Abstract

Nonlinear Riemann-Hilbert ProblemsElias Wegert, TU Bergakademie Freiberg, GermanyThough Bernhard Riemann's thesis is commonly known as the source of thecelebrated Riemann mapping theorem, Riemann himself considered conformalmapping just as an example to illustrate his ideas about a more generalclass of nonlinear boundary value problems for analytic functions.The talks aim on making these Riemann-Hilbert problems more popular, toencourage further research and to find novel applications.In the first part we address the existence and uniqueness of solutionsfor different problem classes and present two applications: potentialflow past a porous object, and a free boundary value problem inelectrochemical machining.In the second part, a connection between Riemann-Hilbert problems and aclass of extremal problems is established. Solutions to Riemann-Hilbertproblems are characterized by an extremal principle which generalizesthe classical maximum principle and Schwarz' lemma. We briefly sketch anapplication to the design of dynamical systems.In the end, a class of nonlinear transmission problems is considered.As a special result, we obtain a hyperbolic version of the Riesz decompositionof functions on the unit circle into an analytic and an anti-analytic part.