×

Strict superharmonicity of Mityuk’s function for countably connected domains of simple structure. (English) Zbl 1372.31006

Summary: Strict superharmonicity of generalized reduced module as a function of a point (we call it Mityuk’s function) is established for the subclass of countably connected domains with unique limit point boundary component. The function just mentioned was first studied in detail by I.P. Mityuk and plays now an important role in the research of the exterior inverse boundary value problems of the theory of analytic functions in the multiply connected domains. At the heart of such a research one can see the fact that the critical points of Mityuk’s function are only maxima, saddles or semisaddles of corresponding surface. This fact is followed from the above strict superharmonicity.

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. T. Nuzhin, “On some inverse boundary value problems and their applications to the definition of sectional shape of twisted rods,” Uch. Zap. Kazan. Univ. 109 (1), 97-120 (1949).
[2] F. D. Gakhov, “On the inverse boundary problems,” Dokl. Akad. Nauk SSSR 86 (4), 649-652 (1952). · Zbl 0053.23901
[3] L. A. Aksent’ev, M. I. Kinder, and S. B. Sagitova, “Solvability of the exterior inverse boundary value problem in the case of multiply connected domain,” Tr. Semin. Kraev. Zadacham 20, 22-34 (1983). · Zbl 0584.30039
[4] M. I. Kinder, “The number of solutions of F. D. Gakhov’s equation in the case of a multiply connected domain,” Izv. Vyssh. Uchebn. Zaved., Mat. 28 (8), 69-72 (1984). · Zbl 0557.30035
[5] M. I. Kinder, “Investigation of F. D.Gakhov’s equation in the case of multiply connected domains,” Tr. Semin. Kraev. Zadacham 22, 104-116 (1985). · Zbl 0622.30040
[6] L. A. Aksent’ev, A. M. Elizarov, and M. I. Kinder, “Inverse boundary value problems for multiply connected domains on Riemann surfaces of genus zero,” Tr. Semin. Kraev. Zadacham 21, 19-32 (1984); Tr. Semin. Kraev. Zadacham 22, 16-29 (1985); Tr. Semin. Kraev. Zadacham 23, 25-36 (1987). · Zbl 0566.30030
[7] Aksent’ev, L. A.; Elizarov, A. M.; Kinder, M. I., Continuation of F. D. Gakhov’s work in inverse boundary value problems, 139-142 (1981) · Zbl 0624.30044
[8] Kinder, M. I., Exterior inverse boundary value problem in multiply connected regions and on Riemann surfaces (1984)
[9] Kazantsev, A. V., Extremal properties of the inner radius and their applications (1990)
[10] L. A. Aksent’ev, A. V. Kazantsev, M. I. Kinder, and A. V. Kiselev, “Classes of uniqueness of an exterior inverse boundary value problem,” Tr. Semin. Kraev. Zadacham 24, 39-62 (1990).
[11] Aksent’ev, L. A.; Kazantsev, A. V.; Kinder, M. I., On classes of uniqueness of an exterior inverse boundary value problem, 61 (1991)
[12] A. V. Kiselev, “Geometric properties of solutions of the exterior inverse boundary value problem,” Izv. Vyssh. Uchebn. Zaved., Mat. 36 (7), 20-25 (1992). · Zbl 0803.30029
[13] A. V. Kazantsev and M. I. Kinder, “Solvability of the exterior inverse boundary value problem in the case of multiply connected domains,” in Proceedings of the 11th International Conference on Algebra and Analysis, Dedicated to the 100th Anniversary of the Birth of N. G. Chebotarev, Kazan, June 5-11, 1994, Vol. 2, pp. 65-66.
[14] I. P. Mityuk, “A generalized reduced module and some of its applications,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 110-119 (1964).
[15] A. V. Kazantsev, “Gakhov set in the Hornich space under the Bloch restriction on pre-Schwarzians,” Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 155 (2), 65-82 (2013). · Zbl 1342.30010
[16] Kazantsev, A. V., Zmorovich’s method in the problem of investigation of Mityuk’s functional, 79-80 (2014)
[17] H. Grötzsch, “Über konforme Abbildung unendlich vielfach zusammenhängender schlichter Bereiche mit endlich vielen Häufungsrandkomponenten,” Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 81, 51-86 (1929). · JFM 55.0792.03
[18] H. Grötzsch, “Das Kreisbogenschlitztheorem der konformen Abbildung schlichter Bereiche,” Ber. Sächs. Akad.Wiss. Leipzig, Math.-Phys. Kl. 83, 238-253 (1931). · JFM 57.0401.02
[19] E. Reich and S. E. Warschawski, “On canonical conformal maps of regions of arbitrary connectivity,” Pacif. J. Math. 10 (3), 965-989 (1960). · Zbl 0091.25503 · doi:10.2140/pjm.1960.10.965
[20] S. Bergman and M. Schiffer, “Kernel functions and conformalmapping,” Compos. Math. 8, 205-249 (1951). · Zbl 0043.08403
[21] G. M. Golusin, Geometric Theory of Functions of a Complex Variable, Vol. 26 of Transl. Math. Monographs (Am.Math. Soc., Providence, 1969).
[22] M. Schiffer, “Hadamard’s formula and variation of domain-functions,” Am. J. Math. 68 (4), 417-448 (1946). · Zbl 0060.23706 · doi:10.2307/2371824
[23] W. Hayman and P. Kennedy, Subharmonic Functions, (Academic, London, 1976), Vol.1. · Zbl 0419.31001
[24] S. Stoilov, The Theory of Functions of a Complex Variable (Fizmatgiz, Moscow, 1962), Vol. 2. · Zbl 0053.23901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.