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Ufimsk. Mat. Zh., 2011, Volume 3, Issue 3, Pages 127–139 (Mi ufa108)  

This article is cited in 8 scientific papers (total in 8 papers)

On estimate of eigenfunctions of the Steklov-type problem with a small parameter in the case of a limit spectrum degeneration

V. A. Sadovnichiia, A. G. Chechkinab

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow, Russia
b The State Uneversity of the Ministry of Finance of the Russian Federation, Moscow, Russia

Abstract: We consider a Steklov-type problem with rapidly alternating boundary conditions (Dirichlet and Steklov) in a bounded two-dimensional domain. The parts of the boundary, where the Dirichlet boundary condition are given, have the length of the order $\varepsilon$ and they alternate with parts of the length of the same order, having the Steklov condition. We prove that the normalized eigenfunctions for a sufficiently small $\varepsilon$ satisfy the Friedrichs-type inequality with the constant of the order $\varepsilon$ and moreover, they converge to zero as $\varepsilon$ tends to zero.

Keywords: spectrum of operator, Steklov-type problem, homogenization, asymptotics.

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Bibliographic databases:
UDC: 517.91
Received: 26.07.2011

Citation: V. A. Sadovnichii, A. G. Chechkina, “On estimate of eigenfunctions of the Steklov-type problem with a small parameter in the case of a limit spectrum degeneration”, Ufimsk. Mat. Zh., 3:3 (2011), 127–139

Citation in format AMSBIB
\by V.~A.~Sadovnichii, A.~G.~Chechkina
\paper On estimate of eigenfunctions of the Steklov-type problem with a~small parameter in the case of a~limit spectrum degeneration
\jour Ufimsk. Mat. Zh.
\yr 2011
\vol 3
\issue 3
\pages 127--139

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    This publication is cited in the following articles:
    1. A. G. Chechkina, V. A. Sadovnichy, “Degeneration of Steklovtype boundary conditions in one spectral homogenization problem”, Eurasian Math. J., 6:3 (2015), 13–29  mathnet
    2. G. A. Chechkin, C. D' Apice, U. De Maio, R. R. Gadyl'shin, “On a Singularly Perturbed Steklov Problem in a Domain Perforated Along the Boundary”, C. R. Mec., 344:1 (2016), 12–18  crossref  isi  elib  scopus
    3. S. T. Erov, G. A. Chechkin, “Vibrations of a fluid containing a wide spaced net with floats under its free surface”, J. Math. Sci. (N. Y.), 234:4 (2018), 407–422  mathnet  crossref
    4. A. G. Chechkina, “Homogenization of spectral problems with singular perturbation of the Steklov condition”, Izv. Math., 81:1 (2017), 199–236  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    5. Chechkina A.G., “Estimate of the Spectrum Deviation of the Singularly Perturbed Steklov Problem”, Dokl. Math., 96:2 (2017), 510–513  crossref  mathscinet  zmath  isi  scopus
    6. Chechkin G.A., Gadyl'shin R.R., D'Apice C., De Maio U., “On the Steklov Problem in a Domain Perforated Along a Part of the Boundary”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 51:4 (2017), 1317–1342  crossref  mathscinet  zmath  isi  scopus
    7. Beliaev A., Krichevets G., “Qualitative Effects of Hydraulic Conductivity Distribution on Groundwater Flow in Heterogeneous Soils”, Fluids, 3:4 (2018), 102  crossref  isi  scopus
    8. Chechkina A.G., D'Apice C., De Maio U., “Rate of Convergence of Eigenvalues to Singularly Perturbed Steklov-Type Problem For Elasticity System”, Appl. Anal., 98:1-2, SI (2019), 32–44  crossref  mathscinet  zmath  isi  scopus
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