Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-70-12-M-7

Sobolev Spaces (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-70-12-M-7 Sobolev Spaces 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers
Area of study [MAT-SPAS] Analysis and Stochastics
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V+2U MAT-70-12-K-7
Sobolev Spaces
P 84 h 186 h - - PL1 9.0 irreg.
  • About [MAT-70-12-K-7]: Title: "Sobolev Spaces"; Presence-Time: 84 h; Self-Study: 186 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 86405 ("Sobolev Spaces")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

  • construction of Sobolev spaces,
  • analysis in Sobolev spaces (convolution, Dirac sequences, partition of unity, dense sets of functions),
  • applications to partial differential equations (Poincaré inequality, fundamental lemma of calculus of variations, weak formulation of boundary problems),
  • Sobolev embedding theorems and trace operator,
  • test function and distribution spaces,
  • analysis in distributions spaces (Fourier transform, differentiation, convolution),
  • dual spaces of Sobolev spaces,
  • fractional Sobolev spaces.

Competencies / intended learning achievements

Upon successful completion of this module, the students have gained in-depth knowledge of a subfield of functional analysis with applications to partial differential equations (PDE). They have acquired a thorough understanding of how Sobolev and distribution spaces are constructed and can apply and critically evaluate important methods and concepts of analysis to Sobolev and distribution spaces. Moreover, they are familiar with applications of the theory of Sobolev spaces to PDE. They understand the proofs presented in the lecture and are able to reproduce and explain them. In particular, they can outline the conditions and assumptions that are necessary for the validity of the statements.

By completing the given exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learnt how to apply these techniques to new problems, analyze them and develop solution strategies independently or by team work.

Literature

  • H.-W. Alt: Lineare Funktionalanalysis,
  • M. Dobrowolski: Angewandte Funktionalanalysis,
  • M. Reed, B. Simon: Functional Analysis I.

Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de).

Requirements for attendance of the module (informal)

Modules:

Courses

Requirements for attendance of the module (formal)

None

References to Module / Module Number [MAT-70-12-M-7]

Module-Pool Name
[MAT-70-MPOOL-7] Specialisation Stochastic Analysis (M.Sc.)
[MAT-AM-MPOOL-7] Applied Mathematics (Advanced Modules M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)