Analysis, Geometry, and Spectral Theory On or Off Fractals

分形或非分形的分析、几何和谱理论

• 批准号: 0070497
• 负责人: Michel Lapidus
• 金额: $ 8.7万
• 依托单位: University of California-Riverside
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070497&HistoricalAwards=false
• 关键词: Analysis; Geometry; Spectral; Theory; Off

ABSTRACTThe PI intends to pursue and amplify his investigations of therelationships between spectral and fractal geometry. We plan to study thevibrations of "fractal drums", both "drums with fractal boundary"(Laplacians on open sets with very irregular boundary) and "drums withfractal membrane" (Laplacians on fractals themselves). The proposedproblems are closely connected to Kac's question "Can one hear the shape ofa drum?" and to its beautiful extensions from the "smooth" to the "fractal"domain by the physicist Michael Berry. Although the proposed theory ismathematically rigorous, it is also naturally physically motivated (with,for example, applications to the scattering of waves by fractal surfacesand the study of porous media), and has recently drawn some of its impetusfrom the use of computer graphics. Moreover, we propose to use and extendthe theory of "complex dimensions" of "fractal strings" (one-dimensionaldrums with fractal boundary)-recently developed extensively by the PI andMachiel van Frankenhuysen in the research monograph [La-vF2] on "FractalGeometry and Number Theory" and motivated in part by the PI's earlier joint work with Carl Pomerance [LaPo] and Helmut Maier [LaMa] on (direct and)inverse spectral problems for fractal strings and the Riemann hypothesis-inorder to study the fascinating oscillatory phenomena occurring in thegeometry and in the spectrum of "drums with fractal boundary" and of "drumswith fractal membrane". ("Complex dimensions" are defined as the poles ofa suitable geometric zeta function. Further, in [La-vF], a detailed studyof their structure is given, for example, in the case of self-similar fractal strings.)We plan to further develop analysis and spectral theory on fractals and onregions with fractal boundary, as well as to investigate problems of a'dynamical nature', of physical significance in condensed matter and solidstate physics; for example, in the study of mechanical or electricaltransport in porous or in random media, as well as of heat diffusions onfractals and in disordered systems. We also intend to pursue ourmathematical and computer graphics-aided study ([LaPa], [LaNRG]) of partialdifferential equations (PDEs)-such as the Laplace, heat and (linear ornonlinear) wave equations-on regions with fractal boundary or on fractalsthemselves. According to appealing experiments and interpretations by thephysicist Bernard Sapoval, this work may help understand the formation offractal structures (for example, coastlines, trees and blood vessels) innature. In the long term, it is hoped that the tools and results developedin this project (and in the PI's earlier investigations) will help us toprobe more deeply than has been so far possible the fine geometricstructure of fractals and of related 'objects' occurring in mathematicsand in physics.

PI打算继续并扩大他对光谱和分形几何之间关系的研究。 我们计划研究“分形鼓”的振动，包括“具有分形边界的鼓”（具有非常不规则边界的开集上的拉普拉斯算子）和“具有分形膜的鼓”（分形本身上的拉普拉斯算子）。 所提出的问题是密切相关的卡茨的问题“一个人能听到鼓的形状吗？以及物理学家迈克尔·贝里从“光滑”域到“分形“域的美丽扩展。 虽然提出的理论在数学上是严格的，但它也有自然的物理动机（例如，应用于分形表面的波散射和多孔介质的研究），并且最近从计算机图形学的使用中获得了一些动力。 此外，委员会认为，我们建议使用和扩展“分形弦”的“复维数”理论，（一维鼓与分形边界）-最近由PI和Machiel货车Frankenhuysen在研究专著[La-vF 2]中广泛开发的“分形几何与数论”，部分动机是PI与Carl Pomerance [LaPo]和Helmut Maier [LaMa]的早期联合工作，分形弦的正、逆谱问题和Riemann假设--为了研究分形弦的几何结构和分形边界鼓和分形膜鼓的谱中的迷人振荡现象。 （“复维数”被定义为一个合适的几何zeta函数的极点。 此外，在[La-vF]中，以自相似分形弦为例，详细研究了它们的结构。我们计划进一步发展分形和分形边界区域的分析和光谱理论，以及研究凝聚态和固态物理中具有物理意义的“动力学性质”问题;例如，在多孔或随机介质中的机械或电输运研究中，以及在分形和无序系统中的热扩散。 我们还打算继续我们的数学和计算机图形辅助研究（[LaPa]，[LaNRG]）的偏微分方程（PDE）-如拉普拉斯，热和（线性或非线性）波动方程-区域分形边界或分形本身。 根据物理学家Bernard Sapoval的实验和解释，这项工作可能有助于理解自然界中分形结构（例如海岸线，树木和血管）的形成。 从长远来看，希望这个项目（以及PI早期的研究）中开发的工具和结果将帮助我们比迄今为止更深入地探索分形的精细几何结构以及在南极学和物理学中发生的相关“对象”。