Abstract

Nonlinear Riemann-Hilbert Problems

Elias Wegert, TU Bergakademie Freiberg, Germany

Though Bernhard Riemann's thesis is commonly known as the source of the
celebrated Riemann mapping theorem, Riemann himself considered conformal
mapping just as an example to illustrate his ideas about a more general
class of nonlinear boundary value problems for analytic functions.
The talks aim on making these Riemann-Hilbert problems more popular, to
encourage further research and to find novel applications.

In the first part we address the existence and uniqueness of solutions
for different problem classes and present two applications: potential
flow past a porous object, and a free boundary value problem in
electrochemical machining.

In the second part, a connection between Riemann-Hilbert problems and a
class of extremal problems is established. Solutions to Riemann-Hilbert
problems are characterized by an extremal principle which generalizes
the classical maximum principle and Schwarz' lemma. We briefly sketch an
application 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 decomposition
of functions on the unit circle into an analytic and an anti-analytic part.