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On the spectrum of an elastic solid with cusps. (English) Zbl 1375.35543

Summary: The spectral problem of anisotropic elasticity with traction-free boundary conditions is considered in a bounded domain with a spatial cusp having its vertex at the origin and given by triples \((x_1,x_2,x_3)\) such that \(x_3^{-2}(x_1,x_2) \in \omega\), where \(\omega\) is a two-dimensional Lipschitz domain with a compact closure. We show that there exists a threshold \(\lambda_!>0\) expressed explicitly in terms of the elasticity constants and the area of \(\omega\) such that the continuous spectrum coincides with the half-line \([\lambda_!,\infty)\), whereas the interval \([0,\lambda_!)\) contains only the discrete spectrum. The asymptotic formula for solutions to this spectral problem near cusp’s vertex is also derived. A principle feature of this asymptotic formula is the dependence of the leading term on the spectral parameter.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35J57 Boundary value problems for second-order elliptic systems
35P05 General topics in linear spectral theory for PDEs
35P15 Estimates of eigenvalues in context of PDEs
74G55 Qualitative behavior of solutions of equilibrium problems in solid mechanics
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Full Text: Euclid