Cauchy Formula, Integral of Cauchy Type and Hilbert Problem for Bianalytic Functions

1999 ◽  
pp. 211-218
Author(s):  
Zhao Zhen
Keyword(s):  
Hilbert Problem ◽  
Cauchy Formula ◽  
Cauchy Type ◽  
Bianalytic Functions

The Riemann–Hilbert Problem in a Domain with Piecewise Smooth Boundaries in Weight Classes of Cauchy Type Integrals with a Density from Variable Exponent Lebesgue Spaces

2009 ◽  
Vol 16 (4) ◽  
pp. 737-755 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Vakhtang Paatashvili
Keyword(s):  
Variable Exponent ◽  
Hilbert Problem ◽  
Domain Boundary ◽  
Smooth Curve ◽  
Piecewise Smooth ◽  
Cauchy Type ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Cauchy Type Integrals ◽  
Riemann Hilbert Problem

Abstract The Riemann–Hilbert problem for an analytic function is solved in weighted classes of Cauchy type integrals in a simply connected domain not containing 𝑧 = ∞ and having a density from variable exponent Lebesgue spaces. It is assumed that the domain boundary is a piecewise smooth curve. The solvability conditions are established and solutions are constructed. The solution is found to essentially depend on the coefficients from the boundary condition, the weight, space exponent values at the angular points of the boundary curve and also on the angle values. The non-Fredholmian case is investigated. An application of the obtained results to the Neumann problem is given.

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