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
Section: Mathematics
Language: English

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

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