Rectifiability Via Curvature and Regularity in Anisotropic Problems

Max Goering

A dissertation submitted in partial fulﬁllment of the requirements for the degree of

Doctor of Philosophy

University of Washington

2021

Reading Committee:

Tatiana Toro, Chair Steﬀen Rohde Stefan Steinerberger

Program Authorized to Oﬀer Degree: Mathematics University of Washington

Abstract

Rectiﬁability via curvature and regularity in anisotropic problems

Max Goering

Chair of the Supervisory Committee: Craig McKibben and Sarah Merner Professor Tatiana Toro Mathematics

Understanding the geometry of rectifiable sets and measures has led to a fascinating interplay of geometry, harmonic analysis, and PDEs. Since Jones’ work on the Analysts’ Traveling Salesman Problem, tools to quantify the flatness of sets and measures have played a large part in this development. In 1995, Melnikov discovered and algebraic identity relating the Menger curvature to the Cauchy transform in the plane allowing for a substantially streamlined story in R2. It was not until the work of Lerman and Whitehouse in 2009 that any real progress had been made to generalize these discrete curvatures in order to study higher-dimensional uniformly rectifiable sets and measures. Since 2015, Meurer and Kolasinski began developing the framework necessary to use discrete curvatures to study sets that are countably rectifiable. In Chapter 2 we bring this part of the story of discrete curvatures and rectifiability to its natural conclusion by producing mul- tiple classifications of countably rectifiable measures in arbitrary dimension and codimension in terms of discrete measures. Chapter 3 proceeds to study higher-order rectifiability, and in

Chapter 4 we produce examples of 1-dimensional sets in R2 that demonstrate the necessity of using the so-called “pointwise” discrete curvatures to study countable rectifiability. Since at least the 1930s with the work of Douglas and Rad‘o, Plateau’s problem has been at the heart of many developments in geometric measure theory and PDEs. After the pioneering work by De Giorgi on the regularity of area minimizing surfaces, studying the regularity of anisotropic energy minimizing surfaces has been an extremely active area of research leading to new interactions between geometry and the fields of PDEs, Calculus of Variations, and Optimal Transport. For anisotropic minimal surfaces all known higher-order regularity results, that is, regularity beyond countable rectifiability, require an assumption on the energy that is similar in effect to the “ellipticity” condition of Almgren in so far as the condition forces the PDE that eventually arises to be a linear elliptic PDE. The second part of this thesis discusses regularity results for anisotropic problems in geometry and PDEs which do not benefit from a naturally arising elliptic PDE. Chapter 5 begins with results in low-dimensions, including a quick proof of regularity in dimension 2 for all anisotropic minimal surfaces, and a

Bernstein-type theorem. Chapter 6 studies k · kp-minimal surfaces by a monotonicity formula, allowing one to consider types of surfaces not covered in Chapter 5. The monotonicity formula ensures that blow-ups of k · kp-stationary surfaces are stationary cones, motivating an exploration of what anisotropic stationary cones in R2 can look like. Chapter 7 produces regularity results for a general family of PDEs, including a maximum principle, a Harnack inequality, and a Liouville theorem. TABLE OF CONTENTS

Page

Chapter 1: Introduction ...... 1 1.1 Rectiﬁability via curvature ...... 2 1.2 Regularity in anisotropic problems ...... 5

Chapter 2: Characterization of countably (Lipschitz,n)-rectiﬁable measures on Rm 10 2.1 An introduction to discrete curvatures and rectiﬁability ...... 11

2.1.1 (Uniform) Rectifiability and βp-coefficients ...... 11 2.1.2 Uniform Rectifiability and integral Menger curvatures ...... 14 2.1.3 Rectifiability and pointwise Menger curvature ...... 16 2.1.4 A new look at rectifiability via Menger curvature ...... 20 2.2 Preliminaries ...... 23 2.2.1 Sets and measures ...... 23 2.2.2 Menger-type curvature, a formal review ...... 26 2.3 Proofs of main results ...... 31 2.3.1 Scaling Menger Curvature ...... 32 2.3.2 Rectifiability from integral Menger curvature ...... 35 2.3.3 Pointwise Menger curvature and β-numbers...... 43

Chapter 3: Sufficient condition for (C1,α, n)-rectifiable sets on Rm ...... 51 3.1 An introduction to higher-order rectifiability ...... 52 3.2 Notation and Background ...... 55 3.3 Proof of Theorem I ...... 57

Chapter 4: Examples of very non-uniformly rectifiable, rectifiable sets ...... 64 4.1 An introduction to local and non-local quantitative flatness ...... 65

4.2 Construction of K0 ...... 67

4.3 Construction of A0 ...... 78

i 4.3.1 Approximations to the 4-corner Cantor set ...... 79

Chapter 5: Regularity in low dimensions ...... 92 5.1 Motivation and history ...... 93 5.2 Preliminaries ...... 94 5.3 Regularity of 1-dimensional minimizing sets of locally ﬁnite perimeter . . . . 97

5.4 The Anisotropic First and Second Variations of Φρ ...... 104 5.5 Monotonicity formula and basic consequences ...... 108 5.6 Anisotropic minimal cones ...... 120 5.7 Appendix: Compactness of energy minimizers and related tools ...... 124

Chapter 6: A Harnack inequality for anisotropic PDEs ...... 132 6.1 Introduction ...... 133 6.2 Notation and Preliminaries ...... 137 6.3 Main results ...... 140 6.3.1 Fundamental Solutions ...... 140 6.3.2 Regularity ...... 143

Bibliography ...... 160

ii ACKNOWLEDGMENTS

I must thank my advisor Tatiana Toro. An advisor is only truly expected to help their students with research and navigating the job market. Tatiana has gone (at least) the extra mile. She is an excellent mentor– leading by example and explaining with patience, how to thrive in academia. She has also ensured I do not lose perspective and always take care of both myself and those closest to me. Tatiana is someone I can turn to for advise beyond simply what to do mathematically, and I am thankful to have I would like to thank Mark Forehand, Steﬀen Rohde, and Stefan Steinerberger for being part of my committee and taking the time to read my thesis, and Jayadev Athreya and Spencer Becker-Kahn for being part of my general exam committee. I have also had the opportunity to work closely with or collaborate with many wonderful people. Thank you Guy, Mariana, Max, Sean, Silvia, Simon, Spencer. Finally, I thank my parents, Jana and Mike, for providing me with a life, with love, and with support that has made all this possible.

iii DEDICATION

To the one with whom I laugh the most. Thanks for all the adventures Sam.

iv 1

Chapter 1 INTRODUCTION 2

This thesis has two major parts. The ﬁrst part, Chapters 2-4, analyzes the relationship between rectiﬁability and discrete curvatures. Chapters 5-7 form the second part, where we turn our focus to the regularity of anisotropic problems. The regularity of two distinct, but closely related, types of anisotropic problems are presented: anisotropic minimal surface problems and anisotropic PDEs.

1.1 Rectiﬁability via curvature

In geometry, manifolds are of particular interest because of their nice, rigid, tangent structure. In the calculus of variations, one can study “surfaces” (i.e., manifolds, currents, varifolds, etc.) that minimize some energy. Unfortunately, the same rigidity of the tangent structure of manifolds that makes studying them so interesting, also means that the limit of manifolds in some ambient space is unlikely to remain a manifold. This lack of “closedness” of the space of submanifolds leads to the desire to ﬁnd more general notions of manifolds that are closed under some appropriate limit. The tangent structure, which depends upon a manifold becoming ﬂat as you zoom in on it, is an important geometric property for any suitable generalization of manifolds.

Countably (Lipschitz, n)-rectiﬁable measures on Rm are precisely the measure-theoretic objects that have a tangent structure. Intuitively, that is they are ﬂat after zooming in far enough and approach a unique tangent. More precisely, for some class of functions C from

Rn → Rm, we say that a Radon measure µ on Rm is countably (C, n)-rectifiable if there m n m exists {fi}i∈N ⊂ C so that µ (R \ ∪ifi(R )) = 0. A set E ⊂ R is said to be countably n (C, n)-rectifiable if the restriction of the n-dimensional Hausdorff measure, H |E is countably (C, n)-rectifiable. Due to Rademacher’s theorem, that is that Lipschitz functions are differentiable almost everywhere, one could suspect that geometrically this means a measure µ is (Lipschitz, n)-rectifiable precisely when µ-a.e. point has an n-dimensional measure theoretic tangent space. While not straightforward, this indeed turns out to be the case, see for instance [51, Theorem 15.19] when considering sets. This demonstrates that reasonable generalizations 3

of manifolds should at least be rectifiable, and in turn shows the need for necessary and/or sufficient conditions to test rectifiability.

Since the Analysts’ Traveling Salesman Theorem, see [42] in R2 and [57] in Rn, tools to quantify how flat a set or measure is at all locations and scales have become a ubiquitious when characterizeing the rectifiability of sets and measures. In fact, these quantitative tools have helped identify a subclass of rectifiable measures, called uniformly rectifiable measures which have interesting roles in geometry and PDEs, see [19, 20].

A measure µ on Rm is said to be uniformly n-rectiﬁiable if it is Ahlfors n-regular, that is if there exists 0 < c < C < ∞ so that crn ≤ µ(B(x, r)) ≤ Crn for all x ∈ sptµ and all 0 < r < diam(sptµ), and if it also has the property that there exists some θ > 0, so that for all x ∈ sptµ and all 0 < r < diam(sptµ), there exists one Lipschitz function f : Rn → Rm with the property that µ(f(B(0, r))) > θµ(B(x, r)) where the dimension of each ball can be understood from context.

The most popular tool to quantify ﬂatness of sets and measures are the β-numbers. For a measure µ, we deﬁne

p 1 1 Z dist(y, L) p n : βµ;p(x, r) = inf n dµ(y) . n-planes L r B(x,r) r

n p Geometrically, one can interpret βµ;p(x, r) as telling you in an L -sense how far µ|B(x,r) is from being supported on an n-dimensional plane. Therefore, assumptions like

Z R n 2 dr βµ;2(x, r) < ∞ (1.1) 0 r imply that as r approaches zero, the support of µ looks more and more like it is contained within an n-dimensional plane, and hence like the measure µ is becoming “ﬂat”.

There are many very useful “parametrization results” which say under suﬃcient ﬂatness assumptions, like (1.1), one can produce a function f parametrizing a set (or measure) and that f is regular. A woefully incomplete list of some of these parametrization results with 4

different levels of regularity includes: continuous [61], Lipschitz [65, 74, 21, 30], C1,α [38]. Quantitative notions of flatness are not the only method to characterize rectifiability. In [19], one of the many characterizations of uniform rectifiability is by the L2-boundedness of singular integral operators. When n = 2, in [54], an algebraic identity was found that directly relates the Menger curvature of triples of points to the Cauchy transform in C. The Menger curvature of three points x1, x2, x3 is defined as the inverse of the radius of the unique circle passing through the three points. In the special case of R2, this identity lead to a dramatic simplification of the equivalences in [19] by passing through the Menger curvature, [53, 52]. Unfortunately, in [32] Farag demonstrates that no algebraic identity of the same form could exist in higher-dimensions to relate generalizations of Menger curvature to the Riesz transforms. Nonetheless, in [45] Leger used geometric methods to show that the Menger curvature can, without Melnikov’s identity, still characterize countably (Lipschitz,1)-rectifiable sets and measures in Rm thanks to the flatness it imposes. Beginning in 2009, [46, 47] classify uniformly n-rectifiable sets by many examples of geometrically motivated generalizations of Menger curvature, which we call discrete curvatures. Later [56] Meurer provides a general definition which encompasses these successful discrete curvatures, and produced a sufficient condition analogous to the work of Leger which guarantees rectifiability of sets. Around the same time, in [43], Kolasinski produced a sufficient condition in terms of some specific discrete curvatures to guarantee countably (C1,α, n)-rectifiability of measures. Chapter 2 is work done solely by the author. It provides two new characterizations of countably (Lipschitz,n)-rectifiable Radon measures on Rm under relatively weak density assumptions, see Theorem 16. We also show a converse to work of Kolasinski, providing a direct quantitative comparison between (centered) β-numbers and Menger curvatures, see Theorem 19. Chapter 3 is joint work with Silvia Ghinassi (a graduate student at the time). We further develop the ideas behind Theorem 19 demonstrating a sufficient condition for a Radon measure to be a countably (C1,α, n)-rectifiable measure, see Theorem I. This is done by demonstrating that finiteness of appropriately modified discrete curvatures imply a set 5

satisfies the hypotheses of a parametrization result [38] by Ghinassi, Theorem 51. Chapter 4 is joint work with Sean McCurdy (a graduate student at the time). We produce two examples of sets in R2 demonstrating that the so-called “integral” Menger curvatures cannot be used to detect countable rectifiability by blowing-up– even for connected sets or Ahlfors regular sets. Due to the fact that β-numbers are more popular than Menger curvatures, this chapter is stated in terms of β-numbers. Theorems 5 and 19 provide a direct translation of these results to results about discrete curvatures. In fact, the original motivation to construct these examples arose from a desire to show the failure of a third attempted characterization of rectifiability by discrete curvatures. That is, we wanted to know if countably (Lipschitz, 1)-rectifiable measures could be characterized by the limiting behavior of integral Menger curvature, i.e., by studying what happens to the expression in (2.33) as r ↓ 0. A major hurdle immediately presents itself: What if for no r > 0 the quantity in (2.33) is finite? Originally, this seemed like just a technicality. However, Theorem 61 (resp. 62) shows that even for countably (Lipschitz,1)-rectifiable sets in R2, even with the restrictive topological (resp. geometric) constraint of connectedness (resp. Ahlfors regularity), one cannot expect the quantity in (2.33) can to be finite for any x in the set or any r > 0.

1.2 Regularity in anisotropic problems

The anisotropic Plateau’s problem asks, “If W is an (n − 1)-dimensional boundary, how nice must the n-dimensional surface S that has the least energy out of all surfaces that span the boundary W be?” There are many ways to interpret this question depending upon how one chooses to define “boundary”, “surface”, and “spans”. We emphasize some of these interpretations during a brief and incomplete survey of the topic. In the 1930s, Douglas [29] and Radó[60] solved the isotropic Plateau problem (area minimizing surfaces) in the plane by studying surfaces parametrized by continuous functions on the disc, with fixed values on the boundary. In the 1950s, De Giorgi [22, 23, 24] introduced another notion of surface, which we now 6

call sets of locally finite perimeter. In 1961 De Giorgi [25] demonstrated regularity of the boundary for area minimizing sets of locally finite perimeter near flat points. To do so, he introduced the “tilt-excess improvement” method. In 1960, Reifenberg [61] used a clever local parametrization argument to produce solutions to the Plateau problem. At the same time, Federer and Fleming generalized De Giorgi’s set of finite perimeter to normal currents, which allow one to study the regularity of n-dimensional energy minimizing surfaces in Rm, ultimately leading to a very satisfying collection of regularity results for anisotropic energy minimizing surfaces of Almgren’s [5] so-called elliptic energies. The epic book by Federer [35, Chapter 5] provides a thorough description of regularity theory for elliptic energy minimizing currents. The general story of regularity for energy minimizing currents, more-or-less originating from [25] is as follows:

(1) Start at a point on a surface where you have zoomed in far enough that the current looks “ﬂat”.

(2) show that this ﬂatness implies the current can largely be written as the graph of a function u. In turn,

(3) setting the ﬁrst variation equal to zero, one discovers this function almost solves some PDE.

(4) an -regularity theorem says that u is close to some v which truly solves the PDE, and

(5) since v solves the PDE, if the PDE is nice, then v is regular, so in fact u is even ﬂatter than you expected.

The way that one defines “flatness” turns out to be quantified by a notion of “tilt”. The more tilt a surface has (i.e., the excess tilt) the less flat it is. Hence, the conclusion in step (5) is often called “tilt-excess decay”. 7

The desire to have a nice PDE in (4) is precisely where Almgren’s ellipticity condition comes from: it is a constraint on the energy that forces the PDE to be a linear uniformly elliptic divergence form PDE. There has been some work attempting to weaken the ellipticity condition, for example [63, 36], for currents. Nonetheless, the weaker assumption still boils down to some sufficient condition to ensure the PDE from (4) is elliptic. The more recent work of Figalli is the only known higher-regularity result for anisotropic energy minimizing currents where it is not assumed that regardless of the orientation, the first variation always produces an elliptic PDE. Still, Figalli assumes that in all “nearby” directions, the resulting PDE is elliptic, and in particular the theory of elliptic PDEs is still the workhorse that yields C1,α-regularity near flat points. Since the PDE, which arises from the first variation of the anisotropic energy, is so important to this technique, it is common to study regularity of surfaces whose first variation is zero. These surfaces are called anisotropic minimal surfaces. Energy minimizing surfaces are always minimal surfaces, but not vice-versa. In the 1970s, Allard [2, 3] introduced the notion of a varifold as a new way to interpret “surface”. The primary tool to study the regularity of area minimal varifolds is through Allard’s monotonicity formula. A nice exposition can be found in [67]. Compared to the regularity theory for energy minimizing currents, the regularity theory for minimal varifolds has lagged behind. The regularity theory for area minimal varifolds is based on a monotonicity formula. When considering n ≥ 2-dimensional varifolds this monotonicity formula holds exclusively for energies that are, up to a constant-coefficient linear change of variables, the area integrand [4]. When the varifold is of dimension n = 1, a monotonic quantity is still known (also [4]) but to the best of the author’s knowledge there has been no work demonstrating regularity for 1-dimensional anisotropic minimal varifolds using this method because the monotonic quantity has no specific formula that has been shown to be beneficial. There has been recent progress moving regularity theory beyond area minimal varifolds to 8

anisotropic energy minimal varifolds [27, 26, 28]. In the absence of a monotonicity formula, attempts at showing the varifold is locally C1,α near ﬂat points, depend once again upon some condition forcing the ﬁrst variation to produce an elliptic PDE.

In chapters 5-6, we study anisotropic minimal surfaces which do not have a first variation that produces an elliptic PDE. The main focus is on surfaces minimizing the k · kp-energy. Specifically, we focus on the local regularity of the boundary of sets of locally finite perimeter

n which locally minimize the k · k`p -energy. To this end, we need an open set A ⊂ R and deﬁne the functional Φρ on sets of locally ﬁnite perimeter by

Z n−1 Φp(E; A) := ρ(νE)dH , ∂∗E∩A where ρ : Rn → Rn is some strictly-convex, positively 1-homogeneous function that is C1 on Rn \{0}. We then study the regularity ∂∗E whenever E satisﬁes

Φp(E; A) ≤ Φp(F ; A) ∀E∆F ⊂⊂ A.

The main focus is when ρ = k · k`p . Any more general regularity statements are most interesting when considering some ρ where the ﬁrst variation does not provide us with an elliptic PDE.

2 Chapter 5 includes a short, but complete, proof that in R , all Φp-energy minimizing sets of locally ﬁnite perimeter have boundaries that are straight edges. Theorem 94 also includes that global minimizers must be halfplanes.

Chapter 6 provides a general suﬃcient condition for a monotonicity formula for Φρ- minimal surfaces in Rn, namely the existence of a positively 1-homogeneous function f, so that hx, ξih∇f(x), ∇ρ(ξ)i ≥ 0 ∀x, ξ. (1.2)

2 We then demonstrate that k · kp-energy minimizing currents in R satisfy a monotonicity formula, which diﬀers from the one in [4]. This monotonicity formula is used to show that 9

blow-ups of energy minimizing currents are global energy minimizing cones. The only speciﬁc property of the monotonicity formula used, is that when n = 2 and f = k · kp = ρ the expression in (1.2) is zero if and only if x · ξ is zero.

2 All of the work done in Chapter 6 could be carried over for k · kp-minimal varifolds in R , ultimately demonstrating that blow-ups of k · kp-minimal varifolds are k · kp-minimal cones. It is known that the only area-minimal cones in the plane are (1) lines and (2) the so-called triple junction, that is where three half-lines join at a point with 120◦ angles. Naturally, this rigidity is lost in the anisotropic problem, but we provide a constructive method to produce all anisotropic stationary triple junctions in the plane. This analysis of triple junctions in the plane can also be extended to triple junctions in higher-dimensions, as long as you know what 2-dimensional plane your triple junction is contained in. Chapter 7 studies the regularity of a wide class of PDEs, which include those that arise as (partially linearized) versions of the PDEs one would get from the ﬁrst variation in the anisotropic minimal surface problem. In particular, this class of PDEs contains lots of non-elliptic PDEs, including the p-Laplacian and the pseudo p-Laplacian. We show that a De Giorgi-Nash-Moser theory holds for these PDEs. In particular, positive subsolutions are locally bounded (Theorem 136), solutions satisfy a maximum principle (Theorem 138), positive solutions satisfy a Harnack inequality (Theorem 139), and a Liouville theorem holds (Theorem 142). 10

Chapter 2 CHARACTERIZATION OF COUNTABLY (LIPSCHITZ,n)-RECTIFIABLE MEASURES ON Rm 11

2.1 An introduction to discrete curvatures and rectiﬁability

In the late 1990s there was a flurry of activity relating 1-rectifiable sets, boundedness of singular integral operators, the analytic capacity of a set, and the integral Menger curvature in the plane. In 1999 Légerextended the results for Menger curvature to 1-rectifiable sets in higher dimension, as well as to the codimension one case. A decade later, Lerman and Whitehouse, and later Meurer, found higher-dimensional geometrically motivated generalizations of Menger curvature that yield results about the uniform rectifiability of measures and the rectifiability of sets respectively. Primarily, these higher-dimensional Menger curvatures have been used to study the regularity of surfaces and knots, for instance, self-avoidance and smoothness of the normal. Herein, we use these tools to find new characterizations of rectifiable Radon measures in arbitrary dimension and codimension (see Theorems 16 and 18). Work in progress indicates the characterization in Theorem 18 (4) and (5) is more likely more useful in practice than the characterization in Theorem 16. We also relate the pointwise Menger curvature to the sum of the β numbers over scales (see Theorem 19). As a consequence of taking tools from knot theory to answer geometric measure theory questions, we include extra details to ensure this paper is sufficiently self-contained for readers from either discipline. Despite this, when classic arguments make repeat appearances, we attempt to avoid repetition by referring the reader to the analogous argument earlier in the paper.

2.1.1 (Uniform) Rectiﬁability and βp-coeﬃcients

Studying rectiﬁable sets and measures (see Deﬁnition 21) is a central topic in geometric measure theory. In his 1990 work on the Analysts’ traveling salesman problem in the plane

[42] (later generalized to 1-dimensional sets in Rm by Okikiolu in [57] and to n-dimensional sets in Rm by Pajot in [58]) Peter Jones introduced what are now called the Jones’ β-numbers, which have dominated the landscape in quantitative techniques relating to rectiﬁability, analytic capacity, and singular integrals. 12

In the joint monograph on the topic [20] David and Semmes laid the framework to understanding the quantitative structures Jones introduced, as well as how to properly generalize these ideas to Ahlfors regular sets and measures higher dimensions. In doing so, they introduced the notion of uniform rectifiability (see Definition 22). David and Semmes gave many equivalent characterization of uniform n-rectifiability. One characterization is related to Jones’ β-numbers the definition of which is included for completeness. For 1 ≤ p < ∞ one defines

p 1 Z p n 1 dist(y, L) βµ;p(x, r) = inf n dµ(y) , (2.1) L r B(x,r) r where the infimum is taken over all n-dimensional affine subspaces L ⊂ Rm. When the dimension n is understood from context, the superscript is typically forgotten. The relevant characterization of uniformly n-rectifiable sets was discovered in [19].

Theorem 1 ([19]). If µ is an n-Ahlfors regular measure on Rm then the following are equivalent. (1) µ is uniformly rectiﬁable.

2n (2) For 1 ≤ p < n−2 there exists some c > 0 depending on p such that the βp-numbers satisfy the following so called Carleson-condition.

Z Z R n 2 dr n βµ;p(y, r) dµ(y) ≤ cR for all x ∈ sptµ, R > 0. (2.2) B(x,R) 0 r

More recently, a much desired characterization of countably n-rectiﬁable measures (a characterization for measures, similar to the characterization for sets from [58]) was discovered by Azzam and Tolsa in a pair of papers [72] and [8]. In particular, the following theorem is from [72].

Theorem 2 ([72]). Let 1 ≤ p ≤ 2. If µ is a ﬁnite Borel measure on Rm which is countably n-rectiﬁable, then Z ∞ n 2 dr m βµ;p(x, r) < ∞ for µ a.e. x ∈ R . (2.3) 0 r 13

The converse has hypothesis on the densities of the measure (see Deﬁnition 24).

Theorem 3 ([8]). Let µ be a ﬁnite Borel measure in Rm such that 0 < Θn,∗(µ, x) < ∞ for µ-a.e. x ∈ Rm. If Z ∞ n 2 dr m βµ;2(x, r) < ∞ µ a.e. x ∈ R , (2.4) 0 r then µ is countably n-rectiﬁable.

Later, [30] (with an alternative proof in [73]) removed the a priori assumption of absolute continuity with respect to the Hausdorﬀ measure. Together, these results can be summarized in the following theorem.

m n,∗ n Theorem 4 ([30]). Let µ be a Radon measure in R with 0 < Θ (µ, x) and Θ∗ (µ, x) < ∞. Then µ is countably n-rectiﬁable if and only if

Z 1 n 2 dr m βµ,2(x, r) < ∞ for µ a.e. x ∈ R . (2.5) 0 r

We note that prior to [30] some partial results on rectiﬁability of measures without the a priori assumption of absolute continuity with respect to the Hausdorﬀ measure were also demonstrated in [9], [10], [11].

In light of the characterization for uniformly rectifiable measures, one might expect that the restriction to p = 2 in Theorem 3 is a consequence of the proof, and potentially a range of values for p could be found. However, in [73] a very flexible construction of 1-rectifiable sets with positive and finite measure was laid out, which, by varying the parameters of this construction demonstrate that for 1 ≤ p < 2 no conclusion can be drawn from µ almost everywhere finiteness of the quantity in (2.3). Under the condition that Θn,∗(µ, x) < ∞, Hölder’sinequality guarantees that if p > 2 then (2.3) implies (2.4), so only the range 1 ≤ p < 2 was of particular interest. 14

2.1.2 Uniform Rectiﬁability and integral Menger curvatures

In 1995 Melnikov discovered a beautiful identity connecting Menger curvature to the analytic capacity in the plane, which kicked off a great deal of research to find curvature-based techniques to classify rectifiable and uniformly rectifiable sets, and further to find relations to singular integrals ([54], [53], [52], [50], [45]).

Unfortunately, all of the progress made applied exclusively to dimension 1 and/or codimension 1 sets. In fact, in [32], Farag showed that in the higher-(co)dimension case there does not exist an algebraic generalization of Melnikov’s identity that could directly relate to Riesz kernels. In particular, this result does not preclude the usefulness of Menger-type curvatures whose integrand take a diﬀerent form.

After a hiatus in progress on this subject, Lerman and Whitehouse showed that a geometrically motivated generalization of Menger curvature was sufficient to characterize uniformly n-rectifiable measures in higher codimension (in fact, their results hold for real separable Hilbert spaces). The work, completed in [46] and [47] was motivated by the original work of David and Semmes [19] and [20], but required a nontrivial and creative idea of “geometric multipoles”. They defined several versions of Menger curvatures of (n + 2)-points in Rm (see [46], [47], and [48]). Among these curvatures, one that is well suited for our setting is

1 n+1 2 2 H (∆(x0, . . . , xn+1)) K1(x0, . . . , xn+1) = (n+1)(n+2) (2.6) diam{x0, . . . , xn+1} where ∆(x0, . . . , xn+1) is the simplex with corners {x0, . . . , xn+1}. Lerman and Whitehouse showed that control of integrals of the integrand K1 has interesting geometric consequence, this is explained in more detail below.

One necessary object to understand and state the results of Lerman and Whitehouse, is a notion of the “space of well-scaled simplices”. More precisely, for some 0 < λ < 1 and 15

m letting X = (x0, . . . , xn+1) denote an (n + 2)-tuple in R , deﬁne

min(X) W (B(x, r)) = {X ∈ B(x, r)n+2 : ≥ λ > 0}. (2.7) λ diam(X)

Where

min(X) = min |xi − xj|. 0≤i

Then, one can think of Wλ(B(x, r)) as the space of well-scaled simplices in B(x, r). We also deﬁne the quantities

Z 2 2 n+2 c1(µ|B(x,r)) = K1(X) dµ (X). (2.8) B(x,r)n+2 and Z 2 2 n+2 c1(µ|B(x,r), λ) = K1(X) dµ (X) (2.9) Wλ(B(x,r))

The measure µn+2 is the measure on (Rm)n+2 resulting from taking (n + 2)-products of µ with itself. The quantity in (2.8) can be thought of as the integral Menger curvature of the measure

µ|B(x,r). The quantity in (2.9) can be thought of as the amount of integral Menger curvature of the measure µ|B(x,r) which arises from “well-scaled simplices”. The main results of [46] and [47] are combined to show the following theorem, which notably holds in the more diﬃcult setting of real-seperable Hilbert spaces.

Theorem 5 ([46], [47]). If µ is an n-Ahlfors regular measure on a possibly inﬁnite dimensional, real, separable Hilbert space, then the following are true:

Z Z 12R 2 n 2 dr c1(µ|B(x,R)) ≤ C2 βµ;2(y, r) dµ(y) B(x,R) 0 r and there exists a λ ∈ (0, 1) such that for all B(x, R) with 2R ≤ diam(sptµ),

Z Z 2R n 2 dr 2 βµ;2(y, r) dµ(y) ≤ C3 · c1(µ|3·B(x,R), λ/2) B(x,R) 0 r 16

Here, both C2 and C3 depend only on n and the Ahlfors regularity constants of µ.

2 In particular, combining the two parts of Theorem 5 with the observation that c1(µ|B(x,R), λ) ≤ 2 c1(µ|B(x,R)), it follows that

Z Z R 2 n n 2 dr ˜ n c1(µ|B(x,R)) ≤ CR ⇐⇒ βµ;2(x, r) ≤ CR , B(x,R) 0 r which by passing through Theorem 1 leads to the following characterization.

Theorem 6 ([19], [46], [47]). If n ≥ 2 and µ is an n-Ahlfors regular measure on Rm then the following are equivalent:

2 n 1) There exists a constant C independent of x and R so that c1(µ|B(x,R)) ≤ CR for all x ∈ sptµ and R > 0. 2) µ is uniformly n-rectiﬁable.

2.1.3 Rectiﬁability and pointwise Menger curvature

In 2015, Meurer found a general class of Menger-type curvatures which satisfy a one-sided comparison to βp-coefficients and proved a sufficient condition for rectifiability of a set based off these integral menger curvatures. The integrand K1 of Lerman and Whitehouse is an example of the general class of Menger-type integrands laid out in [55] (the shortened published version is [56]). So, throughout the remainder this section, on a first read one should interpret the phrase (µ, p)-proper integrand, to mean the integrand K1. The formal definition of a “(µ, p)-proper integrand” is given in Definition 28. A symmetric (µ, p)-proper integrand is defined in Definition 31. A review of notation is also included in the preliminaries in section 2.2.2. One main result of [55] is as follows,

m n Theorem 7 ([55]). Let E ⊂ R be a Borel set and µ = H |E. If K is a (µ, 2)-proper 17

integrand such that

Z Z 2 n+2 MK2 (µ) := ··· K (x0, . . . , xn+1) dµ (x0, . . . , xn+1) < ∞, Rm Rm then E is countably n-rectiﬁable.

n n Remark 8. Meurer also showed that MK2 (H |E) < ∞ implies H |E is locally finite. A similar result does not hold for Borel measures µ who are absolutely continuous with respect to the Hausdorff measure. For instance, consider µ on R2 defined by

1 1 dµ = dH |{x2=0} |x1|

2 where x ∈ R is written as x = (x1, x2). Then sptµ = {x2 = 0} implies K1(y0, y1, y2) =

0 whenever yi ∈ spt(µ) for all i = 1, 2, 3. Consequently, MK1 (µ) = 0. Nonetheless, Z δ dx µ(B(0, δ)) = 1 = ∞ for all δ > 0. −δ |x1|

The next theorem is an interesting intermediate result of [55] which is of similar style to the work of Lerman and Whitehouse. As such, one must again have the correct interpretation of the “space of well-scaled simplices”, namely

t O (x, t) := {(x , . . . , x ) ∈ B(x, kt)n+2 | |x − x | ≥ ∀i 6= j}. (2.10) k 0 n+1 i j k

Given some (µ, p)-proper integrand K, deﬁne the notation

Z p n+2 MKp;k(x, t) := K (x0, . . . , xn+1) dµ (x0, . . . , xn+1) Ok(x,t) so that one can succinctly state:

Theorem 9 ([55]). Let µ be an n-upper Ahlfors regular Borel measure, with upper-regularity

n constant C0 (see Deﬁnition 20). Let 0 < λ < 2 and k > 2, k0 ≥ 1. Then there exist 18

constants

k1 = k1(m, n, C0, k, k0, λ) > 1 and C = C(m, n, K, p, C0, k, k0, λ) ≥ 1

n such that if µ(B(x, t)) ≥ λt , then for every y ∈ B(x, k0t) we have

M p (x, t) M p (y, t) βn (y, kt)p ≤ C K ;k1 ≤ C K ,k1+k0 . µ;p tn tn

A corollary of the previous result, whose relevance to the results of Lerman and White- house is more immediate can be stated as

Corollary 10 ([55]). Let µ be an n-upper Ahlfors regular Borel measure on Rm with upper- n p regularity constant C0 . Fix 0 < λ < 2 , k > 2, k0 ≥ 1 and K any symmetric (µ, p)-proper integrand. Then there exists a constant C = C(m, n, K, p, C0, k, k0, λ) such that

Z Z ∞ n p dt β (x, t) 1 ˜ dµ(x) ≤ CM p (µ), µ;p {δk(B(x,t))≥λ} K Rm 0 t where ˜ µ(B(y, t)) δk(x, kt) = sup n . y∈B(x,kt) t

Remark 11. Although it seemingly goes unmentioned in [55], we note that this corollary implies the following statement:

m n If µ is an n-Ahlfors regular measure on R , and MKp (µ|B(x,r)) . r with suppressed constant independent of x, for all x ∈ sptµ and all 0 < r < diam spt(µ) for some symmetric

2n (µ, p)-proper integrand, and p ∈ [2, n−2 ), then µ is uniformly n-rectiﬁable. This follows directly from Corollary 10 and Theorem 1.

