Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture
Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture
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DOI：
10.1090/s0002-9947-1991-0994168-5
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发表时间：
1991-02
影响因子：
1.3
通讯作者：
M. Lapidus
中科院分区：
文献类型：
--
作者：
M. Lapidus
设E”（n > 1）是一个有界开集，其边界为“分形”边界T .本文推广了Hermann Weyl的经典定理，建立了2 m（m > 1）阶正椭圆算子在λ 2上特征值渐近性的精确余估计.我们考虑Dirichlet和Neumann边界条件。我们的估计，这是表示在闵可夫斯基，而不是Hausdorff尺寸的Y-指定和部分解决了Weyl-Berry猜想的拉普拉斯算子的特征值。贝瑞的分形理论--延伸到了“分形”--与卡茨的问题“人们能听到鼓的形状吗？”此外，它具有重要的物理应用，例如对“分形”表面的波散射或多孔介质的研究。我们还推导出新的剩余估计的渐近的相关的“分区功能”（或跟踪的热半群）。此外，我们提供的例子表明，我们的剩余估计是尖锐的，在每一个可能的“分形”（即，Minkowski）维。在本文中使用的技术属于偏微分方程理论，变分法，近似理论和在较小程度上几何测量理论。这项工作的一个有趣的方面是，它建立了光谱和“分形”几何之间的新联系。
Let Í2 be a bounded open set of E" (n > 1) with "fractal" boundary T . We extend Hermann Weyl's classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order 2m (m > 1) on Í2 . We consider both Dirichlet and Neumann boundary conditions. Our estimate—which is expressed in terms of the Minkowski rather than the Hausdorff dimension of Y—specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry's conjecture—which extends to "fractals" Weyl's conjecture—is closely related to Kac's question "Can one hear the shape of a drum?"; further, it has significant physical applications, for example to the scattering of waves by "fractal" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated "partition function" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible "fractal" (i.e., Minkowski) dimension. The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and—to a lesser extent—geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and "fractal" geometry.