Logarithmic capacity and Riemann and Hilbert problems for generalized analytic functions
The study of the Dirichlet problem with arbitrary measurable boundary data for harmonic functions in the unit disk is due to the famous Luzin dissertation. Later on, the known monograph of Vekua was devoted to boundary-value problems for generalized analytic functions, but only with Hölder continu...
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| Datum: | 2020 |
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| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Видавничий дім "Академперіодика" НАН України
2020
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| Schriftenreihe: | Доповіді НАН України |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/173094 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Logarithmic capacity and Riemann and Hilbert problems for generalized analytic functions / V.Ya. Gutlyanskiĭ, O.V. Nesmelova, V.I. Ryazanov // Доповіді Національної академії наук України. — 2020. — № 8. — С. 11-18. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | The study of the Dirichlet problem with arbitrary measurable boundary data for harmonic functions in the unit disk
is due to the famous Luzin dissertation. Later on, the known monograph of Vekua was devoted to boundary-value
problems for generalized analytic functions, but only with Hölder continuous boundary data. The present paper contains
theorems on the existence of nonclassical solutions of Riemann and Hilbert problems for generalized analy tic
functions with sources whose boundary data are measurable with respect to the logarithmic capacity. Our ap proach
is based on the geometric interpretation of boundary values in comparison with the classical operator approach in
PDE. On this basis, one can derive the corresponding existence theorems for the Poincaré problem on directional
derivatives to the Poisson equations and, in particular, for the Neumann problem with arbitrary boundary data that
are measurable with respect to the logarithmic capacity. These results can be also applied to semilinear equations of
mathematical physics in anisotropic inhomogeneous media. |
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