Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 957–973

ON HILBERT BOUNDARY VALUE PROBLEM FOR BELTRAMI EQUATION

Vladimir Gutlyanskii, Vladimir Ryazanov, Eduard Yakubov and Artyem Yefimushkin

National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics
Generala Batyuka str. 19, 84116 Slavyansk, Ukraine; vgutlyanskii 'at' gmail.com

National University of Cherkasy, Physics Department, Laboratory of Mathematical Physics
and National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics
Generala Batyuka str. 19, 84116 Slavyansk, Ukraine; Ryazanov 'at' nas.gov.ua

Holon Institute of Technology
Golomb St. 52, Holon, 5810201, Israel; yakubov 'at' hit.ac.il

National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics
Generala Batyuka str. 19, 84116 Slavyansk, Ukraine; a.yefimushkin 'at' gmail.com

Abstract. We study the Hilbert boundary value problem for the Beltrami equation in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring–Martio, generally speaking, without (A)-condition by Ladyzhenskaya–Ural'tseva that was standard for boundary value problems in the PDE theory. 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 the generalized regular solutions. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincaré boundary value problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media.

2010 Mathematics Subject Classification: Primary 30C062, 31A05, 31A20, 31A25, 31B25, 35J61; Secondary 30E25, 31C05, 34M50, 35F45, 35Q15.

Key words: Hilbert boundary value problem, Beltrami equation, quasihyperbolic boundary condition, logarithmic capacity, angular limits.

Reference to this article: V. Gutlyanskii, V. Ryazanov, E. Yakubov and A. Yefimushkin: On Hilbert boundary value problem for Beltrami equation. Ann. Acad. Sci. Fenn. Math. 45 (2020), 957–973.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4552