Harmonic Analysis and Applications 1470461277, 9781470461270

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IAS/PARK CITY MATHEMATICS SERIES Volume 27

Harmonic Analysis and Applications Carlos E. Kenig Fang Hua Lin Svitlana Mayboroda Tatiana Toro Editors

American Mathematical Society Institute for Advanced Study Society for Industrial and Applied Mathematics

Harmonic Analysis and Applications

IAS/PARK CITY MATHEMATICS SERIES Volume 27

Harmonic Analysis and Applications Carlos E. Kenig Fang Hua Lin Svitlana Mayboroda Tatiana Toro Editors

American Mathematical Society Institute for Advanced Study Society for Industrial and Applied Mathematics

Ian Morrison, Series Editor Carlos E. Kenig, Fang Hua Lin, Svitlana Mayboroda, and Tatiana Toro, Volume Editors. IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program 2010 Mathematics Subject Classification. Primary 42-06, 53-06, 35-06, 28-06.

Library of Congress Cataloging-in-Publication Data Names: Kenig, Carlos E., 1953– editor. | Lin, Fanghua, 1959– editor. | Mayboroda, Svitlana, 1981– editor. | Toro, Tatiana, 1964– editor. | Institute for Advanced Study (Princeton, N.J.) | Society for Industrial and Applied Mathematics. Title: Harmonic analysis and applications / Carlos E. Kenig, Fang Hua Lin, Svitlana Mayboroda, Tatiana Toro, editors. Description: Providence : American Mathematical Society, [2020] | Series: IAS/Park City mathematics series, 1079-5634 ; volume 27 | Includes bibliographical references. Identifiers: LCCN 2020025415 | ISBN 9781470461270 (hardcover) | ISBN 9781470462819 (ebook) Subjects: LCSH: Harmonic analysis. | Differential equations. | AMS: Harmonic analysis on Euclidean spaces – Proceedings, conferences, collections, etc.. | Differential geometry – Proceedings, conferences, collections, etc.. | Partial differential equations – Proceedings, conferences, collections, etc.. | Measure and integration – Proceedings, conferences, collections, etc. Classification: LCC QA403 .H215 2020 | DDC 515/.2433–dc23 LC record available at https://lccn.loc.gov/2020025415

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

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Contents Preface

vii

Introduction

ix

Lecture Notes on Quantitative Unique Continuation for Solutions of Second Order Elliptic Equations Alexander Logunov and Eugenia Malinnikova

1

Arithmetic Spectral Transitions: A Competition between Hyperbolicity and the Arithmetics of Small Denominators Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

35

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients Zhongwei Shen

73

Stochastic Homogenization of Elliptic Equations Charles K. Smart

131

T 1 and T b Theorems and Applications Simon Bortz, Steve Hofmann, and Jos´e Luis Luna

155

Sliding Almost Minimal Sets and the Plateau Problem G. David

199

Almgren’s Center Manifold in a Simple Setting Camillo De Lellis

257

Lecture Notes on Rectifiable Reifenberg for Measures Aaron Naber

289

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IAS/Park City Mathematics Series S 1079-5634(XX)0000-0

Preface The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the Regional Geometry Institute initiative of the National Science Foundation. In mid-1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS/Park City Mathematics Institute encourages both research and education in mathematics and fosters interaction between the two. The three-week summer institute offers programs for researchers and postdoctoral scholars, graduate students, undergraduate students, high school students, undergraduate faculty, K-12 teachers, and international teachers and education researchers. The Teacher Leadership Program also includes weekend workshops and other activities during the academic year. One of PCMI’s main goals is to make all of the participants aware of the full range of activities that occur in research, mathematics training and mathematics education: the intention is to involve professional mathematicians in education and to bring current concepts in mathematics to the attention of educators. To that end, late afternoons during the summer institute are devoted to seminars and discussions of common interest to all participants, meant to encourage interaction among the various groups. Many deal with current issues in education: others treat mathematical topics at a level which encourages broad participation. Each year the Research Program and Graduate Summer School focuses on a different mathematical area, chosen to represent some major thread of current mathematical interest. Activities in the Undergraduate Summer School and Undergraduate Faculty Program are also linked to this topic, the better to encourage interaction between participants at all levels. Lecture notes from the Graduate Summer School are published each year in this series. The prior volumes are: • Volume 1: Geometry and Quantum Field Theory (1991) • Volume 2: Nonlinear Partial Differential Equations in Differential Geometry (1992) • Volume 3: Complex Algebraic Geometry (1993) • Volume 4: Gauge Theory and the Topology of Four-Manifolds (1994) • Volume 5: Hyperbolic Equations and Frequency Interactions (1995) • Volume 6: Probability Theory and Applications (1996) • Volume 7: Symplectic Geometry and Topology (1997) • Volume 8: Representation Theory of Lie Groups (1998) • Volume 9: Arithmetic Algebraic Geometry (1999) ©2020 American Mathematical Society

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• Volume 10: • Volume 11: try (2001) • Volume 12: • Volume 13: • Volume 14: • Volume 15: • Volume 16: • Volume 17: • Volume 18: • Volume 19: • Volume 20: • Volume 21: • Volume 22: • Volume 23: • Volume 24: • Volume 25: • Volume 26:

Computational Complexity Theory (2000) Quantum Field Theory, Supersymmetry, and Enumerative GeomeAutomorphic Forms and their Applications (2002) Geometric Combinatorics (2004) Mathematical Biology (2005) Low Dimensional Topology (2006) Statistical Mechanics (2007) Analytical and Algebraic Geometry (2008) Arithmetic of L-functions (2009) Mathematics in Image Processing (2010) Moduli Spaces of Riemann Surfaces (2011) Geometric Group Theory (2012) Geometric Analysis (2013) Mathematics and Materials (2014) Geometry of Moduli Spaces and Representation Theory (2015) The Mathematics of Data (2016) Random Matrices (2017)

The American Mathematical Society publishes material from the Undergraduate Summer School in their Student Mathematical Library and from the Teacher Leadership Program in the series IAS/PCMI—The Teacher Program. After more than 25 years, PCMI retains its intellectual vitality and continues to draw a remarkable group of participants each year from across the entire spectrum of mathematics, from Fields Medalists to elementary school teachers. Rafe Mazzeo PCMI Director March 2017

IAS/Park City Mathematics Series S 1079-5634(XX)0000-0

Introduction Carlos Kenig, Fanghua Lin, Svitlana Mayboroda, and Tatiana Toro The origins of the harmonic analysis go back to the ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. His initial interest stemmed from the study of the heat equation, but shortly after, the Fourier series approach became a major tool in the study of diffusion processes and of propagation of waves. Today’s harmonic analysis incorporates elements of geometric measure theory, number theory and probability, and has countless applications in mathematics from data analysis to image recognition, and in physics, from the classical study of sound and vibrations to contemporary problems like properties of matter waves. Due in part to an array of compelling new problems from many sciences and in part to an influx of methods and results from other areas of mathematics, harmonic analysis has seen incredible breakthroughs in the beginning of the 21st century. Hence, the time was ripe for a comprehensive school devoted to these new topics, techniques and results. The most straightforward generalization of Fourier’s decomposition into plane waves is the representation of thermal diffusion processes or wave propagation on a manifold as linear combinations of its Laplace-Beltrami eigenfunctions. It is surprising that some of the most fundamental properties of these eigenfunctions remain imperfectly understood. The first lectures in this book, by Eugenia Malinnikova, give an approachable exposition of spectacular recent progress concerning Hausdorff dimensions of nodal sets, doubling properties of solutions, and the mechanism underlying quantitative unique continuation. The latter, in a sense, describes the limitations on the rate of growth (or decay) of the eigenfunctions of second order elliptic PDEs in a way somewhat analogous to the three sphere theorem. It also leads to an array of results pertaining to Yau’s conjecture and to questions of Nadirashvili regarding different aspects of the structure of the nodal domains. In the context of the Schrödinger operator with a random or otherwise disordered potential, the situation is even more mysterious. The eigenfunctions can exhibit localization or confinement, where away from a small portion of the initial domain, they decay exponentially. In simple terms, the waves that, in a homogeneous environment, look like the familiar sines and cosines in the plane can behave like rapidly decaying exponentials in a disordered medium. This effect, still ©2020 American Mathematical Society

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rather poorly understood, is one of the cornerstones of condensed matter physics, recognized in one of its manifestations by Anderson’s Nobel Prize in Physics. The lectures by Svetlana Jitomirskaya give an exposition of the spectral properties, localization, and structure of eigenfunctions for almost Mathieu operators and for 1D-Schrödinger operators with quasi-periodic potential, as well as their analogues and generalizations. In this context, a delicate mix of PDEs, analysis, and number theory yields beautiful and explicit results, rather rare in the world of Anderson localization, leading to the Cantor structure of the spectrum and detailed patterns of decay of the eigenfunctions. A different direction, just as important for applications, of recent developments in harmonic analysis replaces the random potential discussed above with a random matrix of coefficients of the divergence form elliptic operator, −div A∇. Properties of their solutions can be obtained by techniques of stochastic homogenization. Two sets of lectures treat these themes. The first, by Zhongwei Shen, deals with fundamentals of the periodic homogenization, when the coefficients are actually deterministic, periodic and rapidly oscillating. Here, roughly speaking, are the questions. Does the system behave at large scales like a constant, non-oscillating one? If so, what is the relevant constant coefficient limit? How can the convergence and errors be described? What is the ultimate behavior of eigenvalues, eigenfunctions, and solutions to the boundary value problems? The second set of lectures, by Charles Smart, passes to the case of random coefficients. Here again, the idea is that if A is sufficiently random in a suitable sense then the solutions at large scales behave like harmonic functions. The notes present some cutting-edge techniques in the theory, such as construction of correctors, qualitative and quantitative aspects of approximation, and proof of regularity of solutions. Finally, we pass to the case of a rough geometric environment and the properties of PDEs in the presence of irregularities and structural defects of the boundary and of the environment itself. A lot of effort in the 20th century was devoted to understanding the properties of the solutions of boundary value problems in increasingly complicated geometric scenarios: smooth domains, domains with isolated singularities, Lipschitz domains, and eventually even rougher settings. Simultaneously, the development of the calculus of variations and related techniques led to great advances in the context of free boundary problems, which aim to address the converse question: How do properties of the systems of PDEs affect smoothness of the boundary? For instance, what are the natural domains formed by the minimizers? This has been a very active area of research, which has only recently culminated in optimal, “if and only if” results, identifying the precise structural and regularity assumptions on domain needed for key properties of the solutions to linear and nonlinear PDEs and the exact character of minimizing surfaces.

Carlos Kenig, Fanghua Lin, Svitlana Mayboroda, and Tatiana Toro

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The first lectures exploring this direction, by Steve Hofmann, cover so-called T 1 and T b techniques. One of the major ways harmonic analysis enters questions of the solvability of boundary value problems is through singular integral representation formulas. The Calderón-Zygmund theory says that any reasonable singular integral operator bounded in L2 is also bounded in all Lp spaces, 1 < p < ∞. It is an extremely powerful and perhaps somewhat surprising result that to test L2 boundedness of a singular integral operator it is sufficient to apply it to the identity, or a suitable local substitute, and to show that the value has bounded mean oscillation. This theorem became known as the “T of 1", and of course a detailed statement, especially in contexts more general than that of classical singular integral operators, ends up being quite intricate. The development of this circle of ideas, however, brought a resolution of an array of long-standing conjectures, including the Kato square root problem, and most recently, the proof that absolute continuity of harmonic measure with respect to the Hausdorff measure is equivalent, with some mild topological assumptions, to the geometric property of rectifiability of the boundary. Two sets of lectures attack these questions from a different end, the geometry of minimizers. One can think, for instance, of the Plateau problem, asking what is the geometry of the area-minimizing set spanned by a given curve, or in other words, of a soap film on a given wire frame. The problem can be described as completely resolved or wide open depending on how “spanned" and “area" are defined, what properties of the initial curve are required, and how detailed a description of the minimizing surface is desired. Even problems motivated by actual applications to the real world remain open. Guy David’s lectures present recent progress in the context of Almgren’s notion of minimality. They touch on weaker and more general results (local Ahlfors regularity, rectifiability, limits, monotonicity of density) in the interior of the domain, but ultimately concentrate on regularity near the boundary. The lectures of Camillo De Lellis address area minimizing graphs, both in codimension 1 and in higher co-dimensions, and focus on de Giorgi’s celebrated ε-regularity theorem and a new approach to Almgren’s center manifold due to De Lellis and Spadaro. As the reader has probably gleaned from some of the terminology used above, the beautiful synergy of harmonic analysis, PDEs, and geometric measure theory that leads to the preceding results has brought to the fore some structural, scaleinvariant (rather than pointwise) characterizations of regularity. These roughly speaking require a set, or a measure, to be reasonably close to some other nice flat one instead of imposing conditions, typically more restrictive, such as pointwise smoothness or Lipschitz regularity. One example of such a notion is rectifiability, but there are others. Concentrating on the geometric side of these questions, Aaron Naber’s lectures present two theorems pertaining to the structure of measures and sets. One

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is the classical Reifenberg theorem which says that sets in Euclidean space which are well approximated by affine subspaces on all scales must be homeomorphic to balls. The other is its recent celebrated analogue, the Rectifiable Reifenberg theorem, which says that if a measure μ is summably close on all scales to affine subspaces Lk of dimension k, then μ may be decomposed as μ = μ+ + μk where μk is k-rectifiable with uniform Hausdorff measure estimates, and μ+ has uniform bounds on its mass. Circling back to the questions discussed above, we mention that these results have direct applications to the singular parts of solutions of nonlinear equations and to minimizing surfaces. In concluding this Introduction, let us express our sincere gratitude to Rafe Mazzeo who had the initial idea for the Summer School in Harmonic Analysis and helped us carry out its planning and organization every step of the way. We are also indebted to Michelle Wachs for her help through the years leading to the event and during the program in Park City. We are very thankful to the PCMI staff—Beth Brainard and Dena Vigil—for all their efforts to make the school a successful one. We feel that that this PCMI program had a huge impact on the field, establishing and strengthening collaborative projects at all levels, and seeding a generation of successful young researchers. We are immensely thankful for this opportunity.

IAS/Park City Mathematics Series Volume 27, Pages 1–34 https://doi.org/10.1090/pcms/027/00859

Lecture notes on quantitative unique continuation for solutions of second order elliptic equations Alexander Logunov and Eugenia Malinnikova Abstract. In these lectures we present some useful techniques to study quantitative properties of solutions of elliptic PDEs. Our aim is to outline the proof of a recent result on propagation of smallness. The ideas are also useful in the study of the zero sets of eigenfunctions of the Laplace–Beltrami operator. Some basic facts about second order elliptic PDEs in divergent form are collected in the Appendix at the end of the notes.

Contents 1

2

3

Eigenfunctions of Laplace–Beltrami operators 1.1 Definition 1.2 Courant nodal domain theorem 1.3 More examples 1.4 Bessel functions and Helmholtz equation 1.5 Yau’s conjecture 1.6 Lift of eigenfunctions 1.7 A question of Nadirashvili 1.8 Exercises Doubling index and frequency function 2.1 Frequency function 2.2 Three spheres theorem for elliptic PDEs 2.3 Doubling index 2.4 Doubling index for eigenfunctions 2.5 Cubes 2.6 Remarks on the size of the zero sets of eigenfunctions and the doubling index 2.7 Exercises Small values of polynomials and solutions of elliptic PDEs 3.1 Classical results of Cartan and Polya

2 2 3 4 5 6 6 7 7 9 9 13 14 15 16 17 18 19 20

2010 Mathematics Subject Classification. Primary 35J15; Secondary 31B05. Key words and phrases. Elliptic PDE, unique continuation, Laplace eigenfunctions. This work was completed during the time A.L. served as a Clay Research Fellow. E.M. is supported by Project 275113 of the Research Council of Norway and NSF grant no. DMS1638352. ©2020 American Mathematical Society

1

2

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Quantitative unique continuation

3.2 Remez inequality for polynomials 3.3 Propagation of smallness result 3.4 Base of induction 3.5 Exercises Proof of propagation of smallness result 4.1 On distribution of the doubling indices 4.2 Choosing the right notation 4.3 Recursive inequality implies exponential bound 4.4 Exercises Appendix: Second order elliptic equations in divergence form 5.1 Elliptic operator in divergence form: regularity 5.2 Comparison to harmonic functions

20 21 22 23 25 25 26 27 28 29 29 31

1. Eigenfunctions of Laplace–Beltrami operators 1.1. Definition Let M be an oriented Riemannian manifold with metric tensor g = (gij ), let |g| denote the absolute value of the determinant of the matrix (gij ), and let g−1 = (gij ) be the inverse tensor. The gradient of a C1 function f on M is a vector field locally given by  (gij ∂j f)∂i . gradM f = i,j

The Laplace–Beltrami operator on functions on M is defined as the divergence of the gradient. In local coordinates, it becomes  1 ΔM f =  div( |g|g−1 ∇f), |g| where ∇f = (∂1 f, . . . , ∂n f) in chosen coordinates. The following Green formula holds for functions f, h ∈ W01,2 (M)   hΔM fdVM = − gradM f, gradM hg dVM , M

M

 where the volume form dVM is defined as dVM = |g|dx1 ∧ · · · ∧ dxn in local coordinates . Assume now that M is a compact manifold without boundary. We consider eigenfunctions φλ of the Laplace–Beltrami operator, such that ΔM φλ + λφλ = 0. Then

 M

 | gradM φλ |2g dVM

|φλ |2 dVM .

=λ M

A. Logunov and E. Malinnikova

3

All eigenvalues of −ΔM are real and non-negative. Eigenfunctions corresponding to distinct eigenvalues are orthogonal since    φλ φμ dVM = − (ΔM φλ )φμ dVM = μ φλ φμ dVM . λ M

M

M

The eigenvalues form an increasing sequence that tends to infinity, 0 = λ1 < λ2  λ3  · · ·  λn  · · · . The first eigenfunction φ0 is a constant. There is an orthonormal basis of eigenfunctions for L2 (M). We refer the reader to [6, Chapter 1] for details. Example 1.1.1. (Dirichlet Laplacian for a domain in Rd ) Instead of a compact manifold, we may also consider a bounded domain Ω in Rd and the Laplace operator with the Dirichlet boundary condition Δφ + λφ = 0,

φ|∂Ω = 0.

The first eigenvalue is given by the variational formula  |∇φ|2 , λ1 (Ω) = min φ

Ω

where the minimum is taken over all functions φ ∈ W01,2 (Ω) such that This formula implies that if Ω1 ⊂ Ω2 then



2 Ω |φ|

= 1.

λ1 (Ω1 )  λ1 (Ω2 ). The first eigenfunction does not change sign and can be chosen positive in Ω, while all other eigenfunctions are orthogonal to the first one and therefore change sign in Ω. The eigenvalues can be determined by the min-max formula  |∇φ|2 , (1.1.2) λk (Ω) = min max Ω 2 Ak φ∈Ak Ω |φ| where the minimum is taken over all k-dimensional subspaces of W01,2 (Ω). Alternatively, there is an inductive description of eigenvalues (and eigenfunctions),  |∇φ|2 , (1.1.3) λk (Ω) = min Ω 2 φ Ω |φ| where the minimum is taken over all φ ∈ W01,2 (Ω) which are orthogonal to the first k − 1 eigenfunctions φλ1 , . . . , φλk−1 . The variational characterization of the eigenvalues, (1.1.2) and (1.1.3), also hold for the eigenvalues of the Laplace-Beltrami operator on compact manifolds. 1.2. Courant nodal domain theorem

The zero set Z(φ) of a function φ is

Z(φ) = {x : φ(x) = 0}, and we also refer to it as the nodal set of φ. The connected components of M \ Z(φ) are called the nodal domains of the function φ. The simplest example of a compact manifold is the unit circle T  [0, 2π). Eigenfunctions of the Laplace operator are 2π-periodic solutions of the eigenvalue

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Quantitative unique continuation

problem φ  + λφ = 0. This equation has a 2π periodic solution when λ = n2 for some integer n. The first eigenfunction, corresponding to n = 0 is a constant. For n > 0 the eigenfunctions are linear combinations of φn,1 (θ) = cos(nθ) and φn,2 (θ) = sin(nθ). Each of them has 2n zeros on the circle. The Courant nodal domain theorem gives an upper bound for the number of nodal domains of eigenfunctions on manifolds of arbitrary dimension. Let M be a compact manifold as above and φλn be an eigenfunction of the Laplace-Beltrami operator corresponding to the n-th smallest eigenvalue. Theorem 1.2.1 (Courant). The number of connected components of M \ Z(φλn ) is at most n. For the proof we refer the reader to [8, Chapter 6] and [6]. The proof is beautiful and short except for one non-trivial result on weak unique continuation property of solutions of second order elliptic PDEs. The result says that a non-zero Laplace-Beltrami eigenfunction cannot vanish on an open subset of a manifold. The aim of these notes is to give a new quantitative sharpening of this uniqueness result. 1.3. More examples A first intuition on the geometry of zero sets of eigenfunctions comes from the pictures of nodal domains on the unit sphere and the standard torus, see [18]. Example 1.3.1. The eigenfunctions on the unit sphere Sd in Rd+1 are restrictions of the homogeneous harmonic polynomials which are called spherical harmonics. If P is a polynomial of d + 1 variables, ΔP = 0 and P(x) = |x|n Y(x/|x|), where Y is a function on S = Sd , then ΔS Y + n(n + d − 1)Y = 0. There is a basis of spherical harmonics for L2 (Sd ). Therefore there are no other eigenfunctions of the Laplace-Beltrami operator on the sphere, further details are given in Exercise 1.8.3. Example 1.3.2. Another standard compact manifold, on which we can compute eigenfunctions explicitly, is the torus. Let Td be the d-dimensional torus which we will identify with the rectangle d j=1 [−π, π] glued along each pair of opposite sides. Then we have a basis of eigenfunctions of the form ⎞ ⎛ d d   φ(x) = exp ⎝i nj xj ⎠ , ΔTd φ + n2j φ = 0, j=1

where nj ∈ Z.

j=1

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We notice that if dimension d > 1, there are eigenvalues for the Laplace– Beltrami operators on Sd and Td with arbitrary large multiplicities. This is a source of interesting examples of eigenfunctions. The zero sets of standard spherical harmonics and eigenfunctions on the torus are not hard to visualize, but the structure of the zero sets of linear combinations of these functions (corresponding to the same eigenvalue) may be complicated. 1.4. Bessel functions and Helmholtz equation Another classical example of eigenfunctions are bounded solutions of the Helmholtz equation in Rn , Δφ + λφ = 0. For λ  0 the maximum principle holds and there are no non-trivial bounded solutions. Hence we are interested in the case λ > 0 and, rescaling the variable, we may assume that λ = 1. The Laplace operator in polar coordinates can be written as d−1 1 ∂r φ + 2 ΔS φ. Δφ = ∂2r φ + r r We look for solutions of the equation Δφ + φ = 0 of the form φ(x) = f(|x|)Y(x/|x|). Separating the variables, one can check that Y is an eigenfunction of the Laplace– Beltrami operator on the unit sphere. The eigenvalues on the sphere are given in Example 1.3.1 (see also Exercise 1.8.3 below). Then we find a family of solutions of the Helmholtz equation of the form   x , ΔS Y = −n(n + d − 2)Y, φ(x) = fn (|x|)Y |x| where fn (r) satisfies the following ordinary differential equation r2 f  + (d − 1)rf  + (r2 − n(n + d − 2))f = 0. Writing fn (r) = r1−d/2 gn (r) we see that gn (r) satisfies the Bessel equation r2 g  + rg  + (r2 − (n + d/2 − 1)2 )g = 0. This is a second order ODE with analytic coefficients with a solution Jn+d/2−1 called the Bessel function (of the first kind) which is continuous at the origin. The solution is of the form Jn+d/2−1 (r) = rn+d/2−1 hn+d/2−1 (r) where hn+d/2−1 (r) is an analytic function of r and hn+d/2−1 (0) = 0 (see for example [35, Chapter 4.2]); the second solution has a singularity at r = 0. Thus we get fn (r) = r1−d/2 Jn+d/2−1 (r) = rn hn+d/2−1 (r). We consider positive zeros of Jν (they are simple, since Jν is a non-zero solution do second order ODE) and enumerate them 0 < jν,1 < jν,2 < · · · . Using the obtained description of the solutions of the Helmholtz equation, we can compute eigenfunctions and eigenvalues of the Dirichlet Laplace operator for the unit ball in Rd , see Exercise 1.8.4 below.

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1.5. Yau’s conjecture Examples of eigenfunctions on the torus and sphere show that the number of nodal domains may vary, but is bounded from above as shown by Courant nodal domain theorem. At the same time, there exist eigenfunctions with large eigenvalues and just two nodal domains as was shown already in 1925 in the dissertation of Antonie Stern; see [3] for historical details and references. On the other hand, these examples show that nodal lines become more complicated and dense as the eigenvalue grows. We give a proof of a well known result on the density of the zero sets of eigenfunctions in the next section. First we formulate a deep conjecture of Yau [37]. Conjecture (Yau). Let M be a smooth compact d-dimensional Riemannian manifold. There exist constants C1 and C2 , which depend on M, such that √ √ C1 λ  Hd−1 (Z(φλ ))  C2 λ, for any eigenfunction φλ satisfying ΔM φλ + λφλ = 0. The singular set of a function is the set where both the function and its gradient equal zero. The singular sets of an eigenfunction has Hausdorff dimensions d − 2 and its nodal sets is the union of smooth hypersurfaces with finite (d − 1)-dimensional Hausdorff measure and the singular set. The finiteness of the Hausdorff measure of the nodal set is a non-trivial fact; see [17] for details. The Yau conjecture was proved for the case of real analytic metrics by Donnelly and Fefferman in 1988, [9]. We outline some of the ideas in Section 2.6. 1.6. Lift of eigenfunctions The following lifting trick is used intensively in the study of eigenfunctions. Let M be a d-dimensional manifold and φλ be an eigenfunction, ΔM φλ + λφλ = 0, we define the function √ λt

h(x, t) = φλ (x)e

,

on the product manifold M  = M × R. Then ΔM  h = 0. Locally we view h as a solution of an elliptic equation in divergence form on a subdomain of Rd+1 . The first application of the lifting trick is the proof of the result on the density of the zero sets of eigenfunctions. Proposition 1.6.1. Suppose that M is a compact Riemannian manifold. There exists ρ = ρ(M) such that for any eigenfunction φλ with λ > 0 and any x ∈ M the distance from x to the zero set Z(φλ ) is less than ρλ−1/2 . Proof. Suppose that φλ does not change sign in some ball Br ⊂ M. We assume that r is small enough and consider a chart for M that contains Br . Then the √ function h(x, t) = φλ (x)exp( λt) is a solution of a second order elliptic equation in divergence form and h does not change sign in Br × [−r, r]. By the Harnack inequality, (see Theorem 5.1.6 below) sup |h|  C(M) inf |h|, D

D

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where D = Br/2 × [−r/2, r/2]. That r < ρλ−1/2 then follows from √ √ sup |h| = sup |φλ | exp(r λ/2)  exp(r λ) inf |h|. D

D

Br/2



√ The zero set of h = φλ (x) exp( λt) is the cylinder over Z(φλ ), hence questions about Z(φλ ) can be restated in terms of Z(h). One advantage is that h is a solution of an elliptic second order PDE in divergence form with no lower order terms. 1.7. A question of Nadirashvili Suppose that h is a harmonic function in the unit disc D ⊂ R2 such that h(0) = 0. The zero set of h is the union of analytic curves and by the maximum principle it has no loops. We assume that h(0) = 0 then an elementary geometric argument implies that H1 (Z(h) ∩ D)  2. Nadirashvili asked whether a higher dimensional version of this statement holds. Conjecture (Nadirashvili). There is a constant c > 0 such that for any harmonic function h in the unit ball B of R3 such that h(0) = 0, the following inequality holds H2 (Z(h) ∩ B)  c. The question was formulated for harmonic functions in Rn and remained open for many years. The proof given recently in [26] by the first author is complicated (and beyond the scope of these lectures), it gives the affirmative solution to the version of the Nadirashvili conjecture for solutions of second order elliptic equation in divergence form with smooth coefficients. Theorem 1.7.1 ([26]). Suppose that Lu = div(A∇u) is a uniformly elliptic operator in the unit ball B ⊂ Rd with smooth coefficients. There exists a constant c = c(A) such that for any solution of Lu = 0 with u(0) = 0 satisfies Hd−1 (Z(u) ∩ B)  c. A corollary, also shown in [26], is the lower bound in Yau’s conjecture on compact Riemannian manifolds with smooth metric. A polynomial upper bound Hd−1 (Z(φλ ))  CλAd , where Ad depends only on the dimension of the manifold and C depends on the manifold and the metric was obtained in [25]. 1.8. Exercises Exercise 1.8.1 (Harnack inequality). Let L = div(A∇·) be a uniformly elliptic operator with bounded coefficients. Use the Harnack inequality (Theorem 5.1.6) to prove the following statements. (1) If u is a bounded solution of Lu = 0 in Rd then u is a constant. (2) Let Ct denote the cylinder Ct = {x = (x1 , . . . xd ) ∈ Rd : x21 + · · · + x2d−1  t2 }.

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Quantitative unique continuation

Suppose that Lu + cu = 0, c ∈ R and u is positive in the cylinder C1 and let M(R) = max{u(x) : x ∈ C1/2 , |xd |  R}. Then there exists C such that M(R)  u(0)eCR . Exercise 1.8.2. Suppose that ΔM u + λu = 0 and Ω is a connected component of M \ Z(u). Assume that Ω is a domain with piece-wise smooth boundary and prove that the first Dirichlet Laplace eigenvalue of Ω is λ1 (Ω) = λ. Remark: Careful details can be found in [6], see also [7]. Exercise 1.8.3 (Harmonic polynomials). The restrictions of homogeneous harmonic polynomials on the unit sphere S ⊂ Rd+1 , called spherical harmonics, are the eigenfunctions of the Laplace–Beltrami operator. We denote the eigenspace that corresponds to the eigenvalue λ = n(n + d − 1) by En,d . If Y ∈ En,d then the function P(x) = |x|n Y(x/|x|) is harmonic. (1) Apply Green’s formula in Rd to show that if Yn ∈ En,d and Ym ∈ Em,d with n = m then  Yn Ym = 0. S

(2) Consider the following inner product on the space Pn,d of homogeneous polynomials of degree n in d variables,  α!Pα Qα , [P, Q] = P(D)(Q) = 

|α|=n

 where P(x) = |α|=n Pα Q(x) = |α|=n Qα xα . Show that the space of harmonic polynomials Hn,d ⊂ Pn,d is the orthogonal complement of xα ,

Qn,d = {P ∈ Pn,d : P(x) = |x|2 P1 (x), P1 ∈ Pn−2,d } with respect to this inner product. (3) Show that any homogeneous polynomial F of degree n in Rd can be written as F(x) = Hn (x) + |x|2 Hn−2 (x) + · · · |x|2k Hn−2k , where k = n/2 and Hj is a homogeneous harmonic polynomial of degree j. This implies that spherical harmonics form a basis for L2 (S) and there no other eigenfunctions. (4) Deduce that if Y ∈ Hn,d and F is a polynomial of degree less than n then  S YF = 0. (5) Suppose that P(x) ∈ Hn,d and Q is a factor of P, P = QF for some polynomial F. Show that Q changes sign in Rd . Exercise 1.8.4 (Dirichlet eigenfunctions for balls). Let Jn be the Bessel function such that u(reiθ ) = Jn (r)(a cos nθ + b sin nθ)

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satisfies Δu + u = 0 in R2 , i.e., Jn is a solution of the second order ODE r2 J  + rJ + (r2 − n2 )J = 0. Furthermore, let 0 < jn,1 < jn,2 < · · · be the positive zeros of Jn . (1) Show that there is a constant c such that n  jn,1  cn. (Hint: you may use the equation for the lower bound and the density of zero sets of eigenfunctions for the upper bound.) (2) Show that the following functions φn,k (reiθ ) = Jn (jn,k r)(a cos nθ + b sin nθ) are eigenfunctions of the Dirichlet Laplacian on the unit ball of R2 , and that the smallest eigenvalue is j20,1 . Remark 1: A classical and deep result of Siegel implies that two distinct Bessel functions Jn and Jm with integer n and m have no common zeros and thus all eigenvalues of a disk are simple. Remark 2: Let λd,k be the kth eigenvalue of the Dirichlet Laplace operator on the unit ball B0 ⊂ Rd . Suppose that M is a smooth d-dimensional Riemannian manifold, x ∈ M and let B = B(x, r) be the ball on M of radius r and center x. Let λk (B) be the kth eigenvalue of the Dirichlet Laplace-Beltrami operator for B. Then one can show that (see [6]) λk (B) ∼ r−2 λd,k ,

r → 0.

√ Exercise 1.8.5 (Yau’s conjecture). Prove the lower bound Hd−1 (Z(u))  c λ in the Yau conjecture in dimensions one and two. Hint: for the case d = 2, first use Exercise 1.8.2, then the inequality λ1 (Ω1 )  λ1 (Ω2 ) for Ω1 ⊂ Ω2 , and finally Remark 2 above.

2. Doubling index and frequency function An important tool to study nodal sets of eigenfunctions and growth properties of solutions of elliptic PDEs is the so-called frequency function. The idea goes back to the works of Almgren [2] and Agmon [1], where it was introduced for the case of harmonic functions in Rn . It was generalized to solutions of second order elliptic equations by Garofalo and Lin [13], see also [20] and [31]. 2.1. Frequency function Let A(x) be a symmetric uniformly elliptic matrix with Lipschitz coefficients defined on some ball Br centered at the origin and such that A(0) = I. Define the function μ by (A(x)x, x) μ(x) = , |x|2 then μ(0) = 1, we have

Λ−1  μ(x)  Λ. Moreover, since A has Lipschitz coefficients, A(x) = I + O(|x|)

and μ(x) = 1 + O(|x|).

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Quantitative unique continuation

Let u be a solution to the equation div(A(x)∇u(x)) = 0. We consider weighted averages of |u|2 over spheres:  1−d μ(x)|u(x)|2 ds(x). H(r) = r ∂Br

Denoting by ν = x/|x| the unit outer normal vector for the sphere and applying the divergence theorem, we obtain   (|u|2 A(x)x, ν)ds = r−d div(|u|2 A(x)x). H(r) = r−d ∂Br

Br

In the case of the Laplace operator, A = I and μ(x) = 1, the function t → H(et ) is convex, i.e., H(r)  H(r1 )α H(r2 )1−α ,

1−α r = rα , α ∈ (0, 1). 1 r2

when

This can be proved either using the decomposition of harmonic functions into series of spherical harmonics, or by integration by parts as below, the computations are slightly simplified in this case, see [15]. A similar convexity property was discovered for solution of elliptic equations in [13], we provide a calculation that is a small variation of the one given in [20]. First we compute the derivative of H,  div(|u|2 A(x)x). (2.1.1) H  (r) = −dr−1 H(r) + r−d ∂Br

We rewrite the integral in the second term as  div(|u|2 A(x)x) = ∂Br    2u(∇u, A(x)x) + |u|2 trace(A(x)) + where AD (x) =



∂Br

∂Br

i,j (∂i aij )xj .

|u|2 AD (x),

∂Br

We also note that

μ(x) = 1 + O(|x|), trace(A) = d + O(|x|), and AD (x) = O(|x|). This implies  div(|u|2 A(x)x) = (2.1.2) ∂Br 

 |u|2 μ(x) + O(rd H(r)).

2u(∇u, A(x)x) + d ∂Br

∂Br

We rewrite the first integral in the right-hand side of (2.1.2) using the symmetry of A and then apply the divergence theorem once again to obtain    2u(∇u, A(x)x) = 2u(A(x)∇u, x) = 2r div(uA(x)∇u). ∂Br

∂Br

Br

Next, using the equation div(A∇u) = 0, we have   2u(∇u, A(x)x) = 2r (A(x)∇u, ∇u). (2.1.3) ∂Br

Br

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Finally, combining (2.1.1), (2.1.2), and (2.1.3), we get  (A∇u, ∇u) + O(H(r)). H  (r) = 2r1−d Br

Following [13] and [20], define   (A∇u, ∇u) = r−d I(r) = r1−d Br

(uA∇u, x), ∂Br

and the frequency function of u N(r) =

rI(r) . H(r)

Then

rH  + O(1). 2H Proposition 2.1.5. There exists a constant C that depends only on the ellipticity and Lipschitz constants of matrix A(x) such that for any solution u to div(A∇u) = 0 in a ball BR centered at the origin, the function F(r) = eCr N(r) is increasing on (0, R). (2.1.4)

H  (r) = 2I(r) + O(H(r)),

N(r) =

Proof. We compute N  (r), taking into account that the first derivatives of the coefficients of A are bounded. We already know that H  (r) = 2I(r) + O(H(r)). Next we estimate (rI(r))  . If w is a vector field in Br with (w, x) = r2 on ∂Br , then  (2.1.6) (rI(r))  = (2 − d)I(r) + r2−d (A∇u, ∇u) ∂Br  = (2 − d)I(r) + r1−d div(w(A∇u, ∇u)) Br   = (2 − d)I(r) + r1−d div(w)(A∇u, ∇u) + r1−d (w, ∇(A∇u, ∇u)). Br

Br

We used the divergence theorem in the second equality above. To simplify the last term we note that (w, ∇(A∇u, ∇u)) = 2(w, Hess(u)(A∇u)) + (AD,w ∇u, ∇u),  where AD,w (x) = { k (∂k aij )wk }i,j . Furthermore, the Hessian is a symmetric matrix and (2.1.7)

Hess(u)(w) = ∇(∇u, w) − (Dw)∇u. Thus, we obtain,    (Hess(u)w, A∇u) = (∇(∇u, w), A∇u) − ((Dw)∇u, A∇u) Br B B  r r = (2.1.8) div((∇u, w)A∇u) − ((Dw)∇u, A∇u) Br Br   −1 (∇u, w)(A∇u, x) − ((Dw)∇u, A∇u). =r ∂Br

Br

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Quantitative unique continuation

We used the equation div(A∇u) = 0 for the second identity and the divergence theorem for the third one. Now we choose w(x) = μ(x)−1 A(x)x. Then (w(x), x) = |x|2 ,

div(w) = d + O(|x|).

Dw = I + O(|x|),

We proceed to work with (2.1.8) and rewrite the first integral as   (∇u, w)(A∇u, x) = μ(x)−1 (A∇u, x)2 . ∂Br

∂Br

Now combine the second term in (2.1.6) and the second term in (2.1.8), taking into account the inequalities for Dw and div(w), we get   div(w)(A∇u, ∇u) − 2r1−d ((Dw)∇u, A∇u) = r1−d Br

Br

(d − 2)I(r) + O(rI(r)). Moreover, we have  r1−d

 |(AD,w ∇u, ∇u)|  Cr1−d

Br

r|∇u|2 = O(rI(r)), Br

where C depends on the ellipticity and Lipschitz constants of A and on the dimension of the space. Now (2.1.6), (2.1.7), (2.1.8) and the last two inequalities imply   −d μ(x)−1 (A∇u, x)2 + O(rI(r)). (rI(r)) = 2r ∂Br

Finally, the last inequality and (2.1.4) give N  (r)(N(r))−1 = (rI(r))  (rI(r))−1 − (H  (r))(H(r))−1   2

 (A∇u, x)2 2r−2d 2 + O(1). = μ(x)|u| − (uA∇u, x) I(r)H(r) μ(x) ∂Br ∂Br ∂Br The first term is positive by the Cauchy-Schwarz inequality. Therefore N  (r)  −CN(r) 

and the proposition follows.

Corollary 2.1.9. Suppose that div(A(x)∇u(x)) = 0 in BR0 , where A(x) = I + O(x) as above. Let also N(r) be the frequency of u. Then there exists DN that depends on R0 , N(R0 /2), the ellipticity and Lipschitz constants of the operator, and the dimension of the space, such that   |u|2  DN |u|2 B2r

Br

for any r ∈ (0, R0 /4). Proof. For any r < R0 /2 we write (2.1.4) and apply the proposition H  (r)H(r)−1  2I(r)H(r)−1 + c = 2r−1 N(r) + c  2r−1 N(R0 /2)eC(R0 −2r) .

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Integrating H  (r)/H(r) over an interval [ρ, 2ρ] for ρ < R0 /4, we get   μ(x)|u(x)|2 ds(x)  CN μ(x)|u(x)|2 ds(x), ∂B2ρ

∂Bρ

where CN = exp(C1 + C2 N(R0 /2)) with C2 = C2 (R0 ). Finally, integrating the inequality with respect to ρ from 0 to r, and using that Λ−1  μ  Λ we obtain the required estimate.  2.2. Three spheres theorem for elliptic PDEs Another consequence of the monotonicity of the frequency function is the three sphere theorem. Its simplest version is the classical Hadamard three circle theorem for analytic functions. It states that if f is an analytic function on the unit ball in C and M(r) = max{|f(z)| : |z| = r}, then the following inequality holds M(r)  M(r1 )α M(r2 )1−α ,

1−α where r = rα , r, r1 , r2 < 1. 1 r2

The classical proof is based on the fact that the logarithm of the modulus of an analytic function is subharmonic. It turns out that even without analyticity a version of the Hadamard inequality holds for harmonic functions and more generally for solutions to uniformly elliptic equations. One of the first general results is due to Landis [21]. We derive the three spheres theorem from the properties of the frequency function, following [13]. Proposition 2.1.5 implies the inequality eCr N(2r)  N(r), which, combined with (2.1.4), gives   2rH  (2r) rH  (r)  c+ eCr . H(r) H(2r) Then integrating from r to 2r with respect to dr/r we obtain (2.2.1)

log H(2r) − log H(r)  (c log 2 + log H(4r) − log H(2r))e2Cr .

Proposition 2.2.2. Assume that L = div(A∇·) is a uniformly elliptic operator, A is symmetric and has Lipschitz entries in a domain Ω. Suppose also that A(0) = I and B(0, 4r) ⊂ Ω. There exist α > 0 and C > 0 such that for any solution u of Lu = 0 the following inequality holds  α  1−α  |u|2  C |u|2 |u|2 . ∂B2r

∂Br

∂B4r

Proof. We collect similar terms in (2.2.1) and take the exponent of both sides to obtain  α  1−α  μ|u|2 ds  C1 μ|u|2 ds μ|u|2 ds ∂B2r

∂Br

∂B4r

with α = (1 + so that α can be chosen close to 1/2 as r → 0. This inequality and bounds on μ imply the required estimate.  e4Cr )−1

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Quantitative unique continuation

Assume that A is as above with A(0) = I. Proposition 2.2.2 and the equivalence of Lp -norms for solutions of elliptic equations (see Corollary 5.1.4 below) imply the following three ball inequality for supremum norms

α1

1−α1 sup |u|  C sup |u|

sup |u|

B2r

B8r

Br

,

for some C and α1 ∈ (0, 1) depending on A and r but not on u. We can drop the assumption that A(0) = I applying a local change of variables, balls are replaced by ellipses. Applying the inequality several times and inscribing ellipses in balls we obtain the following statement. (We omit some technical details required for an accurate argument.) Corollary 2.2.3. Let L = div(A∇·) be a uniformly elliptic operator with Lipschitz coefficients in a domain Ω. There exist r0 > 0, k large enough, C and β ∈ (0, 1) such that if B = Br is a ball with r < r0 and Bk2 r ⊂ Ω then

β

1−β sup |u|  C sup |u|

sup |u|

B2r

Bkr

Br

,

for any u that solves the equation Lu = 0 in Ω. The general version of this result can be obtain by the chain argument. Corollary 2.2.4. Let L be as above and B ⊂ K ⊂⊂ Ω, where B is open and K is compact. There exist C and γ ∈ (0, 1) that depend only on K, Ω, B and the ellipticity and Lipschitz constants of L such that for any solution u to Lu = 0 in Ω the following inequality holds  γ  1−γ . sup |u| sup |u|  C sup |u| K

B

Ω

Proof: Chain argument. Assume that supΩ |u| = 1. For each point x ∈ K there is a curve γ connecting x to some fixed point in B. We then can find a finite sequence of balls {Bj }Jj=1 such that r(Bj ) < r0 , B1 ⊂ B, Bj+1 ⊂ 2Bj , k2 Bj ⊂ Ω and x ∈ BJ = B(x). Applying the previous corollary we see that sup |u|  sup |u|  C(sup |u|)β . 2Bj

Bj+1

Bj

Iterating this estimate we obtain sup |u|  C1 (sup |u|)β1 , BJ

B

for some C1 and β1 that depend on C, β and the number J of the iteration steps. Finally, we take a finite cover of K by balls B(x) and get the required estimate.  2.3. Doubling index We prefer to replace the frequency function by a comparable but more intuitive quantity that we call the doubling index. Let h ∈ C(Ω), such that h does not vanish on any open subset of Ω. For any closed ball B such that 2B ⊂ Ω we define max2B |h| . Nh (B) = log maxB |h|

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Note that if p is a homogeneous polynomial of degree n and a ball B is centered at the origin than Np (B) = n log 2. At the same time if we compute the frequency function Np (r) of this polynomial (defined for the case of the Laplace operator, A = I), we get Np (r) = n. In general, if h is a solution to Lh = 0 in the ball BR0 then, using the equivalence of norms (Corollary 5.1.4) and the estimate in the proof of Corollary 2.1.9, we obtain that for r < R0 /4 C−1 1 Nh (r) − C2  Nh (Br )  C1 Nh (4r) + C2 . The inequality above and the almost monotonicity of the frequency implies the following almost monotonicity for the doubling index when 4r < R < R0 , Nh (Br )  C(Nh (BR ) + 1) .

(2.3.1)

2.4. Doubling index for eigenfunctions The monotonicity of the doubling index and three sphere theorem hold for solutions of second order elliptic equations of the form div(A∇h) = 0. For eigenfunctions φλ (x) on compact manifolds there is no monotonicity of the doubling index and the three sphere inequality gets a constant that depends on the eigenvalue. As above, we consider the lift √ h(x, t) = e λt φλ (x) and then apply the results of the previous sections to h that solves an equation of the form div(A∇h) = 0. Donnelly and Fefferman used the doubling indices in their study of nodal sets of eigenfunctions on smooth manifolds. One of their celebrating results for general smooth compact Riemannian manifolds is the following. Proposition 2.4.1. Let M be a smooth compact Riemannian manifold. There exists r0 and C depending on M such that for any eigenfunction φ = φλ , ΔM φλ + λφλ = 0, √ the doubling index Nφ (B)  C λ when B is a ball on M with radius r  r0 . Proof. Let B = B(x, r) be a ball on M. We consider the ball B  on M × [−R, R], R > r, such that the center of B  is (x, 0) and the radius on B  is r. We say √ that B  is the lift of B. We note that Nφ (B)  Nh (B  ) + C λ. It is enough to prove the estimate for the doubling index of h on M × [−R, R]. Assume that maxM |φ| = |φ(x0 )| = 1 and fix r such that for each point x ∈ M the geodesic ball Br (x) is contained in a chart. Let B be any ball of radius r/2k on M and B  be its lift in M × R. We consider a finite chain {Bj }Jj=1 of geodesic balls in M × [−r, r] with centers on M × 0 and equal radii r/2k. We choose the balls such that B1 = B  , Bj+1 ⊂ 2Bj and (x0 , 0) ∈ BJ . Then, since supkBj |h|  er , Corollary 2.2.3 implies sup |h|  c(sup |h|)1/β e−Cr  c(sup |h|)1/β e−Cr . Bj

2Bj

Bj+1

It implies that supB  |h|  c1 , where where c1 depends on r and M (which also determine the number of balls in a chain). Then, √

er

λ

sup |φ|  c1 , B

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Quantitative unique continuation

Thus for any ball B of radius at least r and for the corresponding lifted ball B  √ √ we obtain Nφ (B)  C( λ + 1) and Nh (B  )  C( λ + 1). Finally, the almost monotonicity of the doubling index for h implies similar estimate for balls of radius less than r.  2.5. Cubes A version of the (maximal) doubling index for cubes is used in the next sections. For a given cube Q ⊂ Rd we denote its side length by s(Q). Then the volume of the cube is |Q| = (s(Q))d . Assume that u is a solution to the equation Lu = 0 in a domain Ω ⊂ Rd and for each cube Q with 2Q ⊂ Ω define max2q |u| ∗ . (Q) = sup log (2.5.1) Nu maxq |u| q⊂Q We claim that the almost monotonicity of the usual doubling index implies that the supremum above is finite. By the definition, we have now that if q ⊂ Q then ∗ (q)  N∗ (Q). Nu u ∗ (Q) defined above to the We want to compare the maximal doubling index Nu doubling index log max2Q |u| − log maxQ |u|. Take a cube q ⊂ Q. If q is small, s(q) < cd s(Q), we first apply almost monotonicity inequality for the doubling index (2.3.1). Let b be the largest ball inscribed in q then 2q ⊂ kd b, where √ kd = 2 d and we have maxkd b |u| maxkd B |u| max2q |u|  log  C1 log + C2 , log maxq |u| maxb |u| maxB |u| where B is a ball concentric with b such that kd B ⊂ Q, R = R(B) ∼ s(Q). This implies   maxkd B |u| C1 max2q |u|  C3 . maxq |u| maxB |u| Now, using that R(B) is comparable to s(Q), we repeat the chain argument from the proof of Corollary 2.2.4 to obtain the inequality  γ  1−γ . max |u| max |u|  C max |u| Q

B

2Q

with C and α ∈ (0, 1) which does not depend on B (for B with R(B) ∼ s(Q) the number of balls in the chain is uniformly bounded). Finally,     maxQ |u| maxq |u| γ/C1 maxB |u| γ C C . max2Q |u| maxkd B |u| max2q |u| For large cubes q with s(q)  cd s(Q) the last inequality follows directly from the three balls inequality and the chain argument. Thus we obtain max2Q |u| ∗  a1 Nu (Q) − a2 , (2.5.2) log maxQ |u| where a1 and a2 depend on the ellipticity and Lipschitz constants of the operator only when we assume that s(Q)  1.

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We also consider eigenfunctions on manifolds and define the doubling index for eigenfunctions over cubes in a similar way, to prove that the supremum is finite for this case we can use the monotonicity for the lifted function. 2.6. Remarks on the size of the zero sets of eigenfunctions and the doubling index In this section we first formulate some results that were proved by Donnelly and Fefferman [9]. We assume that M is a real-analytic Riemannian manifold (or that coefficients of the corresponding elliptic operator are real-analytic, see also [23].) Lemma 2.6.1. Let L = div(A∇·) be a uniformly elliptic operator with real analytic coefficients defined in the unit cube Q0 ⊂ Rd+1 . There is constant C = C(L) such that if Lh = 0 in Q0 then for any Q1 such that 4Q1 ⊂ Q0 the size of the zero set of h in Q1 admits the following estimate Hd (Z(h) ∩ Q1 )  CN s(Q1 )d , ∗ (2Q )}. where N = max{1, Nh 1

We don’t know if this lemma remains true for non-analytic case. Suppose that φλ is an eigenfunction on a compact manifold M with real√ analytic metric. Applying Lemma 2.6.1 to the function h(x, t) = φλ (x) exp( λt) on charts and having in mind the bound for the doubling index of h, one obtains the upper bound for Hd (Z(h) ∩ M × [−1, 1]). Moreover, since Z(h) is the cylinder over Z(φ), the upper bound in Yau’s conjecture follows √ Hd−1 (Z(φλ ))  C λ. This part of the conjecture is open for non-analytic manifolds. The best known result, see [25], is based on a non-analytic version of the lemma above, the estimate is Hd (Z(h) ∩ Q1 )  CNA s(Q1 )d for some A = A(d). It implies a polynomial bound in Yau’s conjecture. To obtain the lower bound in Yau’s conjecture on manifolds with real analytic metric, Donnelly and Fefferman proved the following statement. Lemma 2.6.2. Suppose that M is a real-analytic manifold. There exists N0 such that the following is true. If φ = φλ is an eigenfunction on M and M is partitioned into cubes √ −1 with side length ≈ λ , M = ∪q, then for at least half of these cubes q the doubling ∗ (q)  N . index of φ in q is bounded by N0 , Nφ 0 This lemma can be combined with the next one (applied for the lifted function) to give the conjectured lower bound for the size of the zero set of eigenfunctions on real-analytic manifolds. Lemma 2.6.3. Let L = div(A∇·) be a uniformly elliptic operator with smooth coefficients in the unit cube Q0 ⊂ Rd+1 . There exists a function f(N) that depends only on L such

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Quantitative unique continuation

∗ (Q )  N, where Q = 1/4Q , then that if Lh = 0 in Q0 , h(0) = 0 and Nh 0 1 1

Hd (Z(h) ∩ Q1 )  f(N)s(Q1 )d . The last lemma does not require analyticity of the coefficients. A simple quantification of this estimate is known (see remarks in [27]); the statement of 2.6.3 is weaker than Theorem 1.7.1. Detailed discussion of the current state of Yau’s conjecture and related open problems can be found in [29]. We conclude this lecture by formulating an estimate for the size of the zero set from above which is not as precise as the polynomial bound in [25]. It follows from earlier results of Hardt and Simon [17]. Lemma 2.6.4. Let L = div(A∇·) be a uniformly elliptic operator with smooth coefficients in the unit cube Q0 ⊂ Rd+1 . There exists a function F(N) that depends only on L such ∗ (Q )  N, where Q = 1/4Q , then that if Lh = 0 in Q0 , and Nh 0 1 1 Hd (Z(h) ∩ Q1 )  F(N)s(Q1 )d . 2.7. Exercises Exercise 2.7.1. For h harmonic on Rd , define the frequency function of h by rH  (r) , N(r) = 2H(r)  where H(r) = r1−d |x|=r |h(x)|2 ds(x). (1) Show that if h is a homogeneous polynomial of degree n then N(r) = n.  (2) Let h = L k=l pk , where pk is a homogeneous harmonic polynomial of degree k and pl , pL = 0. Show that lim N(r) = l

r→0

and

lim N(r) = L.

r→∞

Remark: l is called the vanishing order of h at the origin. (3) Use the fact that N(r) is a non-decreasing function to prove that  2N(R)  2N(r) R R H(R)   . r H(r) r Exercise 2.7.2 (Applications of the three ball inequality). Suppose that h is a non-constant harmonic function in Rd such that |h|  1 on a half-space {x = (x1 , xd , xd > 0}. Let m(R) = max|x| 0 and α ∈ (0, 1) such that for any R > 0 m(R)  Cm(5R)α . (2) Show that m(R)  c exp(Rβ ) for some β > 0. Exercise 2.7.3 (Log-convex functions). Let m : R+ → R+ be a continuous function. We say that m is log-convex if f(t) = ln(m(exp(t))) is a convex function. (For example if m(x) = xa , a > 0 then f(t) = at and m is log-convex.) Warning: usually a positive function g is called logarithmically convex if log(g) is a convex function.

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(1) Show that if ak are non-negative numbers then n  m(x) = ak xk k=1

is log-convex. Hint: The sum of two log-convex functions is log-convex. (2) Let u be a harmonic function in the unit ball of Rd , we know that ∞  u(x) = |x|k Yk (x/|x|), k=0

where Yk is an eigenfunction of the Laplace-Beltrami operator on the unit sphere S ⊂ Rd . Show that  m(r) = |u(ry)|2 ds(y) S

is log-convex. (3) Let K(x, t) be the heat kernel in Rd , K(x, t) = (4πt)−d/2 exp(−|x|2 /(4t)), and it satisfies the equation ΔK(x, t) = ∂t K(x, t). Suppose that u is a harmonic function in Rd such that u(x) exp(−c|x|2 ) ∈ L2 (Rd ) for any c > 0. Define  |u(x)|2 K(x, t)dt. M(t) = Rd

Compute M  (t) and show that M(m) (t)  0 for any m. Remark: The positivity of all derivatives implies that M(t) is a log-convex function. This convexity was studied by Lippner and Mangoubi [24] for the case of discrete harmonic functions. Exercise 2.7.4 (Reverse Hölder inequality for solutions of elliptic equations). Show that if u is a solution of a uniformly elliptic equation with Lipschitz coefficients, div(A∇u) = 0 in a ball B0 then for some (any) q > 1 there exists Cq (u) such that for any ball B ⊂ 1/2B0  1/q    Cq (u)|B|−1 |u|2 . |B|−1 |u|2q B

B

Remark: It implies that |u|2 is a Muckenhoupt weight and therefore the zero set has zero Lebesgue measure, |Z(u)| = 0. A similar inequality holds for the function  u − |B|−1 B u and together with the Caccioppoli inequality it implies that |∇u|2 is also a Muckenhoupt weight (see [13] for details).

3. Small values of polynomials and solutions of elliptic PDEs Let P be a non-constant polynomial of one complex variable with complex coefficients, P ∈ C[z], P(z) = an zn + an−1 zn−1 + · · · + a1 z + a0 ,

20

Quantitative unique continuation

where aj ∈ C and an = 0. As |z| goes to infinity the behavior of P(z) resembles that of the highest degree term an zn . As we know P(z) has n zeros counting multiplicities and the set {z : |P(z)| < C} is bounded and contains the zeros. We use the following notation Ea (P) = {z : |P(z)| < e−a }. 3.1. Classical results of Cartan and Polya A classical result on the size of the set where a polynomial takes small values is due to H. Cartan. Let Pn denote the set of all polynomials of degree n with leading coefficient 1, Pn = {p(z) = zn + an−1 zn−1 + · · · + a1 z + a0 ∈ C[z]}. Lemma 3.1.1 (Cartan, 1928). Let p ∈ Pn then for any a, α > 0 there exist a finite  −a )α , where r is collection of balls {Bj } such that Ena (p) ⊂ ∪j Bj and j rα j j  e(2e the radius of Bj . In particular, taking α = 2, one obtains that |Ena (p)|  4πe1−2a . This estimate is not sharp as the next result shows. Lemma 3.1.2 (Polya, 1928). Let p ∈ Pn then |Ena (p)|  πe−2a for any a > 0. The last inequality is sharp, the equality is obtained when p(z) = zn . Lemmas of Cartan and Polya deal with polynomials for which the leading coefficient is equal to one and provide estimates of the set of all points of the complex plane where the polynomial is small, the proofs of both lemmas and related results can be found in [30]. We are interested in a local version of such estimates. 3.2. Remez inequality for polynomials Now we consider polynomials with real coefficients on the real line and we do not normalize the leading coefficient. Lemma 3.2.1 (Remez, 1936). Let E be a measurable subset of an interval I of positive measure, |E| > 0. Then for any polynomial Pn ∈ R[x] of degree n   4|I| n max |Pn (x)|  max |Pn (x)| |E| x∈I x∈E More precise inequality and its proof is outlined in the exercises below, see Exercise 3.5.3. The original reference in [33], a proof is also given in a more accessible paper [4]. We reformulate the inequality in the following way   maxx∈E |Pn (x)| 1/n |E|  4|I| , maxx∈I |Pn (x)| for any measurable subset E ⊂ I. We normalize Pn such that maxI |Pn | = 1 and use the notation Ean (Pn ) = {x ∈ R : |Pn (x)| < e−an }. Then the Remez inequality can be written as |Ean (Pn ) ∩ I|  4|I|e−a .

A. Logunov and E. Malinnikova

21

There are interesting generalizations of the Remez inequality, in particular the measure of the set can be replaced by another geometric characteristic; higher dimensional version are also known, we refer the reader to [5, 12]. 3.3. Propagation of smallness result The main result we prove in these lectures is the following quantitative propagation of smallness for solutions of elliptic equation in divergence form. As above we assume that div(A∇·) is a uniformly elliptic operator, A is a symmetric matrix with Lipschitz coefficients on some domain in Rd . We know that a solution to div(A∇h) = 0 cannot vanish on a set of positive measure (see for example Remark after Exercise 2.7.4) and look for a quantitative version of this result. Theorem 3.3.1 ([28]). Let h be a solution of div(A∇h) = 0 in Ω. Assume that |h|  ε

on E ⊂ Ω,

where |E| > 0. Let K be a compact subset of Ω then (3.3.2)

max |h|  C0 sup |h|1−α εα , K

Ω

where C0 > 0 and α ∈ (0, 1) depend on A, |E|, dist(E, ∂Ω), and K. The inequality (3.3.2) can be considered as a version of the three balls theorem where the smallest ball is replaced by a measurable set. The constants in the inequality depend on the measure of the set and the distance from this set to the boundary of Ω but not on the set itself, which could be an arbitrarily wild measurable set. The question whether such inequality holds was asked by Landis, weaker quantitative estimates were obtained by Nadirashvili [32] and Vessella [36]. First, we formulate the following result (Remez inequality for solutions of elliptic PDE, [28]): Claim: Let Q0 be the unit cube in Rd . Assume h is a solution to the equation ˙ div(A∇h) = 0 in 2Q Then for any cube Q ⊂ Q0 and any measurable subset E of Q of positive Lebesgue measure, the following inequality holds   |Q| CN (3.3.3) sup |h|  C sup |h| C , |E| Q E ∗ (Q) is the doubling index defined in (2.5.1). where C depends on A only, and N = Nh

This statement confirms that in some sense solutions of elliptic equations locally behave as polynomials with degree bounded by the multiple of the doubling index. In particular (the lift of) an eigenfunction corresponding to eigenvalue λ √ behaves as a polynomial of degree C λ. This phenomenon was pointed out in the works of Donnelly and Fefferman, see for example [11], where, among other results, an interesting Bernstein type inequality for eigenfunctions is obtained.

22

Quantitative unique continuation

Let us show that (3.3.3) implies Theorem 3.3.1. First we remind that by (2.5.2) exp(a1 N)  ea2 sup |h|(sup |h|)−1 , 2Q

Q

for some a1 , a2 > 0. Suppose that (3.3.3) holds with some constant C, choose C1 = C1 (|E|) such that   |Q| C −1 = ea1 C1 , i.e. C1 = Ca−1 C 1 log(C|Q||E| ). |E| Then

C1

−C1 sup |h|  C sup |h| exp(a1 C1 N)  C2 sup |h| sup |h| Q

E

E

2Q

sup |h|

.

Q

This implies the inequality in the theorem for the case Ω = 2Q and K = Q with α = (C1 + 1)−1 and C0 that depends on |E| and on A but not on h. To obtain the statement of the theorem we use the standard chain argument as in the proof of Corollary 2.2.4. In its turn, the inequality (3.3.3) is equivalent to the following local estimate of the volume of sub-level sets. Lemma 3.3.4. Suppose that div(A∇h) = 0 in 2Q and that supQ |h| = 1. Write ∗ (Q)  1 and N = Nh Ea (h) = {x ∈ Q : |h(x)| < ea }. Then (3.3.5)

|Ea (h)|  Ce−βa/N |Q|,

for some positive C and β that depend on A only. 3.4. Base of induction We prove Lemma 3.3.4 in the next section using double induction on a and N. Now we check the base of the induction, considering two cases a  c0 N and N  N0 . Our aim is to prove the inequality (3.3.5). First we note that for a/N < c0 the inequality holds trivially. Indeed if we choose the constant C = C(β) large enough, we get Ce−βa/N  Ce−βc0  1. Now we want to show that (3.3.5) holds for some β and C if we assume that N is small enough. The lemma below is the base of our induction on N. Lemma 3.4.1. Assume that h satisfies div(A∇h) = 0 in kd Q, supQ |h| = 1 and ∗ (Q)  N . Let E = {x ∈ Q : |h(x)| < e−a }. Then Nh a 0 (3.4.2)

|Ea |  Ce−γa |Q|,

for some γ = γ(N0 , A) and C = C(N0 , A). The estimate on the doubling index implies that sup1/2Q |h|  C(N0 ). We combine this inequality with the oscillation theorem (see Theorem 5.1.5 in Appendix). Recall that oscQ h = supQ h − infQ h.

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23

Theorem 3.4.3. Let L = div(A∇·) be a uniformly elliptic operator in Ω and Lh = 0. There exists τ = τ(s) < 1, depending on s and on the ellipticity constant, such that for any cube Q ⊂ Ω oscsQ h < τ(s) oscQ h

and τ(s) → 0 as s → 0.

Corollary 3.4.4. Assume that h satisfies div(A∇h) = 0 in 2Q, supQ |h| = 1 and ∗ (Q)  N . There exist an integer K, and positive b and m that depend on N , on the Nh 0 0 ellipticity constants of A and on the dimension d such that if Q is partitioned into Kd smaller equal cubes, Q = ∪q, then sup |h|  b

for any q

q

and there exists one cube q0 in the partition such that infq0 |h|  m. ∗ (Q)  N , we get a lower bound on the supremum of |h| on each Proof. Since Nh 0 small cube q. Furthermore, assume that h(x0 ) = maxQ/2 |h|  c(N0 ) (we replace h by −h if necessary) and that K is chosen large enough. We take q0 such that x0 ∈ q0 . Clearly oscQ h  2 and since K/2q0 ⊂ Q by the oscillation theorem we have oscq0 h  2τ(2/K). Then we conclude

inf h = sup h − osc h  c(N0 ) − 2τ(2K−1 )  m, q0

q0

q0

when m < c(N0 )/2 and K is large enough.



In particular, the corollary implies that |{x ∈ Q : |h| < m}|  (1 − K−d )|Q|. Dividing each q once again into smaller cubes, we get on each new cube the supremum of |h| is at least b2 and |{x ∈ Q : |h| < mb}|  (1 − K−d )2 |Q|. Iterating the corollary we see that |{x ∈ Q : |h| < mbl }|  (1 − K−d )l+1 |Q|, when supQ |h| = 1. Thus the estimate (3.4.2) holds for e−a = bl m and γ such that bγ = 1 − K−d , it completes the proof of the Lemma 3.4.1. 3.5. Exercises Exercise 3.5.1. Let f ∈ L2 (T2 ), fL2 = 1. We define the L2 -doubling index of f on a square q by  2 2q |f| n(f, q) = log  . 2 q |f| Assume that T2 is partitioned into K2 equal squares we say that a square is good if n(f, q) < 100. Show that   |f|2  1/2. q good q

Remark: Here the 1/2 is a very rough estimate. Can you can find a better one?

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Quantitative unique continuation

Exercise 3.5.2 (Discrete version of Remez inequality). Use the Remez inequality to show that if P is a polynomial of degree n and S ⊂ I ∩ Z contains n + m points then   4|I| n max |P|. max |P|  m I S Exercise 3.5.3 (Remez inequality for polynomials). Let Tn (x) be the Chebyshev polynomial or degree n, such that Tn (cos θ) = cos(nθ). This sequence can be defined by T0 (x) = 1, T1 (x) = x, Tn+1 (x) = 2xTn (x) − Tn−1 (x). Clearly for each n there is a sequence −1 = xn,0 < xn,1 < · · · < xn,n = 1 such that Tn (xn,k ) = (−1)n−k . Suppose that c > 0 and E ⊂ I = [−1, 1 + c] is a measurable set with |E| = 2. In this exercise we prove that for any polynomial P of degree n max |P|  Tn (1 + c) max |P|. I

E

The equality is obtained for example when E = [−1, 1] and P = Tn . To prove the inequality it is enough to assume that E is open and show that P(1 + c)  Tn (1 + c) max |P|. E

(1) Show that there are points yk in E such that |xn,k − xn,j |  |yk − yj | and 1 + c − xn,k  1 + c − yk for k = 0, . . . , n. (2) Use the Lagrange interpolation formula and the properties of the Chebyshev polynomials to show that P(1 + c)  Tn (1 + c) maxE |P|. (3) Let x > 1, show that Tn (2x − 1)  (4x)n . Remark: This gives a proof of the Remez inequality formulated in the lecture notes. Exercise 3.5.4 (Quantitative unique continuation for harmonic functions). We will use Remez inequality to show the quantitative unique continuation form sets of positive measure for harmonic functions. (1) Suppose that h is a bounded harmonic function in the unit ball B0 . Let r < r0 (d) be small enough. Show that there exists q(r) < 1 and C such that for any integer n there is a polynomial pn having degree at most n such that max |h(x) − pn (x)|  Cq(r)n max |h(x)|.

|x|r

|x|1

Moreover q(r) → 0 as r → 0. (2) Prove that there is r1 = r1 (d) such that if E is a measurable subset of r1 B0 of positive measure, m = |E|, and h is a harmonic function in B0 then max |h|  C(max |h|)α (max |h|)1−α , r1 B0

E

where α depends on m and r1 .

B0

A. Logunov and E. Malinnikova

25

Exercise 3.5.5 (logarithmic capacity). The logarithmic capacity of a compact subset of the complex plane is defined by  1/n min max |p(x)| . cap(K) = lim n→∞

p∈Pn

K

(1) Show that the limit exists. (2) Prove that cap(Ena (p)) = e−a for any p ∈ Pn . (3) Use Polya’s lemma to show that |K|  πcap(K)2 for any compact set K ⊂ C.

4. Proof of propagation of smallness result We now prove Lemma 3.3.4 using double induction on a and N and some iterative argument. We start with some preliminary result on the distribution of the doubling index that will help us to carry on the induction step. 4.1. On distribution of the doubling indices The results on the doubling index that we formulate below are crucial for the proof. Let Q0 be the unit cube in Rd . Assume that f ∈ C(Q0 ) and for any cube q such that 2q ⊂ Q0 define max2q |f| Nf (q) = log . maxq |f| Warning: We have used the notation Nh (r) for the frequency of h in the ball B(0, r) in Section 2. But for the rest of the notes we do not refer to the frequency function and use Nf (q) for the doubling constant of f in a cube q as defined above. Lemma 4.1.1. Let a cube Q ⊂ Q0 be partitioned into Kd equal cubes qi , K  8. Let Nmin = min Nf (qi ) and assume that Nmin is large enough, Nmin  N0 (d). Then i

K Nmin . 8 Proof. Let maxQ/2 |f| = |f(x0 )|, x0 ∈ qi for some i. Then, since Nf (qi )  Nmin , there exists x1 ∈ 2qi such that |f(x1 )|  eNmin |f(x0 )|. At this point, we have x1 ∈ 2qi ⊂ (1/2 + 2/K)Q. We find one of the cubes in the partition for which x1 ∈ q and repeat the step. We end up with a sequence {xj }j such that Nf (Q/2) 

|f(xj )|  ejNmin |f(x0 )| and xj ∈ (1/2 + 2j/K)Q. We repeat this J = K/4 times, such that the last xJ is still in Q. Then max |f|  eKNmin /8 max |f|. Q

which implies the required estimate.

Q/2



For solutions of elliptic equations we can formulate the above result using ∗ (q) the monotonicity of the doubling index and the maximal doubling index Nh defined by (2.5.1).

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Quantitative unique continuation

Corollary 4.1.2. Let L = div(A∇·) be a uniformly elliptic operator in 2Q0 . There exist constants N0 and J0 such that if Lh = 0 in 2Q0 , Q ⊂ Q0 , Q is partitioned into Jd equal cubes qi and J  J0 and N∗ (Q)  N0 , then for at least one cube q in the partition ∗ ∗ (q)  Nh (Q)/2. Nh

We rewrite the inequality (2.5.2) in the following form ∗ (q)  A1 Nh (q) + A2 . Nh (cq)  Nh

Then Corollary 4.1.2 follows immediately from Lemma 4.1.1. Our aim in induction argument is to divide the cube into small cubes and find a sub-cube with small doubling index. 4.2. Choosing the right notation We fix the ellipticity constant Λ > 1 and the Lipschitz constant C and consider second order elliptic operator L = div(A∇·) in the cube 2Q0 , where Q0 is the unit cube in Rd . We vary the parameters N > 1 and a > 0 and aim at proving the estimate (3.3.5). Let m(u, a) = |{x ∈ Q0 : |u(x)| < e−a sup |u|}| Q0

and M(N, a) = sup m(u, a), ∗

where the supremum is taken over all elliptic operators div(A∇·) and functions u satisfying the following conditions in 2Q0 : (i) A(x) = [aij (x)]1i,jd is a symmetric uniformly elliptic matrix with Lipschitz entries and ellipticity and Lipschitz constants bounded by Λ and C respectively, (ii) u is a solution to div(A∇u) = 0 in 2Q0 , ∗ (Q )  N. (iii) Nu 0 Our aim is to show that M(N, a)  Ce−βa/N .

(4.2.1)

The constant β > 0 will be chosen later and will not depend on N. As we remarked in Section 3.4 we can assume that a/N > c0 . By Lemma 3.4.1 we can also assume that N is sufficiently large. The proof now contains two main steps. First, with the help of Corollary 4.1.2 we prove a recursive inequality for M(N, a). Then we show that the recursive inequality implies the exponential bound (3.3.5) by a double induction argument on a and N. 4.2.1. Recursive inequality. (4.2.2)

We show that for some a0 > 0 and s < 1

M(N, a)  (1 − s)M(N/2, a − Na0 ) + sM(N, a − Na0 ).

∗ (Q )  N. Fix a solution u to the elliptic equation div(A∇u) = 0 with Nu 0 d Divide Q0 into J equal subcubes q. Then by Corollary 4.1.2 at least one cube q0

A. Logunov and E. Malinnikova

27

∗ (q )  N/2. We have satisfies Nu 0  |{x ∈ q : |u(x)| < e−a sup |u|}|. m(u, a) = Q0

q

By the definition of the doubling constant we see that sup |u|  c1 J−C1 N sup |u|. q

Q0

Since N is sufficiently large, we can forget about c1 above by increasing C1 and we have sup |u|  e−a0 N sup |u|. q

Q0

Define size(q) := |{x ∈ q : |u(x)| < q |u|}|. We continue to estimate m(u, a) in terms of these sizes   size(q) = size(q0 ) + size(q). m(u, a)  e−a+a0 N sup

q

q =q0

We can estimate the first term by size(q0 )  J−d M(N/2, a − a0 N) using the fact that the restriction of u to the cube 2q corresponds to a solution of another elliptic PDE which can also be written in divergence form with some coefficient matrix which has the same bounds for ellipticity and Lipschitz constants. For the second term, we have  size(q)  (Jd − 1)J−d M(N, a − a0 N) = sM(N, a − a0 N), q =q0

where s = (Jd − 1)J−d < 1. Adding the inequalities for the first and second terms and taking the supremum over u, we obtain the recursive inequality (4.2.2) for M(N, a). 4.3. Recursive inequality implies exponential bound (4.3.1)

We will now prove that

M(N, a)  Ce

−βa/N

for some C large enough and β > 0 small enough by a double induction on N and a. Without loss of generality we may assume N = 2l , where l is an integer number. Suppose that we know (4.3.1) for N = 2l−1 and all a > 0 and now we wish to establish it for N = 2l . By Lemma 3.4.1 we may assume l is sufficiently large. For a fixed l we argue by induction on a with step a0 2l . We may assume that a/N > k0 a0 , where k0 > 0 will be chosen later. For a  k0 a0 N the inequality is true if we choose the constant C large enough. The induction base implies the inequality for k = k0 . We describe the step of the induction from a = (k − 1)a0 2l to a = ka0 2l . By the induction assumption we have M(2l , (k − 1)a0 2l )  Ce−β(k−1)a0 and M(2l−1 , (k − 1)a0 2l )  Ce−2β(k−1)a0 .

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Quantitative unique continuation

We apply the recursive inequality (4.2.2) M(2l , ka0 2l )  C(1 − s)e−2β(k−1)a0 + Cse−β(k−1)a0 . Our goal is to obtain the following inequality (1 − s)e−2β(k−1)a0 + se−β(k−1)a0  e−βka0 for k > k0 and some β > 0. Dividing by e−ka0 β we reduce it to (1 − s)e−βa0 (k−2) + seβa0  1. The last inequality holds with the proper choice of the parameters: once s < 1 and a0 are fixed, we choose β to be small enough so that the second term is less than (1 + s)/2 and then choose k0 sufficiently large that the first term is smaller than (1 − s)/2 when k  k0 . This concludes the induction step and the proof of our main result. More delicate propagation of smallness from sets of codimension smaller then one is discussed in [28]. 4.4. Exercises Exercise 4.4.1. Suppose that Lu = 0 in the unit cube Q0 . (1) Use the oscillation theorem to show that there exists a constant K which depends on the Lipschitz and ellipticity constants for L such that if q is a small cube with Kq ⊂ Q and Z(u) ∩ q = ∅ then maxKq |u|  2. log maxq |u| (2) Show that there exists c and B0 such that if Q0 is partitioned into Bd cubes q, B > B0 and Z(u) ∩ q = ∅ for each q then maxQ |u| Nu (Q/2) = log  cB, maxQ/2 |u| where c depends on K from (1). Exercise 4.4.2. Assume that m : Z+ × Z+ → R+ satisfies m(k, j)  C for j < 4,

m(1, j)  e−j ,

and

1 m(k, j)  m(k − 1, 2(j − 1)) + m(k, j − 1). 4 Prove that m(k, j)  Ce−j . Remark: A similar argument is used to derive the estimate in the lecture notes from the iterative inequality. Exercise 4.4.3 (Remez inequality for eigenfunctions). (1) Let M be a compact manifold. Use the lift and the Remez inequality for solutions of elliptic equations to show that there exists a constant C = C(M) such that for any eigenfunction φλ and any compact set E ⊂ M, we have the following

A. Logunov and E. Malinnikova

inequality.

 max |φλ |  C−1 max |φλ | E

M

|E| C|M|

29

C√λ .

(2) Let M = S2 and B be a small ball on S2 , construct a sequence of eigenon the sphere with λ → ∞ such that supB |φλ |/ supM |φλ | functions φλ √ −c λ. decays as e Exercise 4.4.4. Apply the Remez inequality for solutions of elliptic equations to show that if h is a solution of Lh = 0 in kQ0 then g = log |h| is in BMO and ∗ (Q ). Reminder: A function g is said to have bounded mean gBMO(Q0 )  CL Nh 0 oscillation if there exists a constantC such that 1 |g − cQ |  C |Q| Q for any cube Q and some constants cQ . The smallest C for which the inequality holds is called the BMO-norm of g. In particular if a function g satisfies |{x ∈ Q : |g(x) − cQ | > γ}|  C exp(−Aγ)|Q|, for some cQ then g ∈ BMO and gBMO  c/A.

5. Appendix: Second order elliptic equations in divergence form 5.1. Elliptic operator in divergence form: regularity ond order elliptic equations in divergence form

We study solutions of sec-

Lu := div(A∇u) + cu = 0, where u ∈ W 1,2 (Ω), i.e., |∇u| ∈ L2 (Ω), Ω ⊂ Rd . The matrix A = A(x) is symmetric and uniformly elliptic, i.e., Λ−1 |v|2  (A(x)v, v)  Λ|v|2 for any x ∈ Ω and any v ∈ Rd . First we assume that the elements of A(x) are measurable bounded functions (the boundedness follows from the uniform ellipticity condition). We will assume that c is measurable and bounded, weaker integrability assumptions on c are sufficient for some of the results below. The equation Lu = 0 is understood in the integral sense, similarly, we consider the inequalities Lu  0 and Lu  0. The first classical result is the maximal principle, see for example [14, Theorem 8.1]. We use here the standard notation, u+ = max(u, 0). Theorem 5.1.1 (Maximal principle). Suppose that c  0 and u ∈ W 1,2 (Ω) satisfies Lu  0. Then sup u  sup u+ . Ω

∂Ω

We also use the following classical inequality for gradients of solutions of general elliptic PDEs in divergence form.

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Quantitative unique continuation

Theorem 5.1.2 (Caccioppoli inequality). Suppose that Lu = 0 in Ω, BR ⊂ Ω, r < R. Then    1 ∞ |∇u|2  C + c |u|2 , L 2 (R − r) Br BR where C = C(d, Λ). Classical iteration methods of De Giorgi and Moser imply the following estimates, see [16, Chapter 4] Theorem 5.1.3 (Local boundedness). Suppose that Lu  0 in Ω, 2B ⊂ Ω, then u+ ∈ L∞ loc (Ω) and 1/2   + −1 +2 sup u  C |2B| |u | , 2B

B

where C depends on d, Λ and c∞ . This gives immediately the equivalence of norms Corollary 5.1.4. Suppose that Lu = 0 in 2B0 , where B0 is the unit ball of Rd , then C1 uL2 (B)  uL∞ (B)  C2 uL2 (2B) , where C depends on d, Λ and c∞ . Another part of the regularity theory that goes back to De Giorgi and Moser is the following oscillation theorem (seee [16, Chapter 4]). Theorem 5.1.5 (Oscillation inequality). Let L = div(A∇·) be a uniformly elliptic operator in Ω. There exists q = q(Λ) < 1 such that for any ball B such that 2B ⊂ Ω sup u − inf u < q(sup u − inf u). B

B

2B

2B

The difference supB u − infB u is called the oscillation of the function u in B and denoted by oscB u. A different way to obtain regularity was discovered by Landis (see [21] for details) and developed to elliptic equations is non-divergence form with bounded coefficients by Krylov and Safonov, see [19, 21, 22, 34]. This approach also leads to the oscillation inequality. Finally, we formulate the Harnack inequality of Moser for solutions of elliptic equations in divergence form, see for example [16, Chapter 4]. Theorem 5.1.6 (Harnack inequality). Let u be a non-negative solution to elliptic equation div(A∇u) = 0 in Ω, 2B ⊂ Ω. Then sup u  C inf u, B

B

C = C(d, Λ).

There is a nice proof of the Harnack inequality for solutions of elliptic equations in divergence form that bypasses the classical iteration methods can be found in [34]. Note that in all of the results in this section the constants depend on the ellipticity constant only, thus we may apply the inequalities on small or big scales.

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31

5.2. Comparison to harmonic functions We turn now to elliptic PDEs in divergence form with Lipschitz coefficients. This smoothness assumption allows us to freeze the coefficients and consider the equation as a perturbation of the equation with constant coefficients. Changing coordinates, we can think about constant coefficient elliptic operator as a simple transformation of the usual Laplace operator. More precisely, let u be a solution to div(A∇u) = 0, where A = {aij (x)}, x ∈ Ω and |aij (x) − aij (y)|  C|x − y|. Then for any x0 ∈ Ω, we may choose first a ball Br (x0 ) and then a linear transformation S : Bρ (0) → Br (x0 ) such that f = u ◦ S is a solution of elliptic equation ˜ div(A∇f) = 0 with ˜ A(0) = I, |a˜ ij (y) − δij |  C|y|. Moreover r/ρ is bounded, the bound depends on the ellipticity and Lipschitz constants for A. We mostly study local properties of solutions and then reduce the problem to equation of this specific form. Note that when we apply this idea we get inequalities that hold on small scales, the constants depend on the Lipschitz constants of the coefficients and may grow when we consider large balls. A classical regularity result implies that if u ∈ W 1,2 (Ω) is a weak solution of the divergence form elliptic equation as above (with Lipschitz coefficients) and Ω  ⊂⊂ Ω then u ∈ W 2,2 (Ω  ) and then if ∂Ω  is smooth then by the trace property u, |∇u| ∈ L2 (∂Ω  ). Acknowledgments These notes are based on the lectures given by the second author at Park City Mathematics Institute Summer Program in July 2018. It is a great pleasure to thank the organizers for this great opportunity and wonderful time in Park City. We are grateful to many students and colleagues who attended the lectures and commented on earlier versions on the manuscript. In particular to Paata Ivanisvili, who held problem sessions for these lectures, and to Stine Marie Berge, who carefully read and commented the first version of the lecture notes. Thanks also goes to Jiuyi Zhu and an anonymous referee, whose comments improved the final text.

References [1] Shmuel Agmon, Unicité et convexité dans les problèmes différentiels (French), Séminaire de Mathématiques Supérieures, No. 13 (Été, 1965), Les Presses de l’Université de Montréal, Montreal, Que., 1966. MR0252808 ←9 [2] Frederick J. Almgren Jr., Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), North-Holland, Amsterdam-New York, 1979, pp. 1–6. MR574247 ←9

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[3] P. Bérard and B. Helffer, A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited, Monatsh. Math. 180 (2016), no. 3, 435–468, DOI 10.1007/s00605-015-0788-6. MR3513215 ←6 [4] Borislav Bojanov, Elementary proof of the Remez inequality, Amer. Math. Monthly 100 (1993), no. 5, 483–485, DOI 10.2307/2324304. MR1215537 ←20 [5] A. Brudnyi and Y. Yomdin, Norming sets and related Remez-type inequalities, J. Aust. Math. Soc. 100 (2016), no. 2, 163–181, DOI 10.1017/S1446788715000488. MR3480958 ←21 [6] Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR768584 ←3, 4, 8, 9 [7] Shiu Yuen Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv. 51 (1976), no. 1, 43–55, DOI 10.1007/BF02568142. MR397805 ←8 [8] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR0065391 ←4 [9] Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161–183, DOI 10.1007/BF01393691. MR943927 ←6, 17 [10] Harold Donnelly and Charles Fefferman, Nodal sets for eigenfunctions of the Laplacian on surfaces, J. Amer. Math. Soc. 3 (1990), no. 2, 333–353, DOI 10.2307/1990956. MR1035413 ← [11] H. Donnelly and C. Fefferman, Growth and geometry of eigenfunctions of the Laplacian, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 635–655. MR1044811 ←21 [12] Tamás Erdélyi, Remez-type inequalities and their applications, J. Comput. Appl. Math. 47 (1993), no. 2, 167–209, DOI 10.1016/0377-0427(93)90003-T. MR1237312 ←21 [13] Nicola Garofalo and Fang-Hua Lin, Monotonicity properties of variational integrals, Ap weights and unique continuation, Indiana Univ. Math. J. 35 (1986), no. 2, 245–268, DOI 10.1512/iumj.1986.35.35015. MR833393 ←9, 10, 11, 13, 19 [14] D. Gilbarg and N. Trudinger, Èllipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka (Russian), “Nauka”, Moscow, 1989. Translated from the second English edition by L. P. Kuptsov. MR1063848 ←29 [15] Qing Han, Nodal sets of harmonic functions, Pure Appl. Math. Q. 3 (2007), no. 3, Special Issue: In honor of Leon Simon., 647–688, DOI 10.4310/PAMQ.2007.v3.n3.a2. MR2351641 ←10 [16] Q. Han and F. Lin, Elliptic partial differential equations, American Mathematical Society, 2011. ←30 [17] Robert Hardt and Leon Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989), no. 2, 505–522. MR1010169 ←6, 18 [18] Eric J. Heller. Graphics available at http://ejheller.jalbum.net/EricJHellerGallery/slides/ NodalDomainsI.html. ←4 [19] N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239. MR563790 ←30 [20] Igor Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J. 91 (1998), no. 2, 225–240, DOI 10.1215/S0012-7094-98-09111-6. MR1600578 ←9, 10, 11 [21] E. M. Landis, Some questions in the qualitative theory of second-order elliptic equations (case of several independent variables) (Russian), Uspehi Mat. Nauk 18 (1963), no. 1 (109), 3–62. MR0150437 ←13, 30 [22] E. M. Landis, Second order equations of elliptic and parabolic type, Translations of Mathematical Monographs, vol. 171, American Mathematical Society, Providence, RI, 1998. Translated from the 1971 Russian original by Tamara Rozhkovskaya; With a preface by Nina Uraltseva. MR1487894 ←30 [23] Fang-Hua Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 44 (1991), no. 3, 287–308, DOI 10.1002/cpa.3160440303. ←17 [24] Gabor Lippner and Dan Mangoubi, Harmonic functions on the lattice: absolute monotonicity and propagation of smallness, Duke Math. J. 164 (2015), no. 13, 2577–2595, DOI 10.1215/00127094-3164790. MR3405594 ←19 [25] Alexander Logunov, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Ann. of Math. (2) 187 (2018), no. 1, 221–239, DOI 10.4007/annals.2018.187.1.4. MR3739231 ←7, 17, 18

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[26] Alexander Logunov, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture, Ann. of Math. (2) 187 (2018), no. 1, 241–262, DOI 10.4007/annals.2018.187.1.5. MR3739232 ←7 [27] Alexander Logunov and Eugenia Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, 50 years with Hardy spaces, Oper. Theory Adv. Appl., vol. 261, Birkhäuser/Springer, Cham, 2018, pp. 333–344. MR3792104 ←18 [28] Alexander Logunov and Eugenia Malinnikova, Quantitative propagation of smallness for solutions of elliptic equations, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. III. Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 2391–2411. MR3966855 ←21, 28 [29] A. Logunov and E. Malinnikova, Review of Yau’s conjecture on zero sets of Laplace eigenfunctions, Current developments in mathematics 2018, 2018, pp. 179–212. ←18 [30] D. S. Lubinsky, Small values of polynomials: Cartan, Pólya and others, J. Inequal. Appl. 1 (1997), no. 3, 199–222, DOI 10.1155/S1025583497000143. MR1731340 ←20 [31] Dan Mangoubi, The effect of curvature on convexity properties of harmonic functions and eigenfunctions, J. Lond. Math. Soc. (2) 87 (2013), no. 3, 645–662, DOI 10.1112/jlms/jds067. MR3073669 ←9 [32] N. S. Nadirashvili, Uniqueness and stability of continuation from a set to the domain of solution of an elliptic equation (Russian), Mat. Zametki 40 (1986), no. 2, 218–225, 287. MR864285 ←21 [33] E.J. Remez, Sur une propriété des polynômes de Tchebyscheff, Comm. Inst. Sci. Kharkow 13 (1936), 93–95. ←20 [34] M. V. Safonov, Narrow domains and the Harnack inequality for elliptic equations, Algebra i Analiz 27 (2015), no. 3, 220–237, DOI 10.1090/spmj/1401; English transl., St. Petersburg Math. J. 27 (2016), no. 3, 509–522. MR3570964 ←30 [35] Gerald Teschl, Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, vol. 140, American Mathematical Society, Providence, RI, 2012. MR2961944 ←5 [36] S. Vessella, Quantitative continuation from a measurable set of solutions of elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 4, 909–923, DOI 10.1017/S0308210500000494. MR1776684 ←21 [37] Shing Tung Yau, Problem section, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR645762 ←6 Department of Mathematics, Princeton University, Princeton, NJ, 08544; Email address: [email protected] Current address: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 Department of Mathematical Sciences, Norwegian University of Science and Technology 7491, Trondheim, Norway Email address: [email protected]

IAS/Park City Mathematics Series Volume 27, Pages 35–72 https://doi.org/10.1090/pcms/027/00860

Arithmetic spectral transitions: a competition between hyperbolicity and the arithmetics of small denominators Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang Abstract. These lectures first cover the basics of discrete ergodic Schrodinger operators, with a focus on the 1D quasiperiodic case and the interplay between arithmetics and hyperbolicity. We then present two methods that led to sharp arithmetic spectral transition results. On the localization side, we present a method to prove 1D Anderson localization in the regime of positive Lyapunov exponents, that has, in particular, allowed to solve the arithmetic spectral transition (from absolutely continuous to singular continuous to pure point spectrum) problem for the almost Mathieu operator, in coupling, frequency and phase. On the other end of the arithmetic hierarchy, we present a method to prove quantitative delocalization for 1D operators, leading to sharp arithmetic criterion for the transition to full spectral dimensionality in the singular continuous regime, for the entire class of analytic quasiperiodic potentials.

Contents 1 2

3 4 5 6 7 8

Introduction The basics 2.1 Spectral measure of a selfadjoint operator 2.2 Spectral decompositions 2.3 Ergodic operators 2.4 Schnol’s theorem 2.5 Anderson Localization 2.6 Cocycles and Lyapunov exponents 2.7 Example: The Almost Mathieu Operator 2.8 Continued fraction expansion Basics for the Almost Mathieu Operators First transition line for Diophantine frequencies and phases Asymptotics of the eigenfunctions Universal hierarchical structures Proof of Theorem 5.0.6 Arithmetic criteria for spectral dimension 8.1 m-function and subordinacy theory 8.2 Spectral continuity 8.3 Arithmetic criteria

36 37 37 37 38 38 38 40 42 42 44 45 48 51 55 62 62 64 68

2010 Mathematics Subject Classification. Primary: 47B36; Secondary: 81Q10,37D25. Key words and phrases. Park City Mathematics Institute. ©2020 American Mathematical Society

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Arithmetic spectral transitions

1. Introduction Unlike random, one-dimensional quasiperiodic operators feature spectral transitions with changes of parameters. The transitions between absolutely continuous and singular spectra are governed by vanishing/non-vanishing of the Lyapunov exponent. In the regime of positive Lyapunov exponents there are also more delicate transitions: between localization (point spectrum with exponentially decaying eigenfunctions) and singular continuous spectrum, and dimensional/quantum dynamics transitions within the regime of singular continuous spectrum, governed by the arithmetics. Delicate dependence of spectral properties on the arithmetics is perhaps the most mathematically fascinating feature of quasiperiodic operators, made particularly prominent by Douglas Hofstadter’s famous plot of spectra of the almost Mathieu operators, the Hofstadter’s butterfly [21], see Figure 1.0.1, demonstrating their self-similarity governed by the continued fraction expansion of the magnetic flux.

Figure 1.0.1. Hofstadter’s butterfly This self-similarity is even more remarkable because it appears even in various experimental and quantum computing contexts, see e.g. Figure 1.0.2.

Figure 1.0.2. Photon spectrum simulated using a chain of 9 super-conducting quantum qubits [42]

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37

Remarkably, such self-similarity of both spectra and eigenfunctions were predicted a dozen years before Hofstadter in the work of Mark Azbel [11], which, according to Hofstadter, was way ahead of its time. The self-similar behavior of eigenfunctions reflects the self-similar nature of resonances that are in competition with hyperbolicity provided by the Lyapunov growth. This competition also leads to the sharp transition between pure point (hyperbolicity wins) and singular continuous (resonances win) spectra in the positive Lyapunov exponent regime. In the first three lectures we will outline a method to prove 1D Anderson localization in the regime of positive Lyapunov exponents that has allowed to solve the sharp arithmetic spectral transition problem (from absolutely continuous to singular continuous to pure point spectrum) for the almost Mathieu operator, in coupling, frequency and phase, and to describe the self-similar structure of localized eigenfunctions. The method is an adaptation of [24, 30], but has its roots in [34] and even [32], with an important development in [4]. The last lecture will be devoted to the opposite goal: a method to prove certain delocalization within the regime of singular continuous spectrum (after [27]), that allowed to obtain a sharp arithmetic spectral transition result for the entire class of analytic quasiperiodic potentials.

