Geodesic
Solutions of the Loewner equation with combined driving functions
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 3, pp. 317-325.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{ISU_2021_21_3_a3,
     author = {D. V. Prokhorov and A. M. Zakharov and A. V. Zherdev},
     title = {Solutions of the {Loewner} equation with combined driving functions},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {317--325},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ISU_2021_21_3_a3/}
}
TY  - JOUR
AU  - D. V. Prokhorov
AU  - A. M. Zakharov
AU  - A. V. Zherdev
TI  - Solutions of the Loewner equation with combined driving functions
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2021
SP  - 317
EP  - 325
VL  - 21
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2021_21_3_a3/
LA  - en
ID  - ISU_2021_21_3_a3
ER  - 
%0 Journal Article
%A D. V. Prokhorov
%A A. M. Zakharov
%A A. V. Zherdev
%T Solutions of the Loewner equation with combined driving functions
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2021
%P 317-325
%V 21
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2021_21_3_a3/
%G en
%F ISU_2021_21_3_a3
D. V. Prokhorov; A. M. Zakharov; A. V. Zherdev. Solutions of the Loewner equation with combined driving functions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 3, pp. 317-325. http://geodesic.mathdoc.fr/item/ISU_2021_21_3_a3/

[1] Löwner K., “Untersuchungen über schlichte konforme Abbildungen des Einheitskreses. I”, Mathematische Annalen, 89:1–2 (1923), 103–121 (in Germany) | DOI | Zbl

[2] Lawler G. F., Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, 114, American Mathematical Society, Princeton, 2005, 242 pp. | Zbl

[3] Kager W., Nienhuis B., Kadanoff L. P., “Exact solutions for Loewner evolutions”, Journal of Statistical Physics, 115:3–4 (2004), 805–822 | DOI | Zbl

[4] Lind J. R., “A sharp condition for the Loewner equation to generate slits”, Annales Academiae Scientiarum Fennicae. Mathematica, 30:1 (2005), 143–158 | Zbl

[5] Prokhorov D. V., Zakharov A. M., “Integrability of a partial case of the Loewner equation”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 10:2 (2010), 19–23 (in Russian) | DOI

[6] Prokhorov D., Vasil'ev A., “Singular and tangent slit solutions to the Löwner equation”, Analysis and Mathematical Physics, Trends in Mathematics, eds. Gustafsson B., Vasil'ev A., Birkhäuser, Basel, 2009, 455–463 | DOI | Zbl

[7] Lau K. S., Wu H. H., “On tangential slit solution of the Loewner equation”, Annales Academiae Scientiarum Fennicae. Mathematica, 41 (2016), 681–691 | DOI | Zbl

[8] Wu H. H., Jiang Y. P., Dong X. H., “Perturbation of the tangential slit by conformal maps”, Journal of Mathematical Analysis and Applications, 464:2 (2018), 1107–1118 | DOI | Zbl

[9] Wu H. H., “Exact solutions of the Loewner equation”, Analysis and Mathematical Physics, 10:4 (2020), 59 | DOI