Non-periodic boundary homogenization and "light" concentrated masses
Gregory ChechkinM. PerezEkaterina Yablokova
35B2535P0535B4074Qspectral analysisconcentrated massesboundary homogenizationconcentrated forces
We consider certain spectral problems for the Laplace operator with rapidly alternating boundary conditions in an open bounded domain $\Omega$ of $\mathbb{R}^n$ that contains many \textit{concentrated masses} $B_{\varepsilon}$ near the boundary. The regions $B_{\varepsilon}$ have a diameter $O(\varepsilon)$ and the density takes the value $\varepsilon^{-m}$ in $B_{\varepsilon}$ and $1$ outside. $m$, $n$ and $\varepsilon$ are parameters: $0\le m<2$, $n\ge3$ and $\varepsilon\to0$. We assume small mass of the whole concentrated masses while periodicity of the microstructure is not assumed. We study the asymptotic behavior, as $\varepsilon\to0$, of the eigenelements of the spectral problems. We obtain the homogenized (limit) spectral problems and estimates for the convergence rates of the corresponding eigenelements. Certain associated stationary problems are also considered, and estimates for the convergence rates of the solutions are obtained.
Indiana University Mathematics Journal
2005
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10.1512/iumj.2005.54.2487
10.1512/iumj.2005.54.2487
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Indiana Univ. Math. J. 54 (2005) 321 - 348
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