2. The basics 2.1. Spectral measure of a selfadjoint operator Let H be a selfadjoint operator on a Hilbert space H. The time evolution of a wave function is described in the Schrödinger picture of quantum mechanics by ∂ψ = Hψ. i ∂t The solution with initial condition ψ(0) = ψ0 is given by ψ(t) = e−itH ψ0 . By the spectral theorem, for any ψ0 ∈ H, there is a unique spectral measure μψ0 such that  −itH (2.1.1) (e ψ0 , ψ0 ) = e−itλ dμψ0 (λ). R

2.2. Spectral decompositions

Let H = Hpp



Hsc



Hac , where

Hγ = {φ ∈ H : μφ is γ} and γ ∈ {pp, sc, ac}. Here pp (sc, ac) are abbreviations for pure point (singular continuous, absolutely continuous). The operator H preserves each Hγ , where γ ∈ {pp, sc, ac}. We may then define: σγ (H) = σ(H|Hγ ), γ ∈ {pp, sc, ac}. The set σpp (H) admits a direct characterization as the closure of the set of all eigenvalues σpp (H) = σp (H),

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where σp (H) = {λ : there exists a nonzero vector ψ ∈ H such that Hψ = λψ}. 2.3. Ergodic operators We are going to study discrete one-dimensional Schrödinger operators with potentials related to dynamical systems. Let H = Δ + V be defined by (2.3.1)

(Hu)(n) = u(n + 1) + u(n − 1) + V(n)u(n)

on a Hilbert space H = 2 (Z). Here V : Z → R is the potential. Let (Ω, P) be a probability space. A measure-preserving bijection T : Ω → Ω is called ergodic, if any T -invariant measurable set A ⊂ Ω has either P(A) = 1 or P(A) = 0. By a dynamically defined potential we understand a family Vω (n) = v(T n ω), ω ∈ Ω, where v : Ω → R is a measurable function. The corresponding family of operators Hω = Δ + Vω is called an ergodic family. More precisely, (2.3.2)

(Hω u)(n) = u(n + 1) + u(n − 1) + v(T n ω)u(n).

Theorem 2.3.3 (Pastur [41]; Kunz-Souillard [36]). There exists a full measure set      Ω0 and , pp , sc , ac such that for all ω ∈ Ω0 , we have σ(Hω ) = , and  σγ (Hω ) = γ , γ = pp, sc, ac.  Theorem 2.3.4. [Avron-Simon [10],Last-Simon [38]] If T is minimal, then σ(Hω ) = ,  and σac (Hω ) = ac for all ω ∈ Ω. Theorem 2.3.4 does not hold for σγ (Hω ) with γ ∈ {sc, pp} [26], but whether it holds for σsing (Hω ) = σpp (Hω ) ∪ σsc (Hω ) is an interesting and difficult open problem. 2.4. Schnol’s theorem Let H = Δ + V be a Schrödinger operator on 2 (Z). We say u is a generalized eigenfunction and E is the corresponding generalized eigen1 value if Hu = Eu and |u(n)|  C(1 + |n|) 2 + for some C,  > 0. Theorem 2.4.1 (Schnol’s theorem). Let S be the set of all generalized eigenvalues. For any ψ ∈ 2 (Z), the spectral measure μψ gives full weight to S and σ(Δ + V) = S. Here we modify the definition a little bit to avoid unnecessary notations. We will say that φ is a generalized eigenfunction of H with generalized eigenvalue E, if ˆ + |k|). (2.4.2) Hφ = Eφ, and |φ(k)|  C(1 In the following, we usually normalize φ(k) so that (2.4.3)

φ2 (0) + φ2 (−1) = 1.

2.5. Anderson Localization We say a self-adjoint operator H on 2 (Z) satisfies Anderson localization if H only has pure point spectrum and all the eigenfunctions decay exponentially. By Schnol’s theorem, in order to show the Anderson localization of H, it suffices to prove that all polynomially bounded eigensolutions are exponentially decaying.

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This can be done by establishing exponential off-diagonal decay of Green’s functions. Block-resolvent expansion, a form of which we are about to see, is the backbone of Fröhlich-Spencer’s multi-scale analysis, allowing to pass from smaller to larger scales and from local to global decay. The form we present, first developed for the almost Mathieu operator [32, 34], includes an important modification of multi-scale analysis type arguments, in simultaneously considering shifted boxes. This is the central ingredient in nonperturbative proofs for deterministic potentials [12]. For an interval I ⊂ Z, let GI = (RI (Hx − I)RI )−1 if well defined (GI is called the Green’s function). Definition 2.5.1. Fix τ > 0, 0 < δ < 1/2. A point y ∈ Z will be called (τ, k, δ) regular if there exists an interval [x1 , x2 ] containing y, where x2 = x1 + k − 1, such that |G[x1 ,x2 ] (y, xi )|  e−τ|y−xi | and |y − xi |  δk for i = 1, 2. This definition can be easily made multi-dimensional, with obvious modifications. The following argument is also multi-dimensional but we present a 1D version for simplicity. First note that for Hφ = Eφ, we have φ = GI ΓI φ where ΓI is the decoupling operator at the boundary of I. In one dimensional case this reads (2.5.2)

φ(x) = −G[x1 ,x2 ] (x1 , x)φ(x1 − 1) − G[x1 ,x2 ] (x, x2 )φ(x2 + 1),

where x ∈ I = [x1 , x2 ] ⊂ Z. Theorem 2.5.3. Let h(k) → ∞ as k → ∞. Suppose Hφ = Eφ and φ satisfies (2.4.2). Suppose for any large k ∈ Z, k is (τ, y, δ) regular for some h(k)  y  k. Then H satisfies Anderson localization. Moreover for any eigenfunction, ln |φ(n)|  −τ . lim sup n n Proof. : Under the assumptions, there is some kˆ  δ miny∈[√k,2k] h(y) such that √ for any y ∈ [ k, 2k], there exists an interval I(y) = [x1 , x2 ] ⊂ [−4k, 4k] with y ∈ I(y) such that (2.5.4) dist(y, ∂I(y))  kˆ and (2.5.5)

|GI(y) (y, xi )|  e−τ|y−xi | , i = 1, 2.

Denote by ∂I(y) the boundary of the interval I(y). For z ∈ ∂I(y), let z  be the neighbor of z, (i.e., |z − z  | = 1) not belonging to I(y). √ If x2 + 1 < 2k or x1 − 1 > k, we can expand φ(x2 + 1) or φ(x1 − 1) as (2.5.2). √ We can continue this process until we arrive to z such that z + 1  2k or z − 1  k, or the iterating number reaches [ 2k ˆ ], where [t] denotes the greatest integer less k than or equal to t.

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By (2.5.2),  (2.5.6) φ(k) =



GI(k) (k, z1 )

s zi+1 ∈∂I(z ) i

s 

 GI(z ) (zi , zi+1 ) φ(zs+1 ) , i

i=1

√ where in each term of the summation we have k + 1 < zi < 2k − 1, i = 1, · · · , s, √ 2k / [ k + 2, 2k − 2], s + 1 < [ 2k and either zs+1 ∈ ˆ ]; or s + 1 = [ k ˆ ]. k √ 2k / [ k + 2, 2k − 2], s + 1 < [ kˆ ], by (2.5.5) and noting that we have If zs+1 ∈ |φ(zs+1 )|  (1 + |zs+1 |)C  kC , one has s    GI(k) (k, z1 ) GI(z ) (zi , zi+1 ) φ(zs+1 ) i zi+1 ∈∂I(zi )

(2.5.7)

i=1

 e−τ(|k−z1 |+

s

 i=1 |zi −zi+1 |)

kC

 e−τ(|k−zs+1 |−(s+1)) kC  max{e−τ(k−

√ k−4− 2k ˆ ) k

2k

kC , e−τ(2k−k−4− kˆ ) kC }.

If s + 1 = [ 2k ˆ ], using (2.5.4) and (2.5.5), we obtain k (2.5.8)

ˆ

2k

|GI(k) (k, z1 )GI(z ) (z1 , z2 ) · · · GI(zs ) (zs , zs+1 )φ(zs+1 )|  kC e−τk[ kˆ ] . 1

[ 2k ]

Finally, notice that the total number of terms in ( 2.5.6) is at most 2 kˆ . Combining with (2.5.7) and (2.5.8), since k/kˆ = o(k), we obtain for any ε > 0, |φ(k)|  e−(τ−ε)k for large enough k . For k < 0, the proof is similar. Thus one has (2.5.9)

|φ(k)|  e−(τ−ε)|k| if |k| is large enough.



Therefore we only need to prove that large k ∈ Z, are (τ, h(k), δ) regular for some τ, h, δ. Lemma 2.5.10. Suppose Hφ = Eφ and φ satisfies (2.4.2) and (2.4.3). Then 0 is (τ, k, δ) singular for any τ, δ > 0. 

Proof. It follows from (2.5.2) immediately.

Thus it suffices to show that (τ, k, δ) singular points are sufficiently far apart. 2.6. Cocycles and Lyapunov exponents By a cocycle, we mean a pair (T , A), where an invertible T : Ω → Ω is ergodic, A is a measurable 2 × 2 matrix valued function on Ω and detA = 1. This is what is usually called an SL2 (R) cocycle, but we will simply say “a cocycle”. We can regard it as a dynamical system on Ω × R2 with (T , A) : (x, f) −→ (T x, A(x)f), (x, f) ∈ Ω × R2 . For k > 0, we define the k-step transfer matrix as Ak (x) =

1  l=k

A(T l−1 x).

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41

k For k < 0, define Ak (x) = A−1 −k (T x). Denote A0 = I, where I is the 2 × 2 identity matrix. Then fk (x) = ln ||Ak (x)|| is a subadditive ergodic process. The (non-negative) Lyapunov exponent (LE) for the cocycle (α, A) is given by  1 1 a.e. x ln An (x)dx, ln An (x)dx ===== lim (2.6.1) L(T , A) = inf n n Ω n→∞ n with both the existence and the second equality in (2.6.1) guaranteed by Kingman’s subadditive ergodic theorem. Cocycles with positive Lyapunov exponents are called hyperbolic. Here one should distinguish uniform hyperbolicity where there exists a continuous splitting of R2 into expanding and contracting directions, and nonuniform, where L > 0 but such splitting does not exist. Nevertheless,

Theorem 2.6.2 (Oseledec). Suppose L(T , A) > 0. Then, for almost every x ∈ Ω , there exist solutions v+ , v− ∈ C2 such that ||Ak (x)v± || decays exponentially at ±∞, respectively, at the rate −L(T , A). Moreover, for every vector w which is linearly independent with v+ (resp., v− ), ||Ak (x)w|| grows exponentially at +∞ (resp., −∞) at the rate L(T , A). Suppose u is an eigensolution of Hx u = Eu. Then     u(n + m) u(m) (2.6.3) = An (T m x) , u(n + m − 1) u(m − 1) where An (x) is the transfer matrix of A(x) and

E − v(x) −1 A(x) = . 1 0 Such (T , A(x)) is called the Schrödinger cocycle. Denote by L(E) the Lyapunov exponent of the Schrödinger cocycle (we omit the dependence on T and v). It turns out that (at least for uniquely ergodic dynamics) the resolvent set of H is precisely the set of uniform hyperbolicity of the Schrödinger cocycle. The set σ ∩ {L(E) > 0} is therefore the set of non-uniform hyperbolicity, and is our main interest. Then Oseledec theorem can be reformulated as Theorem 2.6.4. Suppose that L(E) > 0. Then, for every x ∈ ΩE (ΩE has full measure), there exist solutions φ+ , φ− of Hx φ = Eφ such that φ± decays exponentially at ±∞, respectively, at the rate −L(E). Moreover, every solution which is linearly independent of φ+ (resp., φ− ) grows exponentially at +∞ (resp., −∞) at the rate L(E). It turns out that the set where the Lyapunov exponent vanishes fully determines the absolutely continuous spectrum. Theorem 2.6.5 (Ishii-Pastur-Kotani). σac (Hx ) = {E ∈ R : L(E) = 0} every x ∈ Ω.

ess

for almost

The inclusion “⊆” was proved by Ishii and Pastur [22, 41]. The other inclusion was proved by Kotani [35, 43]. Here we give a proof of the Ishii-Pastur part.

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Proof. Denote Z = {E ∈ R : L(E) = 0}. If L(E) > 0, Oseledec’ Theorem says that for almost every x, the eigensolution u(x, E) of Hx u = Eu is either exponentially decaying or exponentially growing. Applying Fubini’s theorem, we see that for almost every x (with respect to P), the set of E ∈ R \ Z for which the property just described fails, has zero Lebesgue measure. In other words, let S1 ⊂ R \ Z be the set with the non-Oseledec behavior. Then S1 has zero Lebesgue measure. It implies that S1 has zero weight with respect to the absolutely continuous part of any spectral measure. Let S2 ⊂ R \ Z be the set with the Oseledec behavior. To prove the Theorem, it suffices to show S2 has zero weight with respect to any ac spectral measure. Indeed, if the solution of Hx u = Eu is exponentially growing at ∞ or −∞, by Schnol’s theorem, such E does not make any contribution to the spectral measure. If the solution of Hx u = Eu is exponentially decaying at both ∞ and −∞, then E is an eigenvalue. The collection of eigenvalues must be countable, which also gives zero weight with respect to the ac spectral measure.  It may seem that positive Lyapunov exponent should imply pure point spectrum with exponentially localized eigenfunctions, since, as above, for every E and a.e. phase a solution, if polynomially bounded, must decay exponentially on both sides. However, this is a flawed argument because, for a given phase, spectral measures may potentially be supported on the zero measure set of E, excluded by the Fubini theorem, for which there may be no such behavior. It turns out this is not a nuisance to disprove in relevant situations, but actually does happen in some of the prominent examples. 2.7. Example: The Almost Mathieu Operator The almost Mathieu operator (AMO) is the (discrete) quasi-periodic Schrödinger operator on 2 (Z): (2.7.1)

(Hλ,α,θ u)(n) = u(n + 1) + u(n − 1) + 2λ cos 2π(θ + nα)u(n),

where λ is the coupling, α is the frequency, and θ is the phase. For the AMO, L(E) can be computed exactly for E on the spectrum, but for now we will just need an estimate L(E)  ln λ for all α ∈ / Q, E (See Theorem 3.0.2 for details). Thus, for λ > 1, Lyapunov exponent is strictly positive on the spectrum. In fact, we will later see that it does not even feel the arithmetics and is constant in the spectrum in both E and α. We now quickly review the basics of continued fraction approximations. 2.8. Continued fraction expansion

Define, as usual, for 0  α < 1,

a0 = 0, α0 = α, and, inductively for k > 0, −1 ak = [α−1 k−1 ], αk = αk−1 − ak .

We define p0 = 0,

q0 = 1,

p1 = 1, q1 = a1 ,

Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

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and inductively, pk = ak pk−1 + pk−2 , qk = ak qk−1 + qk−2 . Recall that {qn }n∈N is the sequence of denominators of best rational approximants to irrational number α, since it satisfies (2.8.1)

for any 1  k < qn+1 , kαR/Z  ||qn α||R/Z .

Moreover, we also have the following estimate, 1 1  Δn  qn αR/Z  . 2qn+1 qn+1 Here, we give several arithmetic conditions on α: (2.8.2)

• α is called Diophantine if there exists κ, ν > 0 such that ||kα||  any k = 0, where ||x|| = min |x − k|. k∈Z • α is called Liouville if − ln ||kα||R/Z ln qn+1 = lim sup (2.8.3) β(α) = lim sup >0 |k| qn n→∞ k→∞

ν |k|κ

for

• α is called weakly Diophantine if β(α) = 0. Clearly, Diophantine implies weakly Diophantine. By Borel-Cantelli lemma, Diophantine α form a set of full Lebesgue measure. Lemma 2.8.4 (Gordon [18], Avron-Simon [9]). Suppose v ∈ C1 (T). There is some constant C such that if β(α) > C, then σpp (Hv,α,θ ) = ∅. Remark: The constant in Lemma 2.8.4 can be estimated in a sharp way [4, 8]. Lemma 2.8.4 is the first indication of the role of arithmetics in the spectral theory of quasiperiodic operators in the regime of positive LE, as it demonstrates the necessity of imposing an arithmetic condition. Let us now denote Pk (x) = det(R[0,k−1] (Hx − E)R[0,k−1] ). It is easy to check by induction that Pk (x) (2.8.5) Ak (x) = Pk−1 (x)

−Pk−1 (T x) −Pk−2 (T x)

.

Thus in the regime of positive L(E), Pk “typically” behaves as ekL(E) . By Cramer’s rule, for given x1 and x2 = x1 + k − 1, with y ∈ I = [x1 , x2 ] ⊂ Z, one has P y+1 x) x2 −y (T (2.8.6) |GI (x1 , y)| = Pk (T x1 x) and (2.8.7)

Py−x1 (T x1 x) . |GI (y, x2 )| = Pk (T x1 x)

44

Arithmetic spectral transitions

Thus if Pk indeed hadn’t deviated much from ekL(E) , we would immediately have exponential decay of both terms. It turns out that for uniquely ergodic T there are no bad deviations for the numerator. Lemma 2.8.8 ([15]). Suppose T is uniquely ergodic, continuous and A is continuous. Then 1 (2.8.9) L(T , A) = lim sup ln An (x). n→∞ x∈Ω n Under the assumptions of Lemma 2.8.8, we have for ε > 0, |Pk (θ)|, Ak (x)  e(L+ε)k , for k large enough.

(2.8.10)

Thus all deviations can only happen on the lower side. We denote the large   deviation set by Ak, = {x : |Pk (x)| < exp (k + 1)(L − ) }. Lemma 2.8.11. Assume x is (L − , k, 14 )-singular. Then, for large k, we can choose / Ak, 1 + for any 1 > 0. j ∈ Ik,x = [x − 3k/4, x − k/4] so that T j+k−1 x ∈ 4

1

1 4 )-singular

Thus, two (L − , k, points x1 , x2 such that Ik,x1 and Ik,x2 do not intersect, produce two long strings of consecutive iterations that fall into the large deviation set.

3. Basics for the Almost Mathieu Operators It is easy to see that Pk (θ) is an even function of θ + 12 (k − 1)α and can be written as a polynomial of degree k in cos 2π(θ + 12 (k − 1)α) : Pk (θ) =

k 

1 1 cj cosj 2π(θ + (k − 1)α)  Qk (cos 2π(θ + (k − 1)α)), 2 2

j=0

where Qk is an algebraic polynomial of degree k. For the almost Mathieu operator, the transfer matrix is given by (3.0.1)

0 

Ak (θ) =

and A(θ) =

A(θ + jα) = A(θ + (k − 1)α)A(θ + (k − 2)α) · · · A(θ)

j=k−1

E − 2λ cos 2πθ −1



. 1 0 By Herman’s trick [12,20], we get the following lower bound estimate for λ > 1,

Theorem 3.0.2. (3.0.3)



 T

(ln |Pk |)dθ  k ln λ;

T

(ln ||Ak ||)dθ  k ln λ.

For the AMO, the Lyapunov exponent on the spectrum actually can be obtained explicitly. Theorem 3.0.4 ([13]). For every α ∈ R\Q, λ > 0 and E ∈ σ(Hλ,α,θ ), one has Lλ,α (E) = max{ln λ, 0}.

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Moreover, one can even compute the Lyapunov exponent L of a complexified cocycle A(x + i). It leads to the following three cases (see [2, 3] for more general definitions). Subcritical: λ < 1. In this case, we have L (E) = 0 for E ∈ σ(Hλ,α,θ ) and ln λ . Hλ,α,θ has purely ac spectrum [1, 5].   −2π Critical: λ = 1. In this case, it can be shown that L(E) = 0 for E ∈ σ(Hλ,α,θ ), but L (E) > 0 for E ∈ σ(Hλ,α,θ ) and  > 0. Hλ,α,θ has purely sc spectrum [6, 7, 23, 37]. Supercritical: λ > 1. L(E) = ln λ > 0 for E ∈ σ(Hλ,α,θ ). In these lectures, we are interested only in the supercritical regime, λ > 1. In the following we always assume

E ∈ σ(Hλ,α,θ ).  The fact that Pk (θ) = Qk cos 2π θ + 12 (k − 1)α , hence is a polynomial in   cos 2π θ + 12 (k − 1)α allows the use of the following Lagrange interpolation trick.   x−xi Note that by Lagrange interpolation, Qk (x) = k j=1 i =j Qk (xj ) xj −xi . Thus if θi , i = 1, ..., k + 1, are in the large deviation set, we must have for some i, max

k+1 

x∈[−1,1]

j=1,j =i

|x − cos 2πθj | > ek . | cos 2πθi − cos 2πθj |

This motivates Definition 3.0.5. We say that the set {θ1 , · · · , θk+1 } is -uniform if (3.0.6) 

max

k+1 

max

x∈[−1,1] i=1,··· ,k+1

j=1,j =i

|x − cos 2πθj |  ek . | cos 2πθi − cos 2πθj |

This is a convenient way of stating that the θi have low discrepancy since ln |a − cos 2πx|dx = − ln 2 for any a ∈ [−1, 1]. We have the following Lemma.

Lemma 3.0.7. Suppose {θ1 , · · · , θk+1 } is 1 -uniform. Then there exists a θi in the set / Ak, , if  > 1 and k is sufficiently large. {θ1 , · · · , θk+1 } such that θi − k−1 2 α∈ We also have Lemma 3.0.8. [4, Lemma 9.7] Let α ∈ R\Q, x ∈ R and 0  0  qn − 1 be such that | sin π(x + 0 α)| = inf0qn −1 | sin π(x + α)|, then for some absolute constant C > 0, (3.0.9)

−C ln qn 

q n −1

ln | sin π(x + α)| + (qn − 1) ln 2  C ln qn .

=0, =0

4. First transition line for Diophantine frequencies and phases We already know that non-Diophantine frequencies are trouble for localization, so let’s fix a Diophantine α. It turns out, somewhat surprisingly, that the phase θ matters as well.

46

Arithmetic spectral transitions

• θ is called Diophantine with respect to α (or just α−Diophantine) if there ν exists κ, ν > 0 such that ||2θ + kα||  |k| κ for any k = 0. • θ is called Liouville with respect to α if − ln ||2θ + kα||R/Z >0 δ(α, θ) = lim sup |k| k→∞ • θ is called weakly Diophantine with respect to α if δ(α, θ) = 0. By Borel-Cantelli lemma, for fixed α, the set of α−Diophantine has full Lebesgue measure. We have Lemma 4.0.1 (J.-Simon [26]). For even functions v ∈ C1 (T), there exists some constant C > 0 such that if δ(α, θ) > C, then σpp (Hv,α,θ ) = ∅. Thus we need Diophantine-type conditions on both α and θ. In this section, we will prove Theorem 4.0.2. Suppose α is Diophantine and θ is Diophantine with respect to α. Then the almost Mathieu operator Hλ,α,θ satisfies Anderson localization. Remark 4.0.3. • Theorem 4.0.2 was proved in [34]. Here the frame of the proof follows [34], with some modifications from [30, 39, 40]. • Actually, the proof of Theorem 4.0.2 holds also for weakly Diophantine frequencies and phases. Let E be a generalized eigenvalue with generalized eigenfunction φ. Without loss of generality, assume φ(0) = 1 (sometimes we assume φ2 (0) + φ2 (1) = 1). Take k > 0. Let n be such that qn  k4 < qn+1 . Set I1 and I2 as follows: I1 = [−qn , qn − 1]

(4.0.4) and (4.0.5)

I2 = [k − qn , k + qn − 1].

The set {θj }j∈I1 ∪I2 consists of 4qn elements, where θj = θ + jα and j ranges through I1 ∪ I2 . Since α is Diophantine, one has qn+1  kC , k  qC n. Theorem 4.0.6. For any ε > 0, the set {θj }j∈I1 ∪I2 is ε-uniform if n is sufficiently large. Proof. We first estimate the numerator in (3.0.6). In (3.0.6), let x = cos 2πa and take the logarithm. One has  ln | cos 2πa − cos 2πθj | j∈I1 ∪I2 ,j =i

(4.0.7)

=



ln | sin π(a + θj )| +

j∈I1 ∪I2 ,j =i

=



+



ln | sin π(a − θj )| + (4qn − 1) ln 2

j∈I1 ∪I2 ,j =i

+



−

+ (4qn − 1) ln 2,

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47

  where, in the final line, + and − are the corresponding sums from the second   line. Both + and − consist of 4 terms of the form of (3.0.9), plus 4 terms of the form (4.0.8)

ln

min

j=0,1,··· ,qn −1

| sin π(x + jα)|,

minus ln | sin π(a ± θi )|. There exists an interval of length qn containing i, in both sums. By the minimality, the minimum over this interval is not more than   ln | sin π(a ± θi )| (). Thus, using (3.0.9) 4 times for each of + and − , one has  (4.0.9) ln | cos 2πa − cos 2πθj |  −4qn ln 2 + C ln qn . j∈I1 ∪I2 ,j =i

The estimate of the denominator of (3.0.6) requires a bit more work. Without loss of generality, assume i ∈ I1 . In (4.0.7), let a = θi . We obtain  ln | cos 2πθi − cos 2πθj | j∈I1 ∪I2 ,j =i

(4.0.10)

=



ln | sin π(θi + θj )| +

j∈I1 ∪I2 ,j =i

=

+

+



(4.0.11)

ln | sin π(θi − θj )| + (4qn − 1) ln 2

j∈I1 ∪I2 ,j =i



where now



+

and



=

−

+ (4qn − 1) ln 2,

ln | sin π(2θ + (i + j)α)|,

j∈I1 ∪I2 ,j =i



(4.0.12)



−



=

ln | sin π(i − j)α|.

j∈I1 ∪I2 ,j =i



. First I1 ∪ I2 can be represented as a disjoint union of We first estimate + four segments Bj , each of length qn . Applying (3.0.9) to each Bj , we obtain   > −4qn ln 2 + ln | sin πθˆ j | − C ln qn − ln | sin 2π(θ + iα)|, (4.0.13) +

j∈J1 ∪J2

where (4.0.14)

| sin πθˆ j | = min | sin π(2θ + ( + i)α)|. ∈Bj

By the fact that θ is Diophantine with respect to α, we have (4.0.15) ln | sin πθˆ j |  −C ln |k|  −C ln qn . Putting (4.0.13)–(4.0.15) together, we have 

(4.0.16) Now let us estimate and i, j ∈ I1 ∪ I2 , (4.0.17)

 −

+

> −4qn ln 2 − C ln qn .

. By the fact that α is Diophantine , we have for i = j,

ln | sin π(θi − θj )|  ln |k|−C  −C ln qn .

48

Arithmetic spectral transitions

Replacing (4.0.15) with (4.0.17) and using the same argument as for a similar estimate, 

(4.0.18)

−

 +

, we get

> −4qn ln 2 − C ln qn .

From (4.0.10), (4.0.16) and (4.0.18), we have for any ε > 0,

 | cos 2πa − cos 2πθj | < e(4qn −1)ε , (4.0.19) max | cos 2πθi − cos 2πθj | i∈I1 ∪I2 j∈I1 ∪I2 ,j =i



for n large enough.

Theorem 4.0.20. Fix any ε > 0. For any large k ∈ Z, k is (ln λ − ε, y, 14 ) regular for 1 some k C  y  k. 1

Proof. Define I1 and I2 as in (4.0.4) and (4.0.5). Take y = 4qn . Then, k C  y  k. By Lemma 3.0.7, there exists some j0 with j0 ∈ I1 ∪ I2 such that 4qn − 1 α∈ / A4qn −1,ε . θj0 − 2 By Lemma 2.8.11, for all j ∈ I1 , θj − 4qn2−1 α ∈ / A4qn −1,ε . Thus we have j0 ∈ I2 . Set I = [j0 − 2qn + 1, j0 + 2qn − 1] = [x1 , x2 ]. By (2.8.6), (2.8.7) and (2.8.10), it is easy to verify |GI (k, xi )|  exp{(ln λ + ε)(4qn − 1 − |k − xi |) − 4qn (ln λ − ε)}. Notice that |k − xi |  qn , so we obtain (4.0.21)

|GI (k, xi )|  exp{−(ln λ − ε)|k − xi |}.



Proof of Theorem 4.0.2. This Theorem now follows by combining Theorems 2.5.3  and 4.0.20.

5. Asymptotics of the eigenfunctions and proof of the second spectral transition line conjecture By Theorem 2.6.5, Hλ,α,θ does not have ac spectrum for λ > 1. Lemmas 2.8.4 and 4.0.1 imply that Hλ,α,θ has purely singular continuous spectrum if δ(α, θ) or β(α) is large, and we proved that there is Anderson localization if β = δ = 0. Is there a sharp transition? The reason large β or δ are trouble is because they lead to resonances: eigenvalues of box restrictions that are too close to each other in relation to the distance between the boxes, leading to small denominators in various expansions. Indeed, large β leads to almost repetitions of the potential, and large δ to almost reflections. In both these cases, the strength of the resonances is in competition with the exponential growth controlled by the Lyapunov exponent. It was conjectured by the author in 1994 [33] that for the almost Mathieu family the two types of resonances discussed above are the only ones that appear, and that the competition between the Lyapunov growth and resonance strength resolves, in both cases, in a sharp way.

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Conjecture 1: 1a: (Diophantine phase) Hλ,α,θ satisfies Anderson localization if λ > eβ(α) and δ(α, θ) = 0, and Hλ,α,θ has purely singular continuous spectrum for all θ if 1 < λ < eβ(α) . 1b: (Diophantine frequency) Suppose β(α) = 0. Then Hλ,α,θ satisfies Anderson localization if λ > eδ(α,θ) , and has purely singular continuous spectrum if 1 < λ < eδ(α,θ) . Conjecture 1a says that without phase resonances, if the Lyapunov exponent beats the frequency resonance, Anderson localization follows. . Conjecture 1b says that without frequency resonances, if the Lyapunov exponent beats the phase resonance, then Anderson localization follows. Otherwise, in both cases, Hλ,α,θ has purely singular continuous spectrum. In order to simplify the presentation, we assume ln qn+1 = β(α). (5.0.1) lim n→∞ qn Given α ∈ R\Q we define functions f, g : Z+ → R+ in the following way. Let qn+1 pn qn qn be the continued fraction approximants to α. For any 2  k < 2 , define f(k), g(k) as follows: for   1, let −(ln λ− r¯ n  =e

Set also (5.0.2)

r¯ n 0

ln qn+1 ln  qn + qn )qn

.

= 1 for convenience. If qn  k < ( + 1)qn with   0, set

 

−|k−(+1)qn | ln λ n + e r¯ +1 , f(k) = e−|k−qn | ln λ r¯ n 

and

q  −|k−(+1)qn | ln λ qn+1 g(k) = e−|k−qn | ln λ n+1 + e . r¯ n r¯ n  +1

(5.0.3)

The graphs of these functions are shown in Figures 5.0.4 and 5.0.5. r¯ n  f(k)

r¯ n +2

r¯ n +4

qn 2

qn

( + 1)qn ( + 2)qn ( + 3)qn ( + 4)qn

Figure 5.0.4. Graph of f(k).

qn+1 2

k

50

Arithmetic spectral transitions qn+1 r¯ n +4

g(k) qn+1 r¯ n +2

qn+1 r¯ n 

qn 2

qn

( + 1)qn ( + 2)qn ( + 3)qn ( + 4)qn

qn+1 2

k

Figure 5.0.5. Graph of g(k). Theorem 5.0.6. [30] Let α ∈ R\Q be such that λ > eβ(α) . Suppose θ is Diophantine with respect to α, E is a generalized eigenvalue of Hλ,α,θ and φ is the generalized  φ(k)  . Then for any ε > 0, there exists K (depending on eigenfunction. Let U(k) = φ(k−1) ˆ λ, α, C, ε and Diophantine constants κ, ν) such that for any |k|  K, U(k) and Ak satisfy (5.0.7)

f(|k|)e−ε|k|  ||U(k)||  f(|k|)eε|k| ,

and (5.0.8)

g(|k|)e−ε|k|  ||Ak ||  g(|k|)eε|k| .

By (2.8.3), Theorem 5.0.6 implies the following Theorem. Theorem 5.0.9. [30]Suppose θ is Diophantine with respect to α. Then 1. Hλ,α,θ has Anderson localization if λ > eβ(α) . 2. Hλ,α,θ has purely singular continuous spectrum if 1 < λ < eβ(α) . 3. Hλ,α,θ has purely absolutely continuous spectrum if λ < 1. Remark 5.0.10. (1) Part 1 of Theorem 5.0.6 holds for δ(α, θ) = 0. (2) Part 2 is known for all α, θ [1] and is included here for completeness. (3) Part 3 is known for all α, θ [8] and is included here for completeness. (4) Parts 1 and 2 of Theorem 5.0.6 verify the frequency half of the conjecture in [33]. The measure theoretic version was proved in [8, 28]. Corollary 5.0.11. Under the conditions of Theorem 5.0.6, we have ln ||Ak || = ln λ. (I) lim sup k k→∞ ln ||Ak || (II) lim inf = ln λ − β. k k→∞ − ln ||U(k)|| = ln λ. (III) lim sup k k→∞ − ln ||U(k)|| (IV) lim inf = ln λ − β. k k→∞

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Now let us move to the Diophantine frequency case. Theorem 5.0.12. [24] Suppose α is Diophantine. We have 1. Hλ,α,θ has Anderson localization if λ > eδ(α,θ) . 2. Hλ,α,θ has purely singular continuous spectrum if 1 < λ < eδ(α,θ) . 3. Hλ,α,θ has purely absolutely continuous spectrum if λ < 1. Remark (1) Parts 1 and 2 of Theorem 5.0.12 hold for weakly Diophantine α. (2) We can prove part 2 for all irrational α, and general Lipschitz v. (3) Parts 1 and 2 of Theorem 5.0.12 verify the phase half of the conjecture stated in [33]. For the Diophantine frequencies case, we can also get the asymptotics of the eigenfunctions and transfer matrices. For simplicity, we only give the asymptotics of eigenfunctions. For any , let x0 (we can choose any one if x0 is not unique) be such that | sin π(2θ + x0 α)| = min | sin π(2θ + xα)|. |x|2||

Let η = 0 if 2θ + x0 α ∈ Z, otherwise let η ∈ (0, ∞) be given by the following equation, (5.0.13)

| sin π(2θ + x0 α)| = e−η|| .

Define fˆ : Z → R+ as follows. ˆ = e−|| Case 1: If x0 ·   0, set f()  ln λ .  − (|x0 |+|−x0 |) ln λ ˆ eη|| + e−|| ln λ . Case 2: If x0 ·  > 0, set f() = e Theorem 5.0.14. [24] Suppose α is Diophantine. Assume ln λ > δ(α, θ). If E is a generalized eigenvalue and φ is the corresponding generalized eigenfunction of Hλ,α,θ , then for any ε > 0, there exists K such that for any ||  K, U() satisfies −ε|| ε|| ˆ ˆ  ||U()||  f()e . (5.0.15) f()e

6. Universal hierarchical structure for Diophantine phases and universal reflective-hierarchical structure for Diophantine frequencies In this section, we will describe the universal hierarchical structure of the eigenfunctions in the Diophantine phase case. For Diophantine frequencies there is another, also universal, structure, conjectured to hold, for a.e. phase for all even functions, that features reflective-hierarchy. We refer the readers to [24] for the description of universal relective-hierarchical structure. Note that Theorem 5.0.6 holds around arbitrary point k = k0 . This implies the self-similar nature of the eigenfunctions: U(k) behaves as described at scale qn but when seen in windows of size qk , qk < qn−1 will demonstrate the same universal behavior around appropriate local maxima/minima.

52

Arithmetic spectral transitions

 φ(k)  To make the above precise, let φ be an eigenfunction, and U(k) = φ(k−1) . j Let Iσ1 ,σ2 = [−σ1 qj , σ2 qj ], for some 0 < σ1 , σ2  1. We will say k0 is a local j-maximum of φ if ||U(k0 )||  ||U(k)|| for k − k0 ∈ Ijσ1 ,σ2 . Occasionally, we will also use terminology (j, σ)-maximum for a local j-maximum on an interval Ijσ,σ . We will say a local j-maximum k0 is nonresonant if κ ||2θ + (2k0 + k)α||R/Z > , qj−1 ν for all |k|  2qj−1 and ||2θ + (2k0 + k)α||R/Z >

(6.0.1)

κ , |k|ν

for all 2qj−1 < |k|  2qj . We will say a local j-maximum is strongly nonresonant if κ (6.0.2) ||2θ + (2k0 + k)α||R/Z > ν , |k| for all 0 < |k|  2qj . An immediate corollary of (the proof of) Theorem 5.0.6 is the universality of behavior at all (strongly) nonresonant local maxima. Theorem 6.0.3. Given ε > 0, there exists j(ε) < ∞ such that if k0 is a local j-maximum for j > j(), then the following two statements hold: If k0 is nonresonant, then ||U(k0 + s)||  f(|s|)eε|s| , (6.0.4) f(|s|)e−ε|s|  ||U(k0 )|| q

for all 2s ∈ Ijσ1 ,σ2 , |s| > j−1 2 . If k0 is strongly nonresonant, then (6.0.5)

f(|s|)e−ε|s| 

||U(k0 + s)||  f(|s|)eε|s| , ||U(k0 )||

for all 2s ∈ Ijσ1 ,σ2 . Theorem 5.0.6 also guarantees an abundance (and a hierarchical structure) of local maxima of each eigenfunction. Let k0 be a global maximum . We first describe the hierarchical structure of local maxima informally. We will say that a scale nj0 is exponential if ln qnj +1 > cqnj .1 Then there is a 0 0 constant scale nˆ 0 thus a constant C := qnˆ 0 +1 , such that for any exponential scale nj and any eigenfunction there are local nj -maxima within distance no more cqnj 0 . Moreover, these are the only local than C of k0 + sqnj for each 0 < |s| < e 0 cqnj cqnj 0 0 ]. The exponential behavior , k0 + e nj0 -maxima in the interval [k0 − e of the eigenfunction in the local neighborhood (of size qnj ) of each such local 0 maximum, normalized by the value at the local maximum is given by f. Note that only exponential behavior at the corresponding scale is determined by f and fluctuations of much smaller size are invisible. 1 Note

that per our simplifying assumption (5.0.1) all scales n are exponential.

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Global maximum b1,1 b1,2

b1,−1

b2,1 b2,2

b−2

b−1

k0

b1

b2

Local maximum of depth 1

b1,1,−1

b1,1,1 b1,1,2 b1,2,1 b1,2,2

b1,−2 b1,−1 b1 b1,1 b1,2 Figure 6.0.6. Universal hierarchical structure of an eigenfunction. Above, a view centered on a global maximum with a window centered on a local maximum of depth 1. Below, a magnified view of this local window looks very much like the global view. Now, let nj1 < nj0 be another exponential scale. Denote the “depth 1” local maximum located near k0 + anj qnj by banj . Near it, we then have a simi0 0 0 lar picture: there are local nj1 -maxima in the vicinity of banj + sqnj for each 1 0 cqnj 1 . Again, this describes all the local qn -maxima within an expo0 < |s| < e j1 nentially large interval. And again, the exponential (for the nj1 scale) behavior

54

Arithmetic spectral transitions

in the local neighborhood (of size qnj ) of each such local maximum, normal1 ized by the value at the local maximum is given by f. Denoting those “depth 2” local maxima located near banj + anj qnj , by banj ,anj we then get the same 1 1 0 0 1 picture taking the magnifying glass another level deeper and so on. At the end we obtain a complete hierarchical structure of local maxima that we denote by banj ,anj ,...,anj with each “depth s + 1" local maximum banj ,anj ,...,anj being s s 0 0 1 1 in the corresponding vicinity of the “depth s" local maximum banj ,anj ,...,anj 0 1 s−1 and with universal behavior at the corresponding scale around each. The accuracy of thge approximations gets lower with each level, yet the depth of the hierarchy that can be so achieved is at least j/2 − C. The upper half of Figure 6.0.6 schematically illustrates the structure of local maxima of depth one and two; the lower half shows that the view around a local maximum appropriately magnified looks like a view of the global maximum. We now describe the hierarchical structure precisely. Suppose κ (6.0.7) ||2(θ + k0 α) + kα||R/Z > ν , |k| for any k ∈ Z\{0}. Fix 0 < σ,  with σ + 2 < 1. Let nj → ∞ be such that ln qnj +1  (σ + 2) ln λqnj . Let c = cj =

(ln qnj +1 − ln |anj |)

−.

ln λqnj

σ ln λqnj

We have cj >  for 0 < anj < e

. Then we have

Theorem 6.0.8. There exists n ˆ 0 (α, λ, κ, ν, ) < ∞ such that for any j0 > j1 > · · · > jk , σ ln λqnj i , i = 0, 1, . . . , k, for all 0  s  k there exists njk  nˆ 0 + k, and 0 < anj < e i n a local njs -maximum banj ,anj ,...,anj on the interval banj ,anj ,...,anj + Icjjs,1 for all s s s 0 0 1 1 0  s  k such that the following holds: I: |banj − (k0 + anj qnj )|  qnˆ 0 +1 , 0 0 0 II: For any 1  s  k, |banj

0

,anj ,...,anj

s

1

− (banj

0

,anj ,...,anj 1

s−1

+ anjs qnjs )|  qnˆ 0 +s+1 .

nj

III: If 2(x − banj ,anj ,...,anj ) ∈ Icj k,1 and |x − banj ,anj ,...,anj |  qnˆ 0 +k , then 0 0 1 k 1 k k for each s = 0, 1, ..., k, (6.0.9)

f(xs )e−ε|xs | 

||U(x)|| ||U(banj

0

where xs = |x − banj

0

,anj ,...,anj

s

1

)||

,anj ,...,anj 1

s

 f(xs )eε|xs | ,

| is large enough.

Moreover, every local njs -maximum on the interval banj ,anj

1

is of the form banj

0

,anj ,...,anj 1

,...,anj

s

 ln λqnj

+ [−e

s−1

for some anjs .

s

 ln λqnj

,e

s

]

Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

55

7. Proof of Theorem 5.0.6 Define bn = qtn with 89  t < 1 (t will be defined later). For any k > 0, we will distinguish two cases with respect to n: (i) |k − qn |  bn for some   1, called n−resonance. (ii) |k − qn | > bn for all   0, called n−nonresonance. Let s be the largest integer such that 4sqn−1  dist(y, qn Z). Theorem 7.0.1. Assume λ > eβ(α) and that θ is Diophantine with respect to α. Suppose that either (1) bn  |y| < Cbn+1 , where C > 1 is a fixed constant, or, (2) 0  |y| < qn . Then for any ε > 0 and n large enough, if y is n−nonresonant, we have y is (ln λ + 8 ln(sqn−1 /qn )/qn−1 − ε, 4sqn−1 − 1, 14 ) regular. ln q

Proof. We again assume for simplicity lim qn+1 = β(α) > 0. Then we have s > 0 n for large n. For an n-nonresonant y in the Theorem, one has (7.0.2)

min

j,i∈I1 ∪I2

ln | sin π(2θ + (j + i)α)|  −C ln qn .

and (7.0.3)

min

i =j;i,j∈I1 ∪I2

ln | sin π(j − i)α|  −C ln qn .

The idea modeled on the proof of Theorem 4.0.6 so we use the same notations.  The upper bound of j∈I1 ∪I2 ,j =i ln | cos 2πa − cos 2πθj | is the same as (4.0.9).  ˆ (4.0.10)-(4.0.12) also hold. However the estimate of j∈J1 ln | sin πθj | is much more difficult in the non-Diophantine case. Here we sketch the argument. Assume that θˆ j+1 = θˆ j + qn α for every j, j + 1 ∈ J1 . Applying the Stirling formula and (7.0.2), one has s   jΔn − C ln qn ln | sin 2πθˆ j | > 2 ln C j∈J1 j=1 (7.0.4) s > 2s ln − Cs ln qn . qn+1 In the other cases, decompose J1 in maximal intervals Tκ such that for j, j + 1 ∈ Tκ we have θˆ j+1 = θˆ j + qn α. Notice that the boundary points of an interval Tκ Δ are either boundary points of J1 or satisfy θˆ j R/Z + Δn  n−1 2 . This follows ˆ ˆ from the fact that if 0 < |z| < qn , then θj + qn αR/Z  θj R/Z + Δn , and θˆ j + (z + qn )αR/Z  zαR/Z − θˆ j + qn αR/Z  Δn−1 − θˆ j R/Z − Δn . Assuming Δ Tκ = J1 , then there exists j ∈ Tκ such that θˆ j R/Z  n−1 2 − Δn . Δn−1 , then If Tκ contains some j with θˆ j R/Z < 10 |Tκ |  (7.0.5) 

Δn−1 2

− Δn − Δn

Δn−1 10

1 Δn−1 − 1  s − 1, 4 Δn

56

Arithmetic spectral transitions

where |Tκ | = b − a + 1 for Tκ = [a, b]. For such Tκ , a similar estimate to (7.0.4) gives  |Tκ | ln | sin πθˆ j | > |Tκ | ln − Cs ln qn qn+1 j∈T κ (7.0.6) s > |Tκ | ln − Cs ln qn . qn+1 Δ

n−1 If Tκ does not contain any j with θˆ j R/Z < 10 , then by (2.8.2)  ln | sin πθˆ j | > −|Tκ | ln qn − C|Tκ |

j∈Tκ

(7.0.7)

> |Tκ | ln By (7.0.6) and (7.0.7), one has  (7.0.8) ln | sin πθˆ j |  2s ln j∈J1

Similarly, (7.0.9)

 j∈J2

ln | sin πθˆ j |  2s ln

s − C|Tκ |. qn+1

s − Cs ln qn . qn+1 s − Cs ln qn . qn+1 



We now turn to estimating the quantities and defined in (4.0.11) + − and (4.0.12). Putting (4.0.13), (7.0.8) and (7.0.9) together, we have  s (7.0.10) > −4sqn ln 2 + 6s ln − Cs ln qn . + qn+1 

, we have the similar Replacing (7.0.2) with (7.0.3) and proceeding as for + estimate,  s > −4sqn ln 2 + 4s ln − Cs ln qn . (7.0.11) − qn+1 From (4.0.10), (7.0.10) and (7.0.11), it follows that  s ln | cos 2πθi − cos 2πθj | > −4sqn ln 2 + 8s ln − Cs ln qn . (7.0.12) qn+1 j∈I1 ∪I2 ,j =i

Combining with (4.0.9), we have for any ε > 0,  | cos 2πa − cos 2πθj | max | cos 2πθi − cos 2πθj | i∈I1 ∪I2 j∈I1 ∪I2 ,j =i

sq

 ) 8 ln( qn−1 n −ε .  exp (4sqn − 1) lnλ + qn−1



Remark 7.0.13. In the nonresonant case, for any ε > 0, 89  t < 1, one has ln λ + 8 ln(sqn−1 /qn )/qn−1  ln λ − 8(1 − t)β − ε > 0. In addition, we have ln λ + 8 ln(sqn−1 /qn )/qn−1  ln λ − 2ε if t is close to 1. Remark 7.0.14. Here, we only use Theorem 7.0.1 with C = 50C , where C is given by (7.0.15) (see below).

Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

57

Clearly, it is enough to consider k > 0. In this section we study the resonant case. Suppose there exists some k ∈ [bn , bn+1 ] such that k is n−resonant. For ε , where C is a large constant (depending on λ, α). any ε > 0, choose η = C Let ln λ ]). (7.0.15) C∗ = 2(1 + [ ln λ − β For an arbitrary solution ϕ satisfying Hϕ = Eϕ, let rn,ϕ = sup |ϕ(jqn + rqn )|, j |r|10η

b 50C∗ qn+1 . n

where |j|  Let φ be the generalized eigenfunction. Denote by n,φ . rn j = rj

Since we keep n fixed in this section we omit the dependence on n from the notation and write rϕ j , Rj , and rj . Note that below we always assume n is large enough. Lemma 7.0.16. Let k ∈ [jqn , (j + 1)qn ] with dist(k, qn Z)  10ηqn , and suppose b . Then for sufficiently large n, further that |j|  48C∗ qn+1 n   |ϕ(k)|  max rϕ j exp −(ln λ − 2η)(dj − 3ηqn ) , (7.0.17)  

rϕ exp −(ln λ − 2η)(d − 3ηq ) , n j+1 j+1 where dj = |k − jqn | and dj+1 = |k − jqn − qn |. Proof. The proof builds on the ideas akin to those used in the proof of Theorem 2.5.3. However it requires a more careful approach. For any y ∈ [jqn + ηqn , (j + 1)qn − ηqn ], apply (i) of Theorem 7.0.1 taking C = 50C . Notice that in this case, we have ln λ + 8 ln(sqn−1 /qn )/qn−1 − η  ln λ − 2η. Thus y is regular with τ = ln λ − 2η. Therefore there can choose an interval I(y) = [x1 , x2 ] ⊂ [jqn , (j + 1)qn ] such that y ∈ I(y), 1 (7.0.18) dist(y, ∂I(y))  |I(y)|  qn−1 4 and (7.0.19)

|GI(y) (y, xi )|  e−(ln λ−2η)|y−xi | , i = 1, 2,

where ∂I(y) is the boundary of the interval I(y) (i.e. {x1 , x2 }), and |I(y)| is the size of I(y) ∩ Z (i.e., |I(y)| = x2 − x1 + 1). For z ∈ ∂I(y), let z  be the neighbor of z, (i.e., |z − z  | = 1) not belonging to I(y). If x2 + 1  (j + 1)qn − ηqn or x1 − 1  jqn + ηqn , we can expand ϕ(x2 + 1) or ϕ(x1 − 1) using (2.5.2). We can continue this process until we arrive to z such that n ] times. z + 1 > (j + 1)qn − ηqn or z − 1 < jqn + ηqn , or we have iterated [ q2q n−1

58

Arithmetic spectral transitions

Thus, by (2.5.2) (7.0.20)

 GI(k) (k, z1 )GI(z ) (z1 , z2 ) · · · GI(zs ) (zs , zs+1 )ϕ(zs+1 ) , |ϕ(k)| = 1

s;zi+1 ∈∂I(zi )

where we have jqn + ηqn + 1  zi  (j + 1)qn − ηqn − 1, i = 1, · · · , s, in each / [jqn + ηqn + 1, (j + 1)qn − ηqn − 1], term of the summation and either zs+1 ∈ 2qn n ], or s + 1 = [ ]. We should mention that zs+1 ∈ [jqn , (j + 1)qn ]. s + 1 < [ q2q qn−1 n−1 If zs+1 ∈ [jqn , jqn + ηqn ], s + 1 < [ q2qn ], this implies n−1

|ϕ(zs+1 )|  rϕ j . By (7.0.19), we have

(7.0.21)

|GI(k) (k, z1 )GI(z ) (z1 , z2 ) · · · GI(zs ) (zs , zs+1 )ϕ(zs+1 )|  1  s ϕ −(ln λ−2η) |k−z1 |+ i=1 |zi −zi+1 |  rj e   −(ln λ−2η) |k−zs+1 |−(s+1)  rϕ j e   2qn ϕ −(ln λ−2η) dj −2ηqn −4− qn−1  rj e .

n ], by the same arguments, we If zs+1 ∈ [(j + 1)qn − ηqn , (j + 1)qn ], s + 1 < [ q2q n−1 have |GI(k) (k, z1 )GI(z ) (z1 , z2 ) · · · GI(zs ) (zs , zs+1 )ϕ(zs+1 )| 1  (7.0.22) −(ln λ−2η) dj+1 −2ηqn −4− q2qn ϕ n−1 .  rj+1 e n If s + 1 = [ q2q ], using (7.0.18) and (7.0.19), we obtain n−1

(7.0.23)

|GI(k) (k, z1 )GI(z ) (z1 , z2 ) · · · GI(zs ) (zs , zs+1 )ϕ(zs+1 )| 1   −(ln λ−2η)qn−1 [ q2qn ] n−1 e |ϕ(zs+1 )|. [

2qn

]

Notice that the total number of terms in (7.0.20) is at most 2 qn−1 and that dj and dj+1 are both at least 10ηqn . By (7.0.21)–(7.0.23), we have   − (ln λ−2η)(dj −3ηqn ) , |ϕ(k)|  max rϕ e j   − (ln λ−2η)(dj+1 −3ηqn ) (7.0.24) rϕ , j+1 e  −(ln λ−2η)q

n |ϕ(p)| max e . p∈[jqn ,(j+1)qn ]

ϕ Now we will show that p ∈ [jqn , (j + 1)qn ] implies |ϕ(p)|  max{rϕ j , rj+1 }. Then (7.0.24) implies case (i) of Lemma 7.0.16. Otherwise, by the definition of  rϕ j , if |ϕ(p )| is maximum over z ∈ [jqn + 10ηqn + 1, (j + 1)qn − 10ηqn − 1] of ϕ  |ϕ(z)|, then |ϕ(p )| > max{rϕ j , rj+1 }. Applying (7.0.24) to ϕ(p ) and noticing that  dist(p , qn Z)  10ηqn , we get    ϕ  |ϕ(p )|  e −7(ln λ−2η)ηqn max rϕ j , rj+1 , |ϕ(p )| . ϕ This is impossible because |ϕ(p )| > max{rϕ j , rj+1 }.



Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

59

By the properties of continued fractions and since θ is α-Diophantine, one can obtain the following estimates: Lemma 7.0.25. For any |i|, |j|  50C∗ bn+1 , the following estimate holds, ln | sin π(2θ + (j + i)α)|  −C ln qn .

(7.0.26)

Lemma 7.0.27. Assume |i|, |j|  50C∗ bn+1 , and i − j = qn Z. Then ln | sin π(j − i)α|  −C ln qn .

(7.0.28) We then have

Theorem 7.0.29. For 1  j  46C (7.0.30)

rϕ j

bn+1 qn ,

the following holds q n+1 exp{−(ln λ − Cη)qn }}.  max{rϕ j±1 j b

Proof. Fix j with 1  j  46C∗ qn+1 and |r|  10ηqn . n Next, define subsets I1 , I2 ⊂ Z as follows 1 1 I1 = [−[ qn ], qn − [ qn ] − 1], 2 2 1 1 I2 = [jqn − [ qn ], (j + 1)qn − [ qn ] − 1]. 2 2 Let θm = θ + mα for m ∈ I1 ∪ I2 . The set {θm }m∈I1 ∪I2 will thus consist of 2qn elements. By Lemmas 7.0.25 and 7.0.27, and following the proof of Theorem 4.0.6, one ln qn+1 −ln j + ε uniform for any ε > 0. Combining with Lemma obtains that {θm } is 2qn 3.0.7, there exists some j0 with j0 ∈ I1 ∪ I2 such that /A θj0 ∈

(2qn −1),(ln λ−

ln qn+1 −ln j −η) 2qn

.

First, we assume j0 ∈ I2 . Set I = [j0 − qn + 1, j0 + qn − 1] = [x1 , x2 ]. In (2.8.10), let ε = η. Combining with (2.8.6) and (2.8.7), it is easy to verify |GI (jqn + r, xi )|

   exp (ln λ + η) 2qn − 1 − |jqn + r − xi |   ln qn+1 − ln j −η . − (2qn − 1) ln λ − 2qn Using (2.5.2), we obtain (7.0.31)

|ϕ(jqn + r)| 

 qn+1 e5ηqn |ϕ(xi )| e−|jqn +r−xi | ln λ , j

i=1,2

where x1 = x1 − 1 and x2 = x2 + 1. Let dij = |xi − jqn |, i = 1, 2. It is easy to check that (7.0.32)

|jqn + r − xi | + dij , |jqn + r − xi | + dij±1  qn − |r|,

and (7.0.33)

|jqn + r − xi | + dij±2  2qn − |r|.

60

Arithmetic spectral transitions

If dist(xi , qn Z)  10ηqn , then we bound ϕ(xi ) in (7.0.31) using (7.0.17). If dist(xi , qn Z)  10ηqn , then we bound ϕ(xi ) in (7.0.31) by some proper rj . Combining with (7.0.32), (7.0.33), we have ϕ qn+1 exp{−(ln λ − Cη)qn }, rϕ j  max rj±1 j qn+1 exp{−(ln λ − Cη)qn }, rϕ j j

qn+1 exp{−2(ln λ − Cη)q } . rϕ n j±2 j However, we cannot have ϕ rϕ j  rj

qn+1 exp{−(ln λ − Cη)qn } j

 rϕ j exp{−(ln λ − β − Cη)qn } so we must have (7.0.34)

ϕ qn+1 exp{−(ln λ − Cη)qn }, rϕ j  max rj±1 j

qn+1 exp{−2(ln λ − Cη)q } . rϕ n j±2 j

In particular, (7.0.35)

ϕ ϕ rϕ j  exp{−(ln λ − β − Cη)qn } max{rj±1 rj±2 }.

If j0 ∈ I1 , then (7.0.35) holds for j = 0. Let ϕ = φ in (7.0.35). We get |φ(0)|, |φ(−1)|  exp{−(ln λ − β − Cη)qn }, which is in contradiction with |φ(0)|2 + |φ(−1)|2 = 1. Therefore j0 ∈ I2 , so (7.0.34) holds for any ϕ. By (2.6.3) and (2.8.10), we have



ϕ(k1 ) ) ϕ(k 2 (7.0.36) || ||  Ce−(ln λ+ε)|k1 −k2 | || ||. ϕ(k1 − 1) ϕ(k2 − 1) This implies

ϕ rϕ j±2  rj±1 exp{(ln λ + Cη)qn },

thus (7.0.34) becomes (7.0.37)

ϕ rϕ j  max{rj±1

for any 1  j  46C∗

qn+1 exp{−(ln λ − Cη)qn }}, j

bn+1 qn .



We now show that by Theorem 2.4.1 exponential growth is not allowed, rj must actually decay. Theorem 7.0.38. For 1  j  10

bn+1 qn ,

the following holds q (7.0.39) rj  rj−1 exp{−(ln λ − Cη)qn } n+1 . j Proof. Let ϕ = φ in Lemma 7.0.29. We must have q (7.0.40) rj  max{rj±1 n+1 exp{−(ln λ − Cη)qn }}, j

Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

61

b

for any 1  j  46C∗ qn+1 . n b , the following holds, Suppose for some 1  j  10 qn+1 n qn+1 exp{−(ln λ − Cη)qn }  rj+1 exp{−(ln λ − β − Cη)qn }. (7.0.41) rj  rj+1 j Applying (7.0.40) to j + 1, we obtain q (7.0.42) rj+1  max{rj , rj+2 } n+1 exp{−(ln λ − Cη)qn }. j+1 Combining with (7.0.41), we must have rj+1  rj+2 exp{−(ln λ − β − Cη)qn }.

(7.0.43)

Generally, for any 0 < p  (C∗ + 1)j − 1, we obtain rj+p  rj+p+1 exp{−(ln λ − β − Cη)qn }.

(7.0.44) Thus

r(C∗ +1)j  rj exp{(ln λ − β − Cη)C∗ jqn }.

(7.0.45)

Clearly, by (7.0.36), one has rj  exp{−(ln λ + Cη)jqn }. Then (7.0.46)

r(C∗ +1)j  exp{((C∗ − 1) ln λ − C∗ β − Cη)jqn }.

By the definition of C∗ , one has (C∗ − 1) ln λ − C∗ β > 0. Thus (7.0.46) is in contradiction with the fact that |φ(k)|  1 + |k|. Now that (7.0.41) can not happen, from (7.0.40), we must have q (7.0.47) rj  rj−1 n+1 exp{−(ln λ − Cη)qn }. j Theorem 7.0.48. For 1  j  10



bn+1 qn ,

the following holds q (7.0.49) rj  rj−1 exp{−(ln λ − ε)qn } n+1 . j Proof. See [30] for details.



We are now ready to complete the proof of Theorem 5.0.6. ε Proof of Theorem 5.0.6. Set t0 = 1 − 8β . Let t = t0 in the definition of resonance, t0 i.e. bn = qn . 0 : Case I:   qtn+1 0 0 , qn+1 − εqtn+1 ) is By case II of Theorem 7.0.1, we know that any y ∈ (εqtn+1 1 (ln λ + 8 ln(sqn /qn+1 )/qn − ε, 4sqn − 1) regular with δ = 4 . Notice that 0 0  εqtn+1 , (s + 1)qn  εqtn+1

thus we have ln λ +

n ) 8 ln(s qqn+1

qn

 ln λ − 8(1 − t0 )β − ε  ln λ − 2ε.

62

Arithmetic spectral transitions

0 0 Thus for any y ∈ (εqtn+1 , qn+1 − εqtn+1 ), y is (ln λ − 2ε, 4sqn − 1) regular. Following the proof of Lemma 7.0.16, one has for qn  k  ( + 1)qn ,

||U(k)||  e−(ln λ−ε)|k| , which implies Theorem 5.0.6 in this case. 0 : Case II: 0    qtn+1 By Theorems 7.0.48 and 7.0.38, and Stirling’s formula, −εjqn εjqn  rj  r¯ n . r¯ n j e j e

Now Theorem 5.0.6 follows from Lemma 7.0.16.



8. Arithmetic criteria for spectral dimension We know that in the regime of positive Lyapunov exponent the spectrum is always singular. Now that we also know (Lemma 2.8.4) that large β implies continuous (and therefore singular continuous) spectrum, it’s natural to ask whether even larger β implies increased continuity. “Continuity” of singular continuous spectrum can be quantified through fractal dimensions. The most popular object is Hausdorff dimension. However Hausdorff dimension is a poor tool for characterizing the singular continuous spectral measures arising in the regime of positive Lyapunov exponents, as it is always equal to zero (a very general theorem of Simon that holds for general ergodic potentials and a.e. phase, see Theorem 8.2.6 [44] (and for every phase for the zero entropy dynamical systems [19] (see also [29, 31]). It turns out that some other dimensions do present good tools to finely distinguish between different kinds of singular continuous spectra appearing in the supercritical regime. The main goal of this lecture is to briefly present a simultaneous quantitative version of two well known statements (1) Periodicity implies absolute continuity. We prove that a quantitative weakening (near periodicity that holds sufficiently long) implies quantitative continuity of the (fractal) spectral measure. (2) Gordon condition (a single/double almost repetition) implies continuity of the spectral measure. Indeed, we prove that a strengthening (with multiple almost repetitions) implies quantitative continuity of the spectral measure. This will allow us to establish a sharp arithmetic criterion for certain dimension of the spectral measure in terms of β, for general analytic potentials. 8.1. m-function and subordinacy theory Let μ be a finite Borel measure on R. Define the Borel transform of μ to be:  1 (8.1.1) dμ(E), z ∈ C. m(z) := E−z It is easy to check that for any finite Borel measure μ on R, its m-function is holomorphic in the upper half plane and satisfies μ(R) z ∈ C+ . m∗ (z) = m(z∗ ), |m(z)|  Imz

Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

63

Remark 8.1.2. Functions with this property are known as Herglotz, Pick or R functions. They map the upper half-plane into itself, but are not necessarily injective or surjective. Note that such an m is holomorphic in C\σ(μ), where σ(μ) := {E ∈ R : μ(E − ε, E + ε) > 0 for all ε > 0}. The boundary behavior of m is linked to the Radon-Nikodym derivative Dμ of μ, which in turn determines the decomposition of μ, see e.g. [45]. Theorem 8.1.3. Let μ be a finite Borel measure and m its Borel transform. Then the limit     (8.1.4) Im m(E) = lim Im m(E + iε) ε↓0

exists a.e. with respect to both μ and Lebesgue measure (finite or infinite) and  1  (8.1.5) Dμ(E) = Im m(E) π   whenever Dμ(E) exists. Moreover, the set {E|Im m(E) = ∞} is a support for the   singular continuous part and {E|Im m(E) < ∞} is a minimal support for the absolutely continuous part. Fractal properties of μ can also be characterized through m. In the rest of this subsection, we briefly review the power-law extension of the Gilbert-Pearson subordinacy theory [16, 17], developed in [29]. For simplicity, consider the right half line operator (2.3.1) on 2 (Z+ ) with boundary condition u(1) = cos ϕ, u(0) = sin ϕ for some ϕ ∈ (−π/2, π/2]. Let μ be the spectral measure. In this case, the Borel transform of μ is also called the Weyl-Titchmarsh m-function. For any function u : Z+ → C and  ∈ R+ , define (8.1.6)

u :=

[] 

|u(n)|2 + ( − [])|u([] + 1)|2

1/2

.

n=1

Suppose u and v solve Hu = Eu with orthogonal boundary conditions

u(1) v(1) = Rϕ , u(0) v(0) where Rϕ is a matrix of rotation by ϕ. Now given any ε > 0, we define a length (ε) ∈ (0, ∞) by requiring the equality 1 (8.1.7) u(ε) · v(ε) = . 2ε The function (ε) is a well defined monotonically decreasing continuous function 1  12 ([] − 1). It turns out which goes to infinity as ε goes to 0, and we also have 2ε

u that the boundary behavior of m(E + iε) is linked in a quantitative way to v (ε) , (ε) thus to the power-law behavior of solutions. Lemma 8.1.8 (J.-Last inequality, [29]). For E ∈ R and ε > 0, √ √ u 5 + 24 5 − 24 < . < (8.1.9) |m(E + iε)| v |m(E + iε)|

64

Arithmetic spectral transitions

From Lemma 8.1.8, one can easily recover the results of Gilbert-Pearson [17] with a simpler proof, while strengthening their theory. The above inequality links the power-law behavior of the generalized eigenfunctions of Hu = Eu and the boundary behavior of the Borel transform of the spectral measure μ in a quantitative way. A particular consequence of Lemma 8.1.8 is Lemma 8.1.10. For any E ∈ R and 0 < γ < 1, suppose there is a sequence of positive numbers εk → 0 and an absolute constant C > 0 so that both u, v satisfy (8.1.11)

2−γ 2 C−1 γ k  uk  Ck

where k = (εk ) is given by (8.1.7). Then (8.1.12)

lim inf ε1−γ |m(E + iε)| < ∞. ε↓0

8.2. Spectral continuity Fix 0 < γ < 1. If (8.1.12) holds for μ a.e. E, we say measure μ is (upper) γ-spectral continuous. Define the (upper) spectral dimension of μ to be  (8.2.1) s(μ) = sup γ ∈ (0, 1) : μ is γ-spectral continuous . In this part, we focus on the quantitative spectral continuity and the lower bound of the spectral dimension. Our spectral continuity result does not necessarily require quasiperiodic structure of the potential and can be generalized to wider contexts (so-called β-almost periodic potential, see [27], - a class, that includes, for example, some skew shift potentials). The general result of [27] only goes in one direction. However, in the important context of analytic quasiperiodic operators this leads to a sharp if-and-only-if result. Let H be defined as in (2.3.2) with quasiperiodic potential: (8.2.2) (Hu)(n) = u(n + 1) + u(n − 1) + v(θ + nα)u(n), θ, α ∈ T, v : T → R. Theorem 8.2.3 ([27]). Let H be as in (8.2.2) with real analytic potential v and μ be the spectral measure2 . Assume L(E) > 0 for all E ∈ R. For any θ ∈ T, s(μ) = 1 if and only if β(α) = ∞. Remark 8.2.4. The theorem also holds locally for any spectral projection onto the subset where the Lyapunov exponent is positive. Remark 8.2.5. The ‘if’ part will be a consequence of Theorem 8.2.7 which can be viewed as a quantitative strengthening of the results of Gordon type (Lemma 2.8.4). The ‘only if’ part follows from the general analytic Theorem 8.3.1 which can be viewed as a weakening/extension of localization type results for large β. Spectral continuity captures the lim inf power-law behavior of m(E + iε), while the corresponding lim sup behavior is linked to the Hausdorff dimension [14]. 2 Discrete Schrödinger operators may have multiplicity two. However, δ , δ always form a cyclic pair, 0 1 so it is enough to consider the so called maximal spectral measure given by μ = μδ0 + μδ1 , where μδ0 and μδ1 are defined as in (2.1.1).

Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

65

One can easily check that dimH (μ)  s(μ)  dimP (μ), where dimH (μ)/dimP (μ) denote the Hausdorff/packing dimension of a measure in the usual sense. Theorem 8.2.6 (Simon, [44]). Suppose H is an ergodic Schrödinger operator as in (2.3.2) with positive Lyapunov exponent. For a.e. phase ω, dimH (μ) = 0. Let H be as in (8.2.2) and μ be the spectral measure. We have the following quantitative lower bound of the spectral dimension. Theorem 8.2.7. Suppose v is Lipschitz continuous. Let 1 ln An (θ). (8.2.8) Λ := sup E∈σ(H),n,θ n There exists an absolute constant C > 0 such that for any θ ∈ T, CΛ s(μ)  1 − . β(α) The general version of Theorem 8.2.7 is actually more robust and only requires some regularity of v, which allows us to obtain new results for other popular models, such as the critical almost Mathieu operator, Sturmian potentials, and others. Lower bounds on spectral dimension also have immediate applications to the lower bounds on packing/box counting dimensions and on quantum dynamics(upper transport exponents). The method developed in [27] for the bounded SL(2, R) case generalizes to the unbounded case (e.g. the Maryland model) and the non-Schrödinger case (e.g. the extended Harper’s model) [46]. For simplicity, we only prove the right half line case and we also assume (5.0.1) holds. According to Lemma 8.1.10, to prove spectral continuity, it is enough to obtain power-law estimate (8.1.11) for half-line solution u of Hu = Eu with any boundary condition ϕ. First, for β large, the system can be approximated by a periodic one exponentially fast, in the following sense. Lemma 8.2.9. Let qn be given as in (5.0.1). For any β < β(α), any θ ∈ T, we have (8.2.10)

Aqn (θ) − Aqn (θ + qn α)  e(−β+2Λ)qn .

The ultimate goal is to estimate ANqn  by the size of qn for N ∼ ecβqn . This eventually leads to the desired power-law for u by (2.6.3). We will conclude this in the end of this part. The standard rational approximation fails here since the cβqn . We need some quantitative error terms may reach the size of eNΛ ∼ ee telescoping arguments. Lemma 8.2.11. Suppose G is a two by two matrix satisfying (8.2.12)

Gj   M < ∞,

for all 0 < j  N ∈ N+ ,

where M  1 only depends on N. Let Gj = G + Δj , j = 1, · · · , N, be a sequence of two by two matrices with (8.2.13)

δ = max Δj . 1jN

66

Arithmetic spectral transitions

If (8.2.14)

NMδ < 1/2,

then for any 1  n  N 

(8.2.15)

n 

Gj − Gn   2NM2 δ.

j=1

Combining (8.2.10) with this lemma, one can show that ANqn is close to AN qn up to the size of AN qn . Now the question is reduced from the dynamical behavior of ANqn to the algebraic properties of AN qn . We need some additional linear algebraic facts about SL(2, R) matrices. Lemma 8.2.16. Suppose G ∈ SL(2, R) with 2 < |Tr G|  6. There exists an invertible matrix B such that

ρ 0 B−1 (8.2.17) G=B 0 ρ−1 where ρ±1 are the two conjugate real eigenvalues of G, |detB| = 1 and  G −1  (8.2.18) B = B  < . |Tr G| − 2 √ 2 G If |Tr G| > 6, then B  √ . |Tr G|−2

Lemma 8.2.19. Suppose G ∈ SL(2, R) has eigenvalues ρ±1 , ρ > 1. For any k ∈ N, if Tr G = 2, then ρk − ρ−k TrG  ρk + ρ−k ·I + · I. · G− (8.2.20) Gk = −1 2 2 ρ−ρ Otherwise, Gk = k(G − I) + I. Assume further that |Tr G| − 2 < τ < 1. Then there exist universal constants 1 < C1 < ∞, c1 > 1/3 such that for 1  k  τ−1 , we have ρk + ρ−k ρk − ρ−k < C 1 , c1 k < < C1 k. 2 ρ − ρ−1 By Lemma 8.2.16, if the trace of Aqn is away from 2, we have the decomposition

 ρ 0 2 Aqn  B−1 , B = B−1    (8.2.22) Aqn = B |Tr Aqn | − 2 0 ρ−1 (8.2.21)

c1
2 + e−10Λqn },

(8.2.25)

n→∞

(8.2.26)

S2 = lim sup {E : |TrAqn | − 2 < e−10Λqn }. n→∞

To estimate the spectral measure of S1 , we use the idea of a Gordon-type argument to estimate the lower bound of the solution. Recall that the key step to prove Lemma 2.8.4 is that for G ∈ SL(2, R) and X ∈ C2 , 1 (8.2.27) max{GX, G−1 X}  |TrG| · X. 2 If E ∈ S1 , roughly speaking, we have a sequence of scales qn such that the trace of Aqn is large. Putting (8.2.10),(8.2.15),(8.2.22) and (8.2.27) together, we can show that there are integer sequences xqn → ∞ independent of E ∈ S1 , such that |u(xqn )| > eqn ,

(8.2.28)

where u solves the half-line problem Hu = Eu with any boundary condition. The following extended Schnol’s Theorem shows that such E must have spectral measure zero. Lemma 8.2.29 (Extended Schnol’s Theorem, [27]). Fix any y > 1/2. For any fixed sequence |xk | → ∞, for spectrally a.e. E, there is a generalized eigenvector u of Hu = Eu, such that |u(xk )| < C(1 + |k|)y . For S2 , note that Aq (E) is a polynomial in E with degree at most q. If the trace is close to 2, the following preimage estimate of a polynomial reduces the set in S2 to several small intervals of width at most e−5Λqn . Lemma 8.2.30 ([25]). Let p ∈ Pn;n (R) with y1 < · · · < yn−1 the local extrema of p. Let (8.2.31)

ζ(p) :=

min

1jn−1

|p(yj )|

and 0  a < b. Then, (8.2.32)

|p−1 (a, b)|  2 diam(z(p − a)) max

b − a  b − a 1

2 , ζ(p) + a ζ(p) + a

where z(p) is the zero set of p and | · | denotes the Lebesgue measure. The definition of m-function implies μ(E − ε, E + ε)  2ε ImM(E + iε), where the right-hand side can be estimated again by subordinacy theory (Lemma 8.1.8) with the help of (8.2.24). Together, these ideas can be used to show that for β large enough, μ({E : |TrAqn | − 2 < e−10Λqn }) < e−Λqn . Then the Borel-Cantelli lemma immediately implies μ(S2 ) = 0. In conclusion, we have the following key estimate for the trace of the transfer matrices.

68

Arithmetic spectral transitions

Theorem 8.2.33. For β > 40Λ and μ a.e. E, there is K(E) such that (8.2.34)

|TraceAqn (E)| < 2 − e−10Λqn , n  K(E).

Combining this trace estimate with previous algebraic facts (8.2.15) and (8.2.24), one has Lemma 8.2.35. There is a sequence of positive integers Nk → ∞ such that for 0 < γ < 1, if 100Λ , (8.2.36) β> 1−γ then N k ·qk (8.2.37) An (E)2  (Nk · qk )2−γ , k  K(E). n=1

Now (8.1.11) follows from (2.6.3) for any boundary condition ϕ. 8.3. Arithmetic criteria In this part, we focus on the spectral singularity and the quantitative upper bound of s(μ). For simplicity, we only state and prove the following upper bound for the right half line AMO. The same result holds for general analytic potentials with positive Lyapunov exponent, which together with Theorem 8.2.7 will complete the proof of Theorem 8.2.3. Theorem 8.3.1. Let H be the AMO given as in (2.7.1). Assume that λ > 1. There exists ϕ ∈ (−π/2, π/2] and an absolute constant c > 0 such that for any θ ∈ T if β(α) < ∞ then for the associated half line spectral measure μ, we have that 1 < 1. (8.3.2) s(μ)  1 + c/β Lemma 8.3.3. For any E there is a n0 such that for any n > n0 , there exists an interval Δn ⊂ T satisfying 1 1 1 , inf ln An (θ) > ln λ. (8.3.4) Leb(Δn )  8n θ∈Δn n 4 Moreover, for all qn large (depending on n0 ), for any θ, and any N ∈ N, there is jN ∈ [2Nqn , 2(N + 1)qn ) such that (8.3.5)

1

AjN (θ, E) > e 36 qn ln λ .

Lemma 8.3.6. For any E ∈ R and β = β(α) < ∞, there is a 0 = 0 (E, β) such that for  > 0 , and any θ ∈ T, the following holds: (8.3.7)

 

2c

Ak (θ, E)2  1+ β .

k=1

Proof of Theorem 8.3.1: For any ϕ, we have (8.3.8)

uϕ 2 + vϕ 2 

 1 Ak (θ)2 . 2 k=1

2c

Therefore, (8.3.7) implies that uϕ 2 + vϕ 2  1+ β for  large.

Svetlana Jitomirskaya, Wencai Liu, and Shiwen Zhang

69

On the other hand, Last and Simon showed in [38] that, for μ-a.e. E, there exist ϕ and C = C(E) < ∞, such that for large , (8.3.9)

uϕ   C1/2 ln .

Combining (8.3.8) and (8.3.9), we have (8.3.10)

vϕ   1/2+c/β

provided β < ∞ and  > 0 (E, β). For any ε > 0 (small), let  = (ε) be given as in (8.1.7). By (8.1.9), one has for any γ ∈ (0, 1), √ 1 vϕ  ε1−γ |mϕ (E + iε)|   1−γ · (5 − 24) ϕ u  2uϕ  vϕ   cγ (1+c/β)γ−1 · ln−2  where cγ > 0 only depends on γ. Now let γ0 = γ > γ0 ,

1 1+c/β

< 1. We have for any

ε1−γ |mϕ (E + iε)|  cγ γ/γ0 −1 · ln−2  → ∞ as ε → 0. Therefore, s(μ)  γ0 , according to the definition (8.2.1).



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Department of Mathematics, University of California, Irvine, California 92697-3875, USA Email address: [email protected] Department of Mathematics, University of California, Irvine, California 92697-3875, USA Current address: Department of Mathematics, Texas A&M University, College Station, TX 778433368, USA Email address: [email protected] Department of Mathematics, University of Minnesota, Minneapolis, MN, 55455 Email address: [email protected]

IAS/Park City Mathematics Series Volume 27, Pages 73–130 https://doi.org/10.1090/pcms/027/00861

Quantitative homogenization of elliptic operators with periodic coefficients Zhongwei Shen Abstract. These lecture notes introduce the quantitative homogenization theory for elliptic partial differential equations. We consider a family of second-order elliptic operators in divergence form with rapidly oscillating periodic coefficients. The qualitative theory of homogenization is presented in Section 1. In Sections 2– 5 we address the basic questions in the quantitative theory: convergence rates and uniform regularity estimates. The material in the lecture notes is taken from the monograph [23], where further results and references as well as additional topics may be found.

Contents 1

2

3

4

5

Introduction and Qualitative Theory 1.1 Correctors and effective coefficients 1.2 H-Compactness 1.3 Homogenization of boundary value problems 1.4 Problems for Section 1 Convergence Rates 2.1 Flux correctors and ε–smoothing 2.2 Convergence rates in H1 2.3 Convergence rates in Lp 2.4 Problems for Section 2 Uniform Lipschitz Estimates—Part I 3.1 Interior estimates 3.2 Boundary Lipschitz estimates for the Dirichlet problem 3.3 Problems for Section 3 Uniform Lipschitz estimates—Part II 4.1 Approximation of solutions at the large scale 4.2 Boundary Lipschitz estimates 4.3 Problems for Section 4 Uniform Calderón-Zygmund Estimates 5.1 The classical Calderón-Zygmund Theorem

74 75 78 80 82 82 82 87 93 98 99 99 105 106 106 107 109 116 117 117

2010 Mathematics Subject Classification. Primary 35B27; Secondary 35J57, 74Q05. Key words and phrases. Homogenization; Regularity; Convergence Rates. Supported in part by NSF grant DMS-1600520. ©2020 American Mathematical Society

73

74

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

5.2 5.3 5.4 5.5

A real-variable argument W 1,p estimates Boundary W 1,p estimates Problems for Section 5

118 123 127 128

1. Introduction and Qualitative Theory Partial differential equations with rapidly oscillating coefficients are used to model various physical phenomena in heterogeneous media, such as composite and perforated materials. Let ε > 0 be a small parameter, representing the scale of the microstructure of an inhomogeneous medium. The local characteristics of the medium are described by functions of form A(x/ε). Since ε is much smaller than the linear size of the domain where the physical process takes place, solving the corresponding boundary value problems for the partial differential equations directly by numerical methods could be costly. To describe the basic idea of homogenization, we consider the Dirichlet problem,

in Ω, − div(A(x/ε)∇uε ) = F (1.0.1) on ∂Ω, uε = f where Ω is a bounded domain in Rd . The d × d coefficient matrix A = A(y) is assumed to be real, bounded measurable, and satisfy the ellipticity condition, (1.0.2)

Aξ, ξ  μ|ξ|2

for any ξ ∈ Rd ,

where μ > 0. Given F ∈ H−1 (Ω) and f ∈ H1/2 (∂Ω), the Dirichlet problem (1.0.1) has a unique weak solution uε in H1 (Ω). Under the additional structure condition that A is periodic with respect to some lattice in Rd , it can be shown that as ε → 0, the solution uε of (1.0.1) converges to u0 weakly in H1 (Ω) and thus strongly in L2 (Ω). Moreover, the limit u0 is the weak solution of the Dirichlet problem,

 − div(A∇u in Ω, 0) = F (1.0.3) u0 = f on ∂Ω,  is constant and satisfies the ellipticity condition where the coefficient matrix A (1.0.2). As a result, one may use the function u0 as an approximation of uε . Since  is constant, the Dirichlet problem (1.0.3) is much easier to handle both analytA ically and numerically than (1.0.1). Similar results may be proved for Neumann type boundary value problems.  which are uniquely determined by A, are The entries of the constant matrix A, given by solving some auxiliary problems in a periodic cell. By taking the inhomogeneity scale ε to zero in limit, we have “homogenized” the microscopically inhomogeneous medium. The limit u0 of the solution uε is called the homogenized or effective solution. The boundary value problem (1.0.3) is referred as

Zhongwei Shen

75

 which describes the the homogenized problem for (1.0.1). The constant matrix A, macroscopic characteristics of the inhomogeneous medium, is called the matrix of homogenized or effective coefficients. See the classical monographs [5, 29] for further details about homogenization of partial differential equations in various settings. These lecture notes provide an introduction to the theory of quantitative homogenization, which is concerned with the convergence rates of uε to u0 and regularity estimates of uε that are uniform in the small parameter ε. We will restrict ourself to the periodic setting and only consider the second-order linear elliptic operator in divergence form Lε , given in (1.1.1). For a comprehensive treatment of and references on quantitative homogenization in the stochastic setting we refer the reader to the monograph [1]. The lecture notes are organized as follows. In this section we establish the qualitative homogenization theory for the elliptic operator − div(A(x/ε)∇). Section 2 is devoted to the question of convergence rates. In Sections 3, 4 and 5, we investigate various regularity estimates that are uniform with respect to ε. Particularly, in Section 3, we prove the uniform interior Lipschitz estimate, using a compactness method of Avellaneda and Lin [4], and give an outline of the proof of the uniform boundary Lipschitz estimate for the Dirichlet problem. In Section 4 we prove the uniform Lipschitz estimate for the Neumann problem. The approach is based on a general scheme for establishing large-scale regularity estimates in homogenization, developed by S. N. Armstrong and C. Smart [3] in the study of stochastic homogenization. In Section 5 we introduce a real-variable method, motivated by a paper of L. Caffarelli and I. Peral [6], for Lp estimates. The method, which is used to establish W 1,p estimates for Lε , may be regarded as a refined and dual version of the celebrated Calderón-Zygmund Lemma. The material in the lecture notes is taken from [23], where further results and references as well as additional topics, including nontangential maximal estimates in Lipschitz domains, may be found. Throughout the notes we will use C and c to denote positive constants that are independent of the parameter ε > 0. They may change from line to line and ffl depend on A and/or Ω. We will use E u to denote the L1 average of a function u over a set E; i.e. ˆ 1 u= u. |E| E E The summation convention that the repeated indices are summed will be used throughout. 1.1. Correctors and effective coefficients Let   (1.1.1) Lε = − div A (x/ε) ∇ ,

ε>0

in Rd . The coefficient matrix (tensor) A in (1.1.1) is given by   A(y) = aαβ ij (y) , with 1  i, j  d and 1  α, β  m.

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

We will always assume that A is real, bounded measurable, and satisfies the conditions, α β μ|ξ|2  aαβ ij ξi ξj

(1.1.2)

and

A∞  μ−1

m×d , where μ > 0. For simplicity, from now on, we for any ξ = (ξα i ) ∈ R shall suppress all super indices (or assume m = 1; however, results and proofs extend to the case m > 1 - the case of elliptic systems). We also assume that A is 1-periodic; i.e., for each z ∈ Zd ,

(1.1.3)

for a.e. y ∈ Rd .

A(y + z) = A(y)

By a linear transformation one may replace Zd in (1.1.3) by any lattice in Rd . k For Y = [0, 1)d and k  1, let Hk per (Y) denote the closure in H (Y) of 1-periodic C∞ functions in Rd . Define   A∇φ · ∇ψ dy (1.1.4) aper φ, ψ = Y

By the Lax-Milgram Theorem, for each 1  j  d, there exists a for φ, ψ ∈ unique function χj in H1per (Y) such that ˆ χj dy = 0, H1 (Y).

Y

and (1.1.5)

    aper χj , ψ = −aper yj , ψ

for any ψ ∈ H1per (Y).

Note that this χj ∈ H1per (Y) is also the unique weak solution of the following problem: ⎧ L(χj ) = −L(yj ) in Rd , ⎪ ⎪ ⎪ ⎨ χj is 1-periodic, (1.1.6) ˆ ⎪ ⎪ ⎪ ⎩ χj dy = 0, Y

where L = − div(A(y)∇). In particular, if ∂aij L(yj ) = − = 0, ∂yi then χj = 0 in Rd . Definition 1.1.7. The vector-valued function χ = (χ1 , χ2 , . . . , χd ) ∈ H1per (Y; Rd ) is called the (first-order) corrector for the operator Lε . It follows from (1.1.5) with ψ = χj that (1.1.8)

χj H1 (Y)  C,

where C depends only on μ (and d, m). Since L(yj + χj ) = 0 in Rd , by rescaling, we obtain  (1.1.9) Lε xj + εχj (x/ε) = 0 in Rd for any ε > 0.

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 = ( Definition 1.1.10. Let A aij ), where 1  i, j  d and   ∂χj ij = aij + aik dy, (1.1.11) a ∂yk Y and define  (1.1.12) L0 = − div(A∇).  is called the matrix of effective or homogenized coefficients. The constant matrix A The operator L0 is called the homogenized operator. The following theorem gives the ellipticity for the homogenized operator L0 . Theorem 1.1.13. Suppose that A is 1-periodic and satisfies (1.1.2). Then  ξ and |A|   μ1 (1.1.14) μ|ξ|2  Aξ, for any ξ ∈ Rd , where μ1 > 0 depends only on μ. Proof. The second inequality in (1.1.14) follows readily from (1.1.8). To see the first, we fix ξ = (ξi ) ∈ Rd and let φ = ξi yi , ψ = ξi χi . Observe that ij = aper (yj + χj , yi ) = aper (yj + χj , yi + χi ). a

(1.1.15) By (1.1.2),

ij ξi ξj = aper (φ + ψ, φ + ψ) a |∇φ + ∇ψ|2 dy

μ Y

|∇φ|2 dy + μ

=μ where we have used the fact

´ Y

Y

|∇ψ|2 dy, Y

∇χi dy = 0. It follows that

ij ξi ξj  μ a

|∇φ|2 dy = μ|ξ|2 .



Y

  Theorem 1.1.16. Let A∗ = a∗ij denote the adjoint of A, where a∗ij = aji . Then  ∗  . ∗ = A A  In particular, if A is symmetric, i.e. aij (y) = aji (y) for 1  i, j  d, so is A.   Proof. Let χ∗ (y) = χ∗j (y) denote the corrector for L∗ε ; i.e. χ∗j is the unique ´ function in H1per (Y) such that Y χ∗j dy = 0 and (1.1.17)

a∗per (χ∗j , ψ) = −a∗per (yj , ψ)

for any ψ ∈ H1per (Y),

where a∗per (φ, ψ) = aper (ψ, φ). Observe that by (1.1.15) and (1.1.17),     ij = aper yj + χj , yi = aper yj + χj , yi + χ∗i a     = a∗per yi + χ∗i , yj + χj = a∗per yi + χ∗i , yj (1.1.18)   = a∗per yi + χ∗i , yj + χ∗j ∗ji , =a for 1  i, j  d.



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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

1.2. H-Compactness Proposition 1.2.1. Let {h } be a sequence of 1-periodic functions. Assume that h L2 (Y)  C

and Y

Let ε → 0. Then h (x/ε )  c0

h dy → c0 ,

as  → ∞.

weakly in L2 (Ω),

as  → ∞, where Ω is a bounded domain in Rd . In particular, if h is 1-periodic and h ∈ L2 (Y), then h(x/ε) 

h dy

weakly in L2 (Ω), as ε → 0.

Y

ffl Proof. By considering the periodic functions h − Y h , one may assume that ´ 2 Y h = 0 and hence c0 = 0. Let u ∈ Hper (Y) be a 1-periodic function such that Δu = h

in Y.

Let g = ∇u . Then h = div(g ) and g L2 (Y)  Ch L2 (Y)  C. Note that

 h (x/ε ) = ε div g (x/ε ) .

It follows that if ϕ ∈ C10 (Ω), ˆ ˆ h (x/ε )ϕ(x) dx = −ε (1.2.2) Ω

g (x/ε ) · ∇ϕ(x) dx → 0, Ω

as ε → 0. To see this, we fix R > 0 so that Ω ⊂ B(0, R). Then ˆ ˆ |g (x/ε )|2 dx  εd |g (y)|2 dy  C g 2L2 (Y)  C,  Ω

B(0,R/ε )

where we have used the periodicity of g for the second inequality (the constant C depends on R). Similarly, h (x/ε )L2 (Ω)  C h L2 (Y)  C;

(1.2.3)

i.e., the sequence {h (x/ε )} is bounded in L2 (Ω). In view of (1.2.2) we conclude  that h (x/ε )  0 weakly in L2 (Ω). Recall that for u = (u1 , u2 , . . . , ud ), div(u) =

d  ∂ui i=1

∂xi

and

 curl(u) =

∂ui ∂uj − ∂xj ∂xi

 . 1i,jd

The following theorem is usually referred as the Div-Curl Lemma, whose proof may be found in [23, 29]. Theorem 1.2.4. Let {u } and {v } be two bounded sequences in L2 (Ω; Rd ). Suppose that (1) u  u and v  v weakly in L2 (Ω; Rd ); (2) curl(u ) = 0 in Ω and div(v ) → f strongly in H−1 (Ω).

Zhongwei Shen

Then

79

ˆ

ˆ (u · v ) ϕ dx → Ω

(u · v) ϕ dx, Ω

as  → ∞, for any scalar function ϕ ∈ C10 (Ω). The next theorem shows that the sequence of operators {Lε } is H-compact in the sense of H-convergence [28]. Theorem 1.2.5. Let {A (y)} be a sequence of 1-periodic matrices satisfying (1.1.2) with the same constant μ. Let F ∈ H−1 (Ω). Suppose that Lε (u ) = F

(1.2.6)

in Ω,

where ε → 0, u ∈ H1 (Ω), and

  Lε = − div A (x/ε )∇ .

We further assume that (1.2.7)

⎧ F → F in H−1 (Ω), ⎪ ⎪ ⎨  u  u weakly in H1 (Ω), ⎪ ⎪ ⎩ A → A0 ,

 denotes the matrix of effective coefficients for A . Then where A (1.2.8)

A (x/ε )∇u  A0 ∇u

weakly in L2 (Ω; Rd ),

A0 is a constant matrix satisfying the ellipticity condition (1.1.14), and u is a weak solution of − div(A0 ∇u) = F

(1.2.9)

in Ω.

 → A0 and A  satisfies (1.1.14), so does A0 . Also, (1.2.9) follows Proof. Since A directly from (1.2.6) and (1.2.8). To see (1.2.8), we let {u } be a subsequence such that A (x/ε )∇u  H

weakly in L2 (Ω; Rd )

for some H ∈ L2 (Ω; Rd ) and show that H = A0 ∇u. This would imply that the full sequence A (x/ε )∇u converges weakly to A0 ∇u in L2 (Ω; Rd ). Without loss of generality we assume that weakly in L2 (Ω; Rd )   for some H = (Hi ) ∈ L2 (Ω; Rd ). Let χ∗ (y) = χ∗k, (y) denote the correctors associated with the matrix A∗ , the adjoint of A . Fix 1  k  d and consider the identity ˆ 

 A (x/ε )∇u , ∇ xk + ε χ∗k, (x/ε ) ψ dx ˆΩ  (1.2.11)

 = ∇u , A∗ (x/ε )∇ xk + ε χ∗k, (x/ε ) ψ dx, (1.2.10)

A (x/ε )∇u  H

Ω

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

where ψ ∈ C10 (Ω). By Proposition 1.2.1, 

(1.2.12) ∇ xk + ε χ∗k, (x/ε ) = ∇xk + ∇χ∗k, (x/ε )  ∇xk ´ weakly in L2 (Ω), using the fact Y ∇χ∗k, dy = 0. Since Lε (uε ) = F in Ω, in view of (1.2.10) and (1.2.12), it follows by Theorem 1.2.4 that the left side of (1.2.11) converges to ˆ ˆ H, ∇xk ψ dx = Hk ψ dx. Ω

Ω

Similarly, note that ∇u  ∇u weakly and

 A∗ (x/ε )∇ xk + ε χ∗k, (x/ε )  lim

 A∗ ∇xk + ∇χ∗k, dy

→∞ Y ∗ ∇xk = lim A →∞  =(A0 )∗ ∇xk

weakly in L2 (Ω), where we have used Proposition 1.2.1 as well as Theorem 1.1.16. Since  ∗ in Rd , L∗ ε xk + ε χk, (x/ε ) = 0 we use Theorem 1.2.4 again to claim that the right side of (1.2.11) converges to ˆ   ∇u, (A0 )∗ ∇xk ψ dx. Ω

As a result, since ψ ∈ (1.2.13)

C10 (Ω)

is arbitrary, it follows that     Hk = ∇u, (A0 )∗ ∇xk = A0 ∇u, ∇xk

in Ω.

This shows that H = A0 ∇u and completes the proof.



We now use Theorem 1.2.5 to establish the qualitative homogenization of the Dirichlet and Neumann problems for Lε . The proof only uses a special case of Theorem 1.2.5, where A = A is fixed. The general case is essential in a compactness argument we will use in Section 3 for the large-scale regularity. 1.3. Homogenization of boundary value problems Assume that A satisfies condition (1.1.2) and is 1-periodic. Let F ∈ L2 (Ω), G ∈ L2 (Ω; Rd ) and f ∈ H1/2 (∂Ω). There exists a unique uε ∈ H1 (Ω) such that Lε (uε ) = F + div(G)

in Ω

and

uε = f

on ∂Ω

(the boundary data is taken in the sense of trace). Furthermore, uε satisfies

uε H1 (Ω)  C FL2 (Ω) + GL2 (Ω) + fH1/2 (∂Ω) , where C depends only on μ and Ω. Let {uε } be a subsequence of {uε } such that as ε → 0, uε  u weakly in H1 (Ω) for some u ∈ H1 (Ω). It follows by Theorem 1.2.5 that  A(x/ε )∇uε  A∇u

Zhongwei Shen

81

and L0 (u) = F + div(G) in Ω. Since f ∈ H1/2 (∂Ω), there exists Φ ∈ H1 (Ω) such that Φ = f on ∂Ω. Using the facts that uε − Φ  u − Φ

weakly in H1 (Ω)

and uε − Φ ∈ H10 (Ω), we see that u − Φ ∈ H10 (Ω). Hence, u = f on ∂Ω. Consequently, u is the unique weak solution to the Dirichlet problem, L0 (u0 ) = F + div(G)

in Ω

and

u0 = f

on ∂Ω.

Since {uε } is bounded in H1 (Ω) and thus any sequence {uε } with ε → 0 contains a subsequence that converges weakly in H1 (Ω), one concludes that as ε → 0,

 A(x/ε)∇uε  A∇u weakly in L2 (Ω; Rd ), 0 (1.3.1) uε  u0 weakly in H1 (Ω). By the compactness of the embedding H1 (Ω) → L2 (Ω), we also obtain (1.3.2)

uε → u0

strongly in L2 (Ω).

Let F ∈ L2 (Ω), G ∈ L2 (Ω; Rd ), and g ∈ H−1/2 (∂Ω), the dual of H1/2 (∂Ω). Consider the Neumann problem ⎧ ⎪ in Ω, ⎨Lε (uε ) = F + div(G) (1.3.3) ∂uε ⎪ = g−n·G on ∂Ω, ⎩ ∂νε where the conormal derivative (1.3.4)

∂uε ∂νε

on ∂Ω is defined by

∂uε ∂uε = ni (x)aij (x/ε) , ∂νε ∂xj

and n denotes the outward unit normal to ∂Ω. Definition 1.3.5. We call uε ∈ H1 (Ω) a weak solution of the Neumann problem (1.3.3) if ˆ ˆ x (1.3.6) A( )∇uε · ∇ϕ dx = (Fϕ − G · ∇ϕ)dx + g, ϕH−1/2 (∂Ω)×H1/2 (∂Ω) ε Ω Ω for any ϕ ∈ C∞ (Rd ). Let {uε } be a subsequence of {uε } such that uε  u0 weakly in H1 (Ω) for some u0 ∈ H1 (Ω). It follows by Theorem 1.2.5 that  weakly in L2 (Ω; Rd ). A(x/ε )∇uε  A∇u 0 By taking limits in (1.3.6) we see that u0 is a weak solution to the Neumann problem: ∂u0 (1.3.7) L0 (u0 ) = F + div(G) in Ω and = g − n · G on ∂Ω, ∂ν0 ´ and that Ω u0 dx = 0, where ∂u ∂u0 ij 0 = ni a (1.3.8) ∂ν0 ∂xj

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

is the conormal derivative associated with the operator L0 . Since such u0 is unique, we conclude that as ε → 0, uε  u0 weakly in H1 (Ω) and thus strongly in L2 (Ω). As in the case of the Dirichlet problem, we also obtain 2 d  A(x/ε)∇uε  A∇u 0 weakly in L (Ω; R ). Remark 1.3.9. The material in this section is standard and may be found in, for example, [5, 7, 9, 29]. 1.4. Problems for Section 1  in the case d = 1. Problem 1.4.1. Find explicit formulas for χ and A Problem 1.4.2. Prove Theorem 1.2.4. Hint: If curl(g) = 0 in a ball B, then there exists G ∈ H1 (B) such that g = ∇G in B. Problem 1.4.3. Assume that F, G and g satisfy the compatibility condition ˆ (1.4.4) F dx + g, 1H−1/2 (∂Ω)×H1/2 (∂Ω) = 0. Ω

Show that the Neumann problem (1.3.3) has a unique (up to a constant in R) ´ solution. Furthermore, if Ω uε dx = 0, then

uε H1 (Ω)  C FL2 (Ω) + GL2 (Ω) + gH−1/2 (∂Ω) , where C depends only on μ and Ω.

2. Convergence Rates

  Let Lε = − div(A(x/ε)∇) for ε > 0, where A(y) = aij (y) is 1-periodic and     = a ij where A satisfies the ellipticity condition (1.1.2). Let L0 = − div(A∇), 2 denotes the matrix of effective coefficients, given by (1.1.11). For F ∈ L (Ω) and ε  0, consider the Dirichlet problem

in Ω, Lε (uε ) = F (2.0.1) uε = f on ∂Ω, and the Neumann problem ⎧ Lε (uε ) = F ⎪ ⎪ ⎪ ⎨ ∂u ε (2.0.2) =g ⎪ ∂ν ε ⎪ ⎪ ⎩ uε ⊥ R in L2 (Ω),

in Ω, on ∂Ω,

where f ∈ H1 (∂Ω), and g ∈ L2 (∂Ω). It is shown in Section 1 that as ε → 0, uε converges to u0 weakly in H1 (Ω) and strongly in L2 (Ω). In this section we investigate the problem of convergence rates in H1 and Lp . For 1  i, j  d, let  ∂  ij , bij (y) = aij (y) + aik (y) χj (y) − a ∂yk

2.1. Flux correctors and ε–smoothing (2.1.1)

Zhongwei Shen

83

where the repeated index k is summed from 1 to d. Observe that the matrix   B(y) = bij (y) is 1-periodic and that BL2 (Y)  C0 for some C0 > 0 depending ij that on μ. Moreover, it follows from the definitions of χj and a ˆ   ∂ (2.1.2) bij (y) dy = 0. bij = 0 and ∂yi Y Proposition 2.1.3. There exist φkij ∈ H1per (Y), where 1  i, j, k  d, such that  ∂  (2.1.4) bij = φkij and φkij = −φikj in Y. ∂yk Moreover, if χ = (χj ) is Hölder continuous, then φkij ∈ L∞ (Y). ´ ´ Proof. Since Y bij dy = 0, there exists fij ∈ H2per (Y) such that Y fij dy = 0 and (2.1.5)

Δfij = bij

in Y.

Moreover, fij H2 (Y)  C bij L2 (Y)  C. Define φkij (y) =

∂   ∂   fij − fkj . ∂yk ∂yi

 ∂ bij = 0, we deduce Clearly, φkij ∈ H1per (Y) and φkij = −φikj . Using ∂y i  ∂ f is a 1-periodic harmonic function and thus is constant. from (2.1.5) that ∂y ij i Hence,   ∂  ∂2  φkij = Δ fij − fkj = Δ fij = bij . ∂yk ∂yk ∂yi Suppose that the corrector χ is Hölder continuous. Recall that   (2.1.6) L χj + yj = 0 in Rd . By Caccioppoli’s inequality, ˆ ˆ C |∇χ|2 dx  2 |χ(x) − χ(y)|2 dx + C rd . r B(y,r) B(y,2r) This implies that |∇χ| is in the Morrey space L2,ρ (Y) for some ρ > d − 2; i.e., ˆ |∇χ|2 dx  C rρ for y ∈ Y and 0 < r < 1. B(y,r)

Consequently, bij ∈ L2,ρ (Y) for some ρ > d − 2 and ˆ |bij (y)| sup dy  C. d−1 x∈Y Y |x − y| In view of (2.1.5), by considering Δ(fij ϕ), where ϕ is a cut-off function, and using a potential representation for Laplace’s equation, one deduces that ˆ |bij (y)| dy  C. (2.1.7) ∇fij L∞ (Y)  C fij L2 (Y) + C sup d−1 x∈Y Y |x − y| It follows that φkij ∈ L∞ (Y).



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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Remark 2.1.8. For scalar elliptic equations in divergence form, it follows from (2.1.6) and the classical De Giorgi - Nash estimate that χ is Hölder continuous. As a result, (φkij ) is bounded if m = 1. The same is true if d = 2 and m > 1.   Remark 2.1.9. A key property of φ = φkij , which follows from (2.1.4), is the identity:   ∂ψ ∂ψ ∂ =ε φkij (x/ε) (2.1.10) bij (x/ε) ∂xi ∂xk ∂xi for any ψ ∈ H2 (Ω). Let uε ∈ H1 (Ω), u0 ∈ H2 (Ω), and wε = uε − u0 − ε χ(x/ε)∇u0 . A direct computation shows that 2  (2.1.11) A(x/ε)∇uε − A∇u 0 − B(x/ε)∇u0 = A(x/ε)∇wε + ε A(x/ε)χ(x/ε)∇ u0 ,   where B(y) = bij (y) is defined by (2.1.1). It follows that  A(x/ε)∇uε − A∇u 0 − B(x/ε)∇u0  2 L (Ω)

(2.1.12)

2

 C ∇wε L2 (Ω) + C ε χ(x/ε)∇ u0 L2 (Ω)  C ∇uε − ∇u0 − ∇χ(x/ε)∇u0 L2 (Ω) + Cε χ(x/ε)∇2 u0 L2 (Ω) ,

where C depends only on μ. This indicates that the 1-periodic matrix-valued function B(y) plays the same role for the flux A(x/ε)∇uε as ∇χ(y) does for ∇uε .     ∂ φkij , the 1-periodic function φ = φkij is called the flux Since bij = ∂y k corrector. To deal with the fact that the correctors χ and φ may be unbounded (for elliptic systems in higher dimensions), we introduce an ε-smoothing operator Sε . ´ Definition 2.1.13. Fix ρ ∈ C∞ 0 (B(0, 1/2)) such that ρ  0 and Rd ρ dx = 1. For ε > 0, define ˆ (2.1.14) Sε (f)(x) = ρε ∗ f(x) = f(x − y)ρε (y) dy, Rd

where ρε (y) =

ε−d ρ(y/ε).

p d d Proposition 2.1.15. Let f ∈ Lp loc (R ) for some 1  p < ∞. Then for any g ∈ Lloc (R ),

1/p

(2.1.16)

g(x/ε) Sε (f)Lp (O)  C sup x∈Rd

where O ⊂ Rd is open,

|g|p B(x,1/2)

 O(t) = x ∈ Rd : dist(x, O) < t ,

and C depends only on p. Proof. By Hölder’s inequality, |Sε (f)(x)|p 

ˆ Rd

|f(y)|p ρε (x − y) dy,

fLp (O(ε/2)) ,

Zhongwei Shen

where we also used the fact

85

ˆ Rd

ρε dx = 1.

This, together with Fubini’s Theorem, gives (2.1.16) for the case O = Rd . The general case follows from the observation that Sε (f)(x) = Sε (fχO(ε/2) )(x) for any x ∈ O.



It follows from (2.1.16) that if g is 1-periodic and belongs to Lp (Y), then g(x/ε)Sε (f)Lp (O)  C gLp (Y) fLp (O(ε/2)) ,

(2.1.17)

where C depends only on p. Proposition 2.1.18. Let f ∈ W 1,p (Rd ) for some 1  p < ∞. Then   g(x/ε) Sε (f) − f Lp (Rd )

1/p (2.1.19) |g(y)|p  Cε sup dy ∇fLp (Rd ) , d−1 B(x,1/2) |y − x| x∈Rd where C depends only on p. Moreover, if q =

2d d+1 , −1/2

Sε (f)L2 (Rd )  Cε

(2.1.20)

fLq (Rd ) ,

Sε (f) − fL2 (Rd )  Cε1/2 ∇fLq (Rd ) .

Proof. To prove (2.1.19), observe that ˆ |Sε (f)(x) − f(x)| 

Rd

By Hölder’s inequality,

ˆ

|Sε (f)(x) − f(x)|  p

It follows that

Rd

|f(y) − f(x)|p ρε (x − y) dy.

ˆ Rd

(2.1.21)

|f(y) − f(x)|ρε (x − y) dy.

|g(x/ε)|p |Sε (f)(x) − f(x)|p dx ˆ ˆ

 Cε−d Using

ˆ |f(x + tz) − f(x)| =

0

t

Rd

|g(x/ε)|p



|f(y) − f(x)|p dy dx. B(x,ε/2)

ˆ 1 ∇f(x + sz) · z ds  t1− p

where |z| = 1 and t > 0, we obtain ˆ ˆ |f(x + tz) − f(x)|p dσ(z)  tp−1 |z|=1

ˆ

ˆ

t

t 0

1/p |∇f(x + sz)|p ds

|∇f(x + sz)|p dsdσ(z)

|z|=1

0

B(x,t)

|∇f(y)|p dy, |y − x|d−1

 tp−1

,

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

where we have used the polar coordinates. Hence, ˆ ˆ ε/2 ˆ |f(y) − f(x)|p dy = |f(x + tz) − f(x)|p td−1 dσ(z)dt |z|=1

0

B(x,ε/2)

ˆ

 Cεd+p−1 B(x,ε/2)

|∇f(y)|p dy, |y − x|d−1

where C depends only on p. This, together with (2.1.21) and Fubini’s Theorem, gives (2.1.19).  (εξ)f(ξ). Applying Next, note that the Fourier transform of Sε (f) is given by ρ 2d Hölder’s inequality and Plancherel’s theorem, if we let q = d+1 , then ˆ ˆ  2 dξ |Sε (f)|2 dx = | ρ(εξ)|2 |f(ξ)| Rd

Rd

1/d

ˆ

2d

  Cε

Rd −1

| ρ(εξ)|

dξ

2  f Lq (Rd )

f2Lq (Rd ) ,

where we have used the Hausdorff-Young inequality  q d  f q d f L (R )

L (R )

in the last step. This gives the first inequality in (2.1.20). Similarly, using ˆ  ρ(0) = ρ dx = 1, Rd

we obtain  2 d ρ(εξ) − 1)f Sε (f) − fL2 (Rd ) = ( L (R ) ˆ 1/(2d)  q d (0)|2d |ξ|−2d dξ C | ρ(εξ) − ρ ∇f L (R ) = Cε

Rd 1/2

 q d (0))|ξ|−1 L2d (Rd ) ∇f ( ρ(ξ) − ρ L (R )

 Cε1/2 ∇fLq (Rd ) , where we have also used the fact (0)|  C|ξ| | ρ(ξ) − ρ for |ξ|  1 in the last step.



It follows from (2.1.19) that if g is 1-periodic and belongs to Lq (Y) for some q > pd, then   (2.1.22) g(x/ε) Sε (f) − f Lp (Rd )  CεgLq (Y) ∇fLp (Rd ) , where C depends only on p and q. Let  (2.1.23) Ωt = x ∈ Ω : dist(x, ∂Ω) < t , where t > 0. The proof for the next two propositions may be found in [23].

Zhongwei Shen

87

Proposition 2.1.24. Let Ω be a bounded Lipschitz domain in Rd and q = for any f ∈ W 1,q (Ω), (2.1.25)

fL2 (Ωt )  Ct1/2 fW 1,q (Ω)

and

2d d+1 .

Then

fL2 (∂Ω)  C fW 1,q (Ω) ,

where C depends only on Ω. Proposition 2.1.26. Let Ω be a bounded Lipschitz domain in Rd and q = g ∈ L2loc (Rd ) be a 1-periodic function. Then for any f ∈ W 1,q (Ω), ˆ (2.1.27) |g(x/ε)|2 |Sε (f)|2 dx  Ct g2L2 (Y) f2W 1,q (Ω) ,

2d d+1 .

Let

Ω2t \Ωt

where t  ε and C depends only on Ω. We end this subsection with the fractional integral estimates. Proposition 2.1.28. For 0 < α < d, define ˆ Tα (f)(x) =

Rd

Then, if 1 < p < (2.1.29)

d α

and

1 q

1 p

=

−

f(y) dy. |x − y|d−α

α d,

Tα (f)Lq (Rd )  C fLp (Rd ) ,

where C depends only on α and p. Proof. The inequality (2.1.29) is referred as the Hardy-Littlewood-Sobolev inequal ity. See [12, pp.162-163] for its proof. 2.2. Convergence rates in H1 Fix a cut-off function ηε ∈ C∞ 0 (Ω) for which we have ⎧ ⎪ 0  ηε  1, |∇ηε |  C/ε, ⎪ ⎨ (2.2.1) ηε (x) = 1 if dist(x, ∂Ω)  4ε, ⎪ ⎪ ⎩η (x) = 0 if dist(x, ∂Ω)  3ε. ε

Let

S2ε

(2.2.2)

= Sε ◦ Sε and define wε = uε − u0 − εχ(x/ε)ηε S2ε (∇u0 ),

where uε ∈ H1 (Ω) is the weak solution of Dirichlet problem (2.0.1) and u0 the homogenized solution. Lemma 2.2.3. Let Ω be a bounded Lipschitz domain in Rd and Ωt be defined by (2.1.23). Then, for any ψ ∈ H10 (Ω), ˆ A(x/ε)∇wε · ∇ψ dx Ω

(2.2.4)  C ∇ψ 2 2 L (Ω) εSε (∇ u0 )L2 (Ω\Ω3ε ) + ∇u0 − Sε (∇u0 )L2 (Ω\Ω2ε ) + C ∇ψL2 (Ω4ε ) ∇u0 L2 (Ω5ε ) , where wε is given by (2.2.2) and C depends only on μ and Ω.

88

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Proof. Since wε ∈ H1 (Ω), by a density argument, it suffices to prove (2.2.4) for any ψ ∈ C∞ 0 (Ω). Note that A(x/ε)∇wε = A(x/ε)∇uε − A(x/ε)∇u0 − A(x/ε)∇χ(x/ε)ηε S2ε (∇u0 )   − εA(x/ε)χ(x/ε)∇ ηε S2ε (∇u0 )    2   = A(x/ε)∇uε − A∇u 0 + A − A(x/ε) ∇u0 − ηε Sε (∇u0 )   − B(x/ε)ηε S2ε (∇u0 ) − εA(x/ε)χ(x/ε)∇ ηε S2ε (∇u0 ) ,  where we have used the fact B(y) = A(y) + A(y)∇χ(y) − A. Next, using ˆ ˆ  A(x/ε)∇uε · ∇ψ dx = A∇u (2.2.5) 0 · ∇ψ dx Ω

Ω

for any ψ ∈ C∞ 0 (Ω), we obtain ˆ A(x/ε)∇wε · ∇ψ dx ˆΩ ˆ C (1 − ηε )|∇u0 | |∇ψ| dx + C ηε |∇u0 − S2ε (∇u0 )| |∇ψ| dx Ω Ω (2.2.6) ˆ 2 + ηε B(x/ε)Sε (∇u0 ) · ∇ψ dx Ω ˆ   + Cε |χ(x/ε)∇ ηε S2ε (∇u0 ) | |∇ψ| dx. Ω

Since ηε = 1 in Ω \ Ω4ε , by the Cauchy inequality, the first term on the right of (2.2.6) is bounded by C ∇u0 L2 (Ω4ε ) ∇ψL2 (Ω4ε ) . Using ηε = 0 in Ω3ε and ∇u0 − S2ε (∇u0 )L2 (Ω\Ω3ε )  ∇u0 − Sε (∇u0 )L2 (Ω\Ω3ε ) + Sε (∇u0 ) − S2ε (∇u0 )L2 (Ω\Ω3ε )  C ∇u0 − Sε (∇u0 )L2 (Ω\Ω2ε ) , we may bound the second term by C ∇u0 − Sε (∇u0 )L2 (Ω\Ω2ε ) ∇ψL2 (Ω) . Also, by the Cauchy inequality and (2.1.17), the fourth term in the right side of (2.2.6) is dominated by C ∇u0 L2 (Ω5ε ) ∇ψL2 (Ω4ε ) + Cε Sε (∇2 u0 )L2 (Ω\Ω2ε ) ∇ψL2 (Ω) . Finally, to handle the third term in the right side of (2.2.6), we use the identity (2.1.10) to obtain   ∂u0 ∂ψ 2 2 ηε B(x/ε)Sε (∇u0 ) · ∇ψ = bij (x/ε)Sε ηε ∂xj ∂xi (2.2.7)     ∂ ∂ψ ∂u0 =ε φkij (x/ε) S2ε ηε . ∂xk ∂xi ∂xj

Zhongwei Shen

89

It follows from (2.2.7) and integration by parts that ˆ ηε B(x/ε)S2ε (∇u0 ) · ∇ψ dx Ω ˆ  Cε ηε |φ(x/ε)| |∇ψ| |S2ε (∇2 u0 )| dx Ω ˆ + Cε |∇ηε | |φ(x/ε)| |∇ψ| |S2ε (∇u0 )| dx Ω

 Cε ∇ψL2 (Ω) Sε (∇2 u0 )L2 (Ω\Ω2ε ) + C ∇ψL2 (Ω4ε ) ∇u0 L2 (Ω5ε ) , where we have used the Cauchy inequality and (2.1.17) for the last step.



Theorem 2.2.8. Assume that A is 1-periodic and satisfies (1.1.2). Let Ω be a bounded Lipschitz domain in Rd . Let wε be given by (2.2.2). Then for 0 < ε < 1,

(2.2.9) ∇wε L2 (Ω)  C ε∇2 u0 L2 (Ω\Ωε ) + ∇u0 L2 (Ω5ε ) . Consequently, (2.2.10)

√ wε H1 (Ω)  C ε u0 H2 (Ω) . 0

The constant C depends only on μ and Ω. Proof. Since wε ∈ H10 (Ω), we may take ψ = wε in (2.2.4). This, together with the ellipticity condition (1.1.2), gives ∇wε L2 (Ω)  Cε ∇2 u0 L2 (Ω\Ω2ε ) + C ∇u0 − Sε (∇u0 )L2 (Ω\Ω2ε )

(2.2.11)

+ C ∇u0 L2 (Ω5ε ) .

ε ∈ C∞ Choose η ηε  1,  ηε = 0 in Ωε ,  ηε = 1 in Ω \ Ω3ε/2 , and 0 (Ω) such that 0   |∇ η|  Cε−1 . It follows that ∇u0 − Sε (∇u0 )L2 (Ω\Ω2ε )   ηε (∇u0 ) − Sε ( ηε ∇u0 )L2 (Rd )  Cε ∇( ηε ∇u0 )L2 (Rd )

 C ε∇2 u0 L2 (Ω\Ωε ) + ∇u0 L2 (Ω2ε ) ,

(2.2.12)

where we have used (2.1.19) for the second inequality. The estimate (2.2.9) now follows from (2.2.11) and (2.2.12). Note that by (2.1.25), √ (2.2.13) ∇u0 L2 (Ω5ε )  C ε u0 H2 (Ω) . The inequality (2.2.10) follows from (2.2.9) and (2.2.13).



Under the additional symmetry condition A∗ = A, i.e., aij = aji , an improved estimate can be obtained using sharp regularity estimates for L0 . Theorem 2.2.14. Suppose that A is 1-periodic and satisfies (1.1.2). Also assume A∗ = A. Let Ω be a bounded Lipschitz domain in Rd . Let wε be given by (2.2.2). Then, for 0 < ε < 1,

√ (2.2.15) wε H1 (Ω)  C ε FLq (Ω) + fH1 (∂Ω) , 0

where q =

2d d+1

and C depends only on μ and Ω.

90

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Definition 2.2.16. For a continuous function u in a bounded Lipschitz domain Ω, the nontangential maximal function of u is defined by

(2.2.17) (u)∗ (x) = sup |u(y)| : y ∈ Ω and |y − x| < C0 dist(y, ∂Ω) for x ∈ ∂Ω, where C0 > 1 is a sufficiently large constant depending on Ω. The proof of Theorem 2.2.14 relies on the following regularity result for solutions of L0 (u) = 0 in Ω. Lemma 2.2.18. Assume that A satisfies the same conditions as in Theorem 2.2.14. Let Ω be a bounded Lipschitz domain in Rd . Let u ∈ H1 (Ω) be a weak solution to the Dirichlet problem: L0 (u) = 0 in Ω and u = f on ∂Ω, where f ∈ H1 (∂Ω). Then (∇u)∗ L2 (∂Ω)  C fH1 (∂Ω) ,

(2.2.19)

where C depends only on μ and Ω.  satisfies the ellipProof. By Lemmas 1.1.13 and 1.1.16, the homogenized matrix A ticity condition (1.1.14) and is symmetric. As a result, the estimate (2.2.19) follows  from [10, 11, 14]. Proof of Theorem 2.2.14. We start by taking ψ = wε ∈ H10 (Ω) in (2.2.4). By the ellipticity condition (1.1.2), this gives (2.2.20)

∇wε L2 (Ω)  Cε Sε (∇2 u0 )L2 (Ω\Ω3ε ) + C ∇u0 L2 (Ω5ε ) + C ∇u0 − Sε (∇u0 )L2 (Ω\Ω2ε ) .

To bound the right side of (2.2.20), we write u0 = v0 + φ, where ˆ Γ0 (x − y)F(y) dy (2.2.21) v0 (x) = Ω

and Γ0 (x) denotes the fundamental solution for the homogenized operator L0 in Rd , with pole at the origin. It follows from the standard Calderón-Zygmund estimates for singular integrals (see [24, Chapter 2]) that ∇2 v0 Lr (Rd )  C FLr (Ω) ,

(2.2.22)

for any 1 < r < ∞. Since |∇Γ0 (x)|  C|x|1−d , by the fractional integral estimates in Proposition 2.1.28, ∇v0 Lp (Rd )  C FLq (Ω) ,

(2.2.23)

where p = and q = p = (2.1.25), yield that 2d d−1

2d d+1 .

These estimates, together with (2.1.20) and

εSε (∇2 v0 )L2 (Ω\Ω3ε )  Cε1/2 ∇2 v0 Lq (Rd )  Cε1/2 FLq (Ω) , and that ∇v0 L2 (Ω5ε )  Cε1/2 ∇v0 W 1,q (Ω)  Cε1/2 FLq (Ω) .

Zhongwei Shen

91

Also, note that by (2.1.20), ∇v0 − Sε (∇v0 )L2 (Ω\Ω2ε )  Cε1/2 ∇2 v0 Lq (Rd )  Cε1/2 FLq (Ω) . In summary we have proved that εSε (∇2 v0 )L2 (Ω\Ω3ε ) + ∇v0 L2 (Ω5ε )

(2.2.24)

+ ∇v0 − Sε (∇v0 )L2 (Ω\Ω2ε )  Cε1/2 FLq (Ω) .

To bound the left side of (2.2.24), with v0 replaced by φ, we first note that L0 (φ) = 0 in Ω and φ = f − v0 on ∂Ω. which lets us apply Lemma 2.2.18. Since v0 H1 (∂Ω)  C v0 W 2,q (Ω)  C FLq (Ω) , where we have used (2.1.25) for the first inequality, we obtain



(∇φ)∗ L2 (∂Ω)  C fH1 (∂Ω) + v0 H1 (∂Ω)  C fH1 (∂Ω) + FLq (Ω) . It follows that

∇φL2 (Ω5ε )  Cε1/2 (∇φ)∗ L2 (∂Ω)

 Cε1/2 fH1 (∂Ω) + FLq (Ω) .

Next, we use the interior estimate for L0 , ˆ C |∇2 φ(x)|2  d+2 |∇φ(y)|2 dy, r B(x,r) where r = dist(x, ∂Ω)/8, and Fubini’s Theorem to obtain ˆ ˆ  −2 |∇2 φ(x)|2 dx  C |∇φ(x)|2 dist(x, ∂Ω) dx Ω\Ωε

Ω\Ωε/2

ˆ

 Cε

−1

|(∇φ)∗ |2 dσ.

∂Ω

Hence, εSε (∇2 φ)L2 (Ω\Ω3ε )  Cε ∇2 φL2 (Ω\Ω2ε )  Cε1/2 (∇φ)∗ L2 (∂Ω)

 Cε1/2 fH1 (∂Ω) + FLq (Ω) . Finally, we observe that as in (2.2.12), (2.2.25)



∇φ − Sε (∇φ)L2 (Ω\Ω2ε )  C ε ∇2 φL2 (Ω\Ωε ) + ∇φL2 (Ω2ε )

 Cε1/2 fH1 (∂Ω) + FLq (Ω) ,

where we have used (2.1.19) for the second inequality. As a result, we have proved that ε Sε (∇2 φ)L2 (Ω\Ω3ε ) + ∇φL2 (Ω5ε ) + ∇φ − Sε (∇φ)L2 (Ω\Ω2ε )

 Cε1/2 fH1 (∂Ω) + FLq (Ω) . This, together with (2.2.20) and (2.2.24), gives (2.2.15).



92

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

 Thus Remark 2.2.26. The proof of Theorem 2.2.14 only uses the symmetry of A. ∗ we may drop the assumption A = A in the case of scalar elliptic equations, as L0 (u) = (1/2)(L0 + L∗0 )(u). Also, since FLq (Ω) = L0 (u0 )Lq (Ω)  C ∇2 u0 Lq (Ω) 2d and fH1 (∂Ω)  C u0 W 2,q (Ω) , where q = d+1 , it follows from (2.2.15) that √ (2.2.27) wε H1 (Ω)  C ε u0 W 2,q (Ω) , 0

where C depends only on μ and Ω. We now consider the two-scale expansions without the ε-smoothing. Theorem 2.2.28. Assume that A is 1-periodic and satisfies (1.1.2). Also assume χ is bounded. Let Ω be a bounded Lipschitz domain in Rd . Let uε ∈ H1 (Ω) be the weak solution of Dirichlet problem (2.0.1) and u0 the homogenized solution. Then, √ (2.2.29) uε − u0 − εχ(x/ε)∇u0 H1 (Ω)  C ε u0 H2 (Ω) , where 0 < ε < 1 and C depends only on μ, χ∞ and Ω. Lemma 2.2.30. If χ is bounded, then



∇χ(x/ε)ψL2 (Ω)  C(1 + χ∞ ) ε ∇ψL2 (Ω) + ψL2 (Ω)

(2.2.31)

for any ψ ∈ H1 (Ω), where C depends only on μ and Ω. Proof. Fix x0 ∈ Rd and let uε (x) = εχj (x/ε) + xj − (x0 )j . Since Lε (uε ) = 0 in Rd , it follows by Caccioppoli’s inequality that ˆ ˆ |∇uε |2 |ϕ|2 dx  C |uε |2 |∇ϕ|2 dx Rd

for any ϕ ∈ ˆ

C10 (Rd ).

B(x0 ,2ε)

Thus, if ϕ ∈

Rd

C10 (B(x0 , 2ε)),

ˆ

ˆ

|∇χ(x/ε)|2 |ϕ|2 dx  C

B(x0 ,2ε)

|ϕ|2 + Cε2

ˆ

+ Cε2 B(x0 ,2ε)

B(x0 ,2ε)

|∇ϕ|2 dx

|χ(x/ε)|2 |∇ϕ|2 dx.

Let ϕ = ψ ηε , where ψ ∈ C10 (Rd ) and  ηε is a cut-off function in C10 (B(x0 , 2ε)) with ε = 1 on B(x0 , ε) and |∇ ηε |  C/ε. We obtain the properties that 0   ηε  1, η ˆ ˆ |∇χ(x/ε)|2 |ψ|2 dx  C (1 + |χ(x/ε)|2 )|ψ|2 dx B(x0 ,ε)

B(x0 ,2ε)

ˆ

+ Cε

2 B(x0 ,2ε)

(1 + |χ(x/ε)|2 )|∇ψ|2 dx.

By integrating the inequality above in x0 over Rd we see that ˆ ˆ |∇χ(x/ε)|2 |ψ|2 dx  C (1 + |χ(x/ε)|2 )|ψ|2 dx d d R R ˆ (2.2.32) + Cε2 (1 + |χ(x/ε)|2 )|∇ψ|2 dx Rd

for any ψ ∈

C10 (Rd ).

Zhongwei Shen

93

If χ is bounded, it follows from (2.2.32) that

∇χ(x/ε)ψL2 (Ω)  C(1 + χ∞ ) ψL2 (Rd ) + ε ∇ψL2 (Rd ) for any ψ ∈ C10 (Rd ). By a limiting argument, the same inequality holds for any ψ ∈ H1 (Rd ). This gives (2.2.31), using the fact that for any ψ ∈ H1 (Ω), one  2 d  C ψ 2  in H1 (Rd ) so that ψ may extend it to a function ψ L (Ω) and L (R )   ψH1 (Rd )  C ψH1 (Ω) [24, Chapter VI]. Proof of Theorem 2.2.28. Suppose χ is bounded. To prove (2.2.29), in view of (2.2.10), it suffices to show that √ (2.2.33) εχ(x/ε)∇u0 − εχ(x/ε)ηε S2ε (∇u0 )H1 (Ω)  C ε u0 H2 (Ω) . To this end, we note that the left side of (2.2.33) is bounded by   Cε χ(x/ε) ∇u0 − ηε S2ε (∇u0 ) L2 (Ω)   + C ∇χ(x/ε) ∇u0 − ηε S2ε (∇u0 ) L2 (Ω)   + Cε χ(x/ε)∇ ∇u0 − ηε S2ε (∇u0 ) L2 (Ω)  Cε ∇(∇u0 − ηε S2ε (∇u0 ))L2 (Ω) + C ∇u0 − ηε S2ε (∇u0 )L2 (Ω) , where we have used (2.2.31). Note that

  ε ∇(∇u0 − ηε S2ε (∇u0 ))L2 (Ω)  ε ∇2 u0 L2 (Ω) + ε ∇ ηε S2ε (∇u0 ) L2 (Ω)  Cε u0 H2 (Ω) + C ∇u0 L2 (Ω5ε ) √  C ε u0 H2 (Ω) ,

and ∇u0 − ηε S2ε (∇u0 )L2 (Ω)  C ∇u0 L2 (Ω5ε ) + ∇u0 − S2ε (∇u0 )L2 (Ω\Ω4ε ) √  C ε u0 H2 (Ω) , where we also used (2.2.25) for the last inequality.



The same estimate also holds for the Neumann problems. Theorem 2.2.34. Assume that A is 1-periodic and satisfies (1.1.2). Let Ω be a bounded Lipschitz domain in Rd . Let uε ∈ H1 (Ω) be the weak solution of the Neumann problem ´ (2.0.2) with Ω uε dx = 0. Let u0 be the homogenized solution. Then, if the corrector χ is bounded and u0 ∈ H2 (Ω), √ (2.2.35) uε − u0 − εχ(x/ε)∇u0 H1 (Ω)  C ε u0 H2 (Ω) for any 0 < ε < 1, where C depends only on μ, χ∞ and Ω. Proof. See [23]



2.3. Convergence rates in Lp In this subsection we establish a sharp O(ε) con2d for Dirichlet problem (2.0.1), under the assumpvergence rate in Lp with p = d−1 tions that A is 1-periodic, satisfies (1.1.2) and A∗ = A. Without the symmetry condition, we obtain an O(ε) convergence rate in L2 .

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Lemma 2.3.1. Let Ω be a bounded Lipschitz domain in Rd . Let wε be given by (2.2.2), where uε is the weak solution of (2.0.1) and u0 the homogenized solution. Assume that 2d . Then, for any ψ ∈ C∞ u0 ∈ W 2,q (Ω) for q = d+1 0 (Ω), ˆ

√ A(x/ε)∇wε · ∇ψ dx  C u0 W 2,q (Ω) ε ∇ψLp (Ω) + ε ∇ψL2 (Ω4ε ) , Rd

where p = q =

2d d−1

and C depends only on μ and Ω.

Proof. An inspection of the proof of Lemma 2.2.3 shows that ˆ A(x/ε)∇wε · ∇ψ dx Ω

 C ∇ψLp (Ω) ε Sε (∇2 u0 )Lq (Ω\Ω3ε ) + ∇u0 − Sε (∇u0 )Lq (Ω\Ω2ε ) + C ∇ψL2 (Ω4ε ) ∇u0 L2 (Ω5ε ) . √ Note that ∇u0 L2 (Ω5ε )  C ε u0 W 2,q (Ω) and that Sε (∇2 u0 )Lq (Ω\Ω3ε )  C ∇2 u0 Lq (Ω) .  0 ∈ W 2,q (Rd ) such that u  0 = u0 in Ω and Since u0 ∈ W 2,q (Ω), there exists u  u0 W 2,q (Rd )  C u0 W 2,q (Ω) . It follows that ∇u0 − Sε (∇u0 )Lq (Ω\Ω3ε )  ∇ u0 − Sε (∇ u0 )Lq (Rd )  0 Lq (Rd )  Cε ∇2 u

(2.3.2)

 Cε u0 W 2,q (Ω) , where we have used (2.1.19) for the second inequality. Combining this with the first inequality above now proves the Lemma.   ∗  = A.  Let Ω be a bounded Lipschitz domain in Rd . Lemma 2.3.3. Assume that A Let u ∈ H10 (Ω) be a weak solution to the Dirichlet problem: L0 (u) = F in Ω and u = 0 on ∂Ω, where F ∈ C∞ 0 (Ω). Then ∇uL2 (Ωt )  Ct1/2 FLq (Ω) ,

(2.3.4) for 0 < t < diam(Ω), and

∇uLp (Ω)  C FLq (Ω) ,

(2.3.5) where p =

2d d−1 ,

q = p =

2d d+1 ,

and C depends only on μ and Ω.

Proof. Write u = φ + v0 , where v0 is defined by (2.2.21). The proof of (2.3.4) is essentially contained in that of Theorem 2.2.14. We leave it as an exercise for the reader. To see (2.3.5), we recall that by the fractional integral estimates, 2d , ψ a continuous ∇v0 Lp (Ω)  CFLq (Ω) . To estimate ∇φ, we fix p = d−1 ∗ function in Ω with nontangential maximal function (ψ) , and use the inequality ˆ 1/p ˆ 1/2 p ∗2 (2.3.6) |ψ| dx C |(ψ) | dσ . Ω

∂Ω

Zhongwei Shen

95

This gives ∇φLp (Ω)  C (∇φ)∗ L2 (∂Ω)  C ∇v0 L2 (∂Ω)  C v0 W 2,q (Ω)  C FLq (Ω) , using Lemma 2.2.18 for the second inequality and (2.1.25) for the third. Finally, to prove (2.3.6), we observe that for any x ∈ Ω and xˆ ∈ ∂Ω with |x − xˆ | = dist(x, ∂Ω) = r, |ψ(x)|  (ψ)∗ (y), if y ∈ ∂Ω and |y − xˆ |  cr. It follows that ˆ C |(ψ)∗ | dσ |ψ(x)|  d−1 r B(x,cr)∩∂Ω ˆ (2.3.7) ˆ (ψ)∗ (y) C dσ(y). d−1 ∂Ω |x − y| Let f ∈ C10 (Ω) and

ˆ g(y) = Ω

|f(x)| dx. |x − y|d−1

Note that by (2.3.7), ˆ ˆ ˆ (ψ)∗ (y)|f(x)| ψ(x)f(x) dx  C dσ(y)dx d−1 Ω Ω ∂Ω |x − y| ˆ (ψ)∗ g dσ =C ∂Ω

 C (ψ)∗ L2 (∂Ω) gL2 (∂Ω)  C (ψ)∗ L2 (∂Ω) gW 1,q (Ω)  C (ψ)∗ L2 (∂Ω) fLq (Ω) , 2d where q = d+1 by using (2.1.25) for the third inequality and singular integral estimates for the fourth. The inequality (2.3.6) follows by a duality argument. 

The next theorem gives a sharp O(ε) convergence rate in Lp with p =

2d d−1 .

Theorem 2.3.8. Assume that A is 1-periodic and satisfies (1.1.2). Also assume A∗ = A. Let Ω be a bounded Lipschitz domain in Rd . Let uε (ε  0) be the weak solution of 2d . Then (2.0.1). Assume that u0 ∈ W 2,q (Ω) for q = d+1 uε − u0 Lp (Ω)  Cε u0 W 2,q (Ω) ,

(2.3.9) where p = q =

2d d−1

and C depends only on μ and Ω.

1 Proof. Let wε be given by (2.2.2). For any G ∈ C∞ 0 (Ω), let vε ∈ H0 (Ω) (ε  0) be the weak solution to the Dirichlet problem,

(2.3.10)

L∗ε (vε ) = G

in Ω

and

vε = 0

on ∂Ω.

Define rε = vε − v0 − εχ(x/ε) ηε S2ε (∇v0 ),

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

ε is a cut-off function in C∞ ε  1, |∇ where η ηε |  C/ε, 0 (Ω) such that: 0  η  ηε (x) = 0 if dist(x, ∂Ω)ε  4ε, and  ηε (x) = 1 if dist(x, ∂Ω)  5ε. Observe that ˆ wε · G dx Ω ˆ = A(x/ε)∇wε · ∇vε dx Ω ˆ ˆ (2.3.11)  A(x/ε)∇wε · ∇rε dx + A(x/ε)∇wε · ∇v0 dx Ω Ω ˆ   + A(x/ε)∇wε · ∇ εχ(x/ε) ηε S2ε (∇v0 ) dx Ω

= I1 + I2 + I3 . To estimate I1 , we note that by (2.2.27) and (2.2.15), √ √ ∇wε L2 (Ω)  C ε u0 W 2,q (Ω) and ∇rε L2 (Ω)  C ε GLq (Ω) , where we have used the assumption A∗ = A. By the Cauchy inequality this gives I1  Cε u0 W 2,q (Ω) GLq (Ω) .

(2.3.12)

Next, to bound I2 , we use Lemma 2.3.1 to obtain

√ I2  C u0 W 2,q (Ω) ε ∇v0 Lp (Ω) + ε ∇v0 L2 (Ω4ε ) (2.3.13)  Cε u0 W 2,q (Ω) GLq (Ω) , where we use Lemma 2.3.3 for the last step. To estimate I3 , we let ηε S2ε (∇v0 ). ϕε = εχ(x/ε) Since ∇χ may not be in Lp (Y), the desired estimate of I3 does not follow directly from Lemma 2.3.1. However, the approach used in its proof does show that I3  Cε u0 W 2,q (Ω) GLq (Ω) ,

(2.3.14)

Indeed, in view of (2.2.6)-(2.2.7). the key is to bound ˆ ηε |∇u0 − S2ε (∇u0 )| |∇χ(x/ε)| |S2ε (∇v0 )| dx ηε  (2.3.15) I31 = Ω

and

ˆ

(2.3.16)

I32 = ε

Ω

ηε (|φ(x/ε)| + |χ(x/ε)|)|∇χ(x/ε)| |S2ε (∇2 u0 )| |S2ε (∇v0 )| dx, ηε 

where φ = (φkij ) is the flux corrector. The other terms may be handled in the same manner as before. To handle I31 , we write 1

|∇χ(x/ε)| = |∇χ(x/ε)| d |∇χ(x/ε)|

d−1 d

and use Hölder’s inequality. This yields  1  d−1 I31   ηε |∇χ(x/ε)| d ∇u0 − S2ε (∇u0 ) Lq (Ω) ηε |∇χ(x/ε)| d S2ε (∇v0 )Lp (Ω)  Cε u0 W 2,q (Ω) ∇v0 Lp (Ω)  Cε u0 W 2,q (Ω) GLq (Ω) ,

Zhongwei Shen

97

where we have used (2.1.22) and (2.1.17). Note that by Sobolev imbedding, we 2d . Thus, by Hölder’s inequality and (2.1.17), have |φ| + |χ| ∈ Lr (Y) for r = d−2 I32  ε ηε |∇χ(x/ε)|S2ε (∇2 u0 )Lq (Ω)  ηε (|φ(x/ε)| + |χ(x/ε)|)S2ε (∇v0 )Lp (Ω)  Cε u0 W 2,q (Ω) GLq (Ω) . In view of (2.3.11)-(2.3.14) we have proved that ˆ wε · G dx  Cε u0 W 2,q (Ω) GLq (Ω) , (2.3.17) Ω

where C depends only on μ and Ω. By duality this implies that wε Lp (Ω)  Cε u0 W 2,q (Ω) . It follows that uε − u0 Lp (Ω)  wε Lp (Ω) + εχ(x/ε)ηε S2ε (∇u0 )Lp (Ω)  Cε u0 W 2,q (Ω) . 

This completes the proof. 2d 2d and q = d+1 , we have Remark 2.3.18. Note that for p = d−1 result, the error estimate (2.3.9) is scale-invariant.

1 q

−

1 p

=

1 d.

As a

Remark 2.3.19. The estimate (2.3.9) was given in [23]. However, the term I3 in [23] is handled incorrectly. The idea for the corrected proof given here is due to W. Niu and Y. Xu, who also extended the scale-invariant estimate to parabolic systems with time-dependent periodic coefficients [18]. The next theorem gives the O(ε) convergence rate in L2 without the symmetry condition, assuming that Ω is a bounded C1,1 domain. The smoothness assumption on Ω ensures the H2 estimate for L0 . Theorem 2.3.20 ([13, 26]). Assume that A is 1-periodic and satisfies (1.1.2). Let Ω be a bounded C1,1 domain in Rd . Let uε ∈ H1 (Ω) (ε  0) be the weak solution of (2.0.1). Assume that u0 ∈ H2 (Ω). Then uε − u0 L2 (Ω)  Cε u0 H2 (Ω) ,

(2.3.21)

where C depends only on μ and Ω. Proof. The proof is similar to that of Theorem 2.3.8. By Theorem 2.2.8, the estimates √ √ wε H1 (Ω)  C ε u0 H2 (Ω) and rε H1 (Ω)  C ε v0 H2 (Ω) 0

0

hold without the symmetry condition. It follows from the proof of Theorem 2.3.8 that ˆ wε · G dx  Cε u0 H2 (Ω) v0 H2 (Ω) . (2.3.22) Ω

is a solution of L0 (v0 ) = G in Ω. Since Ω is C1,1 and Recall that v0 ∈ L0 is a second-order elliptic operator with constant coefficients, it is known that v0 ∈ H2 (Ω) and v0 H2 (Ω)  C GL2 (Ω) (see [8, Chapter 8] or [17, Chapter 4]). H10 (Ω)

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

This, together with (2.3.22), gives ˆ wε · G dx  Cε u0 H2 (Ω) GL2 (Ω) . Ω

By duality we obtain wε L2 (Ω)  Cε u0 H2 (Ω) . Thus, uε − u0 L2 (Ω)  Cε u0 H2 (Ω) + εχ(x/ε)ηε S2ε (∇u0 )L2 (Ω)  Cε u0 H2 (Ω) , which completes the proof.



Analogous results hold for Neumann problems. The proof is almost identical to the case of Dirichlet problems. Theorem 2.3.23. Suppose that A and Ω satisfy the same conditions as in Theorem 2.3.8. Let uε ∈ H1 (Ω) (ε  0) be the weak solution of the Neumann problem (2.0.2). Then, if u0 ∈ W 2,q (Ω), uε − u0 Lp (Ω)  Cε u0 W 2,q (Ω) ,

(2.3.24) where q = p =

2d d+1

and C depends only on μ and Ω.

Theorem 2.3.25. [27] Assume that A is 1-periodic and satisfies (1.1.2). Let Ω be a bounded C1,1 domain in Rd . Let uε ∈ H1 (Ω) (ε  0) be the weak solution of (2.0.2). Assume that u0 ∈ H2 (Ω). Then, for a C depending only on μ and Ω, we have uε − u0 L2 (Ω)  Cε u0 H2 (Ω) .

(2.3.26)

2.4. Problems for Section 2 Problem 2.4.1. Prove the estimate (2.1.7). Problem 2.4.2. Show that if g is 1-periodic, (2.4.3)

g(x/ε)∇Sε (f)Lp (O)  Cε−1 gLp (Y) fLp (O(ε/2)) .

Problem 2.4.4. Let uε be the solution of the Neumann problem (2.0.2) with ´ Ω uε dx = 0, and u0 the homogenized solution. Let wε be defined as in (2.2.2). Show that the inequality (2.2.4) holds for any ψ ∈ H1 (Ω). Problem 2.4.5. Let wε be the same as in Problem 2.4.4. Show that

(2.4.6) wε H1 (Ω)  C ε ∇2 u0 L2 (Ω\Ωε ) + ε ∇u0 L2 (Ω) + ∇u0 L2 (Ω5ε ) for 0 < ε < 1. Consequently, if u0 ∈ H2 (Ω), √ (2.4.7) wε H1 (Ω)  C ε u0 H2 (Ω) . The constant C depends only on μ and Ω. Problem 2.4.8. Suppose that A is 1-periodic and satisfies (1.1.2). Also assume A∗ = A. Let Ω be a bounded Lipschitz domain in Rd . Let wε be the same as in Problem 2.4.4. Show that

√ (2.4.9) wε H1 (Ω)  C ε FLq (Ω) + gL2 (∂Ω) , where q =

2d d+1

and C depends only on μ and Ω.

Zhongwei Shen

99

3. Uniform Lipschitz Estimates—Part I In this section we prove uniform Lipschitz estimates, using a compactness method of Avellaneda and Lin [4]. A very important feature of the family of operators {Lε , ε > 0} is the following rescaling property which plays an essential role in the compactness method as well as in numerous other rescaling arguments:  Claim 3.0.1. If Lε (uε = F, v(x) = uε (rx) and G(x) = r2 F(rx), then L εr (v) = G. The following property of translation is also useful:  Claim 3.0.2. If div A(x/ε)∇uε ) = F and vε (x) = uε (x − x0 ), then we have   −1     − div A(x/ε)∇v ε = F where A(y) = A(y − ε x0 ) and F(x) = F(x − x0 ).  is 1-periodic and satisfies the same ellipticity condition as A. It The matrix A also satisfies the same smoothness condition that we will impose on A, uniformly in ε > 0 and x0 ∈ Rd . 3.1. Interior estimates Theorem 3.1.1. Assume that the matrix A satisfies (1.1.2) and is 1-periodic. Suppose that uε ∈ H1 (B) is a weak solution of Lε (uε ) = F in B, where B = B(x0 , R) for some x0 ∈ Rd and R > 0, and F ∈ Lp (B) for some p > d. Suppose that 0 < ε  r  R. Then

1/2 (3.1.2)

B(x0 ,r)

 Cp

|∇uε |2

⎧ ⎨ ⎩

1/2 B(x0 ,R)

|∇uε |2

+R B(x0 ,R)

|F|p

1/p ⎫ ⎬ ⎭

,

where Cp depends only on μ and p. Estimate (3.1.2) should be regarded as a Lipschitz estimate down to the microscopic scale ε. In fact, under the Hölder continuity condition on A, (3.1.3)

|A(x) − A(y)|  λ|x − y|τ

for any x, y ∈ Rd , where λ > 0 and τ ∈ (0, 1], the full-scale Lipschitz estimate follows from Theorem 3.1.1 by a blow-up argument. Theorem 3.1.4 ([4]). Suppose that A satisfies (1.1.2) and is 1-periodic. Also assume A satisfies the smoothness condition (3.1.3). Let uε ∈ H1 (B) be a weak solution to Lε (uε ) = F in B for some ball B = B(x0 , R), where F ∈ Lp (B) for some p > d. Then

  1/2 1/p  2 p , (3.1.5) |∇uε (x0 )|  Cp |∇uε | +R |F| B

B

where Cp depends only on p, μ and (λ, τ). Proof. We give the proof of Theorem 3.1.4, assuming Theorem 3.1.1. By translation and dilation we may assume that x0 = 0 and R = 1.

100

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

The ellipticity condition (1.1.2), together with the Hölder continuity assumption (3.1.3), allows us to use the following local regularity result: if − div(A(x)∇u) = F

in B(0, 1),

where F ∈ Lp (B(0, 1)) for some p > d, then ⎧

1/2 ⎨ 2 (3.1.6) |∇u(0)|  Cp |∇u| + ⎩ B(0,1)

|F|p B(0,1)

1/p ⎫ ⎬ ⎭

,

where Cp depends only on μ, p, λ and τ (see e.g. [12]). As the case ε  (1/2) follows directly from (3.1.6) and the observation that the coefficient matrix A(x/ε) is uniformly Hölder continuous for ε  (1/2), we may assume 0 < ε < (1/2), . To handle the case 0 < ε < (1/2), we use a blow-up argument and the estimate (3.1.2). Let w(x) = ε−1 uε (εx). Since L1 (w) = εF(εx)

in B(0, 1),

it follows again from (3.1.6) that ⎧

1/2

1/p ⎫ ⎨ ⎬ |∇w(0)|  C |∇w|2 + |εF(εx)|p dx ⎩ B(0,1) ⎭ B(0,1) ⎧ ⎫

1/2

1/p ⎨ ⎬ 1− d 2 p C |∇uε | +ε p |F| , ⎩ B(0,ε) ⎭ B(0,1) where C depends only on d, p, μ, λ and τ. This, together with (3.1.2) with r = ε  and the fact that ∇w(0) = ∇uε (0), gives the estimate (3.1.5). Remark 3.1.7. Let uε = xj + εχj (x/ε). Then Lε (uε ) = 0 in Rd . Since ∇uε = ∇xj + ∇χj (x/ε), no uniform estimate for uε can be expected beyond Lipschitz (unless χ = 0). In the rest of this subsection, we will assume that A satisfies (1.1.2) and that it is 1-periodic. No smoothness condition on A is needed. The proof uses only the interior C1,α estimate for elliptic systems with constant coefficients. Lemma 3.1.8 (One-step improvement). Let 0 < σ < ρ < 1 and ρ = 1 − d p . There exist constants ε0 ∈ (0, 1/2) and θ ∈ (0, 1/4), depending only on μ, σ and ρ, such that

1/2   ∂uε 2 uε − xj + εχj (x/ε) uε (x) − dx B(0,θ) B(0,θ) B(0,θ) ∂xj ⎧ (3.1.9)

1/2

1/p ⎫ ⎨ ⎬  θ1+σ max |uε |2 , |F|p , ⎩ B(0,1) ⎭ B(0,1) whenever 0 < ε < ε0 , and uε ∈ H1 (B(0, 1)) is a weak solution of (3.1.10)

Lε (uε ) = F

in B(0, 1).

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101

Proof. Estimate (3.1.9) is proved by contradiction, using Theorem 1.2.5 and the following observation: for any θ ∈ (0, 1/4), ∂u sup u(x) − u − xj B(0,θ) B(0,θ) ∂xj |x|θ  C θ1+ρ ∇uC0,ρ (B(0,θ)) (3.1.11)

 C θ1+ρ ∇uC0,ρ (B(0,1/4)) ⎧

1/2 ⎨  C0 θ1+ρ |u|2 + ⎩ B(0,1/2)

|F|p B(0,1/2)

1/p ⎫ ⎬ ⎭

,

where u is a solution of − div(A0 ∇u) = F in B(0, 1/2) and A0 is a constant matrix satisfying the ellipticity condition. We note that the last inequality in (3.1.11) is a standard C1,ρ estimate for second-order elliptic systems with constant coefficients, and that the constant C0 depends only on d, μ and ρ [12]. Since σ < ρ, we may choose θ ∈ (0, 1/4) so small that 2d+1 C0 θ1+ρ < θ1+σ . We claim that the estimate (3.1.9) holds for this θ and some ε0 ∈ (0, 1/2), which depends only on μ, σ and ρ. Suppose this is not the case. Then there exist sequences {ε } ⊂ (0, 1/2), {A } satisfying (1.1.2) and (1.1.3), {F } ⊂ Lp (B(0, 1)), and {u } ⊂ H1 (B(0, 1)), such that ε → 0,   − div A (x/ε )∇u = F in B(0, 1),

1/2 2

|u |

(3.1.12) B(0,1)

and



(3.1.13) B(0,θ)

u (x) −

1/p

 1,

|F |

p

 1,

B(0,1)

  u − xj + ε χ,j (x/ε )

B(0,θ)

1/2 ∂u 2 > θ1+σ , dx ∂xj

B(0,θ)

where χ,j denote the correctors associated with the 1-periodic matrix A . Combining (3.1.12) and Caccioppoli’s inequality, we see that the sequence {u } is bounded in H1 (B(0, 1/2)). By passing to subsequences, we may assume that ⎧ ⎪ u  u weakly in L2 (B(0, 1)), ⎪ ⎪ ⎪ ⎪ ⎨u  u weakly in H1 (B(0, 1/2)), ⎪ F  F weakly in Lp (B(0, 1)), ⎪ ⎪ ⎪ ⎪ ⎩ A → A0 ,  denotes the homogenized matrix for A . Since p > d, F  F weakly in where A Lp (B(0, 1)) implies that F → F strongly in H−1 (B(0, 1)). It then follows by Theo  rem 1.2.5 that u ∈ H1 (B(0, 1/2)) is a solution of − div A0 ∇u = F in B(0, 1/2).

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

We now let  → ∞ in (3.1.12) and (3.1.13). This leads to

1/2

1/p |u|2

(3.1.14)

 1,

|F|p

B(0,1)

and



(3.1.15) B(0,θ)

 1,

B(0,1)

u(x) −

1/2 ∂u 2  θ1+σ , dx ∂xj

u − xj B(0,θ)

B(0,θ)

where we have used the fact that u → u strongly in L2 (B(0, 1/2)). Here we also have used the fact that the sequence {χ,j } is bounded in L2 (Y). Finally, we note that by (3.1.15), (3.1.11) and (3.1.14), ⎧

1/2

1/p ⎫ ⎨ ⎬ θ1+σ  C0 θ1+ρ |u|2 + |F|p < 2d+1 C0 θ1+ρ , ⎩ B(0,1/2) ⎭ B(0,1/2) which is in contradiction with the choice of θ. This completes the proof of  Lemma 3.1.8. Remark 3.1.16. Since inf

α∈R E

for any f ∈

L2 (E),

f −

f − α 2 = E

ffl

E

2 f

we may replace B(0,θ) uε in (3.1.9) by the average     ∂uε uε − xj + ε χj (x/ε) dx. B(0,θ) B(0,θ) ∂xj

The observation will be used in the proof of the next lemma. Lemma 3.1.17 (Iteration). Let 0 < σ < ρ < 1 and ρ = 1 − d p . Let (ε0 , θ) be the constants given by Lemma 3.1.8. Suppose that 0 < ε < θk−1 ε0 for some k  1 and uε is   a solution of Lε (uε ) = F in B(0, 1). Then there exist constants E(ε, ) = Ej (ε, ) ∈ Rd for 1    k, such that if   vε = uε − xj + εχj (x/ε) Ej (ε, ), then (3.1.18)

vε −

B(0,θ )

2 1/2

(1+σ) vε θ max

B(0,θ )

Moreover, the constants E(ε, ) satisfy

1/2 |uε | , 2

B(0,1)

|E(ε, )|  C max

1/2 |uε | ,

(3.1.19)

|E(ε,  + 1) − E(ε, )|  C θσ max

B(0,1)

where C depends only on d, μ, σ and ρ.

1/p

 .

B(0,1)

2

B(0,1)

|F|

p

|F|

p

1/p

 ,

B(0,1)

1/2 |uε |2 , B(0,1)

|F|p

1/p

 ,

Zhongwei Shen

103

Proof. We prove (3.1.18)-(3.1.19) by an induction argument on . The case  = 1 follows from Lemma 3.1.8 and Remark 3.1.16, with ∂uε Ej (ε, 1) = B(0,θ) ∂xj (set E(ε, 0) = 0). If now we have the desired constants E(ε, ) for all integers up to some , where 1    k − 1, then to construct E(ε,  + 1), consider the function

w(x) = uε (θ x) − θ xj + εχj (θ x/ε) Ej (ε, ) − B(0,θ )

    uε − yj + εχj (y/ε) Ej (ε, ) dy.

By the rescaling property (3.0.1) and the equation (1.1.9) for correctors, L

ε θ

(w) = F

in B(0, 1),

where F (x) = θ2 F(θ x). Since εθ−  εθ−k+1  ε0 , it follows from Lemma 3.1.8 and Remark  3.1.16 that

∂w w − xj + εθ− χj (θ x/ε) ∂x j B(0,θ) B(0,θ)   2 1/2

 ∂w −  w − yj + θ εχj (θ y/ε) dy dx − (3.1.20) B(0,θ) B(0,θ) ∂xj ⎧

1/2

1/p ⎫ ⎨ ⎬  θ1+σ max |w|2 , |F |p . ⎩ B(0,1) ⎭ B(0,1) By the induction assumption,

1/2 (3.1.21) |w|2  θ(1+σ) max B(0,1)

1/2 |uε |2 , B(0,1)

Also, since 0 < ρ = 1 −

d p,

1/p

|F |

p

(3.1.22)

θ

1/p |F|

(1+ρ)

B(0,1)

|F|p B(0,1)

p

.

B(0,1)

Hence, the right side of (3.1.20) is bounded by ⎧

1/2 ⎨ (+1)(1+σ) 2 θ max |u | , ⎩ B(0,1) ε

|F|p B(0,1)

1/p ⎫ ⎬ ⎭

.

Finally, note that the left side of (3.1.20) may be written as   uε − xj + εχj (x/ε) Ej (ε,  + 1) B(0,θ+1 )



− B(0,θ+1 )

1/2    2 uε − yj + εχj (y/ε) Ej (ε,  + 1) dy dx

with Ej (ε,  + 1) = Ej (ε, ) + θ− B(0,θ)

∂w . ∂xj

1/p

 .

104

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

By Caccioppoli’s inequality,



1/2 2

|∇w|

|E(ε,  + 1) − E(ε, )|  θ

−

⎧ ⎨

 Cθ− max

 Cθ

σ

B(0,θ)

⎩ ⎧ ⎨

max

1/2 |w|2

|F |p

,

B(0,1)

B(0,1)

1/2 2

|uε |

⎩

|F|p

,

B(0,1)

1/2 ⎫ ⎬ ⎭

1/p ⎫ ⎬ ⎭

B(0,1)

,

where we have used estimates (3.1.21) and (3.1.22) for the last inequality. Thus we have established the second inequality in (3.1.19), from which the first follows by summation. This completes the induction argument and thus the proof.  Proof of Theorem 3.1.1. By translation and dilation we may assume that x0 = 0 and R = 1. We may also assume that 0 < ε < ε0 θ, where ε0 , θ are constants given by Lemma 3.1.8. The case ε0 θ  ε < 1 is trivial. Now suppose that 0 < ε < ε0 θ. Choose k  2 so that ε0 θk  ε < ε0 θk−1 . It follows by Lemma 3.1.17 that

2 1/2 uε uε − B(0,θk−1 )

 C θk

B(0,θk−1 )

⎧ ⎨ ⎩

1/2 |uε |2

|F|p

+

B(0,1)

1/p ⎫ ⎬ ⎭

B(0,1)

.

This, together with Caccioppoli’s inequality, gives

1/2

1/2 |∇uε |2

C

B(0,ε)

B(0,ε0 θk−1 )

⎧ ⎨  C θ−k uε − ⎩ k−1 B(0,θ ) ⎧

1/2 ⎨ C |u |2 + ⎩ B(0,1) ε

B(0,θk−1 )

|∇uε |2 2 uε

|F|p

1/2

+ θk B(0,θk−1 )

1/p ⎫ ⎬ ⎭

B(0,1)

|F|2

1/2 ⎫ ⎬ ⎭

.

ffl By replacing uε with uε − B(0,1) uε , we obtain

1/2 |∇uε |2 B(0,ε)

C

⎧ ⎨ ⎩

B(0,1)

uε −

B(0,1)

2 uε

1/2

|F|p

+

1/p ⎫ ⎬

B(0,1)

from which the estimate (3.1.2) follows by the Poincaré inequality.

⎭

, 

Zhongwei Shen

105

3.2. Boundary Lipschitz estimates for the Dirichlet problem Theorem 3.2.1 ([4]). Suppose that A is 1-periodic and satisfies (1.1.2) and (3.1.3). Let Ω be a bounded C1,α domain in Rd . Suppose that

in B(x0 , R) ∩ Ω, Lε (uε ) = F (3.2.2) uε = f on B(x0 , R) ∩ ∂Ω, where x0 ∈ ∂Ω and 0 < R < diam(Ω). Then

(3.2.3)

∇uε L∞ (B(x0 ,R/2)∩Ω) ⎧

1/2 ⎨ 2 C |∇uε | +R ⎩ B(x0 ,R)∩Ω

1/2 |F|

p

B(x0 ,R)∩Ω



+ fL∞ (B(x0 ,R)∩∂Ω) + R ∇tan fC0,σ (B(x0 ,R)∩∂Ω) , σ

where p > d, σ > 0, and C depends only on μ, p, σ, λ, τ, and Ω. Theorem 3.2.1, together with the interior Lipschitz estimate, gives. Theorem 3.2.4 ([4]). Assume that A and Ω satisfy the same conditions as in Theorem 3.2.1. Let uε be the solution of the Dirichlet problem, (3.2.5) Then

Lε (uε ) = F

in Ω

and

uε = f

on ∂Ω.



∇uε L∞ (Ω)  C FLp (Ω) + fC1,σ (∂Ω) ,

where p > d, σ > 0, and C depends only on μ, p, σ, λ, τ, and Ω. Outline of the proof of Theorem 3.2.1. Step 1: Use the compactness method to establish the boundary Hölder estimate in a C1 domain:

1/2 C 2 |uε | , uε C0,σ (B(x0 ,R/2)∩Ω)  σ R B(x0 ,R)∩Ω where σ ∈ (0, 1), x0 ∈ ∂Ω, 0 < R < diam(Ω), Lε (uε ) = 0 in B(x0 , R) ∩ Ω and uε = 0 on B(x0 , R) ∩ ∂Ω. Step 2: Let Gε (x, y) denote the Green function for Lε in a bounded C1 domain Ω. Use the Hölder estimate in Step 1 to prove that for any 0 < σ1 , σ2 < 1, C[δ(x)]σ1 [δ(y)]σ2 |Gε (x, y)|  |x − y|d−2+σ1 +σ2 for any x, y ∈ Ω, where δ(x) = dist(x, ∂Ω). Step 3: Use the estimate for the Green function in Step 2 to see that if Lε (uε ) = 0 in Ω and uε = g on Ω, then for 0 < ε < R < diam(Ω),

1/2

2 |∇uε |  C ∇g∞ + ε−1 gL∞ . B(x0 ,R)∩Ω

106

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Step 4: Let Φε (x) denote the solution of the Dirichlet problem,   (3.2.6) Lε Φε = 0 in Ω and Φε = x on ∂Ω. The vector-valued function Φε (x) is called the Dirichlet corrector for Lε in Ω.   Note that if vε = Φε − x − εχ(x/ε), then Lε vε = 0 in Ω and vε = −εχ(x/ε) on ∂Ω. Use the estimate in Step 3 to show that if Ω is C1,α , ∇Φε L∞ (Ω)  C,

(3.2.7)

where C depends only on μ, λ, τ and Ω. Step 5: Use the compactness method, together with the Lipschitz estimate for the Dirichlet corrector in Step 4, to prove Theorem 3.2.1. A detailed proof may be found in [4, 23]. 3.3. Problems for Section 3 Problem 3.3.1. Let Ω be a bounded Lipschitz domain. Let uε be a weak solution of Lε (uε ) = F + div(f)

in B(x0 , r) ∩ Ω

with uε = g on B(x0 , r) ∩ ∂Ω, where x0 ∈ ∂Ω and 0 < r < r0 . Show that ˆ ˆ ˆ C |∇uε |2 dx  2 |uε |2 dx + Cr2 |F|2 dx r B(x0 ,r)∩Ω B(x0 ,r/2)∩Ω B(x0 ,r)∩Ω ˆ ˆ 2 + C |f| dx + C |∇G|2 dx (3.3.2) B(x0 ,r)∩Ω B(x0 ,r0 )∩Ω ˆ C + 2 |G|2 dx, r B(x0 ,r)∩Ω where G ∈ H1 (B(x0 , r) ∩ Ω) and G = g on B(x0 , r) ∩ ∂Ω. Problem 3.3.3. Prove Theorem 3.2.4, using Theorem 3.2.1 and interior estimates. Problem 3.3.4. Use the compactness method to prove the boundary Hölder estimate.

4. Uniform Lipschitz estimates—Part II In this section we prove uniform Lipschitz estimates for the Neumann problem, ⎧ ⎪ in Ω, ⎨Lε (uε ) = F (4.0.1) ∂uε ⎪ =g on ∂Ω, ⎩ ∂νε   ε where Lε = − div A(x/ε)∇ and ∂u ∂νε denotes the conormal derivative of uε , defined by ∂uε ∂uε = ni aij (x/ε) . (4.0.2) ∂νε ∂xj

Zhongwei Shen

107

Our approach is based on a general scheme for establishing large-scale regularity estimates in homogenization, developed by S. N. Armstrong and C. Smart [3] in the study of stochastic homogenization. Roughly speaking, the scheme states that if a function uε is well approximated by C1,α functions at every scale greater than ε, then uε is Lipschitz continuous at every scale greater than ε. The approach relies on a very weak result on convergence rates and does not involve correctors in a direct manner. In comparison with the compactness method used in Section 3, when applied to boundary Lipschitz estimates, it does not require a-priori Lipschitz estimate for boundary correctors. 4.1. Approximation of solutions at the large scale (4.1.1)

Let

Dr = D(r, ψ)  = (x , ψ(x )) : |x | < r and ψ(x ) < xd < ψ(x ) + 10(M0 + 1)r ,  Δr = Δ(r, ψ) = (x , ψ(x )) : |x | < r ,

where ψ is a Lipschitz function in Rd−1 for which we have both ψ(0) = 0 and ∇ψ∞  M0 . Theorem 4.1.2. Suppose that A is 1-periodic and satisfies (1.1.2). Let uε ∈ H1 (D2r ) be a weak solution to ∂uε = g on Δ2r , Lε (uε ) = F in D2r and ∂νε where F ∈ L2 (D2r ) and g ∈ L2 (Δ2r ). Assume that r  ε. Then there exists w ∈ H1 (Dr ) such that ∂w = g on Δr , L0 (w) = F in Dr and ∂ν0 and 1/2  2 |uε − w| Dr

(4.1.3)  1/2  1/2 1/2 

ε α  2 2 2 2 , C |uε | +r |F| +r |g| r D2r D2r Δ2r where C > 0 and α ∈ (0, 1/2) depend only on μ and M0 . Lemma 4.1.4. Let Ω = Dr for some 1  r  2. Let uε ∈ H1 (Ω) (ε  0) be a ε weak solution to the Neumann problem, Lε (uε ) = F in Ω and ∂u ∂νε = g on ∂Ω, where ´ ´ F ∈ L2 (Ω), g ∈ L2 (∂Ω), and Ω F dx + ∂Ω g dσ = 0. Then there exists p > 2, depending only on μ and M0 , such that

(4.1.5) ∇uε Lp (Ω)  C FL2 (Ω) + gL2 (∂Ω) , where C > 0 depend only on μ and M0 . Proof. Consider the Neumann problem: Lε (vε ) = div(f)

in Ω

and

∂vε =0 ∂νε

on ∂Ω,

108

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

where f ∈ C10 (Ω; Rd ). Clearly, ∇vε L2 (Ω)  CfL2 (Ω) . Using the reverse Hölder inequality and the real-variable argument in Section 5, one may deduce that ∇vε Lp (Ω)  C fLp (Ω) ,

(4.1.6)

for any 2 < p < p. ¯ Since this is also true for the adjoint operator L∗ε , by a duality argument, it follows that

(4.1.7) ∇uε Lp (Ω)  C FLq (Ω) + gB−1/p,p (∂Ω) , for 2 < p < p, ¯ where q1 = p1 + d1 and B−1/p,p (∂Ω) denotes the dual of the Besov  1/p,p (∂Ω). By choosing p ∈ (2, p) ¯ close to 2 so that L2 (Ω) ⊂ Lq (Ω) and space B  L2 (∂Ω) ⊂ B−1/p,p (∂Ω), we see that the estimate (4.1.5) follows from (4.1.7). Lemma 4.1.8. Let uε ∈ H1 (Ω) (ε  0) be the weak solution to the Neumann problem ´ (4.0.1) with Ω uε dx = 0, where Ω = Dr for some 1  r  2. Then

(4.1.9) uε − u0 L2 (Ω)  Cεσ FL2 (Ω) + gL2 (∂Ω) for any 0 < ε < 2, where σ > 0 and C > 0 depend only on μ and M0 . Proof. It follows from (2.4.6) that uε − u0 L2 (Ω)

 C ε ∇2 u0 L2 (Ω\Ωε ) + ε ∇u0 L2 (Ω) + ∇u0 L2 (Ω5ε ) ,

(4.1.10)

where Ωt = {x ∈ Ω : dist(x, ∂Ω) < t}. To bound the right side of (4.1.10), we first use interior estimates for L0 to obtain C |∇2 u0 |2  |∇u0 |2 + C |F|2 2 [δ(x)] B(x,δ(x)/8) B(x,δ(x)/4) B(x,δ(x)/4) for any x ∈ Ω, where δ(x) = dist(x, ∂Ω). We then integrate both sides of the inequality above over the set Ω \ Ωε . Observe that if |x − y| < δ(x)/4, then |δ(x) − δ(y)|  |x − y| < δ(x)/4, which gives (4/5)δ(y) < δ(x) < (4/3)δ(y). It follows ˆ that Ω\Ωε

|∇2 u0 (y)|2 dy

ˆ

ˆ

C

−2

[δ(y)] Ω\Ωε/2

C Ω

 Cε

|∇u0 |

−1− s1

|∇u0 (y)| dy + C

1/s ˆ

ˆ 2s

2

|F(y)|2 dy Ω −2s

dy

[δ(y)] Ω\Ωε/2

ˆ

∇u0 2L2s (Ω)

|F(y)|2 dy,

+ +C Ω

1/s dy

ˆ |F(y)|2 dy

+C Ω

Zhongwei Shen

109

where s > 1 and we have used Hölder’s inequality for the second step. Let p = 2s > 2. We see that 1

1

ε ∇2 u0 L2 (Ω\Ωε )  Cε 2 − p ∇u0 Lp (Ω) + Cε FL2 (Ω)

1 1  Cε 2 − p FL2 (Ω) + gL2 (∂Ω) ,

(4.1.11)

where we have used (4.1.5) for the last inequality. Also, by Hölder’s inequality, 1

(4.1.12)

1

∇u0 L2 (Ω5ε )  Cε 2 − p ∇u0 Lp (Ω)

1 1  Cε 2 − p FL2 (Ω) + gL2 (∂Ω) .

The inequality (4.1.9) with 1 1 − >0 2 p follows from (4.1.10), (4.1.11) and (4.1.12). σ=



Proof of Theorem 4.1.2. By rescaling we may assume that r = 1. It follows from the Caccioppoli inequality (4.3.2) and the co-area formula that there exists some t ∈ (1, 3/2) such that ˆ ˆ ˆ ˆ (4.1.13) |∇uε |2 dσ  C |uε |2 dx + C |F|2 dx + C |g|2 dσ. ∂Dt \Δ2

D2

D2

Δ2

For otherwise, we could integrate the reverse inequality, ˆ ˆ ˆ ˆ |∇uε |2 dσ > C |uε |2 dx + C |F|2 dx + C ∂Dt \Δ2

D2

D2

in t over the interval (1, 3/2) to obtain ˆ ˆ ˆ |∇uε |2 dx > C |uε |2 dx + C D3/2 \D1

D2

D2

|g|2 dσ,

Δ2

ˆ |F|2 dx + C

|g|2 dσ, Δ2

which is in contradiction with (4.3.2). We now let w be the unique solution of the Neumann problem: ∂w ∂uε = on ∂Dt , L0 (w) = F in Dt and ∂ν0 ∂νε ´ ´ ∂w with Dt w dx = Dt uε dx. Note that ∂ν = g on Δ1 , and that by Lemma 4.1.8, 0 uε − wL2 (D1 )  uε − wL2 (Dt )

 Cεσ FL2 (D2 ) + gL2 (Δ2 ) + ∇uε L2 (∂Dt \Δ2 ) , which, together with (4.1.13), yields (4.1.3).



4.2. Boundary Lipschitz estimates The goal of this subsection is to establish uniform boundary Lipschitz estimates in C1,η domains for solutions with Neumann conditions. Throughout the subsection we assume that Dr = D(r, ψ), Δr = Δ(r, ψ), and ψ : Rd−1 → R is a C1,η function satisfying

 ψ(0) = 0, supp(ψ) ⊂ x ∈ Rd−1 : |x | < 3 , (4.2.1) ∇ψ∞  M0 and ∇ψC0,η (Rd−1 )  M0 .

110

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Theorem 4.2.2 (large-scale Lipschitz estimate). Suppose that A is 1-periodic and satisfies (1.1.2). Let uε ∈ H1 (D2 ) be a weak solution of Lε (uε ) = F in D2 with ∂uε p ρ ∂νε = g on Δ2 , where F ∈ L (D2 ), g ∈ C (Δ2 ) for some p > d and ρ ∈ (0, η). Then, for ε  r  1, 1/2  2 |∇uε |  Dr 

(4.2.3)   |∇uε |2

C

1/2

D2

+ FLp (D2 ) + gL∞ (Δ2 ) + gC0,ρ (Δ2 )

,

where C depends only on μ, p, ρ, and (M0 , η) in (4.2.1). No smoothness condition on A is needed for the large-scale estimate (4.2.3). Under the additional Hölder continuity condition (3.1.3), we may deduce the fullscale Lipschitz estimate from Theorem 4.2.2. Theorem 4.2.4 ([2, 15]). Suppose A satisfies (1.1.2), (1.1.3) and (3.1.3). Let Ω be a bounded C1,η domain. Let uε ∈ H1 (B(x0 , r) ∩ Ω) be a weak solution of Lε (uε ) = F in ε B(x0 , r) ∩ Ω with ∂u ∂νε = g on B(x0 , r) ∩ ∂Ω, for some x0 ∈ ∂Ω and 0 < r < r0 . Then ∇uε L∞ (B(x0 ,r/2)∩Ω)

1/2

(4.2.5)

C

2

B(x0 ,r)∩Ω

|∇uε |

1/p

|F|

p

+r B(x0 ,r)∩Ω



+ gL∞ (B(x0 ,r)∩∂Ω) + r gC0,ρ (B(x0 ,r)∩∂Ω) , ρ

where ρ ∈ (0, η), p > d, and C depends only on ρ, p, μ, (λ, τ) in (3.1.3), and Ω. Proof. We give the proof of Theorem 4.2.4, assuming Theorem 4.2.2. By a change of the coordinate system it suffices to prove that if p > d and ρ ∈ (0, η),

  1/2 1/p |∇uε |2 +r |F|p ∇uε L∞ (Dr )  C (4.2.6)

D2r

D2r



+ gL∞ (Δ2r ) + rρ gC0,ρ (Δ2r ) , ε for 0 < r  1, where Lε (uε ) = F in D2r , ∂u ∂νε = g on Δ2r , and C depends only on μ, p, ρ and (M0 , η) in (4.2.1). By rescaling we may assume r = 1. Note that if ε  1, the matrix A(x/ε) is uniformly Hölder continuous in ε. Consequently, the case ε  1 follows from the standard boundary Lipschitz estimates in C1,η domains for elliptic systems with Hölder continuous coefficients. We thus assume that r = 1 and 0 < ε < 1. Let w(x) = ε−1 uε (εx). Then ∂w  2 and 2,  on Δ L1 (w) =  F in D =g ∂ν1

(x) = g(εx), and where  F(x) = εF(εx), g   r = Δ(r, ψ),   r = D(r, ψ), Δ D

  ) = ε−1 ψ(εx ). ψ(x

Zhongwei Shen

111

 satisfies the condition (4.2.1) with the same Since 0 < ε < 1, the function ψ (M0 , η). It follows from the boundary Lipschitz estimates for the operator L1 that

 ∇wL∞ (D  C ∇w +  F +  g  +  g   )  )  )  )  ) . L2 (D Lp (D L∞ (Δ C0,ρ (Δ 2

1

2

2

2

By a change of variables this leads to ∇uε L∞ (Dε )

  1/2 2 C |∇uε | +ε D2ε

|F|

1/p

p

D2ε



+ gL∞ (Δ2ε ) + ε gC0,ρ (Δ2ε ) ρ

(4.2.7)

 2

C

|∇uε |



D2ε 2

C

|∇uε |

1/2

1/2

D2

 + FLp (D2 ) + gL∞ (Δ2 ) + gC0,ρ (Δ2 ) 

+ FLp (D2 ) + gL∞ (Δ2 ) + gC0,ρ (Δ2 )

,

where we have used the fact that p > d and ε < 1 for the second inequality and (4.2.3) for the last. Using (4.2.7) and translation, we may bound |∇uε (x)| by the right side of (4.2.7) for any x ∈ D1 with dist(x, Δ1 )  c ε. Similarly, by combining interior Lipschitz estimates for L1 with (4.2.3), we may dominate |∇uε (x)| by the right side of (4.2.7)  for any x ∈ D1 with dist(x, Δ1 )  c ε. This finishes the proof. Corollary 4.2.8. Suppose A satisfies (1.1.3), (1.1.2) and (3.1.3). Let Ω be a bounded C1,η domain in Rd for some η ∈ (0, 1). Let uε ∈ H1 (Ω) be a weak solution to the Neumann problem: ∂uε (4.2.9) Lε (uε ) = F in Ω and = g on ∂Ω, ∂νε where F ∈ Lp (Ω), g ∈ Cρ (∂Ω) for some p > d and ρ ∈ (0, η), and ˆ ˆ F dx + g dσ = 0. Ω

Then

∂Ω



∇uε L∞ (Ω)  C FLp (Ω) + gCρ (∂Ω) ,

(4.2.10)

where C depends only on p, ρ, μ, (λ, τ) and Ω. The rest of this subsection is devoted to the proof of Theorem 4.2.2. ∂w Lemma 4.2.11. Suppose that L0 (w) = F in Dr and ∂ν = g on Δr for some 0 < r  1. 0 Let  1/2  1/p 1 2 2 p |w − E · x − q| +t |F| I(t) = inf t E∈Rd Dt Dt q∈R

   ∂        1+ρ  ∂ +t +t w−E·x  ∞ w − E · x  0,ρ  L (Δt ) C (Δt ) ∂ν0 ∂ν0

112

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

 for 0 < t  r, where p > d and 0 < ρ < min η, 1 − d p . Then there exists θ ∈ (0, 1/4), depending only on ρ, p, μ and (η, M0 ), such that I(θr)  (1/2)I(r).

(4.2.12)

Proof. The proof uses the boundary C1,ρ estimate in C1,η domains with Neumann conditions for second-order elliptic systems with constant coefficients. By rescaling we may assume r = 1. By choosing q = w(0) and E = ∇w(0), we see that for any θ ∈ (0, 1/4),  1/p 1− d ρ p p |F| I(θ)  Cθ ∇wC0,σ (Dθ ) + Cθ

 Cθ

ρ

D1

 ∇wC0,σ (D1/4 ) +

|F|

p

1/p  ,

D1

1,ρ where we have used the assumption ρ < 1 − d p . It follows from the boundary C estimate for L0 that

 1/2  1/p |w|2 + |F|p ∇wC0,ρ (D1/4 )  C D1

Hence, for any θ ∈ (0, 1/4),

 1/2  ρ 2 (θ)  Cθ |w| + D1

D1

  ∂w   ∂w      + + .   ∂ν0 L∞ (Δ1 ) ∂ν0 C0,ρ (Δ1 )

|F|

p

D1

1/p

  ∂w   ∂w      , + +   ∂ν0 L∞ (Δ1 ) ∂ν0 C0,ρ (Δ1 )

where C depends only on μ, ρ, p and (η, M0 ) in (4.2.1). Finally, since L0 (w − E · x − q) = F for any E ∈

Rd

in D2

and q ∈ R, the inequality above gives I(θ)  C θρ I(1).

We obtain (4.2.12) by choosing θ ∈ (0, 1/4) so small that Cθρ  (1/2).



ε Lemma 4.2.13. Suppose that Lε (uε ) = F in D2 and ∂u ∂νε = g on Δ2 , where 0 < ε < 1. Let  1/2  1/p 1 2 2 p |uε − E · x − q| +t |F| H(t) = inf t E∈Rd Dt Dt

q∈R

     ∂  ∂     1+σ  E·x  ∞ E · x  0,σ + t g − +t , g − L (Δt ) C (Δt ) ∂ν0 ∂ν0  where 0 < t  1 and 0 < ρ < min η, 1 − d p . Then, for ε < t  1,

 1/2

ε α 1 1 2 inf |uε − q| H(θt)  H(t) + C 2 t t q∈R D2t (4.2.14)   1/p p +t |F| + gL∞ (Δ2t ) , D2t

Zhongwei Shen

113

where θ ∈ (0, 1/4) is given by Lemma 4.2.11, α ∈ (0, 1/2) is given by Theorem 4.1.2, and C depends only on ρ, p, μ, and (η, M0 ). Proof. For each t ∈ (ε, 1], let w = wt be the solution of L0 (w) = F in Dt with ∂w ∂ν0 = g on Δt , given by Theorem 4.1.2. Using 1/2  1/2  1/2  2 2 2 |uε − E · x − q|  |uε − w| + |w − E · x − q| Dθt

for any E ∈ (4.2.15)

Dθt

Rd

Dθt

and q ∈ R, we may deduce that  1/2 1 H(θt)  I(θt) + |uε − w|2 . θt Dθt

Similarly, since  1/2  |w − E · x − q|2  Dt

|uε − w|2



1/2

Dt

1/2 ,

Dt

we obtain I(t)  H(t) +

|uε − E · x − q|2

+

1 t

 |uε − w|2

1/2 .

Dt

This, together with (4.2.15) and the estimate I(θt)  (1/2)I(t) in Lemma 4.2.11, gives  1/2 C 1 2 (4.2.16) H(θt)  H(t) + |uε − w| , 2 t Dt 

which, by Theorem 4.1.2, yields (4.2.14). The proof of the next lemma will be given at the end of this section.

Lemma 4.2.17 ([2, 3, 22]). Let H(r) and h(r) be two nonnegative, continuous functions on the interval (0, 1]. Let 0 < ε < (1/4). Suppose that there exists a constant C0 such that (4.2.18)

max H(t)  C0 H(2r)

rt2r

and

max

rt,s2r

|h(t) − h(s)|  C0 H(2r)

for any r ∈ [ε, 1/2]. We further assume that

1 (4.2.19) H(θr)  H(r) + C0 β(ε/r) H(2r) + h(2r) 2 for any r ∈ [ε, 1/2], where θ ∈ (0, 1/4) and β(t) is a nonnegative, nondecreasing function on [0, 1] such that β(0) = 0 and ˆ 1 β(t) (4.2.20) dt < ∞. t 0 Then   (4.2.21) max H(r) + h(r)  C H(1) + h(1) , εr1

where C depends only on C0 , θ, and the function β(t). We now give the proof of Theorem 4.2.2, using Lemmas 4.2.13 and 4.2.17.

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Proof of Theorem 4.2.2. Let uε be a solution of Lε (uε ) = F in D2 with on Δ2 . We define the function H(t) by (4.2.14). It is not hard to see that H(t)  CH(2r)

(4.2.22)

∂uε ∂νε

=g

if t ∈ [r, 2r]

Next, we define h(t) = |Et |, where Et ∈ Rd such that  1/2  1/p 1 2 2 p H(t) = inf |uε − Et · x − q| +t |F| t q∈R Dt Dt      ∂  ∂     1+ρ  + t g − +t . Et · x  ∞ Et · x  0,ρ g − L (Δt ) C (Δt ) ∂ν0 ∂ν0 Let t, s ∈ [r, 2r]. Using C inf |Et − Es |  r q∈R

 2

1/2

|(Et − Es ) · x − q| Dr

 1/2 C 2 = |(Et − Es ) · x − q1 + q2 | inf r q1 ,q2 ∈R Dr  1/2 C inf  |uε − Et · x − q|2 t q∈R Dt  1/2 C inf + |uε − Es · x − q|2 s q∈R Ds   C H(t) + H(s)  CH(2r),

we obtain (4.2.23)

max

rt,s2r

|h(t) − h(s)|  CH(2r).

Furthermore, by (4.2.14),

ε α 1 H(r) + C Φ(2r) 2 r for r ∈ [ε, 1], where α ∈ (0, 1/2) and

  1/2 1/p  1 2 p inf Φ(t) = |uε − q| +t |F| + gL∞ (Δt ) . t q∈R Dt Dt

(4.2.24)

It is easy to see that

H(θr) 

 Φ(t)  C H(t) + h(t) ,

which, together with (4.2.24), leads to

ε α 1 (4.2.25) H(θr)  H(r) + C H(2r) + h(2r) . 2 r Thus the functions H(r) and h(r) satisfy the conditions (4.2.18), (4.2.19) and (4.2.20). As a result, we obtain that for r ∈ [ε, 1],  1/2  1 2 inf |uε − q|  C H(r) + h(r) q∈R r Dr   C H(1) + h(1) .

Zhongwei Shen

115

By taking E = 0 and q = 0, we see that 1/2   |uε |2 + FLp (D1 ) + gL∞ (Δ1 ) + gC0,ρ (Δ1 ) . H(1)  C D1

Also, note that

 2

h(1)  C inf

q∈R

D1

|E1 · x + q|

1/2

1/2  . |uε |

   C H(1) +

2

D1

Hence we have proved that for ε  r  1, 1/2  1 2 |uε − q| inf q∈R r Dr (4.2.26) 1/2   2 C |uε | + FLp (D2 ) + gL∞ (Δ2 ) + gC0,ρ (Δ2 ) . D2

ffl Replacing uε by uε − D2 uε above and using the Poincaré inequality, we obtain  1/2 1 2 inf |uε − q| q∈R r Dr (4.2.27) 1/2   2 C |∇uε | + FLp (D2 ) + gL∞ (Δ2 ) + gC0,ρ (Δ2 ) . D2



This, together with Caccioppoli’s inequality (4.3.2), gives (4.2.3). We end this subsection with the proof of Lemma 4.2.17. Proof of Lemma 4.2.17. It follows from the second inequality in (4.2.18) that h(r)  h(2r) + C0 H(2r) for any r ∈ [ε, 1/2]. Hence, ˆ 1/2 ˆ 1/2 ˆ 1/2 h(r) h(2r) H(2r) dr  dr + C0 dr r r r a a a ˆ 1 ˆ 1/2 h(r) H(2r) = dr + C0 dr, r r 2a a where a ∈ [ε, 1/4]. This implies that ˆ 2a ˆ 1 ˆ 1/2 h(r) h(r) H(2r) dr  dr + C0 dr r r a 1/2 r a ˆ 1/2

H(2r) dr,  C0 H(1) + h(1) + C0 r a which, together with (4.2.18), leads to  ˆ H(a) + h(a)  C H(2a) + h(1) + H(1) +

1

2a

H(r) dr r



for any a ∈ [ε, 1/4]. The first part of (4.2.18) then shows that, for any a ∈ [ε, 1],   ˆ 1 H(r) (4.2.28) H(a) + h(a)  C H(1) + h(1) + dr . r a

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

To bound the integral in the right side of (4.2.28), we use (4.2.19) and (4.2.28) to obtain ˆ 1

H(t) 1 dt H(θr)  H(r) + Cβ(ε/r) H(1) + h(1) + Cβ(ε/r) 2 t r for r ∈ [ε, 1/2]. It follows that ˆ θ ˆ

H(r) 1 1 H(r) dr  dr + C H(1) + h(1) 2 αε r αθε r 

ˆ (4.2.29) ˆ 1 1 dr H(t) +C dt , β(ε/r) t r αε r where α > 1 and we have used the condition (4.2.20) on β(t) for ˆ 1 ˆ 1 ˆ 1 α β(t) β(t) dr = dt  dt < ∞. β(ε/r) r t t αε 0 ε Note that by Fubini’s Theorem, 

ˆ  ˆ 1 ˆ 1 ˆ t 1 H(t) dr dr H(t) dt = dt β(ε/r) β(ε/r) t r r t αε r αε αε 

ˆ ˆ 1 1/α β(s) dt ds = H(t) s t αε ε/t ˆ 1 ˆ 1/α β(s) H(t) ds dt  s 0 αε t ˆ 1 H(t) 1 dt,  4C αε t if α > 1, which only depends on C0 and the function β, is sufficiently large. In view of (4.2.29) this gives ˆ ˆ θ

1 ˆ 1 H(t) 1 1 H(r) H(r) dr  dr + C H(1) + h(1) + dt. 2 αε r 4 αε t αθε r It follows that ˆ θ

H(r) dr  C H(1) + h(1) , αθε r which, by (4.2.18) and (4.2.28), yields, for any r ∈ [ε, 1]. 

