Geometry of Measures

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AbstractToroThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The theme of this proposal is the strong relationship that exists between the questions of how the geometry of a domain can be recovered from the regularity of its harmonic measure, and free boundary regularity problems. Remarkably the analogies become more apparent when examined under a Geometric Measure Theory (GMT) magnifying glass. The core of this proposal addresses three questions. The first one aims to understand domains in higher dimensional Euclidean spaces in terms of their harmonic measure as it has been done in 2 dimensions with great success. The underlying thesis is that in higher dimensions GMT plays the role that complex analysis does in 2 dimensions. The second question is that of the existence and regularity of minimizers for variational problems stated in terms of H\"older continuous metrics rather than smooth metrics. This problem includes the understanding of the structure of the corresponding free boundary. A by-product of this, is a question concerning the regularity of quasi-minimizers of the functional studied by Alt and Caffarelli. The third question goes back to a long term interest of the PI concerning the existence of good parameterization for subsets of Euclidean space. A remarkable feature is that this last project, which is purely in geometry, was motivated by an attempt to answer a question in potential theory. The cross-pollenization between harmonic analysis and GMT has been clearly beneficial to both areas.The theory of calculus of variations has been the main theoretical tool used in the study of variational problems often concerning energy minimization. Energy minimization methods are used to understand the equilibrium configuration of molecules. The basic idea is that a stable state of a molecular system should correspond to a local minimum of their potential energy. The proposed research provides new outlets for GMT, a field of Mathematics that has contributed greatly to the development of the calculus of variations and geometric analysis. The transformative aspect of this grant is the invigoration of this fundamental area of Mathematics. In the last few years, the number of students going into GMT in the US, has greatly diminished while it has increased in Europe. An important feature of the proposed work is that, while some results have already been obtained, there is great potential for expansion. In particular, we expect the active participation of graduate students and junior mathematicians. The field, which has been one of the pillars upon of which some areas of geometric analysis have been built, offers the theoretical framework to study a wide array of variational problems coming from different venues of science.