Gómez, Delfina; Nazarov, Sergei A.; Pérez-Martínez, María-Eugenia Asymptotics for spectral problems with rapidly alternating boundary conditions on a strainer Winkler foundation. (English) Zbl 1458.35041 J. Elasticity 142, No. 1, 89-120 (2020). Summary: We consider a spectral homogenization problem for the linear elasticity system posed in a domain \(\varOmega\) of the upper half-space \(\mathbb{R}^{3+} \), a part of its boundary \(\varSigma\) being in contact with the plane \(\{x_3=0\} \). We assume that the surface \(\varSigma\) is traction-free out of small regions \(T^{\varepsilon} \), where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function \(M(x)\) and a reaction parameter \(\beta (\varepsilon )\) that can be very large when \(\varepsilon \to 0\). The size of the regions \(T^{\varepsilon}\) is \(O(r_{\varepsilon })\), where \(r_{\varepsilon}\ll \varepsilon \), and they are placed at a distance \(\varepsilon\) between them. We provide all the possible spectral homogenized problems depending on the relations between \(\varepsilon, r_{\varepsilon}\) and \(\beta (\varepsilon )\), while we address the convergence, as \(\varepsilon \to 0\), of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on \(\varSigma \). New capacity matrices are introduced to define these strange terms. Cited in 3 Documents MSC: 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 74B05 Classical linear elasticity 74Q20 Bounds on effective properties in solid mechanics 35P05 General topics in linear spectral theory for PDEs 35B25 Singular perturbations in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 35P15 Estimates of eigenvalues in context of PDEs Keywords:boundary homogenization; spectral perturbations; capacity matrices; critical relations PDFBibTeX XMLCite \textit{D. Gómez} et al., J. Elasticity 142, No. 1, 89--120 (2020; Zbl 1458.35041) Full Text: DOI References: [1] Agmon, S.; Douglas, A.; Niremberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Commun. Pure Appl. Math., XVII, 35-92 (1964) · Zbl 0123.28706 [2] Allaire, G., Homogenization of the Naviers-Stokes equations in open sets perforated with tiny holes II. 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