ˆ 1  H(t) dt  C H(1) + h(1) .  H(r) + h(r)  C H(1) + h(1) + t r 4.3. Problems for Section 4 Problem 4.3.1. Suppose that Lε (uε ) = F in D2r and

∂uε ∂νε

= g on Δ2r . Show that

|∇uε |2 dx (4.3.2)

D3r/2



C r2

|uε |2 dx + C r D2r

where C depends only on μ and M0 .

|F|2 dx + C D2r

|g|2 dσ, Δ2r

Zhongwei Shen

117

Problem 4.3.3. Use the method in Section 4 to prove the boundary Lipschitz estimate for the Dirichlet problem in Section 3.

5. Uniform Calderón-Zygmund Estimates In this section we introduce a real-variable method for Lp estimates. The method, which will be used to establish W 1,p estimates for Lε , may be regarded as a refined and dual version of the celebrated Calderón-Zygmund Lemma. 5.1. The classical Calderón-Zygmund Theorem A measurable function K(x, y) in Rd × Rd is called a Calderón-Zygmund kernel, if there exist constants C > 0 and δ ∈ (0, 1] such that |K(x, y)|  C|x − y|−d for any x, y ∈ Rd , x = y, and that |K(x, y + h) − K(x, y)| 

C|h|δ , |x − y|d+δ

|K(x + h, y) − K(x, y)| 

C|h|δ |x − y|d+δ

for any x, y, h ∈ Rd and |h| < (1/2)|x − y| (see e.g. [25]). Definition 5.1.1. We call T a Calderón-Zygmund operator if (1) T is a bounded linear operator on L2 (Rd ); (2) T is associated with a Calderón-Zygmund kernel K(x, y) in the sense that ˆ K(x, y)f(y) dy for x ∈ / supp(f), T (f)(x) = Rd

whenever f is a bounded measurable function with compact support. The following theorem is referred as the Calderón-Zygmund Lemma. Theorem 5.1.2. Let T be a bounded linear operator on L2 (Rd ). Suppose that ˆ |T (b)| dx  C, (5.1.3) Rd \3B

whenever (5.1.4)

ˆ supp(b) ⊂ B,

bL1 (B)  1,

and

b dx = 0. B

Then T is of weak type (1, 1), i.e.,  x ∈ Rd : |T (f)(x)| > t  C f 1 d L (R ) t 1 d for any t > 0 and f ∈ L (R ). Consequently, T is bounded on Lp (Rd ) for 1 < p < 2. Proof. Let f ∈ L1 (Rd ). Use a Calderón-Zygmund decomposition to express f as  bj , f = g+ where g ∈ L2 (Rd ), supp(bj ) ⊂ Bj , and

j

´ Bj

bj dx = 0. See [25].



118

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

Theorem 5.1.5 (Calderón-Zygmund Theorem). Let T be a Calderón-Zygmund operator. Then T is of weak type (1, 1), and is bounded on Lp (Rd ) for 1 < p < ∞. Proof. Step 1: Verify that T satisfies the conditions in Theorem 5.1.2. This gives the boundedness of T on Lp for 1 < p < 2. Step 2: Since the adjoint T ∗ is also a Calderón-Zygmund operator, it follows by Step 1 that T ∗ is bounded on Lp for 1 < p < 2. By duality T is bounded on Lp for 2 < p < ∞.  5.2. A real-variable argument The real-variable argument presented in this subsection is motivated by a paper of L. Caffarelli and I. Peral [6]. For f ∈ L1loc (Rd ), the Hardy-Littlewood maximal function M(f) is defined by   |f| : B is a ball containing x . (5.2.1) M(f)(x) = sup B

The operator M is bounded on for 1 < p  ∞, and is of weak type (1, 1): ˆ  x ∈ Rd : M(f)(x) > t  C |f| dx for any t > 0, (5.2.2) t Rd where C depends only on d (see e.g. [24, Chapter 1] for a proof). For a fixed ball B in Rd , the localized Hardy-Littlewood maximal function MB (f) is defined by   (5.2.3) MB (f)(x) = sup |f| : x ∈ B and B ⊂ B . Lp (Rd )

B

Since MB (f)(x)  M(fχB )(x) for any x ∈ B, it follows that MB is bounded on Lp (B) for 1 < p  ∞, and is of weak type (1, 1). In the proof of Theorem 5.2.6 we will perform a Calderón-Zygmund decomposition. It will be convenient to work with (open) cubes Q in Rd with sides parallel to the coordinate hyperplanes. By tQ we denote the cube that has the same center and t times the side length as Q. We say Q is a dyadic subcube of Q if Q may be obtained from Q by repeatedly bisecting the sides of Q. If Q is obtained from Q by bisecting each side of Q once, we will call Q the dyadic parent of Q . Lemma 5.2.4. Let Q be a cube in Rd . Suppose that E ⊂ Q is open and |E| < 2−d |Q|. Then there exists a sequence of disjoint dyadic subcubes {Qk } of Q such that (1) Qk ⊂ E; (2) the dyadic parent of Qk in Q is not contained in E; (3) |E \ ∪k Qk | = 0. Proof. This is a dyadic version of the Calderón-Zygmund decomposition. To prove the lemma, one simply collects all dyadic subcubes Q of Q with the property that Q ⊂ E and its dyadic parent is not contained in E; i.e. Q is maximal among all dyadic subcubes of Q that are contained in E. Note that since E is open in Q, the set E \ ∪k Qk is contained in the union Z of boundaries of all dyadic  subcubes of Q. Since Z has measure zero, one obtains |E \ ∪k Qk | = 0.

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119

For the proof of the next theorem, we recall the following formula ˆ ˆ ∞  (5.2.5) |f|p dx = p tp−1 x ∈ E : |f(x)| > t dt, 0

E

where E ⊂

Rd

is measurable and p > 0.

Theorem 5.2.6 ([6, 19, 20]). Let B0 be a ball in Rd and F ∈ L2 (4B0 ). Let q > 2 and f ∈ Lp (4B0 ) for some 2 < p < q. Suppose that for each ball B ⊂ 2B0 with |B|  c1 |B0 |, there exist two measurable functions FB and RB on 2B, such that |F|  |FB | + |RB | on 2B, and

 1/q 1/2  1/2   q 2 2 , |RB |  N1 |F| + sup |f| (5.2.7)



2B

2

2B

|FB |

1/2

4B

 N2

4B0 ⊃B ⊃B



2

sup

4B0 ⊃B ⊃B

B

|f|

1/2

B

 2

+η 4B

|F|

1/2 ,

where N1 , N2 > 1, 0 < c1 < 1, and η  0. Then there exists η0 > 0, depending only on p, q, c1 , N1 , N2 , with the property that if 0  η < η0 , then F ∈ Lp (B0 ) and

  1/p 1/2  1/p  p 2 p , (5.2.8) |F| C |F| + |f| 4B0

B0

4B0

where C depends only on N1 , N2 , c1 , p and q. Proof. Let Q0 be a cube such that 2Q0 ⊂ 2B0 and |Q0 | ≈ |B0 |. We will show that

 1/p 1/2  1/p   , (5.2.9) |F|p C |F|2 + |f|p 4B0

Q0

4B0

where C depends only on N1 , N2 , c1 , p, q, and |Q0 |/|B0 |. Estimate (5.2.8) follows from (5.2.9) by covering B0 with a finite number of non-overlapping Q0 of the same size such that 2Q0 ⊂ 2B0 . To prove (5.2.9), let  for t > 0. E(t) = x ∈ Q0 : M4B0 (|F|2 )(x) > t We claim that if 0  η < η0 and η0 = η0 (p, q, c1 , N1 , N2 ) is sufficiently small, it is possible to choose three constants δ, γ ∈ (0, 1), and C0 > 0, depending only on N1 , N2 , c1 , p and q, such that  (5.2.10) |E(αt)|  δ|E(t)| + | x ∈ Q0 : M4B0 (|f|2 )(x) > γt | for all t > t0 , where α = (2δ)−2/p

(5.2.11)

and

t0 = C0

4B0

|F|2 .

Assume the claim (5.2.10) for a moment. We multiply both sides of (5.2.10) by p t 2 −1 and then integrate the resulting inequality in t over the interval (t0 , T ). This leads to ˆ T ˆ ˆ T p p (5.2.12) t 2 −1 |E(αt)| dt  δ t 2 −1 |E(t)| dt + Cγ |f|p dx, t0

t0

4B0

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

where we have used (5.2.5) as well as the boundedness of M4B0 on Lp/2 (4B0 ). By a change of variables in the left side of (5.2.12), we may deduce that for any T > 0, ˆ ˆ T p p p p t 2 −1 |E(t)| dt  C|Q0 |t02 + Cγ |f|p dx. (5.2.13) α− 2 (1 − δα 2 ) 0

Note that (5.2.14)

δαp/2

4B0

= (1/2). By letting T → ∞ in (5.2.13) and using (5.2.5) we obtain ˆ ˆ p |F|p dx  C|Q0 |t02 + C |f|p dx, 4B0

Q0

which, in view of (5.2.11), gives (5.2.9). It remains to prove (5.2.10). To this end we first note that by the weak (1, 1) estimate for M4B0 , ˆ C |F|2 dx, |E(t)|  d t 4B0 where Cd depends only on d. It follows that |E(t)| < δ|Q0 | for any t > t0 , if we choose C0 = 2δ−1 Cd |4B0 |/|Q0 | in (5.2.11) with δ ∈ (0, 1) to be determined. We now fix t > t0 . Since E(t) is open in Q0 , by Lemma 5.2.4, ∪k Qk ⊂ E(t)

and

|E(t) \ ∪k Qk | = 0,

where {Qk } are (disjoint) maximal dyadic subcubes of Q0 contained in E(t). By choosing δ sufficiently small, we may assume that |Qk | < c1 |Q0 |. We will show that if 0  η < η0 and η0 is sufficiently small, it is possible to choose δ, γ ∈ (0, 1) so small that |E(αt) ∩ Qk |  δ|Qk |,

(5.2.15) whenever



x ∈ Qk : M4B0 (|f|2 )(x)  γt = ∅.

(5.2.16)

It is not hard to see that (5.2.10) follows from (5.2.15) by summation. Indeed, |E(αt)| = |E(αt) ∩ E(t)| = |E(αt) ∩ ∪k Qk |

  |E(αt) ∩ Qk | + x ∈ Q0 : M4B0 (|f|2 )(x) > γt k

δ

 k

|Qk | + x ∈ Q0 : M4B0 (|f|2 )(x) > γt

 δ|E(t)| + x ∈ Q0 : M4B0 (|f|2 )(x) > γt , where the summation is taken over only those Qk for which (5.2.16) hold. Finally, to see (5.2.15), we fix Qk that satisfies the condition (5.2.16). Observe that

(5.2.17) M4B0 (|F|2 )(x)  max M2Bk (|F|2 )(x), Cd t

Zhongwei Shen

121

for any x ∈ Qk , where Bk is the ball that has the same center and diameter as Qk . This is because Qk is maximal and so its dyadic parent is not contained in E(t), which in turn implies that |F|2  Cd t

(5.2.18)

B B ∩ B

 for any ball B ⊂ 4B0 for which k = ∅ and diam(B )  diam(Bk ). Clearly we may assume α > Cd by choosing δ small. In view of (5.2.17) this implies that we have  |E(αt) ∩ Qk |  x ∈ Qk : M2Bk (|F|2 )(x) > αt   αt  x ∈ Qk : M2Bk (|FBk |2 )(x) > 4   (5.2.19) αt + x ∈ Qk : M2Bk (|RBk |2 )(x) > 4 ˆ ˆ C Cd d,q  |FBk |2 dx + |RBk |q dx, q αt 2Bk (αt) 2 2Bk

where we have used the assumption |F|  |FBk | + |RBk |

on 2Bk

as well as the weak (1, 1) and weak (q/2, q/2) bounds of M2Bk . By the second inequality in the assumption (5.2.7), we have (5.2.20)

2Bk

|F2Bk |2 dx  2N22

sup

4B0 ⊃B ⊃Bk

B

|f|2 + 2η2

2Bk

|F|2

 2N22 · γt + 2η2 Cd t, where the last inequality follows from (5.2.16) and (5.2.18). Similarly, we may use the first inequality in (5.2.7) and (5.2.18) to obtain

 q 1/2 q q 2 1/2 |RBk | dx  N1 |F| + (γt) 2Bk 4Bk (5.2.21) q/2  Nq . 1 Cd,q t

We now use (5.2.20) and (5.2.21) to bound the right side of (5.2.19). This yields

(5.2.22)

|E(αt) ∩ Qk |

−q/2  |Qk | Cd · N22 · γ · α−1 + η2 · Cd · α−1 + Cd,q · Nq · α 1

q 2 2 −1 −1 2 2 p −1 ,  δ|Qk | Cd · N2 · γ · δ p + Cd · η0 · δ p + Cd,q · Nq 1 ·δ 2

where we have used the fact α = (2δ)− p . Note that since p < q, it is possible to choose δ ∈ (0, 1) so small that q

p −1  (1/4). Cd,q Nq 1δ

After δ is chosen, we then choose γ ∈ (0, 1) and η0 ∈ (0, 1) so small that 2

2

Cd · N22 · γ · δ p −1 + Cd · η20 · δ p −1  (1/4).

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Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

This gives |E(αt) ∩ Qk |  (1/2)δ|Qk | < δ|Qk | 

and finishes the proof. An operator T is called sublinear if there exists a constant K such that  (5.2.23) |T (f + g)|  K |T (f)| + |T (g)| . Theorem 5.2.24 ([19]). Let T be a bounded sublinear operator on L2 (Rd ) with T L2 →L2  C0 . Let q > 2. Suppose that

  1/q q (5.2.25) |T (g)| N B

2

|T (g)|

2B

1/2

 2

+ sup

|g|

B

B ⊃B

1/2 

d d for any ball B in Rd and for any g ∈ C∞ 0 (R ) with supp(g) ⊂ R \ 4B. Then for any ∞ d f ∈ C0 (R ),

T (f)Lp (Rd )  Cp fLp (Rd ) ,

(5.2.26)

where 2 < p < q and Cp depends at most on p, q, C0 , N, and K in (5.2.23). d Proof. Let f ∈ C∞ 0 (R ) and F = T (f). Suppose that supp(f) ⊂ B(0, ρ) for some ρ > 1. Let B0 = B(0, R), where R > 100ρ. For each ball B ⊂ 2B0 with |B|  (100)−1 |B0 |, we define

and

FB = KT (fϕB )

RB = KT (f(1 − ϕB )),

where ϕB ∈ C∞ 0 (9B) such that 0  ϕB  1 and ϕB = 1 in 8B. Clearly, by (5.2.23), |F|  |FB | + |RB | in Rd . By the L2 boundedness of T , we have  1/2 1/2  |FB |2 C |f|2 . 2B

9B

In view of the assumption (5.2.25) we obtain  1/q 1/2  q 2 |RB | C |RB | +C 2B

 C

4B

4B

|T (f)|2

1/2

C

2

4B

sup

B



4B

1/2

|F|

2

4B0 ⊃B ⊃B

+C +C





+C

4B0 ⊃B ⊃B

sup

|f|

|T (fϕB )|2 

sup

4B0 ⊃B ⊃B



B

|f|2

1/2 1/2

2

B

1/2

1/2

|f|

,

where we have used the L2 boundedness of T in the last inequality. It now follows from Theorem 5.2.6 (with η = 0) that (5.2.27) 1/p 1/2 ˆ 1/p ˆ ˆ 1 − 12 p 2 p p |T (f)| dx  C|B0 | |T (f)| dx +C |f| dx B0

4B0

4B0

Zhongwei Shen

123

for 2 < p < q. By letting R → ∞ in (5.2.27) and using the fact that T (f) ∈ L2 (Rd ),  we obtain the estimate (5.2.26). 5.3. W 1,p estimates In this subsection we establish interior W 1,p estimates for solutions of Lε (uε ) = F under the assumptions that A satisfies the boundedness and ellipticity conditions (1.1.2), is 1-periodic, and belongs to VMO(Rd ). In order to quantify the smoothness, we impose the following condition: for 0 < t  1, A  ω(t) (5.3.1) sup A − x∈Rd

B(x,t)

B(x,t)

where ω is a nondecreasing continuous function on [0, 1] with ω(0) = 0. Theorem 5.3.2 (interior W 1,p estimate [4, 6, 21]). Suppose that the matrix A satisfies (1.1.2) and is 1-periodic. Also assume that A satisfies the VMO condition (5.3.1). Let G ∈ Lp (2B; Rd ) for some 2 < p < ∞ and ball B = B(x0 , r). Suppose that uε ∈ H1 (2B) and Lε (uε ) = div(G) in 2B. Then

       1/p

|∇uε |p

(5.3.3)

 Cp

B

2B

|∇uε |2

1/2

+ 2B

|G|p

1/p

,

where Cp depends only on μ, p, and the function ω(t) in (5.3.1). The proof of Theorem 5.3.2 relies on two lemmas. Lemma 5.3.4 (Small-scale estimate). Suppose A satisfies (1.1.2) and (5.3.1). Then let u ∈ H1 (2B) be a weak solution of L1 (u) = 0 in 2B for some B = B(x0 , r) with 0 < r  1. Then for any 2 < p < ∞, 1/p  1/2  p 2 |∇u|  Cp |∇u| , (5.3.5) 2B

B

where Cp depends only on μ, p, and ω(t). Proof. This is a local W 1,p estimate for second-order elliptic systems in divergence form with VMO coefficients. We prove it by using Theorem 5.2.6 with F = |∇u| and f = 0. Let B ⊂ B with |B |  (1/8)d |B|. Let v ∈ H1 (2B ) be the weak solution to the Dirichlet problem:  (5.3.6) div A0 ∇v) = 0 in 3B and v = u on ∂(3B ), where A0 =

3B

A

is a constant matrix. Note that A0 satisfies (1.1.2). Consequently, the Dirichlet problem (5.3.6) is well posed, and the solution satisfies ˆ ˆ 2 |∇v| dx  C |∇u|2 dx, (5.3.7) 3B

3B

where C depends only on μ. Let FB = |∇(u − v)|

and

RB = |∇v|.

124

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

We will show that FB and RB satisfy the conditions in Theorem 5.2.6. Clearly, F = |∇u|  FB + RB on 2B . By the interior Lipschitz estimate for elliptic systems with constant coefficients,    1/2 1/2 1/2 2 2 2 |∇v| C |∇u| C |F| , max RB = max |∇v|  C 2B

2B

3B

3B

4B

4B

where C depends only on μ. Since L1 (u) = 0 in ⊂ 2B, by the reverse Hölder inequality,   1/q 1/2 |∇u|q C |∇u|2 , 3B

4B

where C > 0 and q > 2 depend only on μ. Note that u − v ∈ H10 (3B ) and     in 3B . div A0 ∇(u − v) = div (A0 − A)∇u It follows that 1/2   2 μ |∇(u − v)|  3B

 

0

3B

2

|(A − A )∇u| 

3B

|A − A0 |2p0



 η(r) where p0 = (q/2) > 1 and

4B



η(r) = sup B(x,s)

x∈Rd 0 2, and Lε (uε ) = 0 in B(0, 2). By a simple blow-up argument we may deduce from Lemma 5.3.4 that

1/p

1/2 |∇uε |p

|∇uε |2

C

B(y,ε)

B(y,2ε)

for any y ∈ B(0, 1). This, together with Theorem 3.1.1, gives

1/p

1/2 |∇uε |p

|∇uε |2

C

B(y,ε)

It follows that

.

B(y,1)

ˆ B(y,ε)

|∇uε |p dx  Cεd ∇uε p , L2 (B(0,2)) 

which yields estimate (5.3.9) by an integration in y over B(0, 1).

Proof of Theorem 5.3.2. Suppose that Lε (uε ) = div(G) in 2B0 , with B = B(x0 , r0 ). By dilation we may assume that r0 = 1. We shall apply Theorem 5.2.6 with q = p + 1, η = 0, F = |∇uε |

f = |G|.

and

For each ball B such that 4B ⊂ 2B0 , we write uε = vε + wε in 2B , where vε ∈ H10 (4B ; Rm ) is the weak solution to Lε (vε ) = div(G) in 4B and vε = 0 on ∂(4B ). Let FB = |∇vε |

RB = |∇wε |.

and

Clearly, |F|  FB + RB in 2B . It is also easy to see that  1/2 1/2  |FB |2 C |G|2 4B

 C

4B

2

4B

|f|

1/2 .

To verify the remaining condition in (5.2.7), we note that wε ∈ H1 (4B ) and Lε (wε ) = 0 in 4B . It then follows from Lemma 5.3.8 that   1/q 1/2 q 2 |∇wε | C |∇wε | 2B

 C  C

4B

4B

4B

By Theorem 5.2.6 we obtain 1/p   p |∇uε | C (5.3.10) B

|∇uε |2 |F|2



1/2 +C

1/2 +C

2

4B

4B



|∇uε |

4B

|f|2

1/2

|∇vε |2

1/2

1/2 .

 |f|

p

+C 4B

1/p

126

Quantitative Homogenization of Elliptic Operators with Periodic Coefficients

for any ball B such that 4B ⊂ 2B0 . By a simple covering argument this implies 1/p  |∇uε |p B0



C

(5.3.11)

 C

2B0

2B0

|∇uε |2 |∇uε |2

1/2

1/p

 +C

1/2



2B0

+C 2B0

|f|p 1/p |G|p

, 

where C depends only on μ, p and ω(t) in (5.3.1). Consider the homogeneous Sobolev space

˙ 1,2 (Rd ) = u ∈ L2 (Rd ) : ∇u ∈ L2 (Rd ; Rd ) . W loc

˙ 1,2 (Rd ) are equivalence classes of functions under the relation that Elements of W u ∼ v if u − v is constant. It follows from the boundedness and ellipticity condition 2 d d and the Lax-Milgram Theorem that for any f = (fα i ) ∈ L (R ; R ), there exists a 1,2 d d ˙ unique uε ∈ W (R ) such that Lε (uε ) = div(f) in R . Moreover, the solution satisfies the estimate ∇uε L2 (Rd )  C fL2 (Rd ) , where C depends only on μ. The following theorem gives the W 1,p estimate in Rd . Theorem 5.3.12. Suppose that A satisfies (1.1.2) and is 1-periodic. Also assume that A satisfies the VMO condition (5.3.1). Let f ∈ C10 (Rd , Rd ) and 1 < p < ∞. Then the ˙ 1,2 (Rd ) to Lε (uε ) = div(f) in Rd satisfies the estimate unique solution in W ∇uε Lp (Rd )  Cp fLp (Rd ) ,

(5.3.13)

where Cp depends only on μ, p, and the function ω(t). Proof. We first consider the case p > 2. Let B = B(0, r). It follows from (5.3.3) that 1

1

1

1

∇uε Lp (B)  C |B| p − 2 ∇uε L2 (2B) + C fLp (2B)  C |B| p − 2 ∇uε L2 (Rd ) + C fLp (Rd ) . By letting r → ∞, this gives the estimate (5.3.13). The case 1 < p < 2 follows by a duality argument. Indeed, suppose Lε (uε ) = ˙ 1,2 (Rd ) and f, g ∈ div(f) in Rd and L∗ε (vε ) = div(g) in Rd , where uε , vε ∈ W C10 (Rd ; Rd ). Then ˆ ˆ ˆ f · ∇vε dx = − A(x/ε)∇uε · ∇vε dx = g · ∇uε dx. Rd

Rd

Rd

Since ∇vε Lq (Rd )  C gLq (Rd ) for q = p > 2, we obtain ˆ g · ∇uε dx  C fLp (Rd ) gLq (Rd ) . Rd

By duality this yields ∇uε Lp (Rd )  C fLp (Rd ) .



Zhongwei Shen

127

Remark 5.3.14. Without the periodicity and VMO condition on A, the estimate (5.3.13) holds if 1 1 − < δ, p 2 where δ > 0 depends only on μ. This result, due to N. Meyers [16], follows readily from the proof of Theorem 5.3.12, using the reverse Hölder inequality. 5.4. Boundary W 1,p estimates Let Bα,p (∂Ω) denote the Besov space of order 1 α ∈ (0, 1) and exponent p ∈ (1, ∞) on ∂Ω. In particular, functions in B1− p ,p (∂Ω) are traces of functions in W 1,p (Ω). Theorem 5.4.1 ([4, 21]). Suppose that A satisfies (1.1.2) and is 1-periodic. Also assume that A satisfies the VMO condition (5.3.1). Let Ω be a bounded C1 domain in Rd . Let uε be a weak solution to the Dirichlet problem, (5.4.2)

Lε (uε ) = div(G)

in Ω

and

on ∂Ω.

uε = f

Then for any 1 < p < ∞, (5.4.3)

∇uε Lp (Ω)  C GLp (Ω) + f

1 ,p 1− p

B

(∂Ω)

where C depends only on μ, p, ω(t) and Ω. Proof. Step 1: By extending f to a function in W 1,p (Ω) one may assume that f = 0. Step 2: By a duality argument it suffices to consider the case p > 2. Step 3: By an analog of Theorem 5.2.24 it suffices to prove the boundary reverse Hölder estimate,

1/p

1/2 (5.4.4) B(x0 ,r)∩Ω

|∇uε |p

C

B(x0 ,2r)∩Ω

|∇uε |2

for any 2 < p < ∞, where x0 ∈ ∂Ω, 0 < r < diam(Ω), Lε (uε ) = 0 in B(x0 , 2r) ∩ Ω and uε = 0 on B(x0 , 2r) ∩ ∂Ω. Step 4: Use boundary Hölder estimates and interior W 1,p estimates to prove equation (5.4.4). 

See [23, Section 5.3] for details.

Analogous results hold for the Neumann problems. Let B−α,p (∂Ω) denote the  p . dual of Bα,p (∂Ω), where p = p−1 Theorem 5.4.5 ([15]). Assume that A and Ω satisfy the same condition as in Theorem 5.4.1. Let uε be a weak solution to the Neumann problem, ∂uε (5.4.6) Lε (uε ) = div(G) in Ω and = −n · G + g on ∂Ω. ∂νε Then for any 1 < p < ∞,

(5.4.7) ∇uε Lp (Ω)  C GLp (Ω) + f 1 −1,p , Bp

where C depends only on μ, p, ω(t) and Ω.

(∂Ω)

128

References

5.5. Problems for Section 5 Problem 5.5.1. Suppose that T is associated with a Calderón-Zygmund kernel. Show that if supp(f) ⊂ Rd \ 3B, then for any x, y ∈ B, |T (f)(x) − T (f)(y)|  C sup

B ⊃B B

|f|.

As a consequence, T (f)L∞ (B) 

|T (f)| + C sup B

B ⊃B B

|f|.

Problem 5.5.2. Use Theorem 5.2.24 to prove the Theorem 5.1.5.

References [1] Scott Armstrong, Tuomo Kuusi, and Jean-Christophe Mourrat, Quantitative stochastic homogenization and large-scale regularity, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 352, Springer, Cham, 2019. MR3932093 ←75 [2] Scott N. Armstrong and Zhongwei Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math. 69 (2016), no. 10, 1882–1923, DOI 10.1002/cpa.21616. MR3541853 ←110, 113 [3] Scott N. Armstrong and Charles K. Smart, Quantitative stochastic homogenization of convex integral functionals (English, with English and French summaries), Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 2, 423–481, DOI 10.24033/asens.2287. MR3481355 ←75, 107, 113 [4] Marco Avellaneda and Fang-Hua Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math. 40 (1987), no. 6, 803–847, DOI 10.1002/cpa.3160400607. MR910954 ←75, 99, 105, 106, 123, 127 [5] Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR503330 ←75, 82 [6] L. A. Caffarelli and I. Peral, On W 1,p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), no. 1, 1–21, DOI 10.1002/(SICI)1097-0312(199801)51:11::AIDCPA13.3.CO;2-N. MR1486629 ←75, 118, 119, 123 [7] G. A. Chechkin, A. L. Piatnitski, and A. S. Shamaev, Homogenization, Translations of Mathematical Monographs, vol. 234, American Mathematical Society, Providence, RI, 2007. Methods and applications; Translated from the 2007 Russian original by Tamara Rozhkovskaya. MR2337848 ←82 [8] Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, Translations of Mathematical Monographs, vol. 174, American Mathematical Society, Providence, RI, 1998. Translated from the 1991 Chinese original by Bei Hu. MR1616087 ←97 [9] Doina Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR1765047 ←82 [10] Eugene Fabes, Layer potential methods for boundary value problems on Lipschitz domains, Potential theory—surveys and problems (Prague, 1987), Lecture Notes in Math., vol. 1344, Springer, Berlin, 1988, pp. 55–80, DOI 10.1007/BFb0103344. MR973881 ←90 [11] Wen Jie Gao, Layer potentials and boundary value problems for elliptic systems in Lipschitz domains, J. Funct. Anal. 95 (1991), no. 2, 377–399, DOI 10.1016/0022-1236(91)90035-4. MR1092132 ←90 [12] Mariano Giaquinta and Luca Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, 2nd ed., Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 11, Edizioni della Normale, Pisa, 2012. MR3099262 ←87, 100, 101, 124 [13] Georges Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal. 40 (2004), no. 3-4, 269–286. MR2107633 ←97 [14] David S. Jerison and Carlos E. Kenig, An identity with applications to harmonic measure, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 3, 447–451, DOI 10.1090/S0273-0979-1980-14762-X. MR561530 ←90

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[15] Carlos E. Kenig, Fanghua Lin, and Zhongwei Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc. 26 (2013), no. 4, 901–937, DOI 10.1090/S0894-03472013-00769-9. MR3073881 ←110, 127 [16] Norman G. Meyers, An Lp e-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189–206. MR159110 ←127 [17] Jindˇrich Neˇcas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Neˇcasová and a contribution by Christian G. Simader. MR3014461 ←97 [18] Weisheng Niu and Yao Xu, A refined convergence result in homogenization of second order parabolic systems, J. Differential Equations 266 (2019), no. 12, 8294–8319, DOI 10.1016/j.jde.2018.12.033. MR3944256 ←97 [19] Zhongwei Shen, Bounds of Riesz transforms on Lp spaces for second order elliptic operators (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 55 (2005), no. 1, 173–197. MR2141694 ←119, 122 [20] Zhongwei Shen, The Lp boundary value problems on Lipschitz domains, Adv. Math. 216 (2007), no. 1, 212–254, DOI 10.1016/j.aim.2007.05.017. MR2353255 ←119 [21] Zhongwei Shen, W 1,p estimates for elliptic homogenization problems in nonsmooth domains, Indiana Univ. Math. J. 57 (2008), no. 5, 2283–2298, DOI 10.1512/iumj.2008.57.3344. MR2463969 ←123, 127 [22] Zhongwei Shen, Boundary estimates in elliptic homogenization, Anal. PDE 10 (2017), no. 3, 653–694, DOI 10.2140/apde.2017.10.653. MR3641883 ←113 [23] Zhongwei Shen, Periodic homogenization of elliptic systems, Operator Theory: Advances and Applications, vol. 269, Birkhäuser/Springer, Cham, 2018. Advances in Partial Differential Equations (Basel). MR3838419 ←73, 75, 78, 86, 93, 97, 106, 127 [24] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR0290095 ←90, 93, 118 [25] Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR1232192 ←117, 124 [26] T. A. Suslina, Homogenization of the Dirichlet problem for elliptic systems: L2 -operator error estimates, Mathematika 59 (2013), no. 2, 463–476, DOI 10.1112/S0025579312001131. MR3081781 ←97 [27] Tatiana Suslina, Homogenization of the Neumann problem for elliptic systems with periodic coefficients, SIAM J. Math. Anal. 45 (2013), no. 6, 3453–3493, DOI 10.1137/120901921. MR3131481 ←98 [28] Luc Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. A personalized introduction. MR2582099 ←79 [29] V. V. Jikov, S. M. Kozlov, and O. A. Ole˘ınik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosifyan]. MR1329546 ←75, 78, 82 Department of Mathematics, University of Kentucky, Lexington, KY 40506 Email address: [email protected]

IAS/Park City Mathematics Series Volume 27, Pages 131–154 https://doi.org/10.1090/pcms/027/00862

Stochastic homogenization of elliptic equations Charles K. Smart Abstract. These are notes for a series of lectures given at the IAS/PCMI 2018 graduate summer school. They give a brief introduction to the theory of homogenization of random uniformly elliptic equations in divergence form.

Contents 1

2

3

4

5

Introduction 1.1 Overview 1.2 Harmonic approximation 1.3 Basic examples 1.4 Preliminaries From correctors to homogenization 2.1 A deterministic criterion 2.2 Hodge decomposition 2.3 The easy half of Theorem 2.1.4 2.4 The difficult half of Theorem 2.1.4 2.5 Exercises Qualitative Theory 3.1 Random coefficients 3.2 Analysis on the probability space 3.3 Correctors 3.4 Convergence 3.5 Exercises Quantitative theory 4.1 Periodic coefficients 4.2 Finite range of dependence 4.3 Subadditivity 4.4 Convergence of the expectation 4.5 Convergence of the correctors 4.6 Exercises Large scale regularity 5.1 Capanato iteration 5.2 Exercises

132 132 132 133 133 133 133 134 136 137 139 140 140 140 141 142 143 144 144 144 145 146 149 149 150 150 152

2010 Mathematics Subject Classification. Primary:35B27; Secondary:35B65. Key words and phrases. Periodic homogenization, Regularity. ©2020 American Mathematical Society

131

132

Stochastic homogenization of elliptic equations

1. Introduction 1.1. Overview These notes explore several topics in the theory of homogenization of random uniformly elliptic equations in divergence form. They are intended as a sequel to the lectures given by Shen [12] on the periodic case. Some of what we discuss is fairly old and can be found, for example, in Jikov–Kozlov– Oleinik [8]. Some of what we discuss is quite new and can be found, for example, in Armstrong–Kuusi–Mourrat [2]. We borrow many ideas from these works, but take a slightly different path through the material. Our goal is to provide the reader with sufficient background to appreciate the recent developments in the subject. We take a rather narrow view of elliptic homogenization as large scale harmonic approximation. We also only discuss the L2 theory. This allows us to cover many aspects of the theory while avoiding several technical difficulties. The interested reader should consult Armstrong– Kuusi–Mourrat [2] for a more thorough treatment that includes the parabolic and inhomogeneous cases in addition to optimal rates and stochastic integrability. 1.2. Harmonic approximation We study solutions of the divergence-form elliptic partial differential equation (aij uxj )xi = 0 in Rd ,

(1.2.1)

whose coefficients a ∈ L∞ (Rd , Rd×d ) satisfy the symmetry condition aji = aij

(1.2.2) and uniform ellipticity condition (1.2.3)

Λ−1 ξ2  ξi aij ξj  Λξ2

for ξ ∈ Rd .

The ellipticity constant Λ  1 is fixed throughout the notes. We show that, when the coefficients a are sufficiently random, in a sense to be made precise, the large scale behavior of the solutions of (1.2.1) is similar to the large scale behavior of the solutions of the partial differential equation (a¯ ij uxj )xi = 0 in Rd ,

(1.2.4)

whose coefficients a¯ ∈ Rd×d are constant. More precisely, given an open ball B ⊆ Rd and coefficients a ∈ L∞ (Rd , Rd×d ) satisfying (1.2.2) and (1.2.3), we consider the space A(B) = {u ∈ H1 (B) : (aij uxj )xi = 0 in B} of a-harmonic functions on B. Given constant coefficients a¯ ∈ Rd×d satisfying ¯ using a vari(1.2.2) and (1.2.3), we measure the distance between A(B) and A(B) ¯ ation on Hausdorff distance. We say that A(B) and A(B) are ε-close if, for every ¯ 1 B) such that ¯ u¯ ∈ A(B) and u ∈ A(B), there are v ∈ A( 12 B) and v¯ ∈ A( 2 (1.2.5)

¯ L2 (B) v − u ¯ L2 ( 1 B)  εu 2

and

¯v − uL2 ( 1 B)  εuL2 (B) . 2

¯ R ) and A(BR ) We show that, if a is sufficiently random, then, almost surely, A(B are εR -close with εR → 0 as R → ∞. When such an approximation result holds,

Charles K. Smart

133

we say that (1.2.1) homogenizes to (1.2.4). Since a-harmonic ¯ functions are harmonic up to a change of variables, homogenization tells us that a-harmonic functions behave like harmonic functions on large scales. We thereby conclude many facts about a-harmonic functions that are a priori false for arbitrary coefficients. 1.3. Basic examples There are two simple examples of random coefficients that capture most of the phenomena we are interested in. The reader may choose to keep only these examples in mind and ignore the general case. Example 1.3.1. Deterministic periodic: Any fixed coefficients a ∈ L∞ (Rd , Rd×d ) satisfying (1.2.2), (1.2.3), and a(x + y) = a(x) for x ∈ Rd and y ∈ Zd . Example 1.3.2. Random checkerboard: A random field a ∈ L∞ (Rd , Rd×d ) of the form a(x) = ab(x) , where b : Zd → {0, 1} is a Bernoulli random field, a0 , a1 ∈ Rd×d are symmetric and invertible, and x denotes the coordinatewise rounding of x ∈ Rd . Note that, even though the first example is not random, our theory of stochastic homogenization will still apply. In particular, the results presented in these notes subsume several of those presented by Shen [12]. We will see that there is a natural way to handle the qualitative theory of the periodic and stationary ergodic cases simultaneously. However, we will also see that the quantitative theory of the random case is more difficult. 1.4. Preliminaries We assume the reader is familiar with the basic theory of probability and of divergence form elliptic equations. See for example Billingsley [4] and Evans [5]. We use Hardy notation for constants. This means that C > 1 > c > 0 always denote positive constants that depend only on dimension d and ellipticity Λ and may differ in each instance. We use subscripts to denote additional dependences, so that Cr may also depend on r. Acknowledgements The author was partially supported by the NSF awards DMS-1606670 and DMS-1712841.

2. From correctors to homogenization 2.1. A deterministic criterion We begin by describing a natural criterion for the harmonic approximation result (1.2.5). Throughout this section, we fix coefficients a ∈ L∞ (Rd , Rd×d ) and a¯ ∈ Rd×d that satisfy (1.2.2) and (1.2.3). Since linear functions always solve constant coefficient equations, we see that linear functions are always a-harmonic. ¯ Our criterion for closeness demands that the linear functions are sufficiently well approximated by a-harmonic functions. We say that correctors exist in the open ball B ⊆ Rd if every coordinate function ¯ has an approximation φk ∈ A(B) that satisfies xk ∈ A(B) (2.1.1)

∇φk ≈ ek

in H−1 (B)

134

Stochastic homogenization of elliptic equations

and ¯ k a∇φk ≈ ae

(2.1.2)

in H−1 (B).

¯ We show that the existence of correctors implies A(B) and A(B) are close. To explain why the above criterion is natural, we take a brief detour through physics. Recall that coefficients a can be interpreted as the local conductivity of a heterogeneous medium. From this point of view, the a-harmonic functions are steady-state electric potentials in the heterogeneous medium. There are two physically natural vector fields associated to a steady-state electric potential u. These are the electric field ∇u and the electric flux (or current) a∇u. On the other hand, the constant coefficients a¯ describe the conductivity of a homogeneous medium. In the homogeneous case, there is a natural family of field/flux pairs given by the linear potentials. That is, every constant electric field p ∈ Rd has corresponding electric flux q = ap ¯ in the homogeneous medium. The conditions (2.1.1) and (2.1.2) say that the constant field/flux pairs in the homogeneous medium can be approximated in the heterogeneous medium. Rather than working with the approximations φk ∈ A(B), it is more convenient to work with the differences gk = φk − xk . As we show below, these differences provide a recipe for perturbing a-harmonic ¯ k functions into (almost) a-harmonic functions. For this reason, the g are called correctors. In fact, it is even more convenient to work with the vector fields (2.1.3)

Gk = ∇φk − ek

and Fk = a∇φk − ae ¯ k,

which are called the gradient and flux correctors. In Lemma 2.2.1 below we see how to recover the approximations φk from the correctors. Our main approximation result shows that, if there are gradient and flux cor¯ are close. rectors that are small in H−1 (B), then the spaces A(B) and A(B) Theorem 2.1.4. If there are Gki , Fki ∈ L2 (B1 ) such that (2.1.5)

kj Gki xj − Gxi = 0

(2.1.6)

Fki xi = 0

(2.1.7)

a¯ ik + Fki = aik + aij Gkj

(2.1.8)

and

GH−1 (B1 ) + FH−1 (B1 ) < ε,

¯ 1 ) are Cε-close. then the spaces A(B1 ) and A(B Note that the properties (2.1.5)–(2.1.7) follow from the equation ∇a∇φk = 0 and (2.1.3). This theorem follows from Theorem 2.3.2 and Theorem 2.4.3 below. 2.2. Hodge decomposition We begin our proof of Theorem 2.1.4 by writing the gradient and flux correctors as the derivatives of two auxiliary fields. Of course, we already know that the gradient corrector G can be written Gki = gk xi . However, it is not immediately clear there is such an expression for the flux corrector F.

Charles K. Smart

135

Recall that the derivation ∂ : C∞ (B1 , (Rd )∧k ) → C∞ (B1 , (Rd )∧(k+1) ) defined by (∂h)i0 ,...,in =

n 

i ,...,ik−1 ,ik+1 ,...,in

(−1)k hx0k

k=0

makes C∞ (B1 , R) → C∞ (B1 , Rd ) → C∞ (B1 , Rd ∧ Rd ) → · · · → C∞ (B1 , R) ∂

∂

∂

∂

an exact sequence. Moreover, the L2 adjoint of this sequence is also exact. Assuming that Gk , Fk ∈ C∞ (B1 , Rd ), the conditions (2.1.5) and (2.1.6) are equivalent to ∂∗ Gk = 0 and ∂Fk = 0. From the exactness, it follows that Gk = ∂gk and Fk = ∂∗ fk for some gk ∈ C∞ (B1 , R) and fk ∈ C∞ (B1 , Rd ∧ Rd ). That is, Gk is the gradient of a scalar field and Fk is the divergence of a skew-symmetric matrix field. The following lemma gives an L2 version of this. The properties (2.2.2), (2.2.3), (2.2.4), and (2.2.5) below are an integrated form of the properties (2.1.5), (2.1.6), and (2.1.7) above. In particular, they are implied by the equation ∇ · a∇φk = 0 and (2.1.3). Thus, this lemma allows us to recover the approximations φk from the correctors. Lemma 2.2.1. If Gki , Fki ∈ L2 (B1 ) satisfy (2.1.5)–(2.1.7), then, for r ∈ (0, 1), there are gk , fkij ∈ H1 (Br ) such that (2.2.2)

ki gk xi = G ,

(2.2.3)

fkji = −fkij

(2.2.4)

ki fkij xj = F ,

(2.2.5)

ik ij k a¯ ik + fkij xj = a + a gxj ,

(2.2.6)

gL2 (Br )  Cr GH−1 (B1 ) ,

(2.2.7)

and fL2 (Br )  Cr FH−1 (B1 ) .

Proof. Since k is fixed throughout the proof, we will simplify notation by writing Gi = Gki and Fi = Fki . The equation (2.2.5) is immediate from (2.1.7) once we prove the other statements. Step 1. We first prove a global result on the torus Td = Rd /Zd . Suppose that i G , Fi ∈ L2 (Td ) are mean-zero. Let g, fij ∈ H1 (Td ) denote the unique mean-zero solutions of gxk xk = Gixi

i j and fij x k x k = Fx j − Fx i .

Using (2.1.5) and (2.1.6), compute i (gxi − Gi )xk xk = 0 and (fij xj − F )xk xk = 0.

In particular, gxi = Gi

i and gij xj = F ,

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Stochastic homogenization of elliptic equations

i since the differences gxi − Gi and fij xj − F are harmonic and mean zero on the torus. The standard estimate

hL2 (Td )  C∇hH−1 (Td )

for h ∈ H1 (Td )

implies gL2 (Td )  CGH−1 (Td )

and fL2 (Td )  CFH−1 (Td ) .

The antisymmetry fji = −fij is immediate. Step 2. We obtain the lemma by localizing the global result on the torus. The basic idea is to restrict from the ball B1 to a square and then periodize. By a covering argument, it is enough to construct and estimate g and fij on the unit cube (0, 1)d under the assumption that Gi and Fi are defined on the larger ball Bd . The H1 extension theorem for the cube implies that GH−1 ((0,1)d )  CGH−1 (Bd )

and FH−1 ((0,1)d )  CFH−1 (Bd ) .

We periodize in three steps. First, we restrict G and F to the cube (0, 1]d . Second, we extend G and F to the cube (−1, 1)d by defining x Gi (x1 , ..., xd ) = i Gi (|x1 |, ..., |xd |) |xi | and   xj Fi (x1 , ..., xd ) = Fi (|x1 |, ..., |xd |). |xj | j =i

Third, we extend G and F to Rd by 2Zd -periodicity. Observe that G and F are now curl-free and divergence-free vector fields on the torus that satisfy GH−1 ((−1,1)d )  CGH−1 ((0,1)d )

and FH−1 ((−1,1)d )  CFH−1 ((0,1)d ) .

To invoke the global result on the torus, we need only observe that G and F are mean-zero by our choice of reflections.  2.3. The easy half of Theorem 2.1.4 The next lemma presents the most basic use of the auxiliary fields introduced in the previous section. It is here that we finally justify calling the scalar field gk a corrector. Lemma 2.3.1. Suppose gk , fkij ∈ H1 (B1 ) satisfy (2.2.3) and (2.2.5). If u¯ ∈ C∞ c (B1 ), 1 ij ij u ∈ H0 (B1 ), and (a uxj )xi = (a¯ u¯ xj )xi in B1 , then ¯ C2 (B1 ) . u¯ + u¯ xk gk − uH1 (B1 )  C(gL2 (B1 ) + fL2 (B1 ) )u Proof. Let v = u¯ + u¯ xk gk − u and observe that v ∈ H10 (B1 ). Using (2.2.5), compute aij vxj = a¯ ij u¯ xj − aij uxj + u¯ xk xj aij gk + u¯ xk fkij xj . Using (2.2.3), compute



 B1

vxi u¯ xk fkij xj

Putting these together, compute   ij vxi a vxj = B1

=−

B1

B1

vxi u¯ xk xj fkij .

vxi u¯ xk xj (aij gk − fkij ).

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Using the ellipticity (1.2.3) and Hölder, obtain ∇vL2 (B1 )  C(gL2 (B1 ) + fL2 (B1 ) )uC2 (B1 ) . Since v ∈ H10 (B1 ), the Poincaré inequality gives vL2 (B1 )  C∇vL2 (B1 ) .



¯ 1 ) can be approximated by some u ∈ A(Br ) We now show that any u¯ ∈ A(B for r ∈ (0, 1). The basic idea is to multiply u¯ by a cutoff function and then apply Lemma 2.3.1. Since u¯ is harmonic up to a change of variables, we can control its C2 norm on Br in terms of its L2 norm on B1 . ¯ 1) Theorem 2.3.2. Suppose gk , fkij ∈ H1 (B1 ) satisfy (2.2.3) and (2.2.5). If u¯ ∈ A(B and r ∈ (0, 1), then there is a u ∈ A(Br ) such that ¯ L2 (B1 ) . u¯ + u¯ xk gk − uH1 (Br )  Cr (gL2 (B1 ) + fL2 (B1 ) )u Proof. Write r = 1 − 2s for some s ∈ (0, 1/2). Choose η ∈ C∞ (B1 ) satisfying 1B1−2s  η  1B1−s and ηCk (B1 )  Ck s−k . Let u ∈ H10 (B1 ) denote the unique ¯ xj )xi in B1 . Since η1 in Br , we have u ∈ A(Br ). solution of (aij uxj )xi = (a¯ ij (ηu) Since u¯ is a-harmonic, ¯ standard estimates give ηu¯ ∈ C∞ c (B1 ) and ¯ L2 (B1 ) . ηu ¯ C2 (B1 )  Cr u Conclude by applying Lemma 2.3.1 to ηu¯ and u.



2.4. The difficult half of Theorem 2.1.4 We now show that any u ∈ A(B1 ) can ¯ r ) for r ∈ (0, 1). The situation here is not be approximated by some u¯ ∈ A(B symmetric to Theorem 2.3.2, since a-harmonic functions do not have a priori interior C2 estimates. Instead, we must rely on more sophisticated methods. We first form a guess for u¯ by computing the a-harmonic ¯ extension of the trace of u on ∂B1 . This works modulo a possible boundary layer. Lemma 2.4.1. Suppose gk , fkij ∈ H1 (B1 ) satisfy (2.2.3) and (2.2.5). If u ∈ A(B1 ), ¯ 1 ), u¯ − u ∈ H1 (B1 ), and r ∈ (0, 1), then u¯ ∈ A(B 0

u¯ + u¯ xk g − uH1 (Br ) k

 C∇u ¯ L2 (B1 \Br ) + C(1 − r)−2 (gL2 (B1 ) + fL2 (B1 ) )u ¯ L2 (B1 ) . Proof. Write r = 1 − 2s for some s ∈ (0, 1/2). Choose an η ∈ C∞ (B1 ) satisfying 1B1−2s  η  1B1−s and ηCk (B1 )  Ck s−k . Let v = u¯ + ηu¯ xk gk − u. By applying the cutoff η to the correction u¯ k gk , we obtain that v ∈ H10 (B1 ) is an admissible test function. This allows us to follow the computation of Lemma 2.3.1 with some minor modifications. Using (2.2.5), compute aij vxj = (1 − η)(aij − a¯ ij )u¯ xj + (a¯ ij u¯ xj − aij uxj ) + (ηu¯ xk )xi aij gk + ηu¯ xk fkij xk . Using (2.2.3), compute  B1

 vxi ηu¯ xk fkij xj

=− B1

vxi (ηu¯ xk )xj fkij .

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Stochastic homogenization of elliptic equations

¯ 1 ) ∩ H1 (B1 ), u ∈ A(B1 ), and v ∈ H1 (B1 ), compute Using u¯ ∈ A(B  0  ij ij ij vxi a vxj = vxi (1 − η)(a − a¯ )u¯ xj + vxi (ηu¯ xk )xj (aij gk − fkij ). B1

B1

B1

By a-harmonic ¯ regularity, ¯ L2 (B1 ) . (ηu¯ xk )xj L∞ (B1−s )  Cs−2 u We conclude as in the proof of Lemma 2.3.1.



We use the Meyers [10] estimate to control the boundary layer. This yields a sub-optimal approximation result. Lemma 2.4.2. Suppose gk , fkij ∈ H1 (B1 ) satisfy (2.2.3) and (2.2.5). There is an ε > 0 depending only on d and Λ such that, if u ∈ A(B1 ) and r ∈ (0, 1), then there is a ¯ r ) such that u¯ ∈ A(B u¯ + u¯ xk gk − uH1 (Br )  Cr (gL2 (B1 ) + fL2 (B1 ) )ε uL2 (B1 ) . ¯ r+s ) be the unique Proof. For some s ∈ (0, (1 − r)/2) to be determined, let u¯ ∈ A(B 1 a-harmonic ¯ function that satisfies u¯ − u ∈ H0 (Br+s ). By Lemma 2.4.1, we see that u¯ + u¯ xk gk − uH1 (Br+s )  C∇u ¯ L2 (Br+s \Br ) + Cs−2 (gL2 (B1 ) + fL2 (B1 ) )u ¯ L2 (Br+s ) . To estimate the boundary layer, we recall that the Meyers [10] estimate implies there is a δ > 0 depending only on d and Λ such that u ¯ W 1,2+δ (Br+s )  CuW 1,2+δ (Br+s )  Cr uL2 (B1 ) . The extra integrability allows us to exploit the smallness of Br+s \ Br . Using Hölder’s inequality, we estimate ¯ L2+δ (Br+s ) . ∇u ¯ L2 (Br+s \Br )  Cr sδ/(4+2δ) ∇u We also have, by Cacciopolli, u ¯ L2 (Br+s )  Cr u ¯ L2 (B1 )  Cr uL2 (B1 ) . Combining the above four inequalities gives u¯ + u¯ xk gk − uH1 (Br )  Cr sδ/(4+2δ) uL2 (B1 ) + Cr s−2 (gL2 (B1 ) + fL2 (B1 ) )uL2 (B1 ) . Optimizing s gives the lemma.



To obtain an optimal approximation result, we iterate the above suboptimal result. Our argument follows the kernel-free proof of the Schauder estimates due to Safonov [11]. Theorem 2.4.3. Suppose gk , fkij ∈ H1 (B1 ) satisfy (2.2.3) and (2.2.5). If u ∈ A(B1 ) ¯ r ) such that and r ∈ (0, 1), then there is a u¯ ∈ A(B u¯ + u¯ xk gk − uH1 (Br )  Cr (gL2 (B1 ) + fL2 (B1 ) )uL2 (B1 ) .

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Proof. Write δ = gL2 (B1 ) + fL2 (B1 ) and r = 1 − 2Ns where N > 0 is a large integer to be determined. Throughout the proof, we let the universal constants C > 1 > c > 0 depend on r and N as well. Define the intermediate radii rn = 1 − ns. We recursively define u2n ∈ A(Br2n ) ¯ R ) as follows. and u¯ 2n+1 ∈ A(B 2n+1 Let u0 = u. ¯ R ) such that After u2n is defined, use Lemma 2.4.2 to find u¯ 2n+1 ∈ A(B 2n+1 gk − u2n H1 (Br u¯ 2n+1 − u¯ 2n+1 xk

2n+1

)

 Cδε u2n L2 (Br

2n

).

After u¯ 2n+1 is defined, use Theorem 2.3.2 to find u2n+2 ∈ A(Br2n+2 ) such that gk − u2n + u2n+2 H1 (Br u¯ 2n+1 + u¯ 2n+1 xk

2n+2

)

 Cδu¯ 2n+1 L2 (Br

2n+1

).

Note carefully the presence of u2n on the left-hand side. By Cacciopolli, we also have the estimates u¯ 2n+1 − u2n L2 (Br

2n+1

)

 Cδε u2n L2 (Br

2n

)

and u¯ 2n+1 − u2n + u2n+2 L2 (Br

2n+2

)

 Cδu¯ 2n+1 L2 (Br

2n

).

By induction, the above four inequalities give u2n L2 (Br

2n

)

+ u¯ 2n+1 L2 (Br

2n+1

)

 Cδεn uL2 (B1 ) .

Letting u¯ = u¯ 1 + u¯ 3 + · · · u¯ 2N−1 , we compute u¯ + u¯ xk gk − uH1 (Br  u2N H1 (Br  Cδ

εN

2N

)

+

uL2 (B1 ) +

2N

)

N−1 

u¯ 2n+1 + u¯ 2n+1 gk − u2n + u2n+2 H1 (Br xk

n=0 N−1 

2n+2

)

Cδ1+εn uL2 (B1 ) .

n=0

Letting N  1/ε gives the lemma.



2.5. Exercises Exercise 2.5.1. Consider the space H = L2 (Rd /Zd , Rd ) of L2 integrable vector fields on the torus. Compute the Helmholtz decomposition H = Hc ⊕ Hp ⊕ Hs , where Hc is the space of constant vector fields, Hp is the space of potential (meanzero and curl-free) vector fields, and Hs is the space of solenoidal (mean-zero and divergence-free) vector fields. Hint: consider a Fourier mode V(x) = νe2πIξ·x with ν ∈ Cd and ξ ∈ Zd . Show that V is curl-free if and only if ν and ξ are parallel. Show that V is divergence-free if and only if ν and ξ are orthogonal. Exercise 2.5.2. Give a different proof of first step of Lemma 2.2.1 by examining Fourier modes directly, as in the previous exercise. Exercise 2.5.3. Suppose Gki , Fki ∈ L2 (B1 ) satisfy (2.1.5), (2.1.6), and (2.1.7). Show ¯ 1 ), u ∈ A(B1 ), and r ∈ (0, 1), then there are v ∈ A(Br ) and that, if u¯ ∈ A(B

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Stochastic homogenization of elliptic equations

¯ r ) such that v¯ ∈ A(B u¯ − vL2 (Br ) + u¯ xi + u¯ xk Gki − vxi L2 (Br )  Cr (GH−1 (B1 ) + FH−1 (B1 ) )u ¯ L2 (B1 ) and ¯v − uL2 (Br ) + ¯vxi + v¯ xk Gki − uxi L2 (Br )  Cr (GH−1 (B1 ) + FH−1 (B1 ) )uL2 (B1 ) . This is a stronger version of Theorem 2.1.4.

3. Qualitative Theory 3.1. Random coefficients In order to make precise what we mean by “sufficiently random” coefficients, we need to define a probability space. Let Ω = {a ∈ L∞ (Rd , Rd×d ) : a satisfies (1.2.2) and (1.2.3)} ¯ ⊆ Ω denote the subspace of denote the space of admissible coefficients. Let Ω constant admissible coefficients. We equip the space of admissible coefficients Ω with the weak-∗ topology. For open U ⊆ Rd , let σ(U) denote the σ-algebra generated by the coarsest topology  d on Ω making the functions a → Rd ϕij aij for ϕij ∈ C∞ c (R ) continuous. In d particular, the σ-algebra σ(R ) consists of the Borel sets in the weak-∗ topology on Ω. The group (Rd , +) acts on Ω via the continuous map τ : Rd × Ω → Ω defined by τ(x, a)(y) = a(x + y). We will mostly be concerned with the action of the subgroup (Zd , +). A Borel subset E ⊆ Ω is (Zd , +)-invariant if τx (E) = E for all x ∈ Zd . Here τx : Ω → Ω is defined by τx (a) = τ(x, a). Throughout these notes, we let P : σ(Rd ) → [0, 1] denote a Borel probability measure on Ω that is invariant and ergodic with respect to the action of (Zd , +). That is, P satisfies the invariance condition (3.1.1)

P ◦ τx = P

for all x ∈ Zd ,

and the ergodicity condition (3.1.2)

P(E) ∈ {0, 1} for all (Zd , +)-invariant Borel sets E ⊆ Ω.

These roughly say that P looks the same everywhere and makes no global choices. Henceforth we use a to denote a random coefficient sampled from the (fixed) probability measure P. Objects derived from a, say by solving a boundary value problem for (1.2.1), will also be random. The conditions (3.1.1) and (3.1.2) are what we mean by “suffciently random.” We will see that they are enough to conclude the approximation result (1.2.5). 3.2. Analysis on the probability space The qualitative theory we present makes clever use of a certain dynamical systems formalism. This formalism allows us

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to employ basic functional analysis (the Riesz representation theorem) in the construction of correctors. All of the difficulty lies in crafting the correct definitions. The key insight is a bijection between random variables and stationary random fields. A random variable on (Ω, P) is a Borel measurable function X : Ω → R. A random field is a Borel measurable function X˜ : R × Ω → R. A random field ˜ + y, a) = X(x, ˜ τy a) for all y ∈ Rd . The X˜ : R × Ω → R is stationary if X(x canonical stationary random field is the map (x, a) → a(x). ˜ a) = X(0, ˜ τx a). In Observe that, if X˜ is a stationary random field, then X(x, ˜ a) encodes all of the information in X. ˜ particular, the random variable X(a) = X(0, From this we see that the random variables and stationary random fields are in bijection via the relations ˜ τx a). ˜ a) = X(τx a)) and X(x) = X(0, (3.2.1) X(x, Henceforth, we identify stationary random fields X˜ and random variables X. In particular, given a random variable X, we write X(x) for X ◦ τx . Owing to the Zd -invariance of P, we see that, for any stationary random field X, the random variables X(x) and X(y) have the same law whenever x − y ∈ Zd . Of course, the laws may differ when x − y does not have integer coordinates. This suggests that we work in the space L2 (Ω) of stationary random fields X for which the norm induced by the inner product  2 XY X, YL2 (Ω) = E (0,1)d

is finite. Remark 3.2.2. We have to average over the cube (0, 1)d because we only have Zd -invariance of P. If we had Rd -invariance of P, then our analysis on the probability space would be simpler. However, we would then no longer be generalizing the periodic case and would have to be more careful in constructing our basic examples. We can similarly define H1 (Ω) as the space of stationary random fields in with weak partial derivatives in L2 (Ω). Note that Yk ∈ L2 (Ω) are the weak partial derivatives of X ∈ L2 (Ω) if and only if  d P ϕxk X + ϕYk = 0 = 1 for all ϕ ∈ C∞ c (R ). L2 (Ω)

Rd

H1 (Ω)

inner product is then defined in the usual way in terms of the weak The partial derivatives. 3.3. Correctors Using the above formalism, we construct a¯ and the correctors using the Riesz representation theorem. ¯ and stationary Theorem 3.3.1. There are deterministic constant coefficients a¯ ∈ Ω   ki ki 2 ki random fields G , F ∈ L (Ω) such that E (0,1)d G = 0, E (0,1)d Fki = 0 and, P

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almost surely, kj Gki xj = Gxi ,

Fki xi = 0,

and a¯ ik + Fki = aik + aij Gkj

hold in the sense of distributions. Proof. Let H denote the space of stationary random vector fields L2 (Ω, Rd ). Let Hc ⊆ H denote the fields which are almost surely constant. Let Hp ⊆ H denote  the space of stationary random vector fields G ∈ H such that E (0,1)d G = 0 and Gixj = Gjxi holds in the sense of distributions. Let Hs ⊆ H denote the space of  stationary random vector fields F ∈ H such that E (0,1)d F = 0 and Fixi = 0 holds in the sense of distributions. By the exercises, (3.3.2)

H = Hc ⊕ Hp ⊕ Hs .

By uniform ellipticity, H has an equivalent inner product  V, Ua = E V · aU (0,1)d

Since the map V → V, ek a is bounded on H, the Riesz representation theorem gives a Gk ∈ Hp such that G, ek + Gk a = 0 for all G ∈ Hp . Using the definition of ·, ·a , we see that a(ek + Gk ) ∈ Hc ⊕ Hs . Define a¯ k ∈ Hc and Fk ∈ Hs via  aik + aij Gkj a¯ ik = E [0,1]d

and Fki = aik + aij Gkj − a¯ ik . It remains to verify that a¯ satisfies the symmetry and ellipticity conditions (1.2.2)  and (1.2.3). We leave this to the exercises. 3.4. Convergence Now that we have constructed a¯ and the correctors, it remains ¯ R ) are close for large R. to prove that A(BR ) and A(B We introduce the scaling XR (x) = X(Rx) for random fields X ∈ L2 (Ω) and scales R  1. This scaling is convenient as it allows us to use the deterministic Theorem 2.1.4, which is stated for the unit ball B1 , on larger balls BR . Observe, if G, F are correctors for a, a¯ in BR , then GR , FR ¯ R ) if and only if are correctors for aR , a¯ in B1 . Moreover, u ∈ A(BR ) and u¯ ∈ A(B ¯ 1 ). uR ∈ AR (B1 ) and u¯ R ∈ A(B We invoke the ergodicity (3.1.2) of P to prove the following.  Lemma 3.4.1. If X ∈ L2 (Ω) and X¯ = E (0,1)d X, then  ¯ −1 P lim XR − X H (B1 ) = 0 = 1. R→∞

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Proof. The multiparameter ergodic theorem (see Krengel [9]) implies that    P lim ϕXR = ϕX¯ = 1 for all ϕ ∈ C∞ c (B1 ) R→∞ Rd

and

 P

Rd

 lim

R→∞ B1

 ¯ 2 = |B1 | · E (XR − X)

(0,1)d

 ¯ 2 = 1. (X − X)

The first implies that XR → X¯ in the weak-∗ topology. The second implies that  XR − X¯ is bounded in L2 (B1 ). Together, they imply XR → X¯ in H−1 (B1 ). We use this to prove our main qualitative result. Theorem 3.4.2. The stationary random field ¯ R ) are ε-close} εR = inf{ε > 0 : A(BR ) and A(B satisfies

 P

lim εR = 0 = 1.

R→∞

Proof. By the scaling considerations above, Theorem 2.1.4 implies that εR  C(GR H−1 (B1 ) + FR H−1 (B1 ) ).  Since E (0,1)d G = 0 and E (0,1)d F = 0, we conclude by Lemma 3.4.1. 



3.5. Exercises 1 Exercise 3.5.1. Fix a (deterministic) f ∈ C∞ c (B1 ) and let u ∈ H0 (BR ) denote the unique (random) solution of (aij uxj )xi = f in B1 . Show that the value u(0) is a random variable. That is, the map a → u(0) is measurable with respect to σ(B1 ). In particular, while the weak-∗ topology is quite coarse, it is still rich enough to study the solutions of the random coefficient equation (1.2.1).

Exercise 3.5.2. Show that, if a ∈ Ω is Zd -periodic, then the trivial probability measure P = δ{a} satisfies (3.1.1) and (3.1.2). Exercise 3.5.3. Prove the Helmholtz-Hodge decomposition (3.3.2) of L2 (Ω, Rd ). Exercise 3.5.4. Prove that the deterministic constant coefficients a¯ constructed in Theorem 3.3.1 satisfies the conditions (1.2.2) and (1.2.3). Hint: compute explicit bounds for the linear functional G → G, ek a with respect to the norm induced by the inner product ·, ·a .  Exercise 3.5.5. Prove that a¯ kl = E (0,1)d Gki aij Glj . Exercise 3.5.6. In dimension d = 1, find a formula for a¯ ∈ R. Exercise 3.5.7. Compute a¯ for the deterministic checkerboard in d = 2 with     1 0 2 0 A0 = and A1 = . 0 2 0 1

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Stochastic homogenization of elliptic equations

Hint: Use the fact that A0 R = RA1 and RFk is potential, if   0 −1 R= . 1 0

4. Quantitative theory 4.1. Periodic coefficients We begin our study of the quantitative theory by proving an optimal rate in the periodic setting. This is an immediate corollary of the proof of Theorem 3.4.2. Theorem 4.1.1. If P = δ{a} for some Zd -periodic a ∈ Ω, then the correctors satisfy GR H−1 (B1 ) + FR H−1 (B1 )  CR−1

for R  1.

¯ R ) are CR−1 -close for all R  1. In particular, A(BR ) and A(B Proof. Since a is deterministic, G and F are also deterministic. Since deterministic stationary fields are Zd -periodic, we see that G and F are Zd -periodic. Applying the Poincaré inequality at scale R−1 gives GR H−1 (B1 ) + FR H−1 (B1 )  CR−1 .  Conclude as in Theorem 3.4.2. 4.2. Finite range of dependence Our proof in the qualitative case fails to be quantitative in exactly one place: when we invoke the multiparameter ergodic theorem in Lemma 3.4.1. To obtain quantitative results, we replace the ergodicity condition (3.1.2) with the stronger condition (4.2.1)

σ(U) and σ(V) are P-independent whenever dist(U, V)  1.

That is, if U, V ⊆ Rd are open, dist(U, V)  1, E ∈ σ(U), and F ∈ σ(V), then P(E ∩ F) = P(E)P(F). This is similar to but more general than (see Holroyd– Liggett [7]) assuming the coefficients in each of the unit cubes Zd + (0, 1)d are independent and identically distributed. More generally, given X ∈ L2 (Ω) a stationary random field and U ⊆ Rd an open set, let σ(X, U) denote the σ-algebra generated by the random variables  ∞ U ϕX for all test functions ϕ ∈ Cc (U). We say that the stationary random 2 field X ∈ L (Ω) has range of dependence R > 0 if σ(X, U) and σ(X, V) are Pindependent whenever U, V ⊆ Rd are open and dist(U, V)  R. Thus the assumption (4.2.1) is equivalent to a having a unit range of dependence. Using the finite range of dependence, we quantify the rate of homogenization. Theorem 4.2.2. Suppose d  3 and P satisfies (3.1.1) and (4.2.1). Then there are M > 1 > ε > 0 such that, for all scales R  1, the correctors satisfy   E GR 2H−1 (B ) + FR 2H−1 (B )  MR−ε . 1

1

By Borel–Cantelli, this theorem implies that, almost surely, there is a random finite scale R0  1 such that, for all larger scales R  R0 , GR H−1 (B1 ) + FR H−1 (B1 )  R−δ .

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Here δ > 0 is some universal constant. In particular, this theorem implies a P almost sure algebraic rate of homogenization. Remark 4.2.3. The restriction d  3 is not necessary, as theorem is also true in dimensions d = 1 and d = 2. We leave the d = 1 case to the exercises. The reader can find a proof of the critical d = 2 case in Armstrong–Kuusi–Mourrat [2]. 4.3. Subadditivity To prove Theorem 4.2.2, we revisit some ideas from our de¯ 1) terministic approximation theory. In particular, we recall that A(B1 ) and A(B are close whenever their are sufficiently nice approximations of the coordinate functions in A(B1 ). We attempt to build these linear approximations by solving boundary value problems. An open cube is a set Q = x + (0, L)d ⊆ Rd with corner x ∈ Rd and side length L > 0. To obtain an a-harmonic function close to the linear function p · x, simply compute the unique solution u ∈ p · x + H10 (Q) of the boundary value problem

(aij uxj )xi = 0 in Q u = p·x

on ∂Q.

The solution u is also the minimizer of the variational problem μ(p, Q) =

inf

u∈p·x+H10 (Q)

E(u),



where E(u, Q) =

Q

ij 1 2 uxi a uxj .

We study the large-scale behavior of the energy μ. Lemma 4.3.1. The random function μ has the following properties. (1) Bounded above and below by volume: Λ−1 |p|2 |Q|  μ(p, Q)  Λ|p|2 |Q|. ¯ k , then (2) Subadditive: If Qk are disjoint and Q ⊆ ∪k Q  μ(p, Qk ). μ(p, Q)  k

(3) Quadratic: If u, v ∈

p · x + H10 (Q)

and μ(p, Q) = E(u, Q), then

E(v, Q) − E(u, Q) = E(u − v, Q). Proof. We leave properties (1) and (3) as exercises. To see property (2), choose uk ∈ p · x + H10 (Qk ) satisfying E(uk , Qk ) = μ(p, Qk ). Observe that the function

uk (x) if x ∈ Qk u(x) = p·x otherwise satisfies μ(p, Q)  E(u, Q) =

 k

and u ∈ p · x + H10 (Q).

E(uk , Q) 

 k

E(uk , Qk ) =



μ(p, Qk )

k



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Stochastic homogenization of elliptic equations

We show that the rescaled expectations E|Q|−1 μ(Q) converges as Q → Rd . Lemma 4.3.2. There is a deterministic and constant a¯ ∈ Ω such that μ¯ n (p) = E2−dn μ(p, (0, 2n )d )  μ(p) ¯ = 12 pi a¯ ij pj

as n  ∞

Proof. Lemma 4.3.1 gives Λ−1 |p|2  μ¯ n+1 (p)  μ¯ n (p)  Λ|p|2 . Thus, the limit exists. Let up,Q ∈ p · x + H10 (Q) be the unique function such that μ(p, Q) = E(up,Q , Q). Using the Euler-Lagrange characterization of the minimizer, we see that p → up,Q is linear. It follows that p → μ(p, Q) is a quadratic function with a minimum at zero. In particular, the limiting function ¯ = pi a¯ ij pj for some a¯ ij ∈ R μ(p) ¯ = limn→∞ μ¯ n (p) must have the form μ(p)  satisfying (1.2.2) and (1.2.3). At this point, we could apply the of subadditive ergodic theorem of Akcoglu– Krengel [1] to conclude that, P almost surely, |Q|−1 μ(p, Q) → 12 p · ap ¯ as Q → Rd . Among other things, this implies that the gradients and fluxes of the minimizers converge to the global gradient and flux corrector for the corresponding slope. We take a different approach to make things quantitative. 4.4. Convergence of the expectation The following lemma says that, in dimensions d  3, stationary random fields with a finite range of dependence scale like deterministic periodic fields with respect to the H−1 norm. Lemma 4.4.1. Suppose d  3. If X ∈ L2 (Ω) has a unit range of dependence and satisfies   E (0,1)d X = 0, and E (0,1)d X2  1, then, for all scales R  2, EXR 2H−1 ((0,1)d )  CR−2 . Proof. Recall that, EXR 2H−1 ((0,1)d )  CR−2 + C

 2 (1 + ξ2 )−1 E e2πiξ·x XR . d (0,1) d



ξ∈Z

ξ R

For ξ ∈ Zd with ξ  R, the hypotheses on X give  2  E e2πiξ·x XR   CR−d . (0,1)d

Conclude using



2 −1

ξ R (1 + x )

C

R 1

rd−3 dr.



We obtain a rate of convergence of μ¯ n (p) → μ(p) ¯ by quantifying the subadditivity of μ. The main insight is that, when the difference between successive scale μ¯ n (p) − μ¯ n+1 (p) is small, then there is almost additivity between scale n and all larger scales.

Charles K. Smart

147

Lemma 4.4.2. Suppose d  3 and P satisfies (3.1.1) and (4.2.1). For all m, n  0 and p ∈ Rd , (4.4.3)

0  μ¯ n (p) − μ¯ n+m (p)  C(μ¯ n (p) − μ¯ n+1 (p)).

Proof. Step 1. By scaling, we may assume that n = 0 and |p| = 1. Since p is fixed, we omit it from the notation μ. We recall our proof of subadditivity, which glued together small scale minimizers into a large scale minimizer. For each k ∈ Zd , let uk ∈ p · x + H10 (k + (0, 1)d ) satisfy E(uk , k + (0, 1)d ) = μ(k + (0, 1)d ). We define u ∈ p · x + H10 ((0, 2m )d ) by

uk (x) u(x) = p·x

if x ∈ k + (0, 1)d otherwise.

Let v ∈ p · x + H10 ((0, 2m )d ) satisfy E(v, (0, 2m )d ) = μ((0, 2m )d ). Using Lemma 4.3.1, observe that μ¯ 0 − μ¯ m = E2−md E(u − v, (0, 2m )d ). Step 2. We check how close u is to being a solution of the Euler-Lagrange equation for v. The problem is that u may be quite singular near the boundaries of the cubes k + (0, 1)d for k ∈ Zd where we glued the unit scale minimizers uk together. It is here that we use the smallness of ε = μ¯ 0 − μ¯ 1 . For each k ∈ Zd , let u˜ k ∈ p · x + H10 (k + (−1, 1)d ) satisfy E(u˜ k , k + (−1, 1)d ) = μ(k + (−1, 1)d ). Using Lemma 4.3.1, we see that (4.4.4)

EE(u˜ j − uk , k + (0, 1)d )  Cε whenever k + (0, 1)d ⊆ j + (−1, 1)d .

In particular, we see that any two minimizers u˜ k are compatible when they overd lap. To use this to construct a global vector field, choose η ∈ C∞ c ((−1, 1) ) whose translations ηk (x) = η(x − k) satisfy  ηk = 1 in Rd . k∈Zd

Define the vector field ˜ = G



ηk ∇u˜ k .

k

Observe (see Exercise 4.6.4 below) that  ˜ = 0 for k ∈ Zd (4.4.5) E ∇ · aG k+(0,1)d

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Stochastic homogenization of elliptic equations

and

 E

(4.4.6)

k+(0,1)d

˜ − ∇u|2 + |∇ · aG| ˜ 2  Cε |G

for k ∈ Zd .

Note that (4.4.6) implies, in some sense, that u almost solves ∇ · a∇u = 0. Step 3. We now conclude our probabilistic lemma. Lemma 4.3.1 implies E(u, (0, 2m )d ) − E(v, (0, 2m )d ) = E(u − v, (0, 2m )d )  = (∇u − ∇v) · a(∇u − ∇v) 

(0,2m )d

= (0,2m )d

 ˜ + (∇u − ∇v) · a(∇u − G)

Thus, by Hölder,

(0,2m )d

˜ (v − u)∇ · aG.





E(u − v, (0, 2m )d )  C

(0,2m )d

˜ 2+C |∇u − G|

sup

(0,2m )d



w∈H10 ((0,2m )d )

2 ˜ w∇ · aG

(0,2m )d

|∇w|2

.

By (4.4.6), the first term is bounded by C2dm ε. To estimate the second term, recall that, if w ∈ H10 ((0, R)d ) and X ∈ L2 ((0, R)d ), then  wX  CRd/2+1 ∇wL2 ((0,R)d ) XR H−1 ((0,1)d ) . E (0,R)d

Thus, using (4.4.5) and (4.4.6), we can apply Lemma 4.4.1 to X = ∇ · aG to bound the second term by C2dm ε.  It follows that μ¯ 0 − μ¯ m  C(μ¯ 0 − μ¯ 1 ). The previous lemma implies a quantitative rate of convergence for μ¯ n . Lemma 4.4.7. If d  3 and P satisfies (3.1.1) and (4.2.1), then there are M > 1 > τ > 0 such that ¯  Mτn |p|2 . 0  μ¯ n (p) − μ(p) Proof. Write sn = μ¯ n (p) and s∞ = μ(p). ¯ Recall Lemma 4.4.2 asserts 0  sn − sn+m  C(sn − sn+1 ). Since sn is descending, we that, for every n, there is an n ˜ ∈ [n, n + m) such that  Cm−1 (sn − sn+1 ). sn˜ − sn+1 ˜ For m  1 large universal with Cm−1  1/2, we can find a sequence nk such that nk  mk

and snk − snk +1  2−k (s0 − s1 ).

In particular, whenever n  mk, we have 0  sn − s∞  snk − s∞  C(snk − snk +1 )  C2−k . In particular, −1 n

0  sn − s∞  C2−m which implies the lemma.

|p|2 , 

Charles K. Smart

149

4.5. Convergence of the correctors To prove Theorem 4.2.2, we must connect our subadditive quantity μ to the qualitative correctors G and F. Proof of Theorem 4.2.2. Fix a slope p ∈ Rd with |p| = 1. By Theorem 3.3.1, there  is a unique stationary random field G, F ∈ L2 (Ω, Rd ) such that E (0,1)d G = 0, ˜ ki , F˜ ki ∈ L2 (Ω) are given by ∇ × G = 0, and ∇ · a(p + G) = 0. Indeed, if G i k ki Theorem 3.3.1, then G = p G and we may compute ˜ kj ))x ∇ · a(p + G) = (aij (pj + pk G i

˜ kj )x = pk (aik + aij G i k ik kj ˜ = p (a¯ + F )x i

= 0. Finally, let F = a(p + G) − ap. ¯ As before, we glue together minimizers at scale 2n . For n  0, first choose n u ∈ p · x + H10 ((−2n , 2n )d ) satisfying E(un , (−2n , 2n )d ) = μ(p, (−2n , 2n )d ). Then, extend un to a 2n+1 Zd -invariant random field. By Lemma 4.4.7, there are M > 1 > τ > 0 such that  ∇um − ∇un 2  Mτn for m  n. E2−md (2−m ,2m )d

Moreover, by Exercise 4.6.6, Theorem 2.3.2 and Lemma 3.4.1 imply  ∇un − G2 = 0. (4.5.1) lim E2−nd n→∞

(−2n−1 ,2n−1 )d

Combining the above two estimates, we see that  −nd G − ∇un 2  Mτn . E2 (−2n−1 ,2n−1 )d

Thus, we estimate EG2m H−1 ((−1,1)d )  CMτn + CE(un )2m H−1 ((−1,1)d )  CM(τn + τm−n ). 

A similar argument works for F2m . 4.6. Exercises

Exercise 4.6.1. Explain how the Poincaré inequality is used to prove Theorem 4.1.1. Exercise 4.6.2. Show that the random checkerboard satisfies (3.1.1), and (3.1.2). Show that (3.1.1) and (4.2.1) imply (3.1.2). Exercise 4.6.3. Prove (1) and (3) of Lemma 4.3.1. ˜ = Exercise 4.6.4. Prove (4.4.5) and (4.4.6). Hint: use ∇ · aG  k k ∇η = 0 to handle the second term in (4.4.6). Exercise 4.6.5. Prove that, in d = 2, 0  μ¯ n − μ¯ n+m  Cm(μ¯ n − μ¯ n+1 ). Does this suffice to get an algebraic rate?



k ∇η

k a∇u ˜k

and

150

Stochastic homogenization of elliptic equations

Exercise 4.6.6. Deduce (4.5.1) from Theorem 2.3.2 and Lemma 3.4.1. Do this by ¯ function p · x on the cube (−2m , 2m )d . Note comparing um to the a-harmonic that G is exactly the corrected gradient of p · x.

5. Large scale regularity 5.1. Capanato iteration As an application of the foregoing theory, we prove a large scale regularity result of Gloria–Neukamm–Otto [6] using a Capanato iteration argument from Armstrong–Smart [3]. The basic idea is that, once the correctors are sufficiently small in the H−1 norm, the large scale regularity properties of a-harmonic ¯ functions are inherited by the a-harmonic functions. Throughout this section, we assume that P satisfies (3.1.1) and (3.1.2) and fix qualitative correctors Fki , Gki ∈ L2 (Ω) as in Theorem 3.3.1. We begin with an improvement of flatness estimate. Lemma 5.1.1. For every α ∈ (0, 1), there are M > 1 > ε > δ > 0 such that, if Gs H−1 (B1 ) + Fs H−1 (B1 )  δ for s ∈ [εR, R], then, for all u ∈ A(BR ) and p ∈ Rd , there is a q ∈ Rd such that ∇u − q − qk Gk 2L2 (B

εR )

(εR)d

ε

2α

∇u − p − pk Gk 2L2 (B

R)

Rd

and q − p2  M

∇u − p − pk Gk 2L2 (B

)

R . Rd Proof. By scaling, we may assume that R = 1. Let ε > δ > 0 be small constants to be determined. Fix p ∈ Rd and consider

η = ∇u − p − pk Gk L2 (B1 ) . Since ∇u − p − pk Gk is the gradient of an a-harmonic function in B1 , Theo¯ 2/3 ) such that rem 2.4.3 gives a u¯ ∈ A(B uxj − pj − pk Gkj − u¯ xj − u¯ xk Gkj L2 (B2/3 )  Cδη. Harmonic regularity gives u¯ xj L∞ (B1/3 ) + u¯ xk xj L∞ (B1/3 )  Cη. Our hypothesis and Cacciopolli gives G2L2 (Bε )  Cεd . In particular, if q = ∇u(0), ¯ then u¯ xj + u¯ xk Gkj − qj − qk Gkj 2L2 (Bε )  Cη2 εd+2 and uxj − (pj + qj ) − (pk + qk )Gkj 2L2 (B

δ)

 Cδ2 η2 + Cεd+2 η2 .

If δ2  εd+2 and Cεd+2  εd+2α , then the right-hand side is bounded by εd+2α η2 . Note that we also have q  Cη 

Charles K. Smart

151

Iterating Lemma 5.1.1 yields the following large scale C1,α estimate. Theorem 5.1.2. For any α ∈ (0, 1), there is a δ > 0 such that, if Gs H−1 (B1 ) + Fs H−1 (B1 )  δ for s ∈ [r, R], then there is a p ∈ Rd such that 2

|p|  C

∇u2L2 (B

R)

Rd

and ∇u − p − pk Gk 2L2 (B

r)

r 2α ∇u2 2 L (B

C

)

R . R rd Rd Proof. Let p0 = 0. Let δ, ε > 0 be as in Lemma 5.1.1. Apply Lemma 5.1.1 to obtain a sequence of pn ∈ Rd such that

ηn =

k 2 ∇u − pn − pn k G L2 (B

εn R )

(εn R)d

satisfies ηn+1  ε2α ηn

and |pn+1 − pn |2  Cηn .

The iteration stops at the least n such that εn+1 R < r, at which point |pn |2  Cη0 and ηn  C(r/R)α η0 .  The above estimate implies a Liouville theorem. Theorem 5.1.3. Almost surely, every u ∈ A(Rd ) with sub-quadratic growth in the sense that there is an ε > 0 such that u2L2 (B ) R =0 lim R→∞ Rd+4−2ε has gradient of the form ∇u = p + pk Gk for some p ∈ Rd . Proof. For α = 1 − ε/2, let δ > 0 be as in Theorem 5.1.2. By Lemma 3.4.1, P almost surely, there is an R0  0 such that Gs H−1 (B1 ) + Fs H−1 (B1 )  δ for all s  R0 . By the Cacciopolli inequality, lim

∇u2L2 (B

R)

Rd+2−2ε

R→∞

= 0.

In particular, we may assume that ∇u2L2 (B

)

R  R2−2ε for R  R0 . Rd Thus, for all R  r  R0 , Theorem 5.1.2 gives a p ∈ Rd such that

∇u − p − pk Gk 2L2 (Br )  Cr1−ε/2 R−ε/2 . Sending R → ∞, we see there must be a p ∈ Rd such that ∇u = p + pk Gk in Br .  The same p ∈ Rd works for all Br with r  R0 .

152

References

5.2. Exercises Exercise 5.2.1. Recall the following Schauder estimate: If α ∈ (0, 1), aij uniformly elliptic, aij uxi xj = f in Rd , and a, u, and f are (qualitatively) smooth, then [u]2+α  Cα [a]α [u]2 + Cα [f]α , where |∇k g(x) − ∇k g(y)| |x − y|α x =y

[g]k+α = sup

denotes the Cα seminorm of ∇k g. Complete the following outline of Safonov’s kernel-free proof of this estimate. (1) Show that [u]2+α is equivalent to [u − q]0,Br (x) inf , sup r2+α q∈Q B (x)⊆Rd r

where Q denotes the space of quadratic polynomials. (2) Let Tx u ∈ Q denote the second order Taylor expansion of u at x. For Br (x) ⊆ Rd , write u − Tx u = v + w in Br (x), where v solves the constant coefficient equation

ij ax vxi xj = 0 in Br (x) v = u − Tx u

on ∂Br (x).

Show that w solves

ij ax wxi xj = (ax − a)ij uxi xj + (f − fx ) in Br (x) w=0

on ∂Br (x).

Using what you know about constant coefficients, show that, for s ∈ (0, 1), [u − Tx u − Tx v]0,Brs (x)  [v − Tx v]0,Brs (x) + [w]0,Brs (x)  Cs3 r2+α [u]2+α + Cr2+α [a]α [u]2 + Cr2+α [f]α . (3) Divide the last estimate above by (rs)2+α , select an appropriate s, and compute the supremum over all balls Br (x).

References [1] M. A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes, J. Reine Angew. Math. 323 (1981), 53–67, DOI 10.1515/crll.1981.323.53. MR611442 ←146 [2] Scott Armstrong, Tuomo Kuusi, and Jean-Christophe Mourrat, Quantitative stochastic homogenization and large-scale regularity, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 352, Springer, Cham, 2019. MR3932093 ←132, 145 [3] Scott N. Armstrong and Charles K. Smart, Quantitative stochastic homogenization of convex integral functionals (English, with English and French summaries), Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 2, 423–481, DOI 10.24033/asens.2287. MR3481355 ←150

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[4] Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR1324786 ←133 [5] Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR2597943 ←133 [6] Antoine Gloria, Stefan Neukamm, and Felix Otto, A regularity theory for random elliptic operators, available at arXiv:1409.2678. ←150 [7] Alexander E. Holroyd and Thomas M. Liggett, Finitely dependent coloring, Forum Math. Pi 4 (2016), e9, 43, DOI 10.1017/fmp.2016.7. MR3570073 ←144 [8] V. V. Jikov, S. M. Kozlov, and O. A. Ole˘ınik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosifyan]. MR1329546 ←132 [9] Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR797411 ←143 [10] Norman G. Meyers, An Lp e-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 189–206. MR159110 ←138 [11] M. V. Safonov, Classical solution of second-order nonlinear elliptic equations (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 6, 1272–1287, 1328, DOI 10.1070/IM1989v033n03ABEH000858; English transl., Math. USSR-Izv. 33 (1989), no. 3, 597–612. MR984219 ←138 [12] Zhongwei Shen, Periodic homogenization of elliptic systems, Operator Theory: Advances and Applications, vol. 269, Birkhäuser/Springer, Cham, 2018. Advances in Partial Differential Equations (Basel). MR3838419 ←132, 133 University of Chicago Email address: [email protected]

IAS/Park City Mathematics Series Volume 27, Pages 155–198 https://doi.org/10.1090/pcms/027/00863

T 1 and T b theorems and applications Simon Bortz, Steve Hofmann, and José Luis Luna Abstract. The classical singular integral theory of Calderón and Zygmund, and the related Littlewood-Paley-Stein square function theory, allow one to obtain Lp and endpoint bounds for singular integral operators and square functions, from L2 bounds, given natural regularity properties of the associated kernel. Thus, the question of L2 boundedness is fundamental. In the case of singular integral operators and square functions of convolution type, this question is readily resolved via the Fourier transform and Plancherel’s theorem. In the case of singular integral operators that are not of convolution type, matters are much more problematic. The quest to understand the latter case, and to apply it to the theory of elliptic PDE, has comprised one of the major directions of research in harmonic analysis over the past 40 years. In these notes, we discuss L2 boundedness criteria for singular integrals and square functions, known as “T 1" and “T b" theorems.

Contents 1 2 3 4 5 6

Introduction Generalized Littlewood-Paley Theory T 1 Theorems A T b Theorem for Square Functions and the Cauchy Integral Local T b theory Application: The Kato square root problem

155 159 167 178 182 188

1. Introduction The T 1 and T b Theorems and their generalizations have facilitated the study of several operators of interest in harmonic analysis. In this introductory section we give some examples of operators that may be treated with this theory. We also introduce some of the notation and conventions that will be used throughout this text. We begin with one of the most important examples of a singular integral operator (SIO) not of convolution type: the Cauchy integral on a Lipschitz graph. 2010 Mathematics Subject Classification. Primary 42B20; Secondary 42B25. Key words and phrases. Park City Mathematics Institute, singular integrals, square functions, T1 theorem, Tb theorem, local Tb theorem, Kato problem. ©2020 American Mathematical Society

155

156

T 1 and T b theorems and applications

Example 1.1. Let Γ := {x + iA(x) : x ∈ R} where A : R → R is a Lipschitz function. Then the Cauchy integral of a function f on Γ is given by ˆ f(v) 1 dv , CΓ f(z) := p.v. 2πi Γ z − v or parametrically in the graph co-ordinates by (1.2) CΓ f(x + iA(x))

ˆ 1 1 f(y + iA(y))(1 + A  (y)) dy. 2πi R (x − y) + i(A(x) − A(y)) Here, the abbreviation “p.v." stands for “principal value", and refers to the fact that since the integrand is not absolutely integrable, the integral should be interpreted in some appropriate limiting sense. We return to this matter below. Define for A : R → R Lipschitz ˆ g(y) 1 dy. (1.3) CA (x) := p.v. 2πi R (x − y) + i(A(x) − A(y)) := p.v.

Then since |1 + iA  (x)| ≈ 1 we have that CΓ is bounded as an operator from L2 (Γ ) to L2 (Γ ) if and only if CA is bounded as an operator from L2 (R) to L2 (R). Another operator that may be studied with the techniques of these notes is the gradient of the single layer potential associated to a divergence from elliptic operator (under certain assumptions) on the domain above a Lipschitz graph. The layer potential method reduces the question of solvability of a boundary value problem to that of solvability of an integral equation. For the sake of simplicity, at this stage we consider only at the case where the divergence form operator is the Laplacian. Example 1.4. For A : Rn+1 → R Lipschitz, define Ω := {(x, t) ∈ Rn+1 : t > A(x)}. Clearly, ∂Ω := {(x, t) ∈ Rn+1 : t = A(x)}. For X = (x, t) ∈ Rn+1 , denote by E(X) = cn |X|1−n the fundamental solution to the Laplacian in Rn+1 . We then define the single layer potential of f : ∂Ω → R as ˆ E(X − Y)f(Y) dσ(Y), X ∈ Ω Sf(X) := ∂Ω

where σ is the surface measure on ∂Ω. A fundamental property of the operator S is that its restriction to ∂Ω is a smoothing operator of order 1, mapping ˙ 1,p (∂Ω), a fact which Lp (∂Ω), 1 < p < ∞, into the homogeneous Sobolev space W may be deduced from boundedness on Lp (∂Ω) of the following vector valued singular integral operator (SIO): ˆ ∇X E(X − Y)f(Y) dσ(Y), X ∈ ∂Ω. (1.5) T f(X) := p.v. ∂Ω

For any Lipschitz function A : Rn → R, we may parametrize (1.5) on the graph of A, and write ˆ (x − y, A(x) − A(y)) f(y) dy. TA f(X) := p.v. cn n+1 Rn (|x − y|2 + |A(x) − A(y)|2 ) 2

Simon Bortz, Steve Hofmann, and José Luis Luna

 has components The kernel of TA , K, xj − yj Kj (x, y) = , n+1 (|x − y|2 + |A(x) − A(y)|2 ) 2 Kn+1 (x, y) =

157

1jn

A(x) − A(y) (|x − y|2 + |A(x) − A(y)|2 )

n+1 2

.

The next example is the so-called SIO of “Calderon type". We shall see that Kj , 1  j  n + 1 in Example 1.4 fall into this category. Example 1.6. Suppose A : Rn → R is Lipschitz, F ∈ Ck (R) for some k large enough and Ω : Rn → R is positively homogeneous of degree 0. Suppose further that Ω and F have opposite parity, that is, either Ω is odd and F is even, or vice versa. We then define   ˆ Ω(x − y) A(x) − A(y) f(y) dy. F (1.7) T f(x) = p.v. n |x − y| Rn |x − y| As mentioned above we may realize each Kj , 1  j  n + 1, in Example 1.4 as a SIO of Calderon type if we set xj 1 1  j  n, Ω(x) = , F(s) = n+1 |x| (1 + s2 ) 2 and Ω(x) = 1,

F(s) =

s (1 + s2 )

n+1 2

j = n + 1.

