Geometry of Measures

University Of Washington, Seattle WA

Investigators

Abstract

The title "Geometry of Measures" refers to the study of the regularity and the structure of measures (think of length and area and volume as examples of this concept). Problems both in smooth and fractal geometry, as well as questions arising in partial differential equations, fit under this umbrella. Along these lines, one of the projects the principal investigator studies concerns an energy minimization problem, where noise is taken into account. This provides a more realistic model for natural phenomena. The behavior one expects to verify is that, up to first order approximation, energy minimizers in the noisy setting behave exactly the same way as those in the noise-free environment. This project illustrates the idea that mathematical objects that can be well approximated by more regular ones inherit some of their regularity properties. Several applications of this principle are presented. Four main question are addressed in the project. The first concerns the regularity of almost minimizers associated with free-boundary variational problems involving Holder continuous Riemannian metrics, and the aim of studying th question is to understand the structure of the corresponding free boundary. New results concerning the free-boundary regularity for the minimizing problems with free boundary that were studied earlier by Alt-Caffarelli and Alt-Caffarelli-Friedman are expected. In this case the "good approximating objects" are solutions to the Laplacian in the corresponding Riemannian metric. The second question deals with the regularity of measures that are well approximated by flat measures (i.e., constant multiples of Lebesgue measure on planes) in the Wasserstein distance. Two distinct types of problems are considered, one geometric in nature, another that ties in very closely to the theory of weights in harmonic analysis. The third question concerns the existence of good parameterizations for two different types of subsets of Euclidean space. The fourth question ties in with an important branch of the principal investigator's research, namely, the regularity of elliptic measures associated with divergence-form elliptic operators on nonsmooth domains. The cross-pollination between harmonic analysis and geometric measure theory is one of the pillars of the proposed research.