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

✍ Scribed by Georgii S. Litvinchuk (auth.)

• Publisher: Springer Netherlands
• Year: 2000
• Tongue: English
• Leaves: 387
• Series: Mathematics and Its Applications 523
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages I-XVI
Preliminaries....Pages 1-31
Binomial boundary value problems with shift for a piecewise analytic function and for a pair of functions analytic in the same domain....Pages 33-57
Carleman boundary value problems and boundary value problems of Carleman type....Pages 59-134
Solvability theory of the generalized Riemann boundary value problem....Pages 135-175
Solvability theory of singular integral equations with a Carleman shift and complex conjugated boundary values in the degenerated and stable cases....Pages 177-205
Solvability theory of general characteristic singular integral equations with a Carleman fractional linear shift on the unit circle....Pages 207-249
Generalized Hilbert and Carleman boundary value problems for functions analytic in a simply connected domain....Pages 251-301
Boundary value problems with a Carleman shift and complex conjugation for functions analytic in a multiply connected domain....Pages 303-341
On solvability theory for singular integral equations with a non-Carleman shift....Pages 343-354
Back Matter....Pages 355-378

✦ Subjects

Integral Equations; Functions of a Complex Variable; Operator Theory; Potential Theory; Difference and Functional Equations