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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
\begin{equation} \label{1} \Delta u+\operatorname{grad}\operatorname{div}u=F(x), \end{equation}
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 Franois 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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath
    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  crossref  mathscinet  zmath  isi
    5. C. O. Horgan, “Korn’s Inequalities and Their Applications in Continuum Mechanics”, SIAM Rev, 37:4 (1995), 491  crossref  mathscinet  zmath  isi
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    19. Christian G. Simader, Wolf von Wahl, “Introduction to the Cosserat problem”, Analysis, 26:1 (2006), 1  crossref  mathscinet
    20. 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
    21. Erofeev V.I., “Bratya Kossera i mekhanika obobschennykh kontinuumov”, Vychislitelnaya mekhanika sploshnykh sred, 2:4 (2009), 5–10  elib
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