Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Prokhorov D. V., Ukrainskii D. V. Асимптотическое отношение гармонических мер сторон разреза. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2015, vol. 15, iss. 2, pp. 160-166. DOI: 10.18500/1816-9791-2015-15-2-160-167, EDN: TXMFQT

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
11.06.2015
Full text:
download
(downloads: 93)
Language: 
Russian
Heading: 
Mathematics
UDC: 
517.54
DOI: 
10.18500/1816-9791-2015-15-2-160-167
EDN: 
TXMFQT

Асимптотическое отношение гармонических мер сторон разреза

Autors: 
Prokhorov Dmitri Valentinovich, Saratov State University
Ukrainskii Dmitri Vladimirovich, Saratov State University
Abstract: 

The article is devoted to the geometry of solutions to the chordal Löwner equation which is based on the comparison of singular solutions and harmonic measures for the sides of a slit in the upper half-plane generated by a driving term. An asymptotic ratio for harmonic measures of slit sides is found for a slit which is tangential to a straight line under a given angle, and for a slit with high order tangency to a circular arc tangential to the real axis.

Key words: 
Löwner equation
singular solution
harmonic measure
half-plane capacity
References: 
  1. Löwner K. Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann., 1923, vol. 89, no. 1–2, pp. 103–121.
  2. Lind J., Marshall D. E., Rohde S. Collisions and spirals of Loewner traces. Duke Math. J., 2010, vol. 154, no. 3, pp. 527–573.
  3. Hayman W. K., Kennedy P. B. Subharmonic Functions, vol. 1, London, New York, Academic Press, 1976.
  4. Earle C. J., Epstein A. L. Quasiconformal variation of slit domains. Proc. Amer. Math. Soc., 2001, vol. 129, no. 11, pp. 3363–3372.
  5. Prokhorov D., Zakharov A. Harmonic measures of sides of a slit perpendicular to the domain boundary. J. Math. Anal. Appl., 2012, vol. 394, no. 2, pp. 738–743.
  6. Kager W., Nienhuis B., Kadanoff L. P. Exact solutions for Loewner evolutions, J. Statist. Phys., 2004, vol. 115, no. 3–4, pp. 805–822.
  7. Rad´o T. Sur la repr´esentations conforme de domaines variables. Acta Sci. Math. (Szeged), 1922–1923, vol. 1, no. 3, pp. 180–186.
  8. Голузин Г. М. Геометрическая теория функций комплексного переменного. М. : Наука, 1966.
  9. Markushevich A. I. Sur la repr´esentations conforme des domaines `a fronti`eres variables. Rec. Math. [Mat. Sbornik] N.S., 1936, vol. 1(43), no. 6, pp. 863–886.
  10. Prokhorov D., Vasil’ev A. Singular and tangent slit solutions to the L¨owner equation. Analysis and Mathematical Physics, eds. D. Gustafsson, A. Vasil’ev. Berlin, Birkhauser, 2009, pp. 455–463.
  11. Ivanov G., Prokhorov D., Vasil’ev A. Non-slit and singular solutions to the L¨owner equation. Bull. Sci. Mathem., 2012, vol. 136, no. 3, pp. 328–341.
Received: 
20.01.2015
Accepted: 
29.05.2015
Published: 
30.06.2015
  • 735 reads