Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem
I. Vekua’s integral representations of holomorphic functions, whose m-th derivative (m≥0) is Hӧlder-continuous in a closed domain bounded by the Lyapunov curve, are generalized for analytic functions whose m-th derivative is representable by a Cauchy type integral whose density is from variable expo...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2011/642519 |
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| author | Vakhtang Kokilashvili Vakhtang Paatashvili |
| author_facet | Vakhtang Kokilashvili Vakhtang Paatashvili |
| author_sort | Vakhtang Kokilashvili |
| collection | DOAJ |
| description | I. Vekua’s integral representations of holomorphic functions, whose m-th derivative (m≥0) is Hӧlder-continuous in a closed domain bounded by the Lyapunov curve, are generalized for analytic functions whose m-th derivative is representable by a Cauchy type integral whose density is from variable exponent Lebesgue space Lp(⋅)(Γ;ω) with power weight. An integration curve is taken from a wide class of piecewise-smooth curves admitting cusp points for certain p and ω. This makes it possible to obtain analogues of I. Vekua’s results to the Riemann–Hilbert–Poincaré problem under new general assumptions about the desired and the given elements of the problem. It is established that the solvability essentially depends on the geometry of a boundary, a weight function ω(t) and a function p(t). |
| format | Article |
| id | doaj-art-87c1c680eb8a47a78c0725fb6fc8eafd |
| institution | Kabale University |
| issn | 0972-6802 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-87c1c680eb8a47a78c0725fb6fc8eafd2025-02-03T01:21:41ZengWileyJournal of Function Spaces and Applications0972-68022011-01-019321724410.1155/2011/642519Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problemVakhtang Kokilashvili0Vakhtang Paatashvili1A. Razmadze Mathematical Institute, 1, M. Aleksidze St., Tbilisi 0193, GeorgiaA. Razmadze Mathematical Institute, 1, M. Aleksidze St., Tbilisi 0193, GeorgiaI. Vekua’s integral representations of holomorphic functions, whose m-th derivative (m≥0) is Hӧlder-continuous in a closed domain bounded by the Lyapunov curve, are generalized for analytic functions whose m-th derivative is representable by a Cauchy type integral whose density is from variable exponent Lebesgue space Lp(⋅)(Γ;ω) with power weight. An integration curve is taken from a wide class of piecewise-smooth curves admitting cusp points for certain p and ω. This makes it possible to obtain analogues of I. Vekua’s results to the Riemann–Hilbert–Poincaré problem under new general assumptions about the desired and the given elements of the problem. It is established that the solvability essentially depends on the geometry of a boundary, a weight function ω(t) and a function p(t).http://dx.doi.org/10.1155/2011/642519 |
| spellingShingle | Vakhtang Kokilashvili Vakhtang Paatashvili Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem Journal of Function Spaces and Applications |
| title | Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem |
| title_full | Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem |
| title_fullStr | Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem |
| title_full_unstemmed | Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem |
| title_short | Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem |
| title_sort | generalization of i vekua s integral representations of holomorphic functions and their application to the riemann hilbert poincare problem |
| url | http://dx.doi.org/10.1155/2011/642519 |
| work_keys_str_mv | AT vakhtangkokilashvili generalizationofivekuasintegralrepresentationsofholomorphicfunctionsandtheirapplicationtotheriemannhilbertpoincareproblem AT vakhtangpaatashvili generalizationofivekuasintegralrepresentationsofholomorphicfunctionsandtheirapplicationtotheriemannhilbertpoincareproblem |