FMA305F

Temporary

Potential Theory in the Complex Plane

7.5

Third-cycle course

7151 (Centre of Mathematical Sciences / Mathematics)

 -11-15

Professor Thomas Johansson

English

If sufficient demand

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.

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.

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.

Lectures

Seminars given by participants

Failed, pass

Analytic functions and basic measure theory.

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

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

FMA305F

 -11-15

Professor Thomas Johansson