人们能听到鼓的密度吗？非均匀介质的外尔定律
Can one hear the density of a drum? Weyl's law for inhomogeneous media
摘要：
我们将外尔定律推广到$d$维的非均匀体。利用我们最近在文献\cite{Amore09}中得到的一个扰动格式，我们导出了一个描述$d$维三次封闭区域上负Laplacian特征值的渐近性态的显式公式，该特征值具有Dirichlet或Neumann边界条件.对于均质物体，我们公式中的首项简化为外尔定律的标准表达式。我们还使用Weyl的猜想，以获得非微扰扩展我们的公式，我们比较了我们的分析结果与精确的数值结果使用共形配置方法的参考文献。\cite{Amore08,Amore09}。
英文摘要：
We generalize Weyl's law to inhomogeneous bodies in $d$ dimensions. Using a perturbation scheme recently obtained by us in Ref. \cite{Amore09}, we have derived an explicit formula, which describes the asymptotic behavior of the eigenvalues of the negative laplacian on a closed $d$-dimensional cubic domain, either with Dirichlet or Neumann boundary conditions. For homogeneous bodies, the leading term in our formula reduces to the standard expression for Weyl's law. We have also used Weyl's conjecture to obtain a non-perturbative extension of our formula and we have compared our analytical results with the precise numerical results obtained using the Conformal Collocation Method of Refs. \cite{Amore08,Amore09}.
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