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.

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

NB: If the exam is held at the ITX, the ITX will provide computers. Private computers, tablets or mobile phones CANNOT be brought along to the exam. Books and notes should be brought on paper or saved on a USB key.

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
• Theory exercises
• 16
• Exam
• 20
• Guidance
• 13
• Preparation
• 117
• English
• 206