Abstract:
The vector equation of the static theory of elasticity for a homogeneous isotropic medium is
Δu+graddivu=F(x),
where ω(1−2σ)−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.
\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}
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