V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, Algebra i Analiz, 22:6 (2010), 127–184; St. Petersburg Math. J., 22:6 (2011), 941

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Algebra i Analiz, 2010, Volume 22, Issue 6, Pages 127–184 (Mi aa1217)

This article is cited in 17 scientific papers (total in 17 papers)

Research Papers

The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary

V. A. Kozlov^a, S. A. Nazarov^b

^a Department of Mathematics, Linkopings Universitet, Linkoping, Sweden
^b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

Full-text PDF (581 kB) Citations (17)

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Abstract: Asymptotic expansions are constructed for the eigenvalues of the Dirichlet problem for the biharmonic operator in a domain with highly indented and rapidly oscillating boundary (the Kirchhoff model of a thin plate). The asymptotic constructions depend heavily on the quantity $\gamma$ that describes the depth $O(\varepsilon^\gamma)$ of irregularity ($\varepsilon$ is the oscillation period). The resulting formulas relate the eigenvalues in domains with close irregular boundaries and make it possible, in particular, to control the order of perturbation and to find conditions ensuring the validity (or violation) of the classical Hadamard formula.

Keywords: biharmonic operator, Dirichlet problem, asymptotic expansions of eigenvalues, eigenoscillations of the Kirchhoff plate, rapid oscillation and nonregular perturbation of the boundary.

Received: 15.06.2010

English version:
St. Petersburg Mathematical Journal, 2011, Volume 22, Issue 6, Pages 941–983
DOI: https://doi.org/10.1090/S1061-0022-2011-01178-1

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, Algebra i Analiz, 22:6 (2010), 127–184; St. Petersburg Math. J., 22:6 (2011), 941–983

Citation in format AMSBIB

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https://www.mathnet.ru/eng/aa1217

https://www.mathnet.ru/eng/aa/v22/i6/p127

This publication is cited in the following 17 articles:

D. I. Borisov, R. R. Suleimanov, “Operator estimates for elliptic equations in multidimensional domains with strongly curved boundaries”, Sb. Math., 216:1 (2025), 25–53
D. I. Borisov, R. R. Suleimanov, “Operator Estimates for Problems in Domains with Singularly Curved Boundary: Dirichlet and Neumann Conditions”, Dokl. Math., 2024
Siwar Saidani, Adel Jawahdou, “Asymptotic behaviors for the eigenvalues of the Schrödinger equation”, Applicable Analysis, 2024, 1
Ahlem Jaouabi, Abdessatar Khelifi, “Asymptotic behavior for eigenvalues and eigenfunctions associated to Stokes operator in the presence of a rapidly oscillating boundary”, Math Methods in App Sciences, 47:4 (2024), 1915
Vladimir Lotoreichik, “Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator”, Proc. Amer. Math. Soc., 2024
D. I. Borisov, R. R. Suleimanov, “On operator estimates for elliptic operators with mixed boundary conditions in two-dimensional domains with rapidly oscillating boundary”, Math. Notes, 116:2 (2024), 182–199
Saoussen Boujemaa, Abdessatar Khelifi, “Asymptotic expansion for solution of Maxwell equation in domain with highly oscillating boundary”, Z Angew Math Mech, 103:10 (2023)
Siwar Saidani, Abdessatar Khelifi, “Eigenoscillations of the Maxwell equation in a domain with oscillating boundary”, Complex Variables and Elliptic Equations, 2023, 1
Khelifi A., Jaouabi A., “On the Asymptotic Formulas For Perturbations in the Eigenvalues of the Stokes Equations Due to the Presence of Small Deformable Inclusions”, J. Appl. Anal., 28:1 (2022), 149–164
Kozlov V., Thim J., “Hadamard Asymptotics For Eigenvalues of the Dirichlet Laplacian”, J. Math. Pures Appl., 140 (2020), 67–88
Cardone G., “Waveguides With Fast Oscillating Boundary”, Nanosyst.-Phys. Chem. Math., 8:2 (2017), 160–165
Kozlov V., Thim J., “Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators”, J. Spectr. Theory, 6:1 (2016), 99–135
Haddad J., Montenegro M., “on Differentiability of Eigenvalues of Second Order Elliptic Operators on Non-Smooth Domains”, J. Differ. Equ., 259:1 (2015), 408–421
Thim J., “Asymptotics of Hadamard Type For Eigenvalues of the Neumann Problem on C-1-Domains For Elliptic Operators”, Anal. PDE, 8:7 (2015), 1695–1706
Chechkina A., Pankratova I., Pettersson K., “Spectral Asymptotics For a Singularly Perturbed Fourth Order Locally Periodic Elliptic Operator”, Asymptotic Anal., 93:1-2 (2015), 141–160
Borisov D. Cardone G. Faella L. Perugia C., “Uniform Resolvent Convergence for Strip with Fast Oscillating Boundary”, J. Differ. Equ., 255:12 (2013), 4378–4402
Kozlov V., “Domain Dependence of Eigenvalues of Elliptic Type Operators”, Math. Ann., 357:4 (2013), 1509–1539

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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