In [47, Equation 10.1] Lerman and Whitehouse also introduce the curvature

1 2 2 hmin(x0, . . . , xn+1) K2(x0, . . . , xn+1) = n(n+1)+2 , diam({x0, . . . , xn+1}) 19

where

hmin(x0, . . . , xn+1) = min dist(xi, aff{x0, . . . , xi−1, xi+1, . . . , xn+1}) (2.11) i and aff{y0, . . . , yk} denotes the smallest affine plane containing {y0, . . . , yk}.

Notably, K2 satisﬁes

2 2 K1(x0, . . . , xn+1) ≤ K2(x0, . . . , xn+1) (2.12)

m n+2 for all (x0, . . . , xn+1) ∈ (R ) and is also a (µ, 2)-proper integrand. We deﬁne the pointwise Menger curvature of µ with respect to a (µ, p)-proper integrand K at x and scale r by

Z n p n+1 curvKp;µ(x, r) = K (x, x1, . . . , xn+1) dµ (x1, . . . , xn+1). (2.13) (B(x,r))n+1 Then a simpliﬁed (and strictly weaker) version of the main result of [44] is:

Lemma 12. [44] Let µ be a Borel measure on Rm. Then, there exists Γ = Γ(n, m) such that Z 2R n n n ˆn 2 dr curv 2 (x, R) ≤ Γ Θ (µ, x, r) β (x, r) , K2;µ µ,2 0 r where µ(B(x, r)) Θn(µ, x, r) = rn and 1 Z dist(y, L)p ˆn p : βµ,p(x, r) = inf n dµ(y) (2.14) L3x r B(x,r) r denotes the “centered” βp-numbers. Notably, the inﬁmum is taken over all n-planes passing through the center of B(x, r).

Another useful result which can be found in a more general setting in [44] is

Lemma 13. [44]

m n n,∗ Let µ be a Radon measure on R and x be such that 0 < Θ∗ (µ, x) ≤ Θ (µ, x) < ∞. 20

Then, for p ∈ [1, ∞] and 0 < ρ < ∞,

Z ρ ˆn p dr m βµ;p(x, r) < ∞ µ − a.e. x ∈ R (2.15) 0 r if and only if Z ρ n p dr m βµ;p(x, r) < ∞ µ − a.e. x ∈ R . (2.16) 0 r Moreover, if µ is n-Ahlfors regular, and q ≤ p then there exists C depending on m, n, p, q and the Ahlfors regularity constants such that

Z Z r n q ds βµ;p(y, s) dµ(y) (2.17) B(x,r) 0 s Z Z r ˆn q ds ≤ βµ;p(y, s) dµ(y) B(x,r) 0 s Z Z Cr n q ds ≤ C βµ;p(y, s) dµ(y). B(x,Cr) 0 s

A consequence of Lemma 12 and [72] is:

m n n,∗ Theorem 14 ([72], [44]). Let µ be a Radon measure on R with 0 < Θ∗ (µ, x) ≤ Θ (µ, x) < m n ∞ for µ almost every x ∈ . If µ is countably n-rectiﬁable, then curv 2 (x, 1) < ∞ for µ R K2;µ almost every x ∈ Rm.

Indeed, the equivalence of (i) and (ii) in Lemma 25 ensures that Theorem 14 follows from Lemmata 12, 13, and Theorem 2.

2.1.4 A new look at rectiﬁability via Menger curvature

The ﬁrst result of this paper is a generalization of Theorem 7 to the case of Radon measures with upper-density bounded above and below.

Theorem 15. If µ is a Radon measure on Rm with 0 < Θn,∗(µ, x) < ∞ for µ almost m every x ∈ R and MK2 (µ) < ∞ for some (µ, 2)-proper integrand K, then µ is countably 21

n-rectiﬁable.

In light of Theorem 15, with some additional work we can now answer an open question posed in [47, Section 6] and characterize n-rectiﬁability with respect to Menger-type curvatures.

m n n,∗ Theorem 16. Let µ be a Radon measure on R with 0 < Θ∗ (µ, x) ≤ Θ (µ, x) < ∞ for µ almost every x ∈ Rm. Then the following are equivalent: 1) µ is countably n-rectiﬁable.

m n 2) For µ almost every x ∈ R , curvK2;µ(x, 1) < ∞, where K ∈ {K1, K2}. P∞ 3) µ has σ-ﬁnite integral Menger curvature in the sense that µ can be written as µ = j=1 µj where each µj satisﬁes MK2 (µj) < ∞ where K ∈ {K1, K2}.

Remark 17. Note that Theorem 16 is to Theorem 4 as 5 is to 1, except that unfortunately, the hypothesis in Theorem 16 are strictly stronger than the hypothesis in Theorem 4. The stronger hypothesis, which may also be an artifact of the proof, does suggest that there may be a better way to deﬁne the Menger curvature integrands.

In particular, with some additional work, combining Theorem 4, the equivalence of (2.15) with (2.16) in Lemma 13, and Theorem 16 yields another new characterization of rectiﬁable Radon measures in Theorem 18. Moreover, combining Theorem 1, Theorem 5, and (2.17) yields the characterization of uniformly rectiﬁable measures in Theorem 18.

Theorem 18 ([19], [47], [46], [55], [44], Theorem 16). If µ is a Radon measure on Rm with n n,∗ m 0 < Θ∗ (µ, x) ≤ Θ (µ, x) < ∞ for µ almost every x ∈ R , then the following are equivalent

1. µ is countably n-rectiﬁable.

Z 1 n 2 dr m 2. βµ;2(x, r) < ∞ for µ almost every x ∈ R . 0 r

Z 1 ˆn 2 dr m 3. βµ;2(x, r) < ∞ for µ almost every x ∈ R . 0 r 22

n m (4) curv 2 (x, 1) < ∞ for µ almost every x ∈ . K2;µ R

n m (5) curv 2 (x, 1) < ∞ for µ almost every x ∈ . K1;µ R

m 2n Moreover, if µ is an n-Ahlfors regular Borel measure on R and p ∈ [2, n−2 ) then the following are equivalent

(a) µ is n uniformly-rectiﬁable

Z Z R n 2 dr n (b) βµ;p(y, r) dµ(y) ≤ CR for all R > 0. B(x,R) 0 r

Z Z R ˆn 2 dr ˜ n (c) βµ;p(y, r) dµ(y) ≤ CR for all R > 0. B(x,R) 0 r

0 n (d) M 2 (µ| ) ≤ C R for all R > 0. K1 B(x,R)

00 n (e) M 2 (µ| ) ≤ C R for all R > 0. K2 B(x,R)

n R 1 ˆn 2 dr The ﬁnal main result more directly shows a comparability of curvK2;µ(x, 1) and 0 βµ;2(x, r) r , (see 2.13 and 2.14 respectively) but in the present state requires stronger hypothesis on the density of µ. This is done by proving a converse to Lemma 12

Theorem 19. If µ is an n-Ahlfors upper-regular Radon measure on Rm, K is a (µ, 2)-proper n integrand, and x is such that Θ∗ (µ, x) > 0 then

Z R ˆn 2 dr n β (x, r) curv 2 (x, R). (2.18) µ;2 . K1;µ 0 r

In particular, in conjunction with Lemma 12

Z R Z C1R ˆn 2 dr n n ˆn 2 dr β (x, r) curv 2 (x, R) ≤ curv 2 (x, R) β (x, r) . (2.19) µ;2 . K1;µ K2;µ . µ;2 0 r 0 r 23

In both (2.18) and (2.19) the suppressed constants1 depend on x, R, the upper–regularity constant of µ, m and n.

Since the (suppressed) constant in (2.18) depends on x, it would be interesting to see if µ being n-Ahlfors upper-regular can be weakened to say Θn,∗(µ, x) < ∞.

2.2 Preliminaries

2.2.1 Sets and measures

When comparing two quantities and the precise constant is unimportant, we adopt the notation that

A .x,r,m B means A ≤ CB for some constant C depending on x, r, m. Fewer or more dependencies may be attached to the symbol .. If no dependencies are appended to the symbol, they are explained shortly after the equation appears.

Deﬁnition 20 (Ahlfors-regularity of measures). A measure µ on Rm is said to be n-Ahlfors regular if there exist constants 0 < c, C < ∞ such that

µ(B(x, r)) ≤ Crn ∀x ∈ spt(µ) (2.20) and µ(B(x, r)) ≥ crn ∀0 < r < diam{spt(µ)}, ∀x ∈ spt(µ). (2.21)

A measure µ is said to be n-upper Ahlfors regular if (2.20) holds, and n-lower Ahlfors regular if (2.21) holds. The smallest constant C such that (2.20) holds is called the upper regularity constant for µ, and the largest c such that (2.21) holds is called the lower regularity constant for µ.

1 n The dependence on x comes from λ = λx,R > 0 such that λr ≤ µ(B(x, r)) for all 0 < r < R, see Lemma 25. 24

A measure µ on Rm is said to be absolutely continuous with respect to a measure ν, denoted µ ν if ν(E) = 0 =⇒ µ(E) = 0.

Deﬁnition 21 (Countably Rectiﬁable). In this paper, we follow the convention that a Borel

m n measure µ on R is said to be countably n-rectiﬁable if there exist Lipschitz maps fi : R → m such that R ∞ ! m [ n µ R \ fi(R ) = 0 (2.22) i=1

n n and µ H . A Borel set E is countably n-rectiﬁable if H |E is countably n-rectiﬁable.

Definition 22 (Uniformly rectifiable). A Radon measure µ on Rm is said to be uniformly n-rectifiable if it is n-Ahlfors regular and there exist constants Λ > 0 and 0 < θ < 1 such

n m that for all x ∈ sptµ and all r ≥ 0 there exist a Lipschitz map fx,r : B (0, r) → R such that

n 2 µ (B(x, r) \ fx,r(B (0, r))) ≤ θµ(B(x, r)). (2.23)

m n A Borel set E ⊂ R is said to be uniformly n-rectiﬁable if H |E is uniformly n-rectiﬁable.

Definition 23 (Purely unrectifiable). A Borel measure µ on Rm is said to be n-purely unrectifiable if every Lipschitz map f : Rn → Rm has the property that

n µ(f(R )) = 0. (2.24)

A Borel set E is said to be countably n-rectiﬁable (uniformly n-rectiﬁable/n-purely un-

n rectifiable respectively) if H |E is countably n-rectifiable (uniformly n-rectifiable/n-purely unrectifiable respectively).

Deﬁnition 24 (Density ratios). Given a Borel measure µ on Rm, we deﬁne the function

µ(B(x, r)) Θn(µ, x, r) = . (2.25) rn

2Here and after, Bn(0, r) denotes an n-dimensional ball of radius r centered at 0. Similarly Bm(x, r) is an m-dimensional ball centered at x with radius r. Typically, the dimension is clear from context and hence neglected. 25

Moreover, the n-dimensional upper-density of µ at x, denoted Θn,∗(µ, x) is deﬁned as

n,∗ µ(B(x, r)) Θ (µ, x) = lim sup n (2.26) r→0 r

n and the n-dimensional lower-density of µ at x, denoted Θ∗ (µ, x) is deﬁned by

n µ(B(x, r)) Θ∗ (µ, x) = lim inf (2.27) r→0 rn

n,∗ n If Θ (µ, x) = Θ∗ (µ, x) their common value is called the density of µ at x and is denoted by Θn(µ, x). Notably, µ Hn if and only if Θn,∗(µ, x) < ∞ for µ a.e. x ∈ Rm.

The following lemma is a useful characterization of density properties of measures.

Lemma 25. If µ is a Borel measure on Rm and x ∈ Rm, then for any R > 0, the following are equivalent:

n 1. Θ∗ (µ, x) > 0

2. There exists λ > 0 such that µ(B(x, r)) ≥ λrn for all 0 < r ≤ R.

Similarly, the following are equivalent:

(i) Θn,∗(µ, x) < ∞

(ii) There exists a Λ > 0 such that µ(B(x, r)) ≤ λrn for all 0 < r ≤ R.

Proof. We only discuss the proof that (1) and (2) are equivalent. The proof that (i) and (ii) are equivalent follows the same structure.

n First, note that (2) implies Θ∗ (µ, x) ≥ λ. So, we assume (1) and show (2). Since n Θ∗ (µ, x) > 0 it follows that there exists δ = δ(x) such that for all r ≤ δ, µ(B(x, r)) ≥ n Θ∗ (µ,x) n 2 r . In particular, Θn(µ, x) µ(B(x, δ)) ≥ ∗ δn, 2 26

so for δ ≤ r ≤ R it follows

Θn(µ, x) Θn(µ, x) δn µ(B(x, r)) ≥ µ(B(x, δ)) ≥ ∗ δn ≥ ∗ rn, 2 2 Rn

n n Θ∗ (µ,x) δ n so λ = 2 R ≤ Θ∗ (µ, x)/2 suﬃces.

Finally, given a measure µ on Rm we let µk denote the measure on (Rm)k deﬁned as the k-fold product of µ with itself. Similarly, given a set E ⊂ Rm we let Ek denote the k-fold product of E as a set in (Rm)k.

2.2.2 Menger-type curvature, a formal review

Simplices and Notation

m Given points {x0, . . . , xn} ⊂ R then ∆(x0, . . . , xn) will denote the convex hull of {x0, . . . , xn}.

In particular, if {x0, . . . , xn} are not contained in any (n − 1)-dimensional plane, then

∆(x0, . . . , xn) is an n-dimensional simplex. Moreover, aff{x0, . . . , xn} denotes the smallest affine subspace containing {x0, . . . , xn}. That is aff{x0, . . . , xn} = x0 +span{x1 −x0, . . . , xn − x0}. If ∆ is an n-simplex, it is additionally called an (n, ρ)-simplex if

hmin(x0, . . . , xn) = min dist(xi, aﬀ{x0, . . . , xi−1, xi+1, . . . , xn}) ≥ ρ. i

The next lemma can be found in [56, Lemma 3.7] and quantiﬁes some geometric properties of simplices, which is of particular interest when showing a certain integrand is (µ, p)-proper.

m Lemma 26. Let C ≥ 1, t > 0, x ∈ R , w ∈ B(x, Ct) and S = ∆(x0, . . . , xn) ⊂ B(x, Ct) t be some (n, C )-simplex. Deﬁne Sw = ∆(x0, . . . , xn, w) and choose distinct i, j ∈ {0, . . . , n}. Then

t 1. C ≤ |xi − xj| ≤ diam(Sw) ≤ 2Ct, 27

2. |xi − w| ≤ 2Ct,

tn n (2C)n n 3. Cnn! ≤ H (S) ≤ n! t ,

n+1 H (Sw) 4. dist(w, aﬀ(x0, . . . , xn)) = n Hn(S)

The next lemma is an immediate consequence of repeated applications of [56, Lemma 2.17].

Lemma 27. Let 0 < k ≤ n. If Tx = ∆(x0, . . . , xn) is an (n, ρ)-simplex, and {y0, . . . , yk} are such that for all i ∈ {1, . . . , k} |xi − yi| < δ for some small δ > 0 satisfying (k + 1)δ < ρ, then ∆(y0, . . . , yk, xk+1, . . . , xn) is an (n, ρ − (k + 1)δ)-simplex.

Meurer’s proper integrands

The ability to use a general class of integrands K :(Rm)n+2 → [0, ∞) as a tool to study countably n-rectiﬁable sets and measures in Rm was demonstrated by Meurer in [56]. Below is the deﬁnition of the general class of integrands as laid out by Meurer.

Deﬁnition 28 ((µ, p)-proper integrand). Let n, m ∈ N with 1 ≤ n < m. Let K :(Rm)n+2 → [0, ∞) and 1 < p < ∞. One says that K is a (µ, p)-proper integrand if it fulﬁlls the following four conditions:

1. K is µn+2-measurable, where µn+2 denotes the (n + 2)-product measure of µ.

2. There exists some constants c = c(n, K, p) ≥ 1 and ` = `(n, K, p) ≥ 1 so that, for all

m t t > 0, C ≥ 1, x ∈ R and all (n, C )-simplices ∆(x0, . . . , xn) ⊂ B(x, Ct), it follows

d(w, aﬀ(x , . . . , x ))p 0 n ≤ cC`tn(n+1)Kp(x , . . . , x , w) (2.28) t 0 n

for all w ∈ B(x, Ct). 28

3. For all λ > 0,

n(n+1) p p λ K (λx0, . . . , λxn+1) = K (x0, . . . , xn+1) (2.29)

4. K is translation invariant in the sense that for every b ∈ Rm,

K(x0 + b, . . . , xn+1 + b) = K(x0, . . . , xn+1) (2.30)

Remark 29. The preceding deﬁnition is rather long and written so that expressions show-up in the same form that they do in the proof of [55, Theorem 5.6], a theorem which roughly provides a bound on βp-numbers by Menger curvature. As written above, one may notice that part (2) looks vaguely like one is bounding βp-numbers. However, the relationship becomes more obvious after applying part 3 to re-write part 2 of the deﬁnition of a (µ, p)-proper integrand in the following way:

There exists some constant c = c(n, K, p) ≥ 1 and ` = `(n, K, p) ≥ 1 so that, for all

m t t > 0, C ≥ 1, x ∈ R and all (n, C )-simplices ∆(x0, . . . , xn+1) ⊂ B(x, Ct), it follows

d(w, aff(x , . . . , x ))p x x w 0 n ≤ cC`Kp 0 ,..., n+1 , t t t t for all w ∈ B(x, Ct). In particular, ignoring all details and technicalities it looks like inte- grating the left-hand side over w yields the Lp-distance to a specific plane at scale t is bounded by the Menger curvature integrand “at scale t” when integrated over just one input, while the other inputs span the given affine plane.

Given a Borel measure µ and a (µ, p)-proper integrand K, the integral Menger curvature of µ with respect to K (or simply integral Menger curvature) is

Z p n+2 MKp (µ) = K (x0, . . . , xn+1) dµ (x0, . . . , xn+1). (Rm)n+2 29

The pointwise Menger curvature of x in µ with respect to K at scale r is

Z n p n+1 curvKp;µ(x, r) = K (x, x1, . . . , xn+1) dµ (x1, . . . , xn+1). B(x,r)n+1

We next show why one of the two integrands emphasized throughout this paper does indeed satisfy the deﬁnition of a (µ, 2)-proper integrand. As noted in [55, Lemma 3.9], the computations here are analogous to [55, Lemmata 3.7 and 3.8]. Nevertheless, we include them for the reader’s convenience. To precisely express the two integrands we deﬁne

n+1 X = {(x0, . . . , xn+1): H (∆(x0, . . . , xn+1)) > 0}.

Example 30. The following integrands are (µ, 2)-proper although they ﬁrst appear in the works of Lerman and Whitehouse (see [46], [47], [48]).

n+1 1 H (∆(x0, . . . , xn+1)) K1(x0, . . . , xn+1) := X (x0, . . . , xn+1) (n+1)(n+2) (diam{x0, . . . , xn+1}) 2 and 1 hmin(x0, . . . , xn+1) K2(x0, . . . , xn+1) := X (x0, . . . , xn+1) n(n+1)+2 diam({x0, . . . , xn+1}) 2 where hmin(x0, . . . , xn+1)) = mini dist{xi, aﬀ{x0, . . . , xi−1, xi+1, . . . , xn+1}. Since the expressions for the Menger-type curvatures look like they tend to zero as

c (x0, . . . , xn+1) tend to X , the indicator function is typically neglected. Below is an out- line of why (by considering K1) these examples are proper integrands. The proof of K2 is more straightforward.

Measurability follows due to the fact that X and (Rm)n+2 \ X are open and closed m n+2 n+2 respectively in (R ) . So µ -measurability follows since µ is Borel and K1 is continuous on X and (Rm)n+2 \ X. For the second condition in the deﬁnition of (µ, 2)-proper, consider t > 0,C ≥ 1, x ∈ 30

m t R , ∆ = ∆(x0, . . . , xn) ⊂ B(x, Ct) is an (n, C )-simplex, ﬁx w ∈ B(x, Ct) and let ∆w =

∆(x0, . . . , xn, w). Then,

d(w, aﬀ(x , . . . , x ))2 Hn+1(∆ )2 0 n = n w (26 part (4)) t t Hn(∆) Hn+1(∆ )2 ≤ (n · n! · Cn)2 w (26 part (3)) t · tn Hn+1(∆ ) 2 = (n · n! · Cn)2tn(n+1) w (n+1)+ n(n+1) t 2 n+1 2 n 2 n(n+1) H (∆w) = (n · n! · C ) t (n+1)(n+2) t 2 (n+1)(n+2) n 2 2 n(n+1) 2 ≤ (n · n! · C ) C t K1(∆) (26 parts (1,2))

(n+2)(n+1) 2 hence, the second property holds with ` = 2 + 2n and c = (n · n!) .

For homogeneity, note that if λ > 0, then (x0, . . . , xn+1) ∈ X ⇐⇒ (λx0, . . . , λxn+1) ∈ X.

Moreover, for (x0, . . . , xn+1) ∈ X it follows that

n+1 n+1 n+1 H (λx0, . . . , λxn+1) = λ H (x0, . . . , xn+1).

Consequently

λ2(n+1) K2(λx . . . , λx ) = K2(x , . . . , x ) 1 0 n+1 λ(n+2)(n+1) 1 0 n+1 −n(n+1) 2 = λ K1(x0, . . . , xn+1).

Translation invariance follows from the geometric nature of the deﬁnition.

Deﬁnition 31 (Symmetric (µ, p)-proper integrand). A(µ, p)-proper integrand is said to be symmetric if for all permutations σ ∈ Sn+2

p p K (x0, . . . , xn+1) = K (xσ(0), . . . , xσ(n+1)) 31

The next lemma, due to [56, Lemma 5.1], demonstrates that the restriction to symmetric proper integrands is a non-issue.

Lemma 32. Let µ be a Radon measure on Rm and ﬁx Kp some (µ, p)-proper integrand. Then, ˜ p there exists K a symmetric (µ, p)-proper integrand which satisﬁes MKp (µ∩E) = MK˜p (µ∩E) for all Borel sets E.

The proof is to use Fubini’s theorem to check that the integrand

˜ p 1 X p K (x0, . . . , xn+1) = K (xσ(0), . . . , xσ(n+1)) #|Sn+2| σ∈Sn+2

satisfies MKp (µ ∩ E) = MK˜p (µ ∩ E) for all Borel E. Moreover, it clearly satisfies conditions (1), (3), and (4) in Definition 28. So, it only remains to check that condition (2) holds. But, p ˜ p K ≤ #|Sn+2|K validates condition (2) of Definition (28).

2.3 Proofs of main results

One main tool of Meurer’s work (see [55, Theorem 4.1]), a non-trivial generalization of an analogous result from [45], is the following:

Theorem 33. Let K :(Rm)n+2 → [0, ∞) be a (µ, 2)-proper integrand. If µ is a Borel measure m on R , then there exists some small η = η(K, n, m, C0) > 0, so that if µ satisﬁes

(A) µ(B(0, 2)) ≥ 1 and µ(Rm \ B(0, 2)) = 0

n (B) µ(B) ≤ C0(diam B) for every ball B.

(C) MK2 (µ) ≤ η then there exists some Lipschitz function f : Rn → Rm−n with Lipschitz constant bounded above by some Λ = Λ(K, n, m, C0) such that a rotation of the graph of f, named Γ, satisﬁes

1 µ( m \ Γ) < µ( m). (2.31) R 100 R 32

Moreover, for given K and C0 the Lipschitz constant of f goes to zero as MK2 (µ) approaches zero.

2.3.1 Scaling Menger Curvature

Theorem 33 is one of the main tools for proving Theorem 15. It will also be useful to know how integral Menger curvature scales, and how this impacts Theorem 33.

Proposition 34. Let µ be a Radon measure and K a (µ, p)-proper integrand. Let ν be the

Radon measure deﬁned by ν(A) = λµ(aA + x) for some a, λ > 0 and x ∈ Rm then

n+2 −n(n+1) MK2 (ν) = λ a MK2 (µ). (2.32)

µ(rE+x) m In particular, if µx,r is deﬁned so that µx,r(E) = rn for all E ⊂ R , then

MKp (µ|B(x,r)) M p (µ | ) = (2.33) K x,r B(0,1) rn

Proof. Fix a Borel measure µ on Rm a point, x ∈ Rm and a, λ > 0. Deﬁne f : Rm → Rm by y−x f(y) = a , then ν = λf#µ (See [51, 1.17-1.19] for deﬁnition and use of image measures). Consequently,

Z p MKp (ν) = K (y0, . . . , yn+1) d(λf#µ(y0)) ··· d(λf#µ(yn+1)) (Rm)n+2 Z n+2 p = λ K (ax0 + x, ax1 + x, . . . , axn+1 + x) dµ(x0) ··· dµ(xn+1) (Rm)n+2 Z n+2 −n(n+1) p = λ a K (x0, . . . , xn+1) dµ(x0) ··· dµ(xn+1) (Rm)n+2

where the final line follows by first applying translation invariance and then the homogeneity of Kp (see conditions (2.29) and (2.30) in the definition of a (µ, p)-proper integrand). This 33

proves (2.32). We see (2.33) follows from choosing λ = r−n and a = r−1 combined with the observation

−1 y−x that f (B(0, 1)) = B(x, r) when f(y) = r .

We note that (2.33) is of interest due to the following modiﬁcation of Theorem 33

Theorem 35. Let µ be an n-Ahlfors upper-regular Radon measure with upper-regularity con-

n stant C. Let K be a (µ, 2)-proper integrand. Then, there exists a constant η1 = η1(K, n, m, Θ (µ, x, r),C) > 0 such that M 2 µ| K B(x,r) ≤ η (2.34) rn 1 implies there exists some Lipschitz graph Γ with Lipschitz constant bounded above by some

Λ = Λ(η1) such that 1 µ (B(x, r) \ Γ) < µ(B(x, r)). (2.35) 100

M 2 (µ| ) Moreover, given K, Θn(µ, x, r),C the Lipschitz constant of Γ tends to zero as K B(x,r) rn approaches zero.

An immediate corollary to Theorem 35 is

Corollary 36. Let µ be an n-Ahlfors regular Radon measure with lower-regularity constant c, and upper-regularity constant C. If K is a (µ, 2)-proper integrand and (2.34) is satisﬁed with

η1 = η1(K, n, m, c, C) for all x ∈ spt(µ) and all 0 < r < diam(spt(µ)) then µ is uniformly n-rectiﬁable. µ | Proof. (of Theorem 35). We claim that ν = x,r B(0,1) satisﬁes Θn(µ, x, r)

m 1. ν(B(0, 1)) ≥ 1 and ν(R \ B(0, 1)) = 0.

C 2. ν(B(y, t)) ≤ tn c

3. MK2 µ|B(x,r) M 2 (ν) = . K rncn+2 34

1 µ(B(x,r)) Θn(µ,x,r) Indeed, (1) follows since ν(B((0, 1))) = Θn(µ,x,r) rn = Θn(µ,x,r) = 1 and ν is the restriction of a measure to B(0, 1). To see (2) follows, consider y ∈ B(0, 1) and t > 0. Then,

1 µ (B(x + ry, rt) ∩ B(x, r)) ν(B(y, t)) = Θn(µ, x, r) rn 1 µ(B(x + ry, rt)) 1 C(rt)n ≤ ≤ Θn(µ, x, r) rn Θn(µ, x, r) rn C ≤ tn. Θn(µ, x, r)

Finally, (3) follows since (2.33) in Proposition 34 guarantees

MK2 (µ|B(x,r)) M 2 µ | = K x,r B(0,1) rn then applying (2.32) with λ = Θn(µ, x, r)−1, a = 1 yields

n −1 MK2 (ν) = MK2 (Θ (µ, x, r) µx,r|B(0,1))

n −(n+2) n −(n+2) MK2 (µ|B(x,r)) = Θ (µ, x, r) M 2 µ | = Θ (µ, x, r) K x,r B(0,1) rn

n Consequently, if η1 (K, m, n, Θ (µ, x, r),C) is chosen so that it is at most as large as n n+2 C C Θ (µ, x, r) η(K, m, n, c ) where η(K, m, n c ) is as in Theorem 33, then ν satisﬁes the hy- ˜ m ˜ 1 m pothesis of Theorem 33. Hence, there exists a Lipschitz graph Γ with ν(R \ Γ) < 100 ν(R ). m ˜ But ν(R \Γ) = µ(B(x,r)\Γ) where Γ = rΓ˜ + x. In particular, µ(B(x, r) \ Γ) < 1 µ(B(x, r)) as ν(Rm) µ(B(x,r)) 100 desired.

Remark 37. Note that in the case K = K1 (along with several other speciﬁc integrands studied in [46], [47]) Corollary 36 is already known from the work of Lerman and Whitehouse. In fact, the hypothesis in Theorem 35 is stronger than theirs, because Theorem 35 requires not only a Carleson-type bound on the local integral Menger curvature, but that the bound be by a small constant. 35

To create an eﬀective theory of quantitative, albeit non-uniform rectiﬁability, it would be interesting to address this seemingly unnecessary “smallness” condition being imposed by η1 in (2.34). It would also likely be useful to allow Γ to only satisfy µ(B(x, r) \ Γ) ≤ (1 − )µ(B(x, r)) for some = (η, m, n, K,C, Θn(µ, x, r)).

2.3.2 Rectiﬁability from integral Menger curvature

The goal of this section is to prove Theorem 15 included below for completeness.

Theorem 38. If µ is a Radon measure on Rm with 0 < Θn,∗(µ, x) < ∞ for µ almost m every x ∈ R and MK2 (µ) < ∞ for some (µ, 2)-proper integrand K, then µ is countably n-rectiﬁable.

The proof could be separated into two parts. The first part is a sequence of arguments to show that Radon measures which are mutually absolutely continuous with respect to the Hausdorff measure behave sufficiently similar the Hausdorff measure restricted to sets. The second part follows an argument from Sections 1 and 2 of [45] and is also similar to the case for sets as presented in [56]. Several preparatory lemmas are required.

Lemma 39. Let µ be a Radon measure on Rm with 0 < Θn,∗(µ, x) < ∞ for µ almost every m m 1 m x ∈ R . Then there exists some Borel set E such that ν = µ|E satisﬁes ν(R ) ≥ 2 µ(R ) and

n,∗ m 0 < c ≤ Θ (ν, x) ≤ C < ∞ for ν almost every x ∈ R .

Proof. Without loss of generality, suppose µ(Rm) < ∞. Otherwise, apply the ﬁnite case to a family of disjoint annulli that exhaust Rm. Moreover, suppose µ(Rm) > 0 to avoid trivialities.

m −j n,∗ j c c Deﬁne Ej = {x ∈ R : 2 ≤ Θ (µ, x) ≤ 2 }. Then Ej ⊃ Ej+1 for all j and moreover

∞ \ c m n,∗ n,∗ Ej = {x ∈ R : 0 = Θ (µ, x) or Θ (µ, x) = +∞}. j=1 36

c m Since µ(E1) ≤ µ(R ) < ∞ it follows that

∞ ! c \ c lim µ(Ej ) = µ Ej = 0 j→∞ j=1

1 m −k k so there exists k with µ(Ek) ≥ 2 µ(R ). Fix such k and let c = 2 and C = 2 . Then, deﬁne m 1 m ν = µEk . Since Ek is µ-measurable, it follows ν is Radon and ν(R ) ≥ 2 µ(R ). On the other hand the Lebesgue-Besicovitch diﬀerentiation theorem ensures Θn,∗(ν, x) = Θn,∗(µ, x)

m ν(B(x,r)) r↓0 m −k n,∗ k for ν a.e. x ∈ R since µ(B(x,r)) −−→ 1 for ν a.e. x ∈ R . Consequently 2 ≤ Θ (ν, x) ≤ 2 for ν a.e. x ∈ Rm as desired.

Lemma 40. Let µ be a Radon measure with 0 < c ≤ Θn,∗(µ, x) ≤ C < ∞ for µ a.e. x ∈ Rm such that spt(µ) is bounded. If K is a (µ, p)-proper integrand for some 1 < p < ∞ and

∗ MKp (µ) < ∞, then for all ζ0 > 0 there exists a compact set E ⊂ spt(µ) with

∗ c ∗ n (i) µ(E ) ≥ 2n+2 (diam E )

(ii) For all x ∈ E∗ and all t > 0, µ (E∗ ∩ B(x, t)) ≤ 2Ctn.

∗ n (iii) MKp (µ|E∗ ) ≤ ζ0 (diam E )

Proof. Since µ is Radon, and spt(µ) is bounded, it follows that µ(Rm) < ∞. Deﬁne

m −` n E` = {x ∈ R : ∀t ∈ (0, 2 ), µ(B(x, t)) ≤ 2Ct }. (2.36)

n,∗ Evidently, E` ⊂ E`+1. Moreover, the assumption c ≤ Θ (µ, x) ≤ C for µ almost every x ∈ Rm ensures `→∞ m µ(E`) −−−→ µ(R ).

m 1 m Hence, by the ﬁniteness of µ(R ) there exists some ` such that µ(E`) ≥ 2 µ(R ). Deﬁne

ν = µ|E` . Then, notably

n −` ν(B(x, r)) ≤ 2Cr ∀x ∈ E` ∀0 < r ≤ 2 (2.37) 37

and by Lebesgue-Besicovitch diﬀerentiation theorem [37, Theorem 1.7.1],

µ(B(x, r) ∩ E ) ` −−→r→0 1 µ a.e. x ∈ E µ(B(x, r)) ` which implies

n,∗ c ≤ Θ (ν, x) ν a.e. x ∈ E` (2.38)

µ(B(x, r) m since, by assumption lim sup n ≥ c for µ almost every x ∈ R . r→0 r Deﬁne

n+2 A(τ) = (x0, . . . , xn+1) ∈ (E`) : |x0 − xi| < τ ∀i ∈ {1, . . . , n + 1} . (2.39)

Claim 1: νn+2(A(τ)) −−→τ→0 0. Indeed, note that A(τ) = S {x} × (B(x, τ) ∩ E )n+1. In particular (2.37) ensures that x∈E` ` for all τ ≤ 2−`,

Z n+2 n+1 ν (A(τ)) = ν (B(x, τ) ∩ E`) dν(x) E` Z n (n+1) n (n+1) ≤ (2Cτ ) dν(x) = (2Cτ ) ν(E`). E`

Since ν is finite, the claim follows. Claim 2: I(τ) defined in (2.40) satisfies I(τ) −−→τ→0 0.

