Geometry of Measures

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Most people agree that Euclidean space is flat and that the Lebesgue measure in Euclidean space is doubling (i.e the volume of the ball of radius 2r is bounded above by the volume of the ball of radius r). Toro will investigate the relationship between these two notions: flatness and doubling. She will show the extent to which these two concepts are deeply intertwined. One of the main goals of this project is to show that there exists an equivalence between the doubling properties of a measure and the flatness of its support. Some of the questions discussed here fit naturally into the framework of free boundary regularity problems with weak boundary conditions. In order to address the questions described in this proposal tools from partial differential equations, harmonic analysis and geometric measure theory are needed. 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 geometric measure theory. 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. Many people are concerned with the question of the regularity of the free boundary. In this proposal the investigator addresses the question of the regularity of the free boundary for the obstacle problem. This problem has been studied by Alt and Caffarelli among others. Their hypothesis restrict the sort of obstacles that can be considered. In this project a larger class of obstacles is allowed with the hope that the potential results will cover a broader spectrum of physical problems.