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 $$O(\varepsilon+\omega^{-1})$$.

MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations

Keywords:

existence; uniqueness