Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Course MAT-81-36-K-7

Mathematical Analysis of Linear PDEs (4V+2U, 9.0 LP)

Course Type

SWS Type Course Form CP (Effort) Presence-Time / Self-Study
- K Lecture with exercise classes (V/U) 9.0 CP 186 h
4 V Lecture 56 h
2 U Exercise class (in small groups) 28 h
(4V+2U) 9.0 CP 84 h 186 h

Basedata

SWS 4V+2U
CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Lecturers
+ further Lecturers of the department Mathematics
Area of study [MAT-TEMA] Industrial Mathematics
Additional informations
Livecycle-State [NORM] Active

Contents

  • revision of topics from functional analysis (linear spaces, dual spaces and weak convergence, compact operators, embedding, Fredholm theory, boundaries and regularity); crash course on Sobolev spaces (weak derivatives, definition of Sobolev spaces, approximation by smooth functions, traces, Sobolev inequalities and embedding, Poincaré inequalities),
  • elliptical linear PDE of second order: existence of solutions via the Lax-Milgram theorem, the Fredholm alternative, spectral theory, inner regularity, regularity up to the boundary, maximum principles (weak and strong),
  • parabolic linear PDE of second order: Sobolev spaces involving time, Bochner spaces, existence of weak solutions for parabolic PDE of second order via Galerkin approximations and via the Rothe method, uniqueness, regularity, maximum principles (weak and strong),
  • hyperbolic linear PDE of second order: existence of weak solutions via energy estimates and compactness, uniqueness, regularity.

Literature

  • H. W. Alt: Lineare Funktionalanalysis,
  • L.C. Evans: Partial Differential Equations,
  • H. Brezis: Functional Analysis, Sobolev Spaces, and Partial Differential Equations.

Materials

Further literature will be announced in the lecture; Exercise material is provided.

Registration

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

Requirements for attendance (informal)

Knowledge from the modules [MAT-80-11B-M-4] and [MAT-70-12A-M-7] resp. [MAT-70-12-M-7] is useful but not necessarily required.

Modules:

Courses

Requirements for attendance (formal)

None

References to Course [MAT-81-36-K-7]

Module Name Context
[MAT-81-36-M-7] Mathematical Analysis of Linear PDEs P: Obligatory 4V+2U, 9.0 LP