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
MSc Programme in Statistics
MSc Programme in Mathematics with a minor subject

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

See Absalon for course literature.

Literature may include:

Springer Graduate Text in Mathematics: Gerd Grubb, Distributions and Operators.

A knowledge of Banach and Hilbert spaces corresponding to AdVec or similar.
Knowledge of Functional Analysis is not necessary, but may be helpful.
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
On-site written exam, 3 hours under invigilation
Type of assessment details
The two written 7 days take home assignments count each 20% toward the final grade. The final exam counts 60%.
All aids allowed
Marking scale
7-point grading scale
Censorship form
External censorship

Written exam, 3 hours under invigilation. All aids allowed.

The final grade is the largest of the two numbers: 1) Written exam counts 100% and 2) Written exam counts 60% and the results of the two take home assignments count 20% each.


Criteria for exam assessment

The student should convincingly and accurately demonstrate the knowledge, skills and competences described under Intended 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 limitation – unless you register in the late-registration period (BSc and MSc) or as a credit or single subject student.
Study Board of Mathematics and Computer Science
Contracting department
  • Department of Mathematical Sciences
Contracting faculty
  • Faculty of Science
Course Coordinators
  • Jan Philip Solovej   (7-75716e7178676c426f63766a306d7730666d)
  • Søren Fournais   (8-6972787571646c76437064776b316e7831676e)
phone +45 35330494, office 04.2.14
Saved on the 15-02-2024

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