On a vector potential-theory equilibrium problem with the Angelesco matrix
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.
Unable to display preview. Download preview PDF.
- 5.A. I. Aptekarev and V. A. Kalyagin, “Asymptotic behavior of an nth degree root of polynomials of simultaneous orthogonality, and algebraic functions,” Preprint No. 60 (Keldysh Inst. Appl. Math., Moscow, 1986).Google Scholar
- 8.A. I. Aptekarev, V. G. Lysov, and D. N. Tulyakov, “Three-sheeted Riemann surfaces of genus 0 with fixed projections of the branch points,” Preprint No. 13 (Keldysh Inst. Appl. Math., Moscow, 2007).Google Scholar
- 22.M. A. Lapik, “The extremal functional for vector extremal logarithmic potential problems with external field and Angelesco matrix of interaction,” Preprint No. 83 (Keldysh Inst. Appl. Math., Moscow, 2015).Google Scholar