Potentialteori i det komplexa planet

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

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

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.

*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.

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.

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

**Type of instruction:** Lectures

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

**Assumed prior knowledge:** Analytic functions and basic measure theory.

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

**Course coordinators:**