Geometry of Measures

University Of Washington, Seattle WA

Investigators

Abstract

Abstract: The area of analysis called geometric measure theory (GMT) was formally introduced in the nineteen sixties, when the book ``Geometric Measure Theory'' by H. Federer was published. Since the beginning of the century a large amount of relevant work had been done in this area by many authors including Besicovitch, Carath\'eodory, De Giorgi, Federer himself, Fleming, Marstrand, Morrey, Reifenberg, and Whitney. However it was only when Federer's book was published that it became clear how all these results fit naturally together as a cohesive subject. Unfortunately since its introduction the subject has been perceived as somewhat mysterious and inaccessible. In this proposal the investigator exploits one of the outstanding features of the field, its versatility. The PI intends to apply techniques of GMT to study free boundary regularity problems, to answer questions regarding the existence of good parameterizations for certain metric spaces and to classify sets supporting ``Lebesgue-type'' measures. Free boundary problems arise naturally in physics and engineering. The free boundary may appear as the interface between a fluid and the air, or water and ice. In the filtration problem, which studies how water filtrates from a dam made of a porous medium (say earth), the free boundary separates the wet part from the dry part. The problem of characterizing the regularity of the free boundary has been studied by many authors, among others Alt, Caffarelli and Friedman. In this proposal the investigator addresses this question under very weak free boundary assumptions. By relaxing the regularity hypothesis a broader spectrum of physical problems can be covered. The investigator also proposes to study the structure of sets supporting measures that behave like Lesbegue measure on an affine space. In the context of weak notions of regularity, these sets play the role of tangent planes. In particular the problem of classifying these sets is of primary importance. The ideas discussed here belong to a larger project that intends to establish that weak notions of regularity are for many purposes sufficient to answer basic questions in analysis and geometry.