Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

For citation:

Prokhorov D. V., Samsonova K. A. Integrals of the Loewner equation with exponential driving function. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 4, pp. 98-108. DOI: 10.18500/1816-9791-2013-13-4-98-108

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
25.11.2013
Full text:
download
(downloads: 233)
Language: 
Russian
Heading: 
Mathematics
UDC: 
517.54
DOI: 
10.18500/1816-9791-2013-13-4-98-108

Integrals of the Loewner equation with exponential driving function

Autors: 
Prokhorov Dmitri Valentinovich, Saratov State University
Samsonova Kristina Aleksandrovna, Saratov State University
Abstract: 

We consider the qualitative local behavior of trajectories for the ordinary Loewner differential equation with a driving function which is inverse to the exponential function of an integer power. All the singular points and the corresponding singular solutions are described. It is shown that this driving function generates solutions to the Loewner equation which map conformally a half-plane slit along a smooth curve onto the upper half-plane. The asymptotical correspondence between harmonic measures of two slit sides is derived.

Key words: 
Loewner equation
harmonic measure
singular solutions
driving function
C1-curve
References: 
  1. Lowner K. Untersuchungen ¨ uber schlichte konforme ¨Abbildungen des Einheitskreises. I. Math. Ann., 1923, vol. 89, no. 1–2, pp. 103–121.
  2. Markina I., Vasil’ev A. Virasoro algebra and dynamics in the space of univalent functions. Contemp. Math., 2010, vol. 525, pp. 85–116.
  3. Aleksandrov I. A. Parametric continuations in the theory of univalent functions. Moscow, Nauka, 1976, 344 p. (in Russian).
  4. Lind J., Marshall D. E., Rohde S. Collisions and spirals of Loewner traces. Duke Math. J., 2010, vol. 154(3), pp. 527–573. DOI: 10.1215/00127094-2010-045.
  5. Kufarev P. P. Odno zamechanie ob integralakh uravneniia Levnera. [A remark on integrals of Lowner’s ¨ equation] Doklady Akad. Nauk SSSR, 1947, vol. 57, no. 7, pp. 655—656 (in Russian).
  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. Prokhorov D. V., Zakharov A. M. Integrability of a partial case of the Lowner equation. ¨ Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2010, vol. 10, iss. 2, pp. 19–23 (in Russian).
  8. Marshall D. E., Rohde S. The Loewner differential equation and slit mappings. J. Amer.Math. Soc., 2005, vol. 18, no. 4, pp. 763–778.
  9. Prokhorov D., Vasil’ev A. Singular and tangent slit solutions to the Lowner equation. ¨ Analysis and Mathematical Physics, eds. B. Gustafsson, A. Vasil’ev. Berlin, Birkhauser, 2009, pp. 455–463.
  10. Sansone G. Equazioni differenziale nel campo reale. P. 2 a , 2 a ediz., Bologna, 1949.
  11. Poincare H. ´ Sur les courbes definies par une ´ equation ´ differentielle ´ . J. Math. Pures Appl., 1886, vol. 4, no. 2, pp. 151–217.
  12. Bendixson I. Sur les courbes definies par les ´ equations ´ differentielles. ´ Acta Math., 1901, vol. 24, pp. 1–88.
  13. Golubew W. Differentialgleichungen im komplexen, veb deutsch. Berlin, Verlag Wiss., 1958, 398 p.
  14. Sansone G. Equazioni differenziale nel campo reale. P. 1 a , 2 a ediz., Bologna, 1948.
  15. Borel E. Memoire sur les s ´ eries divergentes. ´ Ann. Sci. Ecole Norm. Sup. ´ , 1899, vol. 16, no. 3, pp. 9–131.
  16. Hayman W., Kennedy P. Subharmonic functions. London, Academic Press, 1976.
  17. Goluzin G. Geometric theory of functions of a complex variable. Transl. Math. Monographs, vol. 26, Providence, RI, AMS, 1969. 676 p. 
Short text (in English):
download
(downloads: 159)
  • 1216 reads