Qualitative Properties of Weak Solutions to Partial Differential Equations - NMMA583

• Title: Kvalitativní vlastnosti slabých řešení parciálních diferenciálních rovnic
• Guaranteed by: Department of Mathematical Analysis (32-KMA)
• Faculty: Faculty of Mathematics and Physics
• Actual: from 2021
• Semester: winter
• E-Credits: 3
• Hours per week, examination: winter s.:2/0, Ex [HT]
• Capacity: unlimited
• Min. number of students: unlimited
• 4EU+: no
• Virtual mobility / capacity: no
• State of the course: not taught
• Language: English, Czech
• Teaching methods: full-time
• Additional information: https://www2.karlin.mff.cuni.cz/~kaplicky/pages/pages/2020z/nmma583.php
• Note: you can enroll for the course repeatedly
• Guarantor: doc. Mgr. Petr Kaplický, Ph.D.; doc. RNDr. Miroslav Bulíček, Ph.D.
• Class: DS, matematické a počítačové modelování; DS, matematická analýza; M Mgr. MA > Volitelné; M Mgr. MOD > Volitelné; M Mgr. NVM > Volitelné
• Classification: Mathematics > Differential Equations, Potential Theory
• Interchangeability : NDIR247

Annotation

English

This course is devoted to the classical results about regularity and qualitative properties of weak solutions to partial differential equations and their systems. We assume the knowledge of basic theory of weak solutions to partial differential equations.

Last update: T_KMA (14.05.2013)

Czech

Přednáška bude věnovaná klasickým výsledkům o regularitě a dalších kvalitativních vlastnostech slabých řešení parciálních diferenciálních rovnic a jejich systémů. Předpokládáme znalost základů teorie slabých řešení parciálních diferenciálních rovnic.

Last update: T_KMA (14.05.2013)

Aim of the course

English

We intend to pique interest of students in beautiful and difficult branch of mathematics and to learn students some of the classical methods of theory of partial differential equations.

Last update: T_KMA (13.05.2013)

Czech

Vzbudit u studentu zájem o krásnou a nárocnou partii matematiky. Naucit se klasické metody teorie parciálních diferenciálních rovnic.

Last update: T_KMA (13.05.2013)

Course completion requirements

Předmět nemá zápočet. Na konci se skládá zkouška. Zkoušet se bude pouze odpřednesená látka.

Last update: Kaplický Petr, doc. Mgr., Ph.D. (19.10.2020)

Literature

[1] Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105, Princeton University Press, Princeton, NJ, 1983.

[2] L. C. Evans: Partial regularity for stationary harmonic maps into spheres. Arch. Ration. Mech. Anal. *116*(2), 101-113 (1991).

[3] M. Bulíček and J. Frehse: /C^\alpha regularity for a class of non-diagonal elliptic systems with p-growth/ , Calc. Var. Partial Differential Equations, *43*, No. 3, 441--462, 2012

[4] M. Bulíček, J. Frehse and M. Steinhauer: /Everywhere C^\alpha -estimates for a class of nonlinear elliptic systems with critical growth/, Adv. Calc.Var, online first, 2013

Last update: T_KMA (14.05.2013)

Teaching methods

The course is taught through ZOOM https://cesnet.zoom.us/j/9924525902.

Its webpage is https://www2.karlin.mff.cuni.cz/~kaplicky/pages/pages/2020z/nmma583.php.

Last update: Kaplický Petr, doc. Mgr., Ph.D. (05.10.2020)

Requirements to the exam

Zkouška bude ústní. Zkoušet budeme látku odpřednesenou na přednášce.

Last update: Kaplický Petr, doc. Mgr., Ph.D. (13.10.2017)

Syllabus

The course has the intention to prepare students for challenges that occur when

instationary PDEs are non-linear. The lecture introduces techniques for existence,

uniqueness and regularity theory which are suitable for non-linear settings; however,

they will be introduced on the most simple model examples. Starting from the

heat equation we will detect fundamental principles and then introduce ways to

generalize these to more sophisticated problems. The generalization shall be made

in accordance to the particular interests of the audience.

Obligatory for the course is the knowledge of the Lebesgue theory of integration.

Some knowledge on weak differentiation and Sobolev spaces is recommended. The

course is intended for Master- and PhD- students that are keen to do research in

mathematics.

Last update: Kaplický Petr, doc. Mgr., Ph.D. (05.10.2016)

Entry requirements

Basic knowledge of weak solutions to PDE's.

Last update: Kaplický Petr, doc. Mgr., Ph.D. (18.04.2018)