Module Handbook

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Module MAT-70-12A-M-7

Introduction to the Theory of Sobolev Spaces (M, 4.5 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-70-12A-M-7 Introduction to the Theory of Sobolev Spaces 4.5 CP (135 h)

Basedata

CP, Effort 4.5 CP = 135 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers
+ further Lecturers of the department Mathematics
Area of study [MAT-SPAS] Analysis and Stochastics
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Notice

This module ist a part of the module [MAT-70-12-M-7] Sobolev Spaces; it can be combined with the module [MAT-71-12A-M-7] Introduction to White Noise Analysis to a module „Analysis in Sobolev and Distribution Spaces“ (9 LP).

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
2V+1U MAT-70-12A-K-7
Introduction to the Theory of Sobolev Spaces
P 42 h 93 h - - PL1 4.5 irreg.
  • About [MAT-70-12A-K-7]: Title: "Introduction to the Theory of Sobolev Spaces"; Presence-Time: 42 h; Self-Study: 93 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 86404 ("Introduction to the Theory of Sobolev Spaces")

Evaluation of grades

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


Contents

  • extension of integration theory (convergence theorems, Lp-spaces, partial integration),
  • 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.

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 understand how Sobolev spaces are constructed and can apply and critically evaluate important methods and concepts of analysis to Sobolev 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.

References to Module / Module Number [MAT-70-12A-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.)