SF2716, Topics in potential theory, spring 2010 Kursansvarig: Björn Gustafsson, 08-790 7418, gbjorn@kth.se Start: Friday, January 29, 10.15-12.00, seminar room 3733.

This side will be regularly updated.

New information:

On Friday May 21 we start presentations by students. This will continue on May 28. On the latter date we have to use another room because of a conference in number theory; the new room is D32, Lindstedtsvägen 5, close to, but outside, the mathematical department. We can use also June 4 (etc) for presentations if there is further interest.

The lecture on April 30 has been cancelled because that date is not an ordinary working day (Valborg).

The old list of homework problems (dated December 9, 2009) will be the official list, valid as examination. To solve about half of them will be enough to pass the course, and suggested latest date for handing in solutions is May 21. The selection of the problems appearing on the list is fairly arbitrary, and it is allowed to exchange problems to other suitable problems (from Ransford's book, for example). Additional examination in form of presentations (30-45 minutes) have been scheduled to take place on May 21 and 28, in case of interest.

General information:

• Credit: 7.5 points (ECTS) in the engineering programme or in the doctoral programme.
• Language: English or Swedish, depending on audience.

Teacher and examinator:

Björn Gustafsson (gbjorn@kth.se), phone 08-790 7418, room 3638.

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Literature:

Basic text:

T. Ransford: Potential Theory in the Complex Plane, Cambridge University Press, 1995.

Other literature (examples):

Books:

O. Kellogg: Foundations of Potential Theory, Springer-Verlag, 1929. (A classical source.)

L. L. Helms: Introduction to Potential Theory, Wiley, 1969.

N. S. Landkoff: Potential Theory, Springer-Verlag, 1972.

J. Doob: Classical Potential Theory and its Probabilistic Counterpart, Springer-Verlag, 1983.

R. Bass: Probabilistic Techniques in Analysis, Springer-Verlag, 1995.

E. Saff, V. Totik: External potentials of Logaritmic Fields, Springer-Verlag, 1997.

D. Armitage, S. Gardiner: Classical Potential Theory, Springer-Verlag, 2001.

(The year refers to the first edition in most cases.)

Articles and summaries:

L. Falk: Varför är Newtons och Coulombs lagar lika? KOSMOS 1997, p. 107-134. (Page 131 is missing in the pdf file.)

A. L. Levin, E. Saff: Potential theoretic tools in polynomial and rational approximation.

B. Gustafsson: Lectures on Balayage , 2003.

T. Sjödin: Potential Theory, 2000 (a short summary of basic concepts in potential theory). Extended version appears as appendix of Doctoral Thesis, 2005.

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Description of the course:

This is a first course in potential theory. Potential theory basically concerns the interaction between mass (or charge) distributions and their potentials. Some key concepts are:

Harmonic, subharmonic and superharmonic functions; potentials; Green functions.

Mass and charge distributions.

Maximum and domination principles.

Energy, capacity.

Balayage (sweeping), harmonic measure.

Potential theory has many applications in physics and serves as a tool in other branches of mathematics, e.g., complex analysis and approximation theory.
Much of the course will be devoted to various applications, exactly which may depend on the audience. The preliminary schedule given below is very tentative,
in particular for the second half of the course (L7-L12).

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Examination:

Essentially by handing in homework problems. Here is a list of homework problems used for a previous version of the course.

To pass the course you should hand in correct and well-written solutions of about 10 problems. For higher grades more will be needed.
Oral examination or presentations may also be used. Examination period: from second half of April till end of May.

 

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Preliminary schedule  
 

Lecture no.	Time and place	Topic	Remarks
L1	Friday, January 29, 10-12, room 3733	Introduction. Physical origins of potential theory	Separate meterial
L2	Friday, February 5, 10-12, room 3733	The language of differential forms; the Newton kernel and the solid angle form	Separate meterial (e.g., Ch 7 in Flanders, "Differential Forms...", AP, 1963)
L3	Friday, February 12, 10-12, room 3733	Sub- and superharmonic functions, meanvalue properties	Ch. 2 in Ransford
L4	Friday, February 19, 10-12, room 3733	Green's functions and Poisson integrals for the ball	Ch. 2, 3 in Ransford
L5	Friday, February 26, 10-12, room 3733	Convergence theorems, regularizations, Green's function	Ch. 2, 3 in Ransford
L6	Friday, March 5, 10-12, room 3733	Dirichlet problem, Perron's method	Ch. 3, 4 in Ransford
L7	Friday, March 19, 10-12, room 3733	Capacity, energy	Ch. 3, 5 in Ransford
L8	Friday, March 26, 10-12, room 3733	Frostman's theorem (on equilibrium distribution)	Ch. 3, 5 in Ransford
L9	Friday, April 9, 10-12, room 3733	Polar sets, computation of capacity	Ch. 5, 6 in Ransford
L10	Friday, April 16, 10-12, room 3733	Transfinite diameter, Fekete points, polynomial approximation	Ch. 6 in Ransford
L11	Friday, April 23, 10-12, room 3733	Continuation of previous lecture
L12	Friday, May 21, 10-12, room 3733	Presentations by students: Andreas Minne (the maximum principle and the first eigenvalue), Martin Strömqvist (partial balayage), Lia Charbit (harmonic functions)
L13	Friday, May 28, 10-12, room D32, Lv 5	Presentations by students: Mahmoudreza Bazarganzadeh (two phase quadrature domains), Antti Haimi (complex dynamics)	Note the change of room
L14	Friday, June 4, 10-12, room 3733	Presentations by students	In case of further interest