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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
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Vakhtang Kokilashvili
Published/Copyright:
March 11, 2010
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.
Key words and phrases:: Cauchy type integrals; the Riemann–Hilbert problem; weighted Lebesgue space with variable exponent; Log-Hölder condition; piecewise smooth boundary; non Fredholmian case
Received: 2008-05-06
Published Online: 2010-03-11
Published in Print: 2009-December
© Heldermann Verlag
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Articles in the same Issue
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- Nonlinear Three-Point Boundary Value Problems for a Class of Impulsive Functional Differential Equations
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- Stability by Fixed Point Theory for Nonlinear Delay Difference Equations
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- On Nonmeasurable Functions of Two Variables and Iterated Integrals
- Bounded and Vanishing at Infinity Solutions of Nonlinear Differential Systems
- On Functional Equations Connected with Quadrature Rules
- 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
- A Counterexample on Embedding of Spaces
- On Solutions of a Singular Viscoelastic Equation with an Integral Condition
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