Geometry of Measures and Applications

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When dipping a frame in a solution of soap suds one produces a thin soap film. Mathematically this object is a constant mean curvature surface. It is closely related to the solution of the Plateau problem, which requires finding a surface of minimal area that spans a given shape in space. This problem is a classical question in the Calculus of Variations. The area is an energy functional, and the expectation is that minimizing it will lead to a stable configuration. In this project the PI addresses questions concerning the minimization of certain energy functionals that take into account noise and small random fluctuations of the phenomena being modeled. The expectation is that this theory will be better suited to reflect actual minimization questions arising in nature. A fundamental feature of the area functional is that it is invariant under rotations of space (if that space is homogeneous). The PI will address geometric and analytic questions in inhomogeneous and crystal-like spaces providing a model that reflects nature more accurately. This project will contribute to US workforce development through training and mentoring of graduate students and post-docs. One of the PI’s goals is to show that “almost minimizers”, which are minimizers to noisy variational problems inherit some of the properties of minimizers of the same functional without noise. This study requires using tools from calculus of variations, harmonic analysis and geometric measure theory. The expectation is that the new ideas developed along the way will find applications in other variational problems with free boundaries. The aim of the project concerning further developing analysis on non-smooth domains is to characterize the geometry of domains in Euclidean space in terms of the properties of solutions to canonical (anisotropic) operators. The project concerning the rectifiability of measures promises to reveal the fine structure of measures defined on crystal-like spaces. The overarching theme of this project brings together tools from Geometric Measure Theory, PDE, Potential Theory, Harmonic Analysis and Calculus of Variations, building bridges between these areas while transforming them by the influx of new ideas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.