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

V. G. Lysov ; D. N. Tulyakov

Informatics and Automation, Complex analysis and its applications, Tome 298 (2017), pp. 185-215.

Résumé

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.

Détail

Keywords: vector equilibrium problem, Angelesco interaction matrix, logarithmic potential, extremal measure, algebraic functions, uniformization of an algebraic curve.

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