Potential Theory in the Complex Plane
Potentialteori i det komplexa planet

FMA305F, 7.5 credits

Valid from: Autumn 2018
Decided by: Professor Thomas Johansson
Date of establishment: 2018-11-15

General Information

Division: Mathematics
Course type: Third-cycle course
Teaching language: English

Aim

To give a postgraduate student in, e.g., complex analysis, harmonic analysis and partial differential equations good knowledge about a number of basic concepts and tools in modern analysis.

Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• be able to state some of the most important properties of harmonic functions of two variables.
• be able to account for the basic theory for subharmonic functions.
• be able to explain the following concepts from the theory of potentials: potential, polar set, equilibrium measure, upper semicontinuous regularization, weak form of the Poisson equation.
• be able to account for the following concepts from the theory for the Dirichlet problem in the plane: Perron function, barrier, regular boundary point, harmonic measure, Green's function, the Poisson-Jensen formula.
• be able to give the definition of the capacity of a set, be able to evaluate it in simple cases, and to estimate it in more general cases.
• be able to describe applications of potential theory to other mathematical fields, e.g. functional analysis, approximation theory, complex analysis or complex dynamics.

Course Contents

Harmonic functions of two variables: Harmonic and holomorphic functions, the Dirichlet problem on the disc, positive harmonic functions. Subharmonic functions: Upper semicontinuous functions, subharmonic functions, the maximum principle, criteria for subharmoniciy, integrability, convexity, smoothing. Potential theory: Potentials, polar sets, equilibrium measures, upper semicontinuous regularization, minus-infinity sets, removable singularities, the generalized Laplacian, thinness. The Dirichlet problem: Solution of the Dirichlet problem, criteria for regularity, harmonic measure, Green's functions, the Poisson-Jensens formula. Capacity: Capacity as a set function, computation of capacity, estimation of capacity, criteria for thinness, transfinite diameter.

Course Literature

Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, 1995. ISBN 9780521466547.

Instruction Details

Type of instruction: Lectures

Examination Details

Examination format: Seminars given by participants
Grading scale: Failed, pass
Examiner:

Admission Details

Assumed prior knowledge: Analytic functions and basic measure theory.

Further Information

Contacts: Jacob Stordal Christiansen (jacob_stordal.christiansen-at-math.lth.se) and Frank Wikström (frank.wikstrom-at-math.lth.se) .

Course Occasion Information

Contact and Other Information

Course coordinators: