Geodesic
Operator estimates for the Steklov problem in an unbounded domain with rapidly changing conditions on the boundary
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 499 (2021), pp. 54-57.

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A spectral problem of the Steklov type for the Laplacian in an unbounded domain with a smooth boundary is considered. The Steklov condition rapidly alternates with the homogeneous Dirichlet condition on a part of the boundary. Operator estimates are obtained, which are used to study the asymptotic behavior of the eigenelements of the original problem as the small parameter tends to zero. The small parameter characterizes the size of the boundary parts with the Dirichlet condition, the distance between which is on the order of the logarithm of the small parameter in a negative power.
Keywords: operator estimates, Steklov problem, boundary homogenization. 
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A. G. Chechkina. Operator estimates for the Steklov problem in an unbounded domain with rapidly changing conditions on the boundary. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 499 (2021), pp. 54-57. http://geodesic.mathdoc.fr/item/DANMA_2021_499_a11/

[1] Mel'nyk T.A., “Asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem in a thick periodic junction”, Nonlinear Oscillations, 4:1 (2001), 91–105 | MR | Zbl

[2] Pérez E., “On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem”, Discrete Contin. Dyn. Syst. Ser. B, 7:4 (2007), 859–883 | MR | Zbl

[3] Nazarov S.A., “Asimptotika resheniya spektralnoi zadachi Steklova v oblasti s zatuplennym pikom”, Mat. zametki, 86:4 (2009), 571–587 | DOI | Zbl

[4] Chechkina A.G., “Usrednenie spektralnykh zadach s singulyarnym vozmuscheniem usloviya Steklova”, Izvestiya RAN, 81:1 (2017), 203–240 | MR | Zbl

[5] Chechkina A.G., D'Apice C., De Maio U., “Rate of Convergence of Eigenvalues to Singularly Perturbed Steklov-Type Problem for Elasticity System”, Applicable Analysis, 98:1-2 (2019), 32–44 | DOI | MR | Zbl

[6] Gadylshin R.R., Chechkin G.A., “Kraevaya zadacha dlya Laplasiana s bystro menyayuschimsya tipom granichnykh uslovii v mnogomernoi oblasti”, Sib. mat. zhurnal, 40:2 (1999), 271–287 | MR | Zbl

[7] Peres M.E., Chechkin G.A., Yablokova (Doronina) E.I., “O sobstvennykh kolebaniyakh tela s “legkimi” kontsentrirovannymi massami na poverkhnosti”, Uspekhi mat. nauk, 57:6 (2002), 195–196 | DOI | MR

[8] Chechkin G.A., “On Vibration of Partially Fastened Membrane with Many “Light” Concentrated Masses on the Boundary”, C. R. Mécanique, 332:12 (2004), 949–954 | DOI | Zbl

[9] Amirat Y., Chechkin G.A., Gadyl'shin R.R., “Asymptotics of the Solution of a Dirichlet Spectral Problem in a Junction with Highly Oscillating Boundary”, C. R. Mécanique, 336:9 (2008), 693–698 | DOI | Zbl

[10] Chechkin G.A., Koroleva Yu.O., Persson L.-E., “On the Precise Asymptotics of the Constant in the Friedrich's Inequality for Functions, Vanishing on the Part of the Boundary with Microinhomogeneous Structure”, Journal of Inequalities and Applications, 2007 (2007), 34138, 13 pp. | DOI | MR | Zbl

[11] Chechkin G.A., Koroleva Yu.O., Meidell A., Persson L.-E., “On the Friedrichs inequality in a domain perforated aperiodically along the boundary. Homogenization procedure. Asymptotics for parabolic problems”, Russ. J. Math. Phys., 16 (2009), 1–16 | DOI | MR | Zbl

[12] Suslina T.A., “Usrednenie zadachi Dirikhle dlya ellipticheskikh uravnenii vysokogo poryadka s periodicheskimi koeffitsientami”, Algebra i analiz, 29:2 (2017), 139–192 | MR

[13] Griso G., “Interior error estimate for periodic homogenization”, Anal. Appl., 4 (2006), 61–79 | DOI | MR | Zbl