St. Petersburg Mathematical Society

Author(s): V. N. Dubinin; E. G. Prilepkina
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 18 (2006), nomer 3.
Journal: St. Petersburg Math. J. 18 (2007), 373-389.
MSC (2000): Primary 30C70, 30C85
Posted: April 10, 2007
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Abstract: Refinements and generalizations of the classical variational principles of conformal mappings are presented; mainly, they follow from potential theory and symmetrization. Part of the results can be viewed as properties of Robin functions and Robin capacities, and also as distortion theorems for univalent functions in finitely connected domains.

1.: G. M. Goluzin, Geometric theory of functions of a complex variable, 2nd ed., ``Nauka'', Moscow, 1966; English transl. of 2nd ed., Transl. Math. Monogr., vol. 26, Amer. Math. Soc., Providence, RI, 1969. MR 0219714 (36:2793); MR 0247039 (40:308)
2.: M. A. Lavrent'ev and B. V. Shabat, Methods of function theory of a complex variable, ``Nauka'', Moscow, 1973. (Russian) MR 1087298 (91k:30003)
3.: G. V. Kuz'mina, Methods of geometric function theory. I, II, Algebra i Analiz 9 (1997), no. 3, 41-103; no. 5, 1-50; English transl., St. Petersburg Math. J. 9 (1998), no. 3, 455-507; no. 5, 889-930. MR 1466796 (98h:30041); MR 1604397 (99c:30047a)
4.: S. Bergman and M. Schiffer, Kernel functions and elliptic differential equations in mathematical physics, Academic Press, New York, 1953. MR 0054140 (14:876d)
5.: V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1, 3-76; English transl., Russian Math. Surveys 49 (1994), no. 1, 1-79. MR 1307130 (96b:30054)
6.: I. I. Lyashko, I. M. Velikoivanenko, V. I. Lavrik, and G. E. Mistetskii, The method of majorant domains in the theory of filtration, ``Naukova Dumka'', Kiev, 1974. (Russian) MR 0464873 (57:4794)
7.: P. Duren, Robin capacity, Computational Methods and Function Theory (Nicosia, 97) (N. Papamichael, St. Ruscheweyh, and E. B. Saff, eds.), Ser. Approx. Decompos., vol. 11, World Sci. Publishing, River Edge, NJ, 1999, pp. 177-190. MR 1700346 (2001b:30040)
8.: P. Duren, J. Pfaltzgraff, and R. Thurman, Physical interpretation and further properties of Robin capacity, Algebra i Analiz 9 (1997), no. 3, 211-219; English transl., St. Petersburg Math. J. 9 (1998), no. 3, 607-614. MR 1466802 (98k:31002)
9.: B. Dittmar and A. Yu. Solynin, Distortion of the hyperbolic Robin capacity under a conformal mapping, and extremal configurations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 263 (2000), 49-69; English transl., J. Math. Sci. (New York) 110 (2002), no. 6, 3058-3069. MR 1756337 (2001f:30030)
10.: M. D. O'Neill and R. E. Thurman, Extremal domains for Robin capacity, Complex Variables Theory Appl. 41 (2000), 91-109. MR 1756782 (2001a:30028)
11.: S. Nasyrov, Robin capacity and lift of infinitely thin airfoils, Complex Variables Theory Appl. 47 (2002), no. 2, 93-107. MR 1892511 (2003c:30019)
12.: M. Stiemer, A representation formula for the Robin function, Complex Variables Theory Appl. 48 (2003), no. 5, 417-427. MR 1974379 (2003m:30056)
13.: A. Yu. Vasil'ev, Robin's modulus in a Hele-Shaw problem, Complex Variables Theory Appl. 49 (2004), no. 7-9, 663-672. MR 2088056 (2006a:30025)
14.: V. N. Dubinin, Generalized condensers and the asymptotics of their capacities under a degeneration of some plates, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 302 (2003), 38-51; English transl., J. Math. Sci. (New York) 129 (2005), no. 3, 3835-3842. MR 2023031 (2004m:31001)
15.: D. Gaier and W. Hayman, On the computation of modules of long quadrilaterals, Constr. Approx. 7 (1991), 453-467. MR 1124970 (93a:30006)
16.: A. Yu. Solynin, Moduli and extremal metric problems, Algebra i Analiz 11 (1999), no. 1, 3-86; English transl., St. Petersburg Math. J. 11 (2000), no. 1, 1-65. MR 1691080 (2001b:30058)
17.: V. M. Miklyukov, Conformal mapping of a nonregular surface and its applications, Volgograd. Gos. Univ., Volgograd, 2005. (Russian)
18.: V. N. Dubinin, Condenser capacities in geometric function theory, Dal'nevost. Univ., Vladivostok, 2003. (Russian)
19.: V. N. Dubinin and E. G. Prilepkina, Preservation of the generalized reduced module under some geometric transformations of the plane domains, Dal'nevost. Mat. Zh. 6 (2005), no. 1-2, 39-56. (Russian)
20.: P. Duren and M. Schiffer, Robin functions and energy functionals of multiply connected domains, Pacific J. Math. 148 (1991), 251-273. MR 1094490 (92c:31003)
21.: -, Robin functions and distortion of capacity under conformal mapping, Complex Variables Theory Appl. 21 (1993), 189-196. MR 1276575 (95g:30031)
22.: V. N. Dubinin and N. V. Èirikh, Some applications of generalized condensers in the theory of analytic functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 314 (2004), 52-75; English transl., J. Math. Sci. (N. Y.) 133 (2006), no. 6, 1634-1647. MR 2119734 (2005i:30036)
23.: Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren Math. Wiss., vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)
24.: N. A. Lebedev, Some estimates for functions regular and univalent in a circle, Vestnik Leningrad. Univ. Ser. Mat. Fiz. Khim. 1955, vyp. 4, 3-21. (Russian) MR 0074514 (17:599c)
25.: V. N. Dubinin, On the Schwarz inequality on the boundary for functions regular in the disk, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 286 (2002), 74-84; English transl., J. Math. Sci. (New York) 122 (2004), no. 6, 3623-3629. MR 1937369 (2003i:30036)
26.: -, Schwarz's lemma and coefficient estimates for regular functions with free definition domain, Mat. Sb. 196 (2005), no. 11, 53-74; English transl., Sb. Math. 196 (2005), no. 11-12, 1605-1625. MR 2216010
27.: A. Yu. Solynin, Functional inequalities via polarization, Algebra i Analiz 8 (1996), no. 6, 148-185; English transl., St. Petersburg Math. J. 8 (1997), no. 6, 1015-1038. MR 1458141 (98e:30001a)
28.: D. Betsakos, Polarization, conformal invariants, and Brownian motion, Ann. Acad. Sci. Fenn. Math. 23 (1998), 59-82. MR 1601843 (99g:31004)
29.: R. W. Barnard and A. Yu. Solynin, Local variations and minimal area problem for Carathéodory functions, Indiana Univ. Math. J. 53 (2004), 135-167. MR 2048187 (2005a:30028)
30.: R. W. Barnard, C. Richardson, and A. Yu. Solynin, Concentration of area in half-planes, Proc. Amer. Math. Soc. 133 (2005), 2091-2099. MR 2137876 (2006b:30048)
31.: G. Pólya and G. Szegö, Isoperimetric inequalities in mathematical physics, Ann. of Math. Stud., no. 27, Princeton Univ. Press, Princeton, NJ, 1951. MR 0043486 (13:270d)
32.: M. Marcus, Radial averaging of domains, estimates for Dirichlet integrals and applications, J. Anal. Math. 27 (1974), 47-78. MR 0477029 (57:16573)
33.: V. N. Dubinin and M. Vuorinen, On conformal moduli of polygonal quadrilaterals, Preprint no. 417, Univ. Helsinki, Dep. Math. and Statist., 2005.
34.: A. W. Goodman, Univalent functions. Vols. I, II, Mariner Publ. Co., Inc., Tampa, FL, 1983. MR 704183 (85j:30035a); MR 0704184 (85j:30035b)
35.: T. Kubo, Symmetrization and univalent functions in an annulus, J. Math. Soc. Japan 6 (1954), no. 1, 55-67. MR 0062230 (15:948a)
36.: J. A. Jenkins, Univalent functions and conformal mapping, Ergeb. Math. Grenzgeb. (N. F.), no. 18, Springer-Verlag, Berlin, 1958. MR 0096806 (20:3288)
37.: M. Ohtsuka, Dirichlet problem, extremal length and prime ends, New York, 1970.
38.: I. A. Aleksandrov, Parametric continuations in the theory of univalent functions, ``Nauka'', Moscow, 1976. (Russian) MR 0480952 (58:1099)
39.: V. V. Goryainov, Semigroups of conformal mapping, Mat. Sb. (N.S.) 129 (1986), no. 4, 451-472; English transl. in Math. USSR-Sb. 57 (1987), no. 2. MR 842395 (87i:30018)
40.: I. A. Aleksandrov and B. G. Tsvetkov, Functions that confromally map the strip into itself, Sibirsk. Mat. Zh. 21 (1980), no. 1, 4-25; English transl., Siberian Math. J. 21 (1980), no. 1, 1-19. MR 562016 (81e:30032)

