Department Mathematik



Website for the lecture
Partial Differential Equations
Winter 2016/17

This webside contains informations concerning the homework sheets and the tutorials
for the lecture Partial Differential Equations, held by

Prof. Dr. Sven Bachmann.
Office: Block B, 4. Floor, 412
Office hours: Thu 10:00-11:00


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.


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.


Mon Tue Wed Thu Fri
B 006
B 006
Tutorial A
B 040
Tutorial B
B 041
Homework Session
B 006
The first lecture takes place on Tuesday, October the 18th. The tutorials and the homework sessions starts one week thereafter.


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.

Homework Sheet
Outline Solution
Tutorial Sheet
Mon, 17/10/2016
Homework 1
Mon, 24/10/2016 16:00st
Outline Solution 1
no tutorials
Mon, 24/10/2016
Homework 2
Mon, 31/10/2016 16:00st
Outline Solution 2
Tutorial 1
Mon, 31/10/2016
Homework 3
Mon, 07/11/2016 16:00st
Outline Solution 3
Tutorial 2
Mon, 07/11/2016
Homework 4
Mon, 14/11/2016 16:00st
Outline Solution 4
Tutorial 3
Mon, 14/11/2016
Homework 5
Mon, 21/11/2016 16:00st
Outline Solution 5
Tutorial 4
Mon, 21/11/2016
Homework 6
Mon, 28/11/2016 16:00st
Outline Solution 6
Tutorial 5
Mon, 28/11/2016
Homework 7
Mon, 05/12/2016 16:00st
Outline Solution 7
Tutorial 6
Mon, 05/12/2016
Homework 8
Mon, 12/12/2016 16:00st
Outline Solution 8
Tutorial 7
Mon, 12/12/2016
Homework 9
Mon, 19/12/2016 16:00st
Outline Solution 9
Tutorial 8
Mon, 19/12/2016
Homework 10
Mon, 09/01/2017 16:00st
Outline Solution 10
Tutorial 9
Mon, 09/01/2017
Homework 11
Mon, 16/01/2017 16:00st
Outline Solution 11
Tutorial 10
Mon, 16/01/2017
Homework 12
Mon, 23/01/2017 16:00st
Outline Solution 12
Tutorial 11
Mon, 23/01/2017
Homework 13
Mon, 30/01/2017 16:00st
Outline Solution 13
Tutorial 12
Mon, 30/01/2017
no homework
Tutorial 13


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.


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;


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.


  • Main Source: L.C. Evans, Partial Differential Equations, Second Edition, AMS, 2010.
  • E. Wienholtz, H. Kalf, T. Kriecherbauer, Elliptische Differentialgleichungen zweiter Ordnung, Springer, 2009.


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.