Website for the lecture

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 noticeboard next to the office B 410. The postexam review takes place on next Monday at 10 o'clock am in our office (B 402). Please send us an email 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 weâ€˜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 noticeboard next to the office B 410. The postexam review takes place on next Monday at 10 o'clock am in our office (B 402). Please send us an email 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
Mon  Tue  Wed  Thu  Fri  
to 10:00 
Lecture B 006 
Lecture B 006 

to 12:00 

to 14:00 
Tutorial A B 040 
Tutorial B B 041 

to 16:00 

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 nonlinear 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 meanvalue properties. 
27/10/2016  Harmonic functions are infinitely differentiable; Integration over spheres and balls; Harnackâ€™s first theorem; Liouvilleâ€™s theorem; Harnackâ€™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; Lebesgueâ€™s spine; Introduction to Poissonâ€™s equation 
17/11/2016  The fundamental solution; Elementary properties; Newtonâ€™s potential solves Poissonâ€™s equation; The Greenâ€™s function: motivation and definition; Existence and uniqueness 
22/11/2016  Estimates on the Greenâ€™s function; Solution of the Dirichlet problem with vanishing B.C.; Symmetry of the Greenâ€™s function; Representation of the solution of the Dirichlet problem for general densities and boundary values 
24/11/2016  The Greenâ€™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  Duhamelâ€™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 HÃ¶lder 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, dâ€™Alembertâ€™s formula; Duhamelâ€™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 halfline; Kirchhoffâ€™s formula for n=3; The method of spherical means 
22/12/2016  Derivation of Kirchhoffâ€™s formula; Discussion; Poissonâ€™s formula for n=2 and the method of descent; Finite propagation speed, Huygenâ€™s principle; Even and odd dimensions 
10/01/2017  The inhomogeneous wave equation: Duhamelâ€™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 HamiltonJacobi equation; Burgerâ€™s equation and shocks; Integral solutions 
26/01/2017  Integral solutions of conservation laws; The RankineHugoniot 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  Sobolevâ€™s inequalities and embeddings; PoincarÃ©â€™s inequality; The LaxMilgram 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 NotesMartin Peev kindly volunteered to share his texed version of the lecture notes here. Please note, that these lecture notes neither official nor necessary uptodate 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.