Differential Operators and Function Spaces (DifFun)

Course content

Differential operators. Distribution theory, Fourier transform of distributions. Function spaces. Applications to concrete differential operator problems.


MSc Programme in Mathematics

Learning outcome


  • Linear differential equations and their relevant side conditions (e.g. boundary, initial)
  • Concept of ellipticity
  • Distributions and their convergence properties
  • Multiplication by smooth functions and derivatives of distributions
  • Fourier transform of distributions
  • Function classes such as Sobolev spaces or Lp spaces and the action on differential operators and  the Fourier transform on these
  • Unbounded operators on Hilbert spaces
  • Solution methods for differential equations such as methods based on the Fourier transform or a variational approach



  • Understand the different realizations of differential operators on relevant function spaces
  • Understand concepts such as existence uniqueness and regularity of solutions to differential equations within the relevant function spaces
  • Determine when a certain solution method applies
  • Calculate with distributions (derivatives, multiplication, ...)
  • Calculate Fourier transform of distributions, and functions in different function classes
  • Know the relations (inclusions) of relevant function spaces



  • Solve classical differential equations
  • Establish existence, uniqueness and regularity of solutions to certain differential equations
  • Describe the different realizations of concrete differential operators on Hilbert spaces
  • Calculate properties (e.g., domain, spectra) of realizations of differential operators

5 hours of lectures and 2 hours of exercises each week for 8 weeks

A knowledge of Banach and Hilbert spaces corresponding to An1 and LIM (alternatively An2 from previous years).
Knowledge of the Fourier transform corresponding to FunkAn is desirable.

Academic qualifications equivalent to a BSc degree is recommended.

Continuous feedback during the course of the semester
7,5 ECTS
Type of assessment
Written assignment, Two 7 days take home assignments
Written examination, 3 hours under invigilation
The two written 7 days take home assignments count each 20% toward the final grade. The final exam counts 60%
In 2022 the exam will be held as an ITX-analogue exam. This means that the exam assignment will be handed out electronically via the ITX-computer, while the students hand-in must be written with pen and paper.
All aids allowed

The University will make computers available to students taking on-site exams at ITX. Students are therefore not permitted to bring their own computers, tablets or mobile phones. If textbooks and/or notes are permitted, according to the course description, these must be in paper format or uploaded through Digital Exam.

Marking scale
7-point grading scale
Censorship form
External censorship
Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome.

Single subject courses (day)

  • Category
  • Hours
  • Lectures
  • 40
  • Preparation
  • 117
  • Theory exercises
  • 16
  • Guidance
  • 13
  • Exam
  • 20
  • English
  • 206


Course number
7,5 ECTS
Programme level
Full Degree Master

1 block

Block 3
No limit
The number of seats may be reduced in the late registration period
Study Board of Mathematics and Computer Science
Contracting department
  • Department of Mathematical Sciences
Contracting faculty
  • Faculty of Science
Course Coordinator
  • Jan Philip Solovej   (7-777370737a696e447165786c326f7932686f)
phone +45 35330494, office 04.2.14
Saved on the 23-03-2022

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