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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© 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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DOI: https://doi.org/10.1007/978-3-7643-9906-1_23
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-9905-4
Online ISBN: 978-3-7643-9906-1
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