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 Uspekhi Mat. Nauk, 1973, Volume 28, Issue 3(171), Pages 43–82 (Mi umn4889)

This article is cited in 24 scientific papers (total in 25 papers)

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

S. G. Mikhlin

Abstract: The vector equation of the static theory of elasticity for a homogeneous isotropic medium is
$$\label{1} \Delta u+\operatorname{grad}\operatorname{div}u=F(x),$$
where $\omega(1-2\sigma)^{-1}$, and $\sigma$ is Poisson's constant, $\omega$ 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 François 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 $\Omega$, 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 $\overset{\circ}W_2^{(1)}(\Omega)$ and $Ł_2(\Omega)$. The eigenvalues condense at the three points $\omega=-1,-2,\infty;$ $\omega=-1$ and $\omega=\infty$ 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 $\omega=0$, and examples show that $\omega=-2$ need not be a point of condensation for eigenvalues of the second problem.

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English version:
Russian Mathematical Surveys, 1973, 28:3, 45–88

Bibliographic databases:

UDC: 517.9:539.3
MSC: 74Bxx, 35J55, 35P05, 35A05
Received: 26.01.1973

Citation: S. G. Mikhlin, “The spectrum of a family of operators in the theory of elasticity”, Uspekhi Mat. Nauk, 28:3(171) (1973), 43–82; 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 Uspekhi Mat. Nauk \yr 1973 \vol 28 \issue 3(171) \pages 43--82 \mathnet{http://mi.mathnet.ru/umn4889} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=415422} \zmath{https://zbmath.org/?q=an:0291.35065} \transl \jour Russian Math. Surveys \yr 1973 \vol 28 \issue 3 \pages 45--88 \crossref{https://doi.org/10.1070/RM1973v028n03ABEH001563} 

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2. Rouben Rostamian, “Internal constraints in linear elasticity”, J Elast, 11:1 (1981), 11
3. G. Geymonat, M. Lobo-Hidalgo, E. Sanchez-Palencia, G. F. Roach, “Spectral properties of certain stiff problems in elasticity and acoustics”, Math Meth Appl Sci, 4:1 (1982), 291
4. Henry C. Simpson, Scott J. Spector, “On the positivity of the second variation in finite elasticity”, Arch Rational Mech Anal, 98:1 (1987), 1
5. C. O. Horgan, “Korn’s Inequalities and Their Applications in Continuum Mechanics”, SIAM Rev, 37:4 (1995), 491
6. Evgueni E. Ovtchinnikov, Leonidas S. Xanthis, “A new Korn's type inequality for thin domains and its application to iterative methods”, Computer Methods in Applied Mechanics and Engineering, 138:1-4 (1996), 299
7. M. A. Ol'shanskii, “On the Stokes problem with model boundary conditions”, Sb. Math., 188:4 (1997), 603–620
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10. Wensen Liu, X. Markenscoff, M. Paukshto, “The Cosserat Spectrum Theory in Thermoelasticity and Application to the Problem of Heat Flow Past a Rigid Spherical Inclusion”, J Appl Mech, 65:3 (1998), 614
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17. 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
18. Manfred Dobrowolski, “On the LBB condition in the numerical analysis of the Stokes equations”, Applied Numerical Mathematics, 54:3-4 (2005), 314
19. Christian G. Simader, Wolf von Wahl, “Introduction to the Cosserat problem”, Analysis, 26:1 (2006), 1
20. E. V. Chizhonkov, “Numerical solution to a stokes interface problem”, Comput. Math. Math. Phys., 49:1 (2009), 105–116
21. Erofeev V.I., “Bratya Kossera i mekhanika obobschennykh kontinuumov”, Vychislitelnaya mekhanika sploshnykh sred, 2:4 (2009), 5–10
22. Martin Sprengel, “Domain robust preconditioning for a staggered grid discretization of the Stokes equations”, Journal of Computational and Applied Mathematics, 2013
23. 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
24. D. A. Zakora, “Model szhimaemoi zhidkosti Oldroita”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 61, RUDN, M., 2016, 41–66
25. 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
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