(in Polish) Dygresyjne wprowadzenie do teorii regularności rozwiązań eliptycznych równań i układów równań 1000-1S19DTR
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.
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