## Spectral properties of certain stiff problems in elasticity and acoustics.(English)Zbl 0561.35059

The perturbation solutions of vibration of an elastic body with small compressibility and the acoustic vibrations of a fluid with small viscosity have been discussed in this paper. The two applied problems have been discussed in an abstract setting. The spectral properties and the asymptotics of the governing elliptic system for the elastic body and its eigenvalues have been studied by means of a uniformly convergent expansion of the stiff type.
After a rescaling of the spectral parameter, the viscous acoustic problem has been shown to be analogous to the elastic problem. It was shown that as the perturbation parameter $$\epsilon$$ $$\to 0$$, infinitely many eigenvalues converge to 0 which is an eigenvalue of the corresponding inviscid problem. The effects of both Dirichlet and mixed boundary conditions have been discussed.
Reviewer: P.Chandran

### MSC:

 35P20 Asymptotic distributions of eigenvalues in context of PDEs 74B05 Classical linear elasticity 76Q05 Hydro- and aero-acoustics 35C20 Asymptotic expansions of solutions to PDEs 35B20 Perturbations in context of PDEs
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