The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries. II (in Russian)

1983 ◽  
Vol 2 (6) ◽  
pp. 523-551 ◽  
Author(s):  
Vladimir Maz'ya ◽  
Boris Plamenevskii
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Mathematical Physics ◽  
Piecewise Smooth ◽  
Equations Of Mathematical Physics

Universal boundary value problem for equations of mathematical physics

2014 ◽  
Vol 285 (1) ◽  
pp. 56-80 ◽  
Author(s):  
I. V. Volovich ◽  
V. Zh. Sakbaev
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Mathematical Physics ◽  
Equations Of Mathematical Physics

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.

First boundary value problem for the equations of hydrodynamics in a domain with a piecewise-smooth boundary

1983 ◽  
Vol 21 (5) ◽  
pp. 777-783 ◽  
Author(s):  
V. G. Maz'ya ◽  
B. A. Plamenevskii
Keyword(s):  
Boundary Value Problem ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Piecewise Smooth ◽  
Piecewise Smooth Boundary

scholarly journals Boundary value problems for partial differential equations with piecewise contant delay

1991 ◽  
Vol 14 (2) ◽  
pp. 363-379 ◽  
Author(s):  
Joseph Wiener
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Mathematical Physics ◽  
Partial Differential ◽  
Equations Of Mathematical Physics

The influence of certain discontinuous delays on the behavior of solutions to some typical equations of mathematical physics is studied.

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