The Hilbert boundary-value problem with infinite index of logarithmic orden in the half-plane

1978 ◽  
Vol 17 (1) ◽  
pp. 1-6 ◽  
Author(s):  
P. Alekna
Keyword(s):  
Boundary Value Problem ◽  
Half Plane ◽  
Boundary Value ◽  
Infinite Index ◽  
Hilbert Boundary Value Problem

scholarly journals Hilbert boundary value problem for generalized analytic functions with a singular line

2021 ◽  
Vol 274 ◽  
pp. 11003
Author(s):  
Pavel Shabalin ◽  
Rafael Faizov
Keyword(s):  
Boundary Value Problem ◽  
Analytic Functions ◽  
Boundary Value ◽  
Singular Line ◽  
Infinite Index ◽  
Number Of Solutions ◽  
Singular Coefficient ◽  
Generalized Analytic Functions ◽  
Complete Study ◽  
Hilbert Boundary Value Problem

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.

On a Riemann Boundary Value Problem with Infinite Index in the Half-plane

2021 ◽  
pp. 221-241
Author(s):  
Hrachik M. Hayrapetyan ◽  
Smbat A. Aghekyan ◽  
Artavazd D. Ohanyan
Keyword(s):  
Boundary Value Problem ◽  
Half Plane ◽  
Boundary Value ◽  
Infinite Index ◽  
Riemann Boundary Value Problem ◽  
Riemann Boundary

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

2021 ◽  
Author(s):  
Vladimir Gutlyanskiĭ ◽  
Vladimir Ryazanov ◽  
Eduard Yakubov ◽  
Artyem Yefimushkin
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Beltrami Equations ◽  
Hilbert Boundary Value Problem

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

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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