Geometry of Measures

翻译后摘要：几何测度理论（GMT）的分析领域正式介绍了在二十世纪六十年代，当书“几何测度理论”由H。Federer出版了。自世纪开始大量的相关工作已经完成了在这一领域的许多authorsincluding贝西科维奇，Carath\'eodory，德乔治，费德勒本人，弗莱明，马斯特兰德，莫里，赖芬贝格，惠特尼。然而，只有当费德勒的书出版时，所有这些结果如何自然地结合在一起成为一个有凝聚力的主题才变得清晰。不幸的是，自从它被介绍以来，这个主题一直被认为有点神秘和难以理解。在这个建议中，调查人员利用了该领域的一个突出特点，即它的多功能性。PI打算应用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良好参数化的问题，并对支持“勒贝格型”测度的集合进行分类。自由边界可能表现为流体和空气，或水和冰之间的界面。在渗流问题中，研究水如何从由多孔介质（比如土）构成的堤坝中渗透出来，自由边界将湿的部分与干的部分分开。自由边界正则性的刻画问题已被许多作者研究过，其中包括Alt、Caffarelli和Friedman。在这个proposalthe调查员解决这个问题下非常弱的自由boundaryassumptions。通过放松正则性假设，可以涵盖更广泛的物理问题. 调查员还建议研究集的结构支持措施，表现得像Lesbeguembrane仿射空间。在弱正则性概念的背景下，这些集合扮演着切平面的角色。 特别是这些集的分类问题是首要的.这里讨论的思想属于一个更大的项目，旨在建立弱的正则性概念，对于许多目的来说，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.