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: 
31.08.2021
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English
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Article type: 
Article
UDC: 
517.54

Solutions of the Loewner equation with combined driving functions

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

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.

Acknowledgments: 
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).
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Received: 
22.03.2020
Accepted: 
27.04.2021
Published: 
31.08.2021