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Geometric Invariants of Spectrum of the Navier–Lamé Operator

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Abstract

For a compact connected Riemannian n-manifold \((\Omega ,g)\) with smooth boundary, we explicitly calculate the first two coefficients \(a_0\) and \(a_1\) of the asymptotic expansion of \(\sum _{k=1}^\infty \mathrm{{e}}^{-t \tau _k^{\mp }}= a_0t^{-n/2} {\mp } a_1 t^{-(n-1)/2} +O(t^{1-n/2})\) as \(t\rightarrow 0^+\), where \(\tau ^-_k\) (respectively, \(\tau ^+_k\)) is the k-th Navier–Lamé eigenvalue on \(\Omega \) with Dirichlet (respectively, Neumann) boundary condition. These two coefficients provide precise information for the volume of the elastic body \(\Omega \) and the surface area of the boundary \(\partial \Omega \) in terms of the spectrum of the Navier–Lamé operator. This gives an answer to an interesting and open problem mentioned by Avramidi in (Non-Laplace type operators on manifolds with boundary, analysis, geometry and topology of elliptic operators. World Sci. Publ., Hackensack, pp. 107–140, 2006). As an application, we show that an n-dimensional ball is uniquely determined by its Navier–Lamé spectrum among all bounded elastic bodies with smooth boundary.

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Acknowledgements

This research was supported by NNSF of China (11671033/A010802) and NNSF of China (11171023/A010801).

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  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Genqian Liu

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Liu, G. Geometric Invariants of Spectrum of the Navier–Lamé Operator. J Geom Anal 31, 10164–10193 (2021). https://doi.org/10.1007/s12220-021-00639-8

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