Geometry of Sets and Measures

Project Description

Principal Investigator: Matthew Badger

Background

Geometric measure theory is a field of mathematics that developed starting in the 1920s and 1930s, growing out of a practical need to describe nonsmooth phenomena such as the formation of corners in soap bubble clusters. The term "measure" refers to an abstract generalization of length, area, or volume, which assigns a size value to every mathematical set. Traditional outlets for geometric measure theory, such as the calculus of variations and geometric analysis, have expanded in recent decades to include partial differential equations and harmonic analysis. The widespread utility and current use of geometric measure theory in different areas of analysis justifies its continued development. The proposed investigation on the geometry of sets and measures seeks to develop new techniques that will expand the toolbox that geometric measure theory provides for researchers in adjacent areas in analysis and geometry.

Objectives

This project focuses on two groups of questions about the geometry of sets and measures in Euclidean space. The first group of questions concerns rectifiable measures, one of the core objects of study in geometric measure theory. Specifically, these questions are aimed at increased understanding of rectifiable measures in the absence of a standing regularity assumption that has been assumed in the past. The main approach entails adapting quantitative techniques developed in the 1990s by Jones and David-Semmes to study the qualitative rectifiability of measures. The second group of questions are designed to examine the geometry of Reifenberg-type sets, which are sets that can be approximated at all locations and scales by one or more kinds of model sets. Instances where Reifenberg-type sets occur include geometric minimization problems and free boundary problems for elliptic partial differential equations. A general goal of this inquiry is to determine in what situations and to what extent good properties of solutions to problems in ideal models (smooth settings) persist under controlled perturbation (weak regularity).

Research Products

Regularity of the singular set in a two-phase problem for harmonic measure with Hölder data
(arXiv:1807.08002)

(with Max Engelstein and Tatiana Toro)

In non-variational two-phase free boundary problems for harmonic measure, we examine how the relationship between the interior and exterior harmonic measures of a domain in n-dimensional Euclidean space influences the geometry of its boundary. This type of free boundary problem was initially studied by Kenig and Toro in 2006 and was further examined in a series of separate and joint investigations by several authors. The focus of the present paper is on the singular set in the free boundary, where the boundary looks infinitesimally like zero sets of homogeneous harmonic polynomials of degree at least 2. We prove that if the Radon-Nikodym derivative of the exterior harmonic measure with respect to the interior harmonic measure has a H\"older continuous logarithm, then the free boundary admits unique geometric blowups at every singular point and the singular set can be covered by countably many C^1,β submanifolds of dimension at most n-3. This result is partly obtained by adapting tools such as Garofalo and Petrosyan's Weiss type monotonicity formula and an epiperimetric inequality for harmonic functions from the variational to the non-variational setting.

Status: accepted. To appear in Rev. Mat. Iberoam.

Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem
(arXiv:1806.01197 | Published Version)

(with Lisa Naples and Vyron Vellis)

We investigate the geometry of sets in Euclidean and infinite-dimensional Hilbert spaces. We establish sufficient conditions that ensure a set of points is contained in the image of a (1/s)-Hölder continuous map f:[0,1] → l², with s>1. Our results are motivated by and generalize the "sufficient half" of the Analyst's Traveling Salesman Theorem, which characterizes subsets of rectifiable curves in R^N or l² in terms of a quadratic sum of linear approximation numbers called Jones' beta numbers. The original proof of the Analyst's Traveling Salesman Theorem depends on a well-known metric characterization of rectifiable curves from the 1920s, which is not available for higher-dimensional curves such as Hölder curves. To overcome this obstacle, we reimagine Jones' non-parametric proof and show how to construct parameterizations of the intermediate approximating curves f_k([0,1]). We then find conditions in terms of tube approximations that ensure the approximating curves converge to a Hölder curve. As an application, we provide sufficient conditions to guarantee fractional rectifiability of pointwise doubling measures in R^N.

Citation: M. Badger, L. Naples, V. Vellis, Hölder curves and parameterizations in the Analyst's Traveling Salesman theorem, Adv. Math. 349 (2019), 564-647. doi:10.1016/j.aim.2019.04.011

Generalized rectifiability of measures and the identification problem
(arXiv:1803.10022 | Published Version)

One goal of geometric measure theory is to understand how measures in the plane or higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz images of subspaces or Lipschitz graphs), but all of them can be encoded using the same framework. In this paper, we describe a framework for generalized rectifiability, review a selection of classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz or Hölder or C^1,α images of Euclidean subspaces, including theorems of Azzam-Tolsa, Badger-Schul, Badger-Vellis, Edelen-Naber-Valtorta, Ghinassi, and Tolsa-Toro.

Note: This survey paper is based on a talk at the Northeast Analysis Network Conference held in Syracuse, New York in September 2017.

Citation: M. Badger, Generalized rectifiability of measures and the identification problem, Complex Anal. Synerg. 5 (2019), 2.

Geometry of measures in real dimensions via Hölder parameterizations
(arXiv:1706.07846 | Published Version)

(with Vyron Vellis)

We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in Rⁿ when s is a real number between 0 and n. This topic in geometric measure theory has been extensively studied when s is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on s-sets by Martín and Mattila from 1988 to 2000. When 0<s<1, we prove that measures with almost everywhere positive and lower density and finite upper density are carried by countably many bi-Lipschitz curves. When 1≤s<n, we identify conditions on the lower density that ensure the measure is either carried by or singular to (1/s)-Hölder curves. The latter results extend part of the recent work of Badger and Schul, which examined the case s=1 (Lipschitz curves) in depth. Of further interest, we introduce Hölder and bi-Lipschitz parameterization theorems for Euclidean sets with "small" Assouad dimension.

Citation: M. Badger, V. Vellis, Geometry of measures in real dimensions via Hölder parameterizations, J. Geom. Anal. 29 (2019), no. 2, 1153-1192. doi:10.1007/s12220-018-0034-2

Multiscale analysis of 1-rectifiable measures II: characterizations
(arXiv:1602.03823 | Published Version)

(with Raanan Schul)

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterizae 1-rectifiable Radon measures in n-dimensional Euclidean space for all n≥2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L² gauge the extent to which the measure admits approximate tangent lines, or has rapidly growing density rations, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between the measure and 1-dimensional Hausdorff measure. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L² variant of P. Jones' traveling salesman construction, which is of indepenedent interest.

Citation: M. Badger, R. Schul, Multiscale analysis of 1-rectifiable measures II: characterizations, Anal. Geom. Metr. Spaces 5 (2017), no. 1, 1-39.

Related: H. Martikainen and T. Orponen (arXiv:1604.04091) have constructed a finite measure in the plane with bounded density-normalized L² Jones function and vanishing lower 1-density. This implies that our use of β^** in Theorem D is sharp and answers a question we posed following Theorem E.

Structure of sets which are well approximated by zero sets of harmonic polynomials
(arXiv:1509.03211 | Published Version)

(with Max Engelstein and Tatiana Toro)

The zero sets of harmonic polynomials play a crucial role in the study of the free boundary regularity problem for harmonic measure. In order to understand the fine structure of these free boundaries a detailed study of the singular points of these zero sets is required. In this paper we study how "degree k points" sit inside zero sets of harmonic polynomials in Rⁿ of degree d (for all n ≥ 2 and 1 ≤ k ≤ d) and inside sets that admit arbitrarily good local approximations by zero sets of harmonic polynomials. We obtain a general structure theorem for the latter type of sets, including sharp Hausdorff and Minkowski dimension estimates on the singular set of "degree k points" (k ≥ 2) without proving uniqueness of blowups or aid of PDE methods such as monotonicity forumlas. In addition, we show that in the presence of a certain topological separation condition, the sharp dimension estimates improve and depend on the parity of k. An application is given to the two-phase free boundary regularity problem for harmonic measure below the continuous threshold introduced by Kenig and Toro.

Citation: M. Badger, M. Engelstein, T. Toro, Structure of sets which are well approximated by zero sets of harmonic polynomials, Anal. PDE 10 (2017), no. 6, 1455-1495.

Rectifiability and elliptic measures on 1-sided NTA domains with Ahflors-David regular boundaries
(arXiv:1507.02039 | Published Version)

(with Murat Akman, Steve Hofmann, and José María Martell)

Consider a 1-sided NTA domain (aka uniform domain) in Rⁿ⁺¹, n≥2, i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume the boundary of the domain is n-dimensional Ahflors-David regular. We characterize the rectifiability of the boundary in terms of absolute continuiuty of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that the boundary can be covered Hⁿ-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains adn to the fact that the boundary possesses exterior corkscrew points in a qualitiative way Hⁿ-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.

Citation: M. Akman, M. Badger, S. Hofmann, J.M. Martell, Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries, Trans. Amer. Math. Soc.369 (2017), no. 8, 2017, 5711-5745.