Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift

The first formulations of linear boundary value problems for analytic functions were due to Riemann (1857). In particular, such problems exhibit as boundary conditions relations among values of the unknown analytic functions which have to be evaluated at different points of the boundary. Singular integral equations with a shift are connected with such boundary value problems in a natural way. Subsequent to Riemann's work, D. Hilbert (1905), C. Haseman (1907) and T. Carleman (1932) also considered problems of this type. About 50 years ago, Soviet mathematicians began a systematic study of these topics. The first works were carried out in Tbilisi by D. Kveselava (1946-1948). Afterwards, this theory developed further in Tbilisi as well as in other Soviet scientific centers (Rostov on Don, Ka zan, Minsk, Odessa, Kishinev, Dushanbe, Novosibirsk, Baku and others). Beginning in the 1960s, some works on this subject appeared systematically in other countries, e. g. , China, Poland, Germany, Vietnam and Korea. In the last decade the geography of investigations on singular integral operators with shift expanded significantly to include such countries as the USA, Portugal and Mexico. It is no longer easy to enumerate the names of the all mathematicians who made contributions to this theory. Beginning in 1957, the author also took part in these developments. Up to the present, more than 600 publications on these topics have appeared.

Klappentext

This book is devoted to the solvability theory of characteristic singular integral equations and corresponding boundary value problems for analytic functions with a Carleman and non-Carleman shift. The defect numbers are computed and the bases for the defect subspaces are constructed. Applications to mechanics, physics, and geometry of surfaces are discussed. The second part of the book also contains an extensive survey of the literature on closely related topics. While the first part of the book is also accessible to engineers and undergraduate students in mathematics, the second part is aimed at specialists in the field.

Inhalt
1 Preliminaries.- 1 On Noether operators.- 2 Shift function.- 3 Operator of singular integration, shift operator, operator of complex conjugation and certain combinations of them.- 4 Singular integral operators with Cauchy kernel.- 5 Riemann boundary value problems.- 6 The Noether theory for singular integral operators with a Carleman shift and complex conjugation.- 2 Binomial boundary value problems with shift for a piecewise analytic function and for a pair of functions analytic in the same domain.- 7 The Hasemann boundary value problem.- 8 Boundary value problems which can be reduced to a Hasemann boundary value problem.- 9 References and a survey of closely related results.- 3 Carleman boundary value problems and boundary value problems of Carleman type.- 10 Carleman boundary value problems.- 11 Boundary value problems of Carleman type.- 12 Geometric interpretation of the conformai gluing method.- 13 References and a survey of closely related results.- 4 Solvability theory of the generalized Riemann boundary value problem.- 14 Solvability theory of the generalized Riemann boundary value problem in the stable and degenerated cases.- 15 References and a survey of similar or related results.- Solvability theory of singular integral equations with a Carleman shift and complex conjugated boundary values in the degenerated and stable cases.- 16 Characteristic singular integral equation with a Carleman shift in the degenerated cases.- 17 Characteristic singular integral equation with a Carleman shift and complex conjugation in the degenerated cases.- 18 Solvability theory of a singular integral equation with a Carleman shift and complex conjugation in the stable cases.- 19 References and a survey of similar or related results.- 6 Solvability theory of general characteristic singular integral equations with a Carleman fractional linear shift on the unit circle.- 20 Characteristic singular integral equation with a direct Carleman fractional linear shift.- 21 Characteristic singular integral equation with an inverse Carleman fractional linear shift.- 22 References and survey of closed and related results.- 7 Generalized Hilbert and Carleman boundary value problems for functions analytic in a simply connected domain.- 23 Noether theory of a generalized Hilbert boundary value problem.- 24 Solvability theory of generalized Hilbert boundary value problems.- 25 Noetherity theory of a generalized Carleman boundary value problem.- 26 Solvability theory of a generalized Carleman boundary value problem.- 27 References and a survey of similar or related results.- 8 Boundary value problems with a Carleman shift and complex conjugation for functions analytic in a multiply connected domain.- 28 Integral representations of functions analytic in a multiply connected domain.- 29 The Noether theory of a generalized Carleman boundary value problem with a direct shift ? = ?+(t) in a multiply connected domain.- 30 The solvability theory of a binomial boundary value problem of Carleman type in a multiply connected domain.- 31 The solvability theory of a Carleman boundary value problem in a multiply connected domain.- 32 The Noether theory of a generalized Carleman boundary value problem with an inverse shift ? = ?_ for a multiply connected domain.- 33 References and a survey of similar or related results.- 9 On solvability theory for singular integral equations with a non-Carleman shift.- 34 Auxiliary Lemmas.- 35 Estimate for the dimension of the kernel of a singular integral operator with a non-Carleman shift having a finite number of fixed points.- 36Approximate solution of a non-homogeneous singular integral equation with a nonCarleman shift.- 37 Singular integral equations with non-Carleman shift as a natural model for problems of synthesis of signals for linear systems with non-stationary parameters.- References.