Analysis on Fractals

关于分数的Strichartz分析研究定义在分数上的函数的性质，以某些类似的微分方程式为中心。在有限类自相似分形上，可以构造起拉普拉斯算子作用的算子，拉普拉斯算子是传统欧氏空间和流形分析的中心对象之一。本项目将尝试将拉普拉斯算子的构造推广到更广泛的分形类，并探索这些拉普拉斯算子和其他微分型算子的基本性质。此外，这个项目将尝试扩大这一新的研究领域与更传统的领域之间的联系，包括全局分析、调和分析、样条和有限元方法、伪微分算子和特殊函数。这个项目包括与本科生(REU)在计算机实验方面的合作，以探索例子和产生猜想。预计这项实验工作将导致对这一主题的更深层次的理论理解。不同领域的科学家已经意识到，现实世界中的许多物体都可以用分形学来建模。为了研究这类分形物体上的物理过程，有必要发展一种数学理论，它将给出关于分形的微分方程式的模拟。这样的理论还处于发展的早期阶段，这个项目将为这一发展做出贡献，将数学其他领域的思想和技术引入到问题中。预计该项目将开发对应用数学家和科学家有用的工具，因为该项目的一部分侧重于解决具体问题的数值方法和算法。本科生对该项目的贡献将是巨大的，反过来对学生来说将是一次积极的教育体验。

AbstractStrichartz Analysis on fractals studies the properties of functions defined on fractals, centering on certain analogs of differential equations. On a limited class of self-similar fractals it is possible to construct operators that play the role of the Laplacian, one of the central objects in traditional analysis on Euclidean spaces and manifolds. This project will attempt to extend the construction of Laplacians to a broader class of fractals, and to explore the fundamental properties of these Laplacians and other differential-type operators. In addition, this project will attempt to broaden the connections between this new area of research and more traditional areas, including global analysis, harmonic analysis, splines and the finite element method, pseudodifferential operators, and special functions. This project includes collaborative work with undergraduate students (REU) on computer experiments to explore examples and generate conjectures. It is expected that this experimental work will lead to deeper theoretical understanding of the subject. Scientists in diverse areas have realized that many objects in the real world can be modeled by fractals. In order to study physical processes on such fractal objects, it is necessary to develop a mathematical theory that will give the analog of differential equations on fractals. Such a theory is in the early stages of development, and this project will contribute to that development, bringing to the problem ideas and techniques from other areas of mathematics. It is expected that this project will develop tools that will be of use to applied mathematicians and scientists, since part of the project focuses on numerical methods and algorithms for solving concrete problems. The contribution of undergraduate students to the project will be significant, and in turn will be a positive educational experience for the students.