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 22, 33, 44, and 66 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.
Keywords:
vector equilibrium problem, Angelesco interaction matrix, logarithmic potential, extremal measure, algebraic functions, uniformization of an algebraic curve.
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
\Bibitem{LysTul17}
\by V.~G.~Lysov, D.~N.~Tulyakov
\paper On a Vector Potential-Theory Equilibrium Problem with the Angelesco Matrix
\inbook Complex analysis and its applications
\bookinfo 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
\serial Trudy Mat. Inst. Steklova
\yr 2017
\vol 298
\pages 185--215
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3829}
\crossref{https://doi.org/10.1134/S037196851703013X}
\elib{https://elibrary.ru/item.asp?id=30727072}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2017
\vol 298
\pages 170--200
\crossref{https://doi.org/10.1134/S008154381706013X}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000416139300013}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85036640188}
Linking options:
https://www.mathnet.ru/eng/tm3829
https://doi.org/10.1134/S037196851703013X
https://www.mathnet.ru/eng/tm/v298/p185
This publication is cited in the following 10 articles:
V. G. Lysov, “Distribution of zeros of polynomials of multiple discrete orthogonality in the Angelesco case”, Russian Math. Surveys, 79:6 (2024), 1101–1103
V. G. Lysov, “Mnogourovnevye interpolyatsii dlya obobschennoi sistemy Nikishina na grafe-dereve”, Tr. MMO, 83, no. 2, MTsNMO, M., 2022, 345–361
V. G. Lysov, “Multilevel interpolations for the generalized Nikishin system on a tree graph”, Trans. Moscow Math. Soc., –
A. I. Aptekarev, R. Kozhan, “Differential equations for the recurrence coefficients limits for multiple orthogonal polynomials from a nevai class”, J. Approx. Theory, 255 (2020), 105409
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
A. I. Bogolyubskii, V. G. Lysov, “Constructive solution of one vector equilibrium problem”, Dokl. Math., 101:2 (2020), 90–92
M. A. Lapik, “Integral formulas for recovering extremal measures for vector constrained energy problems”, Lobachevskii J. Math., 40:9, SI (2019), 1355–1362
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
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
V. G. Lysov, “Ob approksimatsiyakh Ermita–Pade dlya proizvedeniya dvukh logarifmov”, Preprinty IPM im. M. V. Keldysha, 2017, 141, 24 pp.
Citation in format AMSBIB
Contact us: