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A necessary condition for the strong \(G\)-convergence of nonlinear operators of Dirichlet problems with variable domain. (English. Russian original) Zbl 0967.35050

Differ. Equ. 36, No. 4, 599-604 (2000); translation from Differ. Uravn. 36, No. 4, 537-541 (2000).
From the text: Let \(n\in\mathbb{N}\), \(n\geq 2\), \(\Omega\) be a bounded domain in \(\mathbb{R}^n\), and let \(\{\Omega_s\}\) be a sequence of domains in \(\mathbb{R}^n\) lying in \(\Omega\). We consider nonlinear elliptic operators corresponding to Dirichlet problems in the domains \(\Omega_s\). We obtain a necessary condition for the \(G\)-convergence of these operators related to the behavior of some numerical characteristics of the domains \(\Omega_s\) constructed on the basis of solutions of special local variational inequalities. The operator convergence studied here is in general closely related to the convergence of solutions of Dirichlet problems in variable (for example, perforated) domains.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
49J45 Methods involving semicontinuity and convergence; relaxation
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References:

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