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 Trudy Mat. Inst. Steklova, 2017, Volume 298, Pages 185–215 (Mi tm3829)

On a Vector Potential-Theory Equilibrium Problem with the Angelesco Matrix

V. G. Lysovab, D. N. Tulyakovb

a Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141701 Russia
b Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia

Abstract: Vector logarithmic-potential equilibrium problems with the Angelesco interaction matrix are considered. Solutions to two-dimensional problems in the class of measures and in the class of charges are studied. It is proved that in the case of two arbitrary real intervals, a solution to the problem in the class of charges exists and is unique. The Cauchy transforms of the components of the equilibrium charge are algebraic functions whose degree can take values $2$, $3$, $4$, and $6$ depending on the arrangement of the intervals. A constructive method for finding the vector equilibrium charge in an explicit form is presented, which is based on the uniformization of an algebraic curve. An explicit form of the vector equilibrium measure is found under some constraints on the arrangement of the intervals.

 Funding Agency Grant Number Russian Science Foundation 14-21-00025 This work is supported by the Russian Science Foundation under grant 14-21-00025.

DOI: https://doi.org/10.1134/S037196851703013X

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English version:
Proceedings of the Steklov Institute of Mathematics, 2017, 298, 170–200

Bibliographic databases:

UDC: 517.53

Citation: V. G. Lysov, D. N. Tulyakov, “On a Vector Potential-Theory Equilibrium Problem with the Angelesco Matrix”, Complex analysis and its applications, Collected papers. On the occasion of the centenary of the birth of Boris Vladimirovich Shabat, 85th anniversary of the birth of Anatoliy Georgievich Vitushkin, and 85th anniversary of the birth of Andrei Aleksandrovich Gonchar, Trudy Mat. Inst. Steklova, 298, MAIK Nauka/Interperiodica, Moscow, 2017, 185–215; Proc. Steklov Inst. Math., 298 (2017), 170–200

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3829
• https://doi.org/10.1134/S037196851703013X
• http://mi.mathnet.ru/eng/tm/v298/p185

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This publication is cited in the following articles:
1. V. G. Lysov, “Ob approksimatsiyakh Ermita–Pade dlya proizvedeniya dvukh logarifmov”, Preprinty IPM im. M. V. Keldysha, 2017, 141, 24 pp.
2. V. G. Lysov, D. N. Tulyakov, “On the supports of vector equilibrium measures in the Angelesco problem with nested intervals”, Proc. Steklov Inst. Math., 301 (2018), 180–196
3. M. A. Lapik, “Integral formulas for recovering extremal measures for vector constrained energy problems”, Lobachevskii J. Math., 40:9, SI (2019), 1355–1362
4. A. I. Aptekarev, M. A. Lapik, V. G. Lysov, “Direct and inverse problems for vector logarithmic potentials with external fields”, Anal. Math. Phys., 9:3 (2019), 919–935
5. I A. Aptekarev , R. Kozhan, “Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a nevai class”, J. Approx. Theory, 255 (2020), 105409
6. P. D. Dragnev, B. Fuglede, D. P. Hardin, E. B. Saff, N. Zorii, “Constrained minimum Riesz energy problems for a condenser with intersecting plates”, J. Anal. Math., 140:1 (2020), 117–159
7. I A. Bogolyubskii , V. G. Lysov, “Constructive solution of one vector equilibrium problem”, Dokl. Math., 101:2 (2020), 90–92
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