Mathematical Institute of the University of Bonn

RG Analysis and Partial Differential Equations

V5B8 - Selected Topics in Analysis - Sobolev functions on rough domains

Winter Semester 2021/2022

Dr. Olli Saari

Instructor

Lectures

• 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.

Schedule

Week 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.
Week 10, 14 December. Capacity, defintion and basic regularity properties.
Week 11, 21 December. We will finish section 6. Hausdorff dimension and capacity, refined Lebesgue differentiation theorem, removable sets for Sobolev spaces.
Week 12, 11 January. Hardy's inequality: dimension 1, Lipschitz domains, general domains.
Week 13, 18 January. Poincaré's inequality in John domains: definition, Whitney cubes, Boman chains, proof.
Week 14, 25 January. Real interpolation, interpolation space of Lp and Sobolev space. The difficulties have been pushed to the next week and the treatment of the uniform domains.
Week 14, 1 February. Cancelled.

eCampus page of the course.

Exam

The exams are scheduled to take place 7.2.-9.2.2022 and 16.3.-18.3.2022. The questions attempt to check if you have

• learned the basic notions and definitions, formed a good idea on why the definitions stated are useful (roughly grade 3);
• formed an understanding on which results are important and why they are true, learned the outline of their proofs (roughly grade 2);
• mastered several passages of the material in detail and are able to discuss them accordingly (roughly grade 1).

The exam proceeds by a set of questions according to the categories above based on the material of first 13 weeks. For the last item of the list above, be prepared to answer detailed questions about at least two of the following themes:

1. A locally integrable function is weakly differentiable if and only if it satisfies a Poincaré inequality. (Thm 3.1 and Thm 3.4)
2. Trace theorems in half spaces. Trace theorems in Lipschitz domains. (Prop 5.3, Prop 5.4 and Proposition 5.5)
3. Capacity zero and Hausdorff dimension. (Thm 6.8)
4. Hardy's inequality. (Thm 7.2 and Thm 7.3)
5. Poincaré inequality in John domains. (Prop 7.10 and Theorem 7.11)

Literature

The lecture notes and the references therein.