Z p n+2 I(τ) = K (x0, . . . , xn+1) dν (x0, . . . , xn+1) (2.40) A(τ)

p 1 m n+2 n+2 Indeed, MKp (ν) ≤ MKp (µ) < ∞ implies K ∈ L ((R ) , ν ). Then deﬁne

Z Z p n+2 p n+2 I(τ) = 1A(τ)K dν ≤ K dν . (Rm)n+2 (Rm)n+2

Consequently, for any sequence of τk converging to zero, Claim 1 ensures that the correspond- 1 p n+2 1 ing sequence of functions { A(τk)K } converges to zero ν a.e., and is bounded by the L 38

function Kp. So, the dominated convergence theorem validates Claim 2. Consequently given

−` ζ > 0 depending only on c, C and ζ0 to be chosen later, there exists τ0 such that 0 < 2τ0 < 2 and ζ I(2τ ) ≤ ν( m). (2.41) 0 16C R

Next, deﬁne a cover of E` by

n c o G = B(x, τ): x ∈ E , 0 < τ < τ , and Θn(ν, x, τ) ≥ . (2.42) ` 0 2

In particular, (2.38) guarantees G is a ﬁne cover of some ν-measurable set E ⊂ E` with ν-full measure, i.e., ν(E) = ν(Rm). So, the corollary to Besicovitch’s covering theorem, [37,

Corollary 1.5.2] ensures that there exists a countable, disjoint subfamily {Bi} of G with

∞ m [ ν(R \ Bi) = 0. (2.43) i=1

−` In light of (2.36) and (2.42), τ < τ0 < 2 ensures

∞ ∞ n X X diam Bi ν( m) = ν(B ) ≤ (2C) R i 2 i=1 i=1 so that m ∞ n ν( ) X diam Bi R ≤ . (2.44) 2C 2 i=1

n+2 Moreover, (Bi ∩ E`) ⊂ A(2τ0) ∩ Bi so, (2.41) and (2.40) yields

∞ X ζ m M p (ν| ) ≤ I(2τ ) ≤ ν( ). (2.45) K Bi 0 16C R i=1

Deﬁne the index set of “bad” balls, or balls with too much Menger curvature by

n ( diam Bi ) 2 I = i ∈ : M p (ν| ) ≥ ζ (2.46) b N K Bi 4 39

Then,

n X ζ X diam Bi M p (ν| ) ≥ (2.47) K Bi 4 2 i∈Ib i∈Ib

n m Notice that if P diam Bi > ν(R ) then additionally considering (2.45) and (2.47) implies i∈Ib 2 4C

n X ζ m ζ X diam Bi X M p (ν| ) ≤ ν( ) < ≤ M p (ν| ) K Bi 16C R 4 2 K Bi i∈N i∈Ib i∈Ib which is a contradiction. It follows

n m X diam Bi ν( ) ≤ R . (2.48) 2 4C i∈Ib

Now, (2.44) and (2.48) together ensure that Ib 6= N.

From now on, ﬁx i ∈ N \ Ib. The inner-regularity of Radon measures ensures that there exists some compact E∗ with

1 E∗ ⊂ B ∩ E and ν(E∗) ≥ ν(B ). (2.49) i ` 2 i Then evidently, E∗ satisﬁes:

n ∗ 1 1 c diam Bi c ∗ n 1. ν(E ) ≥ 2 ν(Bi) ≥ 2 2 2 ≥ 2n+2 diam(E ) , where the second inequality is

because Bi ∈ G, see (2.42).

∗ −` ∗ 2. For all x ∈ E and for all 0 < t < 2 it follows from (2.36) and E ⊆ E` that

diam (E∗ ∩ B(x, t))n ν (E∗ ∩ B(x, t)) ≤ 2C . (2.50) 2

∗ −` −` On the other hand, E ⊂ Bi and diam(Bi) ≤ 2τ0 < 2 . So, for t ≥ 2 it follows,

n diam B n ν (E∗ ∩ B(x, t)) ≤ ν(B ) ≤ 2C i < 2C 2−` ≤ 2Ctn i 2 40

so that in fact ν|E∗ is n-Ahlfors upper regular with regularity constant 2C, that is (2.50) holds for all t > 0.

3. It follows 1 diam(B )n 2C i ≤ (diam E∗)n. (2.51) 4 2 c Indeed, choose a ball B with

diam B ≤ 2 diam E∗ and E∗ ⊂ B. (2.52)

Combining (2.42), (2.49), (2.50), and (2.52) yields

c diam B n ν(B ) i ≤ i ≤ ν(E∗) = ν(E∗ ∩ B) 4 2 2 diam B n ≤ 2C ≤ 2C diam(E∗)n. 2

Hence, the lower bound on diam E∗ follows.

4. Finally, since i ∈ N \ Ib, (2.46) and (2.51) yields

n ζ diam Bi 2C ∗ n M p (ν| ∗ ) < ≤ ζ (diam E ) K E 4 2 c

cζ0 Choosing ζ = 2C completes the proof.

The next technical lemma is a standard “structure theorem” and is contained in for instance [35, 3.3.12 - 3.3.15]

Lemma 41. If µ is a Radon measure with 0 < Θn,∗(µ, x) for µ almost every x, then one can write µ = µr +µu where µr = µ|E for some µ-measurable E, and µr is countably n-rectiﬁable.

On the other hand, µu is purely unrectiﬁable.

Now, we are prepared to show 41

Lemma 42. Let µ be a Radon measure on Rm with 0 < Θn,∗(µ, x) < ∞ for µ almost every m 2 x ∈ R . Let K a (µ, 2)-proper integrand. After writing µ = µu + µr as in Lemma 41, if m µu(R ) > 0 then MK2 (µ) = +∞.

Notably, Lemma 42 is the contrapositive of Theorem 38.

2 Proof. Let K and µ, µu, µr be as in the statement of Lemma 42. Without loss of generality, m n,∗ m suppose 0 < µ(R ) < ∞. It follows 0 < Θ (µu, x) < ∞ for µu almost every x ∈ R , since the Lebesgue-Besicovitch diﬀerentiation theorem guarantees

n,∗ µu(B(x, r)) µ(B(x, r)) n,∗ Θ (µu, x) = lim sup n = Θ (µ, x) r→0 µ(B(x, r)) r

m for µu a.e. x ∈ R . m Moreover, by assumption µu(R ) > 0. By Lemma 39 it follows that there exists ν0 a n,∗ restriction of µu and some c, C such that 0 < c ≤ Θ (ν0, x) ≤ C < ∞ for ν0 almost every m 1 m m m x ∈ R . Lemma 39 also guarantees that, 0 < 2 µu(R ) < ν0(R ) ≤ µ(R ) < ∞. Since

ν0 is a restriction of µu to some Borel set, it follows that ν0 is a Radon measure satisfying

MK2 (ν0) ≤ MK2 (µu) ≤ MK2 (µ). In the spirit of contradiction, suppose MK2 (µ) < ∞.

m Since ν0(R ) < ∞, without loss of generality, suppose spt(ν0) is bounded. Then, ν0 satisﬁes the hypothesis of Lemma 40. In particular, for

2n+2 −(n+2) ζ < η · where η = η K, n, m, 2n+3Cc−1 is from Theorem 33, (2.53) 0 c

∗ there exists some compact E ⊂ spt(ν0) such that

(i) c ν (E∗) ≥ (diam E∗)n (2.54) 0 2n+2 42

(ii) For all x ∈ E∗ and all t > 0,

∗ n ν0 (E ∩ B(x, t)) ≤ 2Ct . (2.55)

(iii)

∗ n MKp (ν0|E∗ ) ≤ ζ0 (diam E ) (2.56)

Our next goal is to scale and translate ν0 to ﬁnd a measure ν1 which satisﬁes the hypoth- ∗ esis of Theorem 33. To this end, choose x0 ∈ E . Then,

∗ ∗ E ⊂ B(x0, diam(E )). (2.57)

y−x0 Let f(y) = diam(E∗) , so that

−1 ∗ f (B(x0, diam(E ))) = B(0, 1). (2.58)

Deﬁne

n+2 ∗ −n 2 ν = (diam E ) f (ν | ∗ ). (2.59) 1 c ] 0 E

It follows by computations similar to those at the beginning of the proof of Corollary 36 that ν1 satisﬁes the hypotheses of Theorem 33. Consequently, Theorem 33 ensures there exists a Lipschitz graph Γ such that

m ν1(R \ Γ) 1 m < . ν1(R ) 100

But, recalling (2.59), it is clear this implies

m (ν0|E∗ )(R \ f(Γ)) 1 m < . ν0(R ) 100 43

Since f is a translation and scaling, f(Γ) is still a Lipschitz graph, contradicting the fact that ν0 is the restriction of an n-purely unrectiﬁable measure.

2.3.3 Pointwise Menger curvature and β-numbers.

The ﬁrst goal of this section is to prove Theorem 16, included below for convenience.

m n n,∗ Theorem 43. Let µ be a Radon measure on R with 0 < Θ∗ (µ, x) ≤ Θ (µ, x) < ∞ for µ almost every x ∈ Rm. Then the following are equivalent:

1. µ is countably n-rectiﬁable.

m n 2. For µ almost every x ∈ , curv 2 (x, 1) < +∞. R K1;µ

m n 3. For µ almost every x ∈ , curv 2 (x, 1) < +∞. R K2;µ

4. µ has σ-ﬁnite integral Menger curvature in the sense that µ can be written as µ = P∞ µ where each µ satisﬁes M 2 (µ ) < ∞. j=1 j j K1 j

5. µ has σ-ﬁnite integral Menger curvature in the sense that µ can be written as µ = P∞ µ where each µ satisﬁes M 2 (µ ) < ∞. j=1 j j K2 j

Proof. The fact that (1) =⇒ (2) is the content of [44, Lemma 1.1] combined with the characterization by Azzam and Tolsa in Theorem 3. Since K1 ≤ K2 pointwise, it also follows that (2) =⇒ (3). So, it suﬃces to show (3) =⇒ (4) =⇒ (1) and (2) =⇒ (5) =⇒ (1). To this end, ﬁx µ as in the statement of the theorem. Moreover, without loss of generality suppose that

m µ(R ) < ∞ (2.60)

Then, for j ∈ N0 deﬁne

m n Ej = {x ∈ : curv 2 (x, 1) ∈ [j, j + 1)} and µj = µ|Ej . (2.61) R K1;µ 44

Fubini’s theorem [37, Theorem 1.4.1] ensures that the map

Z 2 n+1 n x 7→ K (x, x1, . . . , xn+1) dµ (x1, . . . , xn+1) = curv 2 (x, ∞) 1 K1;µ (Rm)n+1 is measurable. In particular the sets Ej are measurable, as each Ej is the preimage of a

Borel set by a measurable function. For each j ∈ N0 it follows from (2.60) that µj = µ|Ej is a ﬁnite, Borel measure. The Lebesgue-Besicovitch diﬀerentiation theorem ensures that

n,∗ m µj(B(x, r)) m 0 < Θ (µj, x) < ∞ for µj a.e. x ∈ R since lim = 1 for µj a.e. x ∈ R . Since r→0 µ(B(x, r)) 2 K1 is a non-negative function, and µj satisﬁes µj(E) ≤ µ(E) for all µ-measurable sets E it follows

Z n 2 n+1 curv 2 (x, ∞) = K (x, x1, . . . , xn+1) dµ (x1, . . . , xn+1) K1;µj 1 j (Rm)n+1 Z 2 n+1 ≤ K1(x, x1, . . . , xn+1) dµ (x1, . . . , xn+1) (Rm)n+1

≤ (j + 1) ∀x ∈ Ej,

where the ﬁnal line follows from the deﬁnition of Ej, (2.61). Combining the above computation with (2.60) yields,

Z n m MK2 (µj) = curv 2 (x, ∞) dµj(x) < (j + 1)µj( ) < ∞. 1 K1;µj R Rm

Therefore, (3) =⇒ (4). To see (4) =⇒ (1) , note that Theorem 38 ensures each µj P n is n-countably rectifiable. Since µ = j µj satisfies µ H has been decomposed into countably many n-countably rectifiable pieces, it follows µ is countably n-rectifiable.

The fact that (2) =⇒ (5) =⇒ (1) follows identically with K2 in place of K1.

Given the history of the subject, this method of proof is not very satisfying, namely due to the fact that Lerman and Whitehouse’s characterization of uniform rectiﬁability in terms of integral Menger curvature demonstrated an equivalence of a Carleson-type condition for 45

integral Menger curvature and the Carleson condition for the β-numbers. Moreover, the ﬁrst direction in this characterization follows from a direct comparison with β-numbers due to Kolasi´nski. Our next goal is to show Theorem 19, which is equivalent to Theorem 44. We simply use Lemma 25 to restate the theorem for the sake of illuminating the dependencies on x.

Theorem 44. If µ is an n-Ahlfors upper-regular Radon measure on Rm with upper-regularity n constant C0, and there exists λ such that µ(B(x, r)) ≥ λr for all 0 < r ≤ R, then

Z R ˆn 2 dr n βµ;2(x, r) ≤ C3 curvK2;µ(x, R), (2.62) 0 r where C3 = C3(λ, m, n, C0, K), and K is any (µ, 2)-proper integrand.

In fact, if µ, λ, x, r, and R are as above, then for K ∈ {K1, K2}

Z R Z 2R ˆn 2 dr n n ˆn 2 dr βµ;2(x, r) ≤ C curvµ;K2 (x, R) ≤ C · Γ Θ (µ, x, r)βµ;2(x, r) (2.63) 0 r 0 r Z 2R ˜ ˆn 2 dr ≤ C βµ;2(x, r) 0 r with constants C, Γ, C˜ depending on m, n, λ the upper-regularity constant of µ, and K.

This theorem demonstrates a more direct converse to Kolasi´nski’sbound on β-numbers. Alas, notice that in its present form, Theorem 44 requires stronger density conditions than Theorem 43. It would be interesting to try to weaken the density conditions at least as far as they are in Theorem 43. The following technical lemma plays a central role in the proof of Theorem 44. For a review of the notation used in the proof, see Section 2.2.2.

Lemma 45. Let µ be an n-Ahlfors upper-regular Radon measure on Rm with upper-regularity m constant C0. Suppose x ∈ R and λ, R > 0 such that

µ(B(x, r)) ≥ λrn (2.64) 46

holds for all 0 < r ≤ R.

Then, for λ δ = δ(n, λ, C0) = k+2 n−1 (2.65) 2 5 C0 and δ λ η = η(n, λ, C0) = = k+3 n (2.66) 10n 2 5 nC0

n and all 0 < r ≤ R there exist points {xi,r}i=1 ⊂ B(x, r) such that

hmin(x, x1,r, . . . , xn,r) ≥ δr (2.67) and ληm (µ| )(B(x , 5ηr)) ≥ rn = C (m, n, λ, C )rn. (2.68) B(x,r) i,r 2m+1 2 0

In particular, if for each i ∈ {1, . . . , n}, Bi,r := B(xi,r, 5ηr) for any choices of yi ∈ Bi,r it follows that δr h (x, y , . . . , y ) ≥ δr − 5nηr = . (2.69) min 1 n 2

Finally, if Br := B1,r × · · · × Bn,r then

Bδr/3 ∩ Br = ∅. (2.70)

Proof. (of Lemma 56). Let m, n, µ, C0, λ and R be as in the lemma statement. In particular,

λ ≤ C0. Deﬁne δ, η as in (3.9) and (3.10).

Fix 0 < r < R, and suppose there exist {x1, . . . , xk} satisfying

ληm h (x, x , . . . , x ) ≥ δr and µ(B(x , 5ηr)) ≥ rn min 1 k i 2m+1

for all i = 1, . . . , k and assume that k < n. Then, we will ﬁnd a point xk+1 such that ληm n hmin(x, x1, . . . , xk+1) ≥ δr and µ(B(xk+1, 5ηr)) ≥ 2m+1 r . Hence, induction will guarantee 47

the theorem.3

Let Vk = aﬀ{x, x1, . . . , xk} = x + span{x1 − x, . . . , xk − x}. Deﬁne

(Vk)δr = {y ∈ B(x, r) : dist(y, Vk) < δr}. (2.71)

Deﬁne ρ = ρ(λ, C0, n, k, r) by

λ 1 ρ = sr where s = k+1 n < 1. (2.72) C0 2 · 5

Note, s < 1 since λ ≤ C0. Let

G1 = {B(y, 5ρ) | y ∈ Vk ∩ B(x, r)}. (2.73)

5 5ρ We ﬁrst note that G1 is a cover of B(x, r) ∩ (Vk)δr since δ = 2 s implies δr = 2 . So, by Vitali N 0 S we can ﬁnd a subfamily of sets {B(xi, 5ρ)}i=1 such that B(x, r) ∩ (Vk)δr ⊂ i B(xi, 5ρ) and N 0 {B(xi, ρ)}i=1 is disjoint.

0 0 A priori, N could be inﬁnite, but we will see that N ≤ Nk,s = Nn,k,λ,C0 where

2k N = . (2.74) n,k,λ,C0 s

For α a positive integer, letting ωα denote the volume of a k-dimensional unit ball, since

B ∈ G1 implies B ∩ Vk is a k-dimensional ball of radius 5ρ,

N 0 N 0 ! 0 k X k k [ N ωkρ = H (Vk ∩ B (xi, ρ)) = H |Vk B (xi, ρ) i=1 i=1 k k ≤ H |Vk (B(x, 2r)) = ωk(2r)

3 The proof of the inductive step clearly shows that we can also ﬁnd a point x1 with |x1 − x| ≥ δr and ληm n µ(B(x1, 5ηr)) ≥ 2m+1 r . 48

0 k −1 k −1 k so that N ≤ 2 (rρ ) = (2s ) = Nn,k,λ,C0 .

λrn We wish to show that our choice of δ forces µ(B(x, r) ∩ (V ) ) ≤ so that k δr 2

λrn µ(B(x, r) \ (V ) ) ≥ . (2.75) k δr 2

Indeed,

N 0 N 0 X X n n n n µ ((Vk)δr ∩ B(x, r)) ≤ µ(B(xi, 5ρ)) ≤ C0(5ρ) ≤ Nn,k,λ,C0 C05 s r i=1 i=1 2k = C 5nsnrn. s 0

So, it suﬃces to show λrn 2kC 5nsn−krn ≤ 0 2 which holds if and only if

n−k λ s ≤ k+1 n . 2 5 C0 Since s < 1 and k < n this implies that our choice of s in (2.72) suﬃces to ensures (2.75).

Now we claim that (2.75) guarantees the existence of some xk+1 ∈ B(x, r) \ (Vk)δr such that (3.12) holds. The fact that xk+1 ∈/ (Vk)δr will guarantee (3.11).

To this end, let us consider the family of balls

G2 = {B(y, 5ηr) | y ∈ B(x, r) \ (Vk)δr}.

δ Then B ∈ G2 implies B ⊂ B(x, 2r) since λ ≤ C0, (3.9), and (3.10) guarantee 5η < 2n < 1.

Moreover, G2 covers B(x, r) \ (Vk)δr. In particular, Vitali ensures there exists a subfamily M 0 M 0 {B(xi, 5ηr)}i=1 that covers B(x, r) \ (Vk)δr and {B(xi, ηr)}i=1 is a disjoint family. Again, we 49

have no apriori estimate on M 0, but disjointness and containment in B(x, 2r) yields

M 0 m 0 X m m m ωm(rη) M = H (B(xi, rη)) ≤ H (B(x, 2r)) = ωm(2r) . i=1

Consequently, we deﬁne Mm,n,λ,C0 so that

0 −1 m M ≤ (2η ) = Mη,m = Mm,n,λ,C0 . (2.76)

Combining (2.75) and (2.76), we deduce

λrn ≤ µ (B(x, r) \ (V ) ) 2 k δr M 0 ! [ ≤ µ|B(x,r) B(xi, 5ηr) i=1 M 0 X ≤ (µ|B(x,r))(B(xi, 5ηr)) i=1 0 ≤ Mm,n,λ,C0 max (µ|B(x,r))(B(xi, 5ηr)) | i ∈ {1,...,M } . (2.77)

Choosing k + 1 = j such that

0 (µ|B(x,r))(B(xj, 5ηr)) = max (µ|B(x,r))(B(xi, 5ηr)) | i ∈ {1,...,M } , we have from (2.77) that

λrn λrnηm (µ|B(x,r))(B(xk+1, 5ηr)) ≥ ≥ m+1 2Mm,n,λ,C0 2 verifying that xk+1 ∈ B(x, r) \ (Vk)δr satisﬁes (3.12).

It only remains to show (3.14) and (3.13), which follow quickly from the work already done. Indeed, Lemma 27 and (3.11) verify (3.14). On the other hand, (3.13) follows from

Bδr/3 ⊂ B(x, δr/3) and (3.11). 50

Proof. (Of Theorem 44) First, observe that (2.63) follows from (2.62) combined with (2.12) and Lemma 12 and the Ahlfors upper-regularity assumption on µ. Hence, the goal is to verify (2.62). Fix µ, R, λ as in the theorem statement. Let K be some (µ, 2)-proper integrand, and

0 < r ≤ R. Let {xi,r}, Bi,r and Br be as in Lemma 56. Then, first replacing the infimum with an average over fixed planes, and then applying (3.12), yields

Z 2 ˆn 2 1 dist(z, L) βµ;2(x, r) = inf n dµ(z) L3x r B(x,r) r Z Z dist(z, aff{x, y , . . . , y })2 2 dµ(z)dµn(y , . . . , y ) ≤ 1 n 1 n r µn( )rn Br B(x,r) Br Z Z dist(z, aff{x, y , . . . , y })2 dµ(z)dµn(y , . . . , y ) ≤ C 1 n 1 n , r n2+n Br B(x,r) r where C = C(m, n, λ, C0). Since {x, z} ∪ Bi,r ⊂ B(x, r) for all i = 1, . . . , n, (3.14) ensures we can apply (2.28) in the final integral above, so that

Z Z ˆn 2 2 n βµ;2(x, r) ≤ C K(x, z, y1, . . . , yn) dµ(z)dµ (y1, . . . , yn). (2.78) Br B(x,r)

Finally, using the fact that for any 0 < σ < 1,

Z R dr X βˆn (x, r)2 ≤ C βˆn (x, σjR)2 µ;2 r σ µ;2 0 j≥0

δ j when σ = δ/3 and writing rj := 3 R, (3.13) and (2.78) yield

Z R dr Z Z βˆn (x, r)2 ≤ C K(x, z, y , . . . , y )2dµn+1(z, y , . . . , y ), (2.79) µ;2 r 1 n 1 n 0 ∪j≥0Brj B(x,r)

n+1 where C = C(m, n, λ, C0). Since for all j, Brj × B(x, r) ⊂ B(x, r) , (2.62) follows from n+1 non-negativity of the integrand after replacing ∪j≥0Brj ×B(x, r) in (2.79) with B(x, r) . 51

Chapter 3 SUFFICIENT CONDITION FOR (C1,α, n)-RECTIFIABLE SETS ON Rm 52

3.1 An introduction to higher-order rectiﬁability

In 1990 Peter Jones introduced the β-numbers as a quantitative tool to provide control of the length of a rectiﬁable curve and to prove the Analyst’s Traveling Salesman Theorem

[42] in the plane. Kate Okikiolu extended the result to one-dimensional objects in Rn [57]. In order to study the regularity of Ahlfors regular sets and measures of higher dimensions [19, 20], David and Semmes generalized the notion of β-numbers, see (3.4). This was the beginning of quantitative geometric measure theory and has led to lots of activity around characterizing uniformly rectiﬁable measures and their connections to the boundedness of a certain class of singular integral operators.

More recently, rectifiable sets and measures have been studied using the quantitative techniques previously used for uniformly rectifiable measures. For instance, β-numbers can characterize rectifiability of measures, amongst the class of all measures with various density and mass bounds. See for instance [58, 8, 72, 30, 11].

Several other geometric quantities have also proven to be useful in quantifying the regularity of sets and measures. In this paper, we wish to explore how Menger-type curvatures, see Definitions 53 and 55, yield information about C1,α n-rectifiability, see Definition 50, of measures. In 1995, Melnikov discovered an identity for the (1-dimensional or classical) Menger curvature [54] which, in the complex plane, greatly simplified the existing proofs relating rectifiability to the L2-boundedness of the Cauchy integral operator [53, 52]. The dream was that a notion of curvature, and similar identity, could be found in higher dimensions to produce simpler proofs demonstrating the equivalence of uniform rectifiability to the L2-boundedness of singular integral operators. Alas, in 1999 Farag showed that in higher- dimensions no such identity could exist [32]. Nonetheless, geometric arguments made with non-trivial adaptations from [45] have since been used to characterize uniform rectifiability in all dimensions and codimensions in terms of Menger-type curvatures, [46, 47]. A sufficient condition for rectifiability of sets in terms of higher dimensional Menger-type curvatures appears in [56] and was extended to several characterizations of rectifiable measures under 53

suitable density conditions [39]. Menger curvatures have also been used to quantify higher regularity (in a topological sense) of surfaces, see, for instance [68, 14]. Of particular relevance in our context, [43] showed

α 1,α that finiteness of curvµ;p, see Definition 55, is a sufficient condition for C n-rectifiability of measures. A formulation of the theorem is1

m n n,∗ Theorem 46. Let µ be a Radon measure on R , with 0 < Θ∗ (µ, x) ≤ Θ (µ, x) < ∞, for µ-a.e. x ∈ Rm and let 1 ≤ p < ∞, 0 < α ≤ 1. If for µ- a.e. x ∈ Rm

α curvµ;p(x, 1) < ∞, (3.1) then µ is C1,α n-rectiﬁable.

The goal of this article is to prove that for a Radon measure µ satisfying relaxed density assumptions and for any α ∈ [0, 1) if the pointwise Menger curvature with p = 2, see Definition 55, is finite at µ a.e. x then µ is C1,α n-rectifiable. More precisely,

Theorem I. Let µ be a Radon measure on Rm, with 0 < Θn,∗(µ, x) < ∞ for µ-a.e. x ∈ Rm, and let α ∈ [0, 1). If for µ- a.e. x ∈ Rm

α curvµ;2(x, 1) < ∞, (3.2) then µ is C1,α n-rectiﬁable.

The α = 0 case in Theorem I appears in [39]. The case α > 0 is an improvement of a special case of [43] where the lower density assumption is relaxed.

1The familiar reader may be aware that there are two additional parameters in the theorem of [43]. One such parameter is the ability to choose from a small family functions to replace the hmin in the integrand α that deﬁnes curvµ;p. Such choices have previously been shown to be comparable to one another, see [46, 47, 48] or [43, 8.6-8.8]. The second such parameter was originally denoted by l. The l = n + 2 case is written here. Since any other choice of l is a stronger assumption, (changing the parameter ` is equivalent to replacing an Lp bound on some number of components of the integrand with an L∞ bound) we chose to remove this for readability. 54

A rough sketch of the proof is as follows: the condition (3.2) is shown to imply “flatness” of the support of µ in terms of Jones’ square function, and consequently (pieces of) the support of µ can be parametrized by Lipschitz graphs (see [56, Theorem 5.4]) when α = 0 or C1,α images (see [38, Theorem II]) when α > 0. On the other hand, the original proof provided by Kolasińskihad two major parts. First, under the additional assumption that µ is Lipschitz n-rectifiable Kolasińskishowed that the condition (3.1) forced additional flatness on the Lipschitz functions that cover the support of µ, which consequently implied better regularity on each such function (see [62, Lemma A.1]). Second, it remained to show that (3.1) implies Lipschitz n-rectifiability of µ. This was done by appealing to a characterization of rectifiability from [3, §2.8 Theorem 5] which roughly says that if the approximate tangent cone (in the sense of Federer) of µ is contained in an n-plane at almost every point, then µ is Lipschitz n-rectifiable.

Remark 47. Note that Theorem I requires 0 < Θn,∗(µ, x) for µ a.e. x, whereas Theeorem

n 46 requires the stronger assumption that 0 < Θ∗ (µ, x) for µ a.e. x. The stronger density assumption in Theorem 46 allows one to apply Theorem 57. Then Remark 58 and Proposition 59 provide a direct proof that

Z 1 µ 2 dr m β2 (x, r) < ∞ µ a.e. x ∈ R 0 r when working under the hypotheses of Theorem 46. This provides an alternative proof to what we previously called the second major part of the original proof of Theorem 46. The question of whether the proof of Theorem I when α = 0 can be completed by appealing to [8] after controlling Jones’ function as above is an interesting one. Presently, the authors

n do not know how to do this without additionally assuming 0 < Θ∗ (µ, x) for µ almost every x. Another diﬀerence between the two theorems is that Theorem I is stated only when p = 2. Since increasing p only makes condition (3.1) harder to satisfy, the results in Theorem 46 are not sharp in terms of the parameter p. We expect that varying the parameter p would 55

lead to results about rectiﬁability in the sense of Besov spaces, which is beyond the scope of this article.

The proof of Theorem I is divided into two parts. First, we prove the claim for a measure

µ which is n-Ahlfors upper regular on Rm and with positive lower density. Then, we use standard techniques to reduce the general case to the previous one.

3.2 Notation and Background

We begin by stating some deﬁnitions and notation.

Deﬁnition 48. Let 0 ≤ s < ∞ and let µ be a measure on Rm. The upper and lower s-densities of µ at x are deﬁned by

s,∗ µ(B(x, r)) Θ (µ, x) = lim sup s r→0 r

s µ(B(x, r)) Θ∗(µ, x) = lim inf . r→0 rs

If they agree, their common value is called the s-density of µ at x and denoted by

s s,∗ s Θ (µ, x) = Θ (µ, x) = Θ∗(µ, x). (3.3)

m Deﬁnition 49 (βp-numbers). Given x ∈ R , r > 0, p ∈ (1, ∞), an integer 0 ≤ n ≤ m, and a Borel measure µ on Rm deﬁne

p 1 Z p µ 1 dist(y, L) βp (x, r) = inf n d µ(y) , (3.4) L r B(x,r) r where the inﬁmum is taken over all n-planes L.

When we talk about rectifiability and higher-order rectifiability, we mean what Federer would call “countably n-rectifiable”. 56

Deﬁnition 50. A measure µ on Rn is said to be (Lipschitz) n-rectiﬁable if there exist n m countably many Lipschitz maps fi : R → R such that ! m [ n µ R \ fi(R ) = 0. (3.5) i

n 1,α 1,α n A measure µ on R is C n-rectiﬁable if there exist countably many C maps fi : R → Rm such that (3.5) holds.

In [38], a suﬃcient condition for C1,α n-rectiﬁability in terms of β-numbers is provided.

Theorem 51 ([38]). Let µ be a Radon measure on Rm such that n n,∗ m Θ∗ (µ, x) < ∞ and Θ (µ, x) > 0 for µ-almost every x ∈ R , and α ∈ (0, 1). Moreover, suppose, for µ-almost every x ∈ Rm,

Z 1 βµ(x, r)2 dr µ : 2 J2,α(x) = 2α < ∞ (3.6) 0 r r

Then, µ is C1,α n-rectifiable. When α = 1, if we replace r in the left hand side of (3.6) by rη(r), where η(r)2 satisfies the Dini condition, then we obtain that µ is C2 n-rectifiable.

Remark 52. We say that a function ω satisﬁes the Dini condition if R 1 ω(r) 1 1 0 r dr < ∞. A possible choice for η in Theorem 51 is η(r) = log(1/r)γ , for γ > 2 .

Deﬁnition 53 (Classical Menger curvature). Given three points x, y, z ∈ Rm, the (classical) Menger curvature is deﬁned to be the reciprocal of the circumradius of x, y, z. That is,

1 c(x, y, z) = , R(x, y, z) where R(x, y, z) is the radius of the unique circle passing through x, y, z.

In order to work with higher dimensional Menger curvatures, we introduce some notation for simplices in Rm. 57

m Deﬁnition 54 (Simplices). Given points {x0, . . . , xn} ⊂ R , ∆(x0, . . . , xn) denotes the convex hull of {x0, . . . , xn}. In particular, if {x0, . . . , xn} are not contained in any (n − 1)- dimensional plane, then ∆(x0, . . . , xn) is an n-dimensional simplex with corners {x0, . . . , xn}.

Moreover, we denote by aﬀ{x0, . . . , xn} the smallest aﬃne subspace containing {x0, . . . , xn}.

That is aﬀ{x0, . . . , xn} = x0 + span{x1 − x0, . . . , xn − x0}. Then, we deﬁne

hmin(x0, . . . , xn) = min dist(xi, aﬀ{x0, . . . , xi−1, xi+1, . . . , xn}) i to be the minimum height of a vertex over the plane spanned by the opposing face. If ∆ =

∆(x0, . . . , xn) we occasionally abuse notation and write hmin(∆) in place of hmin(x0, . . . , xn). If ∆ as before is an n-simplex, it is additionally called an (n, ρ)-simplex if

hmin(x0, . . . , xn) ≥ ρ.

Deﬁnition 55 (Menger curvatures). For x ∈ Rm, r > 0, α ∈ [0, 1), an integer 0 ≤ n ≤ m, and p ∈ [1, ∞], we deﬁne the curvature of µ at x of scale r to be

Z p α hmin(x, x1, . . . , xn+1) n+1 curvµ;p(x, r) = p(1+α)+n(n+1) dµ , (3.7) B(x,r)n+1 diam ({x, x1, . . . , xn+1}) where µn+1 is the product measure deﬁned by taking (n + 1)-products of µ with itself.

3.3 Proof of Theorem I

We now proceed to prove the theorem in the case where µ is n-Ahlfors upper regular on Rm and has positive lower density µ-almost everywhere. We recall the following Lemma from [39, Lemma 3.13], which says that, under the appropriate density assumptions on a measure µ, given a point x and radius r, the ball B(x, r) contains a large number of eﬀective n-dimensional secant planes through x.

Lemma 56. Let µ be an n-Ahlfors upper-regular Radon measure on Rm with upper-regularity 58

m constant C0. Suppose x ∈ R and λ, R > 0 such that

µ(B(x, r)) ≥ λrn (3.8) holds for all 0 < r ≤ R.

Then, for λ δ = δ(n, λ, C0) = k+2 n−1 (3.9) 2 5 C0 and δ λ η = η(n, λ, C0) = = k+3 n (3.10) 10n 2 5 nC0

n and all 0 < r ≤ R there exist points {xi,r}i=1 ⊂ B(x, r) such that

hmin(x, x1,r, . . . , xn,r) ≥ δr (3.11) and ληm (µ B(x, r))(B(x , 5ηr)) ≥ rn = C (m, n, λ, C )rn. (3.12) i,r 2m+1 2 0

For any 0 < r ≤ R and i ∈ {1, . . . , n}, let Bi,r := B(xi,r, 5ηr) ∩ B(x, r) and Br := B1,r ×

· · · × Bn,r. Then

δr ∩ r = (3.13) B 3 B ∅ and, whenever (y1, . . . , yn) ∈ Br,

δr h (x, y , . . . , y ) ≥ . (3.14) min 1 n 2

m n Theorem 57. If µ is an n-Ahlfors upper-regular measure on R such that Θ∗ (µ, x) > 0 for all x, and

α curvµ;2(x, 1) < ∞ for µ-almost every x, then µ is C1,α n-rectiﬁable. 59

We prove Theorem 57 by showing that for µ as in the statement of the theorem, we can in fact show that Z 1 µ 2 β2 (x, r) dr 2α < ∞ 0 r r for almost every x ∈ Rm, and then appeal to Theorem 51. In the proof we use a slight modification of the usual β-numbers introduced above, the so-called “centered β-numbers”, ˆµ that we denote by β2 . These numbers are defined exactly as the β-numbers, except that the ˆµ infimum in the definition of β2 (x, r) is taken over n-planes passing through x. That is, for x ∈ Rm, r > 0 define

Z dist(z, L)2 dµ(z) ˆµ 2 : β2 (x, r) = inf n . L3x B(x,r) r r

µ In particular, because the inﬁmum in the deﬁnition of β2 (x, r) is taken over a larger class ˆµ µ ˆµ m than the one in β2 (x, r), we have β2 (x, r) ≤ β2 (x, r) for all x ∈ R , r > 0.

n Proof of Theorem 57. Let µ be as in the theorem statement, and x a point so that Θ∗ (µ, x) > 0. Then, there exists some λ > 0 so that for all 0 < r ≤ 1, µ(B(x, r)) ≥ λrn. Now, ﬁx

n n 0 < r ≤ 1. By the deﬁnition of inﬁmum, for any (y1, . . . , yn) ∈ (R ) ,

Z 2 ˆn 2 1 dist(z, L) βµ;2(x, r) = inf n dµ(z) L3x r B(x,r) r Z 2 1 dist(z, aﬀ{x, y1, . . . , yn}) ≤ n dµ(z). (3.15) r B(x,r) r

Choose xi,r,Bi,r, and Br as in Lemma 56. Averaging (3.15) over all (y1, . . . , yn) ∈ Br, and then applying (3.12) yields

Z Z dist(z, aﬀ{x, y , . . . , y })2 dµ(z)dµn(y , . . . , y ) βˆn (x, r)2 ≤ 1 n 1 n µ;2 r µn( )rn Br B(x,r) Br Z Z dist(z, aﬀ{x, y , . . . , y })2 dµ(z)dµn(y , . . . , y ) ≤ C 1 n 1 n , (3.16) r n2+n Br B(x,r) r 60

where C = C(m, n, λ, C0). We now claim that,

2n dist(z, aﬀ{x, y , . . . , y }) ≤ h (x, z, y , . . . , y ). (3.17) 1 n δ min 1 n

Indeed, let ∆ = ∆(x, z, y1, . . . , yn) and ∆w = ∆({x, z, y1, . . . , yn}\{w}), for each w ∈

{x, z, y1, . . . , yn}. Basic Euclidean geometry ensures that

n n+1 dist(z, aﬀ{∆z})H (∆z) = (n + 1)H (∆) (3.18)

n = hmin(x, z, y1, . . . , yn)H (∆w0 ),

where w0 is any vertex such that

dist(w0, aﬀ{{x, z, y1, . . . , yn}\{w0}}) = hmin(x, z, y1, . . . , yn).

On the other hand, since {x, z} ∪ Bi,r ⊂ B(x, r) for all i = 1, . . . , n, Equation (3.14) ensures

n n H (∆w0 ) 2r n ≤ . (3.19) H (∆z) δr

The claim (3.17) now follows from (3.18) and (3.19).

Evidently, diam{x, z, y1, . . . , yn} ≤ 2r. Using this diameter bound, (3.17) and (3.16), we conclude

Z Z h (x, z, y , . . . , y )2 βˆµ(x, r)2 ≤ C min 1 n dµ(z)dµn(y , . . . , y ). (3.20) 2 n2+n+2 1 n Br B(x,r) diam{x, z, y1, . . . , yn}

δ j δ Setting rj = 3 and using the fact that 0 < 3 < 1, it follows that

Z 1 ˆµ 2 ˆµ 2 β (x, r) dr X β (x, rj) 2 ≤ C 2 . (3.21) r2α r δ (r )2α 0 j≥0 j 61

It now follows from (3.21), (3.13), and (3.20) that

Z 1 ˆµ 2 dr β2 (x, r) 1+2α 0 r Z h (x, z, y , . . . , y )2 ≤ C min 1 n dµn+1(y , . . . , y , z) n2+n+2 2α 1 n diam{x, z, y1, . . . , yn} r ∪j Brj ×B(x,rj ) j Z h (x, z, y , . . . , y )2 ≤ C min 1 n dµn+1(y , . . . , y , z) n2+n+2+2α 1 n diam{x, z, y1, . . . , yn} ∪j Brj ×B(x,rj ) Z 2 hmin(x, x1, . . . , xn+1) n+1 ≤ C 2(1+α)+n(n+1) dµ (x1, . . . , xn+1) B(x,1)n+1 diam ({x, x1, . . . , xn+1}) α = C curvµ;2(x, 1),

n+1 where in the penultimate step we used that Brj × B(x, rj) ⊂ B(x, 1) for all j, and the non-negativity of the integrand.

Remark 58. At this point, we brieﬂy focus on the diﬀerence between conditions (3.2) and

α (3.1). The proof of Theorem 57 could be followed out identically, using curvµ;p(x, 1) < ∞ in α place of curvµ;2(x, 1) and by replacing appropriate 2’s with p’s, to obtain

p Z R ˆµ ! βp (x, r) dr α α ≤ C curvµ;p(x, R) < ∞. 0 r r

Consequently, the following proposition is of interest, see Remark 47.

Proposition 59. Let µ be a Radon measure on Rm which is is Ahlfors upper-regular with n,∗ m constant C0, and such that 0 < Θ (µ, x) for µ-almost every x ∈ R . Let p ∈ [1, ∞), and α ∈ (0, 1]. If for µ-a.e. x ∈ Rm,

Z 1 µ p βp (x, r) dr α < ∞, 0 r r then µ is n-rectiﬁable. 62

Proof. We show that the hypotheses imply that for µ- a.e. x ∈ Rm,

Z 1 µ 2 dr β2 (x, r) < ∞ 0 r and then employ [30] to obtain rectiﬁability.

µ p For p ≤ 2, it is enough to observe, from the deﬁnition of βp (x, r) , that

2 p dist(y, P ) 2 dist(y, P ) ≤ 2 p r r

dist(y,P ) µ 2 µ p as r ≤ 2. This immediately implies that β2 (x, r) ≤ βp (x, r) , and hence we are done. Note that in this case we did not use that α > 0. For p > 2, we use H¨olderinquality:

Z 1 Z 1 µ 2 µ 2 dr β2 (x, r) 2α dr β2 (x, r) = 2α · r 0 r 0 r r p 2 p−2 1 µ 2 ! p 1 Z β (x, r) 2 dr Z p dr p 2 2α p−2 ≤ 2α (r ) 0 r r 0 r

2 p−2 1 µ p 1 Z β (x, r) dr p Z 2pα p 2 p−2 −1 ≤ α r dr 0 r r 0 p 2 Z 1 µ p βp (x, r) dr ≤ Cp,α,C0 α , 0 r r where in the last inequality we used the fact that, for p > 2, 1 − 1 µ µ(B(x,r)) 2 p µ β2 (x, r) ≤ rn βp (x, r).

Finally, we reduce Theorem I to Theorem 57.

m n Lemma 60. Let µ be a Radon measure with compact support on R such that 0 < Θ∗ (µ, x) ≤ n,∗ m Θ (µ, x) < ∞ for µ-a.e. x ∈ R . Then there exist an increasing sequence of sets {Ek} such that ∞ ! m [ µ R \ Ek = 0, k=1 63

n m µk := µ Ek is n-Ahlfors upper-regular and Θ∗ (µk, x) > 0 for µk-a.e. x ∈ R .

Proof of Lemma 60. For any positive integer k, let Ek be given by

m n −k Ek = {x ∈ R | µ(B(x, r)) ≤ kr for every r < 2 }.

m ∞ Evidently, Ek ⊆ Ek+1 and µ (R \ ∪k=1Ek) = 0. n −k Notice that µk(B(x, r)) ≤ µ(B(x, r)) ≤ kr , for r < 2 . Since spt(µk) ⊆ spt(µ) has n ﬁnite diameter, it follows that µk is upper n-Ahlfors regular. Moreover Θ∗ (µk, x) > 0 for n almost every x ∈ Ek, by Theorem 2.12(2) in [51], since µk µ and Θ∗ (µ, x) > 0.

Proof of Theorem I. Let µ and α be as in Theorem I. Without loss of generality, by considering µ B(0,Rk) for some sequence of Rk ↑ ∞, we can assume that µ has compact support.

n m To apply Lemma 60 we ﬁrst need to show that Θ∗ (µ, x) > 0 for µ-a.e. x ∈ R . In- deed, when α = 0, µ is n-rectiﬁable (see [39, Theorem 1.19]). Moreover, for α ∈ (0, 1),

α 0 curvµ;2(x, 1) < ∞ implies curvµ;2(x, 1) < ∞. Therefore µ is n-rectifiable and the n- n m rectifiability of µ implies Θ∗ (µ, x) > 0, for µ-a.e. x ∈ R . 1,α Now, define µk as in Lemma 60 and apply Theorem 57 to each µk. Then each µk is C n-rectifiable which implies µ is C1,α n-rectifiable as desired. 64

Chapter 4 EXAMPLES OF VERY NON-UNIFORMLY RECTIFIABLE, RECTIFIABLE SETS 65

4.1 An introduction to local and non-local quantitative ﬂatness

In his solution to the Analyst’s Traveling Salesman Problem [42], Peter Jones introduced a local gauge of flatness which has been generalized by David and Semmes [19] to higher dimensions. These families of local gauges of flatness are called the Jones β-numbers, and they have come to dominate the landscape in quantitative techniques relating rectifiability, potential theory, and boundedness of singular integrals. See, for example the landmark book [20].

For a set E ⊂ Rd, 1 ≤ p < ∞, and an integer 1 ≤ n ≤ d − 1, we write µ = Hn E and deﬁne the Jones β-numbers as follows,

p 1 Z dist(y, L) dµ(y) p βn (x, r) = inf . (4.1) E;p d n L⊂R an n-plane B(x,r) r r

n n n We also write βµ;p(x, r) for βE;p(x, r), when µ = H E is understood. If p = ∞, the β-numbers are defined in terms of the sup-norm instead of the L∞-norm. Various notions of “rectifiability” have been studied over the years and are frequently characterized by β-numbers. We introduce them from most to least regular. The original notion is for 1-dimensional sets in E ⊂ Rd. It is said that E is rectifiable if E can be contained in a curve of finite length. Thanks to [42] in dimension d = 2 and [57] for dimension d ≥ 3, the following characterization is known:

Z Z ∞ d 1 2 dr 1 E ⊂ R is (ﬁnitely) rectiﬁable ⇐⇒ βE;∞(x, r) dH (x) < ∞. (4.2) E 0 r

In addition to generalizing the Jones β-numbers, [19] also introduced the notion of uniform rectifiability. A set E ⊂ Rd is said to be Ahlfors n-regular if there exists 0 < c < C < ∞ such that crn ≤ Hn(E ∩ B(x, r)) ≤ Crn for all x ∈ E and all 0 < r < diam(E). An n-Ahlfors regular E ⊂ Rd is said to be uniformly n-rectifiable if there exist finite constants θ, Λ > 0 such that for all x ∈ E and all 0 < r < diam(E) there is a Lipschitz mapping g : B(0, r) ⊂ Rn → Rd with Lip(g) ≤ Λ such that Hn(E ∩ B(x, r) ∩ g(B(0, r))) ≥ θrn. 66

In [19] the authors show that an n-Ahlfors regular set E ⊂ Rd, is n-uniformly rectiﬁable if and only if the Jones β-numbers satisfy the following Carleson condition for some 1 ≤ p <

2n n−2 ,

Z Z R n n 2 dr n CE;p(x, R) := βE;p(y, r) dµ(y) ≤ cR for all x ∈ E, R > 0. (4.3) B(x,R) 0 r

d n d A set E ⊂ R is said to be countably n-rectiﬁable if there are Lipschitz maps fi : R → R with i = 1, 2,..., such that

n d n H (R \ ∪ifi(R )) = 0.

Recently, Tolsa [72] and Azzam and Tolsa [8] show, as a special case, that E is countably n-rectiﬁable if and only if

Z 1 n n 2 dr n JE(x, 1) = βE;2(x, r) < ∞ for H − a.e. x ∈ E (4.4) 0 r

n where JE(x, 1) is the Jones function at x and scale 1. See [51] for more about countably n-rectiﬁable sets and also [58] and [10] for more about identifying countably n-rectiﬁable sets and measures via β-numbers.

In this paper, we show that sets which satisfy (4.4) can fail to satisfy (4.3) as dramatically as possible. We show this through two examples in the plane. The ﬁrst, Theorem 61, is connected and the second, Theorem 62, is Ahlfors regular.

2 Theorem 61. There exists a rectiﬁable curve (of ﬁnite length), K0 ⊂ R , such that for 1 µ = H K0, for any x ∈ K0, and any δ > 0

Z Z δ 1 2 dr βK0;2(y, r) dµ(y) = ∞. Bδ(x) 0 r

The set K0 arises from unions of modiﬁcations of approximations to snowﬂake-like sets.

Since K0 is a rectiﬁable curve, by the Analyst’s Traveling Salesman theorem, i.e., (4.2), it 67

follows Z Z ∞ 1 2 dr βK0,∞(y, r) dµ(y) < ∞, R2 0 r which indicates that K0 fails to be Ahlfors upper-regular at generic points. We note that this example also gives rise to a curve of finite length (see Remark 73) which has classical tangents nowhere. This is in contrast to the well-known theorem that simple rectifiable curves have tangents almost everywhere, see [31], and also demonstrates the necessity of the “simple” assumption in the main theorem of [17], which states that σ-finite simple curves have classical tangents on a set of positive measure.

Theorem 62. There is a 1-Ahlfors regular, countably 1-rectiﬁable set A0 contained in the 2 1 unit cube in R such that for µ = H A0, for every x ∈ A0, and for every δ > 0,

Z Z δ 1 2 dr βA0;2(y, r) dµ(y) = ∞. Bδ(x) 0 r

The set A0, whose construction was initially motivated by the machinery introduced in [73], is created from scaled unions of approximations to the 4-corner Cantor set. Ultimately the presentation was simpler using the framework of self-similar sets.

Remark 63. These examples can be used to create higher-dimensional ones by taking Carte- sian products with finite intervals. That is, if A ∈ {K0,A0} for any positive integer n < d, define E0 = A × [0, 1]n−1 ⊂ Rn+1. Embedding E0 into the first (n + 1)-dimensions of Rd pre- serves the properties of A. In particular, it is standard that defining β-numbers over cubes

(with sides parallel to the axes in Rd) instead of balls leads to an equivalent definition of the n 1 0 β-numbers. Consequently finiteness of CE;2(x, R) is equivalent to the finiteness of CA;2(x ,R) where x0 is the orthogonal projection of x into R2.

4.2 Construction of K0

To construct a 1-rectiﬁable set, K0, that is connected (hence Ahlfors lower-regular) for which the Jones function is locally non-integrable, we modify approximations to the Koch 68

1 snowﬂake. This set will not be Ahlfors upper regular, i.e., H (B(x, r) ∩ K0) ≤ Cr fails for all C and some (x, r). We begin with an informal description of the technical construction which follows.

The construction splits into two parts. First, build a “base set” E∞ which satisﬁes

CE∞ (0, δ) = +∞. The base set E∞ is designed from modified approximations of the Koch snowflake, see Definition 65 and subsequent discussion. The goal is to build the connected

−i −(i−1) base set E∞ so that within the triadic strips [3 , 3 ]×R the set E∞ looks like successive approximations to the Koch snowflake which arise from more iterations of the “bumping process”. See Figures 5.3a - 4.3 for example sets that could be scaled and set on the triadic intervals [3−1, 2 · 3−1], [3−2, 2 · 3−2], [3−3, 2 · 3−3], as in Figure 4.4, to begin creating the base set E∞. After doing this infinitely many times, and taking care to balance the number of corners with the shallowness of the corners, we create a connected set with finite length such that the infinitely many “bumps” in any neighborhood of the origin give CE∞ (0, δ) = +∞ for all 0 < δ.

After we have constructed the base set E∞, we build the desired ﬁnal set K0. Roughly speaking, this happens by iteratively adding scaled copies of E∞ in a dense way along E∞ itself.

For the remainder of this paper, we only consider E ⊂ R2 and the β-numbers when p = 2. 1 1 1 1 As such, we write βE, βµ, CE, and Cµ in place of βE;2, βµ;2, CE;2, and Cµ;2, see (4.1), (4.3). 2 Moreover, for any set L ⊂ R we write Br(L) = {x : dist(x, L) < r} and Br = Br({0}).

We begin by stating two basic properties of the Jones β-numbers. The ﬁrst, often called ”doubling” despite our choice to scale by a factor of 3, controls how fast the β-numbers can shrink by relating the β-numbers at comparable scales. The second shows how the β-numbers behave under rescaling.

2 Proposition 64. Let E ⊂ R have dimH(E) = 1. 69

1. For any ball Br(y) ⊂ B3r(x),

2 3 2 βE(y, r) ≤ 3 βE(x, 3r)

z,t 2 2. The β-numbers have the following scaling property. If E = tE + z then βEz,t (x, r) = x−z r 2 −1 βE t , t . Consequently, CEz,t (z, r) = tCE(0, t r).

Next, for the reader’s convenience we record some facts about and give a construction of the standard approximations to the Koch snowﬂake.

Definition 65. Let I ⊂ R2 be a line segment, and fix 0 < α < π/2. Define P (I) as the set which results from the following operation:

1. Divide I into three equal subintervals, Ileft ∪ Icenter ∪ Iright.

2. Over the middle interval, Icenter, construct an isosceles triangle with angles α and base

Icenter.

3. Delete Icenter, the base of the isosceles triangle.

We deﬁne S(I) = P (I) \ I, (4.5)

and call S(I) the bump. If qI is the orthogonal projection onto the line containing I ⊥ ⊥ ⊥ and qI is the orthogonal projection onto I , then height(S(I)) = diam{qI (S(I))} and 1 1 width(S(I)) = diam{πI (S(I))} = 3 H (I). We shall abuse our notation slightly by saying that for a collection of line segments, E, the set P (E) is obtained by applying P to each maximal line segment contained in E.

π If I = [0, 1] × {0} and α = 3 , the standard approximations to the Koch snowﬂake are k ∞ k given by {P (I)}k=1, where P denotes applying P iteratively k times. We emphasize a few properties about deformations under the operation P . 70

Proposition 66. For any ﬁnite line segment I ⊂ R2 and positive integer n,

tan(α) height(S(I)) = |I| (4.6) 6 sec(α) H1(S(I)) = |I| (4.7) 3 sec(α) + 2n H1(P n(E)) = H1(E) (4.8) 3

1 n tan(α) 1 o When τ = 20 min 6 , 3 , there exists c0 = c(α) such that for all lines L

1 1 H (S(I) \ Bτ (L)) ≥ c0H (S(I)). (4.9)

Figure 4.1: The set P3(I) when α = π/3. Note: despite the least iterations, this has more length than the following two images.

Figure 4.2: The set P4(I) when α = π/9.

Figure 4.3: The set P5(I) when α = π/27.

Proof. (4.6) and (4.7) follow from planar geometry. The n = 1 case for (4.8) follows by

Figure 4.4: Zoomed in and truncated picture of the 3rd approximation to a base set, made by placing the sets from Figures 5.3a - 4.3 in their appropriate triadic intervals. 71

2 adding back in the unchanged intervals Ileft and Iright, which have total length 3 |I|. The geometric nature of the deﬁnition of P allows us to then iterate this to achieve (4.8).

To verify (4.9) we proceed by contradiction. Suppose no such constant c0 exists. Then, there exists a sequence of lines intersecting S(I) such that

1 −i 1 H (S(I) \ Bτ (Li)) < 2 H (S(I)).

1 After passing to a subsequence, Li converge to some line L with the property that H (S(I) \ Bτ (L)) =

0. Since S(I) is connected, this implies S(I) ⊂ B2τ (L). However, this contradicts the fact 1 that 2τ ≤ 10 min{height(S(I)), width(S(I))}.

Definition 67. Define Pj to be the set operation defined on line-segments by

j−1 [ Pj(I) = P (S(I)) (I \ Icenter) , recalling the deﬁnition of S(I) can be found in (4.5). Loosely speaking, for any line segment,

I, Pj(I) is the set that replaces the center of I with a jth approximation of the Koch curve.

Remark 68. The Hausdorﬀ distance between two compact sets A, F ⊂ R2 is deﬁned by

distH(A, F ) := max sup inf |x − y|, sup inf |x − y| . y∈A x∈F y∈F x∈A

In addition to metrizing the collection of all non-empty compact sets, the Hausdorﬀ distance generates a topology on the collection of non-empty compact sets that is complete, since R2 is complete.

Corollary 69. For any line segment I ⊂ R2 and positive integer n

2 sec(α) + 2n−1 sec(α) H1 (P (I)) = |I| + |I|. (4.10) n 3 3 3 72

Moreover, if α ≤ π/3, tan(α) dist (I,P n(I)) ≤ |I|. (4.11) H 12

Proof. Equations (4.7) and (4.8) verify (4.10). Indeed,

sec(α) + 2n−1 sec(α) + 2n−1 sec(α) H1 P n−1(S(I)) = H1(S(I)) = |I|. 3 3 3

The restriction to α ≤ π/3 ensures the longest line segment of P i(I) has length at most 3−i. Consequently, (4.6) guarantees

n n X X tan(α) dist (P n(I),I) ≤ dist (P i(I),P i−1(I)) ≤ 3−iheight(S(I)) ≤ |I|. H H 12 i=1 i=1

We now deﬁne a sequence of sets Ek which will be instrumental in deﬁning the “base set”

E∞ in our construction.

Deﬁnition 70. Now, we let n be a natural number to be chosen later and E0 = I =

[0, 1] × {0}. We deﬁne E1 = Pn(I). For k ≥ 2 inductively deﬁne

−(k−1) [ −(k−1) Ek = Pkn 0, 3 × {0} 3 , 1 × R ∩ Ek−1 . (4.12)

−(k−1) Notably, for all integers j the operation Pj applied to [0, 3 ]×{0} leaves the segment −k [0, 3 ] × {0} untouched. Consequently, the sequence of sets {Ek} are deﬁned by replacing the “next” triadic interval with a scaled approximation of the Koch snowﬂake. The fact that

−k −(k−1) each triadic strip [3 , 3 ]×R is only modified once in the sequence of sets Ek is ensures the Hausdorff dimension of the final set remains 1. 73

Lemma 71 (Base Set). Fix α ≤ π/3 and any integer n satisfying1

sec(α) + 2n sec(α) + 22n 3−1 < 1 < 3−1 . (4.13) 3 3

Then the sequence of sets Ek from (4.12) converge to a compact and connected Borel set E∞ in the Hausdorﬀ topology on compact subsets. Furthermore, E∞ satisﬁes:

1 1. H (E∞) < ∞

2. For all δ > 0, CE∞ (0, δ) = +∞.

−k Proof. Note that (4.11) ensure that distH(Ek+1,Ek) ∼ 3 . In particular, {Ek} is a Cauchy sequence in the Hausdorﬀ topology. Hence, the existence of the limiting compact set E∞ follows from completeness of the Hausdorﬀ topology on compact sets, see Remark 68.

−k By construction each Ek is connected. In fact, since Ek ∩ B3−k = [0, 3 ] × {0} it follows that Ek \ B3−k (0) is connected for each k. Connectedness of E∞ now follows since

E∞ \ B3−k = Ek \ B3−k . This demonstrates that outside every neighborhood of the origin

E∞ is connected. Consequently, E∞ is connected.

1 To see that E∞ has ﬁnite length we write the H -measure of Ek as the measure of Ek outside B31−k plus the measure of Ek inside the ball B31−k . The two key observations being 1 1 1−k Ek \ B31−k = Ek−1 \ B31−k and H (Ek−1 \ B31−k ) = H (Ek−1) − 3 . Indeed, (4.10) and these observations imply,

1 1 1 H (Ek) = H (Ek ∩ B31−k ) + H (Ek−1 \ B31−k (0)) 2 sec(α) + 2nk−1 sec(α) = |[0, 31−k] × {0}| + 31−k + H1(E ) − 3−k , 3 3 3 k−1 or, equivalently

1Note that for instance, α = π/3 and n ∈ {2, 3} satisﬁes (4.13). 74

" # sec(α) + 2nk H1(E ) − H1(E ) = 3−k sec(α) − 1 . k k−1 3

1 Since H (E0) = 1, iteration yields

k " ni # X sec(α) + 2 H1(E ) = 1 + 3−i sec(α) − 1 . (4.14) k 3 i=1

1 In particular, limk→∞ H (Ek) < ∞ whenever n satisﬁes the lower bound from (4.13). More- 1 1 over H (E∞) = limk→∞ H (Ek) since for all j ≥ k,

∞ ni X sec(α) + 2 H1 (E ∆E ) ≤ 2 3−i sec(α), j k 3 i=k+1

1 which decays to zero as k → ∞. Hence, (4.14) holds for E∞ and 0 < H (E∞) < ∞.

It only remains to show CE∞ (0, δ) = +∞ for all δ > 0. To this end, we ﬁrst note that n −1 sec(α)+2 when r = r(n, α) = 3 3 ,

k+1 −(k+1) 1 −k r 3 H (E ∩ B −k (0)) = 3 + sec(α) − . (4.15) ∞ 3 1 − r 1 − 3−1

Indeed, by (4.14) and the trick used to prove (4.14)

∞ " ni # 1 −k X −i sec(α) + 2 H (E ∩ B −k (0)) = 3 + 3 sec(α) − 1 . ∞ 3 3 i=k+1

Claim: With τ as in Proposition 66 and α ≤ π/3, there exists a constant c1 and integer j0 independent of k such that for any line L, and all k such that nk − 1 − j0 ≥ 0,

nk−1−j0 1 −k sec(α) + 2 H E∞ \ B τ (L) ∩ B3−k ≥ c13 . (4.16) 2·3k 3

Proof of Claim. Writing I0 = [0, 1]×{0}, we will in fact scale by 3k and show the stronger 75

result that

nk−1−j0 1 0 0 sec(α) + 2 0 H Pnk(I ) \ B τ (L) ∩ B30 ≥ c13 |I |. 2·30 3

0 To do so, we ﬁnd a line segment J ⊂ S(I ) \ Bτ (L) such that J has an endpoint in

0 −j0 1 0 common with one of the two line segments of S(I ) and |J| = 3 H (S(I ))/2, where j0 to be chosen later is independent of L. This speciﬁc choice of length and endpoint ensure

nk−1−j0 0 that P (J) ⊂ Pnk(I ). Moreover, the choice of j0 will both guarantee that |J| is large

nk−1−j0 enough and that P (J) remains outside of Bτ/2(L), hence verifying the claim.

0 0 To ﬁnd J, we note that the simple shape of S(I ) guarantees that S(I )\Bτ (L) has at most 0 4 maximal line segments. Hence, there exists a maximal line segment KL ⊂ S(I ) \ Bτ (L) 1 1 1 0 with H (KL) ≥ 4 H (S(I ) \ Bτ (L)). If KL is parallel to L let xL denote either endpoint of

KL. Otherwise, let xL denote the unique endpoint of KL that is not contained in Bτ (L).

−j0 sec(α) 0 Deﬁne J to be the unique subset of KL of length 3 6 |I | with endpoint xL. Now, deﬁne j0 as the smallest integer such that

( −1 ) c0 tan(α) sec(α) τ 3−j0 < min , · |I0| , 4 12 6 2

where c0 is as in Proposition 66. The ﬁrst condition ensures that J ⊂ KL and (4.9) guarantees that the ﬁrst constraint on j0 is independent of L and k. The second constraint combined

nk−1−j0 τ with (4.7) and (4.11) ensure that distH(P (J),J) ≤ 2 . Moreover, choosing j0 to be the

−j0 sec(α) 0 0 0 0 smallest admissible integer guarantees that |J| = 3 6 |I | ≥ c |I | where c is independent of L and k. Finally, (4.8) completes the proof of the Claim since

sec(α) + 2nk−1−j0 H1 P (I0) \ B (L) ≥ H1(P nk−1−j0 (J)) ≥ c |I0|, nk τ/2 1 3

where c1 depends only on α. 76

Whenever nk − 1 − j0 ≥ 0, (6.18) implies

τ 2 nk−1−j0 ! nk 1 k sec(α) + 2 sec(α) + 2 β (0, 3−k)2 ≥ 2·3 c 3−k = c (4.17) E∞ 3−k 3−k 1 3 2 3

−kδ Fix δ > 0 and any integer kδ such that 3 < δ and nkδ − 1 − j0 ≥ 0. Then, with 1 µ = H E∞, repeated applications of Proposition 64, (4.17), and (4.15) yield

Z Z δ ∞ 2 dr −2 X −(k+2) 2 βµ(x, r) dµ(x) ≥ ln(3)3 µ(B3−(k+2) )βµ(0, 3 ) Bδ(0) 0 r k=kδ ∞ nk! X rk+1 3−(k+1) 2 + sec(α) ≥ ln(3)3−2 3−k + sec(α) − c . 1 − r 1 − 3−1 2 3 k=kδ

Due to the lower bound in (4.13), this sum diverges if and only if

∞ nk ∞ X rk+1 1 2 + sec(α) X 3kr2k+1 rk sec(α) − = sec(α) − 1 − r 3k+1 − 3k 3 1 − r 3 − 1 k=kδ k=kδ diverges. Since the lower bound in (4.13) ensures r < 1, this diverges if and only if P∞ (3r2)k diverges which is equivalent to the upper bound in (4.13). k=kδ

2 1 Theorem 72. There exists a connected set, K0 ⊂ R of ﬁnite H -measure such that for any x ∈ K0 and δ > 0

CK0 (x, δ) = ∞.

∞ P Proof of Theorem 61. . Let {ri}i=1 be a sequence of positive numbers such that i ri ≤ 1. x,r 2 x,r Let E ⊂ R be the set E = rE∞ + x. We construct K0 as the union of a countable collection of nested sets {Γi}.

N1 −1−1 Let Γ0 = E∞. Now, let {x1,j}j=1 be a maximal 2 -separated net in Γ0. Let

N1 r1 [ x1,j , N Γ1 = Γ0 ∪ E 1 . j=1 77

i−1 Suppose that we have deﬁned Γi−1, some positive integers {N`}`=1 and a collection of −(i−1)−1 points {x`,j ∈ Γi−2 | 1 ≤ ` ≤ i − 1, 1 ≤ j ≤ N`} that form a maximal 2 -separated net for Γi−2. Then choose Ni ∈ N and points {xi,j}1≤j≤Ni ⊂ Γi−1 so that {x`,j ∈ Γi−1 | 1 ≤ ` ≤ −i−1 i, 1 ≤ j ≤ N`} is a maximal 2 -separated net in Γi−1. Then deﬁne Γi by

 Nj  ri [ xi,j N Γi = Γi−1 ∪  E i  . j=1

∞ We claim that K0 = ∪i=0Γi is the desired set. First note that since each Γi is countably

Ni rectiﬁable, then K0 is countably rectiﬁable. Moreover, {xi,j}j=1 ⊂ Γi−1 for all i ensures K0 inherits connectivity from E∞. Furthermore, since {Γi} is a nested sequence increasing to P K0 and i ri ≤ 1,

∞ Ni ! ∞ ! ri 1 1 [ [ xi,j , N 1 X 1 H (K0) = H E∞ ∪ E i ≤ H (E∞) 1 + ri ≤ 2H (E∞). i=1 j=1 i=1

It only remains to show that for x ∈ K0 and δ > 0 that CK0 (x, δ) = ∞. To this end, ﬁx x ∈ K0, and δ > 0. By deﬁnition of K0, there exists `0 such that x ∈ Γ`0 . Then, by the net −`−1 property of the points {xi,j}, it follows that for ` − 1 ≥ `0 large enough that 2 < δ/4, 1 there exists i ≤ ` with xi,j ∈ Γ`−1 ∩ B(x, δ/2) ⊂ K0 ∩ B(x, δ/2). Writing µ = H K0 and ri 1 xi,j , N µi,j = H E i it follows from monotonicity of the integral that

Z Z δ dr Z Z δ/2 dr β (y, r)2 dµ(y) ≥ β (y, r) dµ (y), (4.18) K0;2 r µi,j ;2 r i,j Bδ(x) 0 Bδ/2(xi,j ) 0

z,t or equivalently CK0 (x, δ) ≥ Cµi,j (xi,j, δ/2). Recalling that E = tE∞ + z, we use (4.18), Proposition 64(2), and Lemma 71 to conclude

δ ri δNi r CK (x, δ) ≥ C x , i xi,j, = CE 0, = ∞. 0 i,j N ∞ E i 2 Ni 2ri 78

Since x ∈ K0 and δ > 0 are arbitrary this ﬁnishes the proof.

1 1 Remark 73. Since K0 from Theorem 61 is connected, H (K0) = H (K0) < ∞ and K0 is compact, see [64, Lemma 3.4, 3.5]. Thus K0 is a rectiﬁable curve by Waz˙ewski’s theorem, see [64, Lemma 3.7] or [1, Theorem 4.4].

The authors thank Matthew Badger for pointing out that K0 coincides with the Hausdorﬀ- limit of {Γi}. So, Go lab’ssemi-continuity theorem and Waz ˙ewski’s theorem suﬃce to ensure

K0 is a rectiﬁable curve. See, for instance, [1] or [31] for relevant theorem statements.

4.3 Construction of A0

The unique compact set ﬁxed by the iterated function system (IFS),

2 2 −2 {Fi,j : R → R |Fi,j(E) = 2 (E + (i, j)), i, j ∈ {0, 3}} is called the 4-corner Cantor set, C. The 4-corner Cantor set is an Ahlfors regular set with positive and ﬁnite H1-measure and is purely unrectiﬁcable. That is, H1 (C ∩ f(R)) = 0 for all Lipschitz functions f : R → R2. Typically, one approximates the 4-corner Cantor set by beginning with the “initial set”

2 [0, 1] in their iteration scheme. However, our motivation for the construction of A0 arises from considering the initial set [0, 1) × {0}. Beginning with a 1-dimensional set allows every approximating set to have positive and ﬁnite H1-measure. This is also critical to produce estimates on the β-numbers of each approximation.