Note that if n = 1 then the operator in (1.7) may be written as   ˆ A(x) − A(y) 1 F . T f(x) = p.v. x−y R x−y The quintessential SIO’s of Calderon type are the Calderon commutators which are related to the symbolic calculus of pseudo-differential operators with nonsmooth (Lipschitz) coefficients, and to the Cauchy integral operator CA above. Example 1.8 (Calderon Commutators [2]). For k ∈ N and A : R → R Lipschitz, the k-th Calderon commutator in dimension one is given by   ˆ A(x) − A(y) k 1 1 (k) f(y) dy. CA f(x) := p.v. 2πi R x − y x−y Observe that, up to normalization, the case k = 0 is the Hilbert transform, and (1) that at least formally, CA is a commutator:   d d (1) −1 d (1.9) CA f = −1 2i dx H, A f = 2i dx (HAf) − A dx (Hf) ; (k)

while CA is a higher order commutator (k = 2, 3, . . .):   k d −1 (k) ... H, A A . . . A f . CA f = k!2i dxk (1)

The operator CA (and its higher dimensional analogues) were introduced by Calderón in his construction of an algebra of SIOs suitable for the treatment of partial differential operators with merely Lipschitz coefficients, thus, a

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sort of pseudo-differential calculus which, in contrast to the classical pseudodifferential calculus, was applicable to operators with rather minimally smooth coefficients [3]. In higher dimensions we may study similar objects. If n  1 and if k ∈ N and Ω is a homogeneous function of degree 0, that is odd if k is even and even if k is odd, we may define   ˆ Ω(x − y) A(x) − A(y) k f(y) dy. T f(x) := p.v. n |x − y! Rn |x − y| Note that if K(x, y) is the kernel of the operator CA in (1.3), then if |A  | < 1, we have that 1 1 1

 = K(x, y) = A(x)−A(y) (x − y) + i(A(x) − A(y)) x−y1+i x−y (1.10)   ∞  A(x) − A(y) k 1 = −i x−y x−y k=0

Then formally we have that CA f(x) =

∞ 

(k)

(−i)k CA f(x).

k=0

The last example we present here is the Kato square root operator. Example 1.11 (Kato Square Root Operator). We say L is a divergence form elliptic operator if it has the form   ∂  ∂ Ai,j (x) L = − div A(x)∇ = − ∂xj ∂xj i,j

where A : Rn → Mn×n (C), with Ai,j (x) ∈ L∞ (Rn ) and there exists λ > 0 such that  Ai,j ξi ξj , ∀x ∈ Rn , ∀ξ ∈ Cn . (1.12) λ|ξ|2  ReA(x)ξ, ξ = i,j

We understand the operator L in the weak sense, since A is not necessarily differentiable; this will be discussed in detail later. The accretivity condition (1.12) √ allows one to define L, and a natural question that arises in this context is whether √ (1.13)  LuL2 (Rn ) ≈ ∇uL2 (Rn ) . √ Note that if Lu = v then (1.13) is equivalent to vL2 (Rn ) ≈ ∇L−1/2 vL2 (Rn ) . Notice then that in the case where L = −Δ, we have that ∇(−Δ)−1/2 = R, the vector Riesz transform, for which the estimate is an easy, and well known, consequence of Plancherel’s Theorem (see [24, Chapter III]). Definition 1.14 (Calderon-Zygmund Kernel). A Calderon-Zygmund (C-Z) kernel is a function K(·, ·) : (Rn × Rn ) \ {(x, y) ∈ Rn × Rn : x = y} → C for which there

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exist constants C > 0 and α ∈ (0, 1] such that C (1.15) |K(x, y)|  , ∀x, y ∈ Rn × Rn \ {x = y} |x − y|n and if h > 0, |x − y| > 2h then |h|α . |x − y|n+α We often refer to (1.15) as the “size condition" and (1.16) as the Hölder continuity (or simply “smoothness") condition. When K is a Calderon-Zygmund kernel we will often write K ∈ C-Z. (1.16)

|K(x + h, y) − K(x, y)| + |K(x, y + h) − K(x, y)|  C

Definition 1.17 (Generalized Calderon-Zygmund Operator). We say that a linn ear operator T mapping D → D  (D = C∞ 0 (R )) is a (generalized) CalderonZygmund singular integral operator (or a singular integral operator in the sense of Coifman and Meyer) if it is associated to a kernel K ∈ C-Z in the sense that for n any f, g ∈ C∞ 0 (R ) with disjoint supports, ˆ ˆ T f, g = K(x, y)f(x)g(y) dy dx. Rn

Rn

If T is a Calderon-Zygmund singular integral operator we often write T ∈ CZO. n Note that T f, g is always well defined, for f, g ∈ C∞ 0 (R ), via the duality pairing. / supp f then Moreover, if f ∈ C∞ 0 with x ∈ ˆ T f(x) = K(x, y)f(y) dy Rn

is well defined as an absolutely convergent integral. The question of the Lp boundedness of Calderon-Zygmund operators can be reduced to that of L2 boundedness by the following theorem. Theorem 1.18. [10] Suppose T ∈ CZO. If T : L2 → L2 then T : L1 → L1,∞ and Lp → Lp for p ∈ (1, ∞). Remark 1.19. The original paper of Calderon and Zygmund was stated only for convolution operators (K(x, y) = K(x − y)) but the proof works for all CZO’s; see, e.g., [25, pp. 18-23].

2. Generalized Littlewood-Paley Theory We begin by recalling basic facts about approximate identities. We use the following notation. For a measurable φ defined on Rn , we set φt (x) := t−n φ(x/t) . Recall that the Hardy-Littlewood maximal function of a locally integrable function f is defined as Mf(x) := sup t>0

|x−y|0

We leave the proof as an exercise. It is elementary, if one does not mind1 introducing a purely dimensional constant on the right hand side of (2.2): just split the integral defining φt ∗ f into a sum of integrals over dyadic annuli of radius ≈ 2k t, and use that φ is decreasing. To obtain the sharp bound as stated is a bit more subtle; see [24, Chapter III] for the details. Lemma 2.3. Let φ ∈ L1 (Rn ) be radial, nonnegative and radially decreasing, such that φ(0) < ∞. Then for any f ∈ Lp (Rn ), 1  p  ∞, and any continuous g with the property |g|  φ we have

 N∗ (gt ∗ f)(x) := sup |gt ∗ f(y)|  C φ(0) + φL1 (Rn ) Mf(x). (y,t): |x−y| 0 such that |x − y| < t. Define the sets A0 := {z ∈ Rn : |x − z|  2t} , We now write



ˆ

|gt ∗ f(y)| 

t−n φ A0

y−z t

A1 := {z ∈ Rn : |x − z| > 2t} .





ˆ |f(z)| dz +

t−n φ A1

y−z t

 |f(z)| dz .

We can estimate the integral over A0 directly by noting φ(s)  φ(0), so that   ˆ |y − z| −n |f(z)| dz  Cφ(0) t φ |h(z)| dz  Cφ(0)Mh(x). r A0 B(x,2t) Note that for z ∈ A1 , we have 2|x − y| < 2t < |x − z|, so by the triangle inequality, 1 |y − z|  |x − z| − |x − y|  |x − z|. 2 Therefore, since φ is radially decreasing,     ˆ ˆ |y − z| |x − z| −n −n |f(z)| dz  |f(z)| dz t φ t φ t 2t A1 Rn  2n φL1 (Rn ) Mf(x), where in the last step we have used Lemma 2.1. 1 Indeed,

we do not mind.



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In the sequel, for an L1 function f, we let ˆ  := e2πiξ·x f(x) dx f(ξ) Rn

denote the Fourier Transform of f; as is well known, by Plancherel’s Theorem, the Fourier transform extend by continuity to L2 , where it is a unitary operator. We let S denote the usual Schwartz class of smooth, rapidly decreasing functions. ´ Proposition 2.4. If ζ ∈ C∞ 0 (B(0, 1)) is real, radial and nontrivial with Rn ζ(x) dx = 0,  = 0, and ζ ∈ S \ {0}. We normalize ζ so that then that ζ(0) ˆ ∞  2 dr = 1 |ζ(r)| r 0  (here we have mildly abused  notation, using that ζ is radial). Set Qt f(x) = ζt ∗ f(x) |z| −n where ζt (z) = t ζ t . We then have the Calderon reproducing formula, that is, ˆ ∞ dt =I where Q2t := Qt ◦ Qt (2.5) Q2t t 0 in the strong operator topology on B(L2 ) (the space of bounded operators on L2 . ´ 1/ 2 n 2 n Proof. We wish to show that lim→0  Q2t f dt t → f in L (R ) for all f ∈ L (R ). 2 n Let f ∈ L (R ), then by Plancherel   ˆ ˆ   1/  1/ dt       2 dt 2  − f =  − f(ξ) Qt f |ζ(t|ξ|)| f(ξ)        t t 2   2 ˆ   |ξ|/   2 dt − 1  = f(ξ) |ζ(t)|   → 0 as  → 0.   t |ξ| 2

We now come to the definition of the so-called Littlewood-Paley g-function, also known as a Vertical Square Function: 1/2 ˆ ∞ 2 dt g(f)(x) = gζ (f)(x) = |Qt f(x)| , t 0 where as before Qt f := ζt ∗ f. Next, we relax the conditions on ζ, requiring only that it be radial and inte grable, and that |ζ(ξ)|  min(|ξ|α , |ξ|−α ) for some α > 0. In particular we get ˆ ∞  2 dt < ∞. μ(ζ) := |ζ(t)| t 0 Proposition 2.6. For ζt as above we have that g : L2 (Rn ) → L2 (Rn ), i.e. ˆ ˆ ˆ ∞ dt dx = μ(ζ) |Qt f(x)|2 |f(x)|2 dx. t Rn 0 Rn Proof. Again this is a simple application of Plancherel’s theorem: ˆ ∞ˆ ˆ ∞ˆ dt   2 dξ dt = |(Qt f)(ξ)|2 dξ |ζ(tξ) f(ξ)| t t 0 0 Rn Rn ˆ ˆ ∞ dt 2  dξ = |f(ξ)|2 |ζ(t|ξ|)| t 0 Rn = μ(ζ) f2L2 (Rn ) .



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Definition 2.7 (Littlewood Paley Kernels and Operators). Consider the family of functions {ψt (x, y)}t , where t > 0 and ψt : Rn × Rn → C. We say this family satisfies the Littlewood-Paley conditions, abbreviated ψt ∈ L-P, if there exist constants C > 0 and α ∈ (0, 1] such that tα (2.8) |ψt (x, y)|  C (t + |x − y|)n+α |h|α , when |h|  t. (t + |x − y|)n+α We refer to (2.8) as the size condition and (2.9) as the Hölder continuity (or simply “smoothness") condition. (2.9)

|ψt (x, y + h) − ψt (x, y)|  C

Associated to a family ψt ∈ satisfying the Littlewood-Paley conditions, we define the operator Θt as follows: ˆ ψt (x, y)f(y) dy. Θt f(x) = Rn

Notice that if we define ψt (x, y) = ζt (x − y), then the Qt operator defined previously is a special case. The next theorem can then be thought of as a generalization of the previous proposition. Theorem 2.10. Suppose that ψt ∈ L-P and the operator Θt f(x) is defined as above. Assume in addition that Θt 1 = 0, i.e. ˆ ψt (x, y) dy = 0, for all x ∈ Rn , t > 0. Rn

Then

ˆ

∞ˆ Rn

0

dxdt  Cψ,n f2L2 (Rn ) . t

|Θt f(x)|2

In other words, defining

ˆ gΘ (f)(x) :=

0

∞

dt |Θt f(x)| t 2

1/2 .

gives a map gΘ : L2 → L2 . Proof. First note that by the size condition for ψt ∈ L-P ˆ tα f(y) dy |Θt f(x)|  n+α Rn (t + |x − y|) ˆ 1 = t−n

n+α f(y) dy. Rn 1 + |x−y| t The RHS is convolution of f and an integrable, radially decreasing function, so the Lemma 2.1 gives that sup |Θt f|  CMf(x). t>0

In particular, by the boundedness of the Hardy-Littlewood maximal function in L2 (Rn ), we get that the operators Θt are linear and uniformly bounded in L2 . Let Qs f := ζs ∗ f be a family of operators satisfying the hypotheses of Proposition 2.4.

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Using Calderón’s reproducing formula we then have ˆ ∞ˆ dxdt I:= |Θt f(x)|2 n t 0 R  2 ˆ ∞ ˆ ˆ ∞ dxdt 2 ds Θt (x) Qs f = (2.11) s t 0 0 Rn 2 ˆ ∞ ˆ ˆ ∞ ds dxdt , = Θt Q2s f(x) n s t 0 0 R where we have used the L2 boundedness of the Θt in the last equality. For any γ > 0 and s, t > 0 we can write

    s t γ/2 s t γ/2 2 2 , , max Θt Qs f(x) . Θt Qs f(x) = min t s t s Now substitute this expression into the right hand side of (2.11) and apply the Cauchy-Schwarz inequality in the innermost integral to get    ˆ ∞ ˆ ˆ ∞ s t γ ds , × min I t s s 0 0 Rn ˆ ∞ ˆ ˆ ∞ γ   dxdt s t 2 2 ds (2.12) , max |Θt Qs f(x)| t s s t 0 Rn 0   ˆ ∞ˆ ˆ ∞ dxdt ds s t γ , , max |Θt Q2s f(x)|2  Cγ t s s t 0 Rn 0 where the last inequality follows since, for any t > 0,   ˆ t γ ˆ ∞  γ ˆ ∞ s t γ ds t s ds ds , = + = C(γ) < ∞. min (2.13) t s s t s s s 0 0 t Next, we make the following Claim 2.14. There exist constants C, β > 0 such that for any h ∈ L2 (Rn ), all s, t > 0 and a.e. x ∈ Rn we have   s t β 2 , M h(x) , (2.15) |Θt Qs h(x)|  C min t s where M2 := M ◦ M denotes the composition of the Hardy-Littlewood maximal operator with itself. Assume the claim for the moment. Integrating in space, using the boundedness in L2 (Rn ) of M and taking the supremum over hL2 (Rn ) = 1 we arrive at   s t 2β 2 , . (2.16) Θt Qs L2 (Rn )→L2 (Rn )  C min t s Let us note that the quasi-orthogonality condition (2.16) is really the heart of the matter here, and that the claimed estimate (2.15) is simply a vehicle to obtain this bound: see Remark 2.20 below. With this we can complete the proof of the theorem as follows: Set γ = β, so that by (2.12), Fubini’s theorem, and then (2.16) and (2.13) (with the roles of s and

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t reversed), we obtain ˆ ∞ ˆ IC 0

ˆ

 Cβ

∞ˆ

min 0

Rn

0



∞

s t , t s

β

|Qs f(x)|2 dx

dt t

ˆ

 ds |Qs f(x)|2 dx s Rn

ds  Cf2L2 (Rn ) , s

where we have used Proposition 2.6 for the last inequality. This gives the result. It remains to prove the claim. Proof of claim. We split this into two cases. Case 1 s  t. Write Qs f = ζs ∗ f, with ζ as in Proposition 2.4. Using Fubini’s theorem and that ζ has mean value zero, we write ˆ ˆ ψt (x, y) ζs (y − z)h(z)dz dy Θt Qs h(x) = Rn Rn ˆ  ˆ (2.17) (ψt (x, y) − ψt (x, z)) ζs (y − z)dy h(z)dz. = Rn

Rn

Notice since ζ is supported in B(0, 1) we integrate only where |y − z| < s. Hence, by the Hölder continuity of ψt , we get ˆ ˆ |y − z|α (ψ  C (x, y) − ψ (x, z))]ζ (y − z)dy |ζ (y − z)|dy t s n t n+α s R Rn (t + |x − z|)

s α tα C ζL1 (Rn ) . t (t + |x − z|)n+α Therefore going back to (2.17) we see that

s α ˆ tα |h(z)|dz |Θt Qs h(x)|  CζL1 (Rn ) n+α t Rn (t + |x − z|)

s α Mh(x),  CζL2 (Rn ) t where we have used Lemma 2.1 for the last step. This gives the claim in case s  t. Case 2 t  s. We exploit the Θt 1 = 0 condition to write  ˆ ˆ Θt Qs h(x) = ψt (x, y) (ζs (y − z) − ζs (x − z)) dy h(z)dz. Rn

Rn

Now we write the innermost integral as ˆ ψt (x, y) (ζs (y − z) − ζs (x − z)) dy + (2.18) I(x, z) + II(x, z) := |x−y|>(st)1/2

ˆ

|x−y|(st)1/2

Notice in particular that Θt Qs h(x) =

ψt (x, y) (ζs (y − z) − ζs (x − z)) dy.

ˆ Rn

(I(x, z) + II(x, z))h(z) dz.

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We first consider II. In the present scenario, |x − y|  (st)1/2  s, so by the support condition on ζ and the triangle inequality, |ζs (y − z) − ζs (x − z)| = |ζs (y − z) − ζs (x − z)|χ{|x−z|2s}

(2.19)

 Cs−n−1 |x − y|χ{|x−z|2s}  C(t/s)1/2 s−n χ{|x−z|2s} , where in the next-to-last step we have used that |∇ζs |  Cs−n−1 . Therefore, using the size condition on ψt , we have  1/2 ˆ tα t s−n χ{|x−z|2s} dy |II(x, z)|  C n+α s Rn (t + |x − y|)  1/2 t s−n χ{|x−z|2s} .  Cn,α s Multiplying by h and integrating we get  1/2 ˆ t |II(x, z)| |h(z)| dz  C |h(z)| dz n s R |x−z|ρ}  C (t + |x − y|)n+α {|x−y|>ρ} ρα tα C α χ ρ |x − y|n+α {|x−y|>ρ}   α/2  t x−y −n =C ρ ϕ s ρ  α/2 t ϕρ (x − y), =C s where ϕ(w) = |w|−(n+α) χ|w|>1 . In particular ϕ has a radially decreasing L1 majorant so that Lemma 2.1 applies. Thus, since |ζs (z)|  Cs−n χ{|z| 0. Remark 2.21. Again for later use, we further note that when ζs , the kernel of the operator Qs , has mean value 0 and is compactly supported in a ball of radius s, and ψt (x, y) satisfies the L-P size condition, then the case s  t of the quasiorthogonality condition (2.16) remains valid when the pointwise L-P smoothness condition is replaced by the following weaker integral condition: there exists β > 0, C < ∞, such that for all x ∈ Rn and all 0 < s  t < ∞, ˆ

s 2β sup |ψt (x, z) − ψt (x, y)| dy  C . t Rn {z;|z−y|s} The fact that the latter condition yields (2.16) is left as an exercise (hint: this is essentially a continuous parameter version of Schur’s test). We conclude this section by reviewing some standard facts concerning BMO and Carleson measures. Recall that BMO is defined as the space of functions b ∈ L1loc modulo constants such that the BMO norm b(x) − [b]Q dx b∗ := sup Q

ffl

Q

is finite, where [b]Q := Q b. One could just as well work with balls as with cubes. By the John-Nirenberg Lemma,

1/p p b(x) − [b]Q dx , 1  p < ∞. b∗ ≈p,n sup Q

Q

Lemma 2.22 (Fefferman-Stein [13]). Suppose b ∈BMO, Θt f = with ψt ∈ L-P, and Θt 1 = 0. Then

´ Rn

ψt (x, y)f(y) dy

dxdt t is a Carleson measure, i.e. there exists a C0 = C0 (n, L-P) such that for every cube Q ⊂ Rn ˆ (Q) ˆ dxdt  C0 b2∗ |Q|, (2.23) μ(RQ ) = |Θt b(x)|2 t 0 Q dμ := |Θt b(x)|2

where RQ := Q × (Q). Sketch of proof. Since Θt 1 = 0, we may assume without loss of generality that b has mean value 0 on 2Q (the concentric double of Q). Write b1 := bχ2Q , b2 := b − b1 . The contribution of b1 is handled by Theorem 2.10 and the JohnNirenberg Lemma for BMO. The contribution of b2 may be handled by the point t α b∗ for x ∈ Q, which in turn follows from wise estimate |Θt b2 (x)|  C (Q)

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the L-P size condition for ψt (with exponent α), plus a telescoping argument. The details are left as an exercise.  Definition 2.24. The infimum of the constants C0 that satisfy (2.23) is called the Carleson norm of μ, and is denoted by μC . Lemma 2.25 (Carleson embedding). Let μ be a Carleson measure. Then for every 0 < p < ∞ and for every function F continuous in Rn+1 + , ˆ ∞ˆ ˆ |F(y, t)|p dμ(y, t)  Cn μC N∗ F(x)p dx. 0

Rn

Rn

We refer to [25, Chapter 2, Section 2] for the proof.

3. T 1 Theorems We now begin our discussion of L2 boundedness criteria for square functions and Calderon-Zygmund operators, in which the question of L2 boundedness of an operator T is reduced to understanding the action of T on some small set of testing functions. The first result of this type for such operators was the David-Journé T 1 Theorem for CZO’s [11]2 , in which there is only one testing function, namely the constant function 1. Christ and Journé later gave an alternative proof of the T 1 Theorem, by first establishing an analogous criterion for square functions [7]. We follow the latter approach here. It should be noted that some of the ideas in [7] were already implicit in the work of Coifman and Meyer [9]. We begin with the T 1 Theorem for square functions. Theorem 3.1 (T 1 Theorem for Square Functions, [7]). Suppose that ψ ∈ L-P with associated Littlewood Paley operator ˆ ψt (x, y)f(y) dy, t > 0. Θt f(x) := Rn

Suppose further that dμ(x, t) := |Θt 1(x)|2

dx dt t

is a Carleson measure, i.e., that ˆ (Q)ˆ 1 dx dt =: μC < ∞. |Θt 1(x)|2 sup t Q Q |Q| 0 Then

ˆ

∞ˆ

dx dt  Cf2L2 (Rn ) t 0 Rn where C depends on dimension, the Littlewood-Paley constants for ψ, and μC . (3.2)

|Θt f(x)|2

Remark 3.3. The converse is also true, as may readily be seen by a slight modification of the proof of Lemma 2.22. 2 Although

there were earlier results of E. Sawyer, of a similar nature, concerning weighted estimates for fractional integrals.

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T 1 and T b theorems and applications

Proof. By Theorem 2.10 we know this holds if Θt 1(x) = 0. Let Pt f(x) = ϕt ∗ f(x) ´ where ϕ ∈ C∞ 0 (B(0, 1)) is positive, radial and Rn ϕ = 1; henceforth, we refer to this as the standard mollifier3 . Using a trick from the argument of [9], we write (3.4)

Θt f(x) = (Θt − (Θt 1(x))Pt )f(x) + Θt 1(x)Pt f(x) =: Rt f(x) + (Θt 1(x))Pt f(x).

Note that Pt 1(x) = 1, so that Rt 1(x) = 0. Also, by the L-P size condition on ψt , we have supt>0 Θt 1∞  Cn,α , so that the kernel of Rt given by  t (x, y) = ψ(x, t) − Θt 1(x)ϕt (x − y) ψ is in the Littlewood-Paley class. We then have that Rt is a Littlewood-Paley operator with Rt 1(x) = 0 and hence by Theorem 2.10 ˆ ∞ˆ dx dt  Cf2L2 (Rn ) . |Rt f(x)|2 (3.5) t 0 Rn On the other hand, by Carleson’s Lemma, we have ˆ ∞ˆ ˆ ∞ˆ dx dt dx dt |Θt 1(x)Pt f(x)|2 |Θt 1(x)|2 |Pt f(x)|2 = t t Rn Rn 0 0 2 (3.6)  CN∗ (Pt f)) 2 n L (R )



CMf2L2 (Rn )

 Cf2L2 (Rn )

where in the second to last inequality we used that N∗ (Pt f(·))(x)  CMf(x) (see  Lemma 2.3). Then (3.2) follows from (3.4), (3.5) and (3.6). To discuss the T 1 Theorem for SIOs, we need the weak boundedness property. Definition 3.7 (Weak Boundedness Property (WBP)). We say an operator T : D → D  satisfies the weak boundedness property, abbreviated WBP, if for all R > 0, x0 ∈ Rn and f, g ∈ C∞ 0 (B(x0 , R)) |T f, g|  CRn (f∞ + R∇f∞ )(g∞ + R∇g∞ ). Proposition 3.8 (Antisymmetry and WBP). Suppose K ∈ CZ is antisymmetric, that is, K(x, y) = −K(y, x), then for Lipschitz f with compact support, ˆ K(x, y)f(y) dy T f(x) := lim T f(x) := lim →0 |x−y|>

→0

exists in the sense of distributions and T satisfies the WBP.   Proof. Let f and g be Lipschitz, with each supported in B(x0 , R) . Then ˆ ˆ T f, g = K(x, y)f(y)g(x) dy dx Rn

ˆ

=−

|x−y|>

Rn

ˆ =− 3 The

Rn

ˆ ˆ

|x−y|>

|x−y|>

K(y, x)f(y)g(x) dy dx K(x, y)f(x)g(y) dx dy.

precise choice of kernel ϕ is not important; we simply choose one with the stated properties.

Simon Bortz, Steve Hofmann, and José Luis Luna

Thus, T f, g =

1 2

=

1 2

(3.9)

169

¨ |x−y|>

K(x, y)(f(y)g(x) − f(x)g(y) dy dx

¨

|x−y|>

K(x, y)([f(y) − f(x)]g(x) − f(x)[g(y) − g(x)]) dy dx

By the C-Z size condition on K, this leads to |T f, g|

ˆ

 (∇f∞ g∞ + ∇g∞ f∞ )

ˆ |x−x0 | 1, where we have abused notation using that ζ is radial. I.e., ϕ is supported in the unit ball.  Returning to the proof of (1) =⇒ (2), we observe that since ˆ ∞ ds Pt := Q2s , s t it follows that Q2 (3.19) −∂t Pt = t . t

172

T 1 and T b theorems and applications

∗ ∞ Also for any f ∈ C∞ 0 , P f → f in D, Pt = Pt . Thus for f, g ∈ C0

T f, g = lim T P f, P g →0

ˆ

1/

= − lim

∂

Pt T Pt f, g dt

∂t →0  ˆ 1/ !

(3.20)

= − lim

→0 

  " " !  ∂ ∂ Pt T Pt f, g + Pt T Pt f, g ∂t ∂t

= lim [I (f, g) + II (f, g)], →0

where we have used the WBP to obtain the the fact that, as R → ∞, |T PR f, PR g|  Rn (PR f∞ + R∇PR f∞ )(PR g∞ + R∇PR g∞ )  R−n f1 g1 → 0. We then observe that by (3.19) ˆ 1/ ˆ 1/ dt dt = −I (f, g) = Q2t T Pt f, g Qt T Pt f, Qt g t t   and ˆ 1/ dt Qt f, Qt T ∗ Pt g −II (f, g) = t  where we used that Q∗t = Qt . By the symmetry of the conditions on T and T ∗ , it is enough to prove that |I |  Cf2 g2

(3.21)

uniformly in , since |II (f, g)| may be handled analogously. Set Θt := Qt T Pt . Then by Cauchy-Schwarz ˆ ∞ˆ dt |Qt T Pt f||Qt g| dx |I (f, g)|  t 0 Rn  1 ˆ ∞ ˆ 1 ˆ ∞ ˆ 2 2 2 dx dt 2 dx dt  |Θt f(x)| |Qt g(x)| t t Rn Rn 0 0 =: A · B. By the classical Littlewood-Paley Theorem (Theorem 2.10) we have B  Cg2 . Thus it remains to show that (3.22)

A  Cf2 .

To this end, we need only verify that Θt satisfies the hypothesis of Theorem 3.1. First we show that the kernel of Θt , ψt , satisfies the Littlewood-Paley conditions. We begin by verifying the size condition, that is, tα (3.23) |ψt (x, y)|  C . (t + |x − y|)n+α

Simon Bortz, Steve Hofmann, and José Luis Luna

Recall that Qt has kernel ζt = t−n ζ( xt ) where ζ ∈ C∞ 0 (B(0, 1) and Thus,

173

´ Rn

ζ(x) dx = 0.

ψt (x, y) = ζt (x − ·), T ϕt (· − y).

(3.24)

We break the proof of (3.23) into two cases. Case 1: |x − y|  100t. In this case we may write the pairing in (3.24) as an ´ absolutely convergent integral. Using the fact that Rn ζ(x) dx = 0 and that ϕ and ζ are supported in B(0, 1) we have ˆ ˆ ζt (x − z)K(z, v)ϕt (v − y) dz dv ψt (x, y) = n n ˆR R ˆ = ϕt (v − y) ζt (x − z)[K(z, v) − K(x, v)] dz dv . |v−y| 0}.

+ Set d := n − 1 so that H+ = Rd+1 and H− = Rd+1 + − . Let f be supported in H . By the size condition on K(x, y), with d = n − 1 it suffices to prove that ˆ p ˆ ˆ |f(y)| |f(x)|p dx . + |x − y|d+1 dy dx  H− H

The proof of the latter fact is left as an exercise (hint: parametrize H± according to (3.40), and use boundedness of the Hardy-type operator ˆ ∞ g→ (t + s)−1 g(s)ds 0

on Lp (0, ∞) (see, e.g., [24, Appendix A.3, p. 271]), and of the d-dimensional  Hardy-Littlewood maximal operator on Lp (Rd )). With the claim in hand we have that for any cube Q and ηQ ∈ C∞ 0 (5Q), with 0  ηQ  1 and ηQ = 1 on 4Q,  1 2 |T (ηQ − 1Q )| dx  |T (ηQ − 1Q )|2 dx Q

Q − 21

 C|Q|

− 21

 C|Q|

ˆ

1 2

2

5Q\Q

|ηQ − 1Q | dx

1

|5Q| 2 = C

where we used Claim 3.39 in the second inequality. This gives uniform control  on (3.37). Of course the same analysis gives uniform control on (3.38).

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4. A T b Theorem for Square Functions and the Cauchy Integral Definition 4.1. A function b ∈ L∞ (Rn ) is accretive if there exists δ > 0 such that Re(b(x))  δ,

for all x ∈ Rn .

Definition 4.2. A function b ∈ L∞ (Rn ) is called pseduo-accretive, sometimes written Ψ-accretive, if there exists a δ > 0 and an approximate identity Pt (i.e. ´ ϕ = 1) such that Pt f = ϕt ∗ f with ϕ ∈ C∞ 0 (B(0, 1)), radial, Rn

(4.3)

|Pt b(x)|  δ,

for all x ∈ Rn , t > 0.

One may also consider the situation that (4.3) holds with Pt replaced by At , the dyadic averaging operator defined below in (5.14). In this case, we say that b is dyadic Ψ-accretive. Note that accretivity implies Ψ-accretivity (and also dyadic Ψ-accretivity), since |Pt b|  Re Pt b = Pt Re(b) (and similarly for At b). Moreover, both of these conditions are strictly stronger than |b(x)|  δ (e.g., consider a complex exponential function). Theorem 4.4 (T b theorem for square functions, Semmes [22]). Suppose that for ´ ψt (x, y)f(y) dy, . Suppose also that there exists some {ψt }t ∈L-P, we have Θt f(x) = Rn a Ψ-accretive b such that dxdt dμ := |Θt b(x)|2 t is a Carleson measure with norm μC . Then ˆ ∞ˆ dxdt  Cf2L2 (Rn ) , |Θt f(x)|2 t 0 Rn where C depends on n, L-P, μC and δ the Ψ-accretive constant for b. Proof. By the Christ-Journé T1 theorem for square functions (Theorem 3.1) it is enough to show that dxdt |Θt 1(x)|2 t is a Carleson measure with norm depending on the allowable parameters, i.e. for every cube Q ⊂ Rn , ˆ (Q)ˆ dxdt 1  C = C(n, L-P, μC , δ). |Θt 1(x)|2 |Q| 0 t Q Since b is Ψ-accretive we have that |Θt 1(x)|  δ−1 |Θt 1(x)Pt b(x)|. We now use the trick of [9] to write   Θt 1(x)Pt b(x) = Θt 1(x)Pt − Θt b(x) + Θt b(x) =: Rt b(x) + Θt b(x). Therefore it’s enough to prove that dxdt , |Rt b(x)|2 t

|Θt b(x)|2

dxdt t

Simon Bortz, Steve Hofmann, and José Luis Luna

179

are Carleson measures. Of course, the second term is a Carleson measure by hypothesis. For the first term, notice that Rt 1(x) = Θt 1(x)Pt 1(x) − Θt 1(x) = 0. As in the proof of Theorem 3.1, the kernel of Rt is in L-P, so by the Fefferman-Stein  lemma (Lemma 2.22), |Rt b(x)|2 dxdt/t is a Carleson measure. Remark. There is also a T b theorem for singular integrals, due to David, Journé and Semmes [12] (a more restrictive, and slightly earlier, version is due to McIntosh and Meyer). Time constraints prevent our presenting this theorem in these lectures, but we note in passing that the idea of its proof is to build a LittlewoodPaley theory adapted to an accretive (or Ψ-accretive) function b, and then following the strategy of the proof of the T 1 theorem. In practice, this is a rather technically delicate business (not as simple as we are making it sound). Let us now proceed to give an important application of the T b theorem for square functions. Recall that we have established the L2 boundedness of the Cauchy integral on Lipschitz graphs with sufficiently small constant. We now remove the smallness condition, using Theorem 4.4. Theorem 4.5 (Coifman-McIntosh-Meyer Theorem [8]). There exist positive constants C and k such that for all A : R → R Lipschitz CA fL2 (R)  C(1 + A  ∞ )k fL2 (R) . The proof we give is not the original one of Coifman, McIntosh and Meyer: theirs was a hands-on argument which preceeded the advent of T 1/T b technology. Instead, we follow the approach of Semmes [22], in which matters are reduced to a certain square function estimate, which in turn is obtained via Theorem 4.4. These square function bounds themselves were originally obtained by C. Kenig in his Ph.D. thesis, again by hands-on methods which preceded the development of T 1/T b techniques. Proof of Theorem 4.5. Let A : R → R and set γ := {x + iA(x) : x ∈ R}. Suppose that Ω := {x + is : s > A(x)} and define for z ∈ Ω ⎧ ´ f(w) −1 ⎨ 2πi γ z−w dw z ∈ Ω Cγ f(z) := ´ f(w) ⎩ −1 2πi p.v. γ z−w dw z ∈ ∂Ω. Now define Ωt := {x + is : s > A(x) + t} and γt := ∂Ωt := {x + i(A(x) + t) : x ∈ R} for t > 0. Let f ∈ Cc (γ) and for z ∈ Ω set F(z) := Cγ f(z). Then ˆ f(w) 1  (4.6) F (z) = dw , z ∈ Ω. 2πi γ (z − w)2 Also for t > 0 we have by the Cauchy integral formula, for z ∈ Ωt ˆ F  (w) 1 (4.7) F  (z) = dw 2πi γt (z − w)2

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T 1 and T b theorems and applications

(here we may use decay at infinity to approximate γt by closed curves surrounding bounded regions which converge to Ωt in the limit). Thus, for z ∈ Ωt  ˆ ˆ 1 f(v) 1 (4.8) F  (z) = dv dw. 2 (2πi)2 γt (z − w)2 γ (w − v) Fix t > 0 and the parameterization given by setting z = x + i(A(x) + 2t) ∈ Ωt , w = y˜ + i(A(y) ˜ + t) ∈ γt and v = y + i(A(y)) ∈ γ. Then   1 ∂ 2 F(x + i(A(x) + 2t)). (4.9) F  (z) = F  (x + i(A(x) + 2t)) = − 4 ∂t Set g(y) := (1 + iA  (y))f(y + iA(y)), then writing (4.8) parametrically we have t2 F  (z) = t2 F  (x + i(A(x) + 2t)) ˆ ˆ   ψt (x, y) ˜ 1 + iA  (y) ˜ ψt (y, ˜ y)g(y)dy dy˜ = R R   = Θt (1 + iA  )(Θt g) (x)

(4.10)

where Θt f(x) =

ˆ R

ψt (x, y)f(y) dy and ψt (x, y) =

t 1 . 2πi (x − y + i(A(x) − A(y) + t))2

Notice that ψt ∈ L-P. Set G(z) ≡ 1, then ˆ 1 t dw = G  (z) = 0. Θt (1 + A  )(x) = 2πi γ (z − w)2 Notice also that 1 + iA  (x) is accretive. Thus by the T b Theorem for square functions (Theorem 4.4), for any h ∈ L2 (R), ˆ ∞ˆ dx dt (4.11) |Θt h(x)|2  C(1 + A  ∞ )4 hL2 (R) , t 0 R where the dependence on 1 + A  ∞ may be gleaned by keeping careful track of the constants as they arise in the proof of Theorem 4.4; we omit the details. For future reference, we note that (4.11) holds also with Θt replaced by Θ−t . Now, let us note that for f ∈ Cc (γ),     Cγ f x + i(A(x) + 2t) + t ∂ Cγ f x + i(A(x) + 2t) f t−1 , (4.12) ∂t as t → ∞. Thus, for f ∈ Cc (γ) and  > 0, we therefore obtain     F(x + i(A(x) + 2)) = Cγ f x + i(A(x) + 2) =: C γ f x + iA(x) ˆ ∞   ∂ Cγ f x + i(A(x) + 2t) dt =− ∂t   2 ˆ ∞   ∂ (t − ) Cγ f x + i(A(x) + 2t) dt = ∂t  ˆ ∞   (t − )dt = −4 Θt (1 + iA  )(Θt g) (x) , t2  where in the last step we have used the identities in (4.9) and (4.10).

Simon Bortz, Steve Hofmann, and José Luis Luna

For h ∈ L2 , set C A h(x) :=

−1 2πi

ˆ R

181

h(y) dy. x − y + i(A(x) − A(y) + 2)

 Recall that g(y) := (1 + iA  (y))f(y + iA(y)), so C A g(x) = Cγ f(x + iA(x)). Hence,  C γ fL2 (γ) ≈ CA gL2 (R) , and fL2 (γ) ≈ gL2 (R) ,

(4.13)

where the implicit constants depend polynomially on 1 + A  ∞ . Now we exploit duality. Let h ∈ Cc (R), and let Θ∗t denote the transpose of Θt . Since Θ∗t = −Θ−t , and since (4.11) applies both to Θt and Θ−t , we have ˆ ∞ˆ   (t − ) dx dt ∗  |h, C g| = 4 (Θ h(x)) (1 + iA )(Θ g) (x) t t A t2  R  1 ˆ ˆ ∞ 1 ˆ ˆ ∞ dx dt 2 dx dt 2  C(1 + A  ∞ ) |Θ∗t h(x)|2 |Θt g(x)|2 t t R  R   C(1 + A  ∞ )5 hL2 (R) gL2 (R) , with constant C independent of  (using (4.12) in the first equality to justify the implicit use of Fubini’s Theorem). Since Cc (R) is dense in L2 (R), we obtain  5 supC A gL2 (R)  C(1 + A ∞ ) gL2 (R) ,

(4.14)

>0

and since (4.13) holds up to a power of 1 + A  ∞ , we also have that for some k,  k supC γ fL2 (γ)  C(1 + A ∞ ) f2 . >0

To prove the theorem, it is enough to show that for all f ∈ C∞ 0 (R), supCA, fL2 (R)  C(1 + A  ∞ )k f2 ,

(4.15)

>0

where CA,

−1 = 2πi

ˆ |x−y|>

1 f(y) dy x − y + i(A(x) − A(y))

Indeed, by antisymmetry of the kernel (x − y + i(A(x) − A(y)))−1 , and Proposition 3.8, CA, f → CA f, as  → 0, in the sense of distributions, so the desired bound for CA (and hence also for Cγ ) follows from (4.15). In turn, to prove (4.15), by (4.14), it suffices to show that, with μ := 1 + A  ∞  |CA, f(x) − Cμ A f(x)|  C(1 + A ∞ )Mf(x) .

(4.16)

Here, as above, M denotes the Hardy-Littlewood maximal operator. Then. |CA, f(x) − Cμ A f(x)|   ˆ f(y) f(y) − dy = x − y + i(A(x) − A(y)) x − y + i(A(x) − A(y) + 2μ) |x−y|>

+

ˆ

|x−y|

2μ |f(y)| dy  CμMf(x) , |x − y|2

by Lemma 2.1. To bound II, note that if |x − y| < , then |A(x) − A(y)|  A  ∞  < μ, which yields that |A(x) − A(y) + 2μ| > μ. Therefore, ˆ 2 1 |f(y)| dy  Mf(x)  2 Mf(x). II  μ |x−y| 1/2.

Define the dyadic averaging operator At by (5.14)

At f(x) =

Q

f(y) dy ,

where Q  is the unique (half-open) dyadic cube containing x, with side length at (Q  ) least t (i.e., 2 < t  (Q)). Observe that for (x, t) ∈ E∗Q , the cube Q  in (5.14) is not contained in any Qj ∈ F, thus by (5.13), Re At bQ (x) > 1/2 , It follows that¨ (5.15)

E∗Q

|Θt 1(x)|2

dx dt 4 t

∀ (x, t) ∈ E∗Q .

¨ |Θt 1(x)At bQ (x)|2 RQ

dx dt . t

It is then enough to show that the right hand side of the inequality above is bounded by a constant times the measure of |Q|. Using a now familar trick, we write (5.16) Θt 1(x)At bQ (x)      = Θt 1(x) At − Pt bQ + (Θt 1(x))Pt − Θt bQ + Θt bQ =: Rt bQ + R˜ t bQ + Θt bQ where Pt is a standard mollifier. The contribution of Θt bQ in (5.15) is bounded t is exactly the same by a constant times |Q| by hypothesis (5.8). Note also that R t 1(x) = 0, and its as the term denoted Rt in the proof of Theorem 4.4; as above, R kernel is in L-P, thus by Theorem 2.10 we obtain ¨ ¨ 2 dx dt  t bQ |2 dx dt  |Rt bQ | |R n+1 t t RQ R (5.17)  CbQ 2L2 (Rn )  CC0 |Q| where we used (5.6) in the last inequality. It remains to deal with Rt . Since ψt ∈ ´ L-P we have |Θt 1(x)|  Rn |ψt (x, y)| dy  C, uniformly in x and t. Consequently, ¨ ¨ dx dt dx dt C . |Rt bQ |2 |(At − Pt )bQ |2 (5.18) n+1 t t RQ R+ Claim 5.19. For all f ∈ L2 (Rn ) ¨ dx dt  Cf2L2 (Rn ) . |(At − Pt )f|2 n+1 t R+ Momentarily assuming the claim, taking f = bQ , we obtain the same bound as in (5.17). Thus, (5.11) holds, and we conclude the proof of the theorem. It therefore remains only to verify the claim.

Simon Bortz, Steve Hofmann, and José Luis Luna

187

Proof of Claim 5.19. As in the proof of the Theorem 2.10 (see also Remark 2.20), it is enough to show that   s t α , (At − Pt )Qs L2 (Rn )→L2 (Rn )  C min t s for some α > 0, where Qs is as usual: Qs f = ζs ∗ f, ζs = s−n ζ( s· ), where ´ ´∞ 2 ζ ∈ C∞ 0 (B(0, 1)), Rn ζ(x) dx = 0 and 0 Qs = I in the strong operator topology. Case 1: t  s. The proof is left as an exercise (but note that (At − Pt )1 ≡ 0; see also the proof of Claim 2.14, although the present situation is actually somewhat simpler, owing to the compact support of the kernels of At and Pt ). Case 2: t > s. In this case we see that (At − Pt )Qs L2 (Rn )→L( Rn )  At Qs L2 (Rn )→L( Rn ) + Pt Qs L2 (Rn )→L( Rn ) = I + II. The bound II  s/t is left as an exercise; again see the proof of Claim 2.14. By Remark 2.21, the bound I  (s/t)1/2 follows from the estimate ˆ s sup |χt (x, z) − χt (x, y)| dy  C , (5.20) n t R {z;|z−y|s} where χt (x, y) is the kernel of the operator At , i.e., 1 1 (y) , χt (x, y) := |Q(x, t)| Q(x,t) where Q(x, t) is the unique half-open dyadic cube containing x with side length at least t. To verify (5.20), we simply observe that if |z − y|  s  t, then we can have χt (x, z) − χt (x, y) non-zero only when dist(y, ∂Q(x, t))  s (here ∂Q(x, t) denotes the boundary of Q(x, t)). Therefore ˆ sup |χt (x, z) − χt (x, y)| dy Rn {z;|z−y|s}

ˆ 

dist(y,∂Q(x,t))s

stn−1 s 1 dy  ≈ . |Q(x, t)| |Q(x, t)| t

This proves the claim, and hence also the theorem.



Section Notes. The first local T b theorem, for singular integral operators, was proved by M. Christ [6], and was related to problems concerning analytic capacity (we will not touch on this subject here). At the heart of his argument was a stopping time construction similar to the one used above. In Christ’s theorem, the testing functions were assumed to be in L∞ . Although it is not immediately obvious, an L∞ testing condition is a stronger assumption than the L2 condition (5.6), since the latter may be localized to, say, 5Q (this is not hard to do). The theorem in the present section, treating square functions with L2 testing conditions, was implicit in the solution of the Kato square root problem [1, 15, 16]. In fact, it may be viewed as a “toy version" of the core of the proof of the Kato square root estimate.

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T 1 and T b theorems and applications

6. Application: The Kato square root problem The square root problem of T. Kato was discussed briefly in the Introduction (Example 1.11), but let us formulate the problem again with a few more details. We begin with a definition. Definition 6.1 (Divergence Form Elliptic Operator). We say a matrix valued function A : Rn → Mn×n is uniformly elliptic if there exist constants λ, Λ > 0 such that (a) A ∈ L∞ with ess supx∈Rn |A(x)ξ, ζ|  Λ|ξ||ζ| for all ξ, ζ ∈ Cn and (b) λ|ξ|2  ess infx∈Rn ReA(x)ξ, ξ for all ξ ∈ Cn where ·, · is the Hermitian inner product on Cn . The constants λ and Λ are called the ellipticity parameters of the matrix A. We say that L is a divergence form elliptic operator if L has the form   ∂  ∂ Ai,j (x) L = − div A(x)∇ = − ∂xj ∂xj i,j

We interpret L in the weak sense by way of a sesquilinear form. In other words, ˙ −1,2 = (W ˙ 1,2 )∗ via ˙ 1,2 we have Lu ∈ W for u, v ∈ W ˆ Lu, v = A∇u, ∇v = A(x)∇u(x) · ∇v(x) dx. Rn

˙ 1,2 = L˙ 1,2 (Rn , C) is the completion Recall that the homogeneous Sobolev space W 2 n of C∞ 0 with respect to the seminorm ∇fL (R ). The ellipticity (aka “accretivity") conditions (a) and (b) above ensure that the operator L = − div A∇, considered as an unbounded operator in the Hilbert space L2 (Rn ), has both its spectrum σ(L) and its numerical range {Lu, u; u, ∇u, Lu ∈ L2 } contained in a sector {ζ ∈ C; | arg ζ|  ω}, for some ω ∈ [0, π/2) (in fact, one λ π < 2 ). Such an operator, called ω-accretive, generates a may take ω := cos−1 Λ contraction semigroup {e−tL }t>0 , and has unique αω-accretive fractional powers Lα when 0 < α < 1. In particular, L has a unique (ω/2)-accretive square root √ √ √ L = L1/2 satisfying L L = L. See [17, 18]. The goal of this section will be to prove the estimate √ (6.2)  LuL2 (Rn )  C∇uL2 (Rn ) , where L is a divergence form elliptic operator and C = C(n, λ, Λ). This is the Kato square root problem. As mentioned in the introduction, if A = 1, the n × n identity matrix, then L = −Δ and (6.2) follows directly from Plancherel’s Theorem. √ In this case, −Δ may be defined via the Fourier transform: for nice enough √ functions f (say f ∈ S), we define −Δf as the inverse Fourier transform of    We also mention that if L is self2π|ξ|f(ξ) (recall that (−Δf)(ξ) = 4π2 |ξ2 |f(ξ)). adjoint (in particular, if the coefficient matrix A is real and symmetric), then (6.2) √ follows trivially from the fact that in that case L is also self-adjoint, hence, √  Lu22 = Lu, u = A∇, ∇u ≈ ∇u22 .

Simon Bortz, Steve Hofmann, and José Luis Luna

189

The point of the problem, then, is to treat the non self-adjoint case. In particular, proving that (6.2) holds for complex elliptic matrices allows one to show that the √ mapping A → L is analytic, and this in turn has application to the initial value problem for time-dependent, variable coefficient wave equations [19]. We remark that the Kato problem in 1 dimension is equivalent to the L2 boundedness of the Cauchy integral on a Lipschitz curve (this is not necessarily obvious, it uses functional calculus); moreover, the 1-D Kato problem was first proved in [8]. In dimensions n > 1, the proof of (6.2) came some 20 years later [1, 15, 16]. In order to minimize some technical issues, we shall prove only a restricted version of (6.2), in which we assume impose an extra hypothesis on the heat kernel of L. The heat kernel, which we shall denote by Wt (x, y), is the kernel of the heat semigroup e−tL ; equivalently, Wt (x, y) = Γ (x, t, y, 0), where Γ is the fundamental solution of the parabolic operator ∂t + L. Remark 6.3. When L = −Δ, Wt (x, y) = Wt (x − y) is the classical Gauss-Weierstrass kernel. We shall assume that Wt (x, y) satisfies a Gaussian upper bound, and Nashtype Hölder continuity, which for future use we formulate after making the change of variable t → t2 :   |x − y|2 −n |Wt2 (x, y)|  Ct exp − Ct2  α   |h| |x − y|2 (6.4) |Wt2 (x, y + h) − Wt2 (x, y)|  Ct−n exp − t Ct2   α  |x − y|2 |h| |Wt2 (x + h, y) − Wt2 (x, y)|  Ct−n exp − , t Ct2 whenever |h|  t, where the positive constants C and α are uniform in x, y, t and h. It can be shown that (6.4) holds always when n = 1, 2, and when A is sufficiently close (depending on dimension and ellipticity), in L∞ norm, to a real, uniformly elliptic matrix. The proof of (6.2) in the presence of (6.4) already involves most of the essential ingredients of the proof in the general case, but the extra hypothesis will allow us to avoid certain technical difficulties that arise in the absence of pointwise heat kernel bounds. √ Remark 6.5. In addition to the precise construction of L itself, we shall also use without proof the following fundamental facts from functional calculus: (1) If (6.4) holds for Wt2 (x, y), then t2 LWt2 (x.y) = −(1/2)t∂t Wt2 (x, y) also satisfies (6.4), with the same exponent α, but possibly with a slightly different (but still uniform) constant C; (2) for g ∈ L2 (Rn ), ˆ ∞ˆ √ 2 dxdt ≈ g2L2 (Rn ) . |t Le−t L g(x)|2 t 0 Rn

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T 1 and T b theorems and applications

√ (3) L and L commute with the semigroup e−τL . (4) e−tL 1 ≡ 1, and therefore ∇x e−tL 1 ≡ 0 and ∂t e−tL 1 = Le−tL 1 = 0. √ (5) Lu is initially well-defined as an element of L2 for u ∈ D(L) (the domain of L), defined as D(L) := {f ∈ L2 (Rn ) : Lf ∈ L2 } . (6) D(L) is dense both in L2 (Rn ) and in the Sobolev space W 1,2 (Rn ). For future reference, let us note two immediate consequences of (6.4) and Remark 6.5 (1): there is a constant C = C(λ, Λ, n, (6.4)) such that (6.6)

2

2

e−t L fL2 (Rn ) + t2 Le−t L fL2 (Rn )  CfL2 (Rn ) ,

and, given two measurable sets E, F, and a function f supported in F, we have the “off-diagonal" decay property   2 2 dist(E, F)2 fL2 (F) . (6.7) e−t L fL2 (E) + t2 Le−t L fL2 (E)  C exp − Ct2 Again for future reference, we now deduce some easy consequences of (6.6) and (6.7). With C = C(λ, Λ, n, (6.4)) as above, (6.8)

2

t∇e−t L fL2 (Rn )  CfL2 (Rn ) . 2

Indeed, set v(x, t) := e−t L f(x), so by ellipticity, the definition of L, and then Cauchy-Schwarz and finally (6.6), 2

t∇e−t L f2L2 (Rn )  Re t2 A∇v, ∇v = Re t2 Lv, v 2

2

 t2 Le−t L fL2 (Rn ) e−t L fL2 (Rn )  f2L2 (Rn ) . Thus, (6.8) holds. Since the adjoint operator L∗ enjoys the same properties as does L, by duality we also have (6.9)

2L

te−t

div fL2 (Rn )  CfL2 (Rn ,Cn ) .

2 t∇e−t L

Next, we observe that also enjoys an “off-diagonal" decay property: for any measurable sets E and F, and for f supported in F,   2 dist(E, F)2 fL2 (F) . (6.10) t∇e−t L fL2 (E)  C exp − Ct2 To see this, note first that we may assume that dist(E, F) > t, otherwise, we may simply invoke the global L2 bound (6.8). Set R := dist(E, F), and let η ∈ C∞ 0 be a bump function adapted to an R/2 neighborhood of E, i.e., 0  η  1, η ≡ 1 on E, with |∇η|  1/R, and dist(supp η, F)  R/2. 2 Then by the definition of η and ellipticity, again setting v = e−t L f, we have

Simon Bortz, Steve Hofmann, and José Luis Luna

−t2 L

t∇e

ˆ

ˆ f2L2 (E)



191

2 2

Rn

2

|t∇v| η dx  Re t

ˆ

= Re t2 Lv, vη2  − Re t2 2

Rn

Rn

A∇v · ∇v η2 dx

A∇v · ∇(η2 )v dx

2

 t2 Le−t L fL2 (supp(η) e−t L fL2 (supp(η)) ˆ t2 +C |∇v| |v| η dx R Rn = I + II . The desired bound holds for term I, by (6.7), applied with E replaced by supp(η). By “Cauchy’s inequality with ε", ˆ ˆ 1 |t∇v|2 η2 dx + |v|2 dx =: II  + II  , II  ε n ε R supp(η) where ε is at our disposal, and we have used that t/R < 1. Choosing ε small enough, we may hide term II  ; with ε now fixed, term II  enjoys the desired bound, by (6.7). Thus, we obtain (6.10). Again by duality, interchanging the roles of L and L∗ , and of E and F, we find that   2 dist(E, F)2 (6.11) te−t L div fL2 (E)  C exp − fL2 (F,Cn ) . Ct2 Remark. Estimates (6.6) - (6.11) do not actually require the pointwise Gaussian bounds in (6.4) and Remark 6.5 (1): it can be shown that they hold for all L as in Definition 6.1. Estimates (6.7) and (6.10) are often referred to as Gaffney estimates (or Davies-Gaffney estimates). At this point, we present one more estimate for future reference, which we state as a lemma. Lemma 6.12. Suppose that St : L2 (Rn ) → L2 (Rn ) satisfies St L2 →L2  C ,

(6.13)

that St 1 = 0 for each t > 0, and that (6.14) whenever E, F ⊂ (6.15)

sup St fL2 (E)

t>0 Rn are

  dist(E, F)2  C exp − fL2 (F) , Ct2

measurable sets, with supp f ⊂ F. Then St hL2 (Rn )  Ct∇hL2 (Rn ) .

The proof of the lemma is left as an exercise. (Hint: write  St h2L2 (Rn ) = St h2L2 (Q) , Q

with the sum running over the dyadic cubes with (Q)/2 < t  (Q). Since ffl  ˜ St 1 = 0, we can replace h by h˜ := h − 2Q h, then write h˜ = ∞ k=0 h1Rk (Q) , where R0 (Q) := 2Q, and Rk (Q) := 2k+1 Q \ 2k Q, k = 1, 2, 3, ...; then use a telescoping argument, (6.13)-(6.14), and Poincare’s inequality to conclude).

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T 1 and T b theorems and applications

Remark 6.16. Note that by Remark 6.5 (4) and (6.8) and (6.10), we may apply the lemma to obtain (6.17)

2

t∇e−t L hL2 (Rn )  Ct∇hL2 (Rn ) .

We now formulate the precise result that we shall prove. Theorem 6.18. Let L = − div A∇ be a divergence form operator in Rn , as in Definition 6.1, with ellipticity parameters λ, Λ. Suppose that the heat kernel for L satisfies the Gaussian property (6.4). Then the estimate (6.2) holds, with C = C(n, λ, Λ, (6.4)) (i.e., with C depending on the “allowable parameters"). √ Remark. The reverse estimate ∇u2   Lu2 follows from the estimate (6.2) for the adjoint operator L∗ : √ √ 1 1 C √ ∇u22  Re A∇u, ∇u = Re  Lu, L∗ u   Lu2 ∇u2 . λ λ λ Proof of Theorem 6.18. We first observe that (6.2) is equivalent to the square function estimate: ¨ 2 dx dt  ∇u22 , |tLe−t L u(x)|2 (6.19) t Rn+1 +

√ because, by Remark 6.5 (2) and (3), with g = Lu, ˆ ∞ √ ¨ √ √ 2 dx dt  −t2 L  tLe t Le−t2 L ( Lu)2 dt ≈  Lu2 = u(x) 2 2 t t 0 Rn+1 + (by Remark 6.5 (5), this makes sense for u ∈ D(L), and by Remark 6.5 (6), it suffices to prove (6.2) for such u). Let us now turn to the proof of (6.19). The first main step is a sort of “T 1" reduction. By Remark 6.5 (3), we have 2

2L

tLe−t L u = −te−t

div A∇u = Θt ∇u ,

where 2L

Θt := −te−t

div A .

Using a now familiar trick, we write (1)

(2)

Θt = Θt (I − Pt ) + (Θt Pt − (Θt 1)Pt ) + (Θt 1)Pt =: Rt + Rt + (Θt 1)Pt where 1 denotes the n × n identity matrix, and Pt is a standard mollifier. Claim 6.20. For i = 1, 2, ¨

(i) R ∇u(x) 2 dx dt  C∇u2 , 2 t t Rn+1 + (i)

i.e., the desired bound holds for the contribution of the two remainder terms Rt , for i = 1, 2. We next want to sketch the proof of this claim, although we will leave some details as exercises.

Simon Bortz, Steve Hofmann, and José Luis Luna

193

We first record some consequences of the heat kernel bounds (6.4), namely that Θt 1 ∈ L∞ (Rn , Cn ) with a uniform bound 2

Θt 1∞ = tLe−t L φ∞  C < ∞

(6.21) (6.22)

−t2 L

(e

− I)f∞  C t ∇fL∞ (Rn )

∀ t > 0 ; and ∀ t > 0 , f Lipschitz.

Here, φ(x) ≡ x is the identity map on Rn , so that ∇φ = 1. We leave it as an exercise to check these facts (hint: (6.21) uses Remark 6.5 (1) and (4); (6.22) uses Remark 6.5 (4) and (6.4)). (i) Consequently, by (6.11) and the definition of Θt , each of Rt , i = 1, 2 satisfies the bound   dist(E, F)2 (i) (6.23) Rt fL2 (E)  C exp − fL2 (F,Cn ) , Ct2 (i)

where f is any Cn -valued function supported on F. In turn, since Rt 1 = 0, by Lemma 6.12, this yields (i)

Rt hL2 (Rn )  Ct∇hL2 (Rn ) . It then follows immediately that, for any “nice" Qs as in Proposition 2.4, t (i) (6.24) Rt Qs 2L2 →L2  C , t  s, s since ∇Qs L2 →L2  1/s. We also have s (i) s  t, (6.25) Rt Qs ∇u2L2 (Rn )  C ∇uL2 (Rn ) , t i.e., when s  t, we have quasi-orthogonality not on all of vector-valued L2 , but on the subspace consisting of L2 gradient fields. For i = 2 this is easy, and in fact in that case, quasi-orthogonality does hold on all of vector-valued L2 , since Pt Qs L2 →L2  s/t (as in the proof of (2.16)). For i = 1, one uses that 2

Θt ∇L2 →L2 = tLe−t L L2 →L2  1/t , by (6.6), and that Qs uL2 (Rn )  s∇uL2 (Rn ) , by Poincare’s inequality, since Qs 1 = 0. We leave the details as an exercise. Claim 6.20 now follows immediately from the quasi-orthogonality estimates (6.24) and (6.25), by familiar techniques. We now turn to the main term, (Θt 1)Pt . By Carleson’s Lemma (Lemma 2.25), ¨ (Θt 1(x))Pt (∇u)(x) 2 dx dt  CμC ∇u2 , 2 n+1 t R+ where dμ(x, t) := |Θt 1(x)|2 dxdt/t, and (in slight variance to earlier notation) μC denotes the dyadic Carleson norm ˆ (Q)ˆ 1 dx dt μC := sup |Θt 1(x)|2 |Q| t 0 Q Q dyadic (we recall that the dyadic Carleson norm controls the Carleson norm over arbitrary cubes). Thus, it remains to show that μC  C. We do this via a local T b

194

T 1 and T b theorems and applications

argument, in the spirit of the proof of Theorem 5.5, but more delicate, because Θt 1 is vector-valued (i.e, Cn -valued), not scalar-valued. For  small, but fixed, we cover Cn by cones of aperture . Enumerating these  ,, where K = K(, n), we see that cones as Γ1 , ..., ΓK ˆ (Q)ˆ K ˆ (Q)ˆ  dxdt dxdt  . |Θt 1|2 |Θt 1|2 1Γk (Θt 1) t t 0 0 Q Q k=1

Thus, it is enough to show that for  chosen small enough, depending only on the allowable parameters, there is a uniform constant C such that ˆ (Q)ˆ dxdt  C, |Θt 1|2 1Γ  (Θt 1) sup |Q|−1 t 0 Q Q for each fixed cone Γ  with  small enough. To this end, we fix a cone  n z (6.26) Γ = {z ∈ C : − ν < }. |z| where ν ∈ Cn is the unit vector in the direction of the central axis of Γ  . In the spirit of Theorem 5.5, we have the following. Lemma 6.27. There is a constant C0 < ∞, and for  > 0 small enough, a constant n n×n , indexed C < ∞ and a system of matrix valued functions bQ = b Q : R → M by the dyadic cubes, such that ˆ (6.28) |bQ (x)|2 dx  C0 |Q| ,  (6.29)

Rn

 bQ (x) dx ξ  |ξ|2 ,

e ξ ·

∀ ξ ∈ Cn ,

Q

ˆ

(Q)ˆ

(6.30) 0

|Θt bQ (x)|2

Q

dxdt  C |Q| . t

Here, the constant C0 is independent of . Eventually, we will take  to be the same parameter as in (6.26). We assume the lemma momentarily, and use it to prove the theorem. To this end, fixing Q, we follow the stopping time argument of Theorem 5.5, in the present case extracting dyadic subcubes Qj ⊂ Q which are maximal with respect to the property that at least one of the following holds:

1 3 or Re ν · (6.31) |bQ |  bQ ν  , 4 4 Qj Qj where ν ∈ Cn is the unit vector in the direction of the central axis of the cone Γ  in (6.26). As in the proof of Theorem 5.5, one may check (and we leave it as an exercise) that |EQ | := |Q \ (∪j Qj )|  η|Q|,

Simon Bortz, Steve Hofmann, and José Luis Luna

195

for some fixed η > 0 (hint: we use here that C0 is independent of ). Moreover, # for (x, t) ∈ E∗Q := RQ \ ( j RQj ), and for z ∈ Γ  , we claim that 1 |z|, 2 where again At denotes the dyadic averaging operator. Indeed, since the opposite inequalities to the two in (6.31) hold in E∗Q , we have that (6.32)

|z · At bQ (x)ν| 

1 3 1 − = 4 4 2 whenever |ω − ν| <  and (x, t) ∈ E∗Q . Taking ω = z/|z|, with z ∈ Γ  , we obtain the inequality in (6.32). Consequently, setting z = Θt 1(x) in (6.32), we obtain ¨ ¨ dxdt dxdt 2  4 , |Θt 1| 1Γ  (Θt 1) |Θt 1 · At bQ ν|2 ∗ ∗ t t EQ EQ ¨ dxdt  4 , |Θt 1 · At bQ |2 t RQ |ω · At bQ (x)ν|  |ν · At bQ (x)ν| − |(ω − ν) · At bQ (x)ν| 

since ν is a unit vector. The rest of the proof follows as in Theorem 5.5 (we leave the details as an exercise). It remains now only to give the proof of Lemma 6.27. Proof of Lemma 6.27. We define bQ : Rn → Mn×n for each dyadic cube Q ⊂ Rn by (6.33)

2 (Q)2 L

bQ (x) := 2 ∇e−

φQ =: ∇FQ ,

with  > 0 to be chosen, small enough, and φQ := ηQ (x)(x − xQ ) , where xQ denotes the centre of Q, and where ηQ ∈ C∞ 0 (5Q) with η ≡ 1 on Q, 0  ηQ  1, and |∇ηQ |  1/(Q). We now show that (6.28)-(6.30) are satisfied by the matrix valued functions bQ defined above, provided that  is chosen sufficiently small: (6.28): bQ 22  C0 |Q| (with C0 independent of ). Applying (6.17) with t = (Q), we have 2

2

bQ 2 = 2∇e− (Q) L φQ 2  ∇φQ 2  |Q|1/2 .

 ´ (6.29): Re ξ · |Q|−1 Q bQ (x) dx ξ  |ξ|2 , ∀ ξ ∈ Cn . By (6.22), applied with t = (Q), we have    −2 (Q)2 L  − I)φQ   C(Q)∇φQ ∞  C(Q) . (e ∞

Consequently, for each 1  j  n, −2 (Q)2 L ∂xj (e − I)φQ (x) dx  C , Q ´ where we have obtained the last inequality by writing Q as an iterated integral, and integrating first in xj .

196

References 2 (Q)2 L

Since 12 bQ − 1 = ∇(e− Consequently,  Re ξ ·



Q

− I)φQ , this implies  1 bQ − 1 (x) dx  C . 2 

  bQ (x) dx ξ = 2|ξ|2 + 2 Re ξ · Q

Q

  1 bQ − 1 (x) dx ξ 2

 2|ξ|2 (1 − C)  |ξ|2 , for  small enough. ´ (Q)´ 2 dxdt  C |Q|. Noting that (6.30): 0  Q |Θt bQ (x)| t 2

2 (Q)2 L

Θt bQ (x) = 2te−t L Le− we have

ˆ

(Q) ˆ 0

Q

|Θt bQ (x)|2

dxdt  4 t

ˆ 0

ˆ

∞

φQ ,

2 +2 (Q)2 )L

Le−(t

φQ 22 t dt

∞

t dt φQ 22 4 (t + (Q)) 0   1 C φQ 22  2 |Q| , C 2 (Q)2   where we have used (6.6) with t replaced by t2 + ((Q))2 in the second inequality. This concludes the proof of the lemma, and hence that of the theorem.   C

References [1] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and P. Tchamitchian, The solution of the Kato Square Root Problem for Second Order Elliptic operators on Rn , Annals of Math. 156 (2002), 633–654. MR1933726 187, 189 [2] A. P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. USA 53 (1965), 1092-1099. MR177312 157, 174 [3] A. P. Calderón, Algebras of singular integral operators, Proc. Sympos. Pure Math. 10 AMS, Providence, RI, 1967 18-55. MR0394309 158 [4] A. P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. USA 74 (1977), 1324-1327. MR466568 174, 175 [5] A. P. Calderón, Commutators, singular integrals on Lipschitz curves and applications. Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 85-96, Acad. Sci. Fennica, Helsinki, 1980. MR562599 174, 175 [6] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloquium Mathematicum LX/LXI (1990) 601-628. MR1096400 182, 187 [7] M. Christ and J. L. Journé, Polynomial growth estimates for multilinear singular integral operators. Acta Math. 159 (1987), no. 1-2, 51-80. 167 MR906525 [8] R. R. Coifman, A. McIntosh and Y. Meyer L’intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes lipschitziennes. (French) [The Cauchy integral defines a bounded operator on L2 for Lipschitz curves] Ann. of Math. 116 (1982), no. 2, 361–387. MR672839 179, 189

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[9] R. R. Coifman and Y. Meyer, Nonlinear harmonic analysis, operator theory and P.D.E. Beijing lectures in harmonic analysis (Beijing, 1984), 3-45, Ann. of Math. Stud., 112, Princeton Univ. Press, Princeton, NJ, 1986. MR864370 167, 168, 178 [10] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals. Acta Math. 88 (1952). 85-139. MR52553 159 [11] G. David and J. L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators. Ann. of Math. (2) 120 (1984), no. 2, 371-397. MR763911 167, 170 [12] G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions paraaccrétives et interpolation. (French) [Calderón-Zygmund operators, para-accretive functions and interpolation] Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56. 179 MR850408 [13] C. Fefferman, and E. M. Stein, Hp spaces of several variables. Acta Math. 129 (1972), no. 3-4, 137–193. MR447953 166, 170 [14] Y. S. Han and E. T. Sawyer, Para-accretive functions, the weak boundedness property and the T b theorem. Rev. Mat. Iberoamericana 6 (1990), no. 1-2, 17-41. MR1086149 174 [15] S. Hofmann, M. Lacey and A. Mc Intosh. The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds. Annals of Math. 156 (2002), pp 623-631. MR1933725 187, 189 [16] S. Hofmann and A. McIntosh, The solution of the Kato problem in two dimensions, Proceedings of the Conference on Harmonic Analysis and PDE held in El Escorial, Spain in July 2000, Publ. Mat. Vol. extra, 2002 pp. 143-160. MR1964818 187, 189 [17] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York 1966. MR0203473 188 [18] T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan 13 (1961), p. 246-274. MR138005 188 [19] A. McIntosh, Square roots of operators and applications to hyperbolic PDE’s, Miniconference on Operator Theory and PDE, Proceedings of the Centre for Mathematical Analysis, ANU Canberra, 5 (1984), 124–136. 189 MR757577 [20] J. Peetre, On convolution operators leaving Lp,λ spaces invariant. Ann. Mat. Pura Appl. (4) 72 (1966) 295-304. MR209917 169 [21] S. Spanne, Sur l’interpolation entre les espaces LpΘ k (French). Ann. Scuola Norm. Sup. Pisa (3) 20 (1966) 625-648. MR209832 169 [22] S. Semmes, Square function estimates and the T (b) Theorem. Proc. Amer. Math. Soc. 110 (1990), 721–726. MR1028049 178, 179 [23] E. M. Stein, Singular integrals, harmonic functions, and differentiability properties of functions of several variables. Singular integrals Proc. Sympos. Pure Math., Chicago, Ill., 1966, pp. 316-335. Amer. Math. Soc., Providence, R.I. 1967. MR0482394 169 [24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princteon University Press, Princeton, NJ, 1970. MR0290095 158, 160, 177 [25] E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. MR1232192 159, 167 Department of Mathematics, University of Washington, Seattle, WA 98195, USA Email address: [email protected] Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Email address: [email protected] Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Email address: [email protected]

IAS/Park City Mathematics Series Volume 27, Pages 199–256 https://doi.org/10.1090/pcms/027/00864

Sliding almost minimal sets and the Plateau problem G. David Abstract. We present some old and recent regularity results concerning minimal and almost minimal sets in domains of the Euclidean space. We concentrate on a sliding variant of Almgren’s notion of minimality, which is well suited in the context of Plateau problems relative to soap films. We are especially interested in regularity properties near a boundary curve, where we would like to get a local C1 description of 2-dimensional almost minimal sets in the spirit of J. Taylor’s theorem, but we first study weaker and more general results (local Ahlfors regularity, rectifiability, limits, monotonicity of density), which we describe far from the boundary for simplicity. There we insist on some simpler techniques, in particular the use of Federer-Fleming projections.

Contents 1 2

3 4

5

Introduction Some Plateau problems 2.1 One dimensional sets, where Plateau is Steiner 2.2 Parameterizations, Radó, and Douglas 2.3 Hausdorff measure, rectifiable sets 2.4 Minimal currents 2.5 Size Minimizing currents 2.6 Reifenberg homology minimizers 2.7 A sliding Plateau problem Almost minimal sets and what we want to do Weak regularity properties for almost minimal sets 4.1 Coral (or reduced) sets 4.2 Local Ahlfors regularity 4.3 Federer-Fleming projections 4.4 A proof of local Ahlfors regularity 4.5 Rectifiability, uniform rectifiability, and projections Limits of almost minimal sets

200 201 201 203 204 205 206 207 207 209 212 213 213 214 217 220 223

2010 Mathematics Subject Classification. Primary 49K99 ; Secondary 49Q20. Key words and phrases. Almgren minimal sets, almost minimal sets, sliding boundary condition, Plateau problem. G. David is/was partially supported by the ANR, programme blanc GEOMETRYA ANR-12-BS010014, the European Community Marie Curie grant MANET 607643 and H2020 grant GHAIA 777822, and the Simons Collaborations in MPS grant 601941, GD.. ©2020 American Mathematical Society

199

200

6

7

8

9

Sliding almost minimal sets and the Plateau problem

Monotonicity of density, near monotonicity, and blow-up limits 6.1 Near monotonicity in the plain case 6.2 The almost-constant density principle 6.3 Blow-up limits 6.4 Monotonicity of density for sliding almost minimal sets? Minimal cones and the Jean Taylor theorem 7.1 Minimal cones of dimension 1 or 2 7.2 Jean Taylor’s theorem at last Sliding almost minimal sets 8.1 Sliding minimal cones 8.2 Regularity attempts near Sliding minimal cones 8.3 Further questions V. Feuvrier and existence results 9.1 Presentation 9.2 The need for a quasiminimal haircut 9.3 Dyadic grids, polyhedral nets, and quasiminimal haircuts 9.4 End of game

228 228 230 231 232 233 233 234 236 236 238 242 243 243 245 247 249

1. Introduction The main objects of these lectures are minimal and almost minimal sets, with the Almgren definition that seems to model best the geometry of soap films and bubbles. Even though this problem will not be addressed directly, we invite the reader to think about the classical Plateau problem, where we are given a smooth closed curve Γ in the Euclidean space R3 , and try to find a surface (a set) spanned by Γ and with minimal area (Hausdorff measure of dimension 2). The lectures will be centered on a fairly small number of techniques, which lead to regularity results for solutions of such a Plateau problem, and more generally of almost minimal sets. In these notes, we shall try to insist a lot on one fundamental tool, the so-called Federer-Fleming projection, which will be used repeatedly. But some other constructions will be described too. The archetype of a good regularity result will be the local C1 description of 2-dimensional minimal sets in R3 (far from the boundary) that was proved by Jean Taylor in [65]. One of the goals of the lecture is to describe recent attempts to give a similar description of minimal sets near a simple boundary of Plateau type. We will use the opportunity given by the Park City lectures to give a simpler description of two otherwise quite long and technical papers [16, 19], and also explain again a scheme of proof for existence results that was introduced by V. Feuvrier [38], and in my opinion not appreciated to its true value. The author wishes to thank the organizers of the event in Park City, and the anonymous referee for a careful reading of these notes.

G. David

201

2. Some Plateau problems There are many different ways to state a Plateau problem, even in the most standard case when the boundary Γ is a smooth curve in R3 . We shall mention a few in this section, but then we shall feel free to concentrate on definitions similar to Almgren’s in [3], especially since they seem to be among the best models for soap films and bubbles and Joseph Plateau himself was interested in soap films (and also interfaces between fluids); see [59]. Soap films and bubbles are composed of two layers of soap molecules with a water-repelling tail and a water-attracting head, which align themselves head to head with a thin layer of water in the middle; the width of the film is roughly equal to the length of two molecules. People usually come up rapidly with a simple formula for the energy of the film, just proportional to the total surface of the film. This sounds vague and imprecise, and the author finds it quite surprising that such a basic modeling actually works so well. The general description for a solution of Plateau’s problem is a set E (or a similar object) spanned by Γ and whose area is minimal, but there are many ways to define the terms “spanned” and “area”; we shall only describe some of them. 2.1. One dimensional sets, where Plateau is Steiner The most reasonable version of Plateau’s problem for 1-d sets is the following. Pick a finite collection of points Ai ∈ Rn , and look for a connected set E that contains the Ai and has minimal length H1 (E) (defined below). This is known as Steiner’s problem. It is rather easy to prove that minimizers exist, by Golab’s theorem on the lower semicontinuity of length among connected sets, and that they are composed of line segments whose endpoints are either points Ai , or some additional points Bj , called Steiner points. Near each Steiner point Bj , E is composed of three line segments that end at Bj with equal 120◦ angles. This angle condition is easily proved by computing the derivative of H1 (E) when we move Bj a little and keep the other vertices fixed. We suggest, as an exercise, to check (or at least guess) what happens when the Ai are the four vertices of a square. Finally, E has no loop (because otherwise we may remove a segment and save some length). Except for the invention of Steiner points and interesting questions about how fast one can compute the minimizers, there is not so much more to be said. Notice however possible ruptures of symmetry (for instance, when the Ai are the four vertices of a square), hence the lack of uniqueness, and even more obviously the fact that some solutions are not smooth away from the Ai . Let us give a more elaborate version of this, with nets and integer multiplicities, which we will use as a first introduction to currents. We are now given a finite collection of points Ai , i ∈ I, and for each one an integer αi ∈ Z; we assume that  αi = 0, (2.1.1) i∈I

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and we look for admissible nets (defined soon) with some minimality property that will be specified later. We see each Ai as a source of electricity, with the intensity αi (negative if αi < 0), and think of admissible nets as electrical nets that satisfy Kirchhoff’s law. That is, an admissible net is a finite collection Ik = [ak , bk ], k ∈ K, of (oriented) intervals together with for each k a multiplicity mk ∈ Z, and that satisfies the following version of Kirchhoff’s law. For each point z ∈ Rn , denote by K+ (z) the set of indices k ∈ K such that z = bk , by K− (z) the set of indices k such that Z = ak , and by I(z) the set of indices i ∈ I such that z = Ai (thus, I(z) has at most one point). Then    mk − mk = αi (2.1.2) k∈K+ (z)

k∈K− (z)

i∈I(z)

Rn

for all z ∈ (but only the nodes really matter). It is not hard to check that (2.1.1) is a necessary and sufficient condition on the αi for the existence of admissible nets. We should probably require that αi = 0 for i ∈ I (otherwise, remove Ai from the discussion), that mk = 0 (otherwise, Ik is useless), and that the intervals Ik have disjoint interiors (otherwise, we can use a finer description where the intersection of the two intervals is an interval of its own, with the sum of the multiplicities). When N is an admissible net and the intervals Ik have disjoint interiors, we shall denote by E(N) = ∪mk =0 [ak , bk ] the support of N. Associated to N is also  the current T = k∈K mk [[Ik ]], where [[Ik ]] is a notation for the one-dimensional current of integration on the oriented segment Ik (but we shall not define this yet). Let us still mention that the Kirchhoff rule (2.1.2) is a complicated way of saying  that ∂T = S, where S = i∈I αi [[Ai ]] is the current of dimension 0 associated to the data. The simplest quantity to minimize on the class of admissible nets is probably the size  |bk − ak |, (2.1.3) S(N) = k∈K;mk =0

where N denotes an admissible net, which is the total length of the useful part of the net. Choosing a net N that minimizes S(N) corresponds roughly to minimizing H1 (E) in the Steiner problem above; let us just make a few observations, and leave their verification as an exercise. For minimal nets, the intervals Ik automatically have disjoint interiors, even if we did not require this initially. If N is a minimizer and E = ∪mk =0 [ak , bk ] denotes its support, then E is as above a finite union of intervals, whose endpoints are the Ai and Steiner points Bj where exactly three intervals Ik end with 120◦ angles. Then the number of Steiner points is at most N − 2, where N is the number of points Ai , and we get a bound on the number of intervals Ik too. We may use this to prove that there is a minimizing net.

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The set E may be different from the solution of the Steiner problem above, because it is not necessarily connected; however, we can easily choose the multiplicities αj so that any minimal net that satisfies the Kirchhoff rule has a connected support that contains the Ai . And also, if E is a solution of the Steiner problem above, we can choose multiplicities mi and αi so that the associated net is supported by E, and even minimizes S if all the supports of admissible nets are connected. This can probably be arranged, but the author was too lazy to check this; notice however that for a higher dimensional minimal set, the construction of a multiplicity on this set so that the associated rectifiable current is a size minimizer is a nontrivial problem. Instead of the size S(N), we may also want to minimize the mass  |mk ||bk − ak |, (2.1.4) M(N) = k∈K

which is a more natural number when we think of N as a current, because it is its norm as a linear form on the set of 1-forms. This is the quantity that most people like to minimize when they talk about minimal currents and surfaces. The reader is invited to play with the size and mass minimizers that arise when the Ai are the four vertices of a square. Let us finally mention that other quantities, such as  |mk |β |bk − ak |, (2.1.5) Mβ (N) = k∈K

with 0 < β < 1 are natural too, in particular in the context of optimal networks, and produce interesting minimal nets, with angles at the Steiner points that now depend on the mi and β. Think about constructing an optimal net of roads to accommodate a flux of cars between some cities, where the cost of construction of a road depends on the intensity of the traffic there, and see [4] for information. 2.2. Parameterizations, Radó, and Douglas Let us now think about the case when Γ is a closed curve in R3 , and we try to minimize the area of a surface E bounded by Γ , in the sense that E = f(D) for some function f defined on the closed unit 2-dimensional disk D and the restriction of f to the unit circle S is a parameterization of Γ . The simplest way to define the area of E is to (assume that  f is Lipschitz and) take A(f) = D Jf (x)dx, where Jf is the appropriate Jacobian of f. This is what Tibor Radó did in 1930 (see [60–62]), with conformal mappings, and Jesse Douglas in 1931, with harmonic parameterizations. [Of course we skip many important contributions, here and below.] The difficulty is that the Lipschitz constant for f may tend to +∞ along a minimizing sequence, which leads to an unpleasant lack of compactness. They have nice solutions to this, where they first select nice parameterizations. In particular, the paper [29] (which the author believes is the main reason for Douglas’ Fields medal) is very clever and easy to read.

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We cannot resist saying two words about it. It makes sense to decide that f will be harmonic in B(0, 1) and continuous on D, because such parameterizations exist. And then the area A(f) can be computed in terms of f|∂D alone. It turns out that the initial problem translates into minimizing   n 2 j=1 |fj (θ) − fj (ϕ)|   (2.2.1) A(f) = dθdϕ, sin2 θ−ϕ 2 where the fj are the coordinates of f, and this is much easier to do. But this way to state the Plateau problem is not entirely satisfactory. First, minimizers of A(f) only give a good description of soap films locally when f is injective. That is, if x, y are interior points of D such that f(x) = f(y), then near f(x), E = f(D) may look like the union of two smooth surfaces that meet transversally; soap films don’t look like this, but rather like the sets of type Y that are described below. And it is difficult to know, given Γ , when the parameterization given by Douglas will be injective. In addition, some of the minimal sets bounded by Γ are often best parameterized by other sets than D, for instance with an additional handle or different topology; it is not clear (to the author) that Douglas’s argument will work in this case. For a little more information on this and the next variants of the Plateau problem, the reader may consult the survey [17]. 2.3. Hausdorff measure, rectifiable sets From now on, all our ways to compute area will rely on the Hausdorff measure, which we define now for the convenience of the reader. The main properties of Hd that we like are that it is a Borel (but not locally σ-finite) measure, and that it coincides with the surface measure on smooth sets. It is defined by (2.3.1)

Hd (E) = lim Hδd (E), δ→0

where (2.3.2)

Hδd (E) = cd inf



diam(Dj )d



j∈N

and the infimum is taken over all coverings of E by a countable collection {Dj } of sets, with diam(Dj )  δ for all j. We may choose the normalizing constant cd so that Hd coincides with the Lebesgue measure on subsets of Rd . See for instance [53] for the important verification that Borel sets are measurable, and also information on rectifiable sets. We shall use a lot of rectifiable sets too. Those are the sets E with σ-finite measure (Federer used to require finite measure, but the standard definitions now allow countable unions) such that E ⊂ Z ∪ G, where Hd (Z) = 0 and G is a countable union of embedded C1 surfaces of dimension d (or equivalently, G is a countable union of images of Rd by Lipschitz mappings). We will recall the

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properties of rectifiable sets as we use them, but let us already say that they have an approximate tangent d-plane at almost every point. 2.4. Minimal currents The most celebrated (and very successful) ways to state Plateau problems are in terms of currents. Existence results are made easier by stating things weakly (i.e., in terms of distributions), because important compactness results can be proved, and the setting is more in terms of differential geometry than for standard PDE’s, relying on the integration of forms and a notion of boundary that comes from exterior derivatives and integration by parts. Generally, a current of dimension d is a continuous linear form on the vector space of smooth d-forms (say, with compact support), but we will restrict here to specific classes of currents (rectifiable currents, integral currents) with a regularity which is essentially the same as the regularity of Radon measures. Finally regularity results for minimizers can often be proved, completing the loop and allowing us to return to smooth minimal sets. Initial and important work was done by Federer, Fleming, De Giorgi, and many others. Out of ignorance and laziness, we just refer to [1, 34, 35, 39, 54] and their references. The simplest example of d-dimensional current is the current of integration on a smooth oriented surface E of dimension d, which acts on a d-form by integrating it on E. But we are interested in the following larger classes of current, with better compactness properties. Now let E be a rectifiable set of dimension d, with locally finite Hausdorff measure Hd (defined below). Also put a measurable orientation τ (i.e., an orientation of the approximate tangent plane to E at x (which exists Hd -almost everywhere), chosen to be a measurable function of x), and choose a measurable multiplicity m(x) on E, with integer values, and integrable against d ; we define the rectifiable current T by H|E  (2.4.1) T , ω = m(x) ω(x) · τ(x) dHd (x), E

where T , ω is our notation for the effect of the current T on the smooth, compactly supported d-form ω, and ω(x) · τ(x) is a notation for the way one uses the orientation τ to integrate a form on E (or a C1 surface to start with). One of the clever ideas behind the use of currents is that we can define boundaries as in differential geometry. The boundary ∂T of the d-dimensional current T is a current of dimension d − 1, defined by duality by (2.4.2)

∂T , ω = T , dω for every (d − 1)-form ω,

where dω denotes the exterior derivative of ω. The point is that when S  is the current of integration on a smooth oriented surface S with boundary Γ , the Green formula says that ∂S  = Γ  , the current of integration on Γ . Notice also that ∂∂ = 0 because dd = 0. An integral current is a rectifiable current T (with an integrable integer multiplicity m) as above, such that ∂T is such a rectifiable current as well.

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The most classical way to state the Plateau problem for currents is to start from a (d − 1)-dimensional current S, with ∂S = 0, and minimize the mass Mass(T ), among d-dimensional currents T that satisfy the boundary equation (2.4.3)

∂T = S.

The mass Mass(T ) is the operator norm of T , where we put a L∞ -norm on forms. When T is a rectifiable current given by (2.4.1),  |m(x)| dHd (x). (2.4.4) Mass(T ) = E

In this setting, there is a very general existence result for these mass minimizing currents; for instance, it is enough to assume that S is rectifiable (as above), compactly supported, and such that ∂S = 0 (necessary for (2.4.3), because ∂∂ = 0). This comes from a quite strong compactness theorem. In addition mass minimizing currents have good regularity properties in general. In particular, in codimension 1 the support of T is a smooth submanifold when n  7. We refer to [58] and its references for loads of information. The whole theory is a great success for weak solutions and Geometric Measure Theory, but the sad news for us is that mass minimizing currents don’t describe most soap films. For one think, there are soap films of dimension 2 in R3 with onedimensional singularities, which therefore cannot come from mass minimizing currents; see the discussion of J. Taylor’s theorem below. 2.5. Size Minimizing currents If we want to describe soap films, it seems that it is better to minimize the size of T among solutions of (2.4.3). The size of T is the Hausdorff measure of its support, i.e., when T is given by (2.4.1),   (2.5.1) Size(T ) = Hd x ∈ E; m(x) = 0 . That is, we no longer count the multiplicity. The difference between mass and size minimizers is essentially the same as in dimension 1 above, and for instance the minimal cones in the description of J. Taylor’s theorem can be realized as supports of size minimizers; we just need to find adequate multiplicities, and the minimality follows from a calibration argument. See [44, 45]. There are some bad news about this variant of the Plateau problem. Some soap films are hard to describe with size minimizers, typically because they are not orientable; in some cases one can circumvent this problem by various algebraic tricks, but altogether it is a little awkward to use orientations on soap films which naturally are not oriented. See [17] for some additional information. The second piece of bad news is that there is no general existence result for size minimizers, even when S is the current of integration on a smooth curve in R3 . One cannot use the same compactness result as for mass minimizers, because we do not control the mass of T along a minimizing sequence. There are interesting partial results by R. Hardt and T. De Pauw [24, 25], with also existence results for functionals between mass and size.

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Some more existence results should follow from Yangqin Fang’s regularity results from [32], but only when d = 2, n = 3, (2.4.3) is replaced by a condition that says that ∂T is homologous to S inside a smooth surface, and the support of T is required to stay on one side of that surface. Another option with an effect similar to using the size is to minimize the mass, but for multiplicities with values in some other, often discrete group. See [52] for an initial description. 2.6. Reifenberg homology minimizers Here and below, we are really considering closed sets E, and want to minimize Hd (E) under a topological constraint that says that E is “spanned by Γ ”, but what exactly should we mean by this? For Reifenberg [64], E is a compact set that contains Γ , and the boundary condiˇ tion is stated in terms of Cech homology on some commutative group G. Specifically we require the inclusion i : Γ → E to induce a trivial homomorphism from Hd−1 (Γ , G) to Hd−1 (E, G). Or we could instead require that it annihilates some given subgroup of Hd−1 (Γ , G). When d = 2, n = 3, and the boundary Γ is a curve, this is a way to say that Γ (or the obvious generator of H1 (Γ , G) associated to Γ ) vanishes in H1 (E, G), or in somewhat more vague terms, that E “fills the hole”. ˇ The choice of Cech homology is not innocent, this homology is more complicated to define, but it has some stability with respect to taking limits, which is good for the existence of minimizers. Reifenberg proved this for compact groups, such as G = Z2 or G = R/Z, under some small regularity for Γ . It is a beautiful (but tough) proof by hands, using minimizing sequences and haircuts. De Pauw obtained the 2-dimensional case when Γ is a curve and G = Z (with currents). In that case the equivalence with the size minimizing problem above is not known yet (multiplicities are hard to construct), but the infimum is the same: see [24]. More recently, essentially optimal existence results were given by Yangqin Fang [30], after a claim by F. Almgren with varifolds [2]. A little later, following U. Menne, Y. Fang and S. Kolasínski [33] gave a proof with flat chains inspired of Almgren’s argument. 2.7. A sliding Plateau problem We finally arrive to the author’s favorite definitions, based on a notion of deformations of a set. Definition 2.7.1. Let a set Γ be given (a priori any closed subset of Rn ). Let E ⊂ Rn and a closed ball B ⊂ Rn be given. A sliding deformation for E in B, with respect to the sliding boundary Γ , is a one-parameter family {ϕt }, 0  t  1, of functions, such that (2.7.2)

(x, t) → ϕ(x, t) = ϕt (x) : E × [0, 1] → Rn is continuous;

(2.7.3)

ϕ0 (x) = x for x ∈ E,

(2.7.4)

ϕt (x) = x for x ∈ E \ B and 0  t  1,

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Sliding almost minimal sets and the Plateau problem

ϕt (E ∩ B) ⊂ B for 0  t  1,

the important sliding condition (2.7.6)

ϕt (x) ∈ Γ when x ∈ E ∩ Γ and 0  t  1

and (2.7.7)

ϕ1 is Lipschitz.

By extension, we also call E1 = ϕ1 (E) a sliding deformation of E in B. Finally, a sliding deformation of E is just a set E1 = ϕ1 (E) which is a sliding deformation of E in some B. There would be similar notions localized in an open set U, but we shall not need them. This is a minor modification of a definition of Almgren [3], to take the boundary into account. The point is to allow E to move, including along Γ , but not to be detached from Γ , a little bit like a shower curtain. Note that ϕ is not required to be injective; in the problems below, if you can pinch E and this way get a new set with less surface, this gives a good competitor. We keep (2.7.7) because Almgren used it and it does not disturb, but we could also drop it, and we should observe that no bound on the Lipschitz constant for ϕ1 is ever required. Because of the extra condition (2.7.6), we really need to state things in terms of a deformation {ϕt }, rather than just the endpoint ϕ1 . Without (2.7.6) (and because we decided to restrict to deformations in a ball; things would be different if we used deformations in a non convex compact set), it would be easy, given the final mapping ϕ1 , to extend by convexity, set ϕt (x) = tϕ1 (x) + (1 − t)x, and observe that it provides a deformation. But here we need to make sure that ϕt (x) ∈ Γ for all t if x ∈ Γ , and when Γ is not convex, an extension as above may be hard to find. It would probably not be natural to demand that ϕt be defined on the whole B (as opposed to E alone), because we do not really want to control the air around the soap film; when there is no boundary condition (or equivalently when we have Γ = Rn ), this makes no difference because we could extend ϕt , but in the present situation we don’t necessarily know how to extend ϕt so that (2.7.6) holds also for x ∈ B \ E. There is a Plateau problem attached to this definition: let E0 be a given closed set, and try to minimize Hd (E) among all the sliding deformations of E0 . In some cases the problem will be uninteresting, either because there is no sliding deformation with Hd (E) < +∞, or on the opposite the infimum is 0. But in general there may be more than one interesting initial set E0 for a given compact set Γ . The author claims that this is probably one of the best ways to model soap films, and also likes the setting for the following reasons. First, the notion of

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sliding competitor may fit the way soap films are created (but the author does not claim any precise knowledge about this). It is nice that we don’t need to say precisely for which topological reason E0 is linked to Γ , or in other words to relate solutions of Plateau problems to specific topological or algebraic reasons that may depend on the problem. The problem is rather insensitive to orientation. And importantly, for each choice of Γ there may be a few different interesting choices of E0 , leading to different minimizers, just as this happens for soap films attached to a wire. Yet there is the usual bad news: no general existence theorem is known, even when d = 2, n = 3, and Γ is a nice curve. Naturally, we do not account here for unrealistic deformations that would extend the film too far: some real life films can be deformed into a point, with a long homotopy that soap is unlikely to discover because this would involve going through a surface with a much larger area. We cannot do much about this, except for mentioning that this may happen. Of course the dynamics of soap films and bubbles is interesting, but this is not the subject of these lectures. We will return extensively to the notion of sliding competitors, but in the mean time let us end this section with the general conclusion that many interesting Plateau problems are still there to solve.

