Boundary value problems with a Carleman shift and complex conjugation for functions analytic in a multiply connected domain

Abstract

In this chapter we extend some results of Chapters 3 and 7 to boundary value problems for a function analytic in a multiply connected domain, bounded by a composite contour Γ consisting of a finite number of simple closed Lyapunov curves. Herein we consider both Noether theory and solvability theory. In the Noether theory, a new quality arises, as a consequence of the fact that the index formula depends not only on the coefficients of the boundary condition but also on the coherence of the domain. In the construction of the Noether theory, certain difficulties arise because there exists no equivalence between the boundary value problems and the corresponding characteristic singular integral equations. There are peculiarities of an essential nature in solvability theory. It turns out that in the case of multiply connected domains the defect numbers of binomial problems depend not only on the index of the problem but also on some non-topological characteristics. In some sense, the transition from a simple connected domain to a multiply connected domain has the same influence in the solvability theory of binomial boundary value problems for functions analytic in a domain as the transition from a binomial problem to a polynomial problem renders in the solvability theory of boundary value problems for functions piecewise analytic in the complex plane. The most complicated situation arises in the solvability theory of polynomial boundary value problems for functions analytic in a multiply connected domain. Meanwhile this theory has not been sufficiently developed.