The general strategy for producing the desired set A0 in Theorem 62 is as follows. We −2i −2i+2 produce a base set Σ0 such that within successive tetradic strips [2 , 2 ] × R the set −2i −2i+2 Σ0 ∩([2 , 2 ] × R) is a scaled version of a higher-iteration approximation to the 4-corner Cantor set. This allows for precise control on the β-numbers in every neighborhood of the origin. Then, following the strategy for Theorem 61 we carefully iterate this set “on itself” in a dense way, taking care to preserve Ahlfors regularity. 79

4.3.1 Approximations to the 4-corner Cantor set

Consider the following sequence of approximations to the 4-corner cantor set, by sets of positive and ﬁnite H1-measure.

Let E0 = [0, 1) × {0} and inductively deﬁne

X i j E = p + 2−2E where p = , . (4.19) k ij k−1 ij 22 22 (i,j)∈{0,3}2

The word similarity is used to refer to any mapping that can be written as a composition of scalings, rotations, reﬂections, and translations. Throughout the rest of the paper, we say that two sets are similar if one is the image of the other by a similarity. In reality the similarities we discuss can always be written as a scaling and translation, as in (4.19).

We let ∆ denote the collection of tetradic half-open cubes in R2, that is

−2k −2k −2k −2k ∆ = {[a2 , (a + 1)2 ) × [b2 , (b + 1)2 ) | a, b, k ∈ Z}.

For some Q ∈ ∆, we let `(Q) denote the sidelength of Q. We partition the tetradic cubes into cubes of ﬁxed sidelength by deﬁning ∆i = {Q ∈ ∆ | `(Q) = 2−2i}.

In general, for a set E ⊂ R2 we respectively denote the length of E and the height of E by

`(E) = diam{πx(E)} and h(E) = diam{πy(E)} where πx and πy denote the orthogonal projection onto the horizontal and vertical axes. In particular, for a cube Q with axis-parallel sides, this notion of length coincides with the cubes sidelength. Hence no confusion with the earlier convention that `(Q) is the sidelength of Q will arise.

Deﬁnition 74 (Clusters and sub-clusters). Any set which is similar to any Ek or Ek ∪[0, 1)× {0} for k ∈ N will be called a cluster. 2k Moreover, for ﬁxed k ∈ N, we will call Ek the 0th sub-cluster of Ek and the 2 line 80

segments that make up Ek are called the kth -subclusters of Ek. For ` ∈ {1, . . . , k − 1}, the 2` 2 -sets contained in Ek which are similar to Ek−` are called the `th sub-clusters of Ek.

Deﬁnition 75 (Root points). We associate to each cluster and each cube a root point. The root point of a cluster E is the lower-most and left-most point in the cluster. Since a subcluster is itself a cluster, the notion of a root point extends to sub-clusters. For a cluster E, we let xE denote its root point. For a tetradic cube Q ∈ ∆ we let xQ denote the lower-most and left-most point of Q and call xQ the root point of Q.

Proposition 76. For ﬁxed non-negative integer k, the set Ek has the following properties.

2k −2k 1 1. Each Ek is a ﬁnite union of 2 intervals each of length 2 . In particular, H (Ek) = 1

and Ek is countably 1-rectiﬁable. Moreover, each connected component I of Ek has ∂I ⊂ `(I)Z2 = 2−2kZ2 and consequently is contained in a line R × {a2−2k} for some

a ∈ N0.

j 2. If j ≥ 0 is an integer and if Q ∈ ∆ is such that Q ∩ Ek is non-empty, then

 xQ + [0, `(Q)) × {0} j ≥ k Q ∩ Ek = (4.20) −2j xQ + 2 Ek−j j ≤ k

3. Each Ek is Ahlfors regular with regularity constant independent of k.

1 −2j 4. For 0 ≤ j ≤ k an integer, each jth subcluster of Ek has H -measure 2 .

−2j 5. For 1 ≤ j ≤ k an integer, the jth subclusters of Ek are 2·2 -separated horizontally and −2j 3 Pk−j −2i −2j at least 2 · 2 -separated vertically. In fact, they are 3 − 4 i=1 2 · 2 -separated vertically.

−2k 6. If J ⊂ Ek is a connected component, then J is a vertical distance of 3 · 2 from the 0 nearest connected component J of Ek. 81

1 7. There exists a universal constant c > 0 such that if k ≥ 2 and µk = H Ek, then for

all x ∈ Ek, Z 1 2 dr βµk (x, r) ≥ c(k − 2) 6·2−2k r

−2 2 Proof. (1) follows immediately from (4.19) since each pij ∈ 2 Z .

To see (2), we ﬁrst note that the case j = 0 is clear for any k ∈ N. Further, the case k = 0 is clear for all j ∈ N. To procede inductively suppose that (4.20) holds for all k ∈ N when j = ` − 1. We will show it holds for all k ∈ N when j = `. Indeed, suppose that Q ∈ ∆` has non-empty intersection with E`. Let xQ be the root of Q. Choose p ∈ {pij}(i,j)∈{0,3}2 such −2 2 ˜ that Q ⊂ p + [0, 2 ) . Then, 4(Q ∩ Ek − p) = (4Q − 4p) ∩ (4Ek − 4p) = Q ∩ Ek−1 where Q˜ := 4Q − 4p ∈ ∆`−1. By the inductive assumption,

 x + [0, `(Q˜)) × {0} ` − 1 ≥ k − 1 ˜  Q˜ Q ∩ Ek−1 =  −2(i−1) xQ˜ + 2 E(`−1)−(i−1) ` − 1 ≤ k − 1.

Translating and scaling this back to what this means about Q ∩ Ek veriﬁes the induction.

1 H (Q∩Ek) (3) follows from (1) and (2) since these imply that `(Q) = 1 for tetradic cubes Q with `(Q) ≤ 1 that intersect Ek. This suﬃces since any ball contains a tetradic cube of comparable sidelength and is contained in 42 tetradic cubes of comparable sidelength.

2k −2k (4) is equivalent to showing that Ek is made of 2 intervals, each of length 2 . (5) The horizontal separation is veriﬁed by an argument similar to the vertical separation. For the vertical separation, we only verify that the vertical separation is at least 2 · 2−2j.

Indeed, this follows since E` is contained in the horizontal strips R × [0, 1/4] ∪ [3/4, 1] for all `. Then, the scaling from (4.19) ensures that the jth subclusters, which arise by applying

−2j 1 −2(j−1) (4.19) j times to the sets Ek−j are vertically 2 · 2 = 2 2 -separated. The reason the height-bound can be improved, is because the jth subclusters are actually contained in smaller strips. See for instance, E1, where the ﬁrst subclusters are contained in lines, and 3 12 15 E2 where the ﬁrst subclusters are contained in the strips R × [0, 16 ] ∪ [ 16 , 16 ]. 82

(6) follows from the fact that vertically-closest connected components in Ek come from −2 −2 the connected components of E1 which are 3 · 2 separated. After being scaled by 2 in (4.19) another (k − 1) times the separation is reduced to a distance of 3 · 2−2k as claimed. This coincides with the precise formula in (5) and could be considered as a base case for induction on j for the interested reader. (7) Throughout the proof of (7), we ﬁx integers 1 ≤ j < k and k ≥ 2.

0 Claim 1: For all x ∈ Ek there exists some x ∈ Ej with

dist(x, x0) ≤ 2−2j (4.21)

Proof of Claim 1. Note that the scaling in (4.19) ensures that for some `, we know that

−2(`+1) every x ∈ E`+1 is within a distance 3 · 2 of a point in E`. Iterating veriﬁes the claim 0 by showing for x ∈ Ek there exists x ∈ Ej such that

k ∞ X X dist(x, x0) ≤ 3 · 2−2` ≤ 3 2−2` = 4 · 2−2(j+1). `=j+1 `=j+1

Claim 2: There exists c independent of j such that for all 5 · 2−2j ≤ r ≤ 11 · 2−2j and all

0 x ∈ Ej, β1 (x0, r)2 ≥ c µj ;2

0 Proof of Claim 2. Let J ⊂ Ej be the connected component containing x . By (4)-(6) of q −2j −2j 2 −2j 2 0 this proposition, it follows that for r ≥ 5 · 2 = (3 · 2 ) + (4 · 2 ) , the ball Br(x ) contains J and 3 other connected components of Ej. Consequently, there are two horizontal u d 0 u d lines, L and L , such that Br(x ) ∩ L ∪ L contains at least 4 connected components of

Ej. Part (1) of this proposition ensures,

u 0 d 0 −2j min{µj (L ∩ Br(x )) , µj L ∩ Br(x ) } ≥ 2 · 2 . (4.22)

Moreover, part (6) ensures that the distance between Lu and Ld is 3 · 2−2j, which combined 83

with (4.22) forces that any line L satisﬁes,

0 −2j−1 −2j µj y ∈ Br(x ) dist(y, L) ≥ 3 · 2 ≥ 2 · 2 . (4.23)

Finally, recalling 5 · 2−2j ≤ r ≤ 11 · 2−2j, (4.23) implies

−2j !2 Z dist(y, L)2 dµ (y) 3·2 2 · 2−2j inf j ≥ 2 ≥ c L 0 Br(x ) r r r r which veriﬁes Claim 2.

0 Claim 3: There exists c such that for all x ∈ Ek and all integers 1 ≤ j < k and ρ such that 6 · 2−2j ≤ ρ ≤ 12 · 2−2j, β1 (x, ρ)2 ≥ c0. (4.24) µk;2

−2j −2j 0 Proof of Claim 3. Claim 1 ensures that for all 5 · 2 ≤ r ≤ 11 · 2 there exists x ∈ Ej 0 d u u d such that Br(x ) ⊂ Bρ(x). As in Claim 2, ﬁx lines L and L such that Br(x) ∩ L ∪ L d u contains at least 4 connected components of Ej. Choose a so that L = R × {a} and L = {a+(0, 3·2−2j)}+R×{0}. Moreover, suppose the left-most connected component of Lu has −2j −2j right-most endpoint with x-value equal to c1. Deﬁne Lv = {c1 +2 }×R and Lh = a+2 .

By Proposition 76(5,6), the neighborhoods Nv = B2−2j (Lv) and Nh = B2−2j (Lh) are disjoint from E` for all ` ≥ j. See Figure 4.5.

Consequently, for any line L the nieghborhood B2−2j−1 (L) can intersect at most 4 of the

“quadrants” made by the neighborhoods of Nv and NL. Making a generous estimate since the ball may cut-oﬀ part of one of the quadrants in Figure 4.5, we conclude

0 −2j−2 −2j−2 µk {y ∈ Br(x ) | dist(y, L) ≥ 2 } ≥ 2 (4.25)

0 ρ where the measure-bound comes Proposition 76(1). Since Br(x ) ⊂ Bρ(x) and 1 ≤ r ≤ C < ∞, Claim 3 follows from (4.25) analogously to how Claim 2 followed from (4.23). 84

Figure 4.5: When j = k − 2, the picture displays a subclusters of equal length for Ej and Ek on the left and right respectively. In Ek, the line Lv and its neighborhood Nv are in green, whereas the line Lh and its neighborhood Nh are drawn where it would pass through both Ej and Ek

Finally, we verify (7) because

Z 1 k Z 11·2−2j 2 dρ X 0 dρ βµk (x, ρ) ≥ c = c(k − 2). −2j ρ −2j ρ 6·2 j=2 6·2

We construct the base set Σ0 from approximations to the 4-corner Cantor set by ﬁrst deﬁning

−2n −2n [ E(n) := (2 , 0) + 2 E22n and Σ0 := E(n) ∪ ([0, 1) × {0}) . (4.26) n

Proposition 77. Σ0 has the following properties.

1 1. 0 < H (Σ0) < ∞ and Σ0 is countably 1-rectiﬁable.

j 2. If j ≥ 0 is an integer and Q ∈ ∆ is such that Q ∩ Σ0 6= ∅, then 85

D C

Figure 4.6: Here we see Σ0 and several examples of Q ∈ ∆. The cube D illustrates the ﬁrst case in Equation (4.27). The cube A illustrates an example of the second case in Equation (4.27). The cubes B and C illustrate examples of the last case in Equation (4.27).

 Σ ∩ [0, `(Q))2 x = (0, 0)  0 Q  −2j Q ∩ Σ0 = xQ + 2 Ek for some k xQ 6= (0, 0) and πy(xQ) 6= 0 (4.27)   −2j xQ + 2 Ek ∪ [0, `(Ek)) × {0} xQ 6= (0, 0) and πy(xQ) = 0.

3. CΣ0 (0, δ) = +∞ for all δ > 0.

Proof. (1) Σ0 has positive and ﬁnite mass due to Proposition 76(1) and the geometric scaling in (4.26). It is also the countable union of countably 1-rectiﬁable sets by Proposition 76(1).

(2) The case when xQ = (0, 0) is clear. Suppose xQ 6= (0, 0). There exists unique a, b such that −2j −2j xQ = a2 , b2 . (4.28)

2 If j = 0, Q ∩ Σ0 6= ∅, and Σ0 ⊂ [0, 1) forces a = b = 0. Therefore, j ≥ 1. Since h(E22n ) < `(E22n ) and the E(n) only use a translation in the positive horizontal direction of

E22n and a homogeneous scaling, it follows that Σ0 ∩ Q 6= ∅ implies 0 ≤ b < a so that a ≥ 1. Since, `(Q) = 2−2j it follows that a2−2j ≥ `(Q). Comparing the translation and scaling sizes 86

in (4.19), a ≥ 22j`(Q) implies

 Q ∩ E(n) b ≥ 1 Σ0 ∩ Q = (4.29) Q ∩ (E(n) ∪ [0, `(E(n))) × {b}) b = 0 for some speciﬁc n ≤ j. For simplicity of writing, assume we are in the ﬁrst case. Then,

2n −2n 2n −2n 2 (Q ∩ E(n) − (2 , 0)) = (2 (Q − (2 , 0))) ∩ E22n or equivalently

−2n −2n 2n −2n Q ∩ E(n) = (2 , 0) + 2 2 Q − 2 , 0 ∩ E22n . (4.30)

In light of (4.30), it follows that (4.20) implies the 2nd case of (4.27) since 22n(Q−(2−2n, 0)) ∈ ∆j−n and n ≤ j. Analogously the b = 0 case corresponds to the 3rd case of (4.27). (3) Fix δ > 0. Choose N large enough that 11 · 2−2N < δ/2. In particular, for all n ≥ N,

1 1 E(n) ⊂ Bδ(0). Then, with µ = H Σ0 and µn = H E(n), it follows from Proposition 76 (1,7), Proposition 64 (2), and the scaling in (4.26) that

−2n X Z Z 2 dr X C (0, δ) ≥ β1 (x, r)2 dµ (x) ≥ c(22n − 2)H1(E(n)), Σ0 µn;2 r n n≥N E(n) 0 n≥N which diverges and completes the proof.

We wish to iterate Σ0 densely along itself while being careful to maintain Ahlfors upper- and lower-regularity. This is attained by scaling, and being careful where we iterate.

Deﬁnition 78 (Tail points). We say a point y is a tail point of E if 0 < H1(E) < ∞ and there exists a tetradic number r and δ > 0 such that

y + rΣ0 ∩ Bδ ⊆ E.

Note, if y ∈ Bδ(x) is a tail point of a set E, then CE(x, δ) ≡ ∞. See Claim 1 from the proof of Theorem 62. 87

Definition 79 (Iterative construction). Let Σ0 be as above. Supposing that Σi−1 has been defined, we define a (possibly empty) special collection of tetradic points,

i −2i 2 −2i 2 −2i D = x ∈ 2 Z x + [0, 2 ) ∩ Σi−1 = x + [0, 2 ) × {0} , (4.31)

and define Σ by i ( ) [ −8i Σi = Σi−1 ∪ x + 2 Σ0 . (4.32) x∈Di Define, [ A0 = Σj. (4.33) j∈N ∞ j ∞ Proposition 80. The sets {Σj}j=0 and {D }j=1 as in Definition 79 have the following properties:

(1) Σj−1 ⊂ Σj for all j ≥ 1,

(2) Σj is contained in countably many horizontal line segments with tetradic heights. (3) There are inﬁnitely many j so that Dj is non-empty.

2 (4) If I is a connected component of Σj then ∂I ⊂ `(I)Z . −2j (5) Σj contains no connected component of length at least 2 that contain no tail point.

Proof. Indeed, (1) follows from (4.32).

(2) Follows by induction. For Σ0 it follows from Proposition 76 (1) combined with the scaling in (4.26). For general Σj induction holds due to the fact that each scaled copy of Σ0 in (4.32) has a tail point on the dyadic lattice Di which is coarser than the tetradic scaling factor of Σ0. (3) folows from (2). (5) follows from (4) and the deﬁnition of Dj in (4.31).

i (4) If I is a connected component of Σj then there exists y ∈ D some i ≤ j such that I −8i 8i is a connected component of y + 2 Σ0. But then, 2 (I − y) is a connected component of −2i 2 8i 8i 2 Σ0. Since y ∈ 2 Z , Propositions 76(1) and 77(2) ensure ∂ (2 (I − y)) ∈ 2 `(I)Z which veriﬁes (4). 88

Deﬁnition 81 (Associated cubes). Any cluster (or subcluster) E has associated to it the

2 dyadic cube QE = xE + [0, `(E)) . In particular, by Proposition 76 (5) it follows that if 0 0 clusters E,E are disjoint with `(E) = `(E ), then QE,QE0 are disjoint cubes. Moreover, for some cluster E, the root point of QE and the root point of E coincide.

Deﬁnition 82. We associate to the base set Σ0 the following family of cubes

−2i 2 QΣ0 = [0, 2 ) : i ≥ 0 ∪ {QE : E is a subcluster of E(n) ⊂ Σ0, n ≥ 1 } (4.34)

i −8i By similarity, for any y ∈ D we associate to y + 2 Σ0 the family of cubes

−8i [ −2k 2 Qy = y + 2 QΣ0 y + {[0, 2 ) : i ≤ k} . (4.35)

We will let

Q = ∪i≥0 ∪y∈Di Qy (4.36) which we stratify by scale in the following sense

Qi = {Q ∈ Q | `(Q) = 2−2i} (4.37) and we enumerate the elements Qi so that

i i N(i) Q = {Qj}j=1 . (4.38)

0 0 0 Finally, for Q ∈ Q and any positive integer ` we let C`(Q) = {Q ∈ Q | Q ⊂ Q, `(Q ) = −2` 2 `(Q)}, and call C`(Q) the `th descendent cubes of Q.

i i i Lemma 83. For all i ≥ 0 and all cubes, Qj ∈ Q , Σi ∩ Qj is similar to one of the following:

−2k 2 1. 2 Σ0 ∪ [0, 1) × {0} ∩ [0, 1) for some integer k.

2. E ∩ QE for some sub-cluster E ⊂ E(n) for some integer n ≥ 1 89

This follows immediately from the explicit deﬁnition of cubes.

j j j Lemma 84. Q ⊂ ∆ and for all Q ∈ ∆ , then either Q ∩ Σj = ∅ or Q ∈ Qj.

This follows from an induction argument similar to the proofs of Propositions 76 (1) and 77 (2). The key observation in the induction is that the scaling in (4.32) ensures that all tail points added in the jth stage have root points in tetradic lattices that are coarser than the length of the scaled copy of Σ0 being added.

Corollary 85. The cubes Q have the following nice properties:

1. Each collection Qi is a disjoint collection of cubes, and for any Q ∈ Q and any integer

` ≥ 0, C`(Q) is a disjoint collection of subcubes of Q.

2. For all non-negative integers i and j,

Σi ⊆ ∪Q∈Qj Q (4.39)

3. In particular, for any Q0 ∈ Qi

\ Σi ∩ Q0 = Σi ∪Q∈C1(Q)Q (4.40)

Proof of Theorem 62. By Lemma 77 (1), Σ0 is 1-rectiﬁable, and A0 is a countable union of scaled translations of Σ0 so A0 is 1-rectiﬁable.

Next, we show that A0 is 1-Ahlfors regular. Indeed, it suﬃces to show that there exists j 0 < c ≤ C < ∞ independent of i such that for for any j ≥ 0, Q ∈ ∆ , and Q ∩ A0 6= ∅,

1 c`(Q) ≤ H (Q ∩ A0) ≤ C`(Q). (4.41)

1 H (Q∩Σj ) j We do this by showing similar bounds for `(Q) for cubes Q ∈ ∆ that intersect Σj, and then proving that not too much additional mass is added to the cube Q. 90

j Due to Lemma 84 the condition that Q ∈ ∆ and Q ∩ Aj 6= ∅ is equivalent to Q ∈ Qj.

Since Q ∈ Qj Lemma 83 characterizes what Q ∩ Σj looks like and we conclude

1 `(Q) ≤ H (Q ∩ Σj) ≤ 3`(Q), (4.42) by considering each of the three cases in Lemma 83. Indeed, each cube either contains its entire bottom portion, or contains a cluster E with `(E) = `(Q). In either case this implies the lower bound in (4.42). On the other hand, we know that a rough upper-bound is to assume that Q ∩ Σj contains a cluster with a line segment at the bottom, and contains Σ0 scaled by 2−2k, then by Proposition 76, the upper bound in (4.42) follows. It remains to show that (4.42) implies (4.41). Due to Proposition 80 (1), the lower-bound in (4.41) is inherited directly from (4.42). The upper-bound follows with the additional observation that for ` ≥ j,

1 −8(`+1) 1 −4(`+1) 1 H (Q ∩ Σ`+1 \ Σ`) ≤ #|D`+1|2 H (Σ0) ≤ 2 H (Σ0).

Summing over ` ≥ j veriﬁes (4.41). It is a standard argument to go from Ahlfors regularity in tetradic/dyadic cubes to in balls, see for instance the brief description in the proof of Proposition 76(3). Since the cubes in Q are all the tetradic cubes with non-empty intersection with A0, we have regularity in tetradic cubes.

Finally, to see that CA0 (x, δ) = ∞ it suﬃces to show the following claim.

Claim 1- If x ∈ A0 and δ > 0, then there is a tail point in A0 ∩ Bδ/2(x).

Brieﬂy assuming that Claim 1 holds, the fact that CA0 (x, δ) = ∞ for all x ∈ A0 and δ > 0 follows since if y is the tail point in Bδ/2(x) then, by Proposition 77 (3) and monotonicity of integrals of non-negative functions:

CA0 (x, δ) ≥ CA0 (y, δ/2) ≥ CΣ0 (0, y) = ∞,

i where y > 0 is some scale dependent on which D the tail point y is in. 91

To verify Claim 1, ﬁx x and δ as in the claim. Adopting the convention that Σ−1 = ∅

ﬁx i0 such that x ∈ Σi0 \ Σi0−1. Choose k to be the smallest natural number such that −8k diam 2 Σ0 ≤ δ/4.

Case 1- Bδ/4(x) ∩ Σk contains a tail. Since Σk ⊂ A0 in this case the claim holds.

Case 2- Otherwise, choose k0 ≥ k such that

  (Σk0−1 \ Σk) ∩ Bδ/4(x) = ∅  (Σk0 \ Σk) ∩ Bδ/4(x) 6= ∅,

that is k0 is the ﬁrst stage after k where something new is added to the ball Bδ/4(x). The way something new is added to the ball Bδ/4(x) in the k0th stage is if there exists y such that,

−8k0 {y + 2 Σ0} ∩ {Σk0 ∩ Bδ/4(x)}= 6 ∅.

But then, y is a tail point of Σk0 and consequently of A0. By our choice of k, we conclude

−4k0 |x − y| < diam(2 Σ0) + δ/4 ≤ δ/2.

Hence the tail point y is indeed in Bδ/2(x). So, by Proposition 64(2)

0 CA0 (x, δ) ≥ CA0 (y, δ/2) ≥ cCΣ0 (0, δ ) = ∞.

This completes the theorem. 92

Chapter 5 REGULARITY IN LOW DIMENSIONS 93

5.1 Motivation and history

Plateau’s original conjecture was an experimentally determined “classification” of the structure of soap bubbles [59]. The classification problem for 2-dimensional area minimal varifolds in R3 was ultimately solved by Taylor [70] confirming Plateau’s experimental results. Area minimizing surfaces are particularly interesting in R3 due to their relationship to so surface tension phenomena. According to Taylor, understanding how thermodynamic forces cause constraints on the shape of surfaces was studied by, amongst others, Maxwell, Laplace, Gauss, Poisson, and Gibbs, before Plateau’s original “laws”.

The techniques used by Taylor were dependent upon a deep understanding of how 1- dimensional minimizers on the sphere behaved, [69]. This type of “dimensionality reduction” is ubiquitous, arising in Federer’s dimension reduction [34], structure theorems for sets [75], and free boundary problems [16], demonstrating the necessity for a deep understanding in the low-dimensional case.

In this chapter, we begin cataloging properties of low-dimensional anisotropic minimizers. First we study general anisotropic energy minimizing sets of locally ﬁnite perimeter in R2 demonstrating that all minimizers are locally straight line segments and producing a Bernstein theorem in the plane, Theorem 91. This proof cannot be reproduced in higher- dimensions because of the existence of saddle points. At a saddle point, one cannot create a competitor by the type of localization argument presented. This observation can be thought of as a qualitative version of the statement that anisotropic minimal surfaces have anisotropic mean curvature zero.

We then turn our attention to k · kp energy minimizing sets of locally finite perimeter. Given the more general results for anisotropic energy minimizing sets in the plane from the previous section, the main interest of this section is the technique, namely the development and exploitation of a new monotonicity formula, Theorem 101 arising from the first-variation of the k · kp-energy. As the only tool used comes fromthe first-variation, the analysis within this section can be repeated for k · kp minimal sets of locally finite perimeter and for minimal 94

varifolds in R2 Since monotonicity formulas demonstrate that blow-ups of minimal sets are minimal cones, Corollary 109, this provides another reason to understand what minimal cones look like in R2. We provide an “algorithm” to construct and classify stationary triple junctions in R2, see Theorem 111 and Figures 5.5a, 5.5b.

5.2 Preliminaries

Suppose A ⊂ Rn is open and ρ ∈ C1(Rn \{0}) satisﬁes

ρ(λx) = λρ(x) ∀ λ > 0 and ρ(x + y) < ρ(x) + ρ(y) ∀ |x| = 1 = |y|. (5.1)

Given an open set A, we wish to study the regularity of the local minimizers of the functional Z n Φρ(E; A) := ρ(νE)dH (5.2) A∩∂∗E when the functional is defined on all sets of locally finite perimeter E ⊂ Rn such that E \ A = Z \ A for some fixed Z ⊂ Rn a fixed set of locally finite perimeter. More precisely,

1 Definition 86. For an open set A0 and a mapping k · k : S → (0, ∞), we say that a set of locally finite perimeter E minimizes Φ(· ; A0) if ∂E = sptµE and for all sets of locally finite perimeter F such that E∆F ⊂⊂ A0 it holds that

Φρ(E; U) ≤ Φρ(F ; U),

where U ⊃ E∆F is a pre-compact, open subset of A0.

We say that E is a (Λ,R0) almost-minimizer of Φρ(·; A0) if ∂E = sptµE and

n−1+α Φρ(E; U) ≤ Φ(F ; U) + Λr ∀E∆F ⊂⊂ U ⊂⊂ A0, diam{E∆F } = r ≤ R0, for any absolute constant α > 0. 95

Figure 5.1: A valid competitor F relative to the set A0.

Remark 87. The purpose of the set A0 in Definition 86 is to define boundary conditions. See fig. 5.1.

The requirement that ∂E = sptµE in Deﬁnition 86 is necessary in order to be able to make topological claims about the boundary of an anisotropic minimizer. Fortunately, given

0 0 any set of locally ﬁnite perimeter E, there exists some Borel set E so that sptµE0 = ∂E . See, for instance, [49, Remark 16.11]. Therefore, this requirement boils down to choosing the “correct representative” of E among all equivalent sets of locally ﬁnite perimeter.

The only speciﬁc energies we consider are

Z n−1 Φp(E; A) := kνEk`p dH 2 ≤ p < ∞. (5.3) ∂∗E∩A

Note Φ2 recovers precisely the perimeter of E inside A.

Deﬁnition 88. An energy is called semi-elliptic if, whenever F is a set of locally ﬁnite perimeter with boundary ∂F ∩ A ⊂ P for some n − 1 dimensional plane P , then for all 96

E 6= F with E∆F ⊂⊂ A,

Φρ(E; A) − Φρ(F ; A) ≥ c [Φ2(E; A) − Φ2(F ; A)] ≥ 0.

The energy Φρ is called elliptic if c > 0.

Z n−1 n−1 ∗ P (E; A) = kνEk2dH = H (∂ E ∩ A). ∂∗E∩A

It is well-known that strict convexity of ρ implies semi-ellipticity, [35, 5.1.2].

Proposition 89. For p > 2, the energy Φp is not elliptic.

Proof. Consider A = {(x, t) ∈ Rn : |x| ≤ 1, t ∈ R} and F = {(x, t) ∈ Rn : t ≤ 0}. For α > 0, 2 deﬁne the surface Cα = {{x, t} : |x| ≤ 1, t ≤ α(1 − |x| )}. Since ∂Cα is the graph of the function u(x) = α(1 − |x|2) it follows that

Z Z 1 n−1 p p p n−1 Φp(Cα,A) = k(−Du, 1)kpdH = 1 + (2α) kxkp dH . |x|≤1 |x|≤1

Hence, for small α > 0,

p Z n−1 (2α) p p+1 Φp(Cα,A) = H (B(0, 1)) + kxkp + o(α ). p |x|<1

In particular, for all p ≥ 2

αp Φ (C ,A) − Φ (F,A) = c + o(αp+1). p α p n,p p

Consequently, if p > 2,

p p+1 Φp(Cα,A) − Φp(F,A) cn,pα + o(α ) lim = lim 2 3 = 0, α↓0 Φ2(Cα,A) − Φ2(F,A) α↓0 cnα + o(α ) verifying the Proposition. 97

2 Remark 90. If E ⊂ R is Φ(· ; A0) minimizing, then ∂E ∩ A0 contains no self-crossings, or else one could reduce the energy Φ by removing the loop formed by ∂E crossing itself.

We follow the convention that if A, B ⊂ R2 then A ≈ B means H1(A∆B) = 0, and 1 A ⊂∼ B means H (A \ B) = 0. Moreover, when considering a set of locally ﬁnite perimeter

A we will always work with a representation of A so that ∂A = sptµA.

For a set of locally ﬁnite perimeter A, let µA denote the Gauss-Green measure associated ∗ to A, νA denote the outward pointing measure theoretic normal, and ∂ A denote the reduced boundary of A.

For any measurable set A ⊂ R2 and a number s ∈ [0, 1] deﬁne

2 (s) 2 H (A ∩ B(x, r)) A = x ∈ R : lim = s . r↓0 H2(B(x, r))

For a set of locally ﬁnite perimeter A ⊂ R2 the essential boundary of A, denoted ∂eA is 2 (0) (1) deﬁned to be the set R \ E ∪ E .

5.3 Regularity of 1-dimensional minimizing sets of locally ﬁnite perimeter

Throughout this section, we use Φ in place of Φρ. It is well-known that strict convexity of the integrand ρ is necessary for there to be a robust regularity theory, see for instance [49, Remark 20.4]. It is also known in Rn that creating competitors by intersecting with half-spaces can only reduce the energy, see for instance [49,

Remark 20.3]. It turns out that for 1-dimensional energy minimizers boundaries in R2 strict convexity is not only necessary, but also suﬃcient for a robust regularity result The heart of the proof boils down to a localized version of the fact that intersections with half-spaces reduce energy.

Theorem 91. Suppose ρ : S1 → (0, ∞) is a lower semicontinuous, bounded, strictly convex 2 function and A0 ⊂ R is an open set of locally ﬁnite perimeter. Then there exists a Φ(· ; A0) minimizer which we denote by E. 98

Moreover, if E minimizes Φ(· ; A0) then there exists a set equivalent to the minimizer, which we also call E,so that whenever ∂E ∩ A0 6= ∅ it follows ∂E ∩ A0 is a non-intersecting 2 collection of line segments. In the case that A0 = R , E must be a half-space.

Remark 92. The existence portion of Theorem 91 is well-known. See, for instance [49, Remark 20.5] and the historical notes and citations therein.

The geometric idea behind the of proof of Theorem 91 can be seen even in Almgren’s definition of an elliptic integrand. The technicalities that arise are primarily due to showing that a point where the boundary is not flat allows one to reproduce a localized version of the half-plane argument from, for instance, [49, Remark 20.3] and create a valid competitor. We first make use of the semicontinuity and boundedness of ρ to make a substantial simplification.

Remark 93 (∂E is locally Lipschitz for anisotropic minimal surfaces). Let ρ be as in the statement of Theorem 91. Since ρ is a positive lower semicontinuous function on S1, it achieves a minimum. Since it is also bounded this means there exist c, C > 0 such that c|ν| ≤ ρ(ν) ≤ C|ν| for all ν ∈ R2 \{0}. By a standard competitor argument, see Lemma 118, and the diﬀerential inequality aﬀorded by the isoperimetric inequality this implies that

∗ if E minimizes Φ(·,A0) and x ∈ ∂ E ∩ A0 then there exists CA = CA(c, C) independent of x such that for all r ∈ (0, dist(x, ∂A0)),

H1(∂∗E ∩ B(x, r)) C−1 ≤ ≤ C . (5.4) A r A

∗ That is, |µE| is Ahlfors regular at small, but locally uniform, scales for points x ∈ ∂ E. This has two immediate consequences: (1) the lower bound ensures that there are no isolated

∗ points in ∂E, and (2) since by our choice of representation ∂E = sptµE = ∂ E, (5.4) ensures

1 ∗ H ((∂E \ ∂ E) ∩ A0) = 0. (5.5)

−1 1 ∗ Indeed, if (5.5) were false one can readily contradict that limr↓0 r H (B(x, r)∩∂ E) = 0 99

for H1-a.e. x∈ / ∂∗E.

If K is a compact subset of A0, Wa˙zewski’stheorem (See, for instance, [1] for a formal theorem statement) therefore ensures that each connected component of ∂E ∩K is a Lipschitz curve since H1(K ∩ ∂E) < ∞ and K ∩ ∂E is compact. In particular, connected components of ∂E ∩ A0 are locally Lipschitz curves.

Theorem 94. If ρ : S1 → (0, ∞) is a lower semicontinuous, bounded, strictly convex func- 2 2 tion, A0 ⊂ R is an open set, and E ⊂ R minimizes Φ(· ; A0) then there exists an equivalent set of locally ﬁnite perimeter which we also call E, so that ∂E ∩ A0 6= ∅ implies ∂E ∩ A0 2 is a collection of non-intersecting line segments. In the case that A0 = R , E must be a half-space.

