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Mat. Sb. (N.S.), 1984, Volume 124(166), Number 3(7), Pages 291–306 (Mi msb2053)  

This article is cited in 1 scientific paper (total in 1 paper)

On the convergence of Galerkin approximations to the solution of the Dirichlet problem for some general equations

G. G. Kazaryan, G. A. Karapetyan


Abstract: The Dirichlet problem with null boundary values is considered for a quasilinear operator of divergence form
$$ Au=\sum_{\alpha\in\mathrm E}D^\alpha A_\alpha(x,D^{\gamma^1}u,…,D^{\gamma^N}u), $$
where $\mathrm E=\{\gamma^1,…,\gamma^N\}$ is a finite collection of multi-indices, and $x$ varies in a domain $\Omega$ when the operator $A$ is in general not elliptic.
Under certain restrictions on the growth of the coefficients $A_\alpha(x,\xi)$ as $|\xi|\to\infty$ and on the domain $\Omega$, it is proved that the Dirichlet problem for the equation $Au=f$ for arbitrary $f\in L_2(\Omega)$ has a weak solution in the class $H$ induced in a natural way by the operator $A$. In addition it is proved that a sequence of Galerkin solutions converges to this solution weakly in $H$.
Bibliography: 30 titles.

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English version:
Mathematics of the USSR-Sbornik, 1985, 52:2, 285–299

Bibliographic databases:

UDC: 517.9
MSC: Primary 35A35, 65N30; Secondary 35J65, 35A05
Received: 16.11.1981 and 16.12.1983

Citation: G. G. Kazaryan, G. A. Karapetyan, “On the convergence of Galerkin approximations to the solution of the Dirichlet problem for some general equations”, Mat. Sb. (N.S.), 124(166):3(7) (1984), 291–306; Math. USSR-Sb., 52:2 (1985), 285–299

Citation in format AMSBIB
\Bibitem{KazKar84}
\by G.~G.~Kazaryan, G.~A.~Karapetyan
\paper On the convergence of Galerkin approximations to the solution of the Dirichlet problem for some general equations
\jour Mat. Sb. (N.S.)
\yr 1984
\vol 124(166)
\issue 3(7)
\pages 291--306
\mathnet{http://mi.mathnet.ru/msb2053}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=752222}
\zmath{https://zbmath.org/?q=an:0578.65114|0554.65081}
\transl
\jour Math. USSR-Sb.
\yr 1985
\vol 52
\issue 2
\pages 285--299
\crossref{https://doi.org/10.1070/SM1985v052n02ABEH002891}


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    This publication is cited in the following articles:
    1. G. A. Karapetyan, H. G. Tananyan, “The small parameter method for regular linear differential equations on unbounded domains”, Eurasian Math. J., 4:2 (2013), 64–81  mathnet  mathscinet  zmath
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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