3. Almost minimal sets and what we want to do Here comes our last section of introduction. Our general goal here will be to study regularity properties of sliding minimal and almost minimal sets; in this section we give some of the relevant definitions and try to justify this goal. Let us directly define sliding almost minimal sets in Rn . Let Γ ⊂ Rn be a closed set, and consider a closed set E ⊂ Rn such that (3.0.1)

Hd (E ∩ B(0, R)) < +∞ for R > 0.

Also, we will let h : (0, +∞) → [0, +∞] be a nondecreasing function such that limr→0 h(r) = 0; we shall call this a gauge function. Definition 3.0.2. We say that E is a sliding almost minimal set, with the sliding boundary Γ and the gauge function h, when (3.0.3)

Hd (E ∩ B(x, r))  Hd (F ∩ B(x, r)) + h(r)rd

whenever F is a sliding competitor for E in any ball B = B(x, r) (as in Definition 2.7.1). When we take Γ = Rn (i.e., forget about the sliding boundary conditions), we get what we’ll call a plain almost minimal set. When h ≡ 0 we get a sliding minimal set (or a plain minimal set). It is easy to localize this definition to an open set U; we say that E is a sliding almost minimal set (with the sliding boundary Γ and gauge function h but we shall not always repeat this) in U when (3.0.3) holds whenever F is a sliding

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competitor for E in a ball B = B(x, r) that is contained in U. Thus Definition 3.0.2 corresponds to U = Rn , and sliding almost minimal set in U are automatically sliding almost minimal in any open set U  ⊂ U. Almost all of our results will be local, i.e., concern sets that are almost minimal in (a neighborhood of) a given ball. Recall that one way to produce deformations of a set E is to pinch them so that two pieces of E come together and we save some Hausdorff measure. Here are some examples, starting with explicit ones, and continuing with institutional ones. The (plain) minimal sets of dimension 1 in a domain U are composed of locally finite unions of line segments, that can only meet by sets of three at their endpoints and with 120◦ angles. This was checked by Morgan [57] as an exercise on currents, and later in [13] with more elementary cut and paste argument. Then, if U = Rn , it can be checked that (modulo sets of vanishing H1 measure), the only minimal sets are the empty set (which we’ll often forget to mention), the lines, and the sets Y (three half lines with the same endpoint that make 120◦ angles there) of Subsection 2.1. As we have seen, the union of two transverse (or even perpendicular) lines is not minimal. It is not almost minimal either. The planes, the cones of type Y and T described below in Subsection 7.1 are also plain minimal sets in Rn . There are a few other explicit examples like this (in higher dimensions), but not so many. Then there are the minimal surfaces, like the catenoid. Those are often locally minimal only, which means that H2 (E)  H2 (F) when F is a deformation of E in a small enough ball B(x, r). For instance, seen from far the catenoid looks a lot like two parallel planes, that we can pinch to get a better competitor. We should also mention that smooth surfaces (like spheres, but not only) are rather easily seen to be locally plain almost minimal, with h(r)  Cr for r small. One of the reasons why we authorized h(r) = +∞ is so that we can say without even thinking that spheres (or objects that are even more irregular at large scales) are almost minimal. This is also a way to make it plain that in some case we do not get any information by comparing E with a deformation in a large ball. We shall give other examples of simple sliding minimal sets later. For the moment, let us only mention the case when d = 2, Γ is a line in R3 , and E is a half plane bounded by Γ , or the union of two half planes bounded by Γ and that make an angle of at least 120◦ there (we’ll call this a set of type V). Let us turn to institutional minimal and almost minimal sets now. If E minimizes Hd among all sliding deformations of a given set E0 , as in Subsection 2.7, it is automatically sliding minimal. But there are other examples. First, it is often possible to prove that the limit E∞ of some minimizing sequence for the sliding Plateau problem above is a sliding minimal set, without being able to show that it is a sliding deformation of E0 , and studying the regularity of E∞ is useful in

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itself, but also could help us prove that E∞ is a sliding deformation of E0 and solve the corresponding Plateau problem. Also, the Reifenberg homology minimizers of Subsection 2.6 are sliding minimal sets; see [17] for the easy verification. In [31], regularity properties for those in a simple context are used to prove that they are also solutions of the Reifenberg Plateau problem with the apparently more friendly singular homology. Similarly, the supports of size-minimizing currents of Subsection 2.5 are sliding minimal (see [17] again) and some regularity for those is useful in itself and could be used for existence results. Concerning almost minimality, this is also a very useful notion, because (local) minimizers of slight modifications of our usual functional Hd are typically almost minimal, with a gauge function like h(r) = Crα (often with α = 1). The simplest example is soap bubbles, which are also subject to different pressures from the two sides of the bubble. The corresponding force is proportional to the surface (and the difference of pressure), and for a deformation of E in B(x, r), we expect a difference of potential energy of order rd+1 , where rd comes from Hd (E ∩ B(x, r)) and the extra power accounts for the displacement. For soap bubbles, we expect E to have a constant mean curvature, proportional to the difference of pressure. Thus the pressure in smaller bubbles is larger, and of course with soap films the pressure is the same on both sides and the mean curvature vanishes. For very small soap bubbles, the pressure is very large and we expect the almost minimality constants for E to deteriorate. Of course we could include other “small” forces too, like the gravity, and another simple example of almost minimal set would be a set that minimizes  d −1  f  C. In E f(x)dH (x) for some Hölder-continuous function f such that C fact, we may also include strongly Euclidean but yet non isotropic elliptic integrands of the form  f(x, TE (x))dHd (x), (3.0.4) J(E) = E

where TE (x) denotes the approximate tangent d-plane to E at x (we may either assume that E is rectifiable, or define J(E) in some other way when it is not rectifiable, and then find out after the fact that minimizers are rectifiable), and f is defined by (3.0.5)

f(x, T ) = Hd (B(0, 1) ∩ A(x)(T )),

where A is a Hölder continuous function with n × n-matrix values (or we should say, linear mappings of Rn ) such that ||A(x)|| and ||A−1 (x)|| are bounded. Of course the standard classes of elliptic integrands are much larger than this (for instance, we could use lp -norms instead of Euclidean ones), but for those we should not expect the corresponding almost minimal sets to be as easy to study as in the Euclidean case. But notice that the quasiminimal sets defined in Subsection 9.2 are the same as long as C−1  f  C.

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The author likes to insist on the extra stability provided by almost minimal sets. While we expect roughly the same low regularity results (say, in the C1 category) for almost minimal sets, we really like the extra flexibility. Other conditions, such as bounds on the first variation for varifolds, seem to be much less flexible. So we want to study the local regularity properties of sliding almost minimal sets (say, with gauge functions h(r)  Crα ), possibly with the hope that existence results may follow. We will present the regularity story in two steps. First we’ll describe general results (Ahlfors regularity, rectifiability, stability under limits, blow-up limits and minimal cones), that can be proved near Γ at the price of longer and more complicated arguments, but which we present only in the plain case (i.e., far from Γ ). Then we will present more recent and precise results specific to 2-dimensional sliding almost minimal sets. We think about variants of J. Taylor’s theorem [65], which will be stated in Section 7 as a best example of what we try to accomplish, and we’ll try to describe attempts to get similar statements near points of a simple boundary Γ (for instance a line). Finally, we shall try to explain a scheme introduced by V. Feuvrier that allows one to use, in the circumstances where we have an appropriate regularity result for sliding minimal sets E (think about the existence of local Lipschitz retractions on E), a minimizing sequence of improved competitors (so that they are quasiminimal) to get an existence result. This fits well with the other results presented here, the author’s impression is that Feuvrier’s argument was not well enough appreciated, and so we want to try once more.

4. Weak regularity properties for almost minimal sets In this section we sketch the proof of some of the weak (but general and useful) properties of almost minimal sets, sliding or not. We rather follow the proofs of [16], not only because they were already written in the sliding context, but also because since time had passed, some of the initial proofs (from sAlMemoir, DSMemoir, for instance) were improved in the mean time. But we’ll do the description in the plain case, because it is less technical and otherwise almost the same. We’ll concentrate on the local Ahlfors regularity of E, its rectifiability, the rectifiability of limits, and some important stability results under limits. The main hero for the part of the proofs that we can describe will be the Federer-Fleming projection on dyadic cubes, also called deformation lemma in some contexts. Our standing assumption now is that E is a (coral, as in the next subsection) almost minimal set (plain to simplify) of dimension d, with a small enough gauge function h (h(r)  Crα for some α > 0 is more than enough), in a domain U ⊂ Rn which contains the balls B(x, r) where we put ourselves.

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Our constants C will be allowed to depend on n, d and h, but not on E, U or B(x, r) and we may have to take the radius r to be small, so that h(r) is small enough. 4.1. Coral (or reduced) sets This will just be a precaution, so that we don’t spend time discussing useless additional sets of vanishing Hd -measure. For a closed subset E ⊂ U, with locally finite Hd -measure in U, denote by E∗ the closed d . That is, support of H|E  (4.1.1) E∗ = x ∈ E ; Hd (E ∩ B(x, r)) > 0 for all r > 0 . We say that E is reduced, or coral when E = E∗ . If E is almost minimal, then E∗ is also almost minimal, with the same gauge h, because it is easy to check that Hd (E \ E∗ ) = 0, and then by direct inspection. With sliding almost minimal sets, this is also true, but it requires a small proof, given in [16]. Incidentally, when we say “this is also true”, we mean under quite general assumptions on the boundary sets Γ (we can authorize more than one at the same time, which is convenient for instance to force E to lie in an initial closed domain). So it is safe to focus on reduced sets. This will simplify our statements; otherwise we would typically have to say that E is composed of a thin part of vanishing measure, plus a locally Ahlfors regular set, say. But we shall keep in mind that E∗ \ E can play a role in some topological problems, even if it does not show up in the nice descriptions below. Anyway, from now on, all our sets will be coral. 4.2. Local Ahlfors regularity We just give a statement for the moment; the proof will be discussed later. Recall the standing assumption on the coral almost minimal set E in U. Theorem 4.2.1. Local Ahlfors-regularity [3, 23]. There exists C  1 such that (4.2.2)

C−1 rd  Hd (E ∩ B(x, r))  Crd

whenever (4.2.3)

x ∈ E and r ∈ (0, 1) are such that B(x, 2r) ⊂ U.

Thus Ahlfors regularity is just a size condition, that says that E is d-dimensional in a very strong and uniform way. Here C depends only on n and h; more precisely we can make sure that C depends only on n, provided that h(r) is small enough, depending on n. This sounds bland, but it is also very useful, in part because many estimates are easier to do with Ahlfors regular sets. The “hard” part seems to be the lower bound (if E is too thin, we can deform it to an even smaller set), but in fact both proofs use the same basic engine, the Federer-Fleming projections described below.

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However, in codimension 1 the proof for the upper bound is quite simple, so we’ll give it now. We assume that d = n − 1, and we want to show that (4.2.4)

Hd (E ∩ B(x, r))  σn−1 rn−1 + h(r)rn−1 ,

where σn−1 = Hn−1 (∂B(0, 1)). As in most of our proofs, we have to find a good competitor for E. Here this will be easy. Select any point x0 ∈ B(x, r/2) \ E; this is possible, since Hd (E ∩ B(x, r/2)) < +∞ and d = n − 1. Then denote by π the radial projection centered at x0 , from B(x, r) \ {x0 } to ∂B(x, r). Extend π to Rn \ {x0 } by setting π(y) = y for y ∈ Rn \ B(x, r). Then, for y ∈ E and 0  t  1, set ϕt (y) = tπ(y) + (1 − t)y; it is easy to check that {ϕt }, t ∈ [0, 1], is a deformation for E in B(x, r). So F = π(E) is a deformation of E, and the definition of an almost minimal set yields Hd (E ∩ B(x, r))  Hd (F ∩ B(x, r)) + h(r)rd , as in (3.0.3). But Hd (F ∩ B(x, r))  Hd (∂B(x, r)) = σn−1 rn−1 because d = n − 1 and π(B(x, r) \ {x0 }) ⊂ ∂B(x, r); (4.2.4) follows. With a sliding boundary condition (with a nice enough boundary Γ ) we would have to be more careful (because π does not necessarily preserve Γ ). We leave the details as an exercise; the same sort of problems arise in higher co-dimensions, making the proof a little less pleasant in this context. In higher co-dimensions, and even with the apparently easier upper bound, how should we proceed? We can still try to project E ∩ B(x, r) on a set with controlled Hd -measure, but we’ll need to be more systematic and persistent. 4.3. Federer-Fleming projections These will be compositions of radial projections (like π above), on faces of various dimensions of cubes. In some arguments, it is useful to replace cubes by convex polyhedra, but for the moment we stick to cubes which are much simpler to organize. We will try to explain the construction in a friendly, but also slightly imprecise way; the reader may consult [23], which is not bad at all. The usual descriptions (such as in [35] (obviously!) or [34]) are usually a little harder to follow, because they are designed for currents. But the construction is the same. We start with some notation. We shall use closed cubes Q ⊂ Rn , which we could take with faces parallel to the axes (this costs nothing), but not necessarily dyadic to start with. For each such cube Q, ∂Q denotes the boundary of Q in Rn . More generally we will be interested in cubes Q ∈ Qk , 0  k  n, the set of k-dimensional (closed) cubes. Such a cube Q is thus contained in some k-dimensional affine subspace H of Rn , and then ∂Q will denote the boundary of Q in H. Notice that (for k  1), ∂Q is composed of 2d cubes of Qk−1 , which we call the faces of Q; we denote by Fk−1 (Q) the set of faces of Q. We also iterate, and (when k  2) call Fk−2 (Q) the set of faces of cubes of Fd−1 (Q), and so on. We even say that F(Q) is the union of all the Fl (Q), 0  l < k; we call these Fl (Q) the subfaces of Q. The l-dimensional skeleton of Q is the set Sl (Q) = ∪S∈Fl (Q) which is a union of l-cubes.

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Return to the Federer-Fleming projection, and start with the main building block. We are given a cube Q ∈ Qk , contained in the k-dimensional affine subspace H, and a point ξ ∈ 12 Q (the cube of H with the same center and half the sidelength); then let πξ denote the radial projection centered at ξ, from Q \ {ξ} to ∂Q. Recall that πξ (y) is the only point of ∂Q such that y ∈ [ξ, y]. We systematically extend π to H \ {ξ} by setting π(y) = y for y ∈ H \ Q; the extended mapping is still Lipschitz away from ξ. In the arguments below, there is also a closed set F ⊂ Q, with Hd (F) < +∞, and our first task is to choose ξ ∈ 12 Q \ F, such that πx (F) is not too large. In fact, often we have other constraints, so we really need to show that for most choices of ξ ∈ 12 Q \ F, πx (F) is not too large. There will be two estimates based on the same comment. Since ξ ∈ 12 Q, all the radii [ξ, z], z ∈ ∂Q, are nicely transverse to ∂Q, and there is a (simple geometric) constant C such that for each y ∈ 12 Q \ {ξ}, the mapping πξ is C|y − ξ|−1 d(Q)Lipschitz near y, where we denote by d(Q) the sidelength of Q. Because of this and by definition of Hd (we don’t even need to disturb the area formula),  d(Q) d (4.3.1) Hd (πξ (F ∩ Q))  C dHd (y). F |y − ξ| The simplest estimate is most useful when we can choose ξ far from F as possible; it says that

d(Q) d Hd (F). (4.3.2) Hd (πξ (F ∩ Q))  C dist(ξ, F) We may use this sort of estimate when we have enough control on F, for instance if we know that it is (a Lipschitz image of) an Ahlfors regular set of dimension d < k, but often we do not know this and F may be roughly dense. Fortunately, if k (the dimension of Q) is larger than d, we can use (4.3.1) and Fubini to show that the average value of Hd (πξ (F ∩ Q)) is under control, and then pick ξ by Chebyshev. That is,   

d(Q) d d H (πξ (F))dξ  C dHd (y)dξ ξ∈ 12 Q\F ξ∈ 21 Q\F y∈F\{ξ} |y − ξ|  

d(Q) d C dξdHd (y) (4.3.3) y∈F ξ∈ 21 Q\{y} |y − ξ|  k = Cd(Q) Hd (y)  Cd(Q)k Hd (F) y∈F

because F has vanishing k-dimensional measure and the integral in ξ converges when d < k. Thus it is easy to pick ξ ∈ 12 Q \ F such that (4.3.4)

Hd (πξ (F))  CHd (F).

This was how we construct one basic block. Now we need to worry about gluing blocks. We start with k = n. When Q is a top dimensional cube, we have defined πξ also on Rn \ Q, by πξ (y) = y. If we have a collection of cubes Qj , disjoint except for their boundaries (and we’ll only use this when they are of the

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same size and belong to the same net) and for each one we pick a center ξj ∈ 12 Qj and define the corresponding mapping πj , we can compose all these mappings (in any order; they commute) and get a mapping which is equal to πj on Qj , and is the identity on Rn \ ∪j Qj . We can also do this at the level of faces of dimension k > d: if we have a collection of faces Qj of the same dimension, disjoint except for their boundaries, and that belong to the same net (this will be clear when we do it; for instance, take any collection of k-faces of dyadic cubes of a given size), we can also compose them and get a Lipschitz mapping π that is equal to πj on Qj , and the identity on the rest of the k-dimensional skeleton of that net. We are now ready to iterate the basic construction and project a set of finite Hd measure on a d-dimensional skeleton. For reasons that will be explained later (basically, reduce a boundary effect), we like to do this on many small cubes at the same time. We start with more notation. Let Q ∈ Qn be given, and let N > 0 be a large integer. We cut Q in the obvious way into Nn almost disjoint (i.e., with disjoint interiors) cubes R, R ∈ Q(Q, N), of sidelength d(R) = N−1 d(Q), and we want to project on the faces of those cubes simultaneously. Let us let Fk (Q, N) = ∪R∈Q(Q,N) Fk (R) (i.e. the set of k-dimensional faces of these cubes), and also let Sk (Q, N) = ∪S∈Fk (Q,N) S = ∪R∈Q(Q,N) Sk (R) (i.e. the corresponding skeleton). Our Federer-Fleming projection will be a deformation for an initial closed set E, with Hd (E ∩ Q) < +∞ (think about our almost minimal set in an open set that contains Q). It will be obtained by composing a collection of deformations gk . We start with k = n, and the set En = E. For each R ∈ Q(Q, N), we apply the basic construction above to the cube R and the set F = En ∩ R. We find a point ξR ∈ 12 R \ En such that, now calling πR the radial projection that we called πξR , (4.3.5)

Hd (πR (En ∩ R))  CHd (En ∩ R).

We compose all these mappings together and find a Lipschitz mapping gn , that maps each En ∩ R ∈ Q(Q, N) to ∂R, and is the identity on En \ Q. We also get that (4.3.6)

Hd (gn (En ∩ Q))  CHd (En ∩ Q) < +∞,

by summing (4.3.5) over R. If n = d + 1, we stop here. Otherwise, we now construct gn−1 , defined on En−1 = gn (En ). We intend to take gn−1 (y) = y for y ∈ En−1 \ Q, so we just need to define gn−1 on En−1 ∩ Q = gn (En ∩ Q) (because gn (y) = y outside of Q). Notice that this set is contained in Sn−1 (Q, N) (the union of faces of cubes R), and we will define gn−1 independently on all the faces S ∈ Fn−1 (Q, N) that compose this skeleton. For the faces S that are contained in ∂Q, we have to take gn−1 (y) = y on S, because we said we want to take gn−1 (y) = y on En−1 \ Q. For each other face S, we apply the basic construction to S, the set F = En−1 ∩ S,

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and we get a point ξS ∈ 12 S \ En−1 such that the analogue of (4.3.4) holds for the radial projection πS = πξS . Notice that πS coincides with the identity on ∂S, so that we do not have a conflict of definition on S ∩ ∂Q = ∂S ∩ ∂Q (since S is not contained in ∂Q). As before, we can now compose all these mappings and get a mapping gn−1 , defined on Sn−1 (Q, N) minus all the centers, hence on En−1 ∩ Q. It is not hard to check that gn−1 is not only defined on En−1 , but in fact Lipschitz there (although maybe with a large constant); there is a little more to say, but not much, and we leave the details. At this point we have a new set En−2 = gn−1 (En−1 ), and the same argument as for (4.3.6) also yields (4.3.7)

Hd (En−2 ∩ Q))  CHd (En ∩ Q) < +∞.

Its image is now composed of E \ Q (where gn and gn−1 coincide with the identity), a piece of ∂Q that we were not allowed to modify, and the rest lies in the smaller skeleton Sn−2 (Q, N). If d = n − 2, we stop. Otherwise we continue, and define gn−2 independently on each face S ∈ Fn−2 (Q, N). We still take gn−2 (y) = y on En−2 \ Q and on En−2 ∩ ∂Q, so we take gn−2 (y) = y on S when S ⊂ ∂Q. For the other faces S, we apply the basic construction with F = En−2 ∩ S, pick a center ξS ∈ 12 S \ En−2 such that the analogue of (4.3.4) holds, and use the radial projection πS = πξS . Then we define gn−2 as before, and continue. Eventually we get a set Ed composed of E \ Q, a piece of ∂Q, and a subset of the d-dimensional skeleton Sd (Q, N). Finally we set f = gn ◦ gn−1 . . . gd+1 . It is easy to see that this is a deformation in Q (because each gn is a deformation in Q, for instance). This will be our standard Federer-Fleming projection, associated to E, Q, and the large integer N. It may happen that the final set En = f(E) is so small inside of Q that for each face S ∈ Sd (Q, N) that is not contained in ∂Q, we can find a point ξS ∈ 12 S \ Ed . When this happens, we can continue the construction one more step, i.e., define  such that gd as above and f = gd ◦ f, and get a new set Ed−1 = f(E) (4.3.8)

Ed−1 ∩ [Q \ ∂Q] ⊂ Sd−1 (Q, N).

This is even better: we essentially managed to kill the interior of Q. Let us end with two observations before we apply this to almost minimal sets. In our construction, each mapping gk maps each k-face S ∈ Fk (Q, N) to itself, so gk , f, and f map every cube R ∈ Q(Q, N) to itself. Hence, if V(R) denotes the collection of all the cubes R  ∈ Q(Q, N) that touch R and N(R) the union of these cubes, an iteration of (4.3.4) yields   Hd (f(E ∩ R  ))  C Hd (E ∩ R  )  C Hd (E ∩ N(R)). (4.3.9) Hd (Ed ∩ R)  R  ∈V(R)

R  ∈V(R)

That is, we also control the measure of the image locally, because we know roughly where each point of Ed comes from. The same remark holds for Ed−1 when f is defined.

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4.4. A proof of local Ahlfors regularity Let us now describe a proof of Theorem 4.2.1 (the local Ahlfors regularity of E when E is almost minimal). We start with the upper bound. Let E be our almost minimal set, and let Q be √ a cube such that 2 nQ ⊂ U. The general idea is that if Hd (E ∩ Q) is too large compared to d(Q)d , the Federer-Fleming projection above gives a deformation Ed of E in Q, which in Q is essentially contained in a d-dimensional skeleton, whose measure is easy to control. A contradiction with the almost minimality of E should ensue. We do the argument with a large N, to be chosen later, and whose effect will be to make the undesirable boundary effects smaller. The difficulty comes from the contribution of Ed ∩ ∂Q in the estimates, which itself is a consequence of the fact that we cannot brutally take two definitions for f, a projection on Sd (Q, M) inside Q and the identity outside. Since the almost minimality of E was written in terms of balls, we use the √ smallest ball B that contains Q, whose radius is r = nd(Q)/2. Notice that √ 2B ⊂ U because 2 nQ ⊂ U. We apply (3.0.3) with this B, remove the contribution of B \ Q which is the same for E and F = Ed , and get that Hd (E ∩ Q)  Hd (Ed ∩ Q) + h(r)rd .

(4.4.1)

Now write Ed ∩ Q = F1 ∪ F2 , where F1 = Ed ∩ ∂Q, and hence F2 ⊂ Sd (Q, N) by construction. The contribution of this part is easily estimated, since  Hd (Sd (R))  CNn (N−1 d(Q))d = CNn−d d(Q)d . (4.4.2) Hd (F2 )  R∈Q(Q,N)

This looks large because of N, but recall that we can pick the Ahlfors regularity constant after we choose N. For F2 , denote by Q1 (Q, N) the set of cubes R ∈ Q(Q, N) that touch ∂Q, and set A2 (Q, N) = ∪R∈Q1 (Q,N) N(R); this is a thin annulus in Q near ∂Q. We observe that F1 ⊂ f(A2 (Q, N)), apply (4.3.9) to each cube R ∈ Q1 (Q, N), notice that the sets N(R) have bounded overlap, and get that   Hd (F1 ∩ R)  C Hd (E ∩ N(R)) Hd (F1 )  C R∈Q1 (Q,N) R∈Q1 (Q,N) (4.4.3)  CHd (E ∩ A2 (Q, N)). Thus by (4.4.1)-(4.4.3), (4.4.4)

Hd (E ∩ Q)  Hd (Ed ∩ Q) + h(r)rd  CHd (E ∩ A2 (Q, N)) + CNn−d d(Q)d + h(r)rd .

Assume that d(Q), then r are so small that h(r)  1, say, and rewrite (4.4.4) as (4.4.5)

Hd (E ∩ Q)  CHd (E ∩ A2 (Q, N)) + C(N)rd .

If Hd (E ∩ Q)  2C(N)rd , we are happy because we get an upper bound on Hd (E ∩ Q). Otherwise, we get the information that (4.4.6)

Hd (E ∩ A2 (Q, N))  (2C)−1 Hd (E ∩ Q),

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which is strange because A2 (Q, N) is as thin as we want, so it should not bring such a large contribution to Hd (E ∩ Q). The standard way to continue the argument (see for instance [23]) would be to apply the argument again to a cube Q1 slightly larger than Q, so that we have Q = Q1 \ A2 (Q1 , N)), find that Hd (E ∩ A2 (Q1 , N)) is even larger, iterate with larger cubes Qk and values Nk of N that get larger each time, and eventually find that if N was chosen large enough, all the Qj are contained in 2Q and then Hd (E ∩ 2Q) = +∞. We do not give the details here because they are a little painful and, once we get to (4.4.6), the significant part of the argument is actually done. For the construction of an appropriate sequence of cubes, we refer to Lemma 4.3 in [16], starting near (4.23). This completes the proof of the upper bound in (4.2.2); for the lower bound, √ we proceed in a similar way. Again we start with a cube Q such that nQ ⊂ U, assume that d(Q)−d Hd (E ∩ Q) is very small, and try to reach a contradiction. Let us again apply the Federer-Fleming argument to Q with the large integer N (to be chosen later). This gives a set Ed , which in the interior of Q is contained in the skeleton Sd (Q, N). For each d-cube S ∈ Fd (Q, N) of that skeleton, if S is not contained in ∂Q, we can apply (4.3.9) to S and find that 1 1 (4.4.7) Hd (Ed ∩ S)  CHd (E ∩ N(S))  CHd (E ∩ Q) < Hd ( S) 2 2 if d(Q)−d Hd (E ∩ Q) is small enough (depending on N). That is, we are in the  and use the set Ed−1 which case when we can apply one more projection, define f, is also a deformation of E in Q (and hence in the smallest ball B that contains Q). We compute as before, but now F2 is contained in a (d − 1)-dimensional skeleton, so we get, instead of (4.4.4), that (4.4.8)

Hd (E ∩ Q)  CHd (E ∩ A2 (Q, N)) + h(r)rd

(4.4.9)

Hd (E ∩ Q)  cd(Q)d ,

√ where r = nd(Q)/2 as before. Again C does not depend on N. This is suspicious, because on average Hd (E ∩ A2 (Q, N)) should be much smaller than Hd (E ∩ Q), and then (4.4.8) would say that Hd (E ∩ Q)  Ch(r)rd , which is even smaller than expected. The standard way to proceed would be (as in [23]) to iterate the argument, find a sequence of cubes Qj that are concentric with Q and whose density tends to 0, and then observe that this is not possible if Q is centered on a point of positive upper density. Here is a hint on how we can proceed otherwise (following the argument below Lemma 4.39 in [16]). If the lower Ahlfors regularity fails, we can find Q as above such that

with c and d(Q) as small as we want, but also 1 (4.4.10) Hd (E ∩ Q)  c2−d d(Q)d . 2

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For this we assume that the center of Q is a point of upper density 1 for E, and replace Q by 12 Q, 14 Q, and so on, until (4.4.9) fails for the first time. Now Hd (E ∩ Q)  2d Hd (E ∩ 12 Q). We use this and Chebyshev (in fact, the pigeon hole principle) to replace Q by a concentric cube Q1 , with 23 Q ⊂ Q1  Q, and for which  2 (Q1 , N))  CN−1 Hd (E ∩ Q1 )  CN−1 Hd (E ∩ Q). (4.4.11) Hd (E ∩ A We still have (4.4.8) for Q1 , so 1 Hd (E ∩ Q)  Hd (E ∩ Q1 )  CHd (E ∩ A2 (Q1 , N)) + h(r)rd 2 (4.4.12)  CN−1 Hd (E ∩ Q) + h(r)rd , with a constant C that does not depend on N, and this contradicts (4.4.9) or  (4.4.10) if h(r) and N−1 are small enough. 4.5. Rectifiability, uniform rectifiability, and projections Recall the definition of rectifiability in Subsection 2.3. It was proved by Almgren [3] that plain almost minimal sets are rectifiable. In fact, plain almost minimal sets are even uniformly rectifiable (UR) [23]. We do not want to say too much about this, but let us at least give a statement with the relevant definition. We still assume that E is a coral almost minimal set in U with gauge function h. Theorem 4.5.1. Uniform rectifiability with BPLG [23]. There exists θ > 0 and M  0 such that for each choice of x ∈ E and r ∈ (0, 1) such that B(x, 2r) ⊂ U, there is a d-dimensional Lipschitz graph G, with Lipschitz constant at most M, such that (4.5.2)

Hd (E ∩ B(x, r) ∩ G)  θrd .

The Lipschitz part means that there is a d-plane P ⊂ Rn and an M-Lipschitz  function ψ : P → P⊥ such that G = y + ψ(y) ; y ∈ P . Provided that h(r)  1, say, the constants M and θ depend only on n and d. The property stated in the theorem (E contains big pieces of Lipschitz graphs locally) is in fact the combination of two properties. First, E is locally uniformly rectifiable, which has many equivalent definitions (see [20, 21]). A simple one is that E locally contains big pieces of Lipschitz images of balls in Rd . That is, there exist θ > 0 and M  0 such that, for x ∈ E and r ∈ (0, 1) such that B(x, 2r) ⊂ U, we can find an M-Lipschitz function g : Rd ∩ B(0, r) → Rn such that (4.5.3)

Hd (E ∩ B(x, r) ∩ g(Rd ∩ B(0, r)))  θrd .

This one is obviously weaker, but in fact not that much. If it is satisfied and in addition E has big projections locally, then it contains big pieces of Lipschitz graphs locally. Big projections mean that we can find θ > 0 such that, for x ∈ E and r ∈ (0, 1) such that B(x, 2r) ⊂ U, we can find a d-plane P ⊂ Rn such that (4.5.4)

Hd (π(E ∩ B(x, r))  θrd ,

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where π denotes the orthogonal projection onto P. The converse (big projections imply BPLG) is clear. See [20, 21]. We will see a proof of the rectifiability of E that also works for sliding almost minimal sets (as soon as Γ is reasonable, and with a little more work). Surprisingly, even though the almost minimality is itself a quantitative notion, the uniform rectifiability of E is really complicated to get, and the author does not know how to extend it to sliding almost minimal sets, except in simple cases where the uniform rectifiability near the boundary is not really significant because it follows too easily from the same thing far from the boundary. It is also worth noticing that in terms of proving other results, uniform rectifiability is not as indispensable as the author once believed. Let us now prove that (4.5.5)

every almost minimal set is rectifiable,

in a way that can be extended to sliding almost minimal sets. The basic tool will be, once again, Federer-Fleming projections. We start with the observation, that the author owes to V. Feuvrier [37] (but may have been known from Almgren), that if F is totally unrectifiable, Q is cube of dimension k > d, and πξ denotes the radial projection on ∂Q with the center ξ (as in the early part of Subsection 4.3), then 1 (4.5.6) Hd (πξ (F ∩ int(Q)) = 0 for Hk -almost every ξ ∈ Q. 2 Recall that we say that the set F is totally unrectifiable when Hd (F ∩ G) = 0 for every rectifiable set G (or equivalently, for every C1 embedded submanifold of dimension d). This looks like the Besicovitch-Federer Projection Theorem, except that the projections are not parallel to (n − d)-planes, and in fact the proof of (4.5.6) uses that theorem (and Fubini). So let us prove (4.5.5). Assume instead that the almost minimal set E is not rectifiable. Write E = Erect ∪ Eirr , where Erect is rectifiable and Eirr is totally unrectifiable (see [53] for this and the density properties below). Since Hd (Eirr ) > 0, we can find x ∈ E such that the upper density of Erect at x vanishes. That is, (4.5.7)

lim r−d Hd (Erect ∩ B(x, r)) = 0.

r→0

Now consider a small cube Q centered at x, and perform a Federer-Fleming projection as before, except that when we choose the centers ξS for the various faces S, we use the standard Chebyshev argument to make sure that the following things happen. Let k denote the dimension of S; thus k  n and we are in the middle of the construction of gk . The set Ek has a part in ∂S, which will not change because πξ (y) = y for y ∈ ∂S. The totally unrectifiable part of Ek ∩ int(S) is sent to a negligible set; this can easily be done because of (4.5.6). Finally, the Hd -measure of the rectifiable part of Ek ∩ int(S) is multiplied by at most C; this last can be arranged by Chebyshev and the proof of (4.3.4). Since we know now

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that E is locally Ahlfors regular, we could even choose ξS far from Ek , so that πξ is C-Lipschitz and the argument looks a little bit simpler, but let us not bother yet. When we proceed like this, then Hd -almost all of the unrectifiable part of E ∩ int(Q) \ Sd (Q, N) disappears, because it is contained in the union of the interiors of the subfaces S of dimensions k > d that are not contained in ∂Q. The unrectifiable part of E ∩ Sd (Q, N) is negligible, because the faces of dimension d are d-rectifiable, so they do not really meet Eirr . We are left with the rectifiable part of E ∩ Q. Here, if Q is small enough, then Hd (Erect ∩ Q)  εd(Q)d , with ε as small as we want, because its center x was chosen so that (4.5.7) holds. Then applying f multiplies this measure by at most C, by choice of the ξS . Altogether, (4.5.8)

Hd (f(E ∩ int(Q)))  Cεd(Q)d .

This is small enough for f(E) not to contain the full 12 S for any face S ∈ Fd (Q, N). Then we can proceed as we did below (4.4.7), compose f with a last mapping gd onto a set Ed−1 which in the interior of Q is contained in a (d − 1)-dimensional skeleton. Then (4.4.8) holds, and we may conclude as above, or more simply observe that since we now know that E is locally Ahlfors regular, we could easily have used Chebyshev to choose a cube Q  , such that 12 Q ⊂ Q  ⊂ Q, and for which (4.4.8) fails because Hd (E ∩ A2 (Q  , N))  CN−1 Hd (E ∩ Q)  CN−1 d(Q)d and Hd (E ∩ Q  )  Hd (E ∩ 12 Q)  C−1 d(Q)d . The desired contradiction comes from  applying the argument above to Q  . We end this section with a remark on how to find big projections. We claim that when E is flat, it has no big hole. Here is the corresponding statement. Theorem 4.5.9. Keep E as above. For each τ  0 we can find ε > 0 and r0 > 0 so that the following holds. Let x ∈ E and r ∈ (0, r0 ] be such that B(x, 2r) ⊂ U. Let P be a d-plane through x and suppose that (4.5.10)

dist(y, P)  εr for y ∈ E ∩ B(x, r).

Let πP denote the orthogonal projection on P. Then (4.5.11)

πP (E ∩ B(x, r)) ⊃ P ∩ B(x, (1 − τ)r).

This stays true (with appropriate modifications) in the sliding case. It applies in many small balls because E is rectifiable and locally Ahlfors Regular, so it has approximate tangent planes almost everywhere (see [53]), and these approximate tangent planes are actual tangent planes (see Exercise 41.21 in [11]). Once we know that E is locally uniformly rectifiable, this also implies that it has big projections (and then that it contains BPLG), because the local uniform rectifiability gives enough balls where E is well approximated by d-planes (look for the WGL in [20]).

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We first give a proof when d = n − 1. Let E, B(x, r), and P satisfy the assumptions. Set B  = B(x, (1 − τ)r). Suppose we can find ξ ∈ P ∩ B  \ πP (E ∩ B(x, r)). We want to deform E in B(x, r) and save some area. Draw the vertical line π−1 P (ξ) that does not meet E ∩ B(x, r). Move points of E ∩ B(x, r) parallel to P, away from the line π−1 P (ξ), and send   ; dist(z, P)  εr . (P ∩ ∂B them radially to the thin vertical wall W = z ∈ π−1 P The measure in the tube of the deformation F is at most Hd (W)  Cεrd . All the measure Hd (E ∩ B(x, r/2))  C−1 rd disappeared. This contradicts the almost minimality if h(r) is small enough. The proof in higher co-dimension is not substantially harder. We first project   E ∩ B ∩ π−1 P (B ) on P ∩ B , and do a nice interpolation of the mapping between that set and ∂B(x, r), where we want our deformation to be the identity. We get a mapping which is nearly 1-Lipschitz, because E stays so close to P in B(x, r). Then only, once the points are sent to P, we push them radially in P ∩ B  , starting from the center ξ which is still not in the image. This way W = P ∩ ∂B  costs  nothing, and the necessary gluing inside of B(x, r) \ π−1 P (B ) costs as little as we want. See Lemma 10.10 of [23] for the proof (of almost the same statement) and  Lemma 7.38 or 9.14 in [16] for the more complicated sliding version.

5. Limits of almost minimal sets There are many things that we would expect to be true, but have to be proved. The main one is the following. Theorem 5.0.1. [10, 16] Let U ⊂ Rn be given, and suppose that each of the sets Ek is a coral (sliding), almost minimal set in E, always with the same reasonably nice sliding boundary Γ and the same gauge function h. Suppose in addition that {Ek } converges, locally in U to a closed set E∞ . Then E∞ is a coral (sliding), almost minimal set in E, with the same sliding boundary and the same gauge function h. Reasonably nice allows Γ to be a C1 surface of any dimension, but more complicated choices are allowed. We will define convergence very soon, but there will be no surprise. Theorem 5.0.1 extends to quasiminimal sets (defined in Subsection 9.2 below). There is also a variant where the sliding boundary for Ek is a set Γk that converges nicely to Γ , but let us skip it for the moment. When Ek is almost minimal with a gauge function hk , and limk→+∞ hk (r) ≡ 0, then E∞ is in fact minimal; this follows rather easily, because the statement shows that it is minimal with any of the functions hN = supkN hk . In fact, there is even a statement that says that as long as the Ek are almost minimal with a fixed gauge function, or even quasiminimal with a fixed constant, and in addition it is a minimizing sequence, then E∞ is minimal. We will use this for (9.2.7) in Section 9.2.

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Theorem 5.0.1 will be the main topic of this section. In the present state of affairs, it is still too complicated to be entirely explained here, but at least we will be able to say something about the lower semicontinuity estimate that is at the center of the argument. Maybe soon Camille Labourie will come up with a simpler argument for the full limiting theorem. First we define the convergence, using the following normalized local Hausdorff distance: for x, r such that B(x, r) ⊂ U, set  dx,r (E, F) = r−1 sup dist(y, E) ; y ∈ F ∩ B(x, r) (5.0.2)  + r−1 sup dist(y, F) ; y ∈ E ∩ B(x, r) . When F ∩ B(x, r) is empty, we let the first supremum be 0, and similarly for the second supremum when E ∩ B(x, r) = ∅. Then let the Ek be closed in U (we don’t need the other case) and E∞ be closed in U (this gives the simplest definitions); we say that {Ek } tends to E∞ (locally in U) when (5.0.3)

lim dx,r (Ek , E) = 0 for every ball B(x, r) ⊂⊂ U.

k→+∞

This definition is nice, because a standard argument with diagonal subsequences shows that given any sequence {Ek } of closed sets in U, we can always extract a subsequence that converges (locally in U) to some closed set E∞ . Our main example will be the blow-up limits of a closed set E. Given x0 ∈ E, a blow-up limit of E at x0 is any limit of a convergent sequence {Ek } (as above), where Ek = r−1 k [E − x0 ] for some sequence {rk } that tends to 0. By what we just said, there is always at least one blow-up limit of E at x0 (start from rk = 2−k and extract a converging subsequence), and in general there may be lots of blow-up limits of E at x0 (think about a spiral). When we work with sliding boundaries, we typically take sequences for which in addition the dilations r−1 k [Γ − x0 ] converge to a limit Γ∞ . So we decided to study the limits of a sequence {Ek } of almost minimal sets in U, all with the same U, the same boundary set Γ (to simplify), and the same gauge function h. An important ingredient is the following. Lemma 5.0.4. Let U, and {Ek } be as above, and suppose that {Ek } converges to E∞ . Then E∞ is rectifiable. The author believes that Almgren [3] probably had this (in the plain case) with essentially the proof below, but he did not check recently. In [23], the authors proved first that the sets Ek are uniformly rectifiable (with uniform bounds), and deduced the lemma from this; this looks reasonable, because uniform rectifiability goes to the limit well, while simple rectifiability does not. That is, it is very easy to find a sequence of rectifiable sets Ek (for instance composed of 4k little squares) that converge to a totally unrectifiable Cantor set E∞ . But of course the sets Ek are not uniformly almost minimal! The author only re-discovered Lemma 5.0.4 (with probably stupid surprise) after spending some time, not being able to prove the uniform rectifiability of

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the sliding almost minimal sets and being upset about it. In addition, the proof is quite simple and it seems that rectifiability of the limit is nearly as useful as uniform rectifiability. So let us give (the idea of) the proof in [16], except that in order to simplify the argument a little, we forget about the sliding boundary and assume that the Ek are plain almost minimal sets. We suppose that E∞ is not rectifiable, and proceed as in the proof of rectifiability. Take a point x0 ∈ E∞ , where the upper density of the rectifiable part Erect ∞ vanishes. Then let Q0 be any small cube centered at x0 . First observe that the Ek satisfy the Ahlfors regularity properties in 3Q0 , uniformly in k. A simple covering argument (with balls of the same size) shows that then E∞ also is Ahlfors regular in 2Q0 . Next we can replace Q0 by a concentric cube Q  , such that 12 Q ⊂ Q   Q, and for which the measure of E∞ near ∂Q is fairly small. More precisely, we first cut Q into Nn subcubes R ∈ Q(N, Q) as we did before, and then we set A+ (N, Q) = (1 + N−1 )Q  \ (1 − 10N−1 )Q  . This set is designed to be a little larger than the thin annulus A2 (N, Q) that was used before; by the pigeon hole principle, we we can easily find Q such that (5.0.5)

Hd (E∞ ∩ A+ (N, Q))  CN−1 Hd (E∞ ∩ 2Q0 )  CN−1 d(Q0 )d .

Next we can use again the local Ahlfors regularity of the Ek and E∞ , and simple coverings by balls, to prove that for k large, (5.0.6)

Hd (Ek ∩ A2 (N, Q))  CHd (E∞ ∩ A+ (N, Q))  CN−1 d(Q0 )d .

Let us now find a Federer-Fleming projection f that essentially kills Eirr ∞ ∩ Q, by at most C. We proceed a and multiplies the (very small) measure of Erect ∞ little differently as for (4.5.5), because it will be better to have some uniformity. So, when we choose the centers ξS of the various faces S in the Federer-Fleming construction, we use the local Ahlfors regularity of E∞ near 2Q to select ξS at distance at least C−1 N−1 d(Q) from the previous image of E∞ . We skip the details again, but the reader may find this argument in Lemma 3.31 of [23], and later in [16]. This way, (4.3.2) says that all the mappings that compose f are C-Lipschitz. Of course f is C-Lipschitz too. Notice that it is also naturally defined and still C-Lipschitz (as a composition of radial projections) in a small neighborhood of E∞ , which in particular contains Ek ∩ 2Q for k large. Then f never multiplies the measure of pieces of Ek inside cubes by more than C, and we get that for k large, (5.0.7)

Hd (f(Ek ) ∩ A(N, Q))  CHd (Ek ) ∩ A2 (N, Q))  CN−1 d(Q0 )d ,

where A(N, Q) is the thinner annulus composed of cubes R ∈ Q(N, Q) that touch ∂Q, and by (5.0.6). This takes care of f(Ek ) ∩ A(N, Q). As in the proof of (4.5.5), we still have some latitude to choose the centers ξS , in particular so that in Q \ A(N, Q), the image of Eirr ∞ is negligible. Since

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d(Q)−d Hd (Erect ∞ ∩ Q) is as small as we want, this implies that f(E∞ ) ∩ Q \ A(N, Q) never fills a half face. This allows us to add one more step to the construction, and get a new mapping, which we shall also call f, so that now f(E∞ ) ∩ (Q \ A(N, Q)) is contained in a (d − 1)-dimensional skeleton, and (again by construction of f), this is also true for f(Ek ) ∩ (Q \ A(N, Q)) for k large. Together with (5.0.7), this yields Hd (f(Ek ) ∩ Q)  CN−1 d(Q0 )d .

(5.0.8)

Recall that Hd (Ek ∩ D)  C−1 d(Q0 )d by local Ahlfors regularity. When Q is √ sufficiently small, so that h( nd(Q)) is very small, all this contradicts the almost  minimality of Ek ; the rectifiability of E∞ follows. We come to the main reason why Theorem 5.0.1 works, which is the lower semicontinuity of Hd . Lemma 5.0.9. Let U, and {Ek } be as above, and suppose that {Ek } converges to E∞ . Then (5.0.10)

Hd (E∞ ∩ V)  lim inf Hd (Ek ∩ V) k→+∞

for every open set V ⊂ U. This even works for quasiminimal sets (see Subsection 9.2), and also when we replace Hd with a large class of elliptic integrands [30]. Here we’ll rapidly discuss the case of plain almost minimal sets, but sliding minimal (or even sliding quasiminimal) sets work as well. For the lower semicontinuity property, it makes sense to take V open. For a closed square V, for instance, it could be that the Ek are lines segments outside V that tend to a side of V. It is worth noting that the result fails miserably without the almost minimality assumption: a sequence of dotted lines may converge to a line. But dotted lines are not almost minimal! In earlier versions of [10] and a first part of [16], the author insisted on using the fact that our almost minimal sets satisfy the “uniform concentration property” of Dal Maso, Morel, Solimini [9]. This property was introduced, in the context of minimizers of the Mumford-Shah functional, precisely to prove the lower semicontinuity of Hn−1 along minimizing sequences, and then possibly get existence results, and it was very tempting to use it, especially because it is an easy consequence of uniform rectifiability. It turns out [16] to be also a consequence of the rectifiability of limits (Lemma 5.0.4), which is lucky because we still cannot prove yet that sliding almost minimal sets are always uniformly rectifiable. But in fact Yangqin Fang [30] discovered that there is a simpler direct proof of (5.0.10), that also uses the rectifiability of the limit, and works in the context of elliptic integrands. Let us say a few words about the proof. Let {Ek }, E∞ , and V be as in the statement. Let ε > 0 and τ > 0 be small, to be chosen later. Observe that since E∞ is rectifiable, for Hd -almost every point of E∞ ∩ V, we have that for r > 0 small

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enough, (5.0.11)

E∞ is εr-close to some d-plane P = P(x, r) in B(x, 2r),

which just means that dist(y, P)  εr for y ∈ E ∩ B(x, 2r), and also (5.0.12)

Hd (E∞ ∩ B(x, r))  (1 + τ)ωd rd i,

where ωd is the Hd -measure of the unit ball in Rd . The balls that satisfy (5.0.11) and (5.0.12) are what is often called a Vitali covering of E ∩ V, and by Vitali’s covering argument (see for instance [53]) we can cover Hd -almost all of E∞ ∩ V by disjoint balls B(xi , ri ) ⊂ V, with small radii ri , and that satisfy (5.0.11) and (5.0.12). Then we can find a finite subcollection {B(xi , ri )}, i ∈ I, that catches most of the mass, so that  (1 + τ)ωd rd (5.0.13) Hd (E∞ ∩ V)  ε + i. i∈I

Now we use Theorem 4.5.9 to estimate each rd i . Notice that there is a finite number of indices i to try, and for each one, if k is large enough the assumptions of Theorem 4.5.9 are satisfied by Ek , with a ball Bi,k centered on Ek (this is needed for the statement), contained in B(xi , ri ) (this will be used soon), and yet of radius (1 − 10ε)ri (this is easy to obtain). The size assumption on the radius r of the theorem is satisfied if we made sure to take the radii small enough in our Vitali collection earlier. And maybe (4.5.10) is only satisfied with the constant 2ε. Anyway, if we choose ε small enough, depending on τ, we get that (5.0.14)

Hd (Ek ∩ B(xi , ri ))  Hd (Ek ∩ Bi,k )  Hd (πP (Ek ∩ Bi,k ))  [(1 − τ)(1 − 10ε)]d ωd rd i,

where P and the last estimate come from (4.5.11). We sum all this, use the fact that the B(xi , ri ) are disjoint and contained in V, and get that for k large,  Hd (E∞ ∩ V)  ε + (1 + τ)ωd rd i i∈I

(5.0.15)

 ε + (1 + τ)[(1 − τ)(1 − 10ε)]−d



Hd (Ek ∩ B(xi , ri ))

i∈I

 ε + (1 + τ)[(1 − τ)(1 − 10ε)]

−d

Hd (Ek ∩ V).

Choosing ε and τ small enough, (5.0.10) and Lemma 5.0.9 follow.



At this point, it really feels like Theorem 5.0.1 should be easy to prove. As these notes are being written, this is not the case. Both in [10] (for the plain case) and [16] (for the sliding case), an additional long and painful construction of competitors is used with lots of special cases and coverings. But the author hopes that Camille Labourie will soon come up with a much more pleasant (and even more general) proof. The conclusion of this section is that, thanks in particular to the (surprising) rectifiability of limits, we have a very nice tool, Theorem 5.0.1, that will allow us to use compactness in many circumstances and make it simpler to think about

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regularity. The story below about blow-up limits and minimal cones depends on this! The author imagines that a large part of people’s preference for weak objects such as currents, flat chains, varifolds, was largely due to the apparent absence of a good limiting theorem for sets. Hopefully Theorem 5.0.1 will also become easy in the near future.

6. Monotonicity of density, near monotonicity, and blow-up limits Here we start with plain almost minimal sets; the situation for sliding almost minimal sets is more complicated and will be discussed later. It is very useful to know, in many circumstances involving minimality (think about minimal sets, surfaces, and currents, but free boundary problems are also concerned by this issue), that some scale-invariant quantity is nondecreasing, or just nearly monotone. Here the quantity of interest will be the density (6.0.1)

θx (r) = r−d Hd (E ∩ B(x, r)),

where we often take the origin x in the (almost) minimal set E. We shall learn with time that low density often rhymes with simplicity, so saying that θx is nondecreasing can also be a way to say that the situation in smaller balls tends to be simpler. The monotonicity of θx for minimal things is a rather ubiquitous fact. Here we just give a small number of statements and hints of proofs, then discuss variants, easy applications, and the sliding case. 6.1. Near monotonicity in the plain case (6.1.1)

We start with the simplest statement:

If E is a plain minimal set in U, then for any x ∈ U r → θx (r) is nondecreasing on (0, dist(x, ∂U)).

But we shall often use the more general near monotonicity of θx for almost minimal sets x. Even though some statements (depending on which definition of almost minimality one takes) also work for x ∈ U \ E, we shall restrict our attention to x ∈ E. Theorem 6.1.2. There is α > 0, that depends on n, such that if E is a plain almost minimal set in U, with a gauge function h that satisfies a Dini condition, then for x ∈ E r dt

(6.1.3) r → θx (r) exp α h(2t) t 0 is a nondecreasing function on (0, 12 dist(x, ∂U)).

r By Dini condition, we just mean that the integral 0 h(2t) dt t converges near 0. When h ≡ 0 and x ∈ E, we recover (6.1.1). Near monotonicity is almost as useful as monotonicity, because the Dini condition says that the integral in (6.1.3) has a limit when r tends to 0, so that for instance, under the assumptions of the

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theorem, we can define the density at x ∈ E by θ(x) = θx (0) = lim θx (r).

(6.1.4)

r→0

 θ(x)  C by Theorem 4.2.1; in fact it is not hard to see that Notice that θ(x)  ωd (the density of a plane). Let us discuss the proof of Theorem 6.1.2. We start with the main case when E is minimal, as in (6.1.1), and the main idea is to compare E with the cone over E ∩ ∂B(x, r). Let us assume for simplicity that x = 0. Since r → Hd (E ∩ B(0, r)) is nondecreasing, it is the integral of its derivative (seen as a Stiljes measure), which is no less than its almost-everywhere derivative. Thus, after a computation that we skip here, it is enough to check that for almost every r ∈ (0, dist(0, ∂U)),  ∂ d H (E ∩ B(0, r)  d r−d−1 Hd (E ∩ B(0, r)). (6.1.5) r−d ∂r But, by the co-area theorem, or rather more directly by approximating the rectifiable set E by C1 surfaces and computing the derivative for each one,  ∂ d (6.1.6) H (E ∩ B(0, r)  Hd−1 (E ∩ ∂B(0, r)) ∂r for almost every r. So it is enough to show that for a.e. r, r (6.1.7) Hd (E ∩ B(0, r))  Hd−1 (E ∩ ∂B(0, r)). d This is beginning to look good, because if X denotes the cone over E ∩ ∂B(0, r)), another application of the co-area formula (or more simply a parameterization of the cone and the area formula) yields r d−1 H (E ∩ ∂B(0, r)) = Hd (X ∩ B(0, r)). (6.1.8) d So the proof would be finished if we knew that X coincides in B(0, r) with a deformation of E in B(0, r). This is not exactly the case, but yet we can approach X (in B(0, r)) by Lipschitz deformations of E in B(0, r), where the Lipschitz mapping is radial, expands a lot an annulus near ∂B(0, r), and contract the rest of B(0, r) brutally to the origin. See Figure 6.1.9 C−1

∂B(0,r)

∂B(0,r) r

0

E before the annulus expansion

Its image

r

The radial mapping

Figure 6.1.9. A deformation of E that is close to the cone. Thus, with a simple limiting argument it is easy to get (6.1.1). In the case of almost minimal sets, the proof is a little longer but the idea remains the same. As before, for almost each r, we can approximate the cone X, construct deformations

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of E in B(0, r), apply the definition of almost minimal sets, and get an estimate that relates Hd (E ∩ B(0, r)) to Hd−1 (E ∩ ∂B(0, r)). We can see this as a differential inequality that relates Hd (E ∩ B(0, r)) and its distribution (or Stieltjes) derivative, which we can integrate to recover (6.1.3). That is, if we differentiate the right-hand side of (6.1.3), the derivative that comes from the Dini integral compensates the error term that comes from the almost minimality of E, if α is large enough. The computations are slightly more unpleasant than in the minimal case, but there is no surprise at the end.  6.2. The almost-constant density principle Here we describe a trick which is often useful, and relies on (unfortunately more than) the proof of Theorem 6.1.2. In rough terms, if E is almost minimal with a small enough gauge h, and in addition θx is nearly constant (instead of nearly monotone) on (0, 2r), then E looks a lot like a minimal cone in B(x, r). This turns out to be quite helpful. Of course a preliminary to this is that if E is a minimal set in B(0, r), and θ0 is constant on (0, r), then E is a minimal cone (we should say, coincides with a minimal cone in B(0, r)). This is true, but the author only knows a surprisingly unpleasant proof [13]. When we look at the proof above, we rapidly find that for almost every x ∈ E, the tangent plane to E at x contains the origin. But to go from this to the conclusion, the author did not find a way that does not use the construction of a complicated deformation of E. Let us now state the almost-constant density principle, and then say how it follows from the preliminary fact. Proposition 6.2.1. For each small δ > 0, we can find ε > 0 such that, if x ∈ E (an almost minimal set in U, with gauge function h), B(x, 2r) ⊂ U, h(2r)  ε, and (6.2.2)

θx (2r) 

inf

0 0. Fix x ∈ E and a blow-up limit Z of E at x. Then there   is a radius r > 0 and C1+a -diffeomorphism Φ with Φ B(0, 2r) ⊂ Φ(B(x, 2r)) ⊂ R3 , Φ(0) = x, and Φ(Z ∩ B(0, 2r)) = E ∩ Φ(B(0, 2r)) ⊃ E ∩ B(x, r). Here a > 0 is a constant that depends only on n and α. Notice that this description implies the uniqueness of the blow-up limit of E at x (in this case, we’ll say E has a tangent cone). We can even say more. For each small τ > 0, we can also make sure that (1 − τ)|y − z|  |Φ(y) − Φ(z)|  (1 − τ)|y − z| for y, z ∈ B(0, 2r), and then naturally Φ(B(0, 2r)) ⊃ B(0, r). It is important that not only E has a C1+a parameterization, but also the parameterization extends to the ambient space. But in the present case, all this simply boils down to the fact that E is composed of the right number of faces, that meet with the correct 2π/3 angles. The story about 2π/3 angles was not announced in the statement, but it easily follows because all of the blow-up limits of E at such points must be sets of type Y (except naturally for the central point when Z is of type T). More regularity (even, all the way to analyticity) could be obtained if we were concentrating on minimal sets. Yet we don’t know the best value of a (but the author would expect α/2). In the other direction, we expect that some (Dini-type) constraint on h is needed for the theorem to hold, we know how to make it work with gauge functions significantly larger than rα , but did not really look for the optimal results. This is a really beautiful result. Recall that the singularities above can be seen in real soap films. The theorem says that (as long as the modeling is correct) you won’t see any other ones. This is what we would love to imitate in more complicated situations. Yet let us mention one drawback: at this time, we do not always know how to estimate r. This is related to the following unpleasant fact. Suppose E is almost minimal in B(0, 1), with a small enough gauge function, and is close enough in B(0, 1) to a plane or a set of type Y. Then our proof shows that the conclusion of Theorem 7.2.1 holds, and that we can even take r = 1/2. But when E looks a lot like a cone of type T, we cannot say that E contains a point of type T (i.e., where the blow-up limits of E are cones of type T). For 2-dimensional sets in Rn , n  4, there is a similar result [13, 14] but weaker in the sense that unfortunately we do not have a precise complete list of minimal cones, and we can prove the full C1+a result only for some blow-up limits; for

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the other ones, we only get a Hölder equivalence and maybe not the uniqueness of the blow-up limit. The author tried in [15] to give a shorter description of the proof, and will use this excuse to say very little here. Surprisingly, for the Hölder equivalence, the main ingredients are merely the monotonicity of density, the almost-constant density principle of Subsection 6.2 and (to find points with high enough density), some small amount of topology. For the proof of the C1 equivalence, we prove a decay property for density, coming from a differential inequality like θx (r)  C−1 r−1 [θx (r) − θx (0)]. Recall that the monotonicity of density (i.e., θ   0) essentially follows by comparing E with a cone. For the differential inequality, we assume that θx (r) > θx (0), and construct a competitor (a deformation of E) which is significantly better than the cone over E ∩ ∂B(x, r), and then hope that we get the right inequality. One of the main ingredient is that for flat enough surfaces, Δf = 0 is a good approximation of the minimal surface equation, hence the graph of a harmonic extension often has a significantly smaller energy than the cone (the graph of the radial extension). We use this, suitable gluing arguments between flat surfaces, and reduce to a length estimate (the “full length property” of X) on small perturbations of the geodesics of K = ∂B(0, 1) ∩ X, where X is the minimal cone that approximates E in B(0, 1). When n = 3 the three known minimal cones satisfy this; when n  4 we don’t know.

8. Sliding almost minimal sets Return to the sliding Plateau problem and almost minimal sets. We would like to extend the theorem of J. Taylor to this context, and the next accessible case seems to be the description of sliding almost minimal sets of dimension 2, near a point of the sliding boundary (because otherwise we may use Theorem 7.2.1). We’ll rapidly restrict to the case when the sliding boundary Γ is a C1 curve, or even a straight line, but we will say a few words about the case when n = 3, Γ is surface (or even a plane) and E has to stay on one side. Again, the dream would be to be able to do the following. First, list all the minimal cones relative to the subject, i.e., all the sliding minimal cones of dimension 2, with a sliding boundary which is a line or a plane (when n = 3). And then, for each such cone X, prove an analogue of Theorem 7.2.1 at points x ∈ E where X is a blow-up limit of E. If we do this we get a classification, up to C1+a diffeomorphisms, of all the singularities of sliding almost minimal sets. But we’ll see that this is a little optimistic 8.1. Sliding minimal cones Recall Definitions 2.7.1 and 3.0.2 for sliding deformations and sliding almost minimal sets relative to a sliding boundary Γ .

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Before we start making lists, observe that every plain minimal set is automatically a sliding minimal set, regardless of its position relative to Γ . This is because we add constraints in the definition of a deformation, so we have less competitors, so it is easier to be sliding minimal. The reader should not be shocked by this in the context of soap films: it is often easy to introduce a (wet!) needle through a soap film, with very little disturbance. Conversely, a sliding minimal set that does not meet Γ is automatically a plain minimal set, because the condition (2.7.6) is void. Here we give examples of sliding minimal cones of dimensions 1 and 2, and we’ll only mention the new (non plain) ones. We start with the simplest problems. When d = 1 and Γ is a line, E can also be a half line with its endpoint on Γ and perpendicular to Γ , or a V-set composed of two half lines that leave from a same point p ∈ Γ , that make the same angle with L, but in opposite directions because they also make an angle at least 2π/3 with each other. That last condition is needed, as before, because otherwise we may replace a piece of V by a piece of Y with the same basis. The verifications are not hard, and neither is the case when d = 1 and Γ is a higher dimensional vector space. The next simple case is when d = 2, n = 3, Γ is the horizontal plane, and E, as well as its deformations, is required to contain Γ and lie in the (closed) upper half space. We’ll call this Fang’s case because what follows comes from [31] and [32]. The sliding minimal cones are then Γ , the union of Γ with a half plane perpendicular to Γ , and the union of Γ with a half cone of type Y perpendicular to Γ . This setting is not so ridiculous. It comes from the following initial problem. We start from a region Ω bounded by a smooth bounded surface Γ , so that Ω is on one side of Γ , and we look at sliding almost minimal sets E ⊂ Ω, associated to the sliding boundary Γ , and with the additional constraint that E ⊃ Γ . That is, sliding deformations as defined as in Section 2.7, but we also required that ϕt (x) ∈ Ω when x ∈ E. Sliding almost minimal sets are defined in terms of these deformations. There is a Plateau problem associated to this. Take a closed set E0 ⊂ Ω that contains Γ , and try to find a sliding deformation E of E0 (that contains Γ ), such that Hd (E) is minimal. If such a set exists, it is clearly sliding minimal. But this is also true when E is a solution of the Reifenberg homology problem associated to the boundary set Γ as in Section 2.6, but with the additional constraint that E ⊂ Ω. Now it turns out that any blow-up limit of a sliding almost minimal set in this setting is one of the minimal cones of Fang’s case (after a rotation, so that the tangent plane to Γ becomes the horizontal plane, and Ω lies above the plane). And this is a case where the boundary regularity program mentioned above works fine: it is possible to prove that if E is a sliding almost minimal set with the constraints above, then near every point of E ∩ Γ , the set E is equivalent through

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a C1+a diffeomorphism to one of the minimal cones above. See [31, 32]; these results also yield new existence results, but let us not elaborate. We should observe that when we do not require E to contain Γ and lie on one side, the situation becomes complicated again, and probably even more difficult than when Γ is a curve, because E may being tangent to Γ in complicated ways. From now on, we assume that d = 2 and Γ is a line. Often we’ll be in R3 , but this is not always needed. There are a few new sliding minimal cones, i.e., that are not plain minimal. The simplest ones are the half planes bounded by Γ ; we’ll call H(Γ ) the collection of these half planes. The verification that every H ∈ H is minimal is not hard. Let π denote the orthonormal projection on the plane P that contains H, and let F be any (sliding) deformation of H in a large ball B centered on Γ . With a little bit of topology (extend the deformation mapping to P by symmetry and use monodromy or degree theory), it is possible to show that π(F ∩ B) contains H ∩ B. Then H2 (F ∩ B)  H2 (π(F ∩ B))  H2 (H ∩ B), and the minimality of E follows. See Section 39 of [19]. Next are the cones of type V, or V-sets. Those are the unions V = H1 ∪ H2 of two half planes Hi ∈ H(Γ ), that make an angle at least 2π/3 along Γ . This is the generalization of the V-sets of dimension 1 above, the reason for the angle condition is the same as above (otherwise, pinch), and the verification of minimality is not so hard either (with a slicing argument; see [7] or Section 39 of [19]). We’ll call V(Γ ) the collection of these cones. A special case of set V ∈ V(Γ ) is a plane that contains Γ . It is also a special case of plain minimal cone, but we expect a different behavior from E near the points where such a plane is a blow-up. The same thing is (sadly) true of the cones Y ∈ Y(Γ ), the cones of type Y whose singular set coincides with Γ . In addition to these cones, Xangyu Liang suggested that the cone Q over the edges of a cube, with great diagonal Γ , is also sliding minimal. Soap experiments suggest that this may be true, but we have no proof in either direction. There may be other sliding minimal cones that we did not think about, but the author would bet that (in R3 ) this is not the case. In Rn , n  4, there are probably many more sliding minimal cones. Anyway, for the discussion below, R3 is enough trouble already. 8.2. Regularity attempts near Sliding minimal cones Here E will be a sliding minimal of dimension 2 in Rn , bounded by a smooth curve Γ . In fact, to keep things simple,we’ll just assume that Γ is a line, and say that the general case would be similar. We want to follow the program mentioned above, and give a good local description of E near any point x ∈ E ∩ Γ , based on the knowledge of a blow-up limit X of E at x. As one could guess, this will be harder to do for some cones X, and we shall not be able to complete our program in all cases. The results below are taken from [19].

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We start with the simplest case when X is a half plane. Then we get a full theorem of Taylor type. Theorem 8.2.1. Let E be a (coral) sliding minimal set in B(0, 2r0 ), with a sliding boundary which is a line through 0. Suppose that h(r)  ε(r/r0 )1+α for 0 < r < 2r0 and d0,2r0 (E, H)  ε for some half plane H ∈ H(L). Then E is C1+a -equivalent to H in B(0, r0 ). Here ε > 0 and a > 0 are small positive constants that depend only on n and on α > 0. Of course, by scale invariance, we could have restricted to r0 = 1. Another way to put the C1+a -equivalence would just have been to say that E ∩ B(0, r0 ) is composed of a single C1+a face, bounded by Γ . Anyway, E has no holes, or complicated topology, in B(0, r0 ). Notice that if 0 ∈ E ∩ Γ and some blow-up limit of E at 0 is H ∈ H(L), then the assumption of the theorem is satisfied for some radii r0 that are as small as we want. Then the conclusion shows that in fact E has a single blow-up limit at 0, and this limit is H. So that in fact the assumption is satisfied for all small r0 . The ingredients of the proof are the following. If you are happy to settle with a bi-Hölder description (rather than C1+a ), a new monotonicity formula [18] for balls B(x, r) centered on E \ L (see (8.2.2) below), some compactness arguments using the almost-constant density principle relative to that monotonicity formula (so that we can approximate E by half planes or planes in many balls), the fact that H is a cone of minimal density, and a Reifenberg parameterization if you want. This goes as in [13] for a bi-Hölder variant of J. Taylor’s theorem. If you want the full C1+a , you have to add a differential inequality, obtained by comparing with minimal cones, some gluing argument to cut E into two slightly smaller faces, and the fact that graphs of harmonic functions are usually closer to minimal than cones. The proof is not so different than the one in [14]. The monotonicity formula from [18] concerns balls B(x, r) that are centered on E \ Γ . For such balls, the usual density θx (r) = r−2 H2 (E ∩ B(x, r)) is not always monotone (for instance if E ∈ H(Γ ) and x ∈ E \ Γ , because θx (r) = π for r small and θx (r) → π/2 when r → +∞). We replace θx (r) with (8.2.2)

Fx (r) = θx (r) + r−2 H2 (Sx ∩ B(x, r)),

where Sx is the shade of L seen from x, i.e. the half plane contained in the plane that contains x and L, and that lies behind L. This new functional is nondecreasing, and is useful for some of the results of this subsection, because it is constant in some interesting cases (half planes or truncated Y-sets). But not for full Y-sets, which is the reason why these will cause trouble later. The next simplest case is when X is a generic cone V of type V. Here generic means that the two half planes that compose V make an angle β ∈ ( 2π 3 , π). The statement is almost the same as above.

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Theorem 8.2.3. Let E be a (coral) sliding minimal set in B(0, 2r0 ), with a sliding boundary which is a line through 0. Suppose that h(r)  ε(r/r0 )1+α for 0 < r < 2r0 and d0,2r0 (E, V)  ε for some generic half plane V ∈ V(Γ ). Then E is C1+a -equivalent to V in B(0, r0 ). Here ε > 0 and a > 0 are two small constants that depend only on n, α > 0, and β ∈ ( 2π 3 , π). In particular, we need to make d0,2r0 (E, V) smaller when β gets close to 2π 3 or to π, because the sharp and flat cases below are a little more complicated. Again, E has no holes or complicated topology in B(0, r0 ); it is composed of two C1+a faces, both bounded by Γ , and that make at points x ∈ Γ an angle β(x) that may vary slowly with x, but stays close to β. The proof of this, and also of the next results, is again similar to what was done in [14], but with the adapted monotonicity formula from [18]. There is no simpler Hölder argument that works here, because part of the proof consists in proving that the angle β(x) that the two faces of E make at x ∈ Γ ∩ B(0, r0 ) varies slowly enough to avoid the more complicated minimal cones studied below. The next case is when X is a sharp cone of type V, which means that the two half planes that compose V = X make an angle β = 2π 3 . This is the first case where although we have a good description of E in the C1+a category, we cannot say that E is equivalent to X, and it even has a different topology in general. We first give an incomplete statement, and then add some information. Theorem 8.2.4. Let E be a (coral) sliding minimal set in B(0, 2r0 ), with a sliding boundary which is a line through 0. Suppose that h(r)  ε(r/r0 )1+α for 0 < r < 2r0 and d0,2r0 (E, V)  ε for some sharp V-set V ∈ V(Γ ). Then in B(0, r0 ), E is composed of two main faces F1 and F2 , plus maybe a third thin set F3 composed of one or more “vertical faces”, that meet along a curve γ that is in general partially contained in Γ . Here ε > 0 and a > 0 depend only on n and α > 0. The two main faces meet all along γ, and F3 is bounded on one side by γ \ Γ , where it meets F1 and F2 with 2π 3 angles, and on the other side by Γ \ γ, where it is attached to Γ as in Theorem 8.2.1.

F1 Γ

0

generic V here Sharp V-sets tangent here Thin triangular face F0 Figure 8.2.5. E near a sharp cone of type V As Figure 8.2.5 shows, the set E looks like a V-set, but which we can pinch a little in some places along Γ and also open slightly in some other places. The general idea is that the two face F1 and F2 can escape from Γ , but have to leave a thin wall F3 that connects them to Γ . Along γ \ Γ , E has a singularity of type

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Y. Along Γ ∩ γ, E is described by Theorem 8.2.3, with anges β(x)  2π 3 that may vary. Finally, γ is tangent to Γ when it leaves it. We do not exclude the possibility that γ may leave Γ and return to it infinitely many times. Yet, for a minimal set, we would not expect anything like this, but just that F1 , F2 , and γ leave Γ frankly. It seems to me that Kenneth Brakke [6] thought that this behavior may happen even when V is a plane that contains L, but I claim that this is not the case. Next we do a small digression. Suppose 0 ∈ E ∩ Γ , and that the blow-up limit X of E at 0 is one of the plain minimal cones (a set of type P, Y, or T), and that X ∩ Γ = {0}. In this case what really happens is that the sliding condition does not really mean much, because in fact the set E is transverse to Γ . Some verification needs to be done, but nothing too hard. Then the proof of regularity for plain almost minimal sets applies in this case. This is not shocking: you may take a soap film, and then quietly put a (wet, this is important!) needle through it, and the set will essentially not be deformed; the needle will just cross E transversally. The tangential case, discussed soon, is a little more delicate, as the film may prefer to follow the needle a tiny bit. We are ready for the next case, when X is a plane P that contains Γ . Keep the same assumptions as in the theorems above, but with d0,2r0 (E, P)  ε for such a plane P. One possibility is just that E is a smooth surface which is tangent to E at 0, but a slightly more general behavior is allowed too, where E is attached to Γ along an open set I ⊂ Γ , as in Theorem 8.2.3. See Figure 8.2.6. That is, E ∩ B(0, r0 ) is still a Lipschitz graph over (its projection on) P, smooth away from Γ , and that may have a small crease along I ⊂ Γ . This happens with soap films (put a needle or a curve tangentially along the film, and you should see some attraction), but an additional reason for this is probably capillarity, which is not the subject here. E lies above Γ here

Possible creases here E U

E

Γ

0

E

U

Γ

Γ

Figure 8.2.6. E near a plane that contains Γ . Let us skip some other cases that are similar to the previous ones (see [19]), and go directly to the main bad case, when X is a cone Y ∈ Y(Γ ), i.e., which contains Γ . There is a natural conjecture about this case, which is that E is the image of Y by a homeomorphism ψ, which sends some curve Γ  = ψ−1 (Γ ) to Γ , but with no special requirement concerning the position of Γ  towards Y. The way E organizes itself around Γ (and in particular the parts of Γ where E looks like a Y-set, a V-set, a H-set, or ∅) would then follow from the position of Γ  relative to Y, as in the the previous case when X ∈ V(Γ ). See Figure 8.2.7.

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Figure 8.2.7. E near cone of Y(Γ ). On the left, the image by ψ−1 , where E becomes Y. On the right, E = ψ(Y) and Γ = ψ(Γ  ). We tried to mark directions along the EY of Y-points of E. Unfortunately, this seems harder to prove than the previous results. The main problem is that the monotonicity formula from [18] does not give good results in the case of Y-sets (it is strictly increasing in the case of interests), and it is not clear that there is a replacement for it. Another way to state the issue is to say that since we expect singularities of type Y in E \ Γ , having a good monotonicity formula for balls centered at these points should help a lot, especially if we intend to apply a form of the almost-constant density principle to control the geometry of E in balls centered at these points. We end this section with a maybe not so easy exercise. We know one way in which the sliding almost minimal set E associated to the smooth boundary curve Γ may leave Γ (i.e., at a point where E las a tangent plane that contains the tangent line to Γ ). Are there other ways to do this? 8.3. Further questions For me the next main question is the regularity of the sliding almost minimal set E near a point where E admits a blow-up limit which is a Y-set Y that contains Γ . If we can do this, there is a good hope that a general regularity result (maybe only in the Hölder category, near some isolated points of E ∩ Γ ) will ensue, and then also that existence results will follow. Making sure that we have a full list of sliding minimal cones in R3 (when Γ is a line) would be nice too. And if this is too easy, do the same thing in R4 . And then we may also worry about the following. Let Γ ⊂ R3 be a smooth closed curve, and choose a closed set E0 , if possible well attached to Γ so that the sliding Plateau problems below are nontrivial. For ε > 0 small, let Tε be a smooth open tube around Γ , of width roughly ε. Consider Fang’s problem where we minimize Hd (F) among sliding deformations F of Eε = ∂Tε ∪ E0 \ (Tε ), inside Ω = R3 \ Tε , and with the boundary ∂Tε . How do the solutions of this problem (which exist by [32]) converge to solutions of the sliding Plateau problem associated to Γ and E0 , assuming that there are some? The reader should not get confused like the author: the extra condition that E ⊃ Γ does not mean much when Γ is one-dimensional, and to some extent it should have less and less importance for the Fang problem on R3 \ ∂Tε , except to give the existence of minimizers. Probably it is better to solve the problem about the bad Y-cone, and then the existence problem for the sliding Plateau problem, before we do this.

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We did not talk at all about what happens to the regularity results when we replace Hd with other elliptic integrands, possibly not even smooth in one of the variables. A simple example of this (but for which only partial descriptions exist at this time) is to use the functional (8.3.1)

J(E) = Hd (E \ Γ ) + αHd (E ∩ Γ )

for some α ∈ [0, 1), for instance in the context where Γ is the horizontal plane in R3 and E is a sliding almost minimal set with boundary Γ , with the additional constraint that E lies in the (closed) upper half space. See [7] for some initial partial results in this context, and [28,30] and probably many others in the general context.

9. V. Feuvrier and existence results In this last section, we present a general scheme for proving existence results. The main tool here will be a construction [36] of adapted polyhedral networks, and then some of the results mentioned above. This is one of the three or four systematic ways to get existence results that the author knows of; maybe not necessarily the simplest (this also depends on people’s personal knowledge), but the author thinks it is quite nice and natural, and was perhaps misunderstood so far. It is somewhat close to Reifenberg’s initial approach in [64], or the proof of [41], but there are significant differences. Another type of proof is through currents (or flat chains, or varifolds) and the celebrated compactness theorem; traditionally they tend not to work so well for sets and size minimizing currents, but there is an approach by Almgren (with varifolds and flat chains, for the Reifenberg homology problem with integrands), which was recently nicely put together by U. Menne, Y. Fang, and S. Kolasínski in [33]. And finally there is a beautiful recent approach by De Lellis, Ghiraldin, and Maggi [26] and De Philippis, De Rosa, and Ghiraldin [27], where they take a d , minimizing sequence of sets Ek , consider a weak limit of the measures μk = H|E k and use it to find a good minimal set. It is interesting to notice that the different approaches have similar or parallel points, but also differences (so that they don’t always work in the same circumstances). 9.1. Presentation The method that we describe here consists in taking a suitably improved minimizing sequence of sets, taking a subsequence that converges for the Hausdorff distance, and hopefully the limit will be the desired minimizer. It was used by V. Feuvrier [37, 38] (to get sliding 2-dimensional minimal sets in a situation without boundary), X. Liang [48] (to get minimal sets under topological constraints [48], and Y. Fang [30] (to get Reifenberg homological minimal sets under general conditions). The first general presentation (too vague?) dates

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from [12], which apparently was not convincing enough. Yet the author does not intend to give all the details. The advantage of Feuvrier’s method is that it looks simple, and also some of the ideas are pleasant. But there are technical problems in some of the ingredients of this proof, more or less as with the other proofs. We’ll do things in the setting of [37, 38], because it is the simplest, and in particular we will not get distracted by boundary problems or topology. Let us work on a flat compact connected manifold M, without boundary, of the following type: M is obtained from a finite number of dyadic cubes, by identifying some faces. The point of this setting (which indeed looks like a toy model) is the following. First, the absence of boundary will allow us to forget about any problem of sliding boundaries on other ways to state Plateau problems. These would add technical annoyances, but not necessarily bad ones. Next, the simple flat structure will allow us to use dyadic grids on M. With a regular manifolds, we would need to worry about how to define polyhedral grids in an environment where the change of chart mappings are not necessarily affine. Apparently this technical difficulty is being taken care of by Feuvrier (constructing adapted grids on a manifold), but this adds to the technical complication of the construction. Also, we’ll work in dimension d = 2; this will allow us to use J. Taylor’s regularity theorem or its extension in [13] in the last step of the verification. But we’ll pretend to work in dimension d for some time, up to the moment when we have to take d = 2. We give ourselves a closed set E0 ⊂ M, with Hd (E0 ) < +∞, and consider the class E of images of E0 by continuous deformations. That is, E ∈ E if there is a (continuous) one parameter family {ϕt } of continuous mappings ϕt : E0 → M, with ϕ0 (x) = x on E0 , and E = ϕ1 (E). And we want to minimize Hd in the class F, i.e., find F ∈ E such that (9.1.1)

Hd (F) = m, where m = inf Hd (E). E∈E

Of course this is more interesting if m > 0, which excludes the case when E0 can be contracted in M to a point (or a lower dimensional object). This is why we want to take a manifold M with some topology. The problem could turn out to be a little too easy to solve with our flat structure, but replacing  Hd (E) with E f(x)dHd (x) for some Hölder-continuous function f on M such that C−1  f  C, or even one of the slightly more general integrands of (3.0.4)(3.0.5), could make the problem less trivial without really changing the proof. We describe a general scheme now, which can work in other circumstances as well. Since we don’t know any other way, we’ll take a minimizing sequence {Ek }, i.e., such that (9.1.2)

Ek ∈ E and

lim Hd (Ek ) = m.

k→+∞

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The logical thing to do next is try to get a subsequence that converges to a limit F, and then try F in (9.1.1). It would be nice if we could use a strong (fine) topology for this, because this would probably make it simpler to prove the minimality of F. For instance, it would be great if we could write Ek = ϕk 1 (E0 ) for some sequence }, 0  t  1, and on top of this the ϕk of one parameter mappings {ϕk t t converge uniformly on E0 × [0, 1], but we should not dream too much: how are we going to get enough compactness for such a sequence? Notice however that this is a little bit what Radó and Douglas did (see Section 2.2), by restricting to smaller classes of parameterizations, but then there are other issues and we no longer want to do this. Anyway, here we decide that we want to make sure that convergent subsequences exist, so we take the weakest topology on E, the topology of local Hausdorff convergence that was defined in near (5.0.3). But here M is compact, so this is just the topology defined by the usual Hausdorff distance. 9.2. The need for a quasiminimal haircut There are two obvious problems with what we just started. The first one is hairs. If we take a limit of a subsequence of {Ek }, it could converge to very large sets, even with infinite measure, that are certainly not minimizers. Indeed, starting with two-dimensional sets in 3-space, k , it is often very easy to replace any good-looking set Ek with an uglier set E which is a deformation of Ek obtained by growing long thin hair from it, in such k is ε-dense in M for any small ε > 0 given in advance, and yet a way that E d  k instead of Ek , it will H (Ek )  Hd (Ek ) + ε. If we have the bad fortune to use E converge to the whole M, something that we want to avoid. So we want to make Ek cleaner before we take a Hausdorff limit. This is what we shall call doing a haircut; in his time Reifenberg did this too, but maybe we want to be more systematic. We’ll find a new minimizing sequence {Fk } in E, obtained from {Ek }, and then only extract a subsequence that converges to a limit F∞ . Our second, in fact related problem, is the convergence of Hausdorff measure. We want to make sure that for this new sequence {Fk } that converges to F∞ , (9.2.1)

Hd (F∞ )  lim inf Hd (Fk ) = lim Hd (Fk ) = m. k→+∞

k→+∞

And in fact the main point of the haircut is to have (9.2.1). Now we have seen such a property before, in Lemma 5.0.9, when we were studying limits of reduced almost minimal sets. Now it will be a little too hard to find a sequence {Fk }, as above, which is composed of almost minimal sets; in fact the best way to get almost minimal sets seems to be to minimize a variant of Hd , and this is precisely what we were trying to do in the first place. But fortunately, the notion of quasiminimal sets (defined soon) is just right: it will be easier to produce a sequence of quasiminimal sets, and yet the notion is sufficiently strong for (9.2.1) to hold as soon as the sets Fk are reduced quasiminimal sets, with quasiminimality constants that do not depend on k. So let us define quasiminimal sets. We keep the

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sliding boundary condition to make a more general condition, but in the setting that we chose, there is no sliding boundary and we can take Γ = ∅. We also localize to an open set U for the same reason. Definition 9.2.2. Let U ⊂ Rn be open and let Γ ⊂ Rn be a closed set. We say that E is a (sliding) quasiminimal set in U, with the sliding boundary Γ , at the size δ0 and with the quasiminimality constant M, when for each closed ball B ⊂ U of radius at most δ0 and each deformation {ϕt }, 0  t  1, for E in B (as in Definition 2.7.1), we have the following estimate. Set  (9.2.3) W = x ∈ E ; ϕ1 (x) = x ⊂ E ∩ B; then (9.2.4)

Hd (E ∩ W)  MHd (ϕ1 (E ∩ W)).

The definition is a little complicated, but makes sense. It says that we may make E smaller by deforming a part of E, but we are not able to save more than 99% of what we modify, say. And it makes sense to allow deformations that modify just a tiny bit of E in a ball. The definition is essentially Almgren’s [3] (he called these sets “restricted sets”), but in [22] and [23] we found out that we wanted exactly that definition. In the plain case, we don’t bother with the existence of a whole one parameter family {ϕt }, because they would be easily reconstructed from ϕ1 , and we state the definition directly in terms of ϕ1 . Also, we could allow other compact sets than balls as the sets where the deformation takes place (this could possibly make a difference on complicated domains U where large closed balls are difficult to find, but so far the author did not see the difference. Finally in [16] the author felt compelled to add one more type of objects, the generalized quasiminimal sets, where one replaces (9.2.4) by (9.2.5)

Hd (E ∩ W)  MHd (ϕ1 (E ∩ W)) + rd h(r),

where r is the radius of B and h is a gauge function as above. This gives a little more generality, and the same proofs apply anyway. As was hinted before, the results of Sections 4 and 5 are still true with quasiminimal sets, with essentially the same proofs. This includes the results about limits, but not the more precise results that involve monotonicity or even less epiperimetric inequalities. This is the main point of [16]. Now how much more flexible is the theory of quasiminimal sets? In dimension d = 1, (connected) quasiminimal sets are essentially the same as Chord-arc curves (length of the arc Γ (a, b) less than C|b − a|), which, when n = 2 and for unbounded curves, are the bilipschitz images of a line. It is easy to see that (in the plain case and, say, in Rn to avoid complications near the boundary) the image of a quasiminimal set by a bilipschitz mapping is also quasiminimal, maybe with a larger constant M and a smaller δ0 . This includes bilipschitz images of (locally) minimal sets, which happen to be (locally)

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quasiminimal with constant M = 1; the verification is unfortunately less pleasant as it should be, due to the strange definition with the set W, but the reader should not be surprised either. Anyway, Lipschitz graphs, for instance, are quasiminimal, and this gives a vague idea of their regularity, since it is known that (away from the sliding boundary) they are uniformly rectifiable. Another way to get a quasiminimal set is to minimize (instead of Hd (E)) a  functional J(E) = E f(x)dHd (x) where we only know that C−1  f(x)  C (and in particular we have no good continuity property for f). We could even let f be an integrand that depends on the approximate tangent plane to E (and is defined in some other way on the unrectifiable part of E). The verification is easy. This may look a little bad, since then we cannot expect quasiminimal sets to have better regularity properties than Lipschitz graphs, but this is also what will give us enough flexibility for the proof of existence that we try to describe here. So let us return to our existence problem. We decide that we’ll do a quasiminimal haircut, i.e., find a new minimizing sequence {Fk }, such that (9.2.6)

the sets Fk are quasiminimal, with some constants δ0 > 0 and M  1 that do not depend on k.

Let F∗k denote the core of Fk ; our life is simpler if the Fk are reduced (i.e., F∗k = Fk , but in fact we do not really expect this, and we’ll have to discuss the difference). Anyway, a small verification (a little less pleasant in the sliding case, but this fact is checked in [16]) shows that the F∗k are quasiminimal, with the same constants δ0 and M. Let us replace {Fk } by a subsequence for which the F∗k converge (locally for the Hausdorff distance, as usual) to a limit F∗∞ . The theorem about limits says that since (9.2.6) holds for the F∗k and at the same time {Fk } is a minimizing sequence (and hence the F∗k cannot be improved by deformation by more than numbers εk that tend to 0; again this takes a small amount of checking, to see that the potential deformation can be extended to Fk \ F∗k , at no cost), we get that F∗∞ is a minimal set

(9.2.7)

(if we were working on a different problem, with a boundary condition, it would be sliding minimal), and, as in Lemma 5.0.9, Hd (F∗∞ )  lim Hd (F∗k ) = lim Hd (Fk ) = m

(9.2.8)

k→+∞

Hd (F

k→+∞

∗ k \ Fk )

= 0 and by (9.2.1)). We claim that our life will be a little (because simpler then, but how do we do our haircut? 9.3. Dyadic grids, polyhedral nets, and quasiminimal haircuts We start with a (slightly too) simple idea: we start from our initial set Ek (from the minimizing sequence), and we perform a Federer-Fleming projection on a dyadic grid of small mesh size. We get a new set E1k , which is contained in a finite union of faces of dimension d. Some faces S are not entirely contained in E1k , and when

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this happens, we can project again on the boundary of S, getting a new set E2k composed of full d-faces and subsets of (d − 1)-faces. This is in fact enough for us, but for the sake of organization, let us continue all the way to the set Ed k , which is still a deformation of Ek (so it lies in E), and in addition lies in the class Gk of finite unions of faces of dimensions at most d of our dyadic grid. We like finite problems, because we can solve them. So we replace Ed k by a 0 d new set Fk , which we choose to minimize H (Fk ) in the class E ∩ Gk . This may be quite a brutal change, so what was the point of the Ed k ? We hope to say that d {Ek } too is a minimizing sequence, i.e., (9.3.1)

lim Hd (Ed k) = m

k→+∞

and then it will follow that {Fk } is minimizing too, since (9.3.2)

Hd (Fk )  Hd (Ed k ).

We will return soon to the issue of whether (9.3.1) holds or not, but first let us say what we win by replacing Ed k by Fk . The point is that (9.2.6) holds, just because d Fk minimize H (Fk ) in the class E ∩ Gk . We can take δ0 = 1 (in fact, provided that earlier, we took a dyadic net of mesh size at most n−1/2 , say), and the constant M does not depend on k, just on n and the fact we use (small enough) dyadic grids. And of course it is important that estimates do not depend on k and do not get worse when we take smaller dyadic grids. Let us see how the proof starts. Let G = ϕ1 (Fk ) be a competitor for Fk , as in Definitions 9.2.2 and 2.7.1. Of course G does not necessarily lie in the class Gk , but we can apply a Federer-Fleming projection to it, as we did earlier to get Ed k, to get a competitor G  for G that lies in Gk . By definition, Hd (Fk )  Hd (G  ). Then we look carefully at how the points move, notice that G  is almost as good a competitor as G was (because whenever we move points from the interior of a face to the boundary of that face, we multiply its measure by at most C), do the algebra, and conclude. See Chapter 11 of [23], and in particular Proposition 11.13 on page 90; in the present situation, Fk is even a restricted minimizer. We return to (9.3.1). Unfortunately, things are not so simple; even for d = 1 in 2-space, Ek could coincide with the first diagonal in a large ball, and when we project it on a dyadic grid (with faces parallel to the axes), we multiply its length √ by roughly 2, which of course is bad and ruins our chances for (9.3.1). This is where the polyhedral nets constructed by V. Feuvrier in [36], and that we shall describe now, will be useful. Here our assumption that our space M is simple (essentially, a Euclidean space with identifications) will simplify our life. Suppose we are given a finite collection of disjoint sets Ui ⊂ M, such that dist(Ui , Uj )  ε for i = j and some small constant ε > 0. Then suppose that we chose for each i a dyadic grid, whose mesh size and direction are allowed to depend on i. What Feuvrier provides is a decomposition of M into small convex polyhedra, with the usual structure of dyadic cubes concerning the decomposition into faces, and whose restriction to an

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ε/10-neighborhood of Ui coincides with a dyadic subnet of the initial net that was given. The size of the polyhedra may be very small, but what is important is that the various angles between faces are bounded from below, by a positive number that depends only on n (Feuvrier says that the faces are uniformly “rotund”). Because of this, we can perform Federer-Fleming projections on these polyhedral nets, and the various constants in the constructions (coming from the constants in (4.3.4), (4.3.5), and the like) are uniform. The construction of these nets is somewhat painful. The result is not too shocking, but this is probably the less amusing part of the proof described here. Now the reader guessed what we shall do. For each k, we want to use a Feuvrier net adapted to our original candidate Ek . Let us assume that Ek is rectifiable (otherwise, we could also project on dyadic cubes in some balls, to make almost all of its unrectifiable part disappear, and modify the rest of the argument below). Then almost-cover Ek (using the Vitali covering argument as for Lemma 5.0.9) by disjoint small balls Bj where Ek is very well approximated by d-planes Pj . For each of these balls, take a dyadic grid such that Pj is a coordinate d-plane, so that when we do a Federer-Fleming projection of Ek ∩ Bj in this dyadic grid, we add almost no measure. Complete the grid as in [36]; in M \ ∪j Bj , we may multiply the measure by C when we project on the d-faces, but we don’t care, because the total mass of Ek there is as small as we want. So, with this adapted grid, we manage to do our first Federer-Fleming projection, replace Ek with Ed k d −k . That as we did before, and in addition make sure that Hd (Ed k )  H (Ek ) + 2 is, we get (9.3.1). The grid has a mesh size that will probably tend to 0 when k tends to +∞, but we don’t care. What is important is that the rotundity, then the Federer-Fleming constants, and finally the quasiminimality constants for Fk in (9.2.6), still obtained by the proof of Proposition 11.13 of [23] (for instance), do not depend on k. At this point we realized our quasiminimal haircut (i.e., replaced {Ek } with a new sequence {Fk } which is still minimizing (by (9.3.1) and (9.3.2)), and satisfies the quasiminimal condition (9.2.6). As was said before, we can replace {Fk } by a subsequence for which the cores F∗k converge to a limit F∗∞ , and we get the two properties (9.2.7) and (9.2.8). 9.4. End of game So far we used very little (the most unpleasant part is our assumption on M, which probably can be removed as soon as Feuvrier constructs his cubes in a manifold). We could even work with elliptic integrands (because Lemma 5.0.9 works in this context). But it is important to notice that we may not be finished yet. Indeed, although (9.2.7) says that F∗∞ is a minimal set (and (9.2.8) that it has the right measure), we do not know yet whether it lies in the class E, or the class where we wanted to minimize Hd . This is where we need to adapt our proof with the specific problem at hand.