Proof. Without loss of generality, assume E = E(1). Suppose for the sake of contradiction that ∂E ∩ A0 6= ∅ is not made up of exclusively straight, non-intersecting line segments. Then, there exists a non-ﬂat curve γ ⊂ ∂E such that the endpoints of γ, denoted by

{x1, x2}, satisfy

|x1 − x2| < dist(γ, ∂A0). (5.6)

By Remark 90, γ has no self-crossings nor does it cross ∂E \ γ.

Let ` be the line segment between x1 and x2. If x ∈ ` then in light of (5.6)

1 dist(x, γ) ≤ dist(x, {x , x }) < dist(γ, ∂A ) 2 1 2 0

Which veriﬁes ` ⊂⊂ A0 and consequently, ` ∪ γ ⊂⊂ A0. If necessary, shorten ` (and then γ accordingly) so that ` ∩ ∂E = γ ∩ ` = {x1, x2}. The fact that “the next crossing” of ` with ∂E exists follows from Remark 93.

In particular, γ ∪` is a Jordan curve. Since `∪γ ⊂⊂ A0, this ensures there exists a unique connected component G of A0 \ (γ ∪ `) whose closure does not meet ∂A0. See Figure 5.3b. At this point there are two cases to consider: when G ⊂ E and when G ⊂ Ec. 1

1If k · k were such that kxk = k − xk for all x ∈ R2 \{0} one could just replace E with Ec to cover both 100

(b) If γ ∩ ` 6= {x1, x2} shorten ` by removing the dashed line segment and redeﬁning γ ac- (a) A valid competitor F relative to the set A0. cordingly.

Figure 5.2

First consider the case where G ⊂ E. Deﬁne the competitor F = E \ G. By choice of G,

E∆F ⊂⊂ A0. So that F is a valid competitor for E in A0.

Moreover, F ⊂ E ensures {νE = −νF } = ∅. Hence, (5.56) implies that G satisﬁes

(0) (1) µG = µE\F = µE F − µF E . (5.7)

(1) (1) (1/2) ∗ (0) (0) (1) Since F ⊂ E is disjoint from E ⊃ ∂ E we have µE F = µE F ∪ F . 1 2 (0) (1) ∗ 1 ∗ Since H R \ F ∪ F ∪ ∂ F = 0 and |µE| = H ∂ E is in particular absolutely continuous with respect to H1, this in turn implies

(0) (0) (1) ∗ ∗ µE F = µE F ∪ F = µE (∂ E \ ∂ F ) . (5.8)

Similarly

(1) ∗ ∗ µF E = µF (∂ F \ ∂ E) . (5.9)

cases simultaneously. However, this additional assumption on k · k is not necessary. 101

Combining (5.7), (5.8), and (5.9) yields

∗ ∗ ∗ ∗ µG = µE (∂ E \ ∂ F ) − µF (∂ F \ ∂ E). (5.10)

Next, we aim to show geometrically evident fact (see Figure 5.3b) that

µG = µE γ − µF `. (5.11)

To this end, ﬁrst note that Remark 93 ensures ∂E ≈ ∂∗E2. But, since ∂F ⊂ (` ∪ ∂E) and H1 (` ∩ ∂E) = 0, it follows from the ﬂatness of ` that ∂F ≈ ∂∗F . Similarly, ∂∗G ≈ ∂G.

By Federer’s theorem and (5.5) this also implies, G(0) ≈ R2 \ G. Therefore, since G = E∆F implies ∂E \ G = ∂F \ G, it follows

(0) ∗ 2 2 (0) ∗ G ∩ ∂ F ≈ (R \ G) ∩ ∂F = (R \ G) ∩ ∂E ≈ G ∩ ∂ E. (5.12)

Moreover, F ∩ G = ∅ implies {νF = νG} = ∅ so that (5.57) implies

∂∗E = ∂∗ (F ∪ G) ≈ F (0) ∩ ∂∗G ∪ G(0) ∩ ∂∗F . (5.13)

Similarly, ∂∗F = ∂∗(E \ G) ≈ E(1) ∩ ∂∗G ∪ G(0) ∩ ∂∗E . (5.14)

However, since ∂∗E ∩ E(1) = ∅ and ∂∗F ∩ F (0) = ∅, (5.12) (5.13) and (5.14) imply

 ∗ ∗ (0) ∗ ∂ E \ ∂ F ≈ F ∩ ∂ G

∂∗F \ ∂∗E ≈ E(1) ∩ ∂∗G .

(0) (1) Since ∂G = γ ∪ ` with γ ⊂∼ F , ` ⊂∼ E and ` ∩ γ ≈ ∅, this veriﬁes (5.11).

Since G ⊂⊂ A0, it follows µG(A0) = 0. Indeed, recalling that E is an energy minimizer

2 2 1 Recall for M1,M2 ⊂ R , M1 ≈ M2 means H (M1∆M2) = 0. 102

2 1 inside A0 ⊂ R , choose ϕ ∈ Cc (A0) such that ϕ ≡ 1 on G ⊃ sptµG and observe

Z Z µG(A0) = ϕdµG = ∇ϕdx = 0. (5.15) A0 A0

Combining (5.11), (5.15), and the fact that νF ` is constant yields

Z Z Z 1 1 1 ρ(νF )dH = ρ νF dH = ρ νEdH . (5.16) ` ` γ

Since ρ is strictly convex and γ is not ﬂat (so νE γ is not constant) we further have

Z Z 1 1 ρ νEdH < ρ(νE)dH . (5.17) γ γ

It now follows from (5.10), (5.11), (5.16), and (5.17) that Φ(E; A0) > Φ(F ; A0). Since F is a valid competitor, this contradicts the Φ(· ; A0) minimality of E, completing Case 1.

c In case G ⊂ E deﬁne F = E ∪ G. Since G = E∆F , is compactly contained in A0, this case follows analogously to previous one.

2 It remains to show that if A0 = R then E is a half-space. Indeed, we just verified that ∂E must be a collection of non-intersecting lines. Therefore, if ∂E contains more than one line, they must be parallel. Since H1 ∂E is locally finite, we can select two consecutive lines in ∂E which we call L1 and L2. Let ~s be a unit vector parallel to L1 and ~t be orthonormal to ~s. The idea is is to build a competitor F , see Figure 5.4, whose boundary is identical to ∂E, except on some rectangle, where on this rectangle, the ~s-directional sides will be in ∂E \ ∂F whereas the ~t-directional sides are in ∂F \ ∂E. By making the ~s-directional sides sufficiently long it will follow that F will have less Φ-energy than E, contradicting that a Φ(· ; R2)-energy minimizing set E can have ∂E containing more than one line. One difficulty that makes the proof more technical, is we need some bounded open set A0 so that making this change on the rectangle above ensures that E∆F is compactly supported in A0. We do this by slightly fattening the rectangle we modify. 103

Figure 5.4: The desired competitor in the case the strip between the consecutive lines is contained in E, in which point E∆F = E \ F . In the case where the strip between the consecutive lines is contained in Ec, the boundaries for E and F remain unchanged, however E∆F = F \ E and similarly E ∩ F should be replaced with Ec ∩ F c.

2 ~ i More precisely, rescale and choose your origin so that Li is the line {x ∈ R : x·t = (−1) } for i ∈ {1, 2}. For each σ, τ > 0 deﬁne the rectangle

2 ~ Rσ,τ = {x ∈ R : −σ ≤ x · ~s ≤ σ, −τ ≤ x · t ≤ τ}

Deﬁne a, b > 0 so that max{ρ(~s), ρ(−~s)} = a and min{ρ(~t), ρ(−~t)} = b. Choose δ > 0 so that R1,1+δ ∩ ∂E = R1,1+δ ∩ (L1 ∪ L2). That is, choose δ so that “fattening” R vertically by b a distance of δ does not meet any new pieces of ∂E. Fix c > a and observe that

Z Z ρ(νE) ≥ 4bc > 4a ≥ ρ(νR). (L1∪L2)∩∂Rc,1 ∂Rc,1\(L1∪L2)

Then, deﬁning F = E \ Rc,1 or F = E ∪ Rc,1 depending on whether or not Rc,1 ⊂ E it follows that Φ(F ; Rc+δ,1+δ) < Φ(E; Rc+δ;1+δ) contradicting the minimality of E and hence 104

verifying ∂E is a single line, so that E is a half-space.

5.4 The Anisotropic First and Second Variations of Φρ

In this section, we derive the first and second variations for the anisotropic energies Φρ 1 n defined in (5.18). To this end, we consider a vectorfield T ∈ Cc (A; R ) and we compute the Taylor expansion of the functional

Z −1∗ n−1 Φρ(ft(E); A) = ρ (∇ft(x)) [νE] Jft(x)dH , (5.18) −1 ∗ ft (A)∩∂ E

p ∗ ft(x) = x + tT (x). In particular, Jft(x) = det((∇ft(x)) (∇ft(x))). We assume that t is −1 small enough that ft (A) = A so that Et = ft(E) is a valid competitor for E inside A. In what follows, we may write ∇T ∗(x) in place of (∇T (x))∗.

Theorem 95. If E is a set of locally ﬁnite perimeter, ρ ∈ C1(Rn \{0}; (0, ∞)) is a positively 1 n 1-homogeneous function, and ft(x) = x + tT (x) for some T (x) ∈ Cc (A; R ) then

Z d ∗ n−1 Φρ(ft(E); A) = ρ(νE) tr(∇T (x)) − hdρ(νE), (∇T (x)) [νE]idH , (5.19) dt t=0 ∂∗E∩A where dρ denotes the diﬀerential of ρ. If additionally ρ, T ∈ C2, then also

2 d 2 ∗ ∗ 2 2 Φρ(ft(E); A) = tr D ρ(νE) [(∇T (x)) [νE] ⊗ (∇T (x)) [νE]] + 2hdρ(νE), (∇T (x)) i dt t=0 (5.20)

∗ 2 2 − hdρ(νE), (∇T (x)) [νE]i tr(∇T (x)) + 2 (tr(∇T (x))) − tr (∇T (x)) .

Proof. We begin with some preliminary observations. First, if a, b, c ∈ Rn and ρ ∈ C2 then

d 2 3 ρ(a + bt + ct + O(t )) = hdρ(a), bi (5.21) dt t=0 105

and 2 d 2 3 2 2 ρ(a + bt + ct + O(t )) = tr D ρ(a)[b ⊗ b] + 2hdρ(a), ci. (5.22) dt t=0

We also recall the Taylor series expansion:

a t2 a2 (1 + at + bt2)1/2 = 1 + t + b − + O(t3) (5.23) 2 2 4

Next we recall

t2 det(Id + tZ) = 1 + t tr(Z) + tr(Z)2 − tr(Z2) + O(ktZk3) (5.24) 2 where kAk2 = tr(A∗A) and that

(Id + tZ)−1 = Id − tZ + t2Z2 + O(ktZk3). (5.25)

We begin by expanding the Jacobian term. Since ft(x) = x + tT (x) then ∇ft(x) = Id + t∇T (x). Hence,

∗ ∗ 2 ∗ (∇ft(x)) (∇ft(x)) = Id + t (∇T (x) + ∇T (x)) + t (∇T (x)) (∇T (x)) = Id + t [∇T ∗(x) + ∇T (x) + t∇T ∗(x)∇T (x)] (5.26) 106

Then, (5.24) and (5.26) imply

∗ ∗ ∗ det (∇ft(x) (∇ft(x))) = 1 + t (tr(∇T (x) + ∇T (x)) + t tr (∇T (x)∇T (x))) t2 + (tr (∇T ∗(x) + ∇T (x)))2 − tr (∇T ∗(x) + ∇T (x))2 2 + O(t3)

= 1 + 2t tr(∇T (x)) t2 + (2 tr (∇T (x)))2 + 2 tr (∇T ∗(x)∇T (x)) 2 − tr (∇T ∗(x))2 + 2∇T ∗(x)∇T (x) + (∇T (x))2 + O(t3)

= 1 + t [2 tr(∇T (x))] (5.27)

+ t2 2(tr(∇T (x)))2 − tr((∇T (x))2) + O(t3)

Applying the expansion (5.23) to (5.27) yields

Jft(x) = 1 + t tr (∇T (x)) " # t2 (2 tr ∇T (x))2 + 2 (tr (∇T (x)))2 − tr (∇T (x))2 − + O(t3) 2 4

= 1 + t tr(∇T (x)) + t2 (tr(∇T (x)))2 − tr (∇T (x))2 + O(t3) (5.28)

Therefore,

2 d d 2 2 Jft(x) = tr(∇T (x)) and 2 Jft(x) = 2 (tr(∇T (x))) − tr (∇T (x)) . dt t=0 dt t=0 (5.29)

−1 It remains to ﬁnd the expansion for ρ ((∇ft(x)) [νE]). Following the convention that 107

A0 = Id for any matrix A, (5.25) implies

∞ −1 X i 2 2 3 (∇ft(x)) = (−t∇T (x)) = Id − t∇T (x) + t (∇T (x)) + O(kt∇T (x)k ), (5.30) i=0 so that

−1∗ ∗ 2 2∗ 3 ρ (∇ft(x)) [νE] = ρ νE − t (∇T (x)) [νE] + t (∇T (x)) [νE] + O(t ) (5.31)

Applying (5.21) to (5.31) veriﬁes

d h −1∗ i ∗ ρ (∇ft(x)) [νE] = −hdρ(νE), (∇T (x)) [νE]i (5.32) dt t=0 and applying (5.22) to (5.31) yields

2 d h −1∗ i 2 ∗ ∗ 2 ρ (∇ft(x)) [νE] = tr D ρ(νE) [(∇T (x)) [νE] ⊗ (∇T (x)) [νE]] + 2hdρ(νE), ci. dt t=0 (5.33)

Finally, combining (5.28) - (5.33) with the product rule we can do the Taylor expansion of the integrand in (5.18). Namely,

−1∗ ∗ ρ (∇ft(x)) [νE] Jft(x) = ρ(νE) + t [ρ(νE) tr(∇T (x)) − hdρ(νE), (∇T (x)) [νE]i]

(5.34) t2 + tr D2ρ(a) [(∇T (x))∗ [ν ] ⊗ (∇T (x))∗ [ν ]] + 2hdρ(a), ci 2 E E ∗ 2 2 − hdρ(νE), (∇T (x)) [νE]i tr(∇T (x)) + 2 (tr(∇T (x))) − tr (∇T (x))

+ O(t3). 108

1 n Consequently, for k = 1, 2 and any T ∈ Cc (A; R ) we can deﬁne the mapping

dk k : δ Φρ(E; A)[T ] = k Φρ(ft(E); A), dt t=0

k where ft is as before. δ Φρ(E; A) is called the kth variation of Φρ(·; A) at E. For future convenience, we observe

Z n−1 δΦρ(E; A)[T ] = ρ(νE) tr(∇T (x)) − hνE, ∇T (x)[dρ(νE)]idH ∂∗E∩A Z = tr ((ρ(νE)Id − (Dρ)(νE) ⊗ νE) ∇T ) . (5.35) ∂∗E∩A

A set E is called stationary Φρ-stationary if it is Φρ-minimal. That is,

δΦρ(E; A) ≡ 0.

5.5 Monotonicity formula and basic consequences

In this section we will produce a sufficient condition for the existence of a specific monotonic quantity on Φρ-minimal sets. We then consider n = 2, ρ = k · k`p , p > 2 to provide a new monotonicity formula, Theorem 101. Monotonicity will follow from plugging an appropriate test function into the first variation. These computations will be done for all n and any ρ, in order to produce the sufficient condition (5.42).

We will then turn our immediate focus to the special case when n = 2 and ρ = k · kp and demonstrate that this special case satisﬁes (5.42). We proceed to record some consequences of this monotonicity formula.

Deﬁnition 96 (f- and p-balls). Given any positively 1-homogeneous function f, we deﬁne

Bf (x, r) := {y : f(y − x) < r}. (5.36) 109

We call Bf (x, r) an f-ball. In the case f = k · kp, we will abuse both notation and language to write Bp(x, r) in place of Bk·kp (x, r) and we then call the set a p-ball.

Theorem 97. If E is Φρ-stationary, then for any positively 1-homogeneous function f ∈ 1 n n C (R \{0}; R ) it follows that for almost every r and any x0 ∈ ∂E,

d −(n−1) t Φρ(E; Bf (x0, r)) (5.37) dt t=r " x−x0 # Z h∇f(x − x0), ∇ρ(νE)ihνE, i d f(x−x0) n−1 = n−1 dH . ∗ dt t=r ∂ E∩Bf (x0,r) f(x − x0)

n Proof. We will prove the case when x0 = 0. Then (5.37) follows by translating. Let ϕ : R → R be a smooth-cutoﬀ, chosen later, and f ∈ C1(Rn \{0}; Rn) a positively 1-homogeneous function. Consider T (x) = ϕ (t−1f(x)) x. Then,

∇T (x) = ϕ t−1f(x) Id + t−1ϕ0(t−1f(x)) (x ⊗ ∇f(x)) .

Hence,

−1 −1 tr ((ρ(νE)Id − (Dρ)(νE) ⊗ νE) ∇T ) = nρ(νE)ϕ(t f(x)) − ρ(νE)ϕ(t f(x))

−1 0 −1 + t ϕ (t f(x)) [ρ(νE)f(x) − h(Dρ)(νE), ∇f(x)ihx, νEi] . (5.38)

Plugging (5.38) into (5.35) implies,

Z −1 −1 0 −1 ρ(νE)(n − 1)ϕ(t f(x)) + t ϕ t f(x) f(x)ρ(νE) Z ϕ0(t−1f(x)) = h∇f(x), ∇ρ(ν )ihν , xi. (5.39) t E E

Since d ϕ t−1f(x) = −t−2f(x)ϕ0(t−1f(x)) dt 110

implies −t d t−1ϕ0(t−1f(x)) = ϕ t−1f(x) , f(x) dt it follows (5.39) is equivalent to

Z d ρ(ν )(n − 1)ϕ(t−1f(x)) − t ϕ t−1f(x) ρ(ν ) E dt E Z t d = − ϕ t−1f(x) h∇f(x), ∇ρ(ν )ihν , xi. (5.40) f(x) dt E E

The LHS of (5.40) is precisely

Z n d −(n−1) −1 n−1 −t t ρ(νE(x))ϕ(t f(x))dH (x) dt ∂∗E so that (5.40) ensures

Z d −(n−1) −1 n−1 t ρ(νE(x))ϕ(t f(x))dH (x) dt ∂∗E Z x = t−(n−1) ϕ0(t−1f(x))h∇f(x), ∇ρ(ν )ihν , i. (5.41) E E f(x)

∞ Replacing ϕ with ϕk ∈ C (R; R) chosen so that

 1 s ≤ 1 − 2−k k 0  −C2 ≤ ϕk ≤ 0 and ϕk(s) = 0 s ≥ 1 for some constant C that depends on n, we pass to the limit taking k → ∞ to discover that for almost every r,

" x # Z h∇f(x), ∇ρ(νE)ihνE, i d −(n−1) d f(x) n−1 t Φρ(E; Bf (0, t)) = n−1 dH . ∗ dt t=r dt t=r ∂ E∩Bf (0,t) f(x) 111

Corollary 98. If f ∈ C1(Rn \{0}; Rn) is positively 1-homogeneous and satisﬁes

n−1 h∇f(x), ∇ρ(ν)ihx, νi ≥ 0 ∀x, ν ∈ S (5.42)

then for all Φρ-stationary sets, any x0 ∈ ∂E,

Φ (E; B (x , r)) r 7→ ρ f 0 rn−1 is monotonically increasing. More speciﬁcally, for almost every 0 < σ < τ

Φ (E; B (x , τ)) Φ (E; B (x , σ)) ρ f 0 − ρ f 0 τ n−1 σn−1 x−x0 Z h∇f(x − x0), ∇ρ(νE)ihνE, i f(x−x0) n−1 = n−1 dH . ∗ ∂ E∩(Bf (x0,τ)\Bf (x0,σ)) f(x − x0)

The corollary is immediate from the Theorem 101, (5.37). It seems unlike that a statement like that in (5.42) will hold for every pair of points {(x, ν) ⊂ R2n}. This is a similar condition to that studied by [4], where Allard uses Morse-theoretic techniques to show that the only ρ where such an f exists are ρ(·) = |A · | for some constant elliptic matrix A. Nonetheless, assuming all

Lemma 99. If n = 2,

sgn(x · y) = sgn(∇kxkp · ∇kykp).

In particular

∇kxkp · ∇kykp = 0 ⇐⇒ x · y = 0. (5.43)

Proof. First compute

p−2 p−2 p−2 p−2 hx, νi = x1y1 + x2y2 and h∇kxkp, ∇kykpi = x1|x1| y1|y1| + x2|x2| y2|y2| . 112

Then  x1y1  > −1 x2y2 > 0  x2y2 x · y > 0 ⇐⇒ x1y1 > −x2y2 ⇐⇒ (5.44) x1y1  < −1 x2y2 < 0, x2y2 and

p−2 p−2 p−2 p−2 hx, yih∇kxkp, ∇kykpi = (x1y1 + x2y2) x1|x1| y1|y1| + x2|x2| y2|y2|

p p p p p−2 p−2 p−2 p−2 = |x1| |y1| + |x2| |y2| + x1x2y1y2 |x1| |y1| + |x2| |y2| .

We see immediately that if x1x2y1y2 ≥ 0 then hx, yih∇kxkp, ∇kνkpi ≥ 0. So, we suppose

x1y1 x1x2y1y2 < 0. This in turn implies < 0. So, within our case, (5.44) implies x2y2

 x1y1  < 1 x2y2 > 0 x · y > 0 ⇐⇒ x2y2 (5.45)

x1y1  > 1 x2y2 < 0. x2y2

On the other hand, still assuming x1x2y1y2 < 0,

 p−2 p−2 x1|x1| y1|y1|  p−2 p−2 > −1 x2y2 > 0  x2|x2| y2|y2| h∇kxkp, ∇kykpi > 0 ⇐⇒ p−2 p−2  x1|x1| y1|y1|  p−2 p−2 < −1 x2y2 < 0 x2|x2| y2|y2|  p−1 x1y1  < 1 x2y2 > 0 ⇐⇒ x2y2 ⇐⇒ x · y > 0. p−1 x1y1  > 1 x2y2 < 0 x2y2

It remains to show x · y = 0 ⇐⇒ h∇kxkp, h∇kykpi = 0.

If x ∈ {e1, e2} then x · y = 0 if and only if y is the other basis direction. In any case,

∇keikp = ei, so if either vector is a basis direction both quantities are zero.

So now suppose hx, yi = 0 and x 6= ei. Then

x1y1 x1y1 6= 0 6= x2y2 and = −1. x2y2 113

But then p−2 x1y1 x1y1 h∇kxkp, ∇kykpi = 0 ⇐⇒ = −1 ⇐⇒ x · y = 0. x2y2 x2y2

Remark 100. For geometric reasons, it would make more sense to study

Φ (E; K (x, r)) r 7→ p q r

−1 −1 where q + p = 1. However, if one tries to show h∇kxkp, ∇kykqihx, yi ≥ 0 for any q 6= p, p q then the proof proceeds identically by replacing powers of yi with yi up until (5.45). But then, in the case x1y1x2y2 < 0,

 p−1 q−1 x1 y1  < 1 x2y2 > 0 h∇kxk , ∇kyk i > 0 ⇐⇒ x2 y2 p q p−1 q−1 x1 y1  > 1 x2y2 < 0 x2 y2  x1y1  < 1 x2y2 > 0 6⇐⇒ x2y2 ⇐⇒ x · y > 0.

x1y1  > 1 x2y2 < 0 x2y2

So, despite the the fact that studying the k · kp-energy ratios over k · kq-balls would be more natural, the desired algebraic inequality does not hold.

Theorem 101. If n = 2, ρ = k · k`p and x0 ∈ ∂E, then

Φ (E; B (x , r)) r 7→ p p 0 r

1 ∗ 1 is monotonically increasing and if 0 < σ < τ satisfy H (∂Bf (x0, σ)∩∂ E) = 0 = H (∂Bf (x0, τ)∩ 114

∂∗E) then

Φ (E; B (x , τ)) Φ (E; B (x , σ)) p f 0 − p f 0 τ σ x Z h∇kxkp, ∇kνEkihνE, i = kxkp dH1. ∗ ∂ E∩{σ≤kx−x0kp≤τ} kxkp

1 ∗ 1 ∗ Consequently, if σ < τ and H (∂Bf (x0, σ) ∩ ∂ E) = 0 = H (∂Bf (x0, τ) ∩ ∂ E), which in particular means for almost every 0 < σ < τ and

Φ (E; B (x , τ)) Φ (E; B (x , σ)) p f 0 = p f 0 τ σ then

1 ∗ νE(x) · (x − x0) = 0 H − a.e. x ∈ (Bf (x0, τ) \ Bf (x0, σ)) ∩ ∂ E.

Proof. The monotonicity formula follows immediately from Theorem 101 and Lemma 99.

The fact that νE(x) · (x − x0) = 0 a.e. Bf (x0, τ) \ Bf (x0, σ) is due to (5.43), as the integrand must be zero almost everywhere.

Deﬁnition 102 (Energy densities). Given x ∈ ∂E ⊂ Rn, and a continuous function f : Rn → R, we deﬁne the f upper- and f lower- energy density of E at x as

n Φp(E; Bf (x, r)) Θf,∗(E, x) = lim inf r↓0 rn−1 and n,∗ Φp(E; Bf (x, r)) Θf (E, x) = lim sup n−1 . r↓0 r

n n,∗ Whenever Θf,∗(E, x) = Θf (E, x) we call the common value the f energy density of E at x, n and denote it Θf (E, x). Whenever f is clear from context, the dependency on f may not be explicitly stated. If

n n,∗ n f = k · kp, we will denote the energy densities by Θp,∗, Θp , Θp and call them the p densities instead of k · kp-densities. 115

The easiest corollary of the monotonicity formula, Theorem 101, is that the energy density

1 Θp(E, x) exists for every x ∈ ∂E.

1 Corollary 103. If 1 < p < ∞, then Θp(E, x0) exists for every x0 ∈ ∂E.

The corollary holds because bounded monotonic sequences always have limits. We next observe the upper semicontinuity of the density:

2 Lemma 104. If {Ej} are Φp-minimizing sets of locally ﬁnite perimeter in R , such that 1 1 Ej → E in L , and xj ∈ Ej for all j are such that xj → x0, then lim supj Θp(Ej, xj) ≤ 1 Θp(E∞, x0).

Proof. First note that by Theorem 121, E is a Φp-minimizing set of locally ﬁnite perimeter. For almost every r ∈ (0, ∞) it follows that

1 1 H (∂Bp(xj, r) ∩ ∂Ej) = 0 = H (∂Bp(x0, r) ∩ ∂E∞) (5.46)

Let > 0 and 0 < r0 be so that (5.46) holds with r ∈ {r0, r0 + }. Fix J large enough 1 that |xj − x0| < for all j ≥ J. If α = lim supj Θp(Ej, xj) then by monotonicity, if j ≥ J

Φ (E ; B (x , r )) α ≤ lim sup p j p j 0 j r0 Φ (E ; B (x , r + )) r + ≤ lim sup p j p 0 0 0 . j r0 + r0

By continuity of Φp on sequences of minimizing sets, Theorem 121, implies

Φ (E; B (x , r + )) r + α ≤ p p 0 0 0 . r0 + r0

Taking ↓ 0 then r0 ↓ 0 completes the proof.

Ultimately, we want to show that the minimizers are ﬂat. The ﬁrst big step to achieve this is to show that the tangents are minimizing cones. The next result shows that constant 116

energy density implies the surface is a cone. A quantitative version of Lemma 99, i.e., something that says

∇kxkp · ∇kykp is small ⇐⇒ x · y is small would lead to a much stronger result that small drops in energy imply the surface is nearly a cone. For a rather recent example, see [30] for similarly quantitative results in diﬀerent settings.

2 Lemma 105. Suppose E ⊂ R is Φp-stationary, x0 ∈ ∂E, τ > 0, and

Φ (E; B (x , τ)) p p 0 − Θ1(E, x ) = 0. (5.47) τ p 0

Then for all τ 0 < τ,

∗ 0 ∗ 0 ∂ E ∩ Bp(x0, τ ) = {(1 − λ)x0 + λy : y ∈ ∂ E ∩ ∂Bp(x0, τ ), λ ∈ [0, 1]}.

In particular, if (5.47) holds for all τ > 0, then ∂∗E must be a cone.

1 Proof. First note (5.47) allows us to apply Theorem 101 to conclude x·νE(x) = 0 for H -a.e. ∗ x ∈ Bp(x0, τ) ∩ ∂ E.

Without loss of generality, let x0 = 0. We next wish to consider Te(x) = h(x)T (x), where T (x) is as in Theorem 97, and h(x) is an arbitrary positively 0-homogeneous function, continuous on S1. However, we may only consider test vector ﬁelds Te ∈ C1, so we currently consider h ∈ C1 and later make an approximation argument. Given h ∈ C1, observe

∇Te(x) = x ⊗ ∇h(x)ϕ(t−1f(x)) + h(x)∇T (x). 117

Therefore,

tr (kνEkpId − (∇kνEkp)(νE) ⊗ νE) ∇Te(x)

= h(x) tr ((kνEkpId − (∇kνEkp ⊗ νE) ∇T (x))

−1 + ϕ(t f(x)) kνEkpx · ∇h(x) − h∇kνEkp, ∇h(x)ihνE, xi .

Hence, one can proceeding identically as in the proof of Theorem 97 (up to multiplying everything that occurs in Theorem 97 by the function h and carrying along the new terms involving ∇h) to compute that for almost every t < τ,

" Z # Z d −1 1 −1 x 0 t kνEkphdH = t h(x)hνE, ih∇kνEkp, ∇kxkpih(x)dH ∗ ∗ dt ∂ E∩Bp(0,t) ∂ E∩∂Bp(0,t) f(x) Z −1 h Dx E Dx Ei + t kνEkp , ∇h(x) − h∇kνEkp, ∇h(x)i , νE . ∗ ∂ E∩Bp(0,t) t t

1 ∗ In fact, because x · νE = 0 for H almost every x ∈ ∂ E ∩ Bτ , it follows that for almost every t < τ, the preceding reduces to

" Z # Z d −1 1 −1 Dx E t kνEkphdH = t kνEkp , ∇h(x) . (5.48) ∗ ∗ dt ∂ E∩Bp(0,t) ∂ E∩Bp(0,t) t

1 In (5.48) we replace h with a sequence of h ∈ C which we let approach an arbitrary 1 2 1 n 0-homogeneous g ∈ C (R \{0}). More precisely, choose h ∈ C (R ) so that

 g(x) kxkp ≥  −1 h(x) = and |x · ∇h(x)| ≤ Ckgk∞ ∀kxkp ≤ . 0 x = 0 118

Then, 0-homogeneity of g ensures

Z Dx E Z Dx E Φ (E; B (0, )) −1 −1 p p t kνEkp , ∇h(x) = t kνEkp , ∇h(x) ≤ C 2 ∗ t ∗ t t ∂ E∩Bp(0,t) ∂ E∩Bp(0,) which, for instance due to (5.47), approaches zero as ↓ 0. On the other hand, whenever < t, the remaining two terms in (5.48) are independent of . Consequently, it follows from (5.48) that for almost every 0 < t < τ

" Z # d −1 1 t kνE(x)kpg(x)dH (5.49) ∗ dt ∂ E∩Bp(0,t) " Z # d −1 1 = lim t kνE(x)kph(x)dH ↓0 ∗ dt ∂ E∩Bp(0,t) Z −1 Dx E = lim t kνE(x)kp , ∇h(x) = 0. ↓0 ∗ ∂ E∩Bp(0,t) t

In other words,

Z 1 1 0 0 g(x)ρ(νE)dH = constant a.e. 0 < τ < τ. 0 τ Bp(0,τ )

Since g ∈ C1(Rn \{0}; R) is an arbitrary 0-homogeneous function, this implies the result, see for [67, Theorem 19.3].

To relate Lemma 105 to information about tangents, we must better understand how tangents relate to the energy density.

n E−x Deﬁnition 106 (Enlargements). If E ⊂ R then Ex,r is the set deﬁned by Ex,r = r .

Lemma 107. If E is a set of locally ﬁnite perimeter, then

Φ (E; K ) Φ (E ; K ) p σr = p x,r σ . σr σ 119

Moreover,

y−x Z h , νEih∇ky − xkp, ∇kνEkpi f(y−x) dHn−1 ∗ ∂ E∩{ρr

Proof. Indeed, since νEx,r (y) = νE(rx + y) it follows that

Z Φp(Ex,r; Kσ) 1 1 = kνEx,r (y)kpdH ∗ σ σ ∂ Ex,r∩Bp(0,σ) 1 Z dH1 = kνE(y)kp ∗ σ ∂ E∩Bp(x,rσ) r Φ (E; B (x, rσ) = p p . rσ

The second claim follows since the integrand is −1-homogeneous, and the measure is 1- homogeneous.

The next corollary follows from the scaling from Lemma 107 and the continuity of the

Φp-energy when restricted to energy minimizing sets, see Theorem 121.

Corollary 108. If E is minimizing and x ∈ ∂E, rj → 0 as j → ∞, and Erj → E∞ then

Φ (E ; B (0, σ)) Θ1(E; x) = p ∞ p ∀σ > 0. (5.50) p 2σ

Finally, we conclude that blow-up limits of energy minimizers are energy minimizing cones. That is,

2 Corollary 109. If E ⊂ R is Φp-energy minimizing, x ∈ ∂E, rj → 0, and Ex,rj → E∞ then E = λE for all λ > 0.

Proof. The fact that E∞ is energy-minimizing follows from Theorem 121. The fact that E∞ is a cone, follows from Corollary 108 and Lemma 105. 120

Remark 110. We reiterate that the above theory can be implemented for sets and varifolds. Indeed, Theorem 101 only requires the ﬁrst variation being zero and does not directly use minimizing. Lemmas 104, 105, and Corollary 108 only uses that the monotonicity formula applies to each set. Here, we merely used the results of Section 5.7 in order to gain monotonicity from the minimizing property, including in limiting sets.

5.6 Anisotropic minimal cones

In this section, we consider global minimal cones in R2. That is, sets of the form

C(E) = {λx : λ > 0, x ∈ E ⊂ ∂Bρ(0, 1)}. (5.51)

First note that E ⊂ S1 must be a ﬁnite collection of points, or else the energy of the cone would be inﬁnite. For area minimal cones in the plane, it’s known the only cones can be: (1) straight lines, and (2) a triple junction, where all three half-lines meet at 120◦ angles. We produce a new anisotropic characterization of triple junctions for anisotropic minimal cones. The key tool to understand anisotropic minimal cones is convex duality. Given ρ as in

(5.1), the convex dual to ρ, denoted ρ∗ is deﬁned by

ρ∗(ξ) = sup{x · ξ : ρ(x) < 1}.

