Logarithmic capacity and Riemann and Hilbert problems for generalized analytic functions

1Gutlyanskii, VYa.
1Nesmelova, OV
1Ryazanov, VI
2Yefimushkin, AS
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
2Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2020, 8:11-18
https://doi.org/10.15407/dopovidi2020.08.011
Section: Mathematics
Language: English
Abstract: 

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.

Keywords: generalized analytic functions, Hilbert and Riemann boundary-value problems, logarithmic capacity
References: 

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