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.

用于紧凑型连接黎曼Ñ -manifold \（（\欧米茄，G）\）具有光滑边界，我们明确地计算所述第一两个系数\（A_0 \）和\（A_1 \）的渐近扩展\（\总和_ {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}）\）为\（t \ rightarrow 0 ^ + \），其中\（\ tau ^ -_ k \）（分别为\（\ tau ^ + _k \））是具有Dirichlet（分别是诺伊曼）边界条件的\（\ Omega \）上的第k个Navier–Lamé特征值。这两个系数为弹性体\（\ Omega \）的体积提供了精确的信息以及根据Navier–Lamé算子的频谱，边界\（\ partial \ Omega \）的表面积。这给出了Avramidi在（非椭圆形算子在具有椭圆算子的边界，分析，几何和拓扑的流形上的非拉普拉斯算子。世界科学出版社，哈肯萨克，第107–140页，2006年）中提到的一个有趣且开放的问题的答案。 。作为一个应用，我们表明，n维球是由其Navier–Lamé谱唯一地确定于所有边界光滑的有界弹性体中的。