On the Dirichlet problem for generalized Cauchy-Riemann equations

• Date: 2025
• Main Authors: Gutlyanskii, V.Yu., Ryazanov, V.I., Salimov, A.R., Salimov, R.R.
• Format: Article
• Language: English
• Published: Видавничий дім "Академперіодика" НАН України 2025
• Series: Доповіді НАН України
• Subjects: Математика
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/206498
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On the Dirichlet problem for generalized Cauchy-Riemann equations / V.Yu. Gutlyanskii, V.I. Ryazanov, A.R. Salimov, R.R. Salimov // Доповіді Національної академії наук України. — 2025. — № 2. — С. 3-10. — Бібліогр.: 15 назв. — англ.

Here we give a survey of consequences of the theory of the Beltrami equations from the complex analysis for the Dirichlet problem to generalized Cauchy-Riemann equations ▽v = B▽u in the real plane R² that describe flows of fluids in anisotropic and inhomogeneous media, where B is a 2 × 2 matrix valued coefficient and the gradients ▽u and ▽v are interpreted as vector columns. Moreover, we clarify the relationships of the latter to the A-harmonic equation div (A▽u) = 0 with matrix valued coefficients A that is one of the main equations of the potential theory, namely, of the hydromechanics (fluid mechanics) in anisotropic and inhomogeneous media in the plane. The survey includes a series of effective integral criteria for existence of regular solutions of the Dirichlet problem with continuous data in arbitrary bounded simple connected domains to generalized Cauchy-Riemann equations with matrix coefficients in the case of anisotropic and inhomogeneous media.