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Partial Differential Equations 1 - NMMA405

Title:	Parciální diferenciální rovnice 1
Guaranteed by:	Mathematical Institute of Charles University (32-MUUK)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2022
Semester:	winter
E-Credits:	6
Hours per week, examination:	winter s.:3/1, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time
Teaching methods:	full-time

Guarantor:	doc. RNDr. Miroslav Bulíček, Ph.D.
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
Comes under:	Doporučené přednášky 2/2
Is interchangeable with:	NDIR045

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Annotation -

Last update: doc. 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: doc. RNDr. Miroslav Bulíček, Ph.D. (27.09.2020)

At the end of the semester there will be a written exam. Students are supposed to provide the knowledge of the theory taught during the semester.

Students are obliged to solve homeworks correctly for passing the tutorials. It is also a necessary condition in order to pass the exam.

Literature -

Last update: doc. 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: doc. RNDr. Miroslav Bulíček, Ph.D. (27.09.2020)

According to sylabus

Syllabus -

Last update: doc. 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: doc. RNDr. Miroslav Bulíček, Ph.D. (27.09.2020)

Basic knowledge of the mathematical analysis, measure theory (including the Lebesgue spaces) and the classical theory of PDEs is needed. Furthermore, starting from the middle of the semester also some basic facts from the functional analysis will be required (Riesz representation theorem for Hilbert spaces, spectrum of selfadjoint compact operators, weak convergence).