On boundaryvalue problems in domains without (A)-condition

1Gutlyanskii, VYa.
1Ryazanov, VI
2Yakubov, E
3Yefimushkin, AS
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
2Holon Institute of Technology, Israel
3Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2019, 3:17-24
https://doi.org/10.15407/dopovidi2019.03.017
Section: Mathematics
Language: English
Abstract: 

We study the Hilbert boundaryvalue problem for the Beltrami equations in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring—Martio, generally speaking, without the standard (A)-condition by Ladyzhenskaya—Ural'tseva. Assuming that the coefficients of the problem are functions of countable bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of its solutions. As consequences, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincaré boundaryvalue problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media.

Keywords: and Poincaré boundaryvalue problems, angular limits, Beltrami equations, Dirichlet, Hilbert, logarithmic capacity, Neumann, quasiconformal functions, quasihyperbolic boundary condition
References: 

1. Lehto O. & Virtanen, K. J. (1973). Quasiconformal mappings in the plane. Berlin, Heidelberg: Springer. doi: https://doi.org/10.1007/978-3-642-65513-5
2. 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. doi: https://doi.org/10.1007/s10958-015-2621-0
3. Gutlyanskii, V., Ryazanov, V., Yakubov, E. & Yefimushkin, A. (2019). On the Hilbert problem for analytic functions in quasihyperbolic domains. Dopov. Nac. acad. nauk Ukr., No. 2, pp. 23-30. doi: https://doi.org/10.15407/dopovidi2019.02.023
4. Gehring, F. W. & Palka, B. P.(1976). Quasiconformally homogeneous domains. J. Anal. Math., 30, pp. 172-199. doi: https://doi.org/10.1007/BF02786713
5. Gehring, F. W. & Martio, O. (1985). Lipschitz classes and quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A. I. Math., 10, pp. 203-219. doi: https://doi.org/10.5186/aasfm.1985.1022
6. Becker, J. & Pommerenke, Ch. (1982). Hölder continuity of conformal mappings and nonquasiconformal Jordan curves. Comment. Math. Helv., 57, No. 2, pp. 221-225. doi: https://doi.org/10.1007/BF02565858
7. Astala, K. & Koskela, P. (1991). Quasiconformal mappings and global integrability of the derivative. J. Anal. Math., 57, pp. 203-220. doi: https://doi.org/10.1007/BF03041070
8. Ladyzhenskaya, O. A. & Ural’tseva, N. N. (1964). Linear and quasilinear elliptic equations. New York, London: Academic Press.
9. Gehring, F. W. & Martio, O. (1985). Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math., 45, pp. 181-206. doi: https://doi.org/10.1007/BF02792549
10. Koosis, P. (1998). Introduction to Hp spaces, Cambridge Tracts in Mathematics. (Vol. 115). Cambridge: Cambridge Univ. Press.
11. Goluzin, G. M. (1969). Geometric theory of functions of a complex variable. Translations of Mathematical Monographs. (Vol. 26). Providence, R.I.: American Mathematical Society. doi: https://doi.org/10.1090/mmono/026
12. Astala, K., Iwaniec, T. & Martin, G. (2009). Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series. (Vol. 48). Princeton: Princeton Univ. Press.
13. Gutlyanskii, V., Ryazanov, V. & Yefimushkin, A. (2016). On the boundaryvalue problems for quasiconformal functions in the plane. J. Math. Sci., 214, No. 2, pp. 200-219. doi: https://doi.org/10.1007/s10958-016-2769-2
14. Iwaniec, T. (1979). Regularity of solutions of certain degenerate elliptic systems of equations that realize quasiconformal mappings in ndimensional space. Differential and integral equations. Boundary value problems. Tbilisi: Tbilis. Gos. Univ., pp. 97-111.
15. Nevanlinna, R. (1944). Eindeutige analytische Funktionen. Michigan: Ann Arbor.