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 30C70, 30C85

V. N. Dubinin
Affiliation: Institute of Applied Mathematics, Far-East Branch, Russian Academy of Sciences, Ul. Radio 7, Vladivostok, 690041, Russia
Email: dubinin@iam.dvo.ru

E. G. Prilepkina
Affiliation: Institute of Applied Mathematics, Far-East Branch, Russian Academy of Sciences, Ul. Radio 7, Vladivostok, 690041, Russia
Email: pril@mail.primorye.ru

DOI: 10.1090/S1061-0022-07-00955-7
PII: S 1061-0022(07)00955-7
Keywords: Variational principles, conformal mapping, distortion theorems, univalent functions, Robin function, Robin capacity, the capacity of a condenser, majorant domains, symmetrization, polarization.
Received by editor(s): 22/FEB/2006
Posted: April 10, 2007
Additional Notes: This research was supported by the ``Leading research school'' program (grant no. Sh-9004.2006.1), by RFBR (grant no. 05-01-00099), and by FEB RAS (grant no. 06-III-A-01-013).
Dedicated: Dedicated to the 100th anniversary of Gennadii Mikhailovich Goluzin's birth
Copyright of article: Copyright 2007, American Mathematical Society

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ISSN: 1547-7371(e) ISSN: 1061-0022(p)

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