1 If ρ is C and strictly convex, then so is ρ∗ and (ρ∗)∗ = ρ. Moreover,

 ρ∗ ((Dρ)(x)) ≡ 1 and ρ ((Dρ∗)(ξ)) ≡ 1 (5.52) (Dρ ) [(Dρ)(x)] = x and (Dρ) [(Dρ )(ξ)] = ξ .  ∗ ρ(x) ∗ ρ∗(ξ)

The ﬁrst observation in (5.52) is why it is convenient to consider E ⊂ ∂Bρ(0, 1) in (5.51). Next, we note a small departure from the typical assumptions in this document and 121

proceed using an abuse of notation. We consider the energy

Z Fρ(V ; A0) := ρ(TxV )dkV k, for a strictly convex norm ρ, defined for either a current, varifold, or closed rectifiable set V , where TxV is the tangent space to V . In the case of currents, it makes sense to consider Fρ for positively one-homogeneous ρ, due to the orientability of currents. Fρ is only well-defined for norms ρ. The considerations of minimal closed rectifiable sets arise, for instance, in the setting of sliding minimizers studied by Almgren and David [6, 18].

Theorem 111. Fix x1 ∈ ∂Bρ(0, 1) and E = {x1, x2, x3}. If ρ is a norm, the following are equivalent.

1. The cone C = C(E) in (5.51) is minimal with respect to the anisotropic energy Fρ

2. For j = 2, 3, xj = (Dρ∗)(yj) where {y2, y3} = ∂Bρ∗ (−(Dρ)(x1), 1) ∩ ∂Bρ∗ (0, 1).

Remark 112. If ρ is strictly convex and C1, but only positively 1-homogeneous, the proof of Theorem 111 will still show that if E is stationary, then for j = 2, 3, xj take the form xj = (Dρ∗)(yj) for some yj satisfying

ρ∗(yj) = 1 = ρ∗(−yj − y1) ⇐⇒ yj ∈ ∂Bρ∗ (0, 1) ∩ ∂ {−y1 − Bρ∗ (0, 1)} .

2 Example 113. Let e2 denote the second standard basis direction in R and deﬁne

 p 2 2  x1 + x2 x2 ≥ 0 ρ∗(x1, x2) = p 2 2  x1 + (10x2) x2 < 0.

1 Then ρ∗ ∈ C is strictly convex and positively 1-homogeneous, and therefore so is ρ = (ρ∗)∗.

Since (Dρ∗)(e2) = e2 it follows (Dρ)(e2) = e2. Note, if ρ∗(y) = 1 then y · e2 ≥ −1/10.

On the other hand, if y ∈ {ρ(−e2 − · ) = 1}, then y · e2 ≤ −9/10. In particular, by Remark 122

2.0 1.0

1.5 0.5

1.0 -1.5 -1.0 -0.5 0.5 1.0

0.5 -0.5

-1.0 -0.5 0.5 1.0 1.5 2.0 -1.0

-0.5 -1.5

-1.0 -2.0

(a) On the left, the set of points with kxk5 = 1 is drawn, with a speciﬁc vector x1 chosen. Its dual vector, y1, is also drawn and is normal to B5(0, 1) at x1. On the right, y1 is translated to the origin. The “dual sphere” ∂B5/4(0, 1) is drawn with a solid line, while the “dual sphere” ∂B5/4(−y1, 1) is drawn with a dashed line. The vectors in green and red point to the unique points in the intersection of both dual spheres. The red and green vectors are also shown to add to the dashed blue vector, representing −y1.

2 1.0

1 0.5

-2 -1 1 2 -1.0 -0.5 0.5 1.0

-1 -0.5

-2 -1.0

(b) On the left, the solid red, blue, and green vectors are translated to the origin. Their respective duals are drawn tangent to B5/4(0, 1). On the right, the respective duals are translated to the origin, drawn inside the ball B5(0, 1). The resulting three vectors create a k · k5-stationary cone. 123

112 it follows there is no minimal triple junction for Fρ containing the point e2.

Of Theorem 111. Observe that

X Fρ(C(E); Bρ(0, 1)) = ρ(x). x∈E

Since we only consider compactly supported variations, and straight line-segments are minimal, the only non-trivial variations one can consider are those which perturb the point of intersection. If said perturbation is in the direction of e, then letting ft(C) denote the set constructed by sliding the intersection point of C(E) a distance t in the direction e, we ﬁnd

X Fρ(ft(C); Bρ(0, 1)) = ρ(x + te). x∈E

Hence,

d X Fρ(ft(C); Bρ(0, 1)) = e · (Dρ)(x). dt t=0 x∈E Since e is arbitrary, this means C(E) is stationary if and only if

X δFρ(C; Bρ(0, 1)) := (Dρ)(x) ≡ 0. (5.53) x∈E

We will now show 1 =⇒ 2. Indeed, assume E = {x1, . . . , x3} and C(E) is stationary.

Denote yi = (Dρ)(xi) for i = 1, 2, 3. Then, (5.52) implies ρ∗(yi) = 1 for all i. On the other hand, (5.53) implies −y1 − y3 = y2 and −y1 − y2 = y3. That is, ρ∗((−y1) − y2) =

ρ∗((−y1) − y2) = 1. Therefore, y2, y3 ∈ ∂Bρ∗ (−y1, 1) ∩ ∂Bρ∗ (0, 1). It now follows from (5.52) and ρ(xj) = 1 that xj (Dρ∗)(yj) = (Dρ∗) [(Dρ)(xj)] = = xj, ρ(xj) as desired.

It remains to show 2 =⇒ 1. First note, by strict convexity of ρ and (−Dρ)(x1) ∈

∂Bρ(0, 1), it follows that the intersection ∂Bρ∗ (0, 1) ∩ ∂Bρ∗ (−(Dρ)(x1), 1) has precisely two 124

points. Let y1 = (Dρ)(x1). For j = 2, 3, translation invariance and symmetry about the origin of ρ∗ implies,

−(y1+yj) ∈ ∂Bρ∗ (0, 1) ⇐⇒ y1+yj ∈ ∂Bρ∗ (0, 1) ⇐⇒ yj ∈ ∂Bρ∗ (−y1, 1) ⇐⇒ −yj ∈ ∂Bρ∗ (y1, 1).

In particular, yj ∈ ∂Bρ∗ (0, 1) ∩ ∂Bρ∗ (−y1, 1) is equivalent to −y1 − yj ∈ ∂Bρ∗ (0, 1) ∩

∂Bρ∗ (−y1, 1).

Strict convexity of ρ∗ and ρ∗(yj) = 1 for all j = 1, 2, 3 ensures that for j = 2, 3, yj 6=

−yj − y1. Since the intersection only has two points, it must be that y2 = −y3 − y1, or equivalently, y2 + y3 = −y1. But then, if for j = 1, 2, 3, yj = (Dρ)(xj) it follows from (5.53) that C(E) is minimal as desired.

We note, that this is a sum of points in ∂Bρ∗ (0, 1) by (5.52). Moreover, if E1 = E \{x1}, then X −(Dρ)(x1) = (Dρ)(x).

x∈E1

So, in the case that E = {x1, x2, x3}, we have not only that ρ∗ ((Dρ)(xi)) = 1, but also that

(Dρ)(x1) + (Dρ)(x2)

5.7 Appendix: Compactness of energy minimizers and related tools

First we’ll recall the relative isoperimetric inequality

Theorem 114. If n ≥ 2 and t ∈ (0, 1), x ∈ Rn and r > 0, then there exists a positive constant c(n, t) such that

1 choosing t = 2 one learns,

n−1 P (E; B(x, r)) ≥ c(n) min {|E ∩ B(x, r)|, |B(x, r) \ E|} n . (5.55)

The isoperimetric inequalities above both are extremely useful for relating information about the smallness of the perimeter of a set (or largeness of a volume of a set, respectively) to the smallness of the volume of the set (or largeness of the perimeter of the set, respectively). It is frequently useful to consider certain representations of sets of locally finite perimeter. One convenient representative is the one defined by the measure-theoretically relevant points in a set. More precisely, for a set of locally finite perimeter E, we define

(s) n |E ∩ B(x, r)| E := x ∈ R : lim = s . r↓0 |B(x, r)|

(1) It is well know that νE = νE(1) and νEc = νE(0) = −νE. That is the set E is a choice of representation of E. Federer deﬁned the essential boundary

∗ n (0) (1) ∂ E = R \ E ∪ E .

The following theorem can be found, for instance, in [49, Theorem 16.2]

Theorem 115 (Federer’s theorem). If E is a set of locally ﬁnite perimeter in Rn, then ∂∗E ⊂ E(1/2) ⊂ ∂eE. In fact, Hn−1(∂eE \ ∂∗E) = 0.

In particular, for any Borel set M ⊂ Rn,

M ≈ M ∩ E(1) ∪ M ∩ E(0) ∪ M ∩ E(1/2) ,

n−1 n−1 where M1 ≈ M2 means M1 and M2 are H -equivalent, that is, H (M1∆M2) = 0.

n (0) (1) n−1 (1/2) ∗ Furthermore, it is known that sptµE = R \ E ∪ E , and H E ∆∂ E = 0. 126

In fact, n−1 n (0) (1/2) (1) H R \ E ∪ E ∪ E = 0.

When considering two sets of locally ﬁnite perimeter E and F , deﬁne the sets

∗ ∗ {νE = νF } := {x ∈ ∂ E ∩ ∂ F : νE(x) = νF (x)}

∗ ∗ {νE = −νF } := {x ∈ ∂ E ∩ ∂ F : νE(x) = −νF (x)}.

The next technical theorem, is geometrically evident, but the technical details of the proof can be found, for instance, in [49, Theorem 16.3].

Theorem 116 (Set operations on Gauss-Green measures). If E and F are sets of locally ﬁnite perimeter, then

(1) (1) n−1 µE∩F = µE F + µF E + νEH {νE = νF },

(0) (1) n−1 µE\F = µE F − µF E + νEH {νE = −νF }, (5.56)

(0) (0) n−1 µE∪F = µE F + µF E + νEH {νE = νF }.

Equivalent statements to the above can be said about the reduced boundary. The only one we use here is,

∗ (0) ∗ (0) ∗ ∂ (E ∪ F ) ≈ F ∩ ∂ E ∪ E ∩ ∂ F ∪ {νE = νF } . (5.57)

Theorem 116 is a technical tool that is useful to compute the energy of modiﬁed sets. For example, in proving Lemma 117 and 118. Lemma 117 will be used anytime we build a competitor. However, it is particularly useful when considering convergent sequences of energy minimizing sets. In those applications, the set F will play the role of a competitor to ˜ the limiting set E and the set E will eventually be replaced with each set Eh in the sequence ˜ of minimizers Eh which converge to E. Finally F will be a desirable competitor for the set E˜. Its desirability will follow from (5.61). 127

The proof of Lemma 117 follows from modifications of the proof [49, Theorem 16.16]. See [66, Proposition 4.3] for an example of necessary modifications, but for a different energy.

˜ n Lemma 117 (Φp-energy of modified competitors). Suppose A0 ⊂⊂ A ⊂ R are both open ˜ ˜ sets, A0 has finite perimeter. If E and F , G are sets of locally finite perimeter satisfying

Hn−1 ∂∗G ∩ ∂∗E˜ = 0 = Hn−1 (∂∗G ∩ ∂∗F ) . (5.58)

Deﬁne F˜ := (F ∩ G) ∪ E˜ \ G (5.59) then F˜ is a set of locally ﬁnite perimeter, and

˜ ˜ ˜ ˜ Φp F ; A0 = Φp E; A0 \ G + Φp(F ; G) + Φp(G; E∆F ). (5.60)

˜ ˜ In particular, if E∆F ⊂⊂ G ⊂⊂ A0 then

˜ ˜ ˜ ˜ Φp(F ; A0) = Φp(F ; A0 \ ∂G) = Φp(F ; A0). (5.61)

Lemma 118 will be used to demonstrate that sequences of minimizers satisfy the properties of a pre-compactness theorem. The proof of Lemma 118 arises as a consequence of the comparability of norms on finite vector spaces and slight modifications of the proof in [49, Theorem ]. To see how this comparison and modifications are done, but for a different energy, see [66, Proposition 4.10].

Lemma 118 (Volume density bounds). Let A be an open set and E be a (Φ, r0)-minimizer in A. For each x ∈ ∂E ∩ A, and all r ∈ (0, dx) where

dx = min{dist(x, ∂A), r0} (5.62) 128

E satisﬁes the following density bounds:

1 Hn(E ∩ B(x, r)) 1 ≤ ≤ 1 − (5.63) p−2 n n p−2 n n 2p + 1 ωnr n 2p + 1 and n−1 n−1 ∗ ωn H (∂ E ∩ B(x, r)) p−2 c(n, p) ≤ ≤ n · n 2p ωn, (5.64) p−2 n−1 rn−1 n 2p + 1 where c(n, p) is the constant c(n, t) from the relative isoperimetric inequality with t satisﬁes

1 t = p−2 n n 2p + 1

In particular, Hn−1(A ∩ (∂E \ ∂∗E)) = 0.

n By Lemma 118, it follows that if E is a Φp-energy minimizer up to scale r0 in R then

E has a representative in the class A(CA(n, p), r0) where CA(n, p) is the Ahlfors (n − 1)- regularity constant implicit in Lemma 118, and

n E is a set of locally ﬁnite perimiter satisfying ∂E = sptµE and its perimeter A(CA, r0) = E ⊂ R measure |µ | is (n − 1)-Ahlfors regular up to scale r with constant C . (5.65) E 0 A

The following compactness theorem can be found in [15, Theorem 3.6].

Theorem 119. If {Ek} ⊂ A(CA, r0) with 0 ∈ ∂Ek for all k ≥ 1, there exists a subsequence

{Ekj }, a set E of locally ﬁnite perimeter, and a non-negative Radon measure, µ, such that

L1 ( n) E −−−−−→loc R E, µ * µ and |µ | * µ. (5.66) kj Ekj E Ekj

Additionally, ∂E = sptµE and µ is (n − 1)-Ahlfors regular up to scale r0 with constant CA.

Furthermore, |µE| ≤ µ and

1. If x ∈ ∂E, then there exists xkj ∈ ∂Ekj such that xkj → x. 129

2. If x ∈ sptµ, then there exists xkj ∈ ∂Ekj such that xkj → x.

3. If xkj ∈ ∂Ekj and xkj → x then x ∈ sptµ.

Combining Lemma 118 and Theorem 119 we achieve our pre-compactness theorem for

Φp-minimizers up to scale r0.

Theorem 120 (Pre-compactness of Φp-minimizers up to scale r0). If {Eh}h∈N is a sequence of Φp-minimizers up to scale r0 in the open set A, then for every A0 ⊂⊂ A with ﬁnite perimeter, there exists a subsequence Ehk and a set of ﬁnite perimeter E ⊂ A0 and a Radon measure µ supported on A0 that is Ahlfors regular up to scale r0 with constant CA, such that

A0 ∩ Eh → E, µA ∩E * µE, |µE | * µ. k 0 hk hk

Additionally, ∂E = sptµE, |µE| ≤ µ, and

1. If x ∈ ∂E, then there exists xhk ∈ ∂Ehk such that xhk → x.

2. If x ∈ sptµ, then there exists xhk ∈ ∂Ehk such that xhk → x.

3. If xhk ∈ ∂Ehk and xhk → x then x ∈ sptµ.

There are two lingering questions after Theorem 120. First, is the limiting set E a

Φp-minimizer up to scale r0? Second, is µ = |µE|? The Theorem 121 provides a positive answer to both questions. The proof of Theorem 121 follows from small modifications of [49, Theorem 21.14]. See [66, Proposition 4.13] for an example of the necessary modifications, but for a different anisotropic energy.

Theorem 121 (Anisotropic closure theorem). If {Eh}h∈N is a sequence of Φp(· ; A) mini- n mizers for the open set A ⊂ R and if A0 ⊂⊂ A is an open set ﬁnite perimeter such that

A0 ∩ Eh → E for some set of ﬁnite perimeter E, then E is a Φp(· ; A0)-energy minimizer. 130

Moreover,

D|µE |µ ≤ 1 µE a.e. x ∈ A0 and Dµ|µE| ≥ cn,p > 0 µ a.e. x ∈ A0. (5.67)

n In particular, if xh ∈ ∂Eh and xh → x then x ∈ spt|µE|. Moreover, for almost every x ∈ R and almost every s small enough, limh→∞ Φp(Eh; B(x, s)) exists and equals Φp(E; B(x, s)).

Lemma 122. If {Eh} are a sequence of Φp-minimizers and Eh → E, then

Φp(E; B(x, s)) = lim Φp(Eh; B(x, s)) (5.68) h→∞ for almost every x ∈ ∂E and s > 0. The set of s depends on x.

Proof. Choose x ∈ ∂E and s < r so that

n−1 ∗ n−1 ∗ H (∂ E ∩ ∂B(x, s)) = 0 = H (∂ Eh ∩ ∂B(x, s)) for all h ∈ N and n−1 (1) (1) lim inf H ∂B(x, s) ∩ Eh ∆E = 0. h→∞

As before, the set of such s is H1-a.e. s ∈ (0, r) Then deﬁne

Fh = (E ∩ B(x, s)) ∪ (Eh \ B(x, s)) .

We know that Φp(E; B(x, s)) ≤ lim infh Φp(Eh; B(x, s)) from lower-seminconinuity. On the other hand, using the minimizing property of Eh and (5.60) we ﬁnd,

Φp(Eh,B(x, s)) ≤ Φp(Fh,B(x, s)) = Φp(E; B(x, s)) + Φp(B(x, s); E∆Eh)

≤ Φp(E; B(x, s)) + P (B(x, s); E∆Eh). 131

Taking the lim inf on both sides yields

lim inf Φp(Eh; B(x, s)) ≤ Φp(E; B(x, s)). h

Since Eh → E, it follows that we could now repeat the argument with any subsequence

Eh(k) (since all such subsequences converge to E). In particular, if we choose Eh(k) such that limk→∞ Φp(Eh(k); B(x, s)) = lim suph Φp(Eh(k); B(x, s)) then (with help from lower- semicontinuity) we learn that

lim sup Φp(Eh; B(x, s)) = lim inf Φp(Eh(k); B(x, s)) ≤ Φp(Eh; B(x, s)) ≤ lim inf Φp(Eh; B(x, s)) h k h 132

Chapter 6 A HARNACK INEQUALITY FOR ANISOTROPIC PDES 133

6.1 Introduction

While uniformly elliptic partial diﬀerential equations are in general well understood, degener- ate elliptic PDEs arising in the case when the lower ellipticity constant is not bounded away from zero present many challenges. These PDEs are often studied from the perspective that they are close to a uniformly elliptic PDE. In this paper we propose a diﬀerent approach for operators that fail to be uniformly elliptic, but instead are γ-homogeneous. More precisely we focus on operators which arise naturally in the study of anisotropic geometric variational problems. That is, for 1 < γ < ∞,

Z 1,γ0 D(u, ϕ) = 0 ∀ϕ ∈ W0 (Ω), (6.1) Ω

1,γ 1,γ0 1,1 where D : W (Ω) × W0 (Ω) → W (Ω) is deﬁned by

D(u, ϕ) = ρ(x, Du)γ−1(Dρ(x, ·))(Du), Dϕ , (6.2) and ρ(x, ·) is 1-homogeneous and strictly convex in the second variable. In particular, PDEs of the form (6.1) (when there is no dependence of ρ on x) arise as the Euler-Lagrange equation of the functional Z ρ(Du)γ. Ω

For this reason we consider the γ in (6.1) as deﬁning the homogeneity of the PDE. A Moser iteration approach allows us to exploit this γ-homogeneity to prove a Harnack inequality and a strong maximum principle, in addition to a Liouville-type theorem for global solutions. These results arise as corollaries of the theorems described below.

0 1 1 For any real number α > 1 we let α denote the Holder conjugate of α, i.e., α + α0 = 1.

Also, whenever ρ(x, ·) is a 1-homogeneous function ρ∗(x, ·) denotes its convex dual deﬁned by

ρ∗(x, ζ) = supρ(x,ξ)<1 ζ ·ξ. Throughout this paper we consider the following PDE generalizing 134

the PDE in (6.1)

Z Z D E 0 ρ(x, (Du))γ−1(Dρ(x, ·))(Du), Dϕ = F~ , Dϕ + fϕ ∀ϕ ∈ W 1,γ (Ω) (6.3) Ω Ω where q 0 n F~ ∈ Lq (Ω) and f ∈ L γ (Ω) for some q > . (6.4) loc loc γ − 1 Theorem 123. Suppose that for some 1 < γ < n, u ∈ W 1,γ(Ω) is a non-negative subsolution of (6.3), that ρ :Ω × Rn \{0} → (0, ∞) satisﬁes (6.7), (6.8), (6.9), and (6.10), and F~ , f, and q satisfy (6.4). If 0 < r < R < 1 and BR ⊂ Ω, then for all 0 < p < ∞, there exists C = C(n, γ, ν, p) so that

n 1 1 − δ γ−1 γ0δ γ−1 p p ~ sup u ≤ C (R − r) kukL (BR) + R kρ∗(x, F )kLq(B ) + R kfk q (6.5) R γ0 Br L

n where δ = 1 − q(γ−1) > 0.

In the case when γ = 2 = γ0, which includes the elliptic case, (6.5) recovers the local- boundedness results of De Giorgi, Nash, and Moser for elliptic divergence form equations. One can verify this includes the elliptic case with symmetric coeﬃcients by considering ρ(x, ξ) = |S(x)ξ| where S(x) = A(x)1/2, which ensures hA(x)ξ, ξi = |S(x)ξ|2. The next example demonstrates that the γ = 2 case is not restricted to the uniformly elliptic setting.

Example 124. If γ = 2 and ρ = k · k`p for some p > 2, then (6.1) with integrand (6.2) 2 becomes the PDE div (D(k · k`p )(Du)) = 0. A computation yields

" p−2 p−2 p−2 # 2 xi xi|xi| xj|xj| (∂i∂jk · k`p )(x) = 2(p − 1) δij − 2(p−1) . kxkp kxkp

2 2 Notably, (D k · k`p )(e) ≡ 0 whenever e is a standard basis vector.

The fact that Example 124 is handled with the methods in this paper highlights that the homogeneity, not the uniform strong convexity, is what matters for studying the 0th order 135

regularity of this type of PDE. The convexity conditions typically imposed when studying elliptic equations additionally requires the degree of homogeneity is ﬁxed at 2.

Convexity has always played a strong role in the Calculus of Variations. It is known that in certain settings speciﬁc convexity conditions are necessary to guarantee higher-order regularity. A natural question is whether there exist weaker convexity conditions depending upon the homogeneity that are suﬃcient to imply higher-order regularity results. One could hope that teasing apart the roles of homogeneity and convexity might recover simpler conditions for higher-order regularity than presently exist.

Example 125. We revisit the previous example in more generality. Consider ρ(·) = kA(·)k`p for p > 2, where A is an orthogonal matrix. This time, we consider any γ > 2. The PDE under consideration takes the form

γ div(D(kA(·)k`p )(Du)) = 0.

We note that when A = I and γ = p this is precisely the so-called pseudo p-Laplacian or anisotropic p-Laplacian. Our results imply a new Harnack inequality for solutions to the pseudo p-Laplacian.

2 γ By the previous example, whenever A(Du) ∈ {±e1, ±e2,..., ±en} the Hessian, D kA(·)k`p (Du), γ n is zero. Performing a Taylor series expansion of kA(·)k`p around an arbitrary ξ1 ∈ R \{0}, ⊥ you ﬁnd for ξ2 ∈ hξ1i small enough,

    γ γ X p X 2 kA(ξ1 + ξ2)kp − kA(ξ1)kp = O  |ξ2 · ei|  + O  |ξ2 · ei| 

ei:A(ξ1)·ei=0 ei:A(ξ1)·ei6=0 recovering diﬀerent growth conditions in diﬀerent directions. This seems to demonstrate that the gains and losses in energy of functional

Z p kA(Du)k`p (6.6) Ω 136

by small perturbations behaves similarly to a “mixed growth” problem. The Euler-Lagrange equation for minimizers of (6.6) is covered as a special case of our results. Finally, without straying too far from this same family of PDEs, we note that the techniques below even cover the case when ρ(x, ξ) = kA(x)[ξ]k`p(x) so long as p(x) ∈ (1, ∞) and A(x) has reasonably bounded eigenvalues.

The second main result of this paper is for non-negative supersolutions.

Theorem 126. Suppose 1 < γ < n and u ∈ W 1,γ(Ω) is a non-negative supersolution to (6.3), for some ρ satisfying (6.7) - (6.10). Assume F~ , f, and q are as in (6.4). If n(γ−1) 0 < r < R < 1 and BR ⊂ Ω, then for all 0 < p < n−γ and all 0 < θ < τ < 1, there exists C = C(n, γ, ν, Λ, q, p, θ, τ) > 0 so that

0 − n δ ~ γ δ p inf u + R kρ∗(F )kLq(B ) + R kfk q ≥ CR kukLp(B ), R γ0 τR BθR L (BR)

n where δ = 1 − q(γ−1) . In the case γ = 2 = γ0 this recovers Moser’s weak Harnack inequality. Moreover, we want to emphasize that the above theorem does not require any a priori boundedness of u. As in the classical case, and addressed in Section 6.3, these two main theorems can be combined to achieve a strong maximum principle and a Harnack inequality. Hence as corollaries these theorems also prove Cα-regularity of solutions and a Liouville-type theorem for bounded solutions on Rn. In the preparation of this paper the author learned about the very recent work in [12]. After discovering that paper we learned that there is also a wealth of literature about minimizers with Orlicz-type growth conditions. See, for instance, [41, 13, 7], and the citations therein. In the special case where F~ , f = 0 the works of [13, 12] recover some of the results of this paper. Using Orlicz spaces they address additional diﬃculties including equations with diﬀerent upper and lower (almost)-homogeneities of the integrand. In the isotropic setting, [7] shows a Harnack inequality similar to the one in Theorem 139. Meanwhile [71] proves a general Harnack inequality, while assuming that the terms F~ , f are in L∞. 137

Also during preparation of this paper [33] proved a Liouville type theorem when γ = 2 and ρ has strictly positive Hessian in the sense that

2 2 2 ⊥ λ |ζ| ≤ (∂ξi ∂ξj ρ)(ξ)ζiζj ≤ Λ|ζ| ∀ζ ∈ ξ .

Their techniques made use of an interesting monotonicity formula.

6.2 Notation and Preliminaries

Throughout we will suppose ρ :Ω × Rn \{0} → (0, ∞) is so that

 ρ(x, ·) ∈ C1(Rn \{0}) ∀x ∈ Ω (6.7) ρ(·, ξ) ∈ L∞(Ω) ∀ξ ∈ Rn \{0}. and that ρ positively 1-homogeneous function in its second variable, i.e.,

n ρ(x, λξ) = λρ(x, ξ) ∀λ > 0, ∀x ∈ Ω, ∀ξ ∈ R \{0}. (6.8)

We further assume that ρ(x, ·) is strictly convex in the sense that

n ρ(x, ξ1 + ξ2) < ρ(x, ξ1) + ρ(x, ξ2) ∀x ∈ Ω, ∀ξ1 6= λξ2 ∈ R \{0}. (6.9)

Finally, we assume there exists 0 < ν ≤ λ < ∞ independent of x so that

ν ≤ ρ(x, ξ) ≤ Λ ∀|ξ| = 1, ∀x ∈ Ω. (6.10)

We will let ρx denote ρ(x, ·) to simplify notation. Namely, (Dρx)(Du) = (Dρ(x, ·))(Du). We say that u ∈ W 1,γ(Ω) is a subsolution (supersolution) to (6.3) if

Z Z D E γ−1 ~ ρx(Du) (Dρx)(Du), Dϕ ≤ (≥) F , Dϕ + fϕ Ω Ω 138

1,γ0 for all non-negative ϕ ∈ W0 (Ω). We say that u is a solution if it is both a subsolution and supersolution. Given a real number, say α, in (1, ∞) or [1, n), we will respectively always let α0 and α∗ denote the Holder and Sobolev exponents. That is,

1 1 nα + = 1 α∗ = . α α0 n − α

1,γ Theorem 127 (Gagliado-Nirenberg-Sobolev). If u ∈ W0 (Ω) then there exists C = C(n, γ) > 0 so that

kukLγ∗(Ω) ≤ CkDukLγ (Ω)

Theorem 128 (Poincare in a ball). For each 1 ≤ γ < n there exists a C = C(n, γ) so that

1 1 Z γ∗ Z γ −n γ∗ 1− n γ r |f − (f)x,r| dy ≤ C2r γ |Df| dy B(x,r) B(x,r)

Remark 129. Since ρ will always satisﬁes (6.7), (6.8), and (6.9), if it also solves the ﬁrst inequality in (6.10) then the Sobolev embedding theorem can be re-written as

−1 kukLγ∗ (Ω) ≤ Cν kρx(Du)kLγ (Ω).

Similarly Poincare in a ball can be re-written as

1 1 Z γ∗ Z γ −n γ∗ −1 1− n γ r |f − (f)x,r| dy ≤ C2ν r γ ρx(Df) dy . B(x,r)

n n Given ρ :Ω×R \{0} → (0, ∞) as in (6.7), (6.8), (6.9), deﬁne ρ∗ :Ω×R \{0} → (0, ∞) by

∗ ∗ ρ∗(x, ξ ) = sup ξ · ξ . ρ(x,ξ)<1

That is, ρ∗(x, ·) is the convex dual of ρ(x, ·) for all x ∈ Ω. We record for the reader’s convenience a few facts that are frequently used: 139

Proposition 130. Let a, b, c, > 0, α ∈ (1, ∞), ρ as in (6.7), (6.8), and (6.9). Suppose

n ξ1, ξ2 ∈ R \{0}.

Young’s inequality says α α0 b 0 c abc ≤ aα + a−α . α α0

Fenchel’s inequality guarantees

ξ1 · ξ2 ≤ ρ(ξ1)ρ∗(ξ2)

It holds,

ρ∗(x, Dρ(ξ1)) ≡ 1. (6.11)

If ρ satisﬁes (6.8) and (6.9) then so does ρ∗.

If ρ satisﬁes (6.10) then for all |ξ| = 1,

−1 −1 Λ ≤ ρ∗(ξ) ≤ ν .

~ q ~ q In particular, F ∈ L (Ω) if and only if ρ∗(·, F ) ∈ L (Ω)

A warm-up exercise is the following Cacciopolli type inequality.

Lemma 131. If u is a subsolution to (6.3) with F~ , f ≡ 0 and 1 < γ < ∞, then

kηρx(Du)kLγ (Ω) ≤ C(n, γ)kuρx(Dη)kLγ (Ω).

Proof. Consider ϕ = ηγu. Then Dϕ = γηγ−1uDη + ηγDu. So, using the 1-homogeneity of ρ 140

and Fenchel’s inequality,

γ−1 ρx(Du) (Dρx)(Du), Dϕ

γ−1 γ γ ≥ −γ(ηρx(Du)) ρ∗(x, (Dρx)(Du))uρx(Dη) + η ρx(Du) .

Using u is a subsolution with F~ , f ≡ 0, and (6.11) yields

Z Z γ γ γ−1 η ρx(Du) ≤ γ (ηρx(Du)) uρx(Dη) Ω Ω 1− 1 1 Z γ Z γ ≤ (ηρ(Du))γ uγρ(Dη)γ . Ω

Dividing completes the proof.

Finally, we recall a technical lemma for later use.

Lemma 132. Let ω, σ be non-decreasing functions in (0,R]. Suppose there exists 0 < τ, δ˜ < 1 so that for all r ≤ R, ω(τr) ≤ δω˜ (r) + σ(r).

Then for any µ ∈ (0, 1) and r ≤ R

n r α o ω(r) ≤ C ω(R) + σ(rµR1−µ) R where C = C(δ,˜ τ) and α = α(δ,˜ τ, µ).

6.3 Main results

6.3.1 Fundamental Solutions

We start with discussing fundamental solutions of the PDE

div ρ(Du)γ−1(Dρ)(du) = 0. 141

Proposition 133. If ρ ∈ C1(Rn \{0}) satisﬁes

1. ρ(λξ) = λρ(ξ) for all λ > 0 and ξ ∈ Rn \{0}

2. ρ(ξ1 + ξ2) = ρ(ξ1) + ρ(ξ2) ⇐⇒ ξ1 = λξ2 for some λ > 0 and c = R |Dρ (x)|−1dHn−1(x), then ρ ρ∗=1 ∗

  γ−1 γ−n  γ−n ρ∗(x) γ − 1 γ 6= n cρv(x) = (6.12) ln(ρ∗(x)) γ = n is a fundamental solution of

Z γ−1 1,γ0 n ρ(Dv) (Dρ)(Dv), Dϕ = 0 ∀ϕ ∈ W0 (R \{0}), (6.13) Rn\0 in the sense that it is both a strong solution, and

Z p.v. ϕ div ρ(Dv)γ−1(Dρ)(Dv) = ϕ(0). Rn

Strict convexity is necessary for the diﬀerentiability of ρ∗. Note also, that we do not know that v ∈ C2 even though it is known to be a strong solution to a 2nd order PDE.

Proof. To see that v is a strong solution, it suﬃces to show in each case that

γ−1 1−γ −n ρ(Dv) (Dρ)(Dv) = cρ ρ∗(x) x.

Indeed, −n −n −n−1 div ρ∗(x) x = tr ρ∗(x) Id − nρ∗(x) x · (Dρ∗)(x) = 0. 142

When γ 6= n we compute:

1−n −1 γ−1 Dv = cρ ρ∗(x) (Dρ∗)(x)

γ−1 1−γ 1−n ρ(Dv) = cρ ρ∗(x) x (Dρ)(Dv) = ρ∗(x) which implies

γ−1 1−γ −n ρ(Dv) (Dρ)(Dv) = cρ ρ∗(x) x.

Now, if γ = n we compute:

Dρ (x) Dv = ∗ ρ∗(x) n−1 1−n −(n−1) ρ(Dv) = cρ ρ∗(x) x (Dρ)(Dv) = , ρ∗(x) again guaranteeing

n−1 1−γ −n ρ(Dv) (Dρ)(Dv) = cρ ρ∗(x) x as desired.

