The vector equation of the static theory of elasticity for a homogeneous isotropic medium is (1)where , and is Poisson's constant, 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 Eug?ne and Fran?ois 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 , 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 and . The eigenvalues condense at the three points ; and 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 , and examples show that need not be a point of condensation for eigenvalues of the second problem.
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