人们可以通过听到Steklov特征值来听到区域边界的面积和曲率
One can hear the area and curvature of boundary of a domain by hearing the Steklov eigenvalues
摘要：
对于光滑黎曼流形$(\mathcal{M},g)$中给定的具有光滑边界的有界区域$Ω$，我们证明了Dirichlet-to-Neumann算子的热核的Poisson型上估计、Sobolev迹不等式、Log-Sobolev迹不等式、Nash迹不等式和Rozenblum-Lieb-Cwikel型不等式都是等价的.将Dirichlet-to-Neumann算子分解为Laplacian算子的平方根和一个伪微分算子，并应用Grubb的符号微积分方法，建立了一个计算Dirichlet-to-Neumann算子的热核迹的渐近展开式的所有系数的方法，即$t\to 0^+$ .特别地，我们明确地给出了这个渐近展开的前四个系数。这些系数给出了精确的信息，关于该领域的边界的Steklov问题的频谱和曲率。
英文摘要：
For a given bounded domain $Ω$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type upper-estimate of the heat kernel associated to the Dirichlet-to-Neumann operator, the Sobolev trace inequality, the Log-Sobolev trace inequality, the Nash trace inequality, and the Rozenblum-Lieb-Cwikel type inequality are all equivalent. Upon decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferntial operator and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as $t\to 0^+$. In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients give precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.
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