NMAK10019U  Differential Operators and Function Spaces (DifFun)

Volume 2018/2019

MSc Programme in Mathematics


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

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
A knowledge of Banach and Hilbert spaces Corresponding to An1 and An2.
Knowledge of the Fourier transform corresponding to FunkAn is desirable.
5 hours of lectures and 2 hours of exercises each week for 8 weeks
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%
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 assesment

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

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