Approximation of solutions of elliptic problems in domains with noncompact boundaries by solutions of exterior or interior problems

Let $\Omega^R$ ($R>0$) be a family of domains approximating a domain $\Omega^\infty$ as $R\to\infty$. For example, $\Omega^R$ can be a family of expanding domains whose union over all $R$ is $\Omega^\infty$, or a family of shrinking domains whose intersection is $\Omega^\infty$. Let $\mathfrak A_R$ be the operator corresponding to a formally symmetric elliptic boundary value problem in $\Omega^R$, and let $u_\varepsilon^R=(\mathfrak A_R+i\varepsilon)^{-1}f$. Conditions are determined under which $u_\varepsilon^R$ converges to a solution of the limit problem as $R\to\infty$, or as $\varepsilon\to0$ and $R\to\infty$ simultaneously.
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