On one exceptional case of the first basic three-element Carleman-type boundary value problem for bianalytic functions in a circle

N. R. Perelman

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 2, pp. 185-192

Résumé

This article considers a non-degenerate (nonreducible to two-element) three-element problem of Carleman type for bianalytic functions in an exceptional case, that is, when one of the coefficients of the boundary condition vanishes at a finite number of contour points. The unit circle is taken as the contour. For this case, an algorithm for solving the problem is constructed, which consists in reducing the boundary conditions of this problem to a system of four Fredholm type equations of the second kind. For this, the boundary value problem for bianalytic functions is represented as two boundary value problems of Carleman type in the class of analytic functions, then, by introducing auxiliary functions, these problems are represented as scalar Riemann problems in the exceptional case. Using the well-known formulas for solving such problems, we reduce each of the boundary conditions of Carleman-type problems for analytic functions to a pair of well-studied equations of the Fredholm type of the second kind.