Metric geometry and functions of bounded variation

度量几何和有界变分函数

• 批准号: 1200915
• 负责人: Nageswari Shanmugalingam
• 金额: $ 23.78万
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1200915&HistoricalAwards=false
• 关键词: Metric; geometry; functions; bounded; variation

Objects occurring in nature rarely are smooth in appearance. From fractal objects to porous media, the equations that govern dissipation of quantities such as heat and pressure have non-smooth components. Such non-smooth objects also occur as limits of certain smooth objects. To understand and exploit the behavior of such objects, we need to remove the smoothness assumptions from Riemannian geometry. For such non-smooth objects we need to study behaviors of non-smooth energy minimizers and their connection to the geometry, that is, the structural properties, of the underlying space or object. This project on metric geometry and functions on bounded variation contributes to this goal by exploring interconnections between the geometry of the underlying metric space equipped with a measure, and the properties of sets of minimal surface areas. The study of objects here are metric spaces equipped with a measure that is doubling (that is, measure of a ball is comparable to the measure of a concentric ball of double the radius) and such that the variance of a Lipschitz continuous function on a ball can be controlled in terms of the average value of its energy (computed using its local oscillation) on that ball. In this setting, this project aims to study properties such as rectifiability, porosity, shape, and natural dimension of sets of locally minimal surface areas. Analysis on metric measure spaces arose from many sources; the study of complex analytic functions, differential equations governing fractures in mixed material and associated mappings of finite distortion, control theory in engineering and associated Carnot-Caratheodory spaces, the study of the famous Poincare conjecture, are some of the roots of this field of study. However, unlike in the Euclidean situation where structural properties of minimal surfaces (surfaces with smallest surface energy) are well understood, in the non-smooth setting that arise in nature and in control theory problems of physics and engineering, the structure of minimal surfaces is poorly understood. This project seeks to explore such structures and expand our knowledge of minimal surfaces in a non-smooth setting. The results of this study will be useful in further understanding the change in the behavior of objects that are transformed due to natural effects such as heat and pressure. In addition to advancing our knowledge of non-smooth objects, the study conducted in this project will also contribute to the ongoing development of the theory of analysis on metric spaces; a great benefit of this developing theory is that since much of the structural tools available in classical Euclidean analysis are not available in the non-smooth setting, new tools and methods have necessarily to be developed, making the theory more accessible to a wider audience.

自然界中发生的物体的外观很少。从分形对象到多孔培养基，控制热量和压力等数量耗散的方程具有非平滑成分。这种非平滑物体也作为某些平滑对象的限制出现。为了理解和利用此类对象的行为，我们需要删除Riemannian几何形状的平滑度假设。对于此类非平滑物体，我们需要研究非平滑能量最小化器的行为及其与几何形状的联系，即基础空间或物体的结构特性。这项关于度量几何形状和有限变化的功能的项目通过探索配备了度量的基础公制空间的几何形状与最小表面积集合的特性之间的互连来促进这一目标。这里对物体的研究是配备了两倍的度量的度量空间（也就是说，球的度量与半径双倍的同心球的度量相媲美），因此，可以根据其能量的平均值（使用其本地振动计算）在球上控制Lipschitz连续函数的差异。在这种情况下，该项目旨在研究局部最小表面区域集的可重新可覆盖，孔隙率，形状和自然维度等属性。 对度量空间的分析来自许多来源。对混合材料中裂缝的复杂分析功能，有限扭曲，工程控制理论以及相关的Carnot-Caratheodory空间的控制理论的研究的研究，对著名的Poincare猜想的研究是该领域的一些根源。然而，与欧几里得的情况不同，在欧几里得的情况下，在自然界中出现的非平滑环境以及物理学和工程的控制理论问题中，最小的表面环境中，人们对最小表面的结构特性（表面最小的表面）不同，最小表面的结构知之甚少。该项目旨在探索这种结构，并在非平滑环境中扩大我们对最小表面的了解。这项研究的结果将有助于进一步理解由于热和压力等自然效应而转化的物体行为的变化。除了促进我们对非平滑物体的了解外，该项目进行的研究还将有助于对度量空间分析理论的持续发展；这种发展理论的一个很大的好处是，由于在非平滑环境中不可用古典欧几里得分析中可用的许多结构工具，因此必须开发新的工具和方法，从而使该理论更容易被更广泛的受众访问。