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Logarithmic Potential and Generalized Analytic Functions

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Abstract

The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Hölder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b\( \overline{h} \) = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ.

The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2.

This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine

    V. Gutlyanskiĭ, O. Nesmelova, V. Ryazanov & A. Yefimushkin

  2. Bogdan Khmelnytsky National University of Cherkasy, Cherkasy, Ukraine

    V. Ryazanov

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  1. V. Gutlyanskiĭ
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  2. O. Nesmelova
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  3. V. Ryazanov
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  4. A. Yefimushkin
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Correspondence to V. Gutlyanskiĭ.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 1, pp. 12–36, January–March, 2021.

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Gutlyanskiĭ, V., Nesmelova, O., Ryazanov, V. et al. Logarithmic Potential and Generalized Analytic Functions. J Math Sci 256, 735–752 (2021). https://doi.org/10.1007/s10958-021-05457-5

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