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.

scholarly journals ABOUT THE APPROXIMATE SOLUTION OF THE USUAL AND GENERALIZED HILBERT BOUNDARY VALUE PROBLEMS FOR ANALYTICAL FUNCTIONS

2000 ◽  
Vol 5 (1) ◽  
pp. 119-126
Author(s):  
V. R. Kristalinskii
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Cauchy Type ◽  
Analytical Functions ◽  
Method Of Solution ◽  
Cauchy Type Integrals ◽  
Hilbert Boundary Value Problem

In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the formproposed by the author. All methods are implemented with the Mathcad and Maple.

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