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The Behaviour of Weak Solutions of Boundary Value Problems for Linear Elliptic Second Order Equations in Unbounded Cone-Like Domains

100%
Wiśniewski D.
Annales Mathematicae Silesianae
|
2016
|
tom 30
|
nr 1
203-217
EN
We investigate the behaviour of weak solutions of boundary value problems (Dirichlet, Neumann, Robin and mixed) for linear elliptic divergence second order equations in domains extending to infinity along a cone. We find an exponent of the solution decreasing rate: we derive the estimate of the weak solution modulus for our problems near the infinity under assumption that leading coefficients of the equations do not satisfy the Dini-continuity condition.
2
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Existence of entropy solutions for nonlinear elliptic degenerate anisotropic equations

61%
Gorban Y.
Open Mathematics
|
2017
|
tom 15
|
nr 1
768-786
EN
In the present article we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L1-right-hand sides in a bounded domain of ℝn(n ⩾ 2) . This class is described by the presence of a set of exponents q1,…, qn and a set of weighted functions ν1,…, νn in growth and coercitivity conditions on coefficients of the equations. The exponents qi characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions νi characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to investigate the existence of entropy solutions of the problem under consideration.
3
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Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

52%
Melnikov Y.
Open Mathematics
|
2010
|
tom 8
|
nr 1
53-72
EN
Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.
4
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Comments on behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities

52%
Bodzioch M., Wiśniewski D., Żyjewski K.
Open Mathematics
|
2017
|
tom 15
|
nr 1
667-678
EN
We have investigated the behaviour of solutions of elliptic quasi-linear problems in a neighbourhood of boundary singularities in bounded and unbounded domains. We found exponents of the solution’s decreasing rate near the boundary singularities.
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