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arXiv · 1512.07552

The vibrational frequencies of the elastic body and its geometric quantities

Genqian Liu
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Abstract

For a bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the trace of the strongly continuous semigroup associated with the Navier-Lamé operator on $Ω$ as $t\to 0^+$. These coefficients (i.e., spectral invariants) provide precise information for the volume of the elastic body $Ω$ and the surface area of the boundary $\partial Ω$ in terms of the spectrum of the Navier-Lamé problem. As an application, we show that an $n$-dimensional ball is uniquely determined by its Navier-Lamé spectrum among all bounded elastic body with smooth boundary.

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Genqian Liu. 2015-12-23. The vibrational frequencies of the elastic body and its geometric quantities. https://arxiv.org/abs/1512.07552

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