# 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 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.

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%
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.
Aid
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
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

### 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 Coordinator
• Jan Philip Solovej   (7-777370737a696e447165786c326f7932686f)
phone +45 35330494, office 04.2.14
Saved on the 23-03-2022

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