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Course, academic year 2019/2020
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Partial Differential Equations 1 - NMMA405
Title in English: Parciální diferenciální rovnice 1
Guaranteed by: Mathematical Institute of Charles University (32-MUUK)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019 to 2019
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:3/1 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English
Teaching methods: full-time
Guarantor: Sebastian Schwarzacher, Dr.
Class: M Mgr. MA
M Mgr. MA > Povinné
M Mgr. MOD
M Mgr. MOD > Povinné
M Mgr. NVM
M Mgr. NVM > Povinné
Classification: Mathematics > Differential Equations, Potential Theory
Annotation -
Last update: RNDr. Miroslav Bulíček, Ph.D. (11.09.2013)
This is the basic course about the theory of partial differential equations. The notion of a weak (distributional) solution and the corresponding function spaces will be introduced and we establish the theory for (linear) elliptic equations.
Course completion requirements
Last update: RNDr. Miroslav Bulíček, Ph.D. (04.10.2018)

Exercises: the students are supposed to solve homeworks (written). The deadlines for each set of homework will be announced at least one week in advance. To obtain credits, students have to solve at least 6 exercises classified as "easy" and ate least oe exrcise classified as "difficult". According to POS, Art. 8, Par. 2 it is not posible to repeat this.

The credit from the exercices is required to participate at the exam.

Exam: written part (based mostly on the material discussed at the exercises and contained in the written homeworks; very basic knowledge from the lectures is also expected); in order to proceed to the oral part, the students must pass the written exam

oral part (based mostly on the material presented during lectures).

Most recommended sources are the book PDE's by L.C. Evans and Lecture notes which will be available on the web page (unfortunately, only in Czech due to historical reasons).

Literature -
Last update: RNDr. Miroslav Bulíček, Ph.D. (10.09.2013)

L. C. Evans: Partial Differential Equations, AMS, 2010

D. Gilbarg, N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer, 2001

Requirements to the exam
Last update: RNDr. Miroslav Bulíček, Ph.D. (04.10.2018)

The credit from the exercises is required to to be allowed to participate at the exam.

The exam will have written part (based on problems studied at the exercises) and oral part (based on material from lectures). In order to proceed to the oral part, the student must pass the written exam. Indeed, everything is combined together and it does not mean that part of the knowledge from the exercises cannot be required at the oral exam. The lectures are either covered by the material of the book PDE's aby L.C. Evans or by the Lecture notes (available unfortunately only in Czech). All this required theoretical material will be covered at the lectures.

Syllabus -
Last update: RNDr. Miroslav Bulíček, Ph.D. (04.10.2018)

General notation of weak solutions

Sobolev spaces: definition and basic overview of its properties, embedding and trace theorems

Weak solutions to linear elliptic equations on bounded domains, various boundary conditions, solution by the use of the Riesz representation theorem and the use of the Lax-Milgram theorem, compactness of the solution operator, eigen-values and eigen-vectors of the solution operator, Fredholm-like theorems and their applications, maximum principle for weak solution, $W^{2,2}$ and higher regularity, symmetric operators and the equivalence with minimizing of a quadratic functional

Bochner spaces: defintion and basic overview of its properties, Gelfand triple, integration by parts formula, embeddding.

Weak solutions to linear parabolic equations, various boundary conditions, construction of a solution via Galerkin method, uniqueness and regularity of solution.

Weak solution to linear hyperbolic equation, various boudary codition, construction of a solution via Galerkin method, uniqueness of solution, finite speed of propagation.

Entry requirements
Last update: RNDr. Miroslav Bulíček, Ph.D. (04.10.2018)

Basic knowledge of the mathematical analysis and measure theory (including the Lebesgue spaces) is needed, but students can learn this from any sources available in the library. Furthermore, starting from the middle of the semester also some basic facts from the functional analysis will be needed (Riesz representation theorem for Hilbert spaces, spectrum of selfadjoint compact operators, weak convergence).

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