Dirichlet problem for general A-harmonic equations in simply connected domains

V.Ya. Gutlyanski; V.I. Ryazanov; R.R. Salimov; E.O. Sevost’yanov

doi:10.15407/dopovidi2026.02.003

Authors

V.Ya. Gutlyanski Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk, Ukraine https://orcid.org/0000-0002-8691-4617
V.I. Ryazanov Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk, Ukraine https://orcid.org/0000-0002-4503-4939
R.R. Salimov Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine https://orcid.org/0000-0001-9395-3334
E.O. Sevost’yanov Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slovyansk, Ukraine https://orcid.org/0000-0001-7892-6186

DOI:

https://doi.org/10.15407/dopovidi2026.02.003

Keywords:

Dirichlet problem, A-harmonic equation, simply connected domains, Beltrami equations

Abstract

The article is devoted to theorems on the existence, representation, and regularity of solutions to the Dirichlet problem with continuous data for general A-harmonic equation div A grad U = 0 in the real plane with matrix valued coefficients A. The equation is one of the main equations of the hydromechanics (fluid mechanics) in anisotropic and inhomogeneous media. Here we present a number of effective integral solvability criteria for this problem of the type of Calderon— Zygmund, Dini—Lavrentiev—Lehto, Orlicz, BMO, FMO and VMO in arbitrary bounded simple connected domains including all Jordan domains. These results are based on the theory of the so-called Beltrami equations with two characteristics in the complex plane and formulated in terms of the corresponding two complex parameters associated with A.

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Dirichlet problem for general A-harmonic equations in simply connected domains

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