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

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 AdVec or similar.
Knowledge of Functional Analysis is not necessary, but may be helpful.
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
Type of assessment details
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 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

### Kursusinformation

Language
English
Course number
NMAK10019U
ECTS
7,5 ECTS
Programme level
Full Degree Master
Duration

1 block

Placement
Block 3
Schedulegroup
C
Capacity
No limit
The number of seats may be reduced in the late registration period
Studyboard
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 28-02-2023

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