RG Analysis and Partial Differential Equations

V5B1: Advanced Topics in Analysis and PDE - Harmonic Measure

Winter Semester 2020/2021

Dr. Olli Saari
Instructor

Lectures

  • Mon recordings in eCampus
  • Wed 14-16 zoom
Starting from 11.1.2021, every week comes with a written lecture as was the custom so far. Every Monday will be replaced by recorded lectures to be found in eCampus (selected proofs). Every Wednesday there will be a zoom session for discussion, questions and summary of the material covered that week.

Week 1, starting 26 Oct.
Week 2, starting 02 Nov.
Week 3, starting 09 Nov.
Week 4, starting 16 Nov.
Week 5, starting 23 Nov.
Week 6, starting 30 Nov.
Week 7, starting 07 Dec.
Week 8, starting 14 Dec.

We have a Christmas break 21.12.2020 - 10.1.2021.

Week 9, starting 11 January. Quasiconformal mappings, read 5.0 to 5.3 (included) in the notes and watch the recordings for Monday 11 Jan in eCampus. The discussion session is on Wednesday as usual.
Week 10, starting 18 January. Quasidisks and chord arc domains, read 5.4 to 5.7 in the notes and watch the recordings for Monday 18 Jan in eCampus. The discussion session is on Wednesday as usual.
Week 11, starting 25 January. Kakutani's theorem, read 6.0 until 6.5 in the notes and watch the recordings for Monday 25 Jan in eCampus. The discussion session is on Wednesday as usual.
Week 12, starting 1 February. The rest of the notes. There are two more recordings in eCampus. No meeting on Wednesday, but please write me if there are questions.

Lecture notes.
eCampus page of the course.

Exam

The first exam period is 15.2.-17.2.2021. The second exam period is 29.3.-31.3.2021. This is an oral exam. You are supposed to master the material listed in the lecture notes. The tentative grading will be: to land between 5 and 3, you are supposed to know the notions and definitions; between 3 and 2, you are supposed to know something about their connections to each other, statements of theorems and such; and between 2 and 1 you are supposed to have solid understanding about the proofs (this does not mean details, but outlines at least).

In particular, prepare to explain the arguments leading to the following eight results: Frostman's theorem, Perron's method (Thm 3.3), Wiener's test, Fatou's theorem (Thm 4.12), Fatou's theorem for Hardy spaces (Thm 4.13), F. and M. Riesz theorem, Sobolev regularity of QS (Lemma 5.8), Kakutani's theorem.

Topics

This course aims at covering the basic theory and fundamental ideas concerning the harmonic measure and its applications to geometric measure theory. The focus will be on the complex plane and on covering a wide array of ideas instead of specializing to a single aspect of the theory. We will discuss classical potential theory (harmonic and superharmonic functions, potentials, capacity, polarity, Dirichlet problem), its probabilistic counterpart (Brownian motion, hitting probability and harmonic measure) as well as connections to geometric measure theory (rectifiability).

Prerequisites

Basic real, complex and functional analysis.

Literature

  • J. Doob, Classical potential theory and its probabilistic counterpart.
  • J. B. Garnett and D. E. Marshall, Harmonic measure.
  • T. Ransford, Potential theory in the complex plane.