论文标题
Navier-Lamé操作员的几何不变
Geometric invariants of spectrum of the Navier-Lamé operator
论文作者
论文摘要
对于紧凑型连接的Riemannian $ n $ -Manifold $（ω，g）$，边界平稳，我们明确地计算了前两个系数$ a_0 $和$ a_1 $的渐近扩张的$ \ sum_ {k = 1} a_1 t^{ - （n-1）/2}+a_2^\ mp t^{ - （n-2）/2}+\ cdots+a_m^\ mp t^{ - （n-m）/2}+o（t^{ - （n-m-1）/2}）是$ k $ -th navier-laméeigenvalue在$ω$上与dirichlet（分别是诺伊曼）边界条件。这两个系数在Navier-Lamé操作员的光谱方面提供了弹性主体$ω$的体积和边界$ \ partialω$的表面积的精确信息。这给出了Avramidi在\ cite {avr10}中提到的有趣而开放的问题的答案。更重要的是，我们的方法在上述渐近扩展中明确计算所有系数$ a_l^\ mp $，$ 2 \ le l \ le m $。作为一种应用，我们表明，$ n $维球由其在具有光滑边界的所有有界弹性物体中的Navier-Lamé光谱决定。
For a compact connected Riemannian $n$-manifold $(Ω,g)$ with smooth boundary, we explicitly calculate the first two coefficients $a_0$ and $a_1$ of the asymptotic expansion of $\sum_{k=1}^\infty e^{-t τ_k^\mp}= a_0t^{-n/2} \mp a_1 t^{-(n-1)/2}+a_2^\mp t^{-(n-2)/2} +\cdots+ a_m^\mp t^{-(n-m)/2} +O(t^{-(n-m-1)/2})$ as $t\to 0^+$, where $τ^-_k$ (respectively, $τ^+_k$) is the $k$-th Navier-Lamé eigenvalue on $Ω$ with Dirichlet (respectively, Neumann) boundary condition. These two coefficients 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 operator. This gives an answer to an interesting and open problem mentioned by Avramidi in \cite{Avr10}. More importantly, our method is valid to explicitly calculate all the coefficients $a_l^\mp$, $2\le l\le m$, in the above asymptotic expansion. 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.
