On boundary-value problems for generalized analytic and harmonic 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, 12:11-18
https://doi.org/10.15407/dopovidi2020.12.011
Section: Mathematics
Language: English
Abstract: 

The present paper is a natural continuation of our last articles on the Riemann, Hilbert, Dirichlet, Poincaré, and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic functions and the so-called A-harmonic functions with arbitrary boundary data that are measurable with respect to the logarithmic capacity. Here, we extend the corresponding results to generalized analytic functions h : D→C with sources   g : ∂z-h = g ∈ Lp , p > 2, and to generalized harmonic functions U with sources G : ΔU =G ∈Lp , p > 2. Our approach is based on the geometric (functional-theoretic) interpretation of boundary values in comparison with the classical operator approach in PDE. Here, we will establish the corresponding existence theorems for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations ΔU =G with arbitrary boundary data that are measurable with respect to the logarithmic capacity. A few mixed boundary-value problems are considered as well. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

Keywords: generalized analytic functions, generalized harmonic functions, logarithmic capacity and potential, Poincaré and Neumann boundary-value problems
References: 

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