Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

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

For citation:

Prokhorov D. V., Zakharov A. M., Zherdev A. V. Solutions of the Loewner equation with combined driving functions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, vol. 21, iss. 3, pp. 317-325. DOI: 10.18500/1816-9791-2021-21-3-317-325

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
Full text:
(downloads: 61)
Article type: 

Solutions of the Loewner equation with combined driving functions

Prokhorov Dmitri Valentinovich, Saratov State University
Zakharov Andrei Mikhailovich, Saratov State University
Zherdev Andrey V., Saratov State University

The paper is devoted to the multiple chordal Loewner differential equation with different driving functions on two time intervals. We obtain exact implicit or explicit solutions to the Loewner equations with piecewise constant driving functions and with combined constant and square root driving functions. In both cases, there is an analytical and geometrical description of generated traces. Earlier, Kager, Nienhuis and Kadanoff integrated the chordal Loewner differential equation either with a constant driving function or with a square root driving function. In the first case, the equation generates a rectilinear slit in the upper half-plane which is orthogonal to the real axis $\mathbb R$. In the second case, a rectilinear slit forms an angle to $\mathbb R$. In our paper, the multiple chordal Loewner differential equation generates more complicated hulls consisting of three rectilinear and curvilinear fragments which can be either intersecting or disjoint. Analytical results of the paper are accompanied by geometrical illustrations.

This work was supported by the Program of development of Regional Scientific and Educational Mathematical Center “Mathematics of Future Technologies” (project No. 075-02-2021-1399).
  1. Lowner K. Untersuchungen uber schlichte konforme Abbildungen des Einheitskreses. I. Mathematische Annalen, 1923, vol. 89, iss. 1–2, pp. 103–121 (in Germany). https://doi.org/10.1007/BF01448091
  2. Lawler G. F. Conformally Invariant Processes in the Plane. Princeton, American Mathematical Society, 2005. 242 p. (Mathematical Surveys and Monographs. Vol. 114).
  3. Kager W., Nienhuis B., Kadanoff L. P. Exact solutions for Loewner evolutions. Journal of Statistical Physics, 2004, vol. 115, iss. 3–4, pp. 805–822. https://doi.org/10.1023/B:JOSS.0000022380.93241.24
  4. Lind J. R. A sharp condition for the Loewner equation to generate slits. Annales Academiæ Scientiarum Fennicæ. Mathematica, 2005, vol. 30, iss. 1, pp. 143–158.
  5. Prokhorov D. V., Zakharov A. M. Integrability of a partial case of the Loewner equation. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2010, vol. 10, iss. 2, pp. 19–23 (in Russian). https://doi.org/10.18500/1816-9791-2010-10-2-19- 23
  6. Prokhorov D., Vasil’ev A. Singular and tangent slit solutions to the Lowner equation. In: Gustafsson B., Vasil’ev A., eds. Analysis and Mathematical Physics. Trends in Mathematics. Birkhauser Basel, 2009, pp. 455–463. https://doi.org/10.1007/978-3-7643-9906-1_23
  7. Lau K. S., Wu H. H. On tangential slit solution of the Loewner equation. Annales Academiæ Scientiarum Fennicæ. Mathematica, 2016, vol. 41, pp. 681–691. http://dx.doi.org/10.5186/aasfm.2016.4142
  8. Wu H. H., Jiang Y. P., Dong X. H. Perturbation of the tangential slit by conformal maps. Journal of Mathematical Analysis and Applications, 2018, vol. 464, iss. 2, pp. 1107–1118. https://doi.org/10.1016/j.jmaa.2018.04.042
  9. Wu H. H. Exact solutions of the Loewner equation. Analysis and Mathematical Physics, 2020, vol. 10, iss. 4, article 59. https://doi.org/10.1007/s13324-020-00403-1