This article is cited in
3 papers
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
$$
Au=\sum_{\alpha\in\mathrm E}D^\alpha A_\alpha(x,D^{\gamma^1}u,\dots,D^{\gamma^N}u),
$$
where
$\mathrm E=\{\gamma^1,\dots,\gamma^N\}$ 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.
UDC:
517.9
MSC: Primary
35A35,
65N30; Secondary
35J65,
35A05 Received: 16.11.1981 and 16.12.1983