The vibrational frequencies of the elastic body and its geometric quantities

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For a bounded domain Ω⊂ Rⁿ 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→ 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 ∂ 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.