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Zap. Nauchn. Sem. POMI, 2003, Volume 302, Pages 52–67 (Mi znsl918)  

This article is cited in 9 scientific papers (total in 9 papers)

The method of extremal metric in extremal decomposition problems with free parameters

G. V. Kuz'mina

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: Let $a_1,…,a_n$ be a system of distinct points on the $z$-sphere $\overline{\mathbb C}$, and let $\mathcal D$ be a system of all non-overlapping simply-connected domains $D_1,…,D_n$ on $\overline{\mathbb C}$ such that $a_k\in D_k$, $k=1,…, n$. Let $M(D_k, a_k)$ be the reduced module of the domain Dk with respect to the point $a_k\in D_k$. In the present paper, we solve some problems concerning the maximum of weighted sums of the reduced modules $M(D_k, a_k)$ in certain families of systems of domains $\{D_k\}$ described above, where the systems of points $\{a_k\}$ satisfy prescribed symmetry conditions. In each case, the proof is based on an explicit construction of an admissible metric of the module problem, which is equivalent to the extremal problem under consideration, from known extremal metrics of simpler module problems.

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English version:
Journal of Mathematical Sciences (New York), 2005, 129:3, 3843–3851

Bibliographic databases:

UDC: 517.54
Received: 17.11.2003

Citation: G. V. Kuz'mina, “The method of extremal metric in extremal decomposition problems with free parameters”, Analytical theory of numbers and theory of functions. Part 19, Zap. Nauchn. Sem. POMI, 302, POMI, St. Petersburg, 2003, 52–67; J. Math. Sci. (N. Y.), 129:3 (2005), 3843–3851

Citation in format AMSBIB
\by G.~V.~Kuz'mina
\paper The method of extremal metric in extremal decomposition problems with free parameters
\inbook Analytical theory of numbers and theory of functions. Part~19
\serial Zap. Nauchn. Sem. POMI
\yr 2003
\vol 302
\pages 52--67
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2005
\vol 129
\issue 3
\pages 3843--3851

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    This publication is cited in the following articles:
    1. G. V. Kuz'mina, “Problems of extremal decomposition of the Riemann sphere. III”, J. Math. Sci. (N. Y.), 133:6 (2006), 1676–1685  mathnet  crossref  mathscinet  zmath  elib  elib
    2. V. N. Dubinin, “Capacities of condensers, generalizations of Grötzsch Lemmas, and symmetrization”, J. Math. Sci. (N. Y.), 143:3 (2007), 3053–3068  mathnet  crossref  mathscinet  zmath  elib  elib
    3. G. V. Kuz'mina, “An extremal metric approach to some problems on extremal decomposition”, J. Math. Sci. (N. Y.), 143:3 (2007), 3124–3136  mathnet  crossref  mathscinet  zmath  elib  elib
    4. G. V. Kuz'mina, “Gennadii Mikhailovich Goluzin and geometric function theory”, St. Petersburg Math. J., 18:3 (2007), 347–372  mathnet  crossref  mathscinet  zmath  elib
    5. A. K. Bakhtin, “Inequalities for Inner Radii of Disjoint Domains and Open Sets”, Proc. Steklov Inst. Math., 261 (2008), 34–43  mathnet  crossref  mathscinet  zmath  isi  elib
    6. J. Math. Sci. (N. Y.), 222:5 (2017), 645–689  mathnet  crossref  mathscinet
    7. S. I. Kalmykov, E. G. Prilepkina, “On the $p$-harmonic Robin radius in the Euclidean space”, J. Math. Sci. (N. Y.), 225:6 (2017), 969–979  mathnet  crossref  mathscinet
    8. G. V. Kuz'mina, “On an extremal metric approach to extremal decomposition problems”, J. Math. Sci. (N. Y.), 225:6 (2017), 980–990  mathnet  crossref  mathscinet
    9. Dubinin V.N., “Some Unsolved Problems About Condenser Capacities on the Plane”, Complex Analysis and Dynamical Systems: New Trends and Open Problems, Trends in Mathematics, ed. Agranovsky M. Golberg A. Jacobzon F. Shoikhet D. Zalcman L., Birkhauser Verlag Ag, 2018, 81–92  crossref  mathscinet  isi  scopus
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