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Russian Mathematical Surveys, 1973, Volume 28, Issue 3, Pages 45–88
DOI: https://doi.org/10.1070/RM1973v028n03ABEH001563
(Mi rm4889)
 

This article is cited in 36 scientific papers (total in 37 papers)

The spectrum of a family of operators in the theory of elasticity

S. G. Mikhlin
References:
Abstract: The vector equation of the static theory of elasticity for a homogeneous isotropic medium is
Δu+graddivu=F(x),
where ω(12σ)1, and σ is Poisson's constant, ω being treated as a spectral parameter. This is then the problem: to examine the spectrum of the family of operators on the left-hand side of (1) for boundary conditions of first or second kind. The problem was first posed at the end of the 19th century by Eugéne and Franзois Cosserat; it has been investigated in recent years by V. G. Maz'ya and the present author. The main results obtained are for an elastic domain Ω, which may be finite, or infinite with a sufficiently smooth finite boundary. In the case of the first boundary-value problem the family operators of the theory of elasticity has a countable system of eigenvectors, orthogonal in the metric of the Dirichlet integral; this system is complete in each of the spaces W(1)2(Ω) and \L2(Ω). The eigenvalues condense at the three points ω=1,2,; ω=1 and ω= are isolated eigenvalues of infinite multiplicity. Similar results are obtained also, for the second boundary-value problem. The essential difference lies in the fact that in this case the eigenvalues have one further condensation point ω=0, and examples show that ω=2 need not be a point of condensation for eigenvalues of the second problem.
Received: 26.01.1973
Bibliographic databases:
Document Type: Article
UDC: 517.9:539.3
Language: English
Original paper language: Russian
Citation: S. G. Mikhlin, “The spectrum of a family of operators in the theory of elasticity”, Russian Math. Surveys, 28:3 (1973), 45–88
Citation in format AMSBIB
\Bibitem{Mik73}
\by S.~G.~Mikhlin
\paper The spectrum of a~family of operators in the theory of elasticity
\jour Russian Math. Surveys
\yr 1973
\vol 28
\issue 3
\pages 45--88
\mathnet{http://mi.mathnet.ru/eng/rm4889}
\crossref{https://doi.org/10.1070/RM1973v028n03ABEH001563}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=415422}
\zmath{https://zbmath.org/?q=an:0291.35065}
Linking options:
  • https://www.mathnet.ru/eng/rm4889
  • https://doi.org/10.1070/RM1973v028n03ABEH001563
  • https://www.mathnet.ru/eng/rm/v28/i3/p43
  • This publication is cited in the following 37 articles:
    1. D. Riccobelli, P. Ciarletta, G. Vitale, C. Maurini, L. Truskinovsky, “Elastic Instability behind Brittle Fracture”, Phys. Rev. Lett., 132:24 (2024)  crossref
    2. Victor A. Eremeyev, “On well‐posedness of the first boundary‐value problem within linear isotropic Toupin–Mindlin strain gradient elasticity and constraints for elastic moduli”, Z Angew Math Mech, 103:6 (2023)  crossref
    3. Henry C. Simpson, “The Complementing and Agmon's Conditions in Finite Elasticity”, J Elast, 139:1 (2020), 1  crossref
    4. Yu. V. Tokovyy, “Cosserat Spectrum of an Axisymmetric Elasticity Problem for a Finite-Length Solid Cylinder”, J. mech., 35:3 (2019), 343  crossref
    5. D. A. Zakora, “Model szhimaemoi zhidkosti Maksvella”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 63, no. 2, Rossiiskii universitet druzhby narodov, M., 2017, 247–265  mathnet  crossref  mathscinet
    6. Alexander E. Kolesov, Petr N. Vabishchevich, “Splitting schemes with respect to physical processes for double-porosity poroelasticity problems”, Russian Journal of Numerical Analysis and Mathematical Modelling, 32:2 (2017), 99  crossref
    7. D. A. Zakora, “Model szhimaemoi zhidkosti Oldroita”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 41–66  mathnet
    8. Costabel M., Crouzeix M., Dauge M., Lafranche Y., “The Inf-Sup Constant For the Divergence on Corner Domains”, Numer. Meth. Part Differ. Equ., 31:2, SI (2015), 439–458  crossref  isi
    9. Martin Sprengel, “Domain robust preconditioning for a staggered grid discretization of the Stokes equations”, Journal of Computational and Applied Mathematics, 2013  crossref
    10. E. V. Chizhonkov, “Numerical solution to a stokes interface problem”, Comput. Math. Math. Phys., 49:1 (2009), 105–116  mathnet  crossref  mathscinet  isi  elib  elib
    11. Erofeev V.I., “Bratya Kossera i mekhanika obobschennykh kontinuumov”, Vychislitelnaya mekhanika sploshnykh sred, 2:4 (2009), 5–10  elib
    12. Christian G. Simader, Wolf von Wahl, “Introduction to the Cosserat problem”, Analysis, 26:1 (2006), 1  crossref  mathscinet
    13. Emil Ernst, “On the Existence of Positive Eigenvalues for the Isotropic Linear Elasticity System with Negative Shear Modulus”, Communications in Partial Differential Equations, 29:11-12 (2005), 1745  crossref
    14. Manfred Dobrowolski, “On the LBB condition in the numerical analysis of the Stokes equations”, Applied Numerical Mathematics, 54:3-4 (2005), 314  crossref
    15. N. S. Bakhvalov, A. V. Knyazev, R. R. Parashkevov, “Extension theorems for Stokes and Lamé equations for nearly incompressible media and their applications to numerical solution of problems with highly discontinuous coefficients”, Numer Linear Algebra Appl, 9:2 (2002), 115  crossref  mathscinet  zmath  isi  elib
    16. M. A. Ol'shanskii, E. V. Chizhonkov, “On the best constant in the inf-sup condition for elongated rectangular domains”, Math. Notes, 67:3 (2000), 325–332  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. Alexander Kozhevnikov, “On a Lower Bound of the Cosserat Spectrum for the Second Boundary Value Problem of Elastostatics”, Applicable Analysis, 74:3-4 (2000), 301  crossref
    18. W. Liu, X. Markenscoff, “The Cosserat spectrum for cylindrical geometries”, International Journal of Solids and Structures, 37:8 (2000), 1165  crossref
    19. W. Liu, X. Markenscoff, “The Cosserat spectrum for cylindrical geometries”, International Journal of Solids and Structures, 37:8 (2000), 1177  crossref
    20. W. Liu, A. Plotkin, “Application of the Cosserat Spectrum Theory to Stokes Flow”, J Appl Mech, 66:3 (1999), 811  crossref  isi  elib
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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