Analysis on Fractals

分形分析是以微分方程的某些类似物为中心，研究定义在分形上的函数的性质。 在有限的一类自相似分形上，可以构造起拉普拉斯算子作用的算子，拉普拉斯算子是欧几里得空间和流形上传统分析的中心对象之一。 这个项目将尝试将拉普拉斯算子的构造扩展到更广泛的分形类，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，该项目将试图扩大这一新的研究领域和更传统的领域，包括全球分析，调和分析，样条和有限元方法，伪微分算子和特殊功能之间的联系。 该项目包括与本科生（REU）合作进行计算机实验，以探索示例并生成图表。 预计这项实验工作将导致更深入的理论理解的主题。 不同领域的科学家已经认识到，真实的世界中的许多物体都可以用分形来建模。 为了研究这种分形物体上的物理过程，有必要发展一种数学理论，它将给出分形微分方程的模拟。 这样的理论处于发展的早期阶段，这个项目将有助于发展，从数学的其他领域带来的问题的想法和技术。 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显着的，反过来将是一个积极的教育经验的学生。

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.