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

S. G. Mikhlin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 28 (1973) no. 3, pp. 45-88

Résumé

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 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 $\Omega$, 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 $\L_2(\Omega)$. 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.