- 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.
Mathematical Analysis of Linear PDEs (M, 9.0 LP)
|Module Number||Module Name||CP (Effort)|
|MAT-81-36-M-7||Mathematical Analysis of Linear PDEs||9.0 CP (270 h)|
|CP, Effort||9.0 CP = 270 h|
|Position of the semester||1 Sem. irreg.|
|Level|| Master (Advanced)|
+ further Lecturers of the department Mathematics
|Area of study||[MAT-TEMA] Industrial Mathematics|
|Reference course of study||[MAT-88.105-SG] M.Sc. Mathematics|
|Type/SWS||Course Number||Title||Choice in |
|SL||SL is |
required for exa.
Mathematical Analysis of Linear PDEs
|P||84 h||186 h||-||-||PL1||9.0||irreg.|
- About [MAT-81-36-K-7]: Title: "Mathematical Analysis of Linear PDEs"; 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: 86288 ("Mathematical Analysis of Linear PDE")
Evaluation of grades
The grade of the module examination is also the module grade.
Competencies / intended learning achievements
They are able to name and to prove the essential propositions of the lecture as well as to classify and to explain the connections.
By solving the given exercises, have gained a skilled and independent handling of the terms and methods of the lecture. They understand the proofs presented in the lecture and are able to reproduce and explain them and to apply them to concrete examples of (initial and) boundary value problems.
- H. W. Alt: Lineare Funktionalanalysis,
- L.C. Evans: Partial Differential Equations,
- H. Brezis: Functional Analysis, Sobolev Spaces, and Partial Differential Equations.
Requirements for attendance (informal)
- [MAT-10-1-M-2] Fundamentals of Mathematics (M, 28.0 LP)
- [MAT-70-11-M-4] Functional Analysis (M, 9.0 LP)