"Far-field interaction" of concentrated masses in two-dimensional Neumann and Dirichlet problems

We study the eigenvalues of the Neumann and Dirichlet boundary-value problems in a two-dimensional domain containing several small, of diameter $O(\varepsilon)$, inclusions of large "density" $O(\varepsilon^{-\gamma})$, $\gamma\geq2$, that is, the "mass" $O(\varepsilon^{2-\gamma})$ of each of them is comparable ($\gamma=2$) or much bigger ($\gamma>2$) than that of the embracing material. We construct a model of such spectral problems on concentrated masses which (the model) provides an asymptotic expansions of the eigenvalues with remainders of power-law smallness order $O(\varepsilon^{\vartheta})$ as $\varepsilon\to+0$ and $\vartheta\in(0,1)$. Besides, the correction terms are real analytic functions of the parameter $|{\ln \varepsilon}|^{-1}$. A "far-field interaction" of the inclusions is observed at the levels $|{\ln \varepsilon}|^{-1}$ or $|{\ln \varepsilon}|^{-2}$. The results are obtained with the help of the machinery of weighted spaces with detached asymptotics and also by using weighted estimates of solutions to limit problems in a bounded punctured domain and in the intact plane.