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

G. G. Kazaryan ; G. A. Karapetyan

Sbornik. Mathematics, Tome 52 (1985) no. 2, pp. 285-299

Résumé

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.