| Title | On boundary-value problems for generalized analytic and harmonic functions |
| Publication Type | Journal Article |
| Year of Publication | 2020 |
| Authors | Gutlyanskii, VYa., Nesmelova, OV, Ryazanov, VI, Yefimushkin, AS |
| Abbreviated Key Title | Dopov. Nac. akad. nauk Ukr. |
| DOI | 10.15407/dopovidi2020.12.011 |
| Issue | 12 |
| Section | Mathematics |
| Pagination | 11-18 |
| Date Published | 12/2020 |
| 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 |
1. Gutlyanskii, V.Ya., Nesmelova, O.V., Ryazanov, V.I. & Yefimushkin, A.S. (2020). Logarithmic capacity and Riemann and Hilbert problems for generalized analytic functions. Dopov. Nac. akad. nauk Ukr., No. 8. pp. 11-18. https://doi.org/10.15407/dopovidi2020.08.011
2. Luzin, N. N. (1915). Integral and trigonometric series. (Unpublished Doctor thesis). Moscow University, Moscow, Russia (in Russian).
3. Luzin, N. N. (1951). Integral and trigonometric series. Editing and commentary by Bari, N. K. & Men’shov, D.E. Moscow, Leningrad: Gostehteoretizdat (in Russian).
4. Efimushkin, A. S. & Ryazanov, V. I. (2015). On the Riemann-Hilbert problem for the Beltrami equations in quasidisks. J. Math. Sci., 211, No. 5, pp. 646-659. https://doi.org/10.1007/s10958-015-2621-0
5. Yefimushkin, A. & Ryazanov, V. (2016). On the Riemann-Hilbert problem for the Beltrami equations. In Complex analysis and dynamical systems VI. Part 2 (pp. 299-316). Contemporary Mathematics, 667. Israel Math. Conf. Proc. Providence, RI: Amer. Math. Soc. https://doi.org/10.5186/aasfm.2020.4552
6. Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2019). To the theory of semilinear equations in the plane. J. Math. Sci., 242, No. 6, pp. 833-859. https://doi.org/10.1007/s10958-019-04519-z
7. Sobolev, S. L. (1963). Applications of functional analysis in mathematical physics. Providence, R.I.: AMS.
8. Bagemihl, F. & Seidel, W. (1955). Regular functions with prescribed measurable boundary values almost everywhere. Proc. Nat. Acad. Sci. USA, 41, pp. 740-743. https://doi.org/10.1073/pnas.41.10.740
9. Vekua, I. N. (1962). Generalized analytic functions. London: Pergamon Press.
10. Gutlyanskii, V., Ryazanov, V. & Yefimushkin, A. (2016). On the boundary-value problems for quasiconformal functions in the plane. J. Math. Sci., 214, No. 2, pp. 200-219. https://doi.org/10.1007/s10958-016-2769-2
11. Gutlyanskii, V., Nesmelova, O. & Ryazanov, V. (2018). On quasiconformal maps and semilinear equations in the plane. J. Math. Sci., 229, pp. 7-29. https://doi.org/10.1007/s10958-018-3659-6
12. Gutlyanskii, V., Ryazanov, V., Yakubov, E. & Yefimushkin, A. (2020). On Hilbert boundary value problem for Beltrami equation. Ann. Acad. Sci. Fenn. Math., 45, pp. 957-973. https://doi.org/10.5186/aasfm.2020.4552
