Convergence of an approximate method of S. A. Khristianovich for solving the Dirichlet problem for an elliptic equation. (English. Russian original) Zbl 0631.30014

Nasyrov, R. M.; Nasyrov, S. R.

Sov. Math., Dokl. 34, 484-488 (1987); translation from Dokl. Akad. Nauk SSSR 291, 294-298 (1986).

The two dimensional boundary value problem \(div(\kappa \nabla u)=0\), \(u| \partial E=f\), E the unit disk, is first reduced in a usual way to the boundary value problem for the Beltrami system \(w_{\bar z}=\mu w_ z\) with a characteristic \(\mu\). If \(\mu\) is embedded in the one- parameter family depending analytically on the parameter \(\lambda\), then expanding the solution in series in \(\lambda\), \[ w(z,\lambda)=\sum w_ k(z)\lambda^ k, \] the authors show that \(\sum w_ k\) converges in \(W^ 1_ p\) to a solution of the Beltrami system. Here \(f\in C^{\alpha}(\partial E)\), \(\alpha >\) and \(2<p<(1-\alpha)^{-1}\). The proof employs the two dimensional Hilbert transformation. Also estimates for uniform convergence and for convergence in Lipschitz spaces are considered.

Reviewer: O.Martio

MSC:

30C62	Quasiconformal mappings in the complex plane
35J25	Boundary value problems for second-order elliptic equations

Keywords:

Beltrami system