On Quaternionic Beltrami Equations

On the Hilbert boundary-value problem for Beltrami equations with singularities

Journal of Mathematical Sciences ◽

10.1007/s10958-021-05309-2 ◽

2021 ◽

Author(s):

Vladimir Gutlyanskiĭ ◽

Vladimir Ryazanov ◽

Eduard Yakubov ◽

Artyem Yefimushkin

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Beltrami Equations ◽

Hilbert Boundary Value Problem

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On the Hilbert boundary-value problem for Beltrami equations with singularities

Ukrainian Mathematical Bulletin ◽

10.37069/1810-3200-2020-17-4-2 ◽

2020 ◽

Vol 17 (4) ◽

pp. 484-508

Author(s):

Vladimir Gutlyanskii ◽

Vladimir Ryazanov ◽

Eduard Yakubov ◽

Artyem Yefimushkin

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Beltrami Equation ◽

General Nature ◽

Countable Collection ◽

Smooth Curves ◽

Beltrami Equations ◽

Jordan Curves ◽

Equations Of Mathematical Physics ◽

Hilbert Boundary Value Problem

We investigate the Hilbert boundary-value problem for Beltrami equations $\overline\partial f=\mu\partial f$ with singularities in generalized quasidisks $D$ whose Jordan boundary $\partial D$ consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves. Generally speaking, generalized quasidisks do not satisfy the standard $(A)-$condition in PDE by Ladyzhenskaya-Ural'tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring-Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations. In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain $D$ of a general nature. As usual, a point in $\overline D$ is called a singularity of the Beltrami equation, if the dilatation quotient $K_{\mu}:=(1+|\mu|)/(1-|\mu|)$ is not essentially bounded in all its neighborhoods. Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincar\'{e} boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.

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On Dirichlet problem for Beltrami equations in finitely connected domains

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2020-34-9 ◽

2021 ◽

Vol 34 ◽

pp. 85-99

Author(s):

Ihor Petkov ◽

Vladimir Ryazanov

Keyword(s):

Dirichlet Problem ◽

Boundary Data ◽

Boundary Behavior ◽

Beltrami Equation ◽

Elliptic Systems ◽

Beltrami Equations ◽

Prime Ends ◽

Wide Circle ◽

Continuous Boundary ◽

Homeomorphic Solution

Boundary value problems for the Beltrami equations are due to the famous Riemann dissertation (1851) in the simplest case of analytic functions and to the known works of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy--Riemann system. Of course, the Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., \cite{Boj} and \cite{Vekua}. Moreover, the corresponding results on the Dirichlet problem for degenerate Beltrami equations in the unit disk can be found in the monograph \cite{GRSY}. In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it was shown that each generalized homeomorphic solution of a Beltrami equation is the so-called lower $Q-$homeomorphism with its dilatation quotient as $Q$ and developed on this basis the theory of the boundary behavior of such solutions. In the next papers \cite{KPR2} and \cite{KPR4}, the latter made possible us to solve the Dirichlet problem with continuous boundary data for a wide circle of degenerate Beltrami equations in finitely connected Jordan domains, see also [\citen{KPR5}--\citen{KPR7}]. Similar problems were also investigated in the case of bounded finitely connected domains in terms of prime ends by Caratheodory in the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}]. Finally, in the present paper, we prove a series of effective criteria for the existence of pseudo\-re\-gu\-lar and multi-valued solutions of the Dirichlet problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains in terms of prime ends by Caratheodory.

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