γ−1 −n Next, using the fact that in either case, ρ(Dv) (Dρ)(Dv) = xρ∗(x) , we observe that 1 n for ϕ ∈ Cc (R )

Z Z γ−1 n −n −n n p.v. ϕ div ρ(Dv) (Dρ)(Dv) dH = p.v. div ϕxρ∗(x) + xρ∗(x) , Dϕ dH n n R Z RZ −n n ϕ n−1 = lim div ϕxρ∗(x) dH = lim n−1 dH ↓0 n ↓0 |Dρ (x)| R \{ρ∗≤} {ρ∗=} ∗ Z ϕ(x) Z = lim dHn−1 = lim − ϕ(x)dν, ↓0 ↓0 {ρ∗=1} |Dρ∗(x)| ρ∗=1

n−1 where ν = H {ρ∗ = 1}. By continuity of ϕ, the result follows. 143

6.3.2 Regularity

In this section we focus on functions u that solve

Z Z D E 0 γ−1 ~ 1,γ ρx(Du) (Dρx)(Du), Dϕ dx = F , Dϕ + fϕ ∀ϕ ∈ W0 (Ω). (6.14) Ω Ω

As a corollary of these results, we can answer further questions about functions u that solve

Z γ−1 ρx(Du) (Dρx)(Du), Dϕ dx = 0. (6.15) Ω

We begin with a Caccioppoli inequality when F~ , f 6≡ 0.

Theorem 134. Suppose u ∈ W 1,γ(Ω) is a subsolution of (6.14) with 1 < γ < ∞ and

ρ :Ω × Rn \{0} satisﬁes (6.7), (6.8), (6.9), and (6.10). Assume F~ , f ∈ Lγ˜(Ω) where 0 γ˜ = max{γ, γ }. If B2R ⊂ Ω and 0 < R ≤ 10 then,

1 1 1 −1 γ−1 γ−1 γ γ ~ γ−1 kρx(Du)kL (BR) ≤ cγ,Λ R kukL (B2R) + kρ∗(x, F )k γ0 + R kfk γ0 . (6.16) L (B2R) L (B2R)

Remark 135. Note, it is necessary for F~ , f ∈ Lγ (resp., F~ , f ∈ Lγ0 ) for the equation (resp., conclusion) to make sense1.

Proof. Suppose without loss of generality, R = 1. Consider the test function ϕ = ηγu for

1 some η ∈ C0 (B2) to be chosen later. Note,

Dϕ = ηγDu + γuηγ−1Dη.

Taking advantage of the 1-homogeneity of ρx, Fenchel’s inequality, Young’s inequality, and

0 ( γ n )0 0 1Technically, by Sobolev embedding we only need f ∈ L n−γ0 ∩ Lγ . 144

γ0 1 γ (6.11) we choose > 0 so that (γ − 1) = 2 ρ(ηDu) and compute

γ−1 γ γ−1 hρx(Du) (Dρx)(Du), Dϕi ≥ ρx(ηDu) − γρx(ηDu) ρ∗(x, (Dρ)(Du))ρx(uDη) γ0 ρ (ηDu)γ ρ (uDη)γ ≥ ρ (ηDu)γ − γ x + x x γ0 γγ ρ (ηDu)γ ≥ x − c ρ (uDη)γ. (6.17) 2 γ x

On the other hand, for > 0 chosen so that γ−1γ = 1/4, Fenchel’s and Young’s inequalities ensure

~ ~ γ−1 γ ~ hF , Dϕi ≤ γ ρ∗(x, F )η ρx(uDη) + η ρ∗(F )ρ(Du)

" 0 # " 0 # ρ (x, F~ )γ ηγ ρ (uDη)γ ρ (x, F~ )γ ρ (Du)γ ≤ γ ∗ + x + ηγ ∗ + γ x γ0 γ γ0 γ0 γ γ ρ (ηDu) 0 = x + c ρ (x, F~ )γ ηγ + ρ (uDη)γ. (6.18) 4 γ ∗ x

Combining (6.17), (6.18), and (6.14) yields,

Z Z Z Z γ ~ γ0 γ γ γ ρx(ηDu) ≤ cγ ρ∗(x, F ) η + ρx(uDη) + fη u Z Z Z ~ γ0 γ γ γ γ γ γ0 ≤ cγ ρ∗(x, F ) η + u (η + ρx(Dη) ) + η f . (6.19)

c Choosing 0 ≤ η ≤ 1, η ≡ 1 on B1, η ≡ 0 on B2 and |Dη| ≤ 2 we ﬁnd

h γ0−1 γ0−1 i γ γ ~ kρx(Du)kL (BR) ≤ cγ,Λ, kukL (B2R) + kρ∗(x, F )k γ0 + kfk γ0 . L (B2R) L (B2R)

Equation (6.16) is recovered by scaling.

We note that in Theorem 134, the fact that ρ can depend on x never needs to be dealt with separately. This is unsurprising because conditions (6.8), (6.9), and Fenchel’s inequality are used at a pointwise level while (6.10) is used at a global level, see Remark 129. Hence, 145

to simplify notation, we only explicitly write-out the dependence of ρ on x in the statements of theorems and suppress this dependence throughout all remaining proofs.

Theorem 136. Suppose ρ :Ω × Rn \{0} satisﬁes (6.7), (6.8), (6.9), and (6.10). Let u be a subsolution to (6.14) and ﬁx 1 < γ < n. If F~ , f and q are as in (6.4), 0 < r < R < 1, and

n BR ⊂ Ω then there exists some C = C(n, ν, Λ, γ, q, p) and δ = 1 − q(γ−1) > 0 so that

+ 1 1 ku k p 0 + L (BR) δ ~ γ−1 γ δ γ−1 sup u ≤ C n + R kρ∗(x, F )kLq(B ) + R kfk q p R γ0 Br (R − r) L (BR)

Proof. We consider the test function ϕ = ηγvβu¯ for β ≥ 0 whereu ¯ = u+ + k and v = min{u, m} for some 0 < k < m < ∞, k to be chosen later. Notice

Dϕ = γηγ−1vβuDη¯ + βηγvβUDv¯ + ηγvβDu.¯

We wish to expand out (6.14) with this choice of ϕ. To this end, ﬁrst observe 1- homogeneity, i.e., (6.8) ensures

ρ(Du)γ−1(Dρ)(Du),βvβ−1uη¯ γDv + ηγvβDu¯

= βρ(Dv)γvβηγ + ρ(Du¯)γηγvβ. (6.20)

Next we apply Fenchel’s and Young’s inequalities in combination with (6.11) for some = (γ) > 0 to be chosen immediately after (6.21),

ρ(Du)γ−1(Dρ)(Du),γηγ−1vβuDη¯

β γ−1 ≥ −γv (ρ(Du¯)η)) ρ∗(Dρ(Du)) (ρ(Dη)¯u) γ γ γ γ 0 ρ(Du¯) η ρ(Dη) u¯ ≥ −γvβ γ + . (6.21) γ0 γγ 146

γ0 γ Choose so that (γ − 1) = 1/2. Since γ0 = γ − 1, this choice of ensures (6.21) becomes

vβρ(Du¯)γηγ ρ(Du)γ−1(Dρ)(Du), γηγ−1vβuDη¯ ≥ − − c vβρ(Dη)γu¯γ, (6.22) 2 γ

where cγ may change depending on the line, but depends only on γ. Now we look at the righthand side. We split this into two pieces and treat the ﬁrst piece in much the same fashion as above.

~ β−1 γ γ β ~ β γ ~ γ β F ,βv uη¯ Dv + η v Du¯ ≤ ρ∗(F ) βv η ρ(Dv) + ρ∗(F ) η v ρ(Du¯) h i γ β ~ ~ = (η v ) (β) ρ∗(F ) (ρ(Dv)) + ρ∗(F )ρ(Du¯)

" 0 ! 0 !# ρ (F~ )γ ρ(Dv)γ ρ (F~ )γ ρ(Du¯)γ ≤ ηγvβ β ∗ + βγ + ∗ + γ γ0 0 1 γ0 0 2 1 γ γ 2 γ γ γ γ γ β ~ γ0 β 1 β1 γ β γ 2 γ β γ = η v ρ∗(F ) 0 + 0 + η v ρ(Dv) + η v ρ(Du¯) . (6.23) γ 0 γ 0 1 γ 2 γ γ 2

Since we want to absorb the last two terms of (6.23) into (6.20) we choose 1, 2 so that −1 γ β −1 γ 1 γ β1 = 2 and γ 2 = 4 . The need for choosing 1/4 for the 2 coeﬃcient is due to the fact that we’ll also be absorbing the ρ(Du¯)-term from (6.22) into (6.20). We note both 1 and 2 depend solely on γ. All-in-all this allows us to re-write (6.23) as

D E 0 β F~ , βvβ−1uη¯ γDv + ηγvβDu¯ ≤ c (1 + β)ηγvβρ (F~ )γ + ηγvβρ(Dv)γ (6.24) γ ∗ 2 1 + ηγvβρ(Du¯)γ. 4

We now deal with the ﬁnal term via Fenchel and Young’s inequalities

D E ~ γ−1 β ~ γ−1 β F , γη v uDη¯ ≤ γρ∗(F )η v uρ¯ (Dη)

" 0 # ρ (F~ )γ ηγ u¯γρ(Dη)γ ≤ γvβ ∗ + γ0 γ

~ γ0 γ β γ γ β = (γ − 1)ρ∗(F ) η v +u ¯ ρ(Dη) v . (6.25) 147

Finally, plugging (6.20), (6.22), (6.24), and (6.25) into (6.14) yields

β Z 1 Z ρ(Dv)γvβηγ + ρ(Du¯)γηγvβ 2 Ω 4 Ω Z Z Z β γ γ γ β ~ γ0 γ β ≤ cγ v ρ(Dη) u¯ + η v ρ∗(F ) + fη v u¯ Ω Ω Ω Z Z γ Z β γ γ γ β u¯ ~ γ0 f γ β γ ≤ cγ v ρ(Dη) u¯ + (1 + β) η v γ ρ∗(F ) + γ−1 η v u¯ . (6.26) Ω Ω k Ω k

The ﬁnal inequality usedu ¯ ≥ k. Set w = vβ/γu¯, we note

β β −1 β β β β Dw = v γ uDv¯ + v γ Du¯ = v γ Dv + v γ Du.¯ γ γ

Since ρ(ξ1 + ξ2) ≤ ρ(ξ1) + ρ(ξ2) for all ξ1, ξ2 it follows that

γ γ γ γ β β β η ρ(Dw) ≤ η v γ ρ(Dv) + v γ ρ(Du¯) γ β γ ≤ 2γ−1 ηγvβρ(Dv)γ + ηγvβρ(Du¯)γ . γ

In particular, this guarantees that for some cγ

Z Z Z γ γ γ−1 β γ β γ 1 γ γ β η ρ(Dw) ≤ cγ(1 + β ) ρ(Dv) v η + ρ(Du¯) η v . (6.27) Ω 2 Ω 4 Ω

Combining (6.26) and (6.27) yields

Z "Z Z ~ γ0 !# γ γ γ γ γ γ ρ∗(F ) f η ρ(Dw) ≤ cγ(1 + β ) w ρ(Dη) + (ηw) γ + γ−1 . (6.28) Ω Ω k k

Due to the observation that ρ(D(ηw))γ ≤ 2γ−1(wγρ(Dη)γ + ηγρ(Dw)γ), (6.28) implies

Z "Z Z ~ γ0 !# γ γ γ γ γ ρ∗(F ) f ρ(D(ηw)) ≤ cγ(1 + β ) w ρ(Dη) + (ηw) γ + γ−1 . (6.29) Ω Ω Ω k k 148

~ γ0 R γ ρ∗(F ) Our next goal is to deal with the term (ηw) kγ . We roughly explain in words how we do this. We ﬁrst use Holder’s inequality to make the Lq norm of F~ appear. To this end,

q 0 q we will introduce the parameter α1 = α(q, γ) = q−γ0 which is equivalent to α1 = γ0 . This is q precisely where the requirement f ∈ L γ0 comes from.

Next, by choosing k appropriately, we will make the term with ρ(F~ ) + f be absorbed into a 1. At this point, the remaining term with ηw will have a strange power. So, we use nγ interpolation, and the fact that γ < γα1 < n−γ , to re-write our strange power as a linear γ γ∗ combination of the L and L norms of ηw. The necessary upper-bound on α1 is satisﬁed n so long as q > γ−1 .

By making the coeﬃcient of the Lγ norm of ηw larger, we can choose the Lγ∗ norm of ηw to be arbitrarily small. This is necessary, as we will apply the Gagliardo-Nirenberg-Sobolev inequality to turn this latter norm into an estimate on the Lγ norm of ρ(D(ηw)) which we can ﬁnally absorb into the left hand side. When we apply Young’s inequality in order to make this

−1 weighted-linear combination of norms appear, we will choose α2 = α(n, q, γ) = θ1 , where

θ1 is the interpolation power. This conveniently makes all exponents outside of integrals disappear, allowing the desired simpliﬁcations to all occur.

We begin the process outlined above by applying Holder’s inequality,

Z ~ γ0 ! ~ γ0 γ ρ∗(F ) f γ ρ∗(F ) f (ηw) + ≤ k(ηω) k α1 + γ γ−1 L (Ω) γ γ−1 Ω k k k k 0 Lα1 (Ω) γ ≤ C(γ, q)kηwkLγα1 (Ω) (6.30)

0 q γ ~ γ0 where α1 = γ0 is as above, and k is chosen so that k = k1 + k2 where k1 = kρ∗(F )kLq(Ω) γ−1 ~ and k2 = kfk q . If F , f ≡ 0 choose k > 0 arbitrary, and you can later send k to L γ0 (Ω) 1 θ1 (1−θ1)(n−γ) zero. Next, we deﬁne θ1 = θ1(q, n, γ) ∈ (0, 1) so that = + . Note that if γα1 γ nγ −1 0 −1 α2 = α(q, n, γ) = θ1 then α2 = (1 − θ1) . Riesz-Thorin interpolation applied to (6.30), 149

Young’s inequality, and the Gagliardo-Nirenberg-Sobolev inequality consecutively ensure

0 Z ~ γ γ γ ρ∗(F ) θ1 1−θ (ηw) γ ≤ kηwkLγ (Ω)kηwkLγ∗ (Ω) Ω k 0 α2 0 1 γθ1α2 γ(1−θ1)α2 ≤ kηwk γ + kηwk γ∗ α2 L (Ω) 0 L (Ω) α2 α2 α0 1 γ 2 γ = kηwk γ + kηwk γ∗ α2 L (Ω) 0 L (Ω) α2 α2 0 −α2 −1 γ α2 γ ≤ α2 kηwkLγ (Ω) + C(q, n, γ, ρ) kρ(D(ηw))kLγ (Ω). (6.31)

See Remark 129 for our non-standard application of Gagliardo-Nirenberg-Sobolev inequality. Next, we want to plug (6.31) into (6.29) and subtract over the ρ(D(ηw)) term, so we γ choose so that the coeﬃcient of kρ(D(ηw))kLγ (Ω) is 1/2. That is, choose = (q, n, γ, ρ) > 0 by

γ α0 1 c (1 + β )C(q, n, γ, ν) 2 = . γ 2 Then using our choice of and plugging (6.31) into (6.29) yields

Z Z Z γ γ γ γ γ α2−1 γ γ ρ(D(ηw)) ≤ cγ(1 + β ) w ρ(Dη) + Cq,n,γ,ν(1 + β ) w η Ω Ω Ω Z γ α2 γ γ γ ≤ Cq,n,γ,ν(1 + β ) w (ρ(Dη) + η ) , (6.32) Ω where as always, α2 = α(n, q, γ) > 0. Finally, the Gagliardo-Nirenberg-Sobolev inequality applied to (6.32) implies

1 α Z γ γ 2 γ γ γ kηwkγχ ≤ Cq,n,γ,ν(1 + β ) γ w (ρ(Dη) + η ) , (6.33) Ω

n where χ = n−γ . Choose the cut-oﬀ function η so that with 0 < r < R < 1 and some 1 BR(x) ⊂⊂ Ω, η ∈ C0 (BR) and

2 η ≡ 1 in B and |Dη| ≤ . (6.34) r R − r 150

Then (6.33) guarantees

1 Z χ γ−1 α2 Z γχ (1 + β ) γ w ≤ Cn,γ,q,ν,Λ γ w Br(x) (R − r) BR where our constant gained a dependence on Λ, which arises by applying (6.34) in the form ρ(Dη) ≤ Λ|Dη| ≤ 2Λ(R − r)−1. Recall the deﬁnition of w and use v ≤ u¯ to discover

1 Z χ γ−1 α2 Z (β+γ)χ (1 + β ) β+γ v ≤ Cn,γ,q,ν,Λ γ u¯ . (6.35) Br(x) (R − r) BR

Taking m ↑ ∞ in (6.35) yields

1 (1 + βγ−1)α2 β+γ ku¯k (β+γ)χ ≤ C ku¯k β+γ , (6.36) L Br(x) n,γ,q,ν,Λ (R − r)γ L (BR(x)) so long as the right hand side is ﬁnite. Note the constant on the righthand side is independent of β, suggesting we iterate. We begin with β = β0 = 0 and for k = 1, 2, 3 ... deﬁne βk = k−1 R−r k γ(χ −1). For k = 0, 1, 2,... we consider rk = r+ 2k+1 . Note, (βk +γ)χ = γχ = (βk+1 +γ) −(k+1) and rk−1 − rk = 2 (R − r).

Writing C = Cn,γ,q,ν,Λ, for k ≥ 0 this choice of βk, rk (6.36) reads

1 γ−1 (1 + β )α2 βk+γ k+1 k γχk−1 ku¯k βk+γ ≤ C γ 2 ku¯k βk−1+γ . L (Brk (x)) (R − r) L (Brk−1 (x))

k−1 Recalling βk = γ(χ − 1) we observe that for k ≥ 2,

1 γ−1 α2 βk+γ k(γ−1)α (1 + β ) −χk 2 k ≤ (R − r) (C(1 + γχ)) γχk−1−1 (R − r)γ k −χk ≤ (R − r) C χk . 151

Consequently, after iterating out the ﬁrst few cases by hand, for all k,

P∞ i P∞ i − i=0 χ i=0 χi−1 ku¯k β +γ ≤ (R − r) C ku¯k β0+γ , k L (Br1 (x)) L (Brk (x))

1,γ where still C = C(n, γ, q, ν, Λ). Since u ∈ W (Ω) and β0 = 0, the right hand side is seen to be ﬁnite, and is independent of k. Taking k → ∞ on the left hand side, and recalling u¯ = u+ + k yields + − n + ∞ γ γ ku kL (Br(x)) ≤ C(R − r) ku kL (BR) + k .

γ0 1 ~ γ γ−1 1 n Recalling k = kρ∗(F )kq + kfk q and noting 1−χ−1 = γ yields the result for the case p ≥ γ. γ0 A classical scaling argument covers the case 0 < p < γ. See, for instance, [40, p. 75].

Theorem 137. Let ρ :Ω × Rn \{0} satisfy (6.7), (6.8), (6.9), and (6.10). Suppose u ∈ 1,γ ~ W (Ω) is a supersolution to (6.14) in Ω and F , f, and q are as in (6.4). If B2R ⊂ Ω and (γ−1)n 0 < p < n−γ , then for any 0 < θ < τ < 1 there exists C = C(n, γ, q, p, ν, Λ, θ, τ) and n δ = 1 − q(γ−1) > 0 so that

1 1 n + δ γ−1 γ0δ γ−1 − + ~ p p inf u + R kρ∗(F )kLq(B ) + R kfk q ≥ CR ku kL (BτR) R γ0 BθR L (BR)

and the dependence on ρ only appears as a dependence on sup|ξ|=1 ρ(ξ) and inf|ξ|=1 ρ(ξ).

Two corollaries of Theorem 136 and Theorem 137 are the strong maximum principle and the weak-Harnack inequality (Theorems 138 and 139), which we will state and prove before proving Theorem 139.

Theorem 138. Let ρ :Ω × Rn \{0} satisfy (6.7), (6.8), (6.9), and (6.10). Suppose u ∈ W 1,γ(Ω) is a super solution to (6.14) in Ω, and F~ , f ≡ 0. Then, if for some ball B ⊂⊂ Ω

sup u = sup u ≥ 0, B Ω then the function u must be constant in Ω. 152

0 Proof. Suppose B ⊂⊂ Ω is so that supB0 u = supΩ u. If necessary there is a much smaller ball B ⊂⊂ Ω so that sup u = sup u. Without loss of generality B ⊂⊂ Ω. Let R BR/3 Ω 2R

M = supΩ u and applying Theorem 139 to the non-negative supersolution M − u it follows that

−n 1 R kM − ukL (B2R/3) ≤ C inf (M − u) = 0. BR/3

Hence, u ≡ M on B2R/3, which implies the theorem.

Theorem 139. Suppose ρ :Ω × Rn \{0} → (0, ∞) satisﬁes (6.7), (6.8), (6.9), and (6.10). If 1 < γ < n and u ∈ W 1,γ(Ω) is a nonnegative solution to (6.14) in Ω for some F~ , f, and q are as in (6.4) and some B3R ⊂ Ω, then there exists some C = C(n, γ, q, ν, Λ) and n δ = 1 − q(γ−1) > 0 so that

1 1 δ ~ γ−1 γ0δ γ−1 sup u ≤ Cn,γ,ν,Λ,q inf u + R kρ∗(F )kLq(B ) + R kfk q . (6.37) B 3R γ0 BR 2R L (B3R)

The preceding Theorem follows readily from Theorem 136 and Theorem 137 by choosing, (γ−1)n ~ for instance, p = 2(n−γ) in both theorems. In the case that F , f ≡ 0 this recovers an inequality of the same form as the classical Harnack inequality.

Proof. (Of Theorem 137.) Let k > 0 to be chosen later,u ¯ = u+ + k , v = min{u,¯ m}. Consider the test function

γ −β p n ϕ = η v u¯ for β ≥ βp > 1, where βp = γ − χ , χ = n−γ . We now proceed in a similar fashion to the proof of Theorem 136. Observe,

ρ(Du)γ−1(Dρ)(Du), − βv−β−1uη¯ γDv + ηγv−βDu¯

= −βρ(Dv)γv−βηγ + ρ(Du¯)γηγv−β

= (1 − β)ρ(Dv)γv−βηγ. (6.38)

We note that (β −1) ≥ 1−βp; this observation will simplify our computations later as, when we iterate β, we may now keep constants independent of β. They will instead depend on βp, 153

which depends on n, γ, p.

Analogous to (6.21) we compute

γ γ γ γ 0 ρ(Du¯) η ρ(Dη) u¯ ρ(Du)γ−1(Dρ)(Du), γηγ−1v−βuDη¯ ≤ γv−β γ + . (6.39) γ0 γγ

Choosing so that (γ − 1)γ0 = −(1 − β)/2 > (1 − γ)/2 > 0 yields

ρ(Du)γ−1(Dρ)(Du),γηγ−1v−βuDη¯

−(1 − β) ≤ v−βηγρ(Dv)γ + c v−βρ(Dη)γu¯γ. (6.40) 2 n,γ,p

Next we treat the F~ term. Proceeding as in (6.23) we discover

h i ~ −β−1 γ γ β γ −β ~ F, − βv uη¯ Dv + η v Du¯ ≥ −(β − 1)η v ρ∗(F ) (ρ(Dv))

" 0 # ηγv−βρ (F~ )γ ηγv−βρ(Dv)γ ≥ (1 − β) ∗ + γ γ0 γ0 γ

(1 − β) 0 ≥ ηγv−βρ(Dv)γ − c ηγv−βρ (F~ )γ (6.41) 4 n,γ,p ∗ where we chose > 0 so that γ−1γ = 1/4. To bound the ﬁnal piece, we compute

D E ~ γ−1 −β ~ γ−1 −β F , γη v uDη¯ ≥ −γρ∗(F )η v uρ¯ (Dη)

" 0 # ρ (F~ )γ ηγ u¯γρ(Dη)γ ≥ −γv−β ∗ + γ0 γ

~ γ0 γ −β γ γ −β = −(γ − 1)ρ∗(F ) η v − u¯ ρ(Dη) v . (6.42)

Using that u is a supersolution and plugging (6.38), (6.40), (6.41), and (6.42) into (6.14) 154

achieves

−(β − 1) Z ρ(Dv)γv−βηγ 4 Ω Z Z Z −β γ γ γ −β ~ γ0 γ −β+1 ≥ −cn,γ,p v ρ(Dη) u¯ − cn,γ,p η v ρ∗(F ) − η v f Ω Ω Ω or after recallingu ¯ ≥ k, β − 1 ≥ βp − 1 > 0 and multiplying by −1,

Z Z γ −β γ −β γ γ ρ(Dv) v η ≤ cn,γ,p v u¯ ρ(Dη) + Ω Ω Z ~ γ0 Z γ −β γ ρ∗(F ) γ −β γ f + η v u¯ γ + η v u¯ γ−1 . (6.43) Ω k Ω k

Note the striking similarity to (6.26). The only difference being the benefit that (6.43) has no constants depending on β. Hence, we follow the process done in the proof of Theorem 136 and choose 1 1 ~ γ−1 γ−1 k = kρ∗(F )kLq + kfk q . L γ0 Recalling 0 < θ < τ < 1, setting w = v−β/γu¯ and proceeding as in the proof of Theorem 136 leads to 1 Z χ Z (γ−β)χ γ−β u¯ ≤ Cn,γ,ν,Λ,p,q,Rθ,Rτ u¯ . (6.44) BRθ BRτ The dependence on Rθ, Rτ comes from the magnitude of the derivative of the cut-off function. Moreover, in (6.44), the dependence on p that is not present in (6.33) is due to having assumed

β ≥ βp. To improve readability, we write C = Cn,γ,ν,Λ,p,q,Rθ,Rτ and suppose R = 1 through the end of (6.50). Now, whenever γ − β < 0 (6.44) ensures

γ−β (γ−β)χ Cku¯kL (Bτ ) ≤ ku¯kL (Bθ). (6.45)

+ Since inf v = limp→−∞ kvkLp , iterating as in Theorem 136, for any β > γ (6.45) implies,

γ−β Cku¯kL (Bτ ) ≤ inf u.¯ (6.46) Bθ 155

On the other hand, whenever 0 < β < γ − 1, (6.44) guarantees

(γ−β)χ γ−β ku¯kL (Bθ) ≤ Cku¯kL (Bτ ). (6.47)

Recalling −β < −1, we can iterate (6.47) ﬁnitely many times depending on p, p0 so that when 0 < p0 < p < (γ − 1)χ, choosing γ − β = p0 and ﬁnitely many iterations of (6.47) ensures

p p ku¯kL (Bθ) ≤ C · C(p0)ku¯kL 0 (Bτ ). (6.48)

We claim the proof is complete once we show that there exists p0 > 0 so that

Z Z u¯−p0 u¯p0 ≤ C. (6.49) Bτ Bτ

Indeed, choosing β = γ + p0 in (6.46) combined with (6.48) and (6.49) yields

inf u¯ ≥ C · C(p0)ku¯kL−p0 (6.50) B1 −1 p = C · C(p0) ku¯kL−p0 ku¯kLp0 ku¯kL 0

≥ C · C(p0)ku¯kLp .

It happens that p0 = p0(n, γ, ν, Λ) so that the constant C(p0) can be absorbed into our universal constant. Recalling thatu ¯ = u+ + k and using scaling, this says

1 1 + δ γ−1 γ0δ γ−1 + ~ p inf u + R kρ∗(F )kLq(B ) + R kfk q ≥ Cn,γ,ν,Λ,q,p,θ,τ ku kL (BRτ ), R γ0 BRθ L (BR) as desired.

ηγ Hence, it only remains to show (6.49). We consider the test function ϕ = u¯γ−1 . Note,

η γ−1 η γ Dϕ = γ Dη − (γ − 1) Du.¯ u¯ u¯ 156

We estimate as usual:

D η γ E ηDu¯γ ρ(Du¯)γ−1(Dρ)(Du¯), −(γ − 1) Du¯ = −(γ − 1)ρ . (6.51) u¯ u¯ and choosing > 0 so that (γ − 1)γ0 = (γ − 1)/2.

0 η γ−1 γ ηDu¯γ ρ(Dη)γ ρ(Du¯)γ−1(Dρ)(Du¯), γ Dη ≤ γ ρ + u¯ γ0 u¯ γγ γ − 1 ηDu¯γ ≤ ρ + c ρ(Dη)γ. (6.52) 2 u¯ γ

On the other hand, sinceu ¯ ≥ k

η γ−1 0 η γ−1 F~ , γ Dη ≥ −γρ (F~ )γ ρ(Dη) u¯ ∗ u¯ " 0 # ρ (F~ )γ ηγ ρ(Dη)γ ≥ −γ ∗ + (6.53) k−γγ0 γ and choosing > 0 so that (γ0)−1γ = 1/4 guarantees

" 0 # η γ η γ γρ(Du¯)γ ρ (F~ )γ hF,~ −(γ − 1) Du¯i ≥ −(γ − 1) + ∗ u¯ u¯ γ γ0 γ0 γ (γ − 1) ηDu¯ 0 ≥ − ρ − c ηγk−γρ (F~ )γ . (6.54) 4 u¯ γ ∗

Finally, note

Z γ −(γ−1) −(γ−1) γ −(γ−1) γ n n fη u¯ ≥ −k kfk kηk nγ ≥ −cγ,n,νk kfk kDηkLγ (6.55) γ L n−γ γ 157

Combining (6.51) - (6.55), with the fact that u is a supersolution yields

" ! 0 # Z Z kfk n Z ~ γ γ γ γ γ ρ∗(F ) γ − η ρ (D(logu ¯)) ≥ −cn,γ,ν ρ(Dη) 1 + γ−1 + γ η Ω k k

Z 0 γ −(γ−1) −γ ~ γ ≥ −cn,γ,ν ρ(Dη) 1 + k kfk n + k kρ∗(F )k n , (6.56) γ L γ−1 where the 2nd inequality follows from the ﬁrst by Holder and Sobolev embedding similar to 1 1 γ−1 ~ γ−1 (6.55). Choosing k = kfk n + kρ∗(F )k γ0 this can be written γ L

Z Z γ γ γ η ρ(D(logu ¯)) ≤ cn,γ,ν ρ(Dη) . (6.57) Ω

0 0 0 2 Now, for all B2r ⊂ B2R, we can choosing η ≡ 1 on Br and η ≡ 0 on Ω \ B2r with |Dη| ≤ r , (6.57) says

Z Z γ γ n−γ |D(logu ¯)| ≤ cn,γ,ν ρ(D(logu ¯)) ≤ cn,γ,ν,Λr 0 0 Br Br

1− n or taking the γ-root and multiplying by r γ that is

1 Z γ 1− n γ r γ |D(logu ¯)| ≤ Cn,γ,ν,Λ. (6.58) 0 Br

1 1− n Noticing (r−n) γ∗ = r γ , we consecutively apply Jensen’s inequality, the Poincare in- quality in a ball, and (6.58) to achieve

1 Z Z γ∗ 1 1 γ∗ | logu ¯ − (logu ¯) 0 | ≤ C | logu ¯ − (logu ¯) 0 | (6.59) n Br n Br r 0 r 0 Br Br

≤ Cn,γ,ν,Λ.

0 0 Since this holds uniformly for all B2r ⊂ B2R, and hence (6.59) holds for all Br ⊂ BR, this ensures logu ¯ ∈ BMO(BR) and consequently, by the John-Nirenberg lemma there exists 158

0 < p0 depending only on n and the constant in (6.59) so that

1 Z sup ep0| logu ¯−(logu ¯)B | < ∞. (6.60) B⊂B3 |B| B

|a−b| p1|a−b| p0|a−b| Finally, since e ≥ 1 we note that e ≤ e if p1 ≤ p0. In particular, without loss (γ−1)n of generality, suppose p0 < n−γ . But then, (6.60) implies (6.49).

Theorem 140 (Improvement of Oscillation). Suppose u, ρ, γ, F~ , f, and q are as in Theorem

139. If B3R ⊂ Ω, then for all 0 < θ < 1,

1 1 α ~ γ−1 γ−1 oscBθR u ≤ Cn,ρ,γ,qθ oscBR u + kρ∗(F )kLq(B ) + kfk q . 2R γ0 L (B2R)

Proof. For 0 < s < 2R let

Ms = sup u and ms = inf u. Bs Bs

Then v ∈ {M2R − u, u − m2r} is a positive solution of (6.14) on B2R. Consider p = 1 and θ = 1/2 < τ < 1 in Theorem 137. That is, for a positive solution v,

Z inf v + G(R) ≥ CR−n v (6.61) BR/2 where 1 1 δ ~ γ−1 γ0δ γ−1 G(R) = R kρ∗(F )kLq(B ) + R kfk q . R γ0 L (B2R)

Note, infBR/2 M2R − u = M2R − MR/2 and infBR/2 u − m2R = mr/2 − m2r. Therefore, applying

(6.61) to M2R − u and u − m2R yields

Z −n M2R − MR/2 + G(R) ≥ CR M2R − u Z −n mR/2 − m2R + G(R) ≥ CR u − m2R. 159

Adding yields

(M2R − m2R) − (MR/2 − mR/2) + 2G(R) ≥ C(M2R − m2R), or equivalently,

1 1 δ γ−1 γ0δ γ−1 ~ q oscBR/2 u ≤ (1 − C) oscB2R u + 2 R kρ∗(F )kLq(B ) + R kfk . R γ0 L (B2R)

Lemma 132 veriﬁes the result by choosing the parameters from Lemma 132 by τ = 1/4,δ˜ =

n (1 − C), and µ(1 − q(γ−1) ) > α. The latter can be done by making α smaller if necessary. Notably α = α(1/4, δ˜) so it has the expected dependencies.

Holder regularity is a classic result of the improvement of oscillation in Theorem 140.

~ α Corollary 141. Suppose u, ρ, γ, F , f and q are as in Theorem 139. Then u ∈ Cloc(Ω) for some α = α(n, γ, ν, Λ, q).

Last, we conclude with a Liouville-type theorem in the case that f, F~ ≡ 0.

Theorem 142 (Liouville Theorem). Let ρ, γ, and u be as in Theorem 139. Suppose additionally that f, F~ ≡ 0. If Ω = Rn and u is bounded from above or below, then u is constant.

n Proof. It suﬃces to assume u ≥ 0 by replacing u with −u+supRn u (u−infR u, resp.) when u is bounded above (below, resp.). Since f, F~ ≡ 0, and Ω = Rn, Theorem 139 implies u is bounded above. Indeed, for x ∈ n, u(x) ≤ sup u ≤ C inf u ≤ Cu(0). In R B|x| n,γ,ν,λ,Q B2|x| particular, kukL∞(Rn) < ∞. Now, iterating Theorem 140 says there exists 0 < θ < 1 so that for any R > 0 and integer k,

k k osc u ≤ θ osc ≤ θ 2kuk ∞ n . BR B2kR L (R )

Taking k and R to inﬁnity consecutively completes the proof. 160

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