Abstract
In this final chapter the general problem of Plateau, usually called the Douglas problem, is studied for minimal surfaces defined on schlicht (i.e. planar) domains which span a given boundary configuration consisting of k closed Jordan curves. By a modification of a method due to Courant it is shown that there exists a solution of Douglas’s problem which minimizes both area and Dirichlet’s integral, if the sufficient Douglas condition is satisfied by the boundary configuration. In particular, this solves the Douglas problem for two linked boundary curves. Finally the connection to Koebe’s mapping theorem is pointed out.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Dierkes, U., Hildebrandt, S., Sauvigny, F. (2010). Introduction to the Douglas Problem. In: Minimal Surfaces. Grundlehren der mathematischen Wissenschaften, vol 339. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11698-8_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-11698-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11697-1
Online ISBN: 978-3-642-11698-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)