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In general we want to minimize Hd (E), or a variant, in a class E such that deformations of E ∈ E automatically lie in E. This way, we start with sets Fk ∈ E. In some cases this may also imply that the core F∗k automatically lies in E, but not always. And then we take a limit of the sets F∗k , and once more it could be that E is not stable under limits. Let us first say how to deal with these two issues in the setting of that was described at the beginning of this section, and then, we’ll rapidly discuss other cases. We decided to take an initial set E0 , and for E the class of (continuous) deformations of E0 in our test manifold M, and also to restrict to d = 2. The obvious difficulty is that since we are in fact dealing with parameterizations (by E0 ), taking limits should be hard, because we have no control on the regularity of the parameterizations. What will save us is the regularity of the limit. By (9.2.7), F∗∞ is a minimal set. We also organized our problem so that there is no boundary Γ to complicate matters. We start with the simpler case when M is of dimension 3. Then we can apply Theorem 7.2.1, and get that locally E is a C1+α variant of a minimal cone of type P, Y, or T. It is also compact (because M is), so we can use this regularity to build a Lipschitz retraction near M, which we can even glue to the identity (far from E), to get a (continuous) one-parameter family of continuous mappings ψt , 0  t  1, from M to M, with the following properties. First, there is an ε > 0 such that (9.4.1)

ψt (x) = x for 0  t  1 when dist(x, F∗∞ )  2ε,

(9.4.2)

ψ1 (x) ∈ F∗∞ when dist(x, F∗∞ )  ε,

and (coming from the C1+α regularity), (9.4.3)

ψ1 is Lipschitz.

The reader imagines also that ψt (x) = x for x ∈ F∗∞ ; we shall not really need that, but we get it from the construction anyway. The proof is not hard; the point is that the existence of a Lipschitz retraction turns out to be local (we can compose retractions). Now we are ready to construct the desired minimizer for H2 in E. Recall that F∗k tends to F∗∞ , hence for k large, F∗k is contained in an ε-neighborhood of F∗∞ . We claim that F = ψ1 (Fk ) does the job. First of all, F ∈ E because F is a deformation of Fk , which itself lies in the class E by construction. Since ψ1 is Lipschitz, Hd (ψ1 (Fk \ F∗k )) = 0 because Fk \ F∗k is a finite union of faces of dimensions at most d − 1. So Hd (F) = Hd (ψ1 (F∗k ))  Hd (F∗∞ )  m, by (9.4.2) and (9.2.8). So F is the desired minimizer. In fact ψ1 (F∗k ) = F∗∞ , because otherwise the inequality above would be strict, so the core of F is our minimal set F∗k , but we did not really need to know this. The argument also works when M is of dimension larger than 2, with minor differences. The main one is that we do not know the exact list of minimal cones

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of dimension 2 in Rn , and [14] only gives a local biHölder description of F∗∞ , with any Hölder exponent smaller than 1. Yet, the combinatorial description of minimal cones (and then of F∗∞ ) in terms of faces that meet with 120◦ angles is still enough to build a retraction near F∗∞ , and then mappings ψt as above, except that (9.4.3) no longer holds, and has to be replaced by a Hölder condition, fortunately with an exponent which is as close to 1 as we want. Then we can complete the argument as above, arguing now that Hd (ψ1 (Fk \ F∗k )) = 0 because ψ1 is Hölder and Fk \ F∗k is at most (d − 1)-dimensional. And in this context, we make sure not to require that the final mappings of deformations are Lipschitz in the definition of E, because otherwise we may have trouble proving that F ∈ E! This completes our sketch of proof of existence, for this specific class E. The proof would also work for the nearly Euclidean elliptic integrands of (3.0.4)(3.0.5), but (as far as the author knows) not with more general integrands, because we do not know a variant of J. Taylor’s theorem in that context. It seems a little unfortunate that we need to reduce to d = 2 just for the sake of finding a local Lipschitz retraction onto F∗k . Yet it makes sense to the author that, especially in the context of minimizing in classes of deformations, the local regularity of the expected minimizers will play a role in the proof of existence. A priori, the existence of retractions looks like a weak regularity property, but the author is not entirely convinced that it is so weak, and anyway does not know how to prove it in higher dimensions. This is also why he does not expect an existence result for the sliding Plateau problem mentioned in Subsection 2.7 before regularity results at the boundary. We conclude this section with some comments on how the argument above works with different problems, related to other choices of classes E. What follows is more a free discussion than anything, and in particular the author did not have enough energy to check all the assertions below carefully, but he included them anyway because he believes that they explain the sort of issues that arise in variants of Plateau problems. First of all, let us not try to apply the method above when we work with a class E which is not stable under deformations (as in Definition 2.7.1). Hopefully this does not remove too many interesting problems. Let us rapidly discuss Reifenberg homology minimizers. The reason why ˇ Reifenberg, and then later authors, restricted to Cech homology, is that this is the one that passes to the limit well. Because of that, if we know that the cores ˇ homology), we can F∗k lien in the desired class E (thus defined in terms of Cech eventually get that F∗k ∈ E too. In some cases, it can be shown that the lowerdimensional part Fk \ F∗k is useless for getting the (d-dimensional) topological constraints in the definition of E, so F∗k ∈ E and we are in business. If I understand well, this is what happens in [28], although in the middle of a very different (and yet beautiful) proof, with the slight disadvantage that the authors need to use homology with a compact group (as Reifenberg did), to be able to cut the

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References

(d − 1)-dimensional piece. The other option, used by Fang [30], is to keep Fk \ F∗k , make its (d − 1)-dimensional part converge as an Ahlfors-regular set of lower dimension away from F∗k , continue with lower dimensions, and this way get a limit F that is not too big. Life is easier than with the d-dimensional piece, because we don’t need to keep precise track of the Hausdorff measure of the lower dimensional pieces, just make sure that it stays controlled so that the Hd measure of the limit vanishes. In the presentation above, we carefully avoided the existence of a boundary set Γ , because this simplified the discussion. For the Reifenberg homology problem, we cannot do that. Yet, due to the fact that we only want to minimize the measure Hd (E \ Γ ) away from Γ , it is possible to limit the discussion about Feuvrier grids and Federer-Fleming projections to what happens on the complement of Γ , because the contribution to a small neighborhood of Γ is as small as we want, and for (9.2.8) a control in compact sets of M \ Γ is enough. See [30] for details. In contrast, existence results for sliding Plateau problems should probably involve a more careful study near Γ , and so would the Reifenberg homology problem if we started to use a boundary Γ with positive Hd -measure, and not require that E ⊃ Γ in the definitions. See [48] for another existence result based on the Feuvrier scheme, [31] for a case where an existence result (this time for Reifenberg singular homology minimizers) is deduced from the regularity properties of sliding minimal sets, and [32] for a similar result for sliding minimal sets of dimension 2 bounded by a smooth surface in R3 .

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IAS/Park City Mathematics Series Volume 27, Pages 257–288 https://doi.org/10.1090/pcms/027/00865

Almgren’s center manifold in a simple setting Camillo De Lellis Abstract. We aim at explaining the most basic ideas underlying two fundamental results in the regularity theory of area minimizing oriented surfaces: De Giorgi’s celebrated ε-regularity theorem and Almgren’s center manifold. Both theorems will be proved in a very simplified situation, which however allows to illustrate some of the most important PDE estimates.

Contents 1

2

3

4

5

Introduction 1.1 Area minimizing graphs 1.2 De Giorgi’s ε-regularity theorem. 1.3 A formula for the excess. 1.4 Codimension 1 and higher codimension. 1.5 Almgren’s regularity theory and the “center manifold” Improved Lipschitz approximation 2.1 Spherical excess and scaling invariance. 2.2 Elementary remarks 2.3 Comparing spherical and cylindrical excess. 2.4 Improved Lipschitz approximation. De Giorgi’s excess decay and the proof of Theorem 1.2.1 3.1 Excess decay. 3.2 Proof of Theorem 1.2.1 3.3 Proof of the excess decay: harmonic blow-up. 3.4 Cylindrical excess decay Center manifold algorithm 4.1 The grid and the πL -approximations. 4.2 Interpolating functions and glued interpolations 4.3 Estimates on the interpolating functions. 4.4 Changing coordinates. C3,β estimates 5.1 Key estimates on the tilted interpolation. 5.2 Proof of estimate (i) in Proposition 4.3.1 5.3 Proof of the estimates (ii) and (iii) of Proposition 4.3.1

258 258 259 259 260 261 263 263 264 264 265 270 271 272 273 275 276 276 277 278 280 281 281 283 285

2010 Mathematics Subject Classification. Primary: 49Q15, 49Q05; Secondary: 35D10. Key words and phrases. Regularity theory, area minimizing, center manifold, excess decay. ©2020 American Mathematical Society

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1. Introduction In these lecture notes I will try to give the core ideas of two fundamental regularity results in geometric measure theory. The subject is rather technical and complicated and it would require at least one monographic semester course of prerequisites before one could even start with the statements. Nonetheless the core of the arguments have a simple analytic (in modern terms “PDE”) nature. These notes are an attempt of conveying them without requiring any knowledge of geometric measure theory. 1.1. Area minimizing graphs Throughout these notes we will thus fix our attention on graphs of Lipschitz maps u : Ω ⊂ Rm → Rn with Lipschitz constant Lip (u). In particular given any Borel set F ⊂ Ω we will denote by gr (u, F) the set gr (u, F) := {(x, y) ∈ F × Rn : y = u(x)} . We will sometimes omit F if it does not play an important role in our discussion. We will often use the simple observation that, if we rotate our system of coordinates by a small angle θ, gr (u, Ω) is still the graph of a Lipschitz function over some domain Ω  in the new coordinates. More precisely we have the following simple lemma. Lemma 1.1.1. There are (dimensional) constants c0 > 0 and C > 0 with the following property. Assume Ω and u are as above with Lip (u)  2 and let A ∈ SO(m + n) with |A − Id|  c0 1 (where Id is the identity map). If we define the coordinate transformation (x  , y  ) = A(x, y), then there is Ω  ⊂ Rm and a Lipschitz u  : Ω  → Rn such that {(x  , y  ) : A−1 (x  , y  ) ∈ gr (u, Ω)} = gr (u  , Ω  ) and Lip (u  )  Lip (u) + C|A − Id|. As it is customary, for F Borel, we will let Volm (gr (u, F)) be the m-dimensional Hausdorff measure of gr (u, F), for which the area formula gives the following identity  1  2 m (det Mαβ (Du))2 , 1 + |Du|2 + (1.1.2) Vol (gr (u, F)) = F

|α|2

Here we use the notation Mαβ (Dh) for the k × k minor of Dh corresponding to the choice of the α1 , . . . , αk lines and β1 , . . . , βk rows and we set |α| := k. As it is obvious from the invariance of the Hausdorff measure under rotations, Volm (gr (u, Ω)) = Volm (gr (u  , Ω  )) when Ω, Ω  , u and u  are as in Lemma 1.1.1 and Ω is Borel (it is elementary to see that then Ω  is Borel as well). In the rest of the paper we will investigate maps u whose graphs are area minimizing in the following sense: Definition 1.1.3. Let u and c0 be as in Lemma 1.1.1. We say that gr (u, Ω) is area minimizing when the following holds for every A ∈ SO(m + n) with |A − Id|  c0 : 1 From

now on |B| will denote the Hilbert-Schmidt norm of the matrix B.

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(AM) If u  is the map of Lemma 1.1.1 and v : Ω  → Rm any other Lipschitz map with {v = u  } ⊂⊂ Ω  , then Volm (gr (v, Ω  ))  Volm (gr (u  , Ω  )) = Volm (gr (u, Ω)) . 1.2. De Giorgi’s ε-regularity theorem. It is well known that area minimizing graphs are in fact real analytic if the Lipschitz constant is sufficiently small 2 . A “classical” path to the statement above is to prove first that u is C1,α and then use Schauder estimates for the Euler-Lagrange equation satisfied by u (which is in fact an elliptic system of partial differential equations) to show that u has higher regularity. The first step is a corollary of a celebrated theorem by De Giorgi. An appropriately general framework for its statement would be that of area minimizing integer rectifiable currents, which however would require the introduction of a lot of terminology and technical tools from geometric measure theory. The first goal of these notes is thus to illustrate De Giorgi’s key idea in the simplified setting of graphs. Theorem 1.2.1 (De Giorgi). For every 0 < α < 1, there are geometric constants ε0 and C > 0 depending only on α, m and n with the following property. Let Ω = B1 ⊂ Rm and u : B1 → Rn be a Lipschitz map with Lip (u)  1 whose graph is area minimizing. Assume (1.2.2)

E := Volm (gr (u, B1 )) − ωm < ε0 ,

where ωm denotes the m-dimensional volume of the unit disk3 B1 = B1 (0) ⊂ Rm . Then 1 u ∈ C1,α (B1/2 ) and in fact DuC0,α (B1/2 )  CE 2 . Observe that obviously the quantity E is nonnegative and that it equals 0 if and only if the function u is constant: E measures thus how close is the surface gr (u, B1 ) to be an horizontal disk B1 (0) × {y}. 1.3. A formula for the excess. De Giorgi proved his theorem in [3] in codimension 1 (namely n = 1) in the framework of reduced boundaries of sets of finite perimeter (which is equivalent to the setting of codimension 1 integral currents, see [22] or [34]). The statement was then generalized to higher codimension (and to minimizers of a general elliptic integrand) in the framework of integral currents by Almgren in [1]. In such generality De Giorgi’s theorem says that if at a certain scale the mass of an area minimizing current is not much larger than that of a disk of the same diameter, then the current is in fact a C1,α graph (at a slightly smaller scale). The interested reader can consult the survey article [5] for a quick and not (too) technical introduction to the topic. 2 While the restriction on the Lipschitz constant is unnecessary when n = 1 (where one can use, for instance, the celebrated De Giorgi-Nash theorem), for n > 1 the situation is much more complicated and it is for instance possible to construct Lipschitz minimal graphs which are not C1 , cf. [31]. The latter examples are, however, critical points and not absolute minimizers. 3 In these notes we will use the term disk for B (x) := {y ∈ R m : |y − x| < r} and the term ball for r Br (p) := {q ∈ Rm+n : |q − p| < r}.

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Before proceeding further we want to highlight an important computation which shows how E in (1.2.2) is essentially an L2 measure of the flatness of gr (u, B1 ). More precisely consider the standard basis e1 , . . . , em , em+1 , . . . , em+n and let π  0 := e1 ∧ . . . ∧ em be the standard unit m-vector orienting π0 = Rm × {0}. If x is a point of differentiability of u and Tp gr (u) is the tangent space to gr (u) at p = (x, u(x)), it is then possible to give a standard orientation to it using the m-vector Tp gr (u) := (e1 + du|p (e1 )) ∧ . . . ∧ (em + du|p (em )) . |(e1 + du|p (e1 )) ∧ . . . ∧ (em + du|p (em ))| In the formula above du|p (ej ) denotes the following vector of {0} × Rn ⊂ Rm+n : n  ∂uk (x) em+k . du|p (ej ) = ∂xj k=1

Moreover, we endow, as customarily, the space of m-vectors with a standard euclidean scalar product, which on simple m-vectors reads as v1 ∧ . . . ∧ vm , w1 ∧ . . . ∧ wm  = det(vj , wk ) .  In particular |v| = v,v is the induced euclidean norm. Elementary computations give then the following identity, which we leave as an exercise (in the rest of the notes we use the notation |Ω| for the Lebesgue mdimensional measure of Ω ⊂ Rm , and we denote by Volm the m-dimensional Hausdorff measure on Rm+n ). Proposition 1.3.1. Let Ω ⊂ Rm be Borel and u : Ω → Rn be a Lipschitz map. Then  1 m |Tp gr (u) − π  0 |2 dVolm (p) . (1.3.2) Vol (u, Ω) − |Ω| = 2 gr (u,Ω) For convenience we stop our discussion to introduce a quantity which will play a fundamental role in the rest of our investigations.  a unit m-vector Definition 1.3.3. Let u : Br (x) → Rn be a Lipschitz map and π orienting the plane π. The cylindrical excess of gr (u) in Cr (x) := Br (x) × Rn with respect to the (oriented) plane π is then given by  1  ) := |Tp gr (u) − π  |2 dVolm (p) . (1.3.4) E(gr (u), Cr (x), π 2 gr (u,Br (x)) If π  =π  0 is the unit m-vector which gives to Rm × {0} the standard orientation,  0 ). we then write E(gr (u), Cr (x)) in place of E(gr (u), Cr (x), π In what follows, if p = (x, y) ∈ Rm × Rn we will often, by a slight abuse of  ) (resp. Cr (p)) in place of Cr (x, π  ) (resp. Cr (x)). notation, write Cr (p, π 1.4. Codimension 1 and higher codimension. Observe that, for a general oriented surface Σ with no boundary in the cylinder Ω × Rn , the right hand side of (1.3.2) can be small even if the left hand side is fairly large. Indeed, if Σ consists

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of N parallel horizontal planes with the same orientation of π0 , the right hand side of (1.3.2) is zero, whereas the left hand side is (N − 1)|Ω|. The example is, moreover, area minimizing. In codimension 1 De Giorgi’s regularity theorem can be considerably strengthened in the following sense. Consider an oriented surface Σ with no boundary in C1 (0) ⊂ Rm × R, with volume bounded by some constant M and which locally minimizes the area. If  1 |Tp Σ − π  0 |2 =: E (1.4.1) 2 Σ is sufficiently small (depending only upon n, m and M), then Σ ∩ C1/2 (0) consists of finitely many disjoint C1,α graphs over B1/2 (and obviously the number N m of such graphs can be bounded by ω−1 m 2 M). Again, the above fact can be conveniently stated and proved in the framework of integral currents. In higher codimension the latter version of De Giorgi’s ε regularity theorem is however false, as witnessed by the following example. Let δ > 0 be small and consider the 2-dimensional surface Σ in R4 = C2 given by

(1.4.2) Σ = (z, w) ∈ B1 × C : δ2 z3 = w2 . A theorem of Federer (based on a computation of Wirtinger) guarantees that Σ is an area minimizing oriented surface (without boundary in the cylinder B1 × Rn ): the Federer-Wirtinger theorem is in fact valid for every holomorphic subvariety of Cn . By choosing δ arbitrarily small we can make (1.4.1) arbitrarily small. On the other hand there is no neighborhood of the origin in which Σ can be described by disjoint C1 graphs over B1 × {0} ⊂ C × {0}. Proving the Federer-Wirtinger theorem requires the introduction of the technology of Federer-Fleming integral currents and goes far beyond the scope of these notes (although the relevant idea is elementary; cf. for instance [5]). The important message is however that in higher codimension area minimizing oriented m-dimensional surfaces can have branching singularities of dimension m − 2, whereas the latter singularities are not present in area minimizing oriented hypersurfaces. This phenomenon creates a wealth of extra difficulties for the regularity theory of area minimizing integral currents in codimension higher than 1. 1.5. Almgren’s regularity theory and the “center manifold” In the seventies and early eighties Almgren wrote a celebrated long monograph, see [2], dedicated to the regularity theory of area minimizing currents in higher codimension, where he was able to finally tackle the presence of branching singularities and prove an optimal dimension bound for them. This complicated theory was recently significantly simplified in a series of joint works by Emanuele Spadaro and the author, see [12–16] and the survey articles [4–6]. The latter works have also sparked considerable further research in the area, going beyond Almgren’s theory and leading to answers to a number of open questions in [2], cf. [7–10, 17–20, 23–30, 32, 35, 36, 39, 40].

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The most difficult part of Almgren’s theory is the construction of what he calls “center manifold”. In a nutshell, if we consider the example of Section 1.4, we can regard it as a “two-sheeted” cover of B1 × C. Although such two-sheeted cover has a singularity in 0, the average of the two sheets is in fact precisely B1 × {0}, so it is a smooth graph over the base. Almgren’s center manifold is a powerful generalization of the latter observation: in an appropriate sense, when E in (1.4.1) is small, the area minimizing surface Σ is close to a multiple cover of the base and the average of the sheets enjoys better properties than the whole object. A nontechnical formulation is that it is possible to construct an efficient C3,β approximation of the average (with β > 0 a small positive dimensional constant): namely, the C3,β norm of such 1 approximation is bounded by E 2 while the distance between the approximation and the average of the sheets is, at every scale, much smaller than the separation between the most distant sheets. In order to illustrare the subtlety of the above claim, consider the following example:

(z, w) ∈ C2 : (w − z2 )2 = z2019 . The latter surface, which has a branching singularity at the origin (as the surface in (1.4.2)), is a double cover of B1 × {0}. For every z = 0 with |z| 0 depending only on m and n with the following property. Let Ω = B1 ⊂ Rm and u : B1 → Rn be a Lipschitz map with Lip (u)  1 whose graph is area minimizing. Assume (1.5.2)

E := Volm (gr (u, B1 )) − ωm < ε0

√ 1 and set σ := (2 m)−1 . Then u ∈ C3,β (Bσ ) and in fact DuC2,β (Bσ )  CE 2 . At a first glance the latter statement is all but surprising. After all, De Giorgi’s Theorem 1.2.1 allows to apply the classical Schauder estimates for ellyptic sys1 tems, hence we can give the estimate DuCk  C(k, m, n)E 2 for all k ∈ N. However, the striking novelty is that Theorem 1.5.1 can be proved without resorting to Schauder estimates and in fact without resorting to any PDE for the

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function u. Although we will use PDE arguments, they will be so elementary and robust that they can be used even in the general (i.e. multisheeted) situation. In [2] the corollary above is observed in the introduction as a mere curiosity. For Spadaro and the author of these notes Almgren’s remark was however a crucial starting point. After finding an elementary proof of the C3,β estimates under the assumptions of De Giorgi’s ε-regularity theorem (cf. [11]), we were able to give in [15] a construction of the center manifold which seems much more efficient than Almgren’s: while Almgren’s proof is almost 500 pages long and occupies half of his monograph, the one of [15] cuts the complexity by a factor 10 and its flexibility has proved to be very useful in other contexts (cf. [8, 18, 36]). In these lecture notes we will follow essentially [11] in the simplified setting of Lipschitz graphs to give a “geometric” proof of Theorem 1.5.1.

2. Improved Lipschitz approximation In order to prove both theorems we will make use of a preliminary important estimate: under the small excess assumption, an area minimizing graph turns out to have a much smaller Lipschitz constant on a rather large subset of its domain. The same statement is still correct in the multisheeted situation and it is a fundamental step in Almgren’s regularity theory, cf. [6]. The proof which we will give in these notes is a simplification of the one given in [13] in the multisheeted situation. 2.1. Spherical excess and scaling invariance. Consider an oriented surface Σ of dimension m in Rm+n and for every p ∈ Σ denote by Tp Σ the unit orienting m-vector of the tangent space Tp Σ. Definition 2.1.1. The spherical excess of Σ in the ball Br (p) ⊂ Rm+n with respect to a unit simple m-vector π  is given by  1 ) = |Tq Σ − π  |2 dVolm (q) . (2.1.2) E(Σ, Br (p), π ωm rm Σ∩Br (p) The spherical excess of Σ in Br (p) is defined as (2.1.3)

E(Σ, Br (p)) =

min E(Σ, Br (p), π ) . π simple, |π| = 1

Before going on with our discussion, we want to introduce a very elementary yet powerful idea. If u : Br (x) → Rn is a Lipschitz map whose graph is area minimizing, then 1 ur (y) := (u(x + ry) − u(x)) r is a Lipschitz map such that • Lip (ur ) = Lip (u); • the graph of ur is area minimizing; • E(gr (u), Bρ (x, u(x))) = E(gr (ur ), Br−1 ρ (0)).

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In other words our problem has a natural invariance under scalings and translations which will be used through the notes to reduce the complexity of several proofs and to gain an intuition on the plausibility of the statements. 2.2. Elementary remarks Note the following obvious fact: if π  is such that  ) = E(Σ, Bτ (p)) , E(Σ, Bτ (p), π then

τ m E(Σ, Bσ (q))  E(Σ, Bσ (q), π )  E(Σ, Bτ (p), π ) σ (2.2.1)

τ m E(Σ, Bτ (p)) ∀Bσ (q) ⊂ Bτ (p) . = σ Similar elementary considerations lead to the following proposition, whose proof is left as an exercise to the reader:

Proposition 2.2.2. Let u : Ω → Rn be a map with Lip (u)  2, p = (x, u(x)) and q = (y, u(y)). Then there are geometric constants C1 , C2 and C3 such that (2.2.3)

(2.2.4)

m  Volm (gr (u) ∩ Br (p))  C1 rm C−1 1 r

if r < dist (x, ∂Ω) , | π1 − π  2 |2  C2 (E(gr (u), B2r (p), π  1 ) + E(gr (u), Bρ (p), π  2 )) if r  ρ  2r < dist (x, ∂Ω)

and (2.2.5)

 2 |2  C3 (E(gr (u), Br (p), π  1 ) + E(gr (u), Br (q), π  2 )) | π1 − π if r = |p − q| < min{dist (x, ∂Ω), dist (y, ∂Ω)} .

2.3. Comparing spherical and cylindrical excess. We now want to compare the two slightly different notions of excess that we have given so far. First, consider p = (x, u(x)). To begin with, since Br (p) ⊂ Cr (x), we have the obvious inequality −m  )  2ω−1 E(gr (u), Cr (x), π ) . E(gr (u), Br (p), π mr

We next wish to show a sort of converse, under the assumption that the domain Ω of u contains B1 (0). By scaling invariance and translation invariance assume x = 0, u(0) = 0 and  0 ) < ε1 is rather small, fix a π1 r = 1. Under the assumption that E(gr (u), B1 , π such that E(gr (u), B1 , π  1 ) = E(gr (u), B1 ). Observe thus that, by Proposition 2.2.2, 1

(2.3.1)

| π0 − π  1 |  Cε12 .

Denote by p the orthogonal projection of Rm+n onto π1 . We claim that, for every η > 0, by choosing ε1 sufficiently small, (2.3.2)

p(gr (u) ∩ B1 ) ⊃ B1−η ∩ π1 .

 If we denote by π⊥ 1 the orthogonal complement of π1 and by C the cylinder

(2.3.3)

C  := (B1−η ∩ π1 ) + π⊥ 1 ,

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(2.3.2) obviously implies that gr (u) ∩ C  ⊂ B1 and allows us to establish the  ,π  1 )  ω2m E(gr (u), B1 ). inequality E(gr (u), C1−η We briefly sketch the proof of (2.3.2). First we know from Lemma 1.1.1 that  gr (u) is the graph of some function u  : Ω  → π⊥ 1 over some domain Ω ⊂ π1 . Moreover, by (2.3.1), Lemma 1.1.1 gives us the estimate 1

Lip (u  )  1 + Cε12  2 and, since (0, 0) ∈ gr (u  , Ω  ), u  (0) = 0. Thus gr (u) ∩ B1 certainly contains gr (u  , B1/4 ), provided ε1 is smaller than a geometric constant. Let now ρ be the maximal radius for which Bρ ⊂ Ω  and gr (u  , Bρ ) ⊂ B1 . First observe that  (2.3.4) |Du  |2  Cε1 , Bρ

because |Du  (x  )|  C|T(x  ,u  (x  )) gr (u  ) − π  1 |. Next we can interpolate between (2.3.4) and Du  L∞  2 to easily conclude 1−

1

1

Du  L2m (Bρ )  CDu  L∞ m Du  Lm2 (B (Bρ )

ρ)

1

 Cε12m .

Using Morrey’s embedding and u  (0) = 0 we thus get 1

u  L∞ (Bρ )  Cε12m . However by Lipschitz regularity of u  and the maximality of ρ, there is a point x  with |x  | = ρ and either |(x  , u  (x  ))| = 1 or limσ→1 (σx  , u(σx  )) ∈ gr (u, Ω). In the second case, consider to have extended the function u to the closure of Ω. Since Ω contains B1 , we then must have that |(x  , u(x  ))|  1, in particular we fall again in the first case. Thus ρ2 + |u  (x)  |2 = |x  |2 + |u  (x)  |2 = 1. Inserting the estimate for u  L∞ , we get 1

ρ2  1 − Cε1m . For ε1 sufficiently small the latter inequality guarantees ρ  1 − η (in fact we can give an effective bound for how small ε1 needs to be in terms of η, namely ε1  C(1 − η)m suffices; these type of bounds are indeed valid in general for suitable generalizations of surfaces which are stationary for the area functional). We summarize our discussion in the following lemma. Lemma 2.3.5. For every η > 0, there is ε¯ > 0 with the following property. Fix a  0 )  ε¯ . Lipschitz map u : B1 → Rn with Lip (u)  1, u(0) = 0 and E(gr (u), B1 , π  1 ) = E(gr (u), B1 ), let π1 be the Let π  1 be the unit m-vector such that E(gr (u), B1 , π m-dimensional plane which is oriented by π  1 and let C be the cylinder in (2.3.3). Then: • C  ∩ gr (u) is the graph of a Lipschitz map v : B1−η ∩ π1 → π⊥ 1

with Lip (v)  2, • gr (v) ⊂ gr (u) ∩ B1 , and ωm E(gr (u), B1 ). • E(gr (v), C  )  2

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2.4. Improved Lipschitz approximation. We now fix our attention on the map  v of the above lemma. Since the cylindrical excess E controls |Dv|2 , the classical Chebyshev’s inequality shows that | {|Dv|  Eγ }  CE1−2γ . for every positive exponent γ. Using a classical tool, namely considering a corresponding upper level set of the maximal function of |Dv| (cf. Section 6.6 of [21] and see below ), we get a stronger result: such set also has measure no larger than CE1−2γ , on the other hand the restriction of v to its complement has Lipschitz constant at most Eγ . For general functions this is the best we can do. However, the minimality assumption on gr (v) allows to give a much better bound, namely to estimate the upper level set of the maximal function with CE1+γ (provided γ is smaller than a geometric constant): this is the content of Proposition 2.4.1 below. The proof of the proposition will introduce two important points: • The fact that the Dirichlet energy and the area integrand are comparable in regions where the tangents to the graph are almost horizontal; • A “cut-and-paste” idea to construct competitors for testing the minimality of the graph of v. Both these two points will be exploited often in the rest of the notes: in the proof of the next proposition we will see all the details, but at later stages we will be less precise and just refer to the arguments of this section. Proposition 2.4.1. There are ε¯ > 0 and γ > 0 geometric constants with the following property. Let v : Br (x) → Rn be a Lipschitz function with Lip (v)  2 whose graph is area minimizing and such that E := r−m E(gr (v), Cr (x)) < ε¯ . If we set ρ := r(1 − Eγ ), then there is a set K ⊂ Bρ (x) such that (2.4.2) (2.4.3)

|Bρ (x) \ K|  rm E1+γ Lip (v|K )  Eγ .

Before coming to the proof we introduce a useful terminology, which will be used often in the rest of the notes Definition 2.4.4. Consider the set K of Proposition 2.4.1 and let w be a Lipschitz extension of v|K to Bρ which does not increase its Lipschitz constant by more than a geometric factor4 . Although w is not unique, we will call it the Eγ -Lipschitz approximation of v. Proof. First of all observe that we can allow a geometric constant C in front of the estimates: then choosing a smaller exponent γ and a sufficiently small ε¯ we can eliminate the constant. Secondly we will focus on the proof of the following 4 Indeed

by Kirszbraun’s Theorem we can require w to have the same Lipschitz constant as v|K . This √ is however a sophisticated theorem, whereas an extension which looses the geometric factor n can be constructed in an elementary way and suffices for our purposes.

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weaker statement: there is an ω > 0 (which depends only upon m and n) and a set L ⊂ Br/2 (x) satisfying (2.4.5)

|Br/2 (x) \ L|  Crm E1+ω

(2.4.6)

Lip (v|L )  CEω ,

for some exponent ω > 0. It can be in fact easily checked that the method of proof allows to pass from r to r(1 − Eμ ) for a suitably small exponent μ and γ can then be chosen to be min {ω, μ}. Finally, by scaling and translating, without loss of generality we can prove (2.4.5)-(2.4.6) when r = 2 and x = 0. Summarizing, we are left with proving the following. Let v : B2 → Rn be a Lipschitz map with Lip (v)  2, whose graph is area minimizing and E := E(gr (v), C2 ) < ε¯ . We look for a set K ⊂ B1 such that |B1 \ K|  CE1+ω

(2.4.7)

Lip (v|K )  CEω .

(2.4.8)

By (1.3.2), (1.1.2) and elementary properties of the integrand in (1.1.2),   −1 2 |Dv|  E  C |Dv|2 C B2

B2

where C is a geometric constant. Let

M|Dv|2

denote the usual maximal function  |Dv|2 . M|Dv|2 (y) := sup r−m Br (y)

r 0 and define z = w ∗ ϕEϑ , where ϕ is a standard smooth mollifier. Observe that   |w − z|2  CE2ϑ |Dw|2  CE1+2ϑ , B5/4 5 At

B3/2

this point the reader can easily check that in fact the estimate is valid with, in place of K,

K  := y ∈ B2−Eμ : M|Dv|2  E2λ

where μ is a suitable positive exponent depending on m and λ. In this and similar considerations one can refine all the arguments to pass from the outer radius 2 to an inner radius 2 − Eμ .

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whereas w − zC0  CEϑ  |Dw|2  CE . B5/4

Using Fubini’s theorem we choose a radius σ ∈]9/8, 5/4[ with the property that Volm−1 (∂Bσ \ K)  CE1−2λ and such that  (2.4.10) (|Dv|2 + |Dw|2 + |Dz|2 )  CE ∂Bσ  (2.4.11) |w − z|2  CE1+2ϑ . ∂Bσ

We fix next a second parameter κ, which we use to define the radii r1 = σ − Eκ r2 = σ − 2Eκ , whereas we set r0 = σ + Eϑ . It is not difficult to see that we can additionally require  |Dv|2  CE1+ϑ . (2.4.12) Br0 \Bσ

The idea is now to define a new function v  such that • v  = v on ∂Bσ ; • v  (x) = w(σ rx ) on ∂Br1 ; 1 • v  (x) = z(σ rx2 ) on Br2 .

In the annuli Bσ \ Br1 and Br1 \ Br2 we wish to define the function v  “interpolating” between the values on the corresponding spheres and keeping the Dirichlet energy under control. It is not difficult to see that this can be done with a linear interpolation along the radii so to have the following estimate on the annulus Bσ \ Br1    2 κ 2 2 −κ |Dv |  CE (|Dw| + |Dv| ) + CE |w − v|2 Bσ \Br1

∂Bσ

∂Bσ

and an analogous one (left to the reader as an exercise) in the annulus Br1 \ Br2 . Now, recall that {w = v} ⊂ K, |∂Bσ \ K|  CE1−2λ and both functions have Lipschitz constant no larger than 2. It is then easy to see that w − vC0 (Bσ )  CE(1−2λ)/(m−1) . We therefore achieve  Bσ \Br1

|Dv  |2  CE1+κ + CE(1−2λ)(1+2(m−1)

−1 )−κ

.

In particular, by choosing λ and then κ sufficiently small we conclude that  |Dv  |2  CE1+κ . (2.4.13) Bσ \Br1

Camillo De Lellis

Similarly, we can estimate 

269

|Dv  |2  CE1+κ + CE1+2ϑ−κ .

Br1 \Br2

Assuming thus that κ < ϑ we actually achieve  (2.4.14) |Dv  |2  CE1+κ . Bσ \Br2

We then leave to the reader to check that the Lipschitz constant of v  is bounded by a constant, thus implying that Volm (gr (v  ), Bσ \ Br2 )  |Bσ \ Br2 | + CE1+κ .

(2.4.15)

Next observe that



|Dv  |2 

 |Dz|2 .

Br2

Bσ

Now we set  = Eϑ and we write |Dz| = |Dw ∗ ϕ |  |Dw| ∗ ϕ  (|Dw|1K ) ∗ ϕ | + (|Dw|1Kc ) ∗ ϕ . Observe that r0 = σ +  and thus   2 |(|Dw|1K ) ∗ ϕ |  (2.4.16)

 2

|Dv| 

Br0 ∩K

Bσ

Bσ ∩K

|Dv|2 + CE1+ϑ .

The remaining part is estimated via  |(|Dw|1Kc ) ∗ ϕ2  CE2λ 1Kc ∗ ϕ 2L2  CE2λ 1Kc 2L1 ϕ 2L2 Bσ (2.4.17)  CE2λ E2(1−2λ) E−mϑ  CE1+κ , provided λ and ϑ are suitably chosen. Combining the last two estimates we easily achieve    2 2 1+κ |Dz|  |Dw| + CE = |Dv|2 + CE1+κ . Bσ ∩K

Bσ

Bσ ∩K

Consider now that, since Lip (v  |Br2 )  expansion of the area functional6 gives

1 Vol (gr (v ), Br2 )  |Br2 | + 2 m

(2.4.18)

Lip (z)  Lip (w)  CEλ , a simple Taylor



 |Br2 | +

1 2

Summing (2.4.15) and (2.4.18) we thus get Volm (gr (v  ), Bσ )  |Bσ | + 6 Recall

that the integrand is

1 + |A|2 +



1 2

 

|Dv  |2 + CE1+κ Br2 Bσ ∩K

|Dv|2 + CE1+κ

 Bσ ∩K

|Dv|2 + CE1+κ .

(det Mαβ (A))2

1 2

|α2

We then just need to observe that | det Mαβ (A)|  |A||α| and that

. √ 1+t = 1+

t 2

+ O(t2 ).

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Almgren’s center manifold in a simple setting

In particular, the minimality of v implies that  1 (2.4.19) Volm (gr (v), Bσ ) − |Bσ |  |Dv|2 + CE1+κ . 2 Bσ ∩K Next, write Volm (gr (v), Bσ ) = Volm (gr (v), Bσ ∩ K) + Volm (gr (v), Bσ \ K)  Volm (gr (v), Bσ ∩ K) + |Bσ \ K| . Recall however that |Dv|  CEλ on K. Hence we can use again the Taylor expansion of the area integrand and conclude  1 |Dv|2 − CE1+κ , Volm (gr (v), Bσ ∩ K)  |Bσ ∩ K| + 2 Bσ ∩K whereas on Bσ \ K we use the crude estimate  C−1 |Dv|2  Volm (gr (v), Bσ\K ) − |Bσ \ K| . Bσ \K

Summarizing (2.4.20)



Volm (gr (v), Bσ ) − |Bσ |  C−1

Bσ \K

|Dv|2 +

1 2

 Bσ ∩K

|Dv|2 − CE1+κ .

Combining (2.4.19) and (2.4.20), we get  |Dv|2  CE1+κ . Bσ \K

K was given as K := {M|Dv|2  E2λ } ∩ B3/2 . We have thus achieved the following: there are constants κ > 0 and λ0 > 0 such that, if λ  λ0 , then the inequality  |Dv|2  CE1+κ B9/8 ∩{M|Dv|2 E2λ }

holds, provided E is sufficiently small. Set now ω := min{ κ3 , λ20 }. We define m|Dv|2 (y) = sup r< 18

1 rm

 |Dv|2 . Br (y)

and L := {y ∈ B1 : m|Dv|2  E2ω }. We next recall the more precise form of the weak L1 estimates for maximal functions, that is  |B1 \ L|  CE−2ω |Dv|2 . B9/8 ∩{m|Dv|2 C−1 E2ω }

where C is a geometric constant. Since, for E small, if m|Dv|2  C−1 E2ω , then M|Dv|2  C−1 E2ω  E2λ0 . Hence we conclude |B1 \ L|  CE1+κ−2ω  CE1+ω .  Since Lip (v|L )  CEω , this completes the proof.

3. De Giorgi’s excess decay and the proof of Theorem 1.2.1 We now examine Theorem 1.2.1. The key idea is that, under its assumptions, the spherical excess decays geometrically at smaller scales: such decay reflects the

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almost harmonicity of u, which in turn is again a consequence the Taylor expansion of the area integrand, computed on the improved Lipschitz approximation. 3.1. Excess decay. De Giorgi’s excess decay is the following proposition (which by scaling and translating we could have equivalently stated with x = 0 and r = 1). Proposition 3.1.1. For every 0 < α < 1 there is a geometric constant ε1 depending only on α, m and n with the following property. Assume u : Br (x) → Rn is a Lipschitz map with Lip (u)  1 whose graph is area minimizing and let p = (x, u(x)). If (3.1.2)

 0 ) < ε1 . E(gr (u), Br (p)) < E(gr (u), Br (p), π

Then (3.1.3)

E(gr (u), Br/2 (p)) 

1 22α

E(gr (u), Br (p)) .

The next idea is that Proposition 3.1.1 can be iterated on “dyadic radii” and combined with (2.2.1) to conclude Corollary 3.1.4. For every 0 < α < 1 there is a geometric constant ε2 depending only on α, m and n with the following property. Assume u : Br (x) → Rn is as in Proposition 3.1.1 with ε2 substituting ε1 . Then

ρ 2α (3.1.5) E(gr (u), Bρ (q))  22m+2α E(gr (u), Br (p)) ∀Bρ (q) ⊂ Br/2 (p) . r Proof of Corollary 3.1.4. By scaling and translating we can assume that q = (0, 0) and r = 2. If ε2 < ε1 , we then have E(gr (u), B1/2 )  2−2α E(gr (u), B1 ) < 2−2α 2m E(gr (u), B2 (p)) < 2m−2α ε2 . Next let π2 and π1 be such that  2 ) = E(gr (u), B1/2 ) E(gr (u), B1/2 , π E(gr (u), B1 , π  1 ) = E(gr (u), B1 ) . Applying Proposition 2.2.2 we easily get ¯ (u), B1 ) | π2 − π1 |2  CE(gr and ¯ | π1 − π  0 |2  CE(gr (u), B1 , π  0) , where C¯ is a geometric constant. In particular, choosing ε2 much smaller than ε1 we can use Proposition 2.2.2 to show that ¯ π2 − π  0 )  C|  0 |2 + 2E(gr (u), B1/2 ) < ε1 E(gr (u), B1/2 , π and we can apply once again Proposition 3.1.1 to estimate E(gr (u), B1/4 )  2−2α E(gr (u), B1/2 )  2−4α E(gr (u), B1 ) . Assume now inductively that you are in the position of applying Proposition 3.1.1 on all radii r = 2−j for j = 0, . . . , k. We then get E(gr (u), B2−k )  2−2kα E(gr (u), B1 )

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and, if E(gr (u), B2−k , π  k+1 ) = E(gr (u), B2−k ), we conclude | πk+1 − π  1| 

k 

1 | πk+1 − π  1 |  C¯ 2

j=1

∞ 

1

1

2−2(j−1)α E(gr (u)B1 ) 2  C(α)ε22 .

j=1

Hence, if ε2 is sufficiently small compared to ε1 (by a factor which depends on  0 ) < ε1 . α but not on k), we can argue as above and conclude that E(gr (u), B2−k , π We are thus in the position of applying Proposition 3.1.1 even with r = 2−k . This proves inductively that E(gr (u), B2−k )  2−2kα E(gr (u), B1 ). Observe now that, given any ρ < 1, if we let k = − log2 ρ , then 2−k−1  ρ  2−k and thus E(gr (u), Bρ )  2m E(gr (u), B2−k )  2m 2−2kα E(gr (u), B1 )  22m+2α ρ2α E(gr (u), B2 (p)) .



3.2. Proof of Theorem 1.2.1 Corollary 3.1.4 leads quickly to Theorem 1.2.1. First, the graphicality lets us compare the spherical excess to the square mean oscillation of Du. Henceforth, we will use the notation (Du)x,ρ to denote the average  1 Du(y) dy . (Du)x,ρ = ωm ρm Bρ (x) Proposition 3.2.1. Let u : Ω → Rn be a map with Lip (u)  1 and let p = (x, u(x)). There is a geometric constant C > 1 such that, if BCr (x) ⊂ Ω, then  (3.2.2) |Du(y) − (Du)x,r |2 dy  Crm E(gr (u), B4r (p)) . Br (x)

Proof. First of all observe that, by the Lipschitz bound on u, gr (u, Br (x)) ⊂ B4r (p).  ) = E(gr (u), B4r (p)). Observe that, again by the LipsNext, let E(gr (u), B4r (p), π chitz bound, the plane oriented by π  is the graph of a linear map Rm ! x → Ax ∈ Rn with |A|  C0 for some geometric constant C0 . An elementary geometric computation then gives that |A − Du(y)|  C| π − T(y,u(y)) gr (u)|. Using again the Lipschitz bound and the area formula we conclude therefore   |A − Du(y)|2 dy  C | π − Tq gr (u)|2 dVolm (q) B4r (p)∩gr (u)

Br (x)

 Cr E(gr (u), B4r (p)) . m

To achieve (3.2.2) recall then that   2 |Du(y) − (Du)x,r | = min Br (x)

A

|A − Du(y)|2 .



Br (x)

Proof of Theorem 1.2.1. From Proposition 1.3.1 and (2.2.1) we easily infer that E(gr (u), B1/2 (p))  2m ε0

for all p = (x, u(x)) with x ∈ B1/2 .

If 2m ε0 is smaller than the threshold ε2 in Corollary 3.1.4 we can then combine it with Proposition 3.2.1 to conclude  1 |Du(y) − (Du)x,r |2 dy  Crm+2α E ∀x ∈ B1/2 , ∀r < . (3.2.3) 8 Br (x)

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273 1

It is a well-known lemma, due to Morrey, that (3.2.3) implies DuCα  CE 2 . We 1 , briefly sketch the proof. First observe that, for r < 16 |(Du)x,r − (Du)x,2r |2   −m 2  Cr |Du(y) − (Du)x,r | + Br (x)

  Cr



|Du(y) − (Du)x,2r |2 Br (x)



2

2

|Du(y) − (Du)x,r | +

−m

|Du(y) − (Du)x,2r |

Br (x)

 Cr2α E .

B2r (x)

In particular iterating on dyadic radii we easily get the estimate  1 1 2−iα E  C2−jα E 2 ∀k  j  3 , |(Du)x,2−k − (Du)x,2−j |  CE 2 jik−1

from which in turn we infer 1

|(Du)x,r − Dux,ρ |  Cρα E 2

∀r  ρ 

1 . 8

If x is a Lebesgue point for Du we then conclude 1

|Du(x) − Dux,ρ |  Cρα E 2

(3.2.4)

Fix now two points x, y ∈ B1/2 (0) with 2ρ := |x − y|  that Bρ (z) ⊂ B2ρ (x) ∩ B2ρ (y) to infer |(Du)x,2ρ − (Du)y,2ρ |2   −m  Cρ |Du − (Du)x,2ρ |2 + 

Bρ (z)

 Cρ

1 8

Observe

|Du − (Du)y,2ρ | Bρ (z)





2

|Du − (Du)x,2ρ | + B2ρ (x)

x+y 2 .

2

2

−m

1 . 8 and let z =

∀ρ 

|Du − (Du)y,2ρ |

 Cρ2α E .

B2ρ (y)

Combining the latter estimate with (3.2.4) we conclude 1

|Du(x) − Du(y)|  C|x − y|α E 2 whenever x, y ∈ B1/2 (0) are Lebesgue points for Du with |x − y|  18 . The conclusion of the theorem follows then from simple calculus considerations.  3.3. Proof of the excess decay: harmonic blow-up. In the rest of the chapter we focus on the proof of Proposition 3.1.1. The considerations of Section 2.3 reduce it to the following decay of the “cylindrical excess”. The simple details of such reduction are left to the reader. Proposition 3.3.1. For every 0 < α < 1 there are ε3 > 0 and η > 0 with the following property. Let v : B1−η (0) → Rn be a Lipschitz function with Lip (v)  2 whose graph is area minimizing and such that E(gr (v), C1−η ) < ε3 . Then there is a unit m-vector π  such that (3.3.2)

E(gr (v), C1/2 , π  )  2−m−2α E(gr (v), C1−η ) .

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Almgren’s center manifold in a simple setting

Using the Eγ -Lipschitz approximation w of v and a Taylor expansion, we easily see that the the Dirichlet energy of w and the excess of u are pretty close, more precisely  1 |Dw|2 = E + O(E1+γ ) 2 Br(1−Eγ ) We next use the above estimate to show Proposition 3.3.3. Let vk : Br (x) → Rn be a sequence of Lipschitz functions with Lip (v)  2 whose graphs are area minimizing and such that Ek := r−m E(gr (vk ), Cr ) ↓ 0 . Consider the rescaled functions fk :=

vk − (vk )0,r(1−Eγ ) 1

.

Ek2 1,2 Then vk converges, up to subsequences, strongly in Wloc (Br ) to an harmonic function.

Proof. Without loss of generality let x = 0 and r = 1 and assume (vk )0,1−Eγ = 0. Note first that up to subsequences we can assume the existence of a weak limit f ∈ W 1,2 (B1 ). Apply the Lipschitz approximation Proposition 2.4.1 and let Kk be the corresponding “good sets”. Moreover let wk be a Eγ k -Lipschitz extension −1 of vk |Kk to Bρk = B(1−Eγ ) . Let fk be the corresponding normalizations Ek 2 wk . k We still conclude that fk converges to f. Assume by contradiction that for some radius ρ < 1, the limit of the fk is weak in the W 1,2 topology. We then must have   |Df|2 < lim inf |Dfk |2 . k

Bρ

Bρ

Now, by using the cut-and-paste argument of Section 2.4 and the Taylor 1expansion of the area functional, we would like to use the graph of gk = Ek2 f as a competitor for gr (vk ), violating the minimality of gr (vk ). We want to achieve this task by first pasting vk with wk over an apprioprate annulus and then gk with wk in a second, slightly smaller, annulus, similarly to what was done in the proof of Proposition 2.4.1 . Note that we have at our disposal the two crucial estimates which were used in the cut-and-paste argument:  |Dgk |2 = O(Ek ) Bρ

and

 |gk − wk |2 = o(Ek ) , Bρ

However, one important issue is that we do not know that Lip (gk ) → 0, which would be crucial to compare the Volm (gr (gk )) to the Dirichlet energy of gk . In order to come around this issue fix a sequence of Lipschitz1 functions hj converging strongly in W 1,2 to f. A suitable diagonal sequence Ek2 hj(k) will have at the same time Lipschitz constants which converge to 0 and will satisfy the estimates needed to use the cut-and-paste argument.

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The harmonicity of f is proved in a similar way: if there is a competitor h for f in some Bσ ⊂⊂ B1 with less Dirichlet energy, we fix an intermediate radius ρ between σ and 1 and we then run the argument above with f replaced by the function

f(x) if |x|  σ  f (x) = h(x) if |x|  σ The cut and paste argument will then be run in annuli contained in Bρ \ Bσ .



3.4. Cylindrical excess decay We are now ready to complete the proof of Proposition 3.3.1. The first ingredient is the following estimate for harmonic functions: Lemma 3.4.1. Consider h : Br (x) → Rn harmonic and let ρ < r. Then 

ρ m+2  2 (3.4.2) |Dh − (Dh)x,ρ |  C |Dh|2 . r Bρ (x) Br (x) Proof. The proof is left to the reader: reduce it by scaling to the case r = 1, use the decomposition of h|∂B1 in spherical harmonics (see for instance [38]) and the mean-value theorem for harmonic functions.  Lemma 3.4.3. Let w : B1/2 → Rn be a Lipschitz function with Lip (w)  2. Set  be the unit vector orienting A = (Dw)0,1/2 , consider the linear map x → Ax and let π its graph according to our definitions. We then have   1 )  |Dw − (Dw)0,1/2 |2 + CLip (w) |Dw|2 . E(gr (w), C1/2 , π 2 B1/2 B1/2 Proof. The proof is left to the reader: it is a simple linear algebra computation combined with a classical Taylor expansion.  Proof of Proposition 3.1.1. First let w be the Eγ -Lipschitz approximation of the map v (see Proposition 2.4.1 and Definition 2.4.4) and observe that, provided E is suf be the unit m-vector orienting the ficiently small, w is defined on B1−2η . Let π graph of the linear map x → Ax with A = (Dw)0,1/2 . By Proposition 2.4.1 and Lemma 3.4.3 we then have  )  E(gr (w), C1/2 , π  ) + CE1+γ E(gr (v), C1/2 , π  1  |Dw − (Dw)0,1/2 |2 + CE1+γ , 2 B1/2 and

1 2

 |Dw|2  E + CE1+γ . B1−2η

Use now Proposition 3.3.3 to infer the existence of a harmonic function h such = o(E). In particular we can estimate that w − h2W 1,2 (B 1−2η )  1 )  |Dh − (Dh)0,1/2 |2 + o(E) E(gr (v), C1/2 , π 2 B1/2

276

and

Almgren’s center manifold in a simple setting

1 2

 |Dh|2  E + o(E) . B1−2η

Using Lemma 3.4.1 we then infer   1 − 2η m+2 E(gr (v), C1/2 , π )  E + o(E) 2

  1 − 2η m+2 + o(1) E(gr (v), C1−η ) ,  2 

which is enough to complete the proof.

4. Center manifold algorithm We next turn our attention to Theorem 1.5.1. We could easily proceed by first using De Giorgi’s theorem to show that u ∈ C1,α , deriving the Euler Lagrange equation for u as a minimizer of the area integrand, and then appealing to the Schauder estimates for elliptic systems. We prefer to ignore Schauder’s estimates: we will introduce an efficient approximation algorithm producing a sequence of regularizations of u which converges uniformly and for which we have uniform C3,β estimates for some positive β. In doing so we will even ignore the fact that u ∈ C1,α and only use some of the corollaries of De Giorgi’s excess decay. 4.1. The grid and the πL -approximations. m

[−σ, σ]

We start by considering the cube

⊂ B1/2 ⊂ B1

and subdividing it in 2mk closed cubes using a regular grid. We require that k  N0 , where N0 will be specified in a moment. We will denote by (L) the sidelength of each cube of the grid and by xL its center. Let pL be the point pL = (xL , u(xL )) and BL be the ball B32M0 (L) (pL ), where √ M0 is another sufficiently large geometric constant. Indeed M0 = n suffices and this choice also dictates the choice of N0 : we want to guarantee that each BL √ is contained in the cylinder C1 and thus we just need 32 nσ2−N0 < 1. Next recall that, by De Giorgi’s excess decay, if we fix any δ > 0 we can assume (4.1.1)

E(gr (u), BL )  C(L)2−2δ E

where C = C(m, n, M0 , N0 , δ). Let now πL be an oriented m-dimensional plane which optimizes the spherical excess7 in BL , namely such that  L ) = E(gr (u), BL ) . E(gr (u), BL , π Recalling the estimates of the previous chapter we get8 (4.1.2)

1

| πL − π  0 |  CE 2

7 We do not discuss here whether such optimizer is unique, since it is irrelevant for the rest of the proof: if there is more than one optimal plane, we just fix an arbitrary choice. 8 Again the constants of the next estimates depend on m, n, M , N and δ: in the rest of the notes 0 0 the constants will depend on these parameters unless we explicitly mention their dependence.

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and in particular (4.1.3)

 0 )  CE . E(gr (u), BL , π

Since we will need it often, we now introduce a special notation to deal with tilted disks and cylinders. First of all we set Br (p, π) := Br (p) ∩ (p + π) and hence we define Cr (p, π) := Br (p, π) + π⊥ , where π⊥ denotes the n-dimensional plane perpendicular to π. Applying Lemma 2.3.5 we conclude that C16M0 (L) (pL , πL ) ∩ gr (u) is in fact the graph of a Lipschitz map vL : B16M0 (L) (pL , πL ) → π⊥ L and we set E(L) := (L)−m E(gr (u), C16M0 (L) (pL , πL )) . Observe that the Lipschitz constant of vL is bounded by 1

πL − π  0 |  1 + CE 2 . Lip (vL )  Lip (u) + C| In particular, if E is sufficiently small, C16M0 (L) (pL , πL ) ∩ gr (u) ⊂ B32M0 (L) (pL , πL ) and so E(L)  CE(L)2−2δ : we can thus apply Proposition 2.4.1. We then denote by fL the E(L)γ -Lipschitz approximation of vL in C8M0 (L) (pL , πL ). fL will be called the πL -approximation. Moreover, recall that the functions fL and vL coincide on a large set, more precisely (4.1.4)

|{vL = fL } ∩ B8M0 (L) (pL , πL )|  C(L)m E(L)1+γ  CE1+γ (L)m+(2−2δ)(1+γ) .

A simple, yet useful, consequence of the latter estimate and the Lipschitz bounds on the two functions is then9 (4.1.5) Volm ((gr (vL )Δ gr (fL )) ∩ C8M0 (L) (pL , πL ))  CE1+γ (L)m+(2−2δ)(1+γ) . 4.2. Interpolating functions and glued interpolations Consider now a standard  smooth function ϕ ∈ C∞ c (B1 ) with ϕ = 1 and let ϕr be the corresponding family of mollifiers. We then set zL := fL ∗ ϕ(L) . zL will be called the tilted interpolating function relative to the cube L. We set conventionally B4M0 (L) (pL , πL ) to be the domain of definition of zL . Clearly Lip (zL )  CE(L)γ  CEγ (L)(2−2δ)γ . Observe therefore that we can use Lemma 1.1.1 to infer the existence of a map gL : B2M0 (L) (xL , π0 ) → π⊥ 0 such that gr (zL ) ∩ C2M0 (L) (pL , π0 ) = gr (gL ) . gL will be called the interpolating function relative to the cube L. 9 AΔB

denotes the symmetric difference of the two sets A and B, namely AΔ B = A \ B ∪ B \ A.

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Almgren’s center manifold in a simple setting

Observe that the domain of gL contains the open cube L  which is concentric to L and has twice its side-length. Consider now a bump function 9 9m ϑ ∈ C∞ c (] − , [ ) 8 8 m which is identically 1 on the cube [−1, 1] . We then let   2(x − xL ) ϑL (x) := ϑ . (L) Obviously ϑL is identically equal to 1 on L and it is supported in a concentric cube of sidelength equal to 98 (L). Denote by Ck all cubes L of the grid (namely of the subdivision of [−σ, σ]m into 2km closed cubes of sidelength 2−k σ). We then define the smooth function  L∈Ck ϑL (x)gL (x)  ζk (x) = L∈Ck ϑL (x) and we call it glued interpolation at scale 2−k . Almgren’s theorem is then a simple corollary of the following Theorem 4.2.1. Fix M0 and N0 as above. If δ > 0 is sufficiently small, then there are positive constants C, β and ε0 (depending on m, n, M0 , N0 and δ) with the following properties. Let u be as in Theorem 1.5.1 and consider for each k the glued interpolation ζk at scale 2−k . Then, 1

(a) Dζk C2,β  CE 2 ; (b) ζk − uC0 ([−σ,σ]m ) → 0 as k ↑ ∞. Clearly the key estimate in the theorem above is (a), since it is rather obvious that each interpolating function gL is in fact very close to u on its own domain of definition: (b) is left to the reader as an exercise. 4.3. Estimates on the interpolating functions. Consider an L ∈ Ck with k > N0 . There is then a unique cube K ∈ Ck−1 which contains it. K will be called the father of L. Analogously, if K ⊃ L, L ∈ Ck , K ∈ Cj and j < k, K will be called an ancestor of L. Finally, if K, L ∈ Ck have nonempty intersection, they will be called neighbors.10 The estimate (a) of Theorem 4.2.1 will then be a consequence of the following ones on the various “pieces” which we glue together. Proposition 4.3.1. Let u be as in Theorem 1.5.1 and the parameters M0 and N0 are fixed as in Theorem 4.2.1. Then, provided δ and E are chosen smaller than appropriate geometric constants, there are β > 0 and C such that the following holds (i) If L ∈ Ck , then (4.3.2)

3  i=1

Di gL C0 (B2M

1

0 (L)

(xL ))

 CE 2

1

D4 gL C0  C2(1−β)k E 2 .

cubes of Ck have disjoint interiors: the intersection of two neighbors is therefore a subset of some m − 1-dimensional plane and might reduce to a single point.

10 Distinct

Camillo De Lellis

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(ii) If K ∈ Cj is an ancestor of L, then (4.3.3)

4 

2(3+β−i)j Di (gL − gK )C0 (B2M

i=0

1

0 (L)

(xL ))

 CE 2 .

(iii) If K, L ∈ Ck are neighbors, then the estimate (4.3.3) holds for gL − gK on its domain of definition B2M0 (L) (xL ) ∩ B2M(0)(L) (xK ). Proof of Estimate (a) in Theorem 4.2.1. Fix k  N0 and define ϑ . θL :=  L J∈Ck ϑJ Observe that we have the obvious estimates Di θL C0  C(i)2ik

∀L ∈ Ck .

Consider now one L ∈ Ck and let N(L) be the set of its neighbors. It is then obvious that  θK (gK − gL ) . (ζk − gL ) = K∈N(L)

In particular for every i ∈ {1, 2, 3} we have11  Di ζk C0 (L)  Di gL C0 + Di (θK (gK − gL ))C0 (L) K∈N(L)



1 2

 CE + C



Dj θK C0 Di−j (gK − gL )C0 (L)

K∈N(L) 0ij 1 2

1

 CE + CE 2



1

2jk 2(i−j−3−β)k  CE 2 .

0ij 1 2

In particular Dζk C2  CE . Note that a very similar computations yields 1

D4 ζk C0  CE 2 2(1−β)k . In particular, if x, y ∈ L we easily conclude that (4.3.4)

1

|D3 ζk (x) − D3 ζk (y)|  |x − y|D4 ζk C0  CE 2 |x − y|β .

Consider next the centers xL and xK of two different cubes K, L ∈ Ck and assume for the moment that J is the first common ancestor of both L and K. Observe that (J)  C|xL − xK |. Moreover, by construction ζk equals gL in a neighborhood of xL and equals gK in a neighborhood of xK . We thus can estimate |D3 ζk (xL ) − D3 ζk (xK )| = |D3 gL (xL ) − D3 gK (xK )|  |D3 gL (xL ) − D3 gJ (xL )| + |D3 gJ (xL ) − D3 gJ (xK )| (4.3.5)

+ |D3 gJ (xK ) − D3 gK (xK )| √  D3 (gL − gJ )C0 (L) + m (J) D4 gJ C0 (J) + D3 (gJ − gK )C0 (K) 1

1

 CE 2 (J)β  CE 2 |xL − xK |β . the estimates we have used the obvious geometric fact that the cardinality of N(L) is bounded by a geometric constant C(m).

11 In

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Almgren’s center manifold in a simple setting

Combining (4.3.4) with (4.3.5) we easily conclude that, for any j  k and any cube M ∈ Cj we have12 1

[D3 ζk ]β,M  CE 2 . In particular the latter estimate holds for every cube J ∈ CN0 . Since however CN0 covers [−σ, σ]m and consists of 2kN0 cubes, we finally get 1

[D3 ζk ]β,[−σ,σ]m  CE 2 .



4.4. Changing coordinates. We will see in the next section that much of the estimates leading to Proposition 4.3.1 will in fact be carried on in the “tilted” systems of coordinates. For this reason we will make heavy use of the following technical lemma. Lemma 4.4.1. There are constants c0 , C > 0 with the following properties. Assume that (i) A ∈ SO(m + n), |A − Id|  c0 , r  1; m n (ii) (x0 , y0 ) ∈ π0 × π⊥ 0 are given and f, g : B2r (x0 ) → R are Lipschitz functions such that Lip (f), Lip (g)  c0

and |f(x0 ) − y0 | + |g(x0 ) − y0 |  c0 r.

Then, in the system of coordinates (x  , y  ) = A(x, y), for (x1 , y1 ) = A(x0 , y0 ), the following holds: (a) gr (f) and gr (g) are the graphs of two Lipschitz functions f  and g  in the tilted system of coordinates, whose domains of definition contain both Br (x1 ); (b) f  − g  L1 (Br (x1 ))  C f − gL1 (B2r (x0 )) ; (c) if f ∈ C4 (B2r (x0 )), then f  ∈ C4 (Br (x1 )), with the estimates

(4.4.2) (4.4.3)

f  − y1 C3 4 

D f C0

 Φ (|A − Id|, f − y0 C3 ) , 

 Ψ (|A − Id|, f − y0 C3 ) 1 + D4 fC0 ,

where Φ and Ψ are smooth functions. Proof. Let P : Rm×n → Rm and Q : Rm×n → Rn be the usual orthogonal projections. Set π = A(π0 ) and consider the maps F, G : B2r (x0 ) → π⊥ and I, J : B2r (x0 ) → π given by F(x) = Q(x, f(x)) and G(x) = Q(x, g(x)), I(x) = P(x, f(x)) and J(x) = P(x, g(x)). Obviously, if c0 is sufficiently small, I and J are injective Lipschitz maps. Hence, gr (f) and gr (g) coincide, in the (x  , y  ) coordinates, with the graphs of the func˜ := J(B2r (x0 )) by tions f  and g  defined respectively in D := I(B2r (x0 )) and D 12 As

it is customary in the PDE literature, [f]α,F denotes the Hölder seminorm of the function f on the subset F of its domain, namely |f(x) − f(y)| sup . [f]α,F := |x − y|α x =y,x,y∈F

Camillo De Lellis

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f  = F ◦ I−1 and g  = G ◦ J−1 . If c0 is chosen sufficiently small, then we can find a constant C such that (4.4.4)

Lip (I), Lip (J), Lip (I−1 ), Lip (J−1 )  1 + C c0 ,

and (4.4.5)

|I(x0 ) − x1 |, |J(x0 ) − x1 |  C c0 r.

Clearly, (4.4.4) and (4.4.5) easily imply (a). Conclusion (c) is a simple consequence of the inverse function theorem. Finally we claim that, for small c0 , (4.4.6)

|f  (x  ) − g  (x  )|  2 |f(I−1 (x  )) − g(I−1 (x  ))| ∀ x  ∈ Br (x1 ),

from which, using the change of variables formula for biLipschitz homeomorphisms and (4.4.4), (b) follows. Claim (4.4.6) is an elementary exercise in classical euclidean geometry and it is left to reader. 

5. C3,β estimates 5.1. Key estimates on the tilted interpolation. In this section we finally come to the core PDE argument which will allow us to derive the estimates of Proposition 4.3.1. We consider however a slightly more general situation. We fix L ∈ Ck but consider any plane π such that in the corresponding cylinder C16M0 (L) (pL , π) we have the estimate13 (5.1.1) E¯ := (L)−m E(gr (u), C16M (L) (pL , π))  CE(L)2−2δ . 0

We then let f¯ be the E¯ γ -Lipschitz approximation and set z¯ := f¯ ∗ ϕ(L) . Proposition 5.1.2. If δ and E are sufficiently small, there is β > 0 and C, C(j) geometric constants such that ¯ 1 ¯z − f (5.1.3)  CE(L)m+3+2β (5.1.4)

L (B4M

0 (L)

(pL ,π))

ΔDj z¯ C0 (B4M

0 (L)

(pL ,π))

 C(j)E(L)1−j+2β

∀j ∈ N .

Proof. In order to simplify the notation we drop pL and π and simply write Bs for Bs (pL , π). Let v be the function whose graph describes gr (u) in the π × π⊥ coordinates. Fix a test function κ ∈ C∞ c (B8M0 (L) ) and consider the first variation of the area functional along κ. By minimality of v $  d 1 + |Dv + sDκ|2 + Mαλ (Dv + sDκ)2 . 0 = [δgr (v)](κ) = ds s=0 |α|2

Moreover, since v and f¯ coincide aside from a set of measure no larger than CE1+γ (L)m+(1+γ)(2−2δ) , cf. (4.1.4) and (4.1.5), we easily conclude ¯ |[δgr (f)](κ)|  CE1+γ (L)m+(1+γ)(2−2δ) Dκ 0 . C

13 In

particular we have already shown that πL falls in such category, but we will need to consider system of coordinates (and in particular cylinders) where the “base plane” π might be tilted, compared to πL , by the same amount which estimates the tilt between πL and the horizontal plane π0 .

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Almgren’s center manifold in a simple setting

Finally we use an explicit computation and a simple Taylor expansion14 to derive    3 γ (2−2δ)γ δgr (f)(κ) ¯ ¯ ¯ − Df : Dκ¯  C |Df| |Dκ|  CE (L) DκC0 |Df|¯ 2  CE1+γ (L)m+(1+γ)(2−2δ) DκC0 (where A : B denotes the Hilbert-Schmidt scalar product of the matrices A and B, namely A : B = tr (AT B)). We next impose that δ is sufficiently small, so that (5.1.5)

(1 + γ)(2 − 2δ) = 2 + 2β

for a positive β. In particular we conclude  Df¯ : Dκ¯  CE1+γ (L)m+2+2β Dκ 0 . (5.1.6) C The gain Eγ is however not important for our considerations and we will therefore neglect it in all our subsequent estimates. If we denote by Kc the complement of the set over which f¯ and v coincide, the same considerations above (the suitable modifications are left to the reader) yield also the estimate   Df(w) ¯ : Dψ(w − x) dw dx B7M

(5.1.7)

0 (L)

 C1Kc ∗ DψL1 + C|Df|¯ 3 ∗ DψL1    c 3  C |K | + |Dv| DψL1  CE(L)m+2+2β DψL1

for every test ψ ∈ C∞ c (BM0 (L) ). The latter are the fundamental estimates from which (5.1.3) and (5.1.4) are (respectively) derived. We start with (5.1.4) considering that   Δ¯zC0 (B4M (L) ) = sup ψΔ¯z = sup Df¯ : D(ψ ∗ ϕ(L) ) 0

ψ L1 1

ψ L1 1

(5.1.6)

 CE(L)m+2+2β

sup

ψ L1 1

D(ψ ∗ ϕ(L) )C0

 CE(L)m+2+2β Dϕ(L) C0  CE(L)1+2β . As for the higher derivative estimates, they are obvious consequences of Dj z¯ = f¯ ∗ Dj (ϕ(L) ) . 14 The

Taylor expansion is the expansion of DA F, where

1  2 F(A) = 1 + |A|2 + (det Mαβ (A))2 |α|2

is the area integrand. It is then elementary that, if Aij denotes the ij-entry of the matrix A, then ∂Aij F(A) = Aij + O(|A|3 ) .

Camillo De Lellis

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However, the inequality (5.1.3) is more subtle. Write  ¯ − f(x)) ¯ ¯ dy . z¯ (x) − f(x) = ϕ(L) (x − y)(f(y) In order to simplify our notation assume for the moment x = 0 and compute    |y| ¯  y ∂f ¯ z¯ (0) − f(0) = ϕ(L) (y) τ dτ dy |y| 0 ∂r     |y| y y = ϕ(L) (y) · dτ dy ∇f¯ τ |y| |y| 0  1 ¯ = ϕ(L) (y) ∇f(σy) · y dσ dy 0

 1 ϕ(L)

=  =

0

w

¯ ∇f(w) ·w 

¯ More generally, z¯ (x) − f(x) =

σ  1 0

w dσ dw σm+1

w −m−1 σ ϕ(L) dσ dw. σ  

¯ ∇f(w) ·

=:Φ(w)



¯ ∇f(w) · Φ(w − x) dw and   ¯ = ∇f(w) · Φ(w − z) dw dz.

¯ 1 ¯z − f L (F)

F

Since ϕ is radial, the function Φ is a gradient. Indeed, it can be easily checked that, for any ψ, the vector field ψ(|w|) w is curl-free. Moreover, the support of Φ is compactly contained in B(L) . Thus we can apply (5.1.7) to derive ¯ 1  C E1+γ (L)m+2+2β Φ 1 . (5.1.8) ¯z − f L (B4M

Since ΦL1 

 1 0

 |w| ϕ

w (L)σ

0 (L)

L

)

 (L)

−m −m−1

σ

1  dσ dw = (L) 0

|y| ϕ(y) dy dσ ,

we conclude ΦL1  C(L). Inserting this in (5.1.8) concludes the proof.



5.2. Proof of estimate (i) in Proposition 4.3.1 Consider now L ∈ Ck as in the statement of the Proposition and let L = Lk ⊂ Lk−1 ⊂ . . . ⊂ LN0 be its “ancestry”. Fix the plane π = πL and fix a natural number j ∈ [N0 , k − 1]. Recall that, by the De Giorgi’s excess decay, |πL − πLj |  CE(Lj )2−2δ : arguing as for estimating E(gr (u), C16M0 (L) (pLj , πLj )), we easily conclude that (Lj )−m E(gr (u), C16M0 (Lj ) (pLj , πL ))  CE(Lj )2−2δ2 . Hence we are in the position to apply the estimate of Proposition 5.1.2 in the cylinder Cj = C8M0 (Lj ) (pLj , π). If f¯j are the corresponding Lipschitz approximations and z¯ j = f¯j ∗ ϕ(Lj ) , we then conclude from Proposition 5.1.2 Δ¯zj C0  CE(Lj )1+2β ¯zj − f¯j L1  CE(Lj )m+3+2β .

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Almgren’s center manifold in a simple setting

Consider now two consecutive maps z¯ j and z¯ j+1 . The domain of definition of the second map is B = B4M0 (L)j+1 (pLj+1 , π) and is contained in the domain of definition of the first. Moreover, recalling (4.1.4), in such common domain B both f¯j and f¯j+1 coincide with the same function (and hence they are equal) except for a set of measure at most CE1+γ (Lj )m+2+2β . In particular, they coincide on a nonempty set and, since they are both Lipschitz, f¯j − f¯j+1 C0 (B)  C(Lj ). Combining the latter two bounds we are able to estimate the L1 norm of f¯j − f¯j+1 . We thus conclude the two estimates (5.2.1) (5.2.2)

Δ(¯zj − z¯ j+1 )C0 (B)  CE(Lj )1+2β ¯zj − z¯ j+1 L1 (B)  CE(Lj )m+3+2β .

We leave to the reader the proof that, classical estimates for the Laplacian and classical interpolation inequalities (cf. for instance [33]), imply then (5.2.3)

D(¯zj − z¯ j+1 )C0 (B  )  CE(Lj )2+2β ,

for B  = B 7 M (L ) . We can next estimate Di (z¯j − z¯ j+1 C0 for i > 1 interpolating 0 j 2 between (5.2.3) and (5.2.4)

ΔDi−1 (¯zj − z¯ j+1 )C0 (B)  CE(Lj )2−i+2β ,

where the latter follows from the estimates of the higher derivatives of the Laplacian given in Proposition 5.1.2. We can thus conclude that for i ∈ {0, 1 . . . , 4} and in B  = B3M0 (Lj ) Di (¯zj − z¯ j+1 )C0 (B  )  CE(Lj )3+2β−i  CE2−(3+2β−i)j . By interpolation we achieve then [D3 (¯zj − z¯ j+1 )]β,B   CE(Lj )β . Summing the corresponding geometric series and recalling that zL = z¯ k we easily achieve Di (zL − z¯ N0 )C0 (B3M

0 (L)

[D3 (zL − z¯ N0 )]β,B3M

(pL ,πL ))

 CE

(pL ,πL )

 CE ,

0 (L)

for i ∈ {0, 1, 2, 3},

and D4 (zL − z¯ N0 )C0 (B3m

0 (L)

(pL ,πL ))

 CE(L)2β−1  CE2(1−2β)k .

Now, since z¯ N0 is the convolution at a scale comparable to 1 of a Lipschitz func1 1 tion f¯N0 with Df¯N0 L2  CE 2 , we have D¯zN0 Cs  C(s)E 2 for every s ∈ N. We therefore easily conclude15 DzL C2,β (B3M D4 zL C0 (B3M

1

(pL ,π)) 0 (L)

 CE 2 1

0 (L)

(pL ,π))

 CE 2 2(1−β)k .

Using now Lemma 4.4.1 we achieve estimate (i) in Proposition 4.3.1. 15 Indeed

the proof gives the better exponent 1 − 2β in the bound for the C4 norm.

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5.3. Proof of the estimates (ii) and (iii) of Proposition 4.3.1 First of all we observe that in order to show (ii) in full generality, it suffices to show it when K is the father of L and then sum the corresponding estimates over the relevant ancestry of L in the general case. As such, the estimates (ii) and (iii) are then very similar and they can in fact be proved using the same idea. The key is again a suitable L1 estimate. Lemma 5.3.1. Assume u and δ satisfy the assumptions of Proposition 5.1.2 and consider a pair of cubes (K, L) which consists of either father and son or of two neighboring cubes of the same cubical partition Ck . If F is the intersection of the domain of definitions of gL and gK , then gL − gK L1 (F)  CE(L)m+3+β .

(5.3.2)

The lemma will be proved below and we now show how to derive the estimates (ii) and (iii) from it. First observe that the case of fourth derivatives in (4.3.3) is an obvious consequence of the estimate (4.3.2). As for the the other derivatives, ob1 serve that we know from Part (i) of Proposition 5.1.2 that gL − gK C3,β (F)  CE 2 . Combining the latter bound with (5.3.2) we then conclude the proof applying the following lemma. Lemma 5.3.3. For every m, 0 < r < s and κ > 0 there is a positive constant C (depending on m, κ and sr ) with the following property. Let f be a C3,κ function in the disk Bs ⊂ Rm , taking values in Rn . Then (5.3.4) Dj fC0 (Br )  Cr−m−j fL1 (Bs ) + Cr3+κ−j [D3 f]κ,Bs

∀j ∈ {0, 1, 2, 3} ,

where C is a constant depending only on m, n and κ. Proof. A simple covering argument reduces the lemma to the case s = 2r. Moreover, define fr (x) := f(rx) to see that we can assume r = 1 and, arguing componentwise, we can assume n = 1. So our goal is to show (5.3.5)

3 

|Dj f(y)|  CfL1 + C[D3 f]κ

∀y ∈ B1 , ∀f ∈ C3,κ (B2 , R) ,

j=0

By translating it suffices then to prove the estimate (5.3.6)

3 

|Dj f(0)|  CfL1 (B1 ) + C[D3 f]κ,B1

∀f ∈ C3,κ (B1 ) .

j=0

Consider now the space of polynomials R in m variables of degree at most 3,  which we write as R(x) = 0|j|3 Aj xj , where we use the convention that: • j = (j1 , . . . , jm ) denotes a multi-index; • |j| = j1 + . . . + jm ; j m . • xj = x11 xj22 . . . xjm This is a finite dimensional vector space, on which we can define the norms   |R| := 0|j|3 |Aj | and R := B1 |R(x)| dx. These two norms must then be equivalent, so there is a constant C (depending only on m), such that |R|  CR

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for any such polynomial. In particular, if P is the Taylor polynomial of third order for f at the point 0, we conclude  3  |Dj f(0)|  C|P|  CP = C |P(x)| dx B1

j=0

 CfL1 (B1 ) + Cf − PL1 (B1 )  CfL1 + C[D3 f]κ .



In order to complete our task we are only left with proving Lemma 5.3.1. Proof of Lemma 5.3.1. Since the two cases are analogous, we consider the one in which K is the father of L. We let zL and zK be the corresponding tilted interpolating functions, which come from the convolutions of the functions fL and fK . Now, the graph of zK is contained in the cylinder C4M0 (K) (pK , πK ). We can however apply Lemma 4.4.1 and find functions fˆ and zˆ defined on B3M0 (K) (pK , πL ) → π⊥ L with the properties that gr (ˆz) = gr (zK ) ∩ C3M0 (K) (pK , πL ) ˆ = gr (fK ) ∩ C3M (K) (pK , πL ) . gr (f) 0 Now, by Lemma 4.4.1 we have ˆ 1  zK − fK  ˆz − f L (B3M

0 (K)

L1 (B4M

(pK ,πL ))

0 (K)

(pK ,πK ))

 CE(K)m+3+β ,

where we have used Proposition 5.1.2 in the last inequality. Consider now that B = B4M0 (L) (pL , πL ), which is the domain of zL , is conˆ and gr (fL ) coincide with gr (u) tained in B3M0 (K) (pK , πL ). Moreover, both gr (f) except for a set of m-dimensional volume bounded by C(K)m+2+β , cf. (4.1.4). In particular, |{fˆ = fL } ∩ B|  CE(K)m+2+β , and thus the two functions agree on a nonempty set. Since they are both Lipschitz with bounded Lipschitz constant,  CE(K)m+3+β . fˆ − fL  1 L (B)

We can thus use again Proposition 5.1.2 to infer ˆz − zL L1 (B)  CE(K)m+3+β . Now, consider that gr (gL ) = gr (zL ) ∩ C2M0 (L) (xL , π0 ) gr (gK ) ∩ C2M0 (L) (xL , π0 ) = gr (ˆz) ∩ C2M0 (L) (xL , π0 ) . Thus we can use Lemma 4.4.1 one last time to derive gK − gL L1 (B2M

0 (L)

(xL ,π0 ))

 Cˆz − zL L1 (B)  CE(K)m+3+β

and conclude the proof of the lemma.



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References [1] F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87 (1968), 321– 391, DOI 10.2307/1970587. MR225243 ←259 [2] Frederick J. Almgren Jr., Almgren’s big regularity paper, World Scientific Monograph Series in Mathematics, vol. 1, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. Q-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2; With a preface by Jean E. Taylor and Vladimir Scheffer. MR1777737 ←261, 262, 263 [3] Ennio De Giorgi, Frontiere orientate di misura minima (Italian), Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico Scientifica, Pisa, 1961. MR0179651 ←259 [4] Camillo De Lellis, Almgren’s Q-valued functions revisited, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1910–1933. MR2827872 ←261 [5] Camillo De Lellis, The regularity of minimal surfaces in higher codimension, Current developments in mathematics 2014, Int. Press, Somerville, MA, 2016, pp. 153–229. MR3468252 ←259, 261, 262 [6] Camillo De Lellis, The size of the singular set of area-minimizing currents, Surveys in differential geometry 2016. Advances in geometry and mathematical physics, Surv. Differ. Geom., vol. 21, Int. Press, Somerville, MA, 2016, pp. 1–83. MR3525093 ←261, 262, 263 [7] Camillo De Lellis, Two-dimensional almost area minimizing currents, Boll. Unione Mat. Ital. 9 (2016), no. 1, 3–67, DOI 10.1007/s40574-016-0057-1. MR3470822 ←261 [8] Camillo De Lellis, Guido De Philippis, Jonas Hirsch, and Annalisa Massaccesi, Boundary regularity of mass-minimizing integral currents and a question of Almgren, 2017 MATRIX annals, MATRIX Book Ser., vol. 2, Springer, Cham, 2019, pp. 193–205. MR3931066 ←261, 263 [9] Camillo De Lellis, Matteo Focardi, and Emanuele Nunzio Spadaro, Lower semicontinuous functionals for Almgren’s multiple valued functions, Ann. Acad. Sci. Fenn. Math. 36 (2011), no. 2, 393–410, DOI 10.5186/aasfm.2011.3626. MR2757522 ←261 [10] Camillo De Lellis, Andrea Marchese, Emanuele Spadaro, and Daniele Valtorta, Rectifiability and upper Minkowski bounds for singularities of harmonic Q-valued maps, Comment. Math. Helv. 93 (2018), no. 4, 737–779, DOI 10.4171/CMH/449. MR3880226 ←261 [11] Camillo De Lellis and Emanuele Spadaro, Center manifold: a case study, Discrete Contin. Dyn. Syst. 31 (2011), no. 4, 1249–1272, DOI 10.3934/dcds.2011.31.1249. MR2836351 ←263 [12] Camillo De Lellis and Emanuele Nunzio Spadaro, Q-valued functions revisited, Mem. Amer. Math. Soc. 211 (2011), no. 991, vi+79, DOI 10.1090/S0065-9266-10-00607-1. MR2663735 ←261, 262 [13] Camillo De Lellis and Emanuele Spadaro, Regularity of area minimizing currents I: gradient Lp estimates, Geom. Funct. Anal. 24 (2014), no. 6, 1831–1884, DOI 10.1007/s00039-014-0306-3. MR3283929 ←261, 262, 263 [14] Camillo De Lellis and Emanuele Spadaro, Multiple valued functions and integral currents, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 4, 1239–1269. MR3467655 ←261, 262 [15] Camillo De Lellis and Emanuele Spadaro, Regularity of area minimizing currents II: center manifold, Ann. of Math. (2) 183 (2016), no. 2, 499–575, DOI 10.4007/annals.2016.183.2.2. MR3450482 ←261, 262, 263 [16] Camillo De Lellis and Emanuele Spadaro, Regularity of area minimizing currents III: blow-up, Ann. of Math. (2) 183 (2016), no. 2, 577–617, DOI 10.4007/annals.2016.183.2.3. MR3450483 ←261 [17] Camillo De Lellis, Emanuele Spadaro, and Luca Spolaor, Regularity theory for 2-dimensional almost minimal currents II: Branched center manifold, Ann. PDE 3 (2017), no. 2, Paper No. 18, 85, DOI 10.1007/s40818-017-0035-7. MR3712561 ←261 [18] C. De Lellis, E. Spadaro, and L. Spolaor, Regularity Theory for 2-Dimensional Almost Minimal Currents II: Branched Center Manifold, Ann. PDE 3 (2017), no. 2, 3:18. MR3712561 ←261, 263 [19] Camillo De Lellis, Emanuele Spadaro, and Luca Spolaor, Uniqueness of tangent cones for twodimensional almost-minimizing currents, Comm. Pure Appl. Math. 70 (2017), no. 7, 1402–1421, DOI 10.1002/cpa.21690. MR3666570 ←261 [20] Camillo De Lellis, Emanuele Spadaro, and Luca Spolaor, Regularity theory for 2-dimensional almost minimal currents I: Lipschitz approximation, Trans. Amer. Math. Soc. 370 (2018), no. 3, 1783–1801, DOI 10.1090/tran/6995. MR3739191 ←261

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[21] Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR1158660 ←266, 267 [22] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR0257325 ←259 [23] Matteo Focardi, Andrea Marchese, and Emanuele Spadaro, Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result, J. Funct. Anal. 268 (2015), no. 11, 3290–3325, DOI 10.1016/j.jfa.2015.02.011. MR3336726 ←261 [24] Francesco Ghiraldin and Luca Spolaor, On the number of singular points for planar multivalued harmonic functions, Manuscripta Math. 154 (2017), no. 3-4, 513–525, DOI 10.1007/s00229-017-09200. MR3713924 ←261 [25] Jonas Hirsch, Partial Hölder continuity for Q-valued energy minimizing maps, Comm. Partial Differential Equations 41 (2016), no. 9, 1347–1378, DOI 10.1080/03605302.2016.1204313. MR3551461 ←261 [26] Jonas Hirsch, Boundary regularity of Dirichlet minimizing Q-valued functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 4, 1353–1407. MR3616337 ←261 [27] Jonas Hirsch, Examples of holomorphic functions vanishing to infinite order at the boundary, Trans. Amer. Math. Soc. 370 (2018), no. 6, 4249–4271, DOI 10.1090/tran/7192. MR3811527 ←261 [28] Jonas Hirsch, Salvatore Stuvard, and Daniele Valtorta, Rectifiability of the singular set of multiplevalued energy minimizing harmonic maps, Trans. Amer. Math. Soc. 371 (2019), no. 6, 4303–4352, DOI 10.1090/tran/7595. MR3917224 ←261 [29] B. Krummel and N. Wickramasekera, Fine properties of branch point singularities: Two-valued harmonic functions, ArXiv e-prints (November 2013), available at arXiv:1311.0923. ←261 [30] B. Krummel and N. Wickramasekera, Fine properties of branch point singularities: Dirichlet energy minimizing multi-valued functions, ArXiv e-prints (November 2017), available at arXiv:1711.06222. ←261 [31] H. B. Lawson Jr. and R. Osserman, Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system, Acta Math. 139 (1977), no. 1-2, 1–17, DOI 10.1007/BF02392232. MR452745 ←259 [32] Philippe Logaritsch and Andrea Marchese, Kirszbraun’s extension theorem fails for Almgren’s multiple valued functions, Arch. Math. (Basel) 102 (2014), no. 5, 455–458, DOI 10.1007/s00013-014-0642-4. MR3254787 ←261 [33] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 733–737. MR208360 ←284 [34] Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR756417 ←259 [35] Emanuele Nunzio Spadaro, Complex varieties and higher integrability of Dir-minimizing Q-valued functions, Manuscripta Math. 132 (2010), no. 3-4, 415–429, DOI 10.1007/s00229-010-0353-5. MR2652440 ←261 [36] Luca Spolaor, Almgren’s type regularity for semicalibrated currents, Adv. Math. 350 (2019), 747–815, DOI 10.1016/j.aim.2019.04.057. MR3948685 ←261, 263 [37] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR0290095 ←267 [38] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR0304972 ←275 [39] Salvatore Stuvard, Multiple valued Jacobi fields, Calc. Var. Partial Differential Equations 58 (2019), no. 3, Paper No. 92, 83, DOI 10.1007/s00526-019-1545-9. MR3948282 ←261 [40] S. Stuvard, Multiple valued sections of vector bundles: the reparametrization theorem for Q-valued functions revisited, ArXiv e-prints (April 2017), available at arXiv:1705.00054. ←261 Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA Institut für Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland Email address: [email protected]

IAS/Park City Mathematics Series Volume 27, Pages 289–246 https://doi.org/10.1090/pcms/027/00866

Lecture notes on rectifiable Reifenberg for measures Aaron Naber Abstract. These series of notes serve as an introduction to some of both the classical and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is about studying general sets or measures which can be, in one sense or another, approximated on all scales by well behaved spaces, typically just Euclidean space itself. Such sets and measures turn out not to be arbitrary, and often times come with special structure inherited from what they are being approximated by. We will begin by recalling and proving the standard Reifenberg theorem [20], which says that sets in Euclidean space which are well approximated by affine subspaces on all scales must be homoemorphic to balls. These types of results have applications to studying the regular parts of solutions of nonlinear equations. The proof given is designed to move cleanly over to more complicated scenarios introduced later. The rest of the lecture notes are designed to introduce and prove the Rectifiable Reifenberg Theorem [12], including an introduction to the relevant concepts. The Rectifiable Reifenberg Theorem roughly says that if a measure μ is summably close on all scales to affine subspaces Lk of dimension k, then μ = μ+ + μk may be broken into pieces such that μk is k-rectifiable with uniform Hausdorff measure estimates, and μ+ has uniform bounds on its mass. Such results have applications to studying the singular parts of solutions of nonlinear equations. The proof given is designed to give a simple introduction to ways of thinking in more modern PDE analysis, including an introduction to Neck regions and their Structure Theory as in [14, 18].

Contents Introduction 1 Discussion of Reifenberg Methods and Outline of Notes 2 Required Technical Background for these Notes 2.1 Implicit Function Theorem 2.2 Elementary Measure Theory 2.3 Vitali Covering Lemma 2.4 Submanifolds of Euclidean Space 2.5 Regularity of Submanifolds 2.6 Projections to Submanifolds

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2010 Mathematics Subject Classification. Primary 35-02; Secondary 35J60. Key words and phrases. Reifenberg, Rectifiable, Nonlinear Equations. ©2020 American Mathematical Society

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Lecture 1: Classical Reifenberg 3 Overview 4 Examples 5 Proof of Reifenberg Theorem 5.1 Submanifold Approximation Theorem 5.2 Distance Approximation Theorem 6 Proof of Distance Approximation Theorem

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Lecture 2: Rectifiable Reifenberg for Measures 7 Applying the classical Reifenberg theorem 8 Hausdorff, Minkowski, and Packing Content 8.1 Rectifiability of Sets 9 Jones β-numbers 10 Rectifiable Reifenberg Theorem for Measures

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Lecture 3: Outline Proof of Rectifiable Reifenberg 11 Neck Regions and their Structure and Decomposition 12 Using the Neck Decomposition Theorem 12.1 Proof of Corollary 10.0.6

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Lecture 4: Proof of Neck Structure and Decomposition Theorems 13 Proof of Neck Structure Theorem 13.1 Best Subspaces on Neck Regions 13.2 Submanifold Approximation Theorem 13.3 Using the Submanifold Approximation Theorem 13.4 Proof of Submanifold Approximation Theorem 13.5 Subspace Selection Lemma 13.6 Distance Approximation Theorem 14 Proof of Neck Decomposition Theorem 14.1 Proof Outline and Induction Step 14.2 Notation and Ball Types 14.3 Collapsing and d-Ball Covering 14.4 Symmetry and e-Ball Covering 14.5 Neck Regions and c-Ball Covering 14.6 Inductive Proof of the Neck Decomposition Theorem

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Introduction 1. Discussion of Reifenberg Methods and Outline of Notes We begin these notes by listing a few types of Reifenberg results which exist, as well as the their primary applications. We will not try and be overly precise at this point, and will return to details after. Much more complete introductions are given to those topics discussed in these notes at the beginning of each lecture.

