(in Polish) Dygresyjne wprowadzenie do teorii regularności rozwiązań eliptycznych równań i układów równań

General data

Course ID:	1000-1S19DTR
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	(unknown)
Name in Polish:	Dygresyjne wprowadzenie do teorii regularności rozwiązań eliptycznych równań i układów równań
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Seminars for Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Type of course:	elective seminars
Short description:	The goal of this seminar is to present in an orderly manner the introduction to the theory of elliptic problems. We will follow the exposition presented in Lisa Beck's book. We plan to offer a number of digression from the main course to less standard topics, including the calculus of variations, the "bubbling off" phenomenon or the size of the critical set.
Full description:	The goal of this seminar is to present in an orderly manner the introduction to the theory of elliptic problems. We shall do this following the exposition presented in [B]. We have in mind covering the following topics: the basic methods for equations (finite differences); de Giorgi's method and Moser's iterative technique. We are interested in the partial regularity for systems and among others the blow-up technique, the method of A-harmonic approximations. Our ultimate goal is the regularity theory for the quasi-linear systems. We will spend some time discussing estimates of the Hausdorff dimension of the singular set. As promised the main course will be accompanied by digressions. One of the side trips will be devoted to the calculus of variations, see [DG], and equations with the right-hand-side in L^1. There, we can learn about "bubbling off" phenomenon. We will talk about functionals with nonstandard growth. Another planned digression is related to the theory of harmonic maps, see [MY]. We have to explain what are the harmonics maps. During this side trip we will learn how we can estimate the size of the singular set, [HL], [SU]. It is easy to multiply digressions. Their number and directions we take depend on the interests of the audience.
Bibliography:	(in Polish) [B] L.Beck, Elliptic Regularity Theory, A First Course, Springer, Cham, 2016 [DG] F.Duzaar, J.Grotowski, Existence and regularity for higher-dimensional H-systems. Duke Math. J. 101 (2000), no. 3, 459-485. [HL] R.Hardt, F.-H.Lin, Stability of singularities of minimizing harmonic maps J. Differential Geom. 29 (1989), no. 1, 113-123. [M] P.Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions. Arch. Rational Mech. Anal. 105 (1989), no. 3, 267-284. [MY] L.Mou, P.Yang, Regularity for n-harmonic maps. J. Geom. Anal. 6 (1996), no. 1, 91-112 [SU] R.Schoen, K.Uhlenbeck, A regularity theory for harmonic maps. J. Differential Geom. 17 (1982), no. 2, 307–335.

This course is not currently offered.

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