Geometry of Measures

摘要：几何测量理论（GMT）这一分析领域是在20世纪60年代H.费德勒出版的《几何测量理论》一书中正式引入的。自本世纪初以来，许多作者在这一领域做了大量的相关工作，包括贝西科维奇、卡拉多里、德乔吉、费德勒本人、弗莱明、马斯特兰德、莫雷、赖芬伯格和惠特尼。然而，直到费德勒的书出版后，人们才明白所有这些结果是如何自然地结合在一起，形成一个有凝聚力的主题。不幸的是，自从这门学科被引入以来，它一直被认为是有点神秘和难以理解的。在这个建议中，研究者利用了该领域的一个突出特征，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。自由边界问题在物理学和工程学中自然出现。自由边界可以表现为流体与空气或水与冰之间的界面。在过滤问题中，研究水如何从多孔介质（如土）筑成的大坝中过滤出来，自由边界将湿部分与干部分分开。许多作者研究了自由边界正则性的表征问题，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。通过放宽规则假设，可以涵盖更广泛的物理问题。研究者还提出了在仿射空间上研究与lesbeguem测度行为相似的集支持测度的结构。在弱正则性概念的背景下，这些集合扮演切平面的角色。特别是对这些集合进行分类的问题是最重要的。这里讨论的想法属于一个更大的项目，旨在建立弱规则概念在许多目的下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.