Justification of fictitious domains method of solving mixed boundary value problems for quasilinear elliptic equations. (English. Russian original) Zbl 0909.35046

Mosc. Univ. Math. Bull. 51, No. 3, 12-17 (1996); translation from Vestn. Mosk. Univ., Ser. I 1996, No. 3, 16-23 (1996).
The paper deals with homogeneous mixed boundary value problems in a domain \(G\) with complex geometry for quasilinear elliptic equations with certain conditions on the order of the coefficient growth. The coefficients are extended to the domain \(D\) with simple geometry by means of a large parameter \(\omega\) through the segment of the boundary \(G\) with the first boundary condition or a small parameter \(\varepsilon\) through the segment of the boundary \(G\) with the second boundary condition. For the solution \(u_{\omega,\varepsilon}\) of the problem of the fictitious domain method an a priori estimate is established in the norm \(W^1_2(D)\) uniform in \(0\leq \varepsilon\leq 1\) and \(1\leq \omega\leq \infty\).
The existence and uniqueness theorem is proved for all \(0\leq \varepsilon\leq 1\), \(\omega \geq 1\). Also the convergence of \(u_{\omega,\varepsilon}\) is shown in norm \(W^1_2(G)\) to the solution of the initial problem as \(\omega\to\infty\), \(\varepsilon\to 0\) with velocity \(O(\varepsilon+\omega^{-1})\).


35J65 Nonlinear boundary value problems for linear elliptic equations