# Differential Operators and Function Spaces (DifFun)

### Course content

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

Education

MSc Programme in Mathematics

Learning outcome

Knowledge:

• 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

Competences:

• 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

Skills:

• 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 An2.
Knowledge of the Fourier transform corresponding to FunkAn is desirable.

Academic qualifications equivalent to a BSc degree is recommended.

Written
Continuous feedback during the course of the semester
ECTS
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%
Aid
All aids allowed
Marking 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