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Differential Operators and Function Spaces (DifFun)
Provider: Faculty of Science

Activity no.: 5607-19-07-31 
Enrollment deadline: 30/01/2019
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time04.02.2019, at: 09:00 - 07.04.2019, at: 16:00
Regular seats50
ECTS credits7.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserJan Philip Solovej    E-mail address: solovej@math.ku.dk
Written languageEnglish
Teaching languageEnglish partially in English
Semester/BlockBlock 3
Scheme groupC
Exam formWritten assignment
Exam formWritten examination
Exam detailsWritten 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%
Exam aidsAll aids allowed
Internal/external examiner
Exam re-examinationWritten 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.
Course workload
Course workload categoryHours
Lectures40.00
Theory exercises16.00
Exam20.00
Guidance13.00
Preparation117.00
Sum206.00
Content
Differential operators. Distribution theory, Fourier transform of distributions. Function spaces. Applications to concrete differential operator problems.

Formel requirements
A knowledge of Banach and Hilbert spaces Corresponding to An1 and An2.
Knowledge of the Fourier transform corresponding to FunkAn is desirable.

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

Teaching and learning methods
.5 hours of lectures and 2 hours of exercises each week for 8 weeks

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