Valid from: Autumn 2018
Decided by: Professor Thomas Johansson
Date of establishment: 2018-11-15
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
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
Assumed prior knowledge: Analytic functions and basic measure theory.
Contacts: Jacob Stordal Christiansen (jacob_stordal.christiansen-at-math.lth.se) and Frank Wikström (frank.wikstrom-at-math.lth.se) .