V5B8 - Selected Topics in Analysis - Sobolev functions on rough domains
Winter Semester 2021/2022
- Dr. Olli Saari
- Tue 10-12, N 0.003. Lectures can only take place if the participants can present a proof of immunity or a valid negative test result.
ScheduleWeek 1, 12 October. Test functions, distributions, density of smooth functions, basic calculus of Sobolev functions and compact embedding of Sobolev space into Lp.
Week 2, 19 October. Weak Lp, Marcinkiewicz interpolation, covering lemma, maximal functions.
Week 3, 26 October. Poincaré's inequality, Sobolev's inequality with good lambda argument.
Week 4, 2 November. Truncation argument and strong type Sobolev inequality, corollaries of Poincaré's inequality, converse of Poincaré's inequality.
Week 5, 9 November. Fourier transform, Besov spaces, Gagliardo seminorm.
Week 6, 16 November. Equivalence of Gagliardo seminorm and Besov seminorm.
Week 7, 23 November. Remaining inequality from the previous equivalence, remarks on metric spaces and inhomogeneous norms.
Week 8, 30 November. No lecture! Study the sections 5 and 6 in the notes. Those sections will be discussed in the lecture of the next week but with less detail. In particular, prepare to ask about what remained unclear when reading, if anything.
Week 9, 7 December. Traces. Notes.
All lecture notes.
eCampus page of the course.
OverviewIn your first courses on analysis or PDE, you have already encountered the notion of weak differentiability and learned many important theorems about Sobolev functions when the ambient domain is nice enough. In this course, we study domains that are not that nice. Studying such rough domains can be motivated in many ways, for instance by applications. A fluid flow in a porous medium can create a free boundary that is a fractal; the boundary of a water wave can exhibit a cusp shape, and non-linear phenomena in general do not like to fit in the framework of Lipschitz domains. Sobolev spaces are however indispensable for studying any of those. When moving away from the classical assumptions on domains, a part of the Sobolev theory stays unchanged, a part of it completely falls apart, and a part continues its life in a weaker form. The ultimate goal of these lectures is to give good understanding of the interrelation of (geometric) theory of Sobolev functions and the properties of rough domains.
- Review of the basics, definitions, Poincaré inequalities, embeddings, capacity
- Fractional spaces, Hajłasz Sobolev spaces, interpolation.
- Traces and extensions, Lipschitz domains and beyond.
- Calderón-Zygmund theory, Poincaré and Bogovskij integrals, Korn's inequality.
- Hardy's inequalities.
- John domains and their characterizations.