Graduate course in Potential Theory

I intend to give an introduction course on potential theory in
Period 3, January 22 - March, 2007


Potential theory has its stable roots in the physics around gravity and electromagnetism. There are several central concepts in the theory that have names relating to this early connection, such as capacity etc. However, since the theory started to live a mathematical life of its own over hundred years ago (Schwartz solved Dirichlet's problem in the plane using the Poisson kernel 1872), there have been many beautiful results proven.
                Rather unexpectedly, potential theory turned out to have a strong connection with probability theory, eg. Kakutani 1944, which highly vitalized the field. In this course we will go over the classical background of the theory, but also investigate some more modern concept such as Balayage and some relations to probability theory for example via conditional Brownian motions.

Course book
Potential Theory: Selected Topics  (ISBN 3-540-61583-0), H. Aikawa and M. Essén, Lecture Notes in Mathematics 1633, Springer Verlag, 1996. Shortly, we will have one copy of this book in the library as a reference copy, i.e. not for loan.  [I will also update this site with more precise information on how you might get hold of this text.]

Contents
We will follow parts of the book above (see for example its list of contents), but perhaps also add some other features, such as discrete potential theory, boundary behavior, depending of the interest in the group.

Reference books
Classical Potential Theory, D. Armitage and S.Gardiner, Springer Verlag, 2001.
Potential Theory in the Complex Plane, T. Ransford, London Mathematical Society student texts 28, 1995.

Schedule
There will be two lectures a week during this period: Mondays 10-12 and Thursdays 13.15-15.00 in MVL14.

Examination
There will be home-assignments, and an oral examination at the end.


Welcome,
Torbjörn Lundh
Jan. 10 12.30, Organization meeting. MLV15

We discuss possible time slots for the course. And I give a very short overview of  some of the suggested contents. I will also inform you about the literature and the examination in more detail.
                       
Mon. Jan. 22, MLV14, 10-12.
  • Some more historical/physical background to potential theory.
  • Introduction of Green's functions.
  • Definitions (3 different) of Analytic sets, and some of their properties.

 Thur. Jan. 25, 1-3 pm.

  • Capacities
  • Hausdorff measures
Mon. Jan. 29, MLV14, 10-12.

  •  Frostman's Theorem

 Thur. 1 Feb, 1-3 pm.

  • Frostman's Theorem
  • Is m_h an outer mesure?
Mon. Feb. 5, MLV14, 10-12.

  • Lemma 2.12.
  • m_h is a capacity
  • Generalized potentials
  • Energies

 Thur. Feb 8, 1-3 pm.
Sportlov

Sports-break
Mon, Feb. 19, 10-12

  • Back to basics II
  • Harnack's inequality

 Wed. Feb. 21, 9.00-9.45, MLV14

  • Back to basics III
  • Perron solutions
  • Barriers

 Thursday, Feb. 22, 13-15
  • Geometric conditions for solving the Dirichlet problem including the Wiener criterion.
  • Simple random walk
  • Discrete potential theory
Mon, Feb. 26, 10-12.
  • SRW / discret pot. theo. continuation
  • Brownian motion, and harmonic  measures (with an experiment - if all goes well).
 Wed. Feb. 28, 9.00-9.45

 Thursday, March 1, 13-15.
Mon. March 5, 10-12
  • Connection BM and potential theory
 Wed. March 7, 9.00-9.45
  • Lemma 4.2
 Thursday, March 8, 13-15

  • The strong maximum principle (Theorem 4.3)
Mon. March 12, 10-12

  • Theorem 6.1
  • Capacity
  • Equilibrium distributions
 Thursday, March 15, 13-15, final lecture!
  • Reduced functions
  • Balayage
  • Thinness
  • Minimal thinness

OBS: Dead line for the home-assignments