Boundary Value Problems of Analytic and Harmonic Functions in a Domain with Piecewise Smooth Boundary in the Frame of Variable Exponent Lebesgue Spaces

2011 ◽  
pp. 17-39
Author(s):  
Vakhtang Kokilashvili
Keyword(s):  
Boundary Value Problems ◽  
Harmonic Functions ◽  
Variable Exponent ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Piecewise Smooth ◽  
Piecewise Smooth Boundary ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces

The Riemann boundary value problem in the class of Cauchy type integrals with densities of grand variable exponent Lebesgue spaces

2016 ◽  
Vol 23 (4) ◽  
pp. 551-558 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi ◽  
Vakhtang Paatashvili
Keyword(s):  
Boundary Value Problem ◽  
Variable Exponent ◽  
Boundary Value ◽  
Function Spaces ◽  
Cauchy Type ◽  
Riemann Boundary Value Problem ◽  
Lebesgue Spaces ◽  
Standard Type ◽  
Riemann Boundary ◽  
Variable Exponent Lebesgue Spaces

AbstractThe present paper deals with the Riemann boundary value problem for analytic functions in the framework of the new function spaces introduced by the first two authors, the so-called grand variable exponent Lebesgue spaces which unify two non-standard type function spaces: variable exponent Lebesgue spaces and grand Lebesgue spaces.

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.

Sign in / Sign up

Export Citation Format

Share Document