula for n=3; The method of spherical means |
| 22/12/2016 | Derivation of Kirchhoff’s formula; Discussion; Poisson’s formula for n=2 and the method of descent; Finite propagation speed, Huygen’s principle; Even and odd dimensions |
| 10/01/2017 | The inhomogeneous wave equation: Duhamel’s Principle, the retarded potential; Energy methods: local energy conservation and domain of influence, global energy conservation and equipartition; Uniqueness; |
| 12/01/2017 | Introduction to characteristics; Geometric derivation of the characteristic equations for quasilinear PDEs; Examples; |
| 17/01/2017 | The transversality condition; Solution to the characteristic equations provide a solution of the PDE; The characteristic equations for general first order PDEs; An example solution |
| 19/01/2017 | The noncharacteristic condition; Existence and uniqueness of a local solution; Examples |
| 24/01/2017 | Examples of the method of characteristics: conservation laws, the Hamilton-Jacobi equation; Burger’s equation and shocks; Integral solutions |
| 26/01/2017 | Integral solutions of conservation laws; The Rankine-Hugoniot condition for shock curves; Uniqueness and the entropy condition |
| 31/01/2017 | Weak solutions of PDEs: Introduction; Weak derivatives: definition, first examples and properties; Definition of Sobolev spaces; Radial singularities |
| 07/02/2017 | Sobolev’s inequalities and embeddings; Poincaré’s inequality; The Lax-Milgram theorem; |
| 09/02/2017 | Weak solutions for uniformly elliptic equations; Existence and uniqueness of weak solutions; Elliptic regularity; |
Skript
You can download the lecture notes here: Lecture Notes Martin Peev kindly volunteered to share his texed version of the lecture notes here. Please note, that these lecture notes neither official nor necessary up-to-date and corrected.Literature
- Main Source: L.C. Evans, Partial Differential Equations, Second Edition, AMS, 2010.
- E. Wienholtz, H. Kalf, T. Kriecherbauer, Elliptische Differentialgleichungen zweiter Ordnung, Springer, 2009.
Team
Lecturer. The lecture is held by Prof. Dr. Sven Bachmann.
Assistants. The tutorials are held by Ruth Schulte and Adrian Dietlein.
Corrector. The correction of the homework sheets is done by Wolfgang Bliemetsrieder.