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One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains

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Abstract

We suggest an approximate method of finding a conformal mapping of an annulus onto an arbitrary bounded doubly connected polygonal domain. The method is based on the parametric Loewner–Komatu method. We consider smooth one parameter families \(\mathcal{F}(z,t)\) of conformal mappings of concentric annuli onto doubly connected polygonal domains \(\mathcal{D}(t)\) which are obtained from a fixed doubly connected polygonal domain \(\mathcal{D}\) by making a finite number of rectilinear slits of variable lengths; meanwhile we do not require the family of domains \(\mathcal{D}(t)\) to be monotonous. The integral representation for the conformal mappings \(\mathcal{F}(z,t)\) has unknown (accessory) parameters. We find a PDE for this family and then deduce a system of PDEs describing dynamics of accessory parameters and conformal modulus of \(\mathcal{D}(t)\) when changing the parameter \(t\). We note that the right-hand sides of equations in the system of ODEs involve functions standing for speeds of movement of the end points of the slits. This allows us to control completely dynamics of the slits and to seek their agreed change, if we have more than one slit. We also give some results of numerical calculations which show the efficiency of the suggested method.

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REFERENCES

  1. I. A. Aleksandrov, Parametric Extensions in the Theory of Univalent Functions (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  2. N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Vol. 79 of AMS Transl. Math. Monographs (Fizmatlit, Moscow, 1970; AMS, Rhode Island, 1990).

  3. N. I. Akhiezer, ‘‘Aerodynamic studies,’’ Tr. Fiz.-Mat. Otdel. AN Ukr. SSR 7, 223–231 (1928).

    Google Scholar 

  4. G. M. Goluzin, ‘‘On the parametric representation of functions univalent in a ring,’’ Mat. Sb. (N.S.) 29 (71), 469–476 (1951).

  5. G. M. Goluzin, ‘‘On conformal mapping of doubly connected domains bounded by rectilinear and circular polygons,’’ in Conformal Mapping of Simply and Multiply Connected Domains, Sci. Studies of Inst. Math. Mech. Leningr. State Univ. (ONTI, Glav. Red. Fiz.-Mat. Lit., Moscow, 1937), pp. 90–97.

  6. V. V. Goryainov, ‘‘Semigroups of analytic functions in analysis and applications,’’ Russ. Math. Surv. 67, 975–1021 (2012).

    Article  MathSciNet  Google Scholar 

  7. V. Ya. Gutlyanskii and A. O. Zaidan, ‘‘On conformal mappings of polygonal domains,’’ Ukr. Mat. Zh. 45, 1484–1494 (1993).

    Article  MathSciNet  Google Scholar 

  8. M. R. Kuvaev, ‘‘On Loewner type equation for multiply connected domains,’’ Dissertation (Tomsk. State Univ., Tomsk, 1952).

  9. M. R. Kuvaev and P. P. Kufarev, ‘‘On Loewner type equation for multiply connected domains,’’ Uch. Zap. Tomsk. Univ., No. 25, 19–34 (1955).

  10. P. P. Kufarev, ‘‘On a method of determining parameters in the Schwarz–Christoffel integral,’’ in The Collection of Works by P. P. Kufarev: To the 100th Anniversary, Ed. by I. A. Aleksandrov (Izd. Nauch.-Tech. Lit., Tomsk, 2009), pp. 86–88 [in Russian].

    Google Scholar 

  11. N. N. Nakipov and S. R. Nasyrov, ‘‘A parametric method of finding accessory parameters for the generalized Schwarz–Christoffel integrals,’’ Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 158, 202–220 (2016).

    Google Scholar 

  12. S. R. Nasyrov, ‘‘Uniformization of one-parametric families of complex tori,’’ Russ. Math. (Iz. VUZ) 61 (8), 36–45 (2017).

  13. L. Yu. Nizamieva, ‘‘Interior and exterior mixed inverse boundary value problems with respect to the parameter \(x\),’’ Dissertation (Kazan State Univ., Kazan, 2011).

  14. U. V. Chistyakov, ‘‘On a method of approximate calculation of function conformall mapping a disk onto domain boupnded by circular arcs and rectilinear segments,’’ Uch. Zap. Tomsk. Univ., No. 14, 143–151 (1960).

  15. M. Bauer, D. Bernard, and K. Kytola, ‘‘Multiple Schramm–Loewner evolutions and statistical mechanics martingales,’’ J. Stat. Phys. 120, 1125–1163 (2005).

    Article  MathSciNet  Google Scholar 

  16. U. J. Bauer, ‘‘Conformal mappings onto simply and multiply connected circular arc polygon domains,’’ Dissertation (Julius-Maximilians-Univ., Würzburg, 2015).

  17. L. Bieberbach, ‘‘Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln,’’ S.-B. Preuss. Akad. Wiss., 940–955 (1916).

  18. Ch. Böhm and W. Lauf, ‘‘A Komatu–Loewner equation for multiple slits,’’ Comput. Methods Funct. Theory 14, 639–663 (2014).

    Article  MathSciNet  Google Scholar 

  19. F. Bracci, M. D. Contreras, S. Diaz-Madrigal, and A. Vasilev, ‘‘Classical and stochastic Loewner–Kufarev equations,’’ in Harmonic and Complex Analysis and its Applications, Ed. by A. Vasil’ev, Part of Trends in Mathematics (Springer Int., Switzerland, 2014), pp. 39–134.

  20. L. de Branges, ‘‘A proof of the Bieberbach conjecture,’’ Acta Math., No. 1–2, 137–151 (1985).

  21. P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, New York, 1971).

    Book  Google Scholar 

  22. D. Crowdy, ‘‘The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains,’’ Proc. R. Soc. A 461(2061), 2653–2678 (2005).

    Article  MathSciNet  Google Scholar 

  23. D. Crowdy, ‘‘Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions,’’ Math. Proc. Cambridge Phil. Soc. 142, 319–339 (2007).

    Article  MathSciNet  Google Scholar 

  24. D. G. Crowdy, A. S. Fokas, and Ch. C. Green, ‘‘Conformal mappings to multiply connected polycircular arc domains,’’ Comput. Methods Funct. Theory 11, 685–706 (2011).

    Article  MathSciNet  Google Scholar 

  25. D. G. Crowdy and J. S. Marshall, ‘‘Conformal mappings between canonical multiply connected domains,’’ Comput. Methods Funct. Theory 6, 59–76 (2006).

    Article  MathSciNet  Google Scholar 

  26. D. Dautova, S. Nasyrov, and M. Vuorinen, ‘‘Conformal module of the exterior of two rectilinear slits,’’ Comput. Methods Funct. Theory 21, 109–130 (2021).

    Article  MathSciNet  Google Scholar 

  27. T. K. DeLillo, A. R. Elcrat, and J. A. Pfaltzgraff, ‘‘Schwarz–Christoffel mapping of the annulus,’’ SIAM Rev. 43, 469–477 (2001).

    Article  MathSciNet  Google Scholar 

  28. T. K. DeLillo and E. H. Kropf, ‘‘Numerical computation of the Schwarz-Christoffel transformation for multiply connected domains,’’ SIAM J. Sci. Comput. 33, 1369–1394 (2011).

    Article  MathSciNet  Google Scholar 

  29. T. K. DeLillo and E. H. Kropf, ‘‘Slit maps and Schwarz–Christoffel maps for multiply connected domains,’’ Electron. Trans. Numer. Anal. 36, 195–223 (2010).

    MathSciNet  Google Scholar 

  30. T. K. Delillo, T. A. Driscoll, A. R. Elcrat, and J. A. Pfaltzgraff, ‘‘Computation of multiply connected Schwarz–Christoffel maps for exterior domains,’’ Comput. Methods Funct. Theory 6, 301–315 (2006).

    Article  MathSciNet  Google Scholar 

  31. T. A. Driscoll and L. N. Trefethen, Schwarz–Christoffel Mapping (Cambridge Univ. Press, Cambridge, 2002).

    Book  Google Scholar 

  32. J. Dubedat, ‘‘Commutation relations for Schramm–Loewner evolutions,’’ Comm. Pure Appl. Math. 60, 1792–1847 (2007).

    Article  MathSciNet  Google Scholar 

  33. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs (AMS, Providence, RI, 1969).

    Book  Google Scholar 

  34. K. Graham, ‘‘On multiple Schramm–Loewner evolutions,’’ J. Stat. Mech., No. 3, P03008 (2007).

  35. H. Hakula, A. Rasila, and M. Vuorinen, ‘‘On moduli of rings and quadrilaterals: Algorithms and experiments,’’ SIAM J. Sci. Comput. 33, 225–242 (2011).

    Article  MathSciNet  Google Scholar 

  36. P. Henrici, Applied and Computational Complex Analysis (Wiley, New York, 1986), Vol. 3.

    Google Scholar 

  37. T. R. Hopkins and D. E. Roberts, ‘‘Kufarev’s method for determining the Schwarz–Christoffel parameters,’’ Numer. Math. 33, 353–365 (1979).

    Article  MathSciNet  Google Scholar 

  38. V. Y. Komatu, ‘‘Darstellungenn der in einem Kreisringe analytischen Funktionen nebst den Anwendungen auf kanforme Abbildung uber Polygonalringgebiete,’’ Jpn. J. Math. 19, 203–215 (1945).

    Article  MathSciNet  Google Scholar 

  39. V. Y. Komatu, ‘‘Untersuchungen über konforme Abbildung von zweifach zusammenhängenden Gebieten,’’ Proc. Phys. Math. Soc. Jpn., III Ser. 25, 1–42 (1943).

    Google Scholar 

  40. S. Lang, Elliptic Functions (Springer, New York, 1987).

    Book  Google Scholar 

  41. S. R. Nasyrov, ‘‘Uniformization of simply-connected ramified coverings of the sphere by rational functions,’’ Lobachevskii J. Math. 39, 252–258 (2018).

    Article  MathSciNet  Google Scholar 

  42. A. Posadskii and S. Nasyrov, ‘‘One-parameter families of conformal mappings of the half-plane onto polygonal domains with several slits,’’ Lobachevskii J. Math. 44, 1439–1454 (2023).

    Article  MathSciNet  Google Scholar 

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Funding

This research is supported by the Russian Science Foundation, grant no. 23-11-00066.

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  1. Kazan Federal University, 420008, Kazan, Tatarstan, Russia

    A. Dyutin & S. Nasyrov

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  1. A. Dyutin
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Dyutin, A., Nasyrov, S. One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains. Lobachevskii J Math 45, 390–411 (2024). https://doi.org/10.1134/S1995080224010128

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