Givi Khuskivadze, Vakhtang Kokilashvili, and Vakhtang Paatashvili
abstract:
In the present monograph, on the basis of the Cauchy type
integral theory discontinuous boundary value problems for analytic
functions with oscillating conjugacy coefficients and boundaries
are studied. For analytic functions from Smirnov classes, the complete
solution of the Riemann-Hilbert problem in domains with arbitrary piecewise
smooth boundaries is presented. On the basis of the investigation of the linear
conjugation problem, the boundary properties of derivatives of functions
conformally mapping the unit circle onto a domain admitting a boundary with
tangential oscillation less than $\pi$, are studied. From new representations
derived for the above-mentioned functions, some well-known results of
Lindel\"{o}f, Kellog and Warschawski as well as their generalizations are
obtained; the Dirichlet and Neumann problems for harmonic functions from
Smirnov classes are investigated; the picture of solvability is described
completely; the non-Fredholm case is exposed; an influence
of geometric properties of boundaries on the solvability
is revealed; in all cases of
solvability explicit formulas for the solutions in terms of Cauchy type
integrals and conformally mapping functions are given.
Mathematics Subject Classification: 30E20, 30E25, 35Q15, 45E05, 42B20, 46E40, 30C35, 26A39, 35C05, 35J25.
Key words and phrases: Analytic and harmonic functions, discontinuous boundary value problem of linear conjugation, oscillating conjugacy coefficients, Riemann-Hilbert problem, factorization of functions, piecewise smooth boundary, angular points, cusps, Carleson curves, Cauchy type integrals, singular integrals, the Muckenhoupt $A_p$ condition, two-weight inequality, $A$- and $B$-integrals, $\wt{L}$-integral, Cauchy type $\wt{L}$ -integral, conformal mapping, Hardy and Smirnov classes, Dirichlet and Neumann problems.