Geodesic
Asymptotic ratio of harmonic measures of slit sides
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 2, pp. 160-167 
D. V. Prokhorov; D. V. Ukrainskii. Asymptotic ratio of harmonic measures of slit sides. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 2, pp. 160-167. http://geodesic.mathdoc.fr/item/ISU_2015_15_2_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Löwner K., “Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I”, Math. Ann., 89:1–2 (1923), 103–121 | DOI | MR | Zbl

[2] Lind J., Marshall D. E., Rohde S., “Collisions and spirals of Loewner traces”, Duke Math. J., 154:3 (2010), 527–573 | DOI | MR | Zbl

[3] Hayman W. K., Kennedy P. B., Subharmonic Functions, v. 1, Academic Press, London–New York, 1976 | MR | Zbl

[4] Earle C. J., Epstein A. L., “Quasiconformal variation of slit domains”, Proc. Amer. Math. Soc., 129:11 (2001), 3363–3372 | DOI | MR | Zbl

[5] Prokhorov D., Zakharov A., “Harmonic measures of sides of a slit perpendicular to the domain boundary”, J. Math. Anal. Appl., 394:2 (2012), 738–743 | DOI | MR | Zbl

[6] Kager W., Nienhuis B., Kadanoff L. P., “Exact solutions for Loewner evolutions”, J. Statist. Phys., 115:3–4 (2004), 805–822 | DOI | MR | Zbl

[7] Radó T., “Sur la représentations conforme de domaines variables”, Acta Sci. Math. (Szeged), 1:3 (1922–1923), 180–186

[8] Goluzin G. M., Geometric Theory of Functions of Complex Variables, Nauka, M., 1966 | MR

[9] Markushevich A. I., “Sur la représentations conforme des domaines à frontières variables”, Rec. Math. N.S., 1(43):6 (1936), 863–886

[10] Prokhorov D., Vasil'ev A., “Singular and tangent slit solutions to the Löwner equation”, Analysis and Mathematical Physics, eds. D. Gustafsson, A. Vasil'ev, Birkhauser, Berlin, 2009, 455–463 | DOI | MR | Zbl

[11] Ivanov G., Prokhorov D., Vasil'ev A., “Non-slit and singular solutions to the Löwner equation”, Bull. Sci. Mathem., 136:3 (2012), 328–341 | DOI | MR | Zbl