Geometry of Measures

摘要：20世纪60年代，当费德勒的著作《几何测量理论》出版时，被称为几何测量理论(GMT)的分析领域正式引入。自本世纪初以来，许多作家在这一领域做了大量的相关工作，包括贝西科维奇、卡拉斯艾奥里、德乔吉、费德勒本人、弗莱明、马斯特兰德、莫里、莱芬伯格和惠特尼。然而，直到费德勒的书出版后，人们才清楚地知道，所有这些结果是如何自然地结合在一起，成为一个有凝聚力的主题。不幸的是，自问世以来，这个主题一直被认为有点神秘和难以理解。在这项建议中，研究人员利用了该领域的突出特征之一，即其多功能性。PI的目的是应用GMT技术来研究自由边界正则性问题，回答关于某些度量空间是否存在好的参数化的问题，并对支持‘’勒贝格‘型’的集合进行分类。自由边界问题在物理和工程中是自然产生的。自由边界可能表现为流体与空气或水与冰之间的界面。在过滤问题中，研究水如何从由多孔介质(比如土壤)组成的大坝中过滤出来，自由边界将湿部分与干部分分开。刻画自由边界正则性的问题已被许多作者研究过，其中包括Alt，Caffarelli和Friedman。在这个提议中，研究者在非常弱的自由边界假设下解决了这个问题。通过放松规律性假设，可以涵盖更广泛的物理问题。研究者还建议研究支撑度量的集合的结构，这些度量的行为类似于仿射空间上的Lesbeguemure。在弱正则性概念的背景下，这些集合扮演着切平面的角色。特别是，对这些集合进行分类的问题是首要的。这里讨论的想法属于一个更大的项目，该项目旨在确立正则性的弱概念对于许多目的足以回答分析和几何中的基本问题。

Abstract:The area of analysis called geometric measure theory (GMT) was formallyintroduced in the nineteen sixties, when the book ``Geometric MeasureTheory'' by H. Federer was published. Since the beginning of the century alarge amount of relevant work had been done in this area by many authorsincluding Besicovitch, Carath\'eodory, De Giorgi, Federer himself,Fleming, Marstrand, Morrey, Reifenberg, and Whitney. However it was onlywhen Federer's book was published that it became clear how all theseresults fit naturally together as a cohesive subject. Unfortunately sinceits introduction the subject has been perceived as somewhat mysterious andinaccessible. In this proposal the investigator exploits one of theoutstanding features of the field, its versatility. The PI intends toapply techniques of GMT to study free boundary regularity problems, toanswer questions regarding the existence of good parameterizations forcertain metric spaces and to classify sets supporting ``Lebesgue-type''measures.Free boundary problems arise naturally in physics and engineering. Thefree boundary may appear as the interface between a fluid and the air, orwater and ice. In the filtration problem, which studies how waterfiltrates from a dam made of a porous medium (say earth), the freeboundary separates the wet part from the dry part. The problem ofcharacterizing the regularity of the free boundary has been studied bymany authors, among others Alt, Caffarelli and Friedman. In this proposalthe investigator addresses this question under very weak free boundaryassumptions. By relaxing the regularity hypothesis a broader spectrum ofphysical problems can be covered. The investigator also proposes to studythe structure of sets supporting measures that behave like Lesbeguemeasure on an affine space. In the context of weak notions of regularity,these sets play the role of tangent planes. In particular the problem ofclassifying these sets is of primary importance. The ideas discussed herebelong to a larger project that intends to establish that weak notions ofregularity are for many purposes sufficient to answer basic questions inanalysis and geometry.