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.

自然界中的物体很少是外观光滑的。从分形物体到多孔介质，控制热量和压力等量耗散的方程都具有非光滑成分。这种非光滑物体也作为某些光滑物体的限制而出现。为了理解和利用此类对象的行为，我们需要从黎曼几何中删除平滑度假设。对于这种非光滑物体，我们需要研究非光滑能量最小化器的行为及其与几何结构的联系，即底层空间或物体的结构特性。这个关于度量几何和有界变分函数的项目通过探索配备有度量的基础度量空间的几何形状与最小表面积集合的属性之间的互连来实现这一目标。这里研究的对象是配备有加倍测度的度量空间（即球的测度与半径两倍的同心球的测度相当），并且使得球上 Lipschitz 连续函数的方差可以根据该球上的能量平均值（使用其本地振荡计算）进行控制。在此背景下，该项目旨在研究局部最小表面积组的可整流性、孔隙率、形状和自然尺寸等属性。 对度量测度空间的分析有很多来源；复杂解析函数的研究、混合材料中控制裂缝的微分方程和有限变形的相关映射、工程控制理论和相关的卡诺-卡拉特奥多里空间、著名的庞加莱猜想的研究，是该研究领域的一些根源。然而，与欧几里得情况不同，在欧几里得情况下，最小表面（具有最小表面能的表面）的结构特性已被很好地理解，在自然界中以及物理和工程的控制理论问题中出现的非光滑环境中，最小表面的结构是不太了解。该项目旨在探索此类结构并扩展我们对非光滑环境中最小表面的知识。这项研究的结果将有助于进一步了解由于热和压力等自然效应而发生变化的物体的行为变化。除了增进我们对非光滑物体的了解之外，该项目中进行的研究还将有助于度量空间分析理论的不断发展；这种发展理论的一个巨大好处是，由于经典欧几里得分析中可用的许多结构工具在非光滑环境中不可用，因此必须开发新的工具和方法，使该理论更容易为更广泛的受众所理解。