Geometry of Measures

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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.