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Singular and Tangent Slit Solutions to the Löwner Equation

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Book cover Analysis and Mathematical Physics

Part of the book series: Trends in Mathematics ((TM))

Abstract

We consider the Löwner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by Hölder continuous driving terms with exponent 1/3 in the Löwner equation. Singular solutions are described, and the critical value of the norm of driving terms generating quasisymmetric slits in the disk is obtained.

The first author was partially supported by the Russian Foundation for Basic Research (grant 07-01-00120) and the second by the grants of the Norwegian Research Council #177355/V30, the European Science Foundation RNP HCAA, and the Scandinavian Network Analysis and Applications (Nordforsk).

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics and Mechanics, Saratov State University, Astrakhanskay Str. 83, 410012, Saratov, Russia

    Dmitri Prokhorov

  2. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, Bergen, 5008, Norway

    Alexander Vasil’ev

Authors
  1. Dmitri Prokhorov
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  2. Alexander Vasil’ev
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology (KTH), 100 44, Stockholm, Sweden

    Björn Gustafsson

  2. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008, Bergen, Norway

    Alexander Vasil’ev

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© 2009 Birkhäuser Verlag Basel/Switzerland

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Prokhorov, D., Vasil’ev, A. (2009). Singular and Tangent Slit Solutions to the Löwner Equation. In: Gustafsson, B., Vasil’ev, A. (eds) Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9906-1_23

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