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Estimation of solutions of boundary-value problems in domains with concentrated masses located periodically along the boundary: case of light masses. (English. Russian original) Zbl 1076.35014
Math. Notes 76, No. 6, 865-879 (2004); translation from Mat. Zametki 76, No. 6, 928-944 (2004).
Summary: We study the asymptotic behavior of solutions and eigenelements of boundary value problems with rapidly alternating type of boundary conditions in the domain \(\Omega\subset\mathbb{R}^n\). The density, which depends on a small parameter \(\varepsilon\), is of the order of \(O(1)\) outside small inclusions, where the density is of the order of \(O((\varepsilon\delta)^{-m}\). These domains, i.e., concentrated masses of diameter \(O(\varepsilon\delta)\), are located near the boundary at distances of the order of \(O(\delta)\) from each other, where \(\delta= \delta(\varepsilon)\to 0\). We pose the Dirichlet condition (respectively, the Neumann condition) on the parts of the boundary \(\partial\Omega\) that are tangent (respectively, lying outside) the concentrated masses. We estimate the deviations of the solutions of the limit (averaged) problems from the solutions of the original problems in the norm of the Sobolev space \(W^1_2\) for \(m< 2\).

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J25 Boundary value problems for second-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
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