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(1) The classical Reifenberg [20]. (a) Background: Given two sets S1 and S2 we define their Hausdorff distance dH (S1 , S2 ) ≡ inf{ : S1 ⊆ B (S2 ) andS2 ⊆ B (S1 )}, see (3.0.2) for more. (b) Statement: Assume S ⊆ B2 (0n ) is a set such that for each Br (x) ⊆ B2 there exists an affine subspace Lk = Lk x,r with the property that that   dH S ∩ Br (x), L ∩ Br (x) < (n)r. Then S ∩ B1 is actually homeomorphic to a ball of dimension k. (c) Application: This is used to study the manifold structure of the regular sets of minimal surfaces. (2) Reifenberg Theorem for Metric Spaces by Cheeger-Colding [4]. (a) Background: Given two metric spaces X1 and X2 we say their Gromov Hausdorff1 distance dGH (X1 , X2 ) <  if there exists -dense subsets {xi1 } ⊆ X1 and {xi2 } ⊆ X2 such that |d(xi1 , xj1 ) − d(xi2 , xj2 )| < . (b) Statement: Assume a metric space X has the property that that each ball Br (x) ⊆ B2 (p) is Gromov Hausdorff close to Euclidean space:   dGH X ∩ Br (x), Br (0n ) < (n)r. Then X ∩ B1 (p) is actually homeomorphic to a k-dimensional ball. (c) Application: This is used to show the manifold structure of the regular sets of limits of spaces with lower Ricci curvature bounds. See also [9] for a more general study of such spaces. (3) Uniform Rectifiability and Ahlfor’s regular Measures (a) Background: We say a measure μ ⊆ B1 (0n ) is Ahlfor’s regular if crk  μ(Br (x))  Crk for all x ∈ suppμ and r  2. We define the Jones β-numbers of any measure μ by  2 −2−k inf d(x, Lk )2 dμ[x] , βk (x, r) ≡ r Lk

Br (x)

where the inf is taken over all affine subspaces, see Section 9 for a much more complete introduction. (b) Setup: For an Ahlfor’s regular measure μ one sees that uniform rectifiability2 is equivalent to the measures support being summably close 2 on all scales to affine subspaces: i.e. 0 βk (x, s)2 ds s < δ, see Jones [15], David-Semmes [7], and Toro [23]. (c) These ideas were used by Jones to solve the traveling salesman problem, and refinements were used by David-Semmes to prove estimates on Calderon-Zygmund operators constructed from μ. Local refinements were used by Tolsa [22] and Tolsa-Azzam [1] to characterize when measures with upper and lower density bounds are rectifiable. 1 This

definition is not equal to, but it is uniformly equivalent to, the Gromov Hausdorff distance. the sake of the introduction view k-rectifiable as being a k-manifold away from a set of measure zero. Precise definitions and statements are given in Section 8. Uniform rectifiability roughly means that on all balls one can cover most of the support of μ by a single chart, see [8]. 2 For

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(4) Rectifiable Reifenberg Theorem by Edelen-Naber-Valtorta [12], NaberValtorta [19]. (a) Statement: If the support of a general measure μ is summably close 2 on all scales to affine subspaces Lk , i.e. 0 βk (x, s)2 ds s < Γ where βk are the Jones β-numbers, then μ = μ+ + μk may be broken into pieces such that μk is k-rectifiable with uniform Hausdorff measure estimates and μ+ has uniform bounds on the measure. (b) Application: This can be viewed as effective versions of the previous setup, as one concludes measures bounds instead of assuming them, and is used to study the rectifiable structure and volume bounds of singular sets of nonlinear equations. (5) Reifenberg Theorem to Subset of Subsets. (a) Setup: Consider a closed subset C of the space of all subsets. Assume for each ball that S ∩ Br (x) is close to some element C ∈ C. Then in many special cases, the set S will inherit special properties itself from C. See the work of Badger-Lewis [3]. (b) Application: Take C to be the zero sets of harmonic polynomials. Such subsets enjoy a frequency monotonicity, which weakly transfers to the set S itself and builds a certain stratified structure on S, see the work of Badger-Engelstein-Toro [2]. See also [6] for an application of similar ideas to minimal cones. (6) Canonical Rectifiable Reifenberg Theorems. (a) Setup: Most Reifenberg results rely on the same basic construction to build the Reifenberg maps. In more complicated situations, as when the underlying space itself is twisted, this construction leads to additional errors and does not allow for rectifiable and finite measure control. Instead, one builds Reifenberg maps canonically by letting the maps themselves solve an equation. (b) Application: The main application of this is to study the singular sets of spaces with lower Ricci curvature bounds by Cheeger-JiangNaber [5]. In that case, to approximate the singular set by a kdimensional Euclidean space also requires approximating the underlying manifold. These errors are worse and fundamentally not controllable using Reifenberg constructions. One instead solves for harmonic mappings into Rk , and proves that on the (approximate) singular sets that these mappings are automatically Reifenberg and even rectifiable Reifenberg. These notes will focus primarily on (1) and (4) above. Here is an outline. In Lecture 1 we will study and prove the classical Reifenberg Theorem. Our proof of the classical Reifenberg Theorem is designed with the rectifiable Reifenberg in mind, so that many of the technical complications which appeared previously in the literature, see [19] for instance, may be avoided.

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In Lecture 2 we will give the necessary background needed so that we may end with a statement of the rectifiable Reifenberg Theorem. This includes an introduction to the Jones β-number, which measures on a given ball how far away the support of a measure μ is to being contained in an affine subspace. The rectifiable Reifenberg theorem roughly states that if a measure μ has appropriate integral control on its β-numbers, then it must be decomposable into pieces μ = μ+ + μk , where μk has k-rectifiable support with finite Hausdorff measure and μ+ is a uniformly finite measure. The proof in these notes is different from [12] and has been designed as a simple case of how one approaches singularity analysis in general. In Lecture 3 we will introduce the notion of Neck regions and state the Neck Structure and Neck Decomposition Theorems. Neck regions are roughly those regions for which a weak version of a Reifenberg type rigidity hold for μ, and which the techniques of Lecture 1 will apply. The Neck Decomposition Theorem will tell us how to decompose B1 , in a crucially effective way, into pieces which are either Neck regions or into regions which already have mass bounds. After stating and discussing the Neck Structure and Neck Decomposition Theorems we will use them in Lecture 3 to prove the rectifiable Reifenberg Theorem itself. In Lecture 4 we will prove the Neck Structure and Neck Decomposition Theorems, thus completing the proof of the rectifiable Reifenberg Theorem. The proof of the Neck Structure Theorem will follow a very similar line of attack as our proof of the classical Reifenberg in Lecture 1, once some suitable technical complications are addressed. The proof of the Neck Decomposition Theorem is an involved covering argument, the idea of which originates in the papers [14, 19]. It is worth taking a moment to mention that the proof structure of these notes are designed to be as widely applicable as possible. Our proof of the classical Reifenberg is not just designed to be applied to the rectifiable Reifenberg, but in the process will build a variety of structure which is used in a lot of applications itself. Likewise our construction of Neck regions together with the Neck Structure and Neck Decomposition theorem is precisely what appears in many of the recent applications of this type of analysis. Neck regions first appeared in the proof of the n − 4 finiteness conjecture for manifolds with bounded Ricci curvature [14], and the proof of the energy identity conjecture for Yang Mills [18]. Neck regions in those cases are quite a bit more subtle (as one cannot directly assume β-number control), and thus the neck structure theorems there take a lot more work. However the neck decomposition theorems are almost verbatim. In both cases the reifenberg context makes for an excellent test case.

2. Required Technical Background for these Notes These notes are designed to be almost entirely self-contained. It goes without saying that the more background one has in some basic geometric measure theory

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the more comfortable you may feel, but strictly speaking this is not needed as we will build by hand almost all of the structure we require. The following is meant to list some theorems and basic technical tools which will get used frequently. The reader is free, and indeed encouraged, to skip this section for now and come back when appropriate, as many of the technical constructions will be easier to follow when there is a context. The exercises of this section, and indeed these notes as a whole, are meant to be clear to a reader familiar with the ideas involved. If any tricks are needed, these are basically always stated in a hint, as the goal of the exercises is to familiarize the reader with the basic technical building blocks. 2.1. Implicit Function Theorem The Implicit Function Theorem is typically formulated in an ineffective manner, however since we will care about the estimates let us state for convenience the effective result (whose proof is verbatim the Implicit Function Theorem itself): Theorem 2.1.1 (Implicit Function Theorem). Let f : B2 (0n ) × B2 (0m ) → Rm be a C1 function and assume f(0, 0) = 0 and |∂x f(x, y)|, |∂y f(x, y) − Id|, |∂2 f| < δ. Then there exists g : B1 (0n ) → Rm such that |g|, |∂i g|, |∂i ∂j g| < C(n)δ with f(x, g(x)) = 0 for all x ∈ B1 (0n ). The above not only tells us that the zero set of f is a graphical manifold near (0, 0), but also gives good estimates on the structure of that manifold. 2.2. Elementary Measure Theory You will need to understand the definition of a Borel measure. Some knowledge of the Hausdorff measure is not required, as we will review this in Section 8, but it would be very helpful. 2.3. Vitali Covering Lemma We may not quote the Vitali Covering Lemma, however its proof will be implicit in a lot of constructions. Let us state the classical result: Lemma 2.3.1 (Vitali Covering Lemma). Let {Brα (xα )} be any collection of balls with rα  A < ∞. Then there exists a countable disjoint subcollection {Bri (xi )} such that # # Brα (xα ) ⊆ B5ri (xi ). In general, the reader (and indeed any aspiring analyst) needs to get very comfortable with ways of covering sets by balls in controlled manners. Let us give a handful of exercises which will help in this direction. In practice one proves effective content estimates on a well behaved collection of balls by taking the collection, identifying it with a collection of balls in Euclidean space, and then estimating there. The following exercise teaches us the minimal structure we need on these balls in Euclidean space in order to conclude content estimates: Exercise 2.3.2. Let {Bri (xi )} ⊆ B2 (0k ) be a collection of balls in Euclidean Space Rk . Show the following:

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 k (a) If {Bri (xi )} are disjoint then ri  C(k). #  k (b) If B1 (0 ) ⊆ Bri (xi ) then C(k)  rk i. We will often need to build controlled covers of regions by balls of some predetermined size. The next exercise is a gentle introduction into how one takes a covering and refines from it a ‘well behaved’ covering: Exercise 2.3.3. For S ⊆ B1 (0n ) let rx : S → R+ with rx > r0 > 0 be an assigned radius function. Then (a) Show one can choose a maximal subcollection {Bri (xi )} ⊆ {Brx (x)} such that {Bri /5 (xi )} are disjoint. Here by maximal we mean that if y ∈ B1 then Bry /5 (y) ∩ Bri /5 (xi ) = ∅ for some i. # (b) Show S ⊆ Bri (xi ). (c) Argue as in the last Exercise to see #{Bri (xi )}  N(n, r0 ). Our last exercise is our most technical, however constructions of this type are used almost continuously. The idea is similar to the last exercise, but we drop our assumed lower bound on the radius and replaced it instead with Lipschitz control on the radius function. The next exercise can be used to build well behaved partitions of unity, and we will later use a very similar construction to do just that: Exercise 2.3.4. Let rx : B1 (0n ) → R be a nonnegative radius function for which |∇rx |  τ−1 for some τ > 0. Let A0 ≡ {x : rx = 0} and let A+ ⊆ {rx > 0} a maximal subset such that {B10−3 τrx (x)} are disjoint. # (a) Show B1 ⊆ A0 ∪ A+ B10−2 τrx (x). (b) Show if B10−1 τrx (x) ∩ B10−1 τry (y) = ∅ then 10−1 ry  rx  10rx . (c) Show for y ∈ B1 that #{x ∈ A+ : B10−1 τrx (x) ∩ B10−1 τry (y) = ∅}  C(n). 2.4. Submanifolds of Euclidean Space As all submanifolds will be built explicitly, one may manage without any real previous knowledge of submanifolds of Euclidean space, however the reader will find the learning curve shortened somewhat is time is spent on this first. Let us very quickly recall a few definitions, and then present some exercises which are relevant to technical constructions which will appear: Definition 2.4.1 (Submanifolds). Recall the following: (1) We call a differentiable map f : U ⊆ Rk → Rn an immersion if for each x ∈ U the linear map dx f : Rk → Rn given by dx f[v] ≡ ∂i f(x)vi is injective. (2) We say a subset S ⊆ Rn is a submanifold if for all y ∈ S, there is a neighborhood y ∈ V and an immersion f : U ⊆ Rk → Rn such that S ∩ V = f(U). We call the pair (U, f) a chart. In practice, the submanifolds of these notes will always come from one chart, as we will only be interested in local constructions. As defined, one might also

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call S above an embedded submanifold, which is again perfectly acceptable for the constructions of these notes. We will be interested in tangent spaces of submanifolds: Definition 2.4.2 (Tangent Spaces). If we are given S ⊆ Rn a submanifold and f : U ⊆ Rk → Rn a chart with f(x) = y, we define the tangent space to be the affine subspace Ty S ≡ {w ∈ Rn : w = y + dx f[v] for v ∈ Rk }. Let us give a few useful exercises on tangent spaces: Exercise 2.4.3. Show Ty S is independent of the chart. I.e. if f  : U  ⊆ Rk → Rn is another chart of S with f  (x  ) = y, then the tangent space defined from f  is the same as that defined from f. If w ∈ Ty S then we define the norm of w by |w| ≡ |w − y|. That is, if we view Ty S as a linear subspace by moving y to the origin and then we define the norm there. yi −y Exercise 2.4.4. Show that if yi ∈ S → y ∈ S with w = y + lim |y , then w ∈ Ty S i −y| is a unit tangent vector, when the limit exists.

Let us now consider a few simple examples: Example 2.4.5. Let f : Rk → Rn be a linear isometric immersion, then the submanifold f(Rk ) = S = L is an affine subspace. In this case, the tangent space Tx S = L is also L for each x ∈ S. Example 2.4.6. Let L ⊆ Rn be an affine subspace and let Lˆ ⊥ be the perpendicular subspace3 . Let f : L → Lˆ ⊥ be a smooth mapping, then the map g : L → Rn given by g(x) = x + f(x) is a chart and gives rise to a graphical submanifold S = g(L) = Graph(L). 2.5. Regularity of Submanifolds Although the submanifolds of these notes will come from a single chart, it turns out that it may be much more convenient when working locally to build a new chart tailored to the local structure of the submanifold. This is related to considering notions of regularity for submanifolds. Locally, every submanifold looks like the last example and can be written as a graph over an affine subspace, thus let us formalize this into a notion of regularity: Definition 2.5.1. We say S ⊆ Rn is (δ, r)-graphical if for each x ∈ S there exists −1 an affine subspace Lx ⊆ Rn and fx : Lx → Lˆ ⊥ x with r |fx |, |∂i fx |, r|∂i ∂j fx |  δ such that Graph(Lx ) ∩ Br (x) = S ∩ Br (x). Remark 2.5.2. The factors of r are scale invariant factors. Thus if we rescale and translate Rn so that Br (x) → B1 (0) then S becomes (δ, 1)-graphical. ˆ to represent a point of notation we will use L to represent affine subspaces and we will put hat’s L ˆ goes through the origin while L need not. linear subspaces, i.e. L

3 As

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Remark 2.5.3. We can also let δx and rx be functions on S and say S is (δx , rx )graphical. Let us present some technical exercises which will build an intuition. If the reader is not familiar with the notion of Hausdorff distance between sets then we refer them to the definition given in (3.0.2): Exercise 2.5.4. Let S be (δ, r)-graphical with fx : Lx → Lˆ ⊥ x a graphing function. Then show that dH (S ∩ Br (x), Lx ∩ Br (x)) < δr. Exercise 2.5.5. Let S be (δ, r)-graphical with fx : Lx → Lˆ ⊥ x a graphing function. Assume Lx is an affine subspace such that dH (Lx ∩ Br (x), Lx ∩ Br (x)) < δr. Then show for δ sufficiently small there exists a graphing function fx : Lx → (Lˆ x )⊥ with r−1 |fx |, |∂i fx |, r|∂i ∂j fx |  C(n)Aδ such that Graph(Lx ) ∩ Br (x) = S ∩ Br (x). Hint: Let fx be the composition of the projection map from Lx to Lx and fx . The above exercise tells us we have some flexibility on which affine subspaces we pick. The next exercise tells us that the tangent spaces of graphical submanifolds are well approximated: Exercise 2.5.6. Let S be (δ, r)-graphical with fx : Lx → Lˆ ⊥ x a graphing function. Show for each y ∈ S ∩ Br (x) that dH (Ty S ∩ Br (x), Lx ∩ Br (x)) < C(n)δr. 2.6. Projections to Submanifolds Finally, let us discuss a little the natural projection map associated to each submanifold. Composing these will form a key technical tool in the construction of the Reifenberg maps later in these notes. To begin, given an affine subspace L let πL : Rn → L be the projection map to L, and let πˆ L : Rn → Lˆ be the projection map to the associated linear subspace. We wish to build projection maps to general submanifolds: Theorem 2.6.1. Let S ⊆ Rn be a (δ, r)-graphical submanifold. Then the closest point projection mapping πS : Br (S) → S ⊆ Rn defined by πS (x) = arg miny∈S 12 |x − y|2 is well defined and satisfies (1) πS ∩ S = Id, (2) |∂i πS (y) − πˆ Lx | < C(n)δ where y ∈ Br (x) with x ∈ S. (3) r|∂i ∂j πS | < C(n)δ.

4

Proof. We at least outline the proof. There are actually several approaches to this, including more geometric ones which I personally prefer, however we will outline a proof using the Implicit Function Theorem so we can stick with ideas more consistent with these notes. Thus let f : Lx → Lˆ ⊥ x be a δ-graphing function for S on Br (x). Let us consider on Br (x) the function G : Lx × Lˆ ⊥ x × Lx → Lx given by5   1 1 G(y, z, y ), v   = ∂v  |y − y  |2 + |z − f(y  )|2 = ∂v  |(y, z) − (y  , f(y  ))|2 , 2 2 that ∂i πS is an n × n matrix. partial derivative uses the  notation in order to signify that we are taking the partial derivative in the y  direction.

4 Note 5 The

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so that G(y, z, y  ) is the horizontal derivative of the square distance between (y, z) ∈ Br (x) and (y  , f(y  )) ∈ S. In particular, if πS (y, z) = (y  , f(y  )) then G(y, z, y  ) = 0, and thus for each (y, z) there exists y  such that G(y, z, y  ) = 0. Using our estimates on f let us also see that: |∂y G(y, z, y  ) + Id| , |∂z G(y, z, y  )| , |∂y  G(y, z, y  ) − Id| < C(n)δ . We can use the Implicit Function Theorem 2.1.1 in order to find g : Br/2 (x) → Lx such that G(y, z, g(y, z)) = 0 with r−1 |g(y, z) − y| , |∂y g − Id| , |∂z g|  C(n)δ , r|∂∂g|  C(n)δ . For each (y, z) ∈ Br/2 (x) we then have, by the above estimates, that y  = g(y, z) is the unique point such that G(y, z, g(y, z)) = 0, and thus we must, in turn, have πS (y, z) = (g(y, z), f(g(y, z))). The estimates on g and f therefore prove the  desired estimates on πS . Let us end now with an intuitive exercise: Exercise 2.6.2. Show using (2) that |πS (x) − x|  (1 + C(n)δ)d(x, S) for x ∈ Br (S).

Lecture 1: Classical Reifenberg 3. Overview The classical Reifenberg theorem describes sets which may be approximated on all points and scales by affine subspaces. The basic claim is that such sets must in fact be homeomorphic to Euclidean balls, and thus are quite rigid. To describe this in more detail let us recall the Hausdorff distance between sets: Definition 3.0.1. Let A, B ⊆ Rn be subsets, then we define their Hausdorff distance (3.0.2)

dH (A, B) ≡ inf{r > 0 : A ⊆ Br (B) and B ⊆ Br (A)} .

It is worth observing that the Hausdorff distance is a complete metric on closed subsets. There are many references, see for instance [13], for a much more complete description of the Hausdorff distance. Exercise 3.0.3. Check the following: (1) Let S ⊆ B1 (0n ) be any closed subset, and let {xi } ∈ S be an δ-dense subset. Then dH (S, {xi })  δ. (2) For S ⊆ B1 (0n ) × {0} ⊆ B1 (0n ) × R2 any closed subset, let Sδ ≡ S × S1 (δ), where S1 (δ) ⊆ R2 is the circle of radius δ. Then dH (S, Sδ )  δ. By letting δ → 0 in the above examples one sees that Hausdorff distance certainly does not preserve either upper or lower dimensional bounds in any manner. Let us now define carefully what it means for a set to satisfy the Reifenberg condition:

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Definition 3.0.4. Let S ⊆ B2 ⊆ Rn be a closed set: (1) We define the L∞ Jones β-number as the infimum over all k-dimensional −1 inf 6 affine subspaces Lk of β∞ Lk dH (S ∩ Br (x), L ∩ Br (x)) . k (x, r) ≡ r (2) S satisfies the δ-Reifenberg condition if for all x ∈ S with Br (x) ⊆ B2 we have β∞ (x, r) < δ. It turns out the Reifenberg situation can occur naturally, in one guise or another, in a variety of situations. Reifenberg was the first to prove that this forces a very strict rigidity on S. In particular, S must be a topological manifold: Theorem 3.0.5 (Reifenberg’s Theorem). Let S ⊆ B2 ⊆ Rn satisfy the δ-Reifenberg condition. Then for every 0 < α < 1 if δ < δ(n, α), there is a φ : S ∩ B1 (0n ) → B1 (0k ) which is a Cα -bi-Hölder map. Precisely: 1 |x − y|1+α < |φ(x) − φ(y)| < 2|x − y|1−α . 2 The first lecture in these notes will focus on describing some basic examples of the above, in particular to see that the result is sharp as stated, and to go through a careful proof. Our proof is maybe not quite the standard one, and instead is designed so that it will easily generalize later to more complicated situations with minimal additional work.

4. Examples We begin in this section by discussing a handful of examples. We will build up the complexity of these examples with the goal of seeing that the Reifenberg Theorem is sharp. Example 4.0.1 (Trivial Example). Let Lk be any k-dimensional subspace and then let S = Lk ∩ B2 . Then S satisfies the δ-Reifenberg condition for every δ > 0. Exercise 4.0.2 (Graphical). Let f : L → L⊥ be compactly supported with f(0) = 0 and |∇f| < δ. If S = Graph(f) ∩ B2 = {(x, f(x)) : x ∈ L} ∩ B2 then show that S satisfies the δ-Reifenberg condition. The next example is our main one, and the first nontrivial example. It will show the sharpness of the bi-Hölder condition. Additionally, the proof of the properties of the example will purposefully be done so as to motivate how the general proof of Reifenberg should go. Roughly, the proof of the general Reifenberg theorem is more or less just a reverse engineering of the following example: Example 4.0.3 (Von-Koch Snowflake). The construction is in iterative steps, and we will build a sequence of piecewise linear Si which will converge to our final example. Let us begin with the iterative construction: 6 Note

this is not really the usual Jones β-number, we make this definition for comparison later. A better name might be the Reifenberg number or two-sided Jones β-number.

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Construction for Iteration: Let a,b be the line segment between a, b ∈ R2 and let δ ∈ R. We will alter a,b to be the union of four line segments δa,b ≡ a,c ∪ c,d ∪ d,e ∪ e,b , which are well defined by the properties that a,c , e,b ⊆ a,b with c,d ∪ d,e ∪ c,e forming an oriented isosoceles triangle , |a,c | = |c,d | = |d,e | = |e,b | with |δa,b |2 = (1 + δ2 )|a,b |2 , where || is the length of the given line segment. Let θδ be the angle between c,d and c,e , and let dδ ≡ |a,b |−1 d(d, a,b ). Note that both satisfy Exercise 4.0.4. Show C−1 δ  dδ , θδ  Cδ. In particular this then gives (4.0.5)

dH (a,b , δa,b ) = O(δ)|a,b | , |δa,b |2 = (1 + δ2 )|a,b |2 .

Now our iterative construction is as follow. Let a0−1 = (−2, 0), a01 = (2, 0) with # S0 = a0 ,a0 the associated interval. Inductively, if Si = ai ,ai is piecewise j j+1 −1 1 linear with 4i edges then let % % δai ,ai = ai+1 ,ai+1 Si+1 = j

be piecewise linear with one of these edges by

4i+1

j+1

j

j+1

edges. By (4.0.5) we can compute the length of any

|i | ≡ |iai ,ai | = |aij+1 − aij | = 4 · 4−i (1 + δ2 )i/2 ≡ 4 · 4−αi , j

j+1

√ 2 . Note that α → 1 as δ → 0. In where 0 < α < 1 is given by 4−α = 1+δ 4 particular, using (4.0.5) we can then compute (4.0.6)

i/2 |Si | = 4 1 + δ2 ,

dH (Si , Sj )  O(δ)

j √  1 + δ2 k  O(δ) 4−α i = O(δ)|i | . 4

k=i

Thus the sequence Si is Cauchy in the Hausdorff topology and hence there exists a limit S = lim Si , see [13] and the remark after (3.0.2). Note that the angle between each segment in Si is given by θδ = O(δ), from which one can conclude i Exercise 4.0.7. For each x ∈ Si show that β∞ Si (x, | |) < O(δ).

It is then immediate from the above exercise and the Hausdorff estimate in (4.0.6) that for i < j: i ∞ i β∞ Sj (x, | |)  O(δ) + βSi (x, | |)  O(δ) .

In particular, we get that S is a O(δ)-Reifenberg space. Now that we have built the example, let us study the structure of this example a little. In particular, imagine what the Reifenberg map Φ : [−2, 2] → S might look like. Note already that we know from the volume estimate of (4.0.6) that S cannot be uniformly bi-Lipschitz to an interval, and thus best case scenario is

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bi-Hölder. In that sense we have already shown that the Reifenberg theorem must be sharp. However, let us go to the trouble of building the bi-Hölder Reifenberg map Φ : [−2, 2] → S. Our construction will mimic the proof of Theorem 3.0.5 itself, however will be technically much less involved for the example, and therefore a good place to build intuition for the general case. Our strategy will be to build maps Φi : [−2, 2] → Si with uniform bi-Hölder estimates, and then limit. We first consider the projection maps πi : Si+1 → Si . Note that for δ > 0 small this map is clearly well defined and indeed bi-Lipschitz with the estimates:  |πi (x) − πi (y) |dπi |(x) − 1 = lim − 1  1 + δ2 − 1 = O(δ2 ) , |x − y| y∈S →x i+1 (4.0.8) |πi (x) − x|  dδ |i | = O(δ)|i | . Observe that the first estimate has an δ2 . We will actually not use this square improvement here, however it appears again (crucially) when going from classical to rectifiable Reifenberg results, which is why we emphasize it here. Now we have defined a mapping from Si+1 to Si , so let us compose these mappings in order to define Πi,j : Si → Sj and Πi = Πi,0 : Si → S0 = [−2, 2] by Πi,j = πj ◦ · · · ◦ πi−1 . Note that although each map Πi is bi-Lipschitz, we see that the bi-Lipschitz constants are becoming increasing large, so that we cannot hope to preserve that estimate. If we can show the maps Πi are uniformly bi-Hölder then Φi = Π−1 i are as well. So let x, y ∈ Si and let j  i be the largest j such that |x − y|  |j |. If j < i √ then we also have 1 + δ2 |x − y|  |j |. Then to estimate |Πi (x) − Πi (y)| we write Πi = Πj,0 ◦ Πi,j . We will estimate each factor separately, using different estimates from (4.0.8). First let us write xk = Πi,k (x) and yk = Πi,k (y) where j < k, then using the second estimate from (4.0.8) we can get |xk+1 − xk | = |πk+1 (xk ) − xk |  δ|k | = O(δ) 4−α|k−j| |j | , |yk+1 − yk |  O(δ) 4−α|k−j| |j | , √ 2 where recall 4−α ≡ 1+δ . We then have 4 |Πi,j (x) − x| 

i−1 

|xk+1 − xk |  O(δ)|j |  O(δ)|x − y| ,

j

|Πi,j (y) − y|  O(δ)|x − y| . Combining and using the triangle inequality gives |Πi,j (x) − Πi,j (y)| − |x − y|  |Πi,j (x) − x| + |Πi,j (y) − y|  O(δ)|x − y| . Note we have proved that if |x − y| ≈ |j |, then Πi,j is a uniformly bi-Lipschitz map when comparing x and y. To move from Sj to S0 we now rely on the first estimate from (4.0.8) in order to obtain the following estimate.

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Πi (x) − Πi (y) = Πj,0 (Πi,j (x)) − Πj,0 (Πi,j (y))  (1 + δ2 )j/2 |Πi,j (x) − Πi,j (y)|  (1 + O(δ))(1 + δ2 )j/2 |x − y|  (1 + O(δ))(1 + δ2 )j/2 |j |

 √1 + δ2  j  1 + δ2 j γ ≡ 4(1 + O(δ)) 4 4

 √1 + δ2  β j  (1 + O(δ)) 4  (1 + O(δ))|j |γ 4  (1 + O(δ))|x − y|γ ,  √1+δ2 γ 2 where γ < 1 was defined by = 1+δ and thus is as close to 1 as we 4 4 wish as δ → 0. A verbatim argument shows the opposite inequality, and thus this proves the uniform bi-Hölder estimate. = 4(1 + O(δ))

5. Proof of Reifenberg Theorem We will now focus on giving a proof of the classical Reifenberg Theorem. Our proof is designed to motivate how we will be approaching the more general and challenging cases. As such it differs from other proofs in the literature, see for instance [10, 21]. We have also gone to some effort to make the general scheme one which applies in seemingly very different scenarios in geometric analysis, albeit in often much more complicated ways.7 It will be convenient in the construction to make the following notation. For each Br (x) ⊆ B2 with r  10 d(x, S) let us fix a choice of k-dimensional subspace Lx,r = Lx,r [S] satisfying8 (5.0.1)

Lx,r ∈ arg min dH (S ∩ Br (x), L ∩ Br (x)) ≡ β∞ k (x, r) · r . L

Notation: Let us discuss some notation which will be in effect throughout these lectures: (1) Given an affine subspace Lk ⊆ Rn let Lˆ ⊆ Rn denote the linear subspace associated to L. (2) We let πL : Rn → L ⊆ Rn and πˆ L : Rn → Lˆ ⊆ Rn denote the orthogonal projection maps. (3) In the case of the subspaces Lx,r we will write the projection maps as πx,r and πˆ x,r . The following exercise is a key observation in the Reifenberg theorem: 7 The

reason for this change is subtle and easy to miss in the current setup, as the goal is to avoid rather serious technical challenges in the case where holes of arbitrary size and distribution can appear. For instance in [14] a more standard Reifenberg type construction is used, but in exchange one is stuck proving a much more refined version of a neck region, which causes a variety of headaches. The proof presented in this section is meant to avoid these issues when the time comes. 8 Recall the distance function d(y, S) ≡ inf x∈S d(y, x).

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Exercise 5.0.2. Assume S ⊆ B2 satisfies the δ-Reifenberg condition, and then choose Br (x), Bs (y) ⊆ B2 with r  10d(x, S) and s  10d(y, S). For a  10 let Ba−1 s (y) ⊆ Bar (x) ⊆ Ba2 s (y) 9 . Then show dH (Lx,r ∩ Br (x), Ly,s ∩ Br (x)) < C(n)δr and ||πˆ x,r − πˆ y,s || < C(n)δ , where || · || is the matrix norm. The above exercise is telling us that the ‘best’ subspaces Lx,r cannot change too quickly from scale to scale, and is the basis for our ability to glue them together in a controlled fashion. The outline of the proof of the Reifenberg Theorem is as follows. In Section 5.1 we begin by stating the Submanifold Approximation Theorem 5.1.1, which builds a family of smooth manifolds Sr which approximate the set S on scale r. One can take S1 = L to be an affine subspace without any loss, and if we consider the sequence Si ≡ S2−i then a key observation will be that Si and Si+1 are smoothly close. Thus we can consider the projection maps πi : Si+1 → Si and we will build our final bi-Hölder mapping Π : S → L = S1 from S to k-dimensional Euclidean space by simply composing the projection maps πi . Compare this outline to Example 4.0.3. The most direct route to understanding the existence of Sr is to use Exercise 5.0.2 to glue together the subspaces Lx,r . One can indeed make this rigorous, but primarily because of how things will work in future sections, we take a different approach. Instead we will construct a smooth function Φr : B2 → R which behaves like a distance function to Sr . A little more precisely, Φr will be a Morse Bott function whose zero level set Sr ≡ Φ−1 r (0) will consist of nondegenerate critical points, and thus is a smooth manifold. Estimates on Φr will then translate to estimates on Sr . In the end this proof strategy requires a little more work than simply gluing together the subspaces Lx,r , say in the spirit of [21] or [10], however this comes with a key advantage down the road. In the proof of the Rectifiable Reifenberg Theorem 10.0.2, and in particular in the proof of the associated Neck Structure Theorem 11.0.11, a very similar construction and proof will be needed, however it will be done on a discrete set of balls. This discreteness can cause a major technical headache, and previous arguments [14] have used quite involved covering arguments to deal with it. Instead, we will see our approach for the classical Reifenberg Theorem will pass over almost verbatim, with only minimal extra work. This is one of those situations where there are 10 different possible approaches for the simplified setup, and the question is the choose the one which minimizes complications in the more challenging cases. 9 This

is saying that the balls Bs (y) and Br (x) are comparable on scale a  10.

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5.1. Submanifold Approximation Theorem of smooth manifold approximations to S.

Now we begin by building a series

Theorem 5.1.1 (Reifenberg Submanifold Approximation). Let S ⊆ B2 ⊆ Rn satisfy the δ-Reifenberg condition. Then for each 0 < r < 1 there exists a smooth submanifold Sr ⊆ B2 which satisfies (1) (2) (3) (4)

dH (Sr , S) < C(n)δ r, dH (Sr ∩ Bs (x), Lx,s ∩ Bs (x)) < C(n)δs for s  r. Sr is a (C(n)δ, r)-graphical submanifold, see Definition 2.5.1. There is a smooth πr : Br (S) → Sr ⊆ Rn with πr ∩ Sr = Id, and with the estimates |∂i πr (y) − πˆ y,10r | , r|∂i ∂j πr | < C(n)δ. 10 (5) dH (Sr/2 , Sr ) < C(n)δr with |πr (x) − x| < C(n)δr for x ∈ Sr/2 . (6) For x ∈ Sr/2 and a unit vector v ∈ Lx,r we have ||dπr [v]| − 1| < C(n)δ2 . 11 Remark 5.1.2. Note that if we take S1 to be a linear subspace with π1 the orthogonal projection map then the above holds with r = 1. It will be convenient to make this choice. Remark 5.1.3. Observe in (6) the square gain on δ in the error. This will not be used in this section, but in the rectifiable Reifenberg, see also [7, 10, 15], it is an important gain in order to conclude mass bounds and rectifiable structure. Let us first see that most of the conclusions of Theorem 5.1.1 follow from (1) and (3): Exercise 5.1.4. Show (2) follows from (1) and the Reifenberg property of S. Exercise 5.1.5. Define πr : Br (S) → Sr to be the closest point projection map πr (x) = arg miny∈Sr |x − y|. Show (4) using (3) and Theorem 2.6.1. Exercise 5.1.6. Show the first part of (5) follows from (1). Show the second part of (5) follows from Exercise 2.6.2. Thus we see (1)—(5) follow from (1) and (3). To show (6) follows from (1)—(5) is done in two steps. First, observe that the tangent spaces of Sr and Sr/2 must be close: Exercise 5.1.7. Show first that dH (Ty Sr ∩ Br (x), Tz Sr ∩ Br (x)) < C(n)δr and then that dH (Ty Sr ∩ Br (x), Ty Sr/2 ∩ Br (x)) < C(n)δr for y, z ∈ B2r (x) with x ∈ S. Hint: Use (2), (3) and Exercise 2.5.6. Exercise 5.1.8. Use (2) and Exercise 5.1.7 to show (6). Hint: Observe first that 1 = |v|2 = |dπr [v]|2 + |dπr [v] − v|2 and then , by Taylor √ expansion, that 1− x ≈ 1 − 12 x + O(x2 ) in order to derive the chain of inequalities 1  |dπr [v]| = 1 − |dπr [v] − v|2  1 − C(n)δ2 . since πr maps Rn to Rn that ∂i πr is a matrix, and thus our norm |∂i πr (y) − πˆ x,10r | is the matrix norm. 11 Recall dπ [v] = ∂ π vi ∈ T r i r πr (x) Sr . 10 Recall

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Finally, we add one more complication to the mix by doing the above slightly less locally: Exercise 5.1.9. Use the last two exercises to show that for any x ∈ S and for any z−y satisfies πˆ x,r [v] − v < C(n)δ. y, z ∈ Sr ∩ B2r (x) that the unit vector v ≡ |z−y| Hint: Consider the curve γ(t) = tz + (1 − t)y and begin by using (3) to show d2   2 πr (γ(t)) − γ(t)  C(n)δ|z − y|. Next use the fundamental theorem to get dt   d πr (γ(t)) − γ(t)  C(n)δ|z − y|. Finally use that |dπr − πˆ x,r | < C(n)δ to dt conclude the final estimate. Let us now see how to prove the Reifenberg Theorem given the Submanifold Approximation Theorem: Proof of Theorem 3.0.5 given Theorem 5.1.1. Consider the radii ri = 2−i and the submanifolds Si = Sri from Theorem 5.1.1. Let πi = πri ∩ Si+1 : Si+1 → Si be the δ-submersion from Theorem 5.1.1 restricted to Si+1 . Our main claims are the following: Claim 5.1.10. Fix x, y ∈ Si+1 . (1) If |x − y| < ri , then |πi (x) − πi (y)| − |x − y|  C(n)δ |x − y|. (2) If |x − y|  ri , then |πi (x) − πi (y)| − |x − y|  C(n)δri . Proof. To prove (1) consider the straight line xt = tx + (1 − t)y which connects y to x. Let Lt ≡ Txt Sr , then by Exercise 5.1.7, Exercise 5.1.9 and Theorem 5.1.1.(2) we have for each t ∈ [0, 1] that d πr (xt ) − (x − y) = |dπr (xt )[x − y] − (x − y) dt    | dπr (xt ) − πˆ Lt [x − y] + |πˆ Lt [x − y] − (x − y)  C(n)δ |x − y| . By integrating we in particular have shown Claim 5.1.10, indeed we have the stronger estimate (πi (x) − πi (y)) − (x − y)  C(n)δ |x − y|. To prove (2) first observe that since Si+1 ⊆ BCδri (S) ⊆ B2Cδri (Si ) we have by Exercise 5.1.7 the estimate |π(x) − x|, |π(y) − y| < C(n)δri . Using the triangle inequality we get (x − y) − (π(x) − π(y))  C(n)δri , which finishes the proof12 .  With the claim in hand let us build the maps which will connect S to L. Let us first define the maps Πi,j ≡ πj ◦ · · · ◦ πi−1 : Si → Sj Πi ≡ Πi,0 : Si → S0 ≡ L , where recall that as in the remark following Theorem 5.1.1 we have taken S0 to be a linear subspace. We claim the following: Claim 5.1.11. Let x, y ∈ Si , then |x − y|1+C(n)δ  |Πi (x) − Πi (y)|  |x − y|1−C(n)δ . 12 Recall

the analyst’s convention that C(n) changes from line to line, but is always a dimensional constant.

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Proof. To prove the claim let d = di ≡ |x − y| and define dj ≡ |Πi,j (x) − Πi,j (y)|. Note that if dj  rj then by (1) of Claim 5.1.10 we have that (5.1.12)

|dj−1 − dj |  C(n)δ dj =⇒ dj−1  (1 + C(n)δ) dj =⇒ dj−1  rj−1 =⇒ dk < rk for all k  j .

On the other hand if dj > rj , then by (2) of Claim 5.1.10 we have that |dj+1 − dj |  C(n)δ rj . In particular, using from above that dk > rk for all k > j this then gives dj  di +

i−1 

|dk+1 − dk |  di + C(n)δ

j

j 

rk  di + C(n)δ dj ,

i

and hence dj  (1 + C(n)δ)d. Now let j be the smallest integer such that d  rj . The above tells us that dj  (1 + C(n)δ)d, and then using (5.1.12) j − i times we obtain |Πi (x) − Πi (y)| ≡ d0  (1 + C(n)δ)j−i d  (1 + C(n)δ)ln d d = d1−C(n)δ = |x − y|1−C(n)δ , which provides one direction of the claim. The other direction is the same.



To finish the proof of the Reifenberg Theorem 3.0.5 we need to simply limit Πi → Π : S → L by combining Claim 5.1.11 with the Ascoli theorem and Theo rem 5.1.1.(1). 5.2. Distance Approximation Theorem and Proof of Theorem 5.1.1 We have now seen how to prove the Reifenberg Theorem 3.0.5 given the Submanifold Approximation Theorem 5.1.1. Our focus now becomes the proof of the Submanifold Approximation Theorem itself. Our basic strategy for the proof of Theorem 5.1.1 will be to build a smooth function Φr : B2r (S) → R which roughly behaves as smooth approximation to the distance function to Sr . In reality, we will build Φr  0 first and then define Sr ≡ Φ−1 r (0) to be the zero level set. We will see that sufficiently strong estimates hold on Φr in order to conclude our estimates on Sr . We will build Φr in two steps. First, we will build a smoothly varying distribution on B2 which will assign to each y ∈ B2 a k-dimensional affine subspace Lr (y) which acts as a Reifenberg approximation on the scale (5.2.1)

ry ≡ 10d(y, S) ∨ r ,

where recall s ∨ t ≡ max{s, t}. This assignment has a variety of useful applications in its own right. We will then use these affine subspaces to build Φr directly. We begin with the statement of the subspace selection lemma: Lemma 5.2.2 (Subspace Selection Lemma). Let S ⊆ B2 ⊆ Rn satisfy the δ-Reifenberg condition, and let 0 < r < 1 be fixed with ry defined in (5.2.1). Then for each y ∈ B2

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there exists a k-dimensional affine subspace Ly where if πˆ y = πˆ Ly and my ≡ πy [y] then: (1) Ly varies smoothly in y when we take ry |∇πˆ y |, |∇i my − πˆ y |  C(n)δ and when we take r2y |∇2 πˆ y |, ry |∇2 my |  C(n)δ. (2) We have dH (S ∩ B10ry (y), Ly ∩ B10ry (y)) < C(n)δry . (3) We have dH (Ly ∩ B10r¯ y (y), Ly,105 r¯ y ∩ B10r¯ y (y)) < C(n)δ. We will prove the above in the next section, and simply take it for granted now. Morally, it is nothing more than an averaging procedure, though requires a little technical work to check the details. Given Lemma 5.2.2 we define our approximate distance function Φr : B2 → R as follows: 1 1 1 Φr (y) ≡ d(y, Ly )2 = |y − πy [y]|2 = |y − my |2 . (5.2.3) 2 2 2 Let us collect together the main properties of this approximate distance function: Theorem 5.2.4 (Approximate Distance Function). Let Φr be defined in (5.2.3) with ry from (5.2.1). Then for each y ∈ B2 the following is satisfied: ⊥ (1) For each x ∈ S and  ∈ Lx ∩ Br (x) ∃! z ∈ Lˆ x +  such that Φr (z ) = 0. 2 (2) |∇Φr | − 4Φr (y)  C(n)δΦr (y). 2 (3) |∇2 Φr (y) − πˆ ⊥ y | < C(n)δ . (k) 2 2−k for k  3. (4) |∇ Φ|(y)  C(n, k)δ r Remark 5.2.5. Building a function which satisfies just (2)-(4) is a little easier and one does not need the Subspace Selection Lemma. Our construction is primarily designed so that (1) is also easily satisfied. Without (1) defining Sr = Φ−1 r (0) may not be a reasonable definition. We will also prove the above in the next subsection, it is mostly a direct application of the definition of Φr combined with the properties of Ly . In this section we want to use Theorem 5.2.4 in order to finish the proof of the Submanifold Approximation Theorem 5.1.1: Proof of Theorem 5.1.1 given Theorem 5.2.4. Let us define the set Sr ≡ Φ−1 r (0). Let us make some first observations about this set: Exercise 5.2.6. Use Lemma 5.2.2.(2), Theorem 5.2.4.(1) and the definition of Φr to show dH (Sr , S) < C(n)δr. The exercise thus proves Theorem 5.1.1.(1). Now we will prove some regularity results on Sr , and then use this regularity to prove Theorem 5.1.1.(3). Both will eventually be consequences of the Implicit Function Theorem. Let x ∈ S and let us write B2r (x) in coordinates (y, z) where y ∈ Lx and z ∈ Lˆ ⊥ x, where Lx is the subspace given in Lemma 5.2.2. Let us consider the derivative ⊥ mapping F : Lx × Lˆ ⊥ x → Lx given by F(y, z), w = ∂w Φr (y, z) .

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Using the definition of Φr , Theorem 5.2.4.(2) and Theorem 5.2.4.(3) we have in B2r (x) that: (5.2.7)

|F(y, z) − z| < C(n)δ r , |∂y F(y, z)| , |∂z F(y, z) − Id| < C(n)δ .

First note that the zeros of F describe Sr : Exercise 5.2.8. Show Sr ∩ Br (x) = {|∇Φr | ≡ 0} = {(y, z) ∈ Br (x) : F(y, z) = 0}. Hint: Note that (5.2.7) says that F(y, z) has a unique zero on each z-slice, use Theorem 5.2.4.(1) to see that the zero of F must be a zero of Φr . Remark 5.2.9. This exercise is the main place we use condition (1) of Theorem 5.2.4. Now by (5.2.7) and Theorem 5.2.4.(3) we may use the Implicit Function Theorem 2.1.1 in order to find a smooth function f : Br (x) ∩ Lx → Lˆ ⊥ x such that |∇f|  C(n)δ , r|∇2 f|  C(n)δ , Sr ∩ Br (x) = {(y, z) ∈ Br (x) : F(y, z) = 0} = {(y, f(y))} ∩ Br (x) . Thus we have seen that Sr is locally a smooth graphical submanifold and thus proved Theorem 5.1.1.(3). We have seen in Section 5.1 that (1) − (6) of Theorem 13.2.1 follow from (1) and (3), and therefore we have completed the proof of  Theorem 5.1.1.

6. Proof of Distance Approximation Theorem We now complete the proof of the Reifenberg Theorem by completing the proof of the Subspace Selection Lemma 5.2.2 and the Distance Approximation Theorem 5.2.4. Let us begin some technical results, in particular we first build a useful covering of B2 : # d(xα ,S)∨r , Lemma 6.0.1. There exists a covering B2 ⊆ α Br˜ α (xα ), where r˜ α = r˜ xα = 100 and smooth nonnegative functions φα such that (1) (2) (3) (4)

{B 1 r˜ α (xα )} are disjoint. 4 For each y ∈ B2 we have #{xα : y ∈ B4r˜ α (xα )} < C(n).  φα = 1 on B2 with supp φα ⊆ B4r˜ α (xα ). 13 |∂(k) φα |  C(n, k)˜r−k α .

Proof. Let {xα } ∈ S be any maximal subset so that {Br˜ α /4 (xα )} are disjoint. By maximal we mean if y ∈ B2 then Br˜ y /4 (y) ∩ Br˜ α /4 (xα ) = ∅ for some α. Now let # us show that B2 ⊆ α Br˜ α (xα ). So for y ∈ B2 , we may choose an α b such that 1 we have that r˜ y  2˜rα . In Br˜ y /4 (y) ∩ Br˜ α /4 (xα ) = ∅. Observing that |∇˜ry |  100 # particular then gives us y ∈ Br˜ α (xα ), and thus we have shown B2 ⊆ α Br˜ α (xα ). The proof of (2) follows from a volume estimate. Indeed, for y ∈ B2 consider the subset {xβ }N 1 such that y ∈ B4r˜ β (xβ ). Observe as in the last paragraph that 13 Recall

|∂(k) f| is the matrix norm of the kth derivative of f.

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1 by using |∇˜ry |  100 we have for any such β that 12 ry  rβ  2ry . In particular, Br˜ y /10 (xβ ) ⊆ B8r˜ y (y) and are disjoint. Thus we can estimate % Nωn 10−n r˜ n Br˜ y /10 (xβ ))  Vol(B8r˜ y (y)) = ωn 8n r˜ n y = Vol( y,

which gives N  80n , as claimed. To build the partition of unity first let φ  : B4 (0n ) → R be a fixed smooth, compactly supported nonnegative function with φ  ≡ 1 on B1 . Let us define  (x) ≡ φ  (˜ r−1 φα α (x + xα )), and with this the partition of unity itself by φ  (x) . φα (x) ≡  α  α φα (x)  Exercise 6.0.2. Use (2) to prove c(n)  φα (x)  C(n) for x ∈ B2 . Use this to prove (3) and (4).  Let us now first complete the proof of the Subspace Selection Lemma: Proof of Subspace Selection Lemma 5.2.2. Let {Br˜ α (xα )} and φα be the covering and partition of unity from Lemma 6.0.1. For each α let Lα ≡ Lxα ,104 r˜ α ,

 be as in (5.0.1). Morally, we simply want to define Ly ≡ φα Lα and check what estimates hold. Of course, one needs a well defined way of averaging affine subspaces in order to do this. Indeed, using the notion of nonlinear averages this is possible, but since we want these notes to be self-contained (and that is a rather technical procedure) we will do this by hand. However, it is helpful to keep in mind that the remainder of the proof is nothing other than some technical work in order to average nonlinear objects. Now we wish to define subspaces Ly as in the lemma. To define an affine subspace we need a linear subspace πˆ y and a point y ∈ Ly . Let us begin by writing these out, and then we will move on to estimating them. We define the point y simply by  φα (y)πα [y] . y ≡ α

The definition of πˆ y is a bit more involved. Let us begin by defining the matrix valued function  φα (y)πˆ α . My ≡ α

One sees from Exercise 5.0.2 and Lemma 6.0.1 that |My − πˆ β | < C(n)δ for any y ∈ B8r˜ β (xβ ), and in particular My is close to a projection map. If e1 (y), . . . , en (y) are the eigenvectors of My , in decreasing order, we then define our linear subspace πˆ y ≡ span{e1 (y), . . . , ek (y)} . Using our estimate on My we have |πˆ y − πβ | < C(n)δ for any y ∈ B8r˜ β (xβ ). Let us state our first Claims on the regularity of y and My :

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Claim 6.0.3. We have the estimates |∂i y − πˆ y |, r˜ y |∂2 y |  C(n)δ. Claim 6.0.4. We have the estimates |My − πˆ β | , r˜ 2y |∂2 My |  C(n)δ. Proof. We prove the gradient estimate of Claim 6.0.3. The other estimates are all the same. We first compute   ∂i φα (y)πα (y) + φα πα [ei ] . ∂i y = α

Now by using Exercise 5.0.2 and that  ∂i φ α = 0 ,



α φα

= 1 we obtain the following:

α

|πˆ α − πˆ β |, |πˆ ⊥ ˆ⊥ rβ if B8r˜ α (xα ) ∩ B8r˜ β (xβ ) = ∅ . α −π β | < C(n)δ˜ Choosing β so that xβ ∈ B4r˜ y (y) we then have  ∂i φα (y)(πα − πβ )|  C(n)δ˜r−1 |∂i y − πβ | = | y , α

where we have used that our partition estimates on φα and that r˜ y ≈ r˜ α for any  ball B4r˜ α (xα ) which contains y. Estimating the subspaces πˆ y takes a bit more technical work, as it is not just a partition of unity argument: Claim 6.0.5. r˜ y |∂πˆ y |, r˜ 2y |∂2 πˆ y |  C(n)δ. Proof. We will focus on the gradient estimate, the Hessian estimate is the same. The key is that we need to convert the estimates on My into estimates on πˆ y . The important point in this estimate is that there is a gap between the k largest eigenvalues and the n − k smallest eigenvalues, otherwise the estimate would not even be correct. Let us begin by using the Rellich characterization to write  My (ei , ei ) , πˆ y ≡ arg sup trL (My ) = arg sup ˆk L

ˆk L

where the sup is taken over all k-dimensional subspaces and ei are an arbitrary orthonormal basis of Lˆ k . Now observe from Claim 6.0.4 that for y ∈ B4r˜ β (xβ ) we have |πˆ y − πˆ β | , |My − πˆ β | < C(n)δ . Now consider the spaces of linear maps V = {v : Lˆ β → Lˆ ⊥ β} , −1 Vs = {v : Lˆ β → Lˆ ⊥ β s.t. ||v|| < 10 } ,

where Vs ⊆ V is the subset of maps with small norm. For v ∈ Vs we denote by Lˆ v = Graph(f) = {(, v()) :  ∈ Lˆ β } the associated linear subspace, and thus we may view Vs as the open set of linear subspaces which are close to Lˆ β . Then we define the smooth mapping F : B4r˜ β (xβ ) × Vs → V by F(y, v), w ≡ ∂w trLˆ v (My ) .

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Note that F(y, πˆ y ) = 0. We wish to use the Implicit Function Theorem 2.1.1 to give estimates on πˆ y . Thus using Claim 6.0.4 and the eigenvalue gap we have the estimates r|∂yi F| , r2 |∂yi ∂yj F| , |∂v F, w − v, w| < C(n)δ . In particular, by the Implicit Function Theorem 2.1.1 we may choose a function πˆ y : B2r˜ β (xβ ) → Vs such that {(y, v) : F(y, v) = 0} = {(y, πˆ y )} which thus satisfies the estimates of the claim.  Having constructed Ly we need only see that it satisfies the desired estimates from the Lemma to conclude the proof. Exercise 6.0.6. Show the following: (1) Using πy [v] = πˆ y [v − y ] + y , show that |∂my − πˆ y |, ry |∂2 my |  C(n)δ. (2) Show dH (Ly ∩ B10r˜ y (y), Lα ∩ B10r˜ y (y)) < C(n)δ ry . Use this to prove  estimate dH (Ly ∩ B10r˜ y (y), S ∩ B10r˜ y (y)) < C(n)δ ry . With the Subspace Selection Lemma in hand, we now prove Theorem 5.2.4: Proof of the Approximate Distance Function Theorem 5.2.4. Recall that we define Φr explicitly by the formula 1 1 1 Φr (y) ≡ d(y, Ly )2 = |y − πy [y]|2 = |y − my |2 , 2 2 2 as in (5.2.3). We begin by proving (1). Thus let x ∈ S and  ∈ Lx ∩ Br (x). Let ˆ⊥ φ : Lˆ ⊥ x → Lx be a smooth cutoff function with φ ≡ 1 in Br (0) and φ ≡ 0 outside of B2r (0). Note that for each y ∈ Br (x) there exists a unique point in the intersection Ly ∩ Lˆ ⊥ x since they are transverse. This point moves smoothing since Ly moves ˆ⊥ smoothly. Consider the smooth mapping π : Lˆ ⊥ x → Lx given by π(y) ≡ y − φ(y) · Ly ∩ Lˆ ⊥ x . Note that π = y outside B2r and by applying our estimates on Ly we obtain the inequalities |π(y) − y| < C(n)δ r ,

|dπ − Id| < C(n)δ .

We see then that π is a degree 1 mapping which fixes the boundary of B2r (0), and hence there exists y ≈ 0 for which π(y) = 0. At this point we then have 1 y ∈ Ly =⇒ Φr (y) = |y − πy (y)|2 = 0 , 2 as claimed. Estimates (2)–(4) are now relatively straight forward computations. To prove Theorem 5.2.4.(2) we first compute the derivative of Φr : ˆ y − ∂my )[ei ], y − my  . ∂i Φr = πˆ ⊥ y [ei ], y − my  + (π Squaring this gives |∂Φr |2 − |y − my |2  C(n)δ |y − my |2  C(n)δ Φr ,

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as claimed. To compute Theorem 5.2.4.(3) we similarly first compute the Hessian ˆ⊥ ˆ y − ∂my )[ej ], πˆ ⊥ ˆ⊥ ∂i ∂j Φr = πˆ ⊥ y [ei ], π y [ej ] + (π y [ei ] + ∂j π y [ei ], y − my  + (∂j πˆ y − ∂j ∂my )[ei ], y − my  + (πˆ y − ∂my )[ei ], πˆ ⊥ y [ej ] + (πˆ y − ∂my )[ei ], (πˆ y − ∂my )[ej ] , which gives

∂i ∂j Φr − πˆ ⊥  C(n)δ , y

as claimed. Theorem 5.2.4.(4) is the same.



Lecture 2: Rectifiable Reifenberg for Measures 7. Applying the classical Reifenberg theorem Let us now explore the various issues that arise in attempting to use the classical Reifenberg theorem in applications (for instance singular sets of nonlinear equations). To summarize we need to deal with the following three issues: (I) The Hausdorff distance used in Reifenberg condition behaves as a pointwise L∞ bound, and in practice we will have more integral control than pointwise control. (II) In applications our sets or measures can have holes and need not satisfy the Reifenberg condition! Best we can do is force symmetry on some special regions. (III) bi-Hölder control is simply too weak. Lack of gradient control prevents understanding of volume or rectifiable structure. To deal with (I) it becomes more natural to discuss controlling measures than sets and one works with the Jones β-numbers of these sets instead of the Hausdorff distance. Though this adds some technical complication, it is a relatively minor issue by itself. Dealing with (II) is a more serious, and one introduces k-neck regions to help deal with this, see Section 11. One is able to gain back a weak version of the Reifenberg control in this case, but only on certain regions and in a discrete sense. Dealing with these neck regions then becomes similar to the classical Reifenberg case, though dealing with the holes presents some subtle points in the construction. One also has to then prove such neck regions exist and are even fairly common, which is the content of the the Neck Decomposition Theorem in Section 11. Dealing with (III) is again a serious issue, and will require both the neck region ideas of (II) and a more refined collection of hypotheses. One issue at hand is the snowflake example of the previous section, which shows that the assumptions of the Reifenberg theorem cannot give better than bi-Hölder control. One therefore needs more than just scalewise control on the Jones β-numbers,

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and we will require a Dini condition be satisfied. To understand this a little better let us begin by revisiting the snowflake example: Example 7.0.1 (Von-Koch Snowflake 2). We are refining the snowflake construction of Example 4.0.3, so that much of our terminology originates there. As before let a0−1 = (−2, 0), a01 = (2, 0) with S0 = a0 ,a0 the associated interval, but now also −1 1 choose a sequence 0 < δi  18 δ. Similar to the original construction, we define Si # inductively in the following way. If Si = ai ,ai is piecewise linear, then let j j+1 % δ % δi i ai ,ai ≡ ai+1 ,ai+1 . Si+1 = Si ≡ j

j+1

j

j+1

j

Note that we must have dH (Si , Sj )  4  2−i O(δ), and in particular there exists S = lim Si and S is an O(δ)-Reifenberg set as in Example 4.0.3. As before, use the pythogorean theorem to compute the length of Si to be   |Si |2 = 16 1 + δ2j . −k O(δ ) k k=i 2

ji

As observed previously, for |Si | to remain uniformly bounded it is not sufficient for δj to remain uniformly small. One sees from the above that |Si | remains uni 2 formly bounded iff δj < ∞. In particular, to control the volume and Lipschitz structure of S one requires not only that the Reifenberg constant of S ∩ Br (x) tend to zero as r tends to zero, but that the Reifenberg constants be square summable in the scales. Let us note that the above example was generalized into a theorem in [10], where they show that a Reifenberg set S for which the sum 2  −i 2 2 ds (x, 2 ) ≈ β∞ β∞ k k (x, s) s 0 is uniformly bounded at each point is bi-Lipschitz to B1 (0k ). They push this one step further to handle a simplified version of (II) where one assumes uniform tilting control over the holes. A similar setup will be present in these notes in the Neck Regions of Section 11. Most of the work of these notes will be about being able to produce such Neck regions effectively, which becomes a bit subtle in the context of general measures. We want in this lecture to work toward stating the main generalization of the Reifenberg Theorem which will interest us and solve the issues (I), (II), (III). The context is more involved now, and thus we will need to begin by describing some structure.

8. Hausdorff, Minkowski, and Packing Content In this section we give a brief review of the notions of Hausdorff, Minkowski, and packing content. For a more detailed reference, we refer the reader to [13, 16]. Let us begin with the notions of content:

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Definition 8.0.1 (Content). Given a set S ⊆ Rn and r > 0 we define the following: (1) The k-dimensional Hausdorff r-content of S is given by 14 

% Hrk (S) ≡ inf ωk rk Bri (xi ) and ri  r . i :S⊆ (2) The k-dimensional Minkowski r-content of S is given by 

% k Mk (S) ≡ inf ω r : S ⊆ B (x ) ≈ rk−n Vol(Br (S)) . r i k r (3) The k-dimensional packing r-content of S is given by 

Pk ωk rk r (S) ≡ sup i : xi ∈ S with {Bri (xi )} disjoint, and ri  r . Exercise 8.0.2. Let S = L ∩ B1 where L is an -dimensional subspace. Show there exists 0 < c(n) < C(n) < ∞ such that for all 0 < r < 1:  k− , Hrk (S ) , Mk r (S ) ≈ r  Pk r (S ) = ∞ if  > k ,  k− Pk if k   . r (S ) ≈ r

Example 8.0.3. Let S = Qn ∩ B1 (0n ) be the rationals. Then for all 0 < r < 1 we have r→0

Hrk (S) = 0 if k > 0 with Hrk (S) −→ ∞ if k = 0 . In particular, Hk (S) = 0 for k > 0 and so dimH (S) = 0. However, the Minkowski and Packing content are quite badly behaved: r→0

k−n −→ ∞ for k < n , Mk r (S) ≈ r

Pk r (S) = ∞ for all k . Morally, this is because the closure S = B1 is an n-dimensional set, and so from a packing and minkowski point of view S itself is treated as an n-dimensional set. Note then that controlling the Hausdorff content amounts to finding some covering of S which is well behaved, controlling the Minkowski content amounts to saying the covering S by balls of radius r is well behaved, and controlling the packing content amounts to saying every covering is well behaved. In particular, bounding the Hausdorff content is less powerful than bounding the Minkowski content, which is itself less powerful than bounding the packing content. Thus we have the relations k Hrk (S)  Mk r (S)  Pr (S) ,

where  means the inequality holds up to a dimensional constant. One can use these notions in the classical manner to define measures and dimensions. In particular, for completeness sake let us recall the definition of Hausdorff measure: Definition 8.0.4 (Hausdorff Measure). Given S ⊆ Rn we define its Hausdorff measure Hk (S) = limr→0 Hrk (S). 14 The

constant ωk is the volume of a unit ball in Rk .

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8.1. Rectifiability of Sets Let us now discuss the notion of rectifiable sets. In essence, these are sets which are manifolds away from a set of measure zero, though this set of measure zero need not be closed. We begin with a definition: Definition 8.1.1. Let S ⊆ Rn be a set, then we say S is k-rectifiable if there exists a countable collection of Lipschitz maps fi : Si ⊆ Rk → Rn such that # Hk (S \ fi (Si )) = 0. Sometimes the above is referred to as countably rectifiable, and one additionally assumes Hk (S) < ∞ in order to call S rectifiable. Example 8.1.2. Any k-dimensional submanifold Sk ⊆ Rn is k-rectifiable. Further, if S˜ k ⊆ Sk is an arbitrary subset, then S˜ k is also k-rectifiable and if we define # S ≡ q∈Qn (S˜ k + q), then S is also k-rectifiable. Note that the example S above is dense in Rn , so the notation of rectifability depends heavily on the ability to decompose the set. The notion of a rectifiable measure is very similar: Definition 8.1.3. Let μ be a measure on B1 (0n ). We say μ is k-rectifiable if there exists a k-rectifiable set S such that μ(B1 \ S) = 0 and μ ∩ S is absolutely continuous with respect to the Hausdorff measure Hk ∩ S.

9. Jones β-numbers The Rectifiable Reifenberg Theorem 10.0.2 we will be introducing will be for a measure μ instead of a set S. As such, let us discuss the Jones β-numbers to estimate how close the support of μ is to a k-plane in a more L2 sense, as in (I). Because of the possibility of holes as in (II) we will only be concerned with how closely the support of μ is to living inside a k-plane, without care for how dense the support is inside Lk . Precisely: Definition 9.0.1 (Jones β-numbers). Given a measure μ and integer k ∈ N we define the L2 β-numbers  2 2 −2−k βk (x, r; μ) = βk (x, r) ≡ inf r d(y, L)2 dμ[y] , Lk

where the infimum is taken over all k-planes

Br (x)

Lk .

When no confusion arises we will simply write β(x, r) and drop the dependence on the measure. Let us state the following example, which shows in particular how control on β(x, r) does not stop the existence of ‘holes’ in supp μ: Example 9.0.2. Let Lk ⊆ Rn be a fixed subspace and let μk be an arbitrary measure with supp μk ⊆ Lk ∩ B2 . Then βk (x, r) = 0 for all Br (x) ⊆ B2 . Let us now give a series of examples which illustrate how βk behaves when the support of μ takes its support on sets of various dimensions:

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Example 9.0.3. As a more specific example, consider μk ≡ α0 δ0 + αk Hk ∩ Lk ∩ B2 , where Hk ∩ Lk is the k-dimensional Hausdorff measure restricted to the subspace Lk , δ0 is the Dirac delta measure at the origin, and α0 , αk are arbitrary. Then βk (x, r) = 0 for all Br (x) ⊆ B2 . We see from the above example, by taking α0 , αk very large, that the measure μk does not need to have any a priori bounds, even if βk (x, r) = 0 is identically zero. The example also illustrates how even if the support suppμk is k-rectifiable, the measure itself may not be. The next example studies what happens for measures supported on higher dimensional subsets: Example 9.0.4. Consider μ+ = δHn ∩ B2 , where Hn is the n-dimensional Hausdorff measure. Then we can compute βk (x, r)2 ≈ ωn δ rn−k . In particular, we have that βk (x, r) is always δ-small, and indeed is decaying polynomially. The above example shows that βk (x, r) may be uniformly small, even decaying, but that the support of μk need not live on a k-dimensional object. The following exercises are fairly straightforward but very instructive in building an intuition for the behavior of the β-numbers: Exercise 9.0.5. Show that: (1) If Bs (y) ⊆ Br (x) ⊆ Bas (y), then βk (y, s)  C(n, a)βk (x, r). 1 2  2 2 ds (2) If ri = 2−i , then r β(x, s)2 ds rri 1 β(x, ri )  r β(x, s) s . s  2 2 (3) If μ = μ1 + μ2 , then β(x, r; μi )  β(x, r; μ) .

10. Rectifiable Reifenberg Theorem for Measures We are now in a position to deal with the general case and state the Rectifiable Reifenberg Theorem, which is designed to handle the issues (I), (II) and (III). Let us now combine the examples of the last section in order to illustrate all the subtle issues involved in trying to use the βk -numbers to study a completely general measure: Example 10.0.1 (Varying Dimensions Example). Consider the measure μ ≡ μk + μ+ , where μk is from Example 9.0.2 and μ+ is from Example 9.0.4. Then by considering the subspace Lk from the examples we may estimate βk (x, r)2  4ωn δ rn−k . In particular, for arbitrary α0 , αk we have that β(x, r) is always δ-small, and indeed is decaying polynomially.

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The above example shows that even when one has extremely strong conditions on βk , that for a general measure we cannot make a general statement about the rectifiability of μ or its total mass. This may seem like endgame, however the example also suggests that maybe we can decompose the measure into pieces where we do have such control. Indeed, this is exactly the case and the content of the Rectifiable Reifenberg Theorem: Theorem 10.0.2 (Rectifiable Reifenberg [12]). Let μ be a nonnegative Borel-regular measure supported in B1 (0n ). Suppose  2 dr (10.0.3) β(x, r)2 dμ  Γ . r B1 0 Then we can write μ = μk + μ+ into a sum of measures such that (1) μ+ (B1 )  C(n)Γ . (2) If K ≡ supp μk then K is k-rectifiable with Hk (K) < C(n), and indeed we have the Minkowski and packing content estimates: Vol(Br (K))  c(n)rn−k ,

Pk r (K)  C(n) .

Remark 10.0.4. This result holds for measures in Hilbert spaces, see [11], and in particular the c(n) constants above are turned into c(k) constants. The result is also generalizable to Banach Spaces, see [11], however this is more subtle. The above result is for a general measure. We may obtain some stronger results if we strict ourselves to measures with either upper or lower density bounds. First let us precisely define this: Definition 10.0.5. Let μ be a nonnegative measure. Then we define the upper and lower densities: μ(Br (x)) θ∗ (μ, x) ≡ lim sup , k r→0 ωk r μ(Br (x)) . ωk rk Let us discuss some corollaries of Theorem 10.0.2: θ∗ (μ, x) ≡ lim inf

r→0

Corollary 10.0.6. Let μ be a nonnegative Borel-regular measure supported in B1 (0n ) and let (10.0.3) hold. Then we have the following:

 (1) If θ∗ (μ, x)  A then μ(B1 )  C(n) Γ + A . (2) If a  θ∗ (μ, x) then K

≡ supp μ is k-rectifiable with the Hausdorff measure bound Hk (K)  C(n) 1 + a−1 Γ . (3) If a  θ∗ (μ, x) and θ ∗ (μ, x) A then μ is k-rectifiable and we have the mass bound μ(B1 )  C(n) Γ + A .

Remark 10.0.7. Note that we are requiring the relatively weak conditions of an upper bound on the lower density in (1), and conversely a lower bound on the upper density in (2). This is directly due to the packing estimates on K. If one only had weaker Hausdorff measure estimates on K, one would have to make

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the stronger assumptions of upper bounds on upper density and lower bounds on lower density. Tolsa and Azzam [1] have proved the following related result: If μ is a measure whose upper and lower densities are bounded almost everywhere, then μ is k2 rectifiable if 0 βk (x, s)2 ds s < ∞ for a.e. x. In comparison the primary goal of Theorem 10.0.2 is therefore to remove the assumption of density bounds, and instead conclude the existence of mass bounds (either in the form of μ-measure bounds or Hausdorff measure bounds as in Theorem 10.0.2). It is interesting to note when working under the assumption of density bounds as in [1], that Tolsa [22] and is also able to prove the opposite direction. That is, 2 if μ is a k-rectifiable measure with density bounds, then 0 βk (x, s)2 ds s < ∞ for a.e. x. On the surface this might be an unexpected result, and I think provides great intuition about the connection of rectifiability and the Jones β-numbers.

Lecture 3: Outline Proof of Rectifiable Reifenberg We now want to begin the proof of the Rectifiable Reifenberg Theorem 10.0.2. This lecture will focus on introducing Neck regions and their associated Structure and Decomposition theorems. After we discuss these we will use them to prove Theorem 10.0.2. In subsequent lectures we will prove the Neck Structure and Neck Decompositions themselves. In the process we will build quite a bit more information than is present in Theorem 10.0.2, which itself is quite useful in applications.

11. Neck Regions and their Structure and Decomposition This section is dedicated to introducing the reader to the notion of a neck region, and we will be stating the basic structure theory and decomposition theorem associated to Neck Regions. Neck regions first made their appearance in [14] during the proof of the n − 4 finiteness conjecture and again in [18] in the proof of the energy identity conjecture for Yang-Mills. They are also used in [5, 17] in order to to prove the rectifiability of singular sets of harmonic maps and spaces with lower Ricci curvature, respectively. The notion of a neck region developed in these notes is very analogous, though in some manners quite a bit easier to work with than in the last references due to the technical conditions involved. Neck Regions will be regions which we can control in a manner analogous to our control in the classical Reifenberg theorem. There are many subtle points, including the fact that we cannot get a true Reifenberg condition to hold. That is, when we restrict ourselves to well behaved points they may not be dense in some kplane on each scale. However, we can replace it with a weaker notion by making sure at each scale there are enough well controlled points to at least weakly span a k-dimensional plane. This will be enough to get the control we desire in the end.

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To make this more precise let us introduce the notions of linear independence and noncollapsing. Recall a set of points {xi }k 0 is called linearly independent if no point lives in the span of any of the others. More effectively: Definition 11.0.1 (-Linear Independence). We call a set {xi }k 0 ∈ Br -linearly   independent if xi+1 ∈ Br x0 + span{x1 − x0 , . . . , xi − x0 } for each i. We say a set S ⊆ Br is a (k, )- linearly independent set if there is a -linearly independent set of points {xi }k 0 ∈ S. The notion of -independence can be viewed as a very weak version of the Reifenberg condition. That is, a set of points may not densely span an affine space, but they at least effectively span such a space. In practice what we will need to be independent are noncollapsing points, which is to say points with lower mass bounds. Precisely: Definition 11.0.2 (Noncollapsing). Let μ be a measure, then we say a ball Br (x) is (k, , ν)-noncollapsed if there exists a 2-linearly independent {xi }k 0 ∈ Br (x) such that we have the lower mass bounds μ(Br (x)) > ν(r)k . Remark 11.0.3. The condition that Br (x) be (k, , ν)-noncollapsed guarantees not only that there are balls with definite mass in Br (x), but that there are k + 1 such balls which effectively span a k-dimensional affine subspace. Remark 11.0.4. The condition implies that if yi ∈ Br (xi ) then {yi }k 0 are -linearly independent. Let us now give our formal definition of Neck Regions: Definition 11.0.5 (Neck Regions). Let μ be a measure on Br with C ⊆ Br a closed subset and rx : C → R+ a radius function such that the closed balls {Bτ2 rx (x)} are disjoint 15 . We call N = Br \ Brx (C) a (k, δ, , ν)-neck region if 16 (n1) For x ∈ C and rx  s  r, there is an affine Lk such that L ∩ Bs ⊆ Bτs (C) and C ∩ Bs ⊆ Bδs (L). (n2) For each x ∈ C with τ−1 rx  r  1, then Br (x) is (k, , ν)-noncollapsed.  2r 2 (n3) rx βk (x, s)2 ds s < δ for each x ∈ C. Remark 11.0.6. We call C0 ≡ {x ∈ C : rx = 0} and C+ ≡ {x ∈ C : rx > 0}. One should imagine δ 0 very small, then by the above we can find r > 0 such that if  2r ds Uη,r ≡ {x ∈ UΛ :  η} , βk (x, s)2 s 0 then μΛ (B1 \ Uη,r ) < η . Let us now use the Hausdorff measure estimate on Uη,r ⊆ UΛ ⊆ suppμ previously proved in order to find a covering %  Bri (xi ) with ri  r and rk Uη,r ⊆ i  C(n) . i

Let μi ≡ μΛ ∩ Uη,r ∩ Bri (xi ). Then we can apply Theorem 10.0.2 in order to write − μi = μ+ i + μi such that k μ+ i (Bri (xi ))  C(n)η ri ,

μk i is k-rectifiable . that since P(suppμk ) < C(n) we have that suppμk is compact, thus this is a reasonable constraint on the radius.

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In particular, we have that μΛ is k-rectifiable away from a set of measure   μΛ (B1 \ Uη,r ) + μ+ rk i  C(n)η . i (Bri (xi ))  η + C(n)η i

As η > 0 was arbitrary, this proves that μΛ is k-rectifiable. As Λ was arbitrary, this proves that μ is k-rectifiable.

Lecture 4: Proof of Neck Structure and Decomposition Theorems We focus in this lecture on the meat of the argument, and prove the neck structure theorem and the neck decomposition theorem.

13. Proof of Neck Structure Theorem Our proof of the classical Reifenberg Theorem 3.0.5, given in Lecture 1, was designed precisely to pass over to the context of the Neck Structure Theorem. Though the basic outline will remain the same, in the context of Theorem 11.0.11 most of the results are a bit more refined and there are a handful of technical challenges and nuances beyond the classical result, which we will describe. In particular, we begin with a few technical preliminaries in order to deal with this: 13.1. Best Subspaces on Neck Regions Exercise 5.0.2 told us in the context of the classical Reifenberg theorem that the best approximating subspaces do not change much from scale to scale. For a general measure, even with well controlled β-numbers, this will not need to be the case. We will study this phenomena some in this section and aim toward proving that at least on Neck regions, one can indeed control the best subspaces in a manner analogous to Exercise 5.0.2. Let us begin with some notation and definitions. Recall that for each x ∈ B1 and 0 < r  10 we have fixed a choice of affine k-dimensional subspace Lx,r = Lx,r [μ] satisfying  −2−k (13.1.1) d2 (y, L)dμ[y] , Lx,r ∈ arg min r L

so that

Br (x)

 βk (x, r)2 = r−2−k

d2 (y, Lx,r )dμ[y] . Br (x)

Given an affine subspace Lx,r we denote πx,r : Rn → Lx,r ⊆ Rn to be the projection map, Lˆ x,r the associated linear subspace, and πˆ x,r : Rn → Lˆ x,r ⊆ Rn the linear projection map. The first point that is worth making is that unlike Exercise 5.0.2, if μ is arbitrary then there is no reason the subspaces need to be comparable, even if the β-numbers are small: Example 13.1.2. Let p, q ∈ B1 be points with |p − q| = 10−1 μ = δp + δq be the sum of dirac deltas. Note that for every x ∈ B1 and r > 0 we have that β1 (x, r) = 0. If p, q ∈ Br (x) are both points in the ball then the best subspace Lx,r is the line connecting p and q. If Bs (y) contains only one of the points, say p, then Ly,s

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can be any line which contains p. In particular, Lx,r and Ly,s need not be at all comparable, even if Br (x) and Bs (y) are comparable. Our main goal in this section is to see that in a Neck region the above does not happen, namely the subspaces Lx,r are indeed comparable, in the spirit of Exercise 5.0.2. The following is the main result of this section, which we will prove by its completion: Proposition 13.1.3 (Best Subspace Behavior on Neck Regions). Let N = B1 \ Brx (C) be a (k, δ, , ν)-neck region with δ < δ(n, , ν). Consider subspaces {Lx,r } defined as in (13.1.1) and let r  102 rx . Then for x, y ∈ C the following hold: (1) dH (Lx,r ∩ Br (x), Ly,s ∩ Br (x)) < C(n, , ν, a)βk(x, 10a r) r if |x − y|  10r and a−1  sr  a. (2) If Br (x) and Bs (y) are as above then |πˆ x,r [v]| − 1  C(n, , ν, a)βk(x, 10a r)2 for each v ∈ Lˆ y,s with |v| = 1,. Estimates (1) and (2) above tell us that if Br (x) and Bs (y) are comparable balls, then Lx,r and Ly,s are comparable affine subspaces. As in Example 13.1.2 let us note that we need to be in a neck region for this, otherwise such results are false. Let us also emphasize the square gain in (2), as it is key to the volume and rectifiability estimates on μ later. To prove Proposition 13.1.3 we need to learn how to compare the best approximating subspaces of a measure in an accurate manner. The following definition of a center of mass, which originates in [12], will be used to tell us how to control at least one point in a given ball: Definition 13.1.4. [Center of Mass] We define the generalized μ-center of mass X of  a ball Br (x) as follows. If μ(Br (x)) < ∞, let X = μ(Br1(x)) Br (x) zdμ(z) be the usual center of mass. If μ(Br (x)) = ∞, we let X be any point in the intersection

  & k 2 X ∈ Br (x) ∩ affine V : d(z, V) dμ(z) < ∞ . Br (x)

Exercise 13.1.5. Show if μ(Br (x)) = ∞ but βk (x, r) < ∞ then such a point X exists. Exercise 13.1.6. More generally, chose Bs (y) ⊆ Br (x) with μ(Bs (y)) = ∞ but with βk (x, r) < ∞, then show the center of mass Y of Bs (y) lives in Lx,r . Let us now see how on a given ball we can at least control how far away the center of mass is from a best subspace. The main result is the following: Lemma 13.1.7 (Center of Mass and Best Subspaces). Suppose Bs (y) ⊂ Br (x) and let Lx,r be a best subspace as in (13.1.1). Let Y be the generalized center of mass for Bs (y). Then rk βk (x, r)2 r2 . (13.1.8) d(Y, Lx,r )2  μ(Bs (y))

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Remark 13.1.9. In this business it is key how various quantities depend on one another. Let us emphasize that the above formula depends inverse linearly on μ(Bs (y)) and quadratically on βk (x, r)2 . This relationship will be crucial to the mass bounds on Neck Regions later. Proof. We can assume βk (x, r) < ∞, otherwise there is nothing to show. From Exercise 13.1.6 we have if μ(Bs (y)) = ∞ then Y ∈ Lx,r . Otherwise we can check the claim by using Jensen’s inequality to calculate,  1 d(z, Lx,r )2 dμ(z) d(Y, Lx,r )2  μ(Bs (y)) Bs (y)  rk+2 rk+2 r−k−2 βk (x, r)2 .  d(z, Lx,r )2 dμ(z) =  μ(Bs (y)) μ(Bs (y)) Br (x) Now we need to move from our ability to control best subspaces at a single point to being able to control the whole best subspace. The first step in this direction is to see that two k-dimensional affine subspaces are close iff they are close at k + 1 independent points. This is quite intuitive as an affine subspace is well defined by such a collection. Before reading the next lemma recall from Definition 11.0.1 the notion of (k, )-linearly independence: Lemma 13.1.10 (Subspace Distance Estimates). Let L1 , L2 be two k-dimensional affine subspaces and let {xi }k 0 ⊆ B1 be a (k, )-linearly independent set. Then we have the estimate     d(xi , L1 ) + d(xi , L2 ) . dH L1 ∩ B1 , L2 ∩ B1  C(n, ) i

  Proof. We may assume i d(xi , L1 ) + d(xi , L2 ) < 10−3 , as otherwise by choosing C(n, ) large the estimates trivially hold. Let L = x0 + span{xi − x0 }, and we will prove the result    d(xi , L1 ) , dH L1 ∩ B1 , L ∩ B1  C(n, ) i

the general case then follows from a triangle inequality. Now with all of this given we define {x1i }k 0 = {πL1 (xi )} and see that this set is −1 1 (k, 10 )-linearly independent with d(xi , xi ) < 10−2 . Now let 1 be any point in L1 ∩ B1 . Then we can uniquely write 1 − x10 =

k 

i1 (x1i − x10 ) .

1

An instructive exercise, which depends strongly on the -linear independence of the set {x1i }, is to show the following: Exercise 13.1.11. Show |i1 | < C(n, )|1 − x10 | < C(n, ).  i Now if we define  ≡ x0 + k 1 1 (xi − x0 ) ∈ L then we have the estimate   d(xi , L1 ) . d(1 , ) < C(n, ) x1i − xi | = C(n, ) i

i

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Since 1 was arbitrary this proves that every point of L1 ∩ B1 lives within at most  C(n, ) i d(xi , L1 ) of a point in L. The verbatim argument works with L1 and L switched, thus proving the Hausdorff estimate and hence the Lemma.  The above Lemma’s will be the key point in proving Proposition 13.1.3.(1). In order to prove Proposition 13.1.3.(2) we provide one more general lemma:   Lemma 13.1.12. Let L1 , L2 be two linear subspaces with d ≡ dH L1 ∩ B1 , L2 ∩ B1 . Then for each v ∈ L2 with |v| = 1 we have that |π1 [v]| − 1 < d2 . Proof. The proof is a simple application of the pythagorean theorem, however we emphasize here the important square gain on d here, as this is crucial for future applications. To prove the result let v  = π1 [v] with w ≡ v  − v. Note the two estimates: |w| = d(v, L1 )  d , |v|2 = |v  |2 + |w|2 . With these in hand we obtain '  |v | − 1  1 − 1 − |w|2  |w|2  d2 , 

as claimed. Finally we are now in a position to prove Proposition 13.1.3:

Proof of Proposition 13.1.3. We first prove Proposition 13.1.3.(1). Let us fix Br (x) with r  102 rx , and by using (n2) let {zi }k 0 ∈ Br/4 (x) be a (k, 2)-linearly independent set with ri = r/4 such that μ(Bri (zi )) > νrk i . Now let Zi ∈ Bri (zi ) be the generalized μ-center of mass as in (13.1.4). Note then that the {Zi } are (k, )linearly independent. Thus by using Lemma 13.1.7 and that μ(Bri (zi )) > νrk i by (n2) we get that d(Zi , Lx,r ) < C(n, ν, )βk (x, r)r , d(Zi , Lx,10ar ) < C(n, , a, ν)βk (x, 10ar) r , We find that dH (Lx,r ∩ B10ar (x), Lx,10ar ∩ B10ar (x)) < C(n, , ν, a)βk (x, 10ar) r by applying Lemma 13.1.10. Note that Bs (x) ⊆ B10ar (x), so that the same argument gives dH (Ly,s ∩ B10ar (x), Lx,10ar ∩ B10ar (x)) < C(n, )βk (x, 10ar) r. The triangle inequality then proves Proposition 13.1.3.(1). Now to prove Proposition 13.1.3.(2) we simply use Lemma 13.1.12 together with Proposition 13.1.3.(1).  13.2. Submanifold Approximation Theorem Recall that a primary goal in the proof of the Neck Structure Theorem 11.0.11 is to build a submanifold T which contains C and is bi-Lipschitz to the ball B1 (0k ). In the spirit of the classical Reifenberg, we will build a family Tr of smooth submanifolds which live near C and are scale invariantly smooth. To state the Approximating Submanifold

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Theorem more precisely, let us begin by introducing some notation. Recall from (11.0.7) our extension of the radius function rx to all of B1 by a regularity scale procedure. In the construction of the submanifolds Tr we will not want to look below scale r, and as such we also consider the following: ry ≡ ry ∨ r ≡ max{ry , r} . Note that ry  d(y, C) ∨ r. We can now state the Approximating Submanifold Theorem, which is one of the central technical results needed in the proof of Theorem 11.0.11: Theorem 13.2.1 (Approximating Submanifold Theorem). Let N = B1 \ Brx (C) be a (k, δ, , ν)-neck region with with subspaces {Lx,r } defined as in (13.1.1). For each r > 0 there exists a smooth submanifold Tr ⊆ B2 with Tr = T1 = L0,2 on the annulus A3/2,2 (0) and which satisfy (1) C ⊆ BC(n)δry (Tr ), Tr ⊆ BC(n)τry (C).

 −1  τ s (2) dH (Tr ∩ Bs (x), Lx,s ∩ Bs (x)) < C(n) r¯ x βk (x, t)dt  C(n)δs for s  rx and x ∈ Tr . (3) Tr is a (C(n)βk (x, τ−1 rx ), rx )-graphical submanifold, see Definition 2.5.1. (4) There is a smooth πr : Bry (Tr ) → Tr ⊆ Rn with πr ∩ Tr = Id and the estimate |∇2 πr |(y) < C(n)β(y, τ−1 ry ). (5) dH (Tr/2 , Tr ) < C(n)δr with |πr (x) − x| < C(n)δr for x ∈ Tr/2 . (6) For x ∈ Tr/2 and v ∈ Lx,r¯ x a unit vector, ||dπr [v]| − 1| < C(k)βk (x, τ−1 rx )2 . Remark 13.2.2. Note that if we take T1 to be the affine subspace L0,2 with π1 the orthogonal projection map then the above holds with r = 1. It will be convenient to make this choice. Remark 13.2.3. Note the square gain in (6) on the β-number estimate. Let us begin with some exercises. If the reader completed the exercises of Section 2.4 and Section 5.1 then these are almost the same: Exercise 13.2.4. Use Exercise 2.5.6, Theorem 13.2.1.(2) and Theorem 13.2.1.(3) to see that for each x ∈ Tr if L ≡ Tx Tr is the tangent space at x, then dGH (L ∩ B1 (x), Lx,rx ∩ B1 (x))  C(n)βk (x, τ−1 r). Exercise 13.2.5. Use the last exercise, Theorem 13.2.1.(6) and Theorem 13.2.1.(4) to see that for each x ∈ Tr if v ∈ Tx Tr is a unit tangent vector21 at x, then we have ||dπr [v]| − 1| < C(n)βk (x, τ−1 rx )2 . Exercise 13.2.6. Let x, y ∈ Tr with |x − y|  10rx . Let σ : [0, 1] → Tr be the curve   connecting x and y defined by σ(t) = πr (1 − t)x + ty . Use Theorem 13.2.1.(4) to show the length |σ| of σ satisfies (1 − C(n)δ)|x − y|  |σ|  (1 + C(n)δ)|x − y|. Tx Tr is an affine subspace, we say v ∈ Tx Tr is a unit vector if |v − x| = 1. That is, as an element of the associated linear subspace v is a unit norm vector.

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Hint: Write the length |σ| = estimate |σ|. ˙

0

331

|σ| ˙ and argue as is outlined in Exercise 5.1.9 to

The above exercise is telling us that locally the intrinsic and extrinsic geometry of Tr are the same. The proof of Theorem 11.0.11 in the next subsection will implicitly prove that this holds globally. 13.3. Proof of Neck Structure Theorem 11.0.11 given the Approximating Manifold Theorem 13.2.1 The beginning of the proof is similar to the proof of the classical Reifenberg Theorem 3.0.5 given Theorem 5.1.1. However, the key estimates are different as we now need to conclude bi-Lipschitz control over the submanifolds Tr , not just bi-Hölder, and we need to conclude a mass bound on the neck region μ(N). Begin by considering the radii ri = 2−i and the submanifolds Ti = Tri from Theorem 13.2.1. Let πi = πri ∩ Ti+1 : Ti+1 → Ti be the projection map from Theorem 13.2.1 restricted to Ti+1 . Let us define the maps Πi,j ≡ πi−1 ◦ · · · ◦ πj : Ti → Tj Πi ≡ Πi,0 : Ti → T0 ≡ L . Note that the Πi are necessarily diffeomorphisms which equal the identity in A3/2,2 (0). Note first the weak estimate, which follows from Exercise 2.6.2 and Theorem 13.2.1.(5): (13.3.1) |Πi,j (x) − x| 



j    |π Πi,+1 (x) − Πi,+1 (x)| < C(n)δ r  C(n)δrj . i

With this in hand we turn to our first main Claim of the result: Claim 13.3.2. If x ∈ Ti and v ∈ Tx Ti has length 1, then ||dΠi [v]| − 1| < C(n)δ. Proof. Note by the chain rule that dΠi,j [v] = dπj [dΠi,j−1 [v]]. Thus using (13.3.1), (1) of Exercise 9.0.5 and Exercise 13.2.5 we have that ||dΠi,j [v]| − |dΠi,j−1 [v]||  C(n)βk (x, τ−1 rj )2 |dΠi,j−1 [v]| , and hence

  1 − C(n)βk (x, τ−2 rj )2 |dΠi,j−1 [v]|  |dΠi,j [v]|    1 + C(n)βk (x, τ−1 rj )2 |dΠi,j−1 [v]| .

Composing the upper bounds gives us |dΠi,j [v]|
0 with  < (n) and δ < δ(n, , ν), there is a covering B1 ⊆ S− ∪ Sk ∪ S+ with % %  % C0,a , S+ = Na ∩ Bra ∪ Brb (xb ) and Sk = a

b

a

Aaron Naber

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and such that (1) Na = B2ra (xa ) \ Bra,x (Ca ) are (k, δ, , ν)-neck regions. In particular, we have μ(Na )  C(n)δ rk a and the C0,a are k-rectifiable by Theorem 11.0.11. (2) Brb (xb ) satisfies the measure constraint μ(B2rb ) < C(n)ν rk b .  k  k (3) We have the content estimates ra + rb < C(n, δ, , ν, Γ ) and packing estimate Pk (S− ∪ Sk ) < C(n, δ, , ν, Γ ), (4) We have the Hausdorff measure estimate Hk (S− ) = 0. We begin by discussing a variety of notation which will be convenient throughout the proof: Definition 14.0.1. We define the k-dimensional distortion Dk (x, r) of a measure μ by r ds Dk (x, r) = βk (x, s)2 . s 0 The following short exercises give some good intuition for the basic properties and behavior of Dk : Exercise 14.0.2. Show the following: (1) Dk (x, r) is monotone in r and Dk (x, r) = Dk (x, s) for some s < r iff suppμ ∩ Br (x) ⊆ Lk for some k-dimensional affine subspace. (2) C(n)−1 βk (x, r)2  Dk (x, 2r) − Dk (x, r)  C(n)βk (x, 2r)2 .   (3) C(n)−1 ri r βk (x, ri )2  Dk (x, r)  C(n) ri 2r βk (x, ri )2 if we take ri = 2−i . There are two primary pieces of information to keep track of during the proof. The first is the distortion drop from scale to scale, the second is a lower mass bound on balls. We formalize this with the following noncollapsing set:23 Definition 14.0.3 (Noncollapsing Set). Let , ν > 0 be fixed, then we define the set of noncollapsed points: V(x, r) ≡ {y ∈ Br (x) : μ(Bs (y)) > ν sk for r  s  r} . Remark 14.0.4. Recall the definition of noncollapsing as in Definition 11.0.2. The following tells us that V(x, r) must always live close to a best approximating subspace: Exercise 14.0.5. If βk (x, 2r) < δ2 and L = Lx,2r is a best affine subspace obtaining βk (x, 2r), then for each y ∈ V(x, r) we have d(y, L) < ( + C(n, )ν−1/2 δ2 )r. Hint: Apply Lemma 13.1.7 to the center of mass Y ∈ Br (y) and use the triangle inequality. 14.1. Proof Outline and Induction Step The Proof of Theorem 11.0.15 will be done inductively on the size of the distortion. 23 Recall

ωn is the volume of the unit ball in Rn .

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Lecture Notes on Rectifiable Reifenberg for Measures

14.2. Notation and Ball Types In our notation for the proof during this section the subscript we use to denote a ball will always designate a special structure of that ball. All ball types will fall into the following categories: (a) A ball Bra (xa ) will always be associated with a (k, δ, , ν)-neck region Na = B2ra (xa ) \ Bra,x (Ca ). (b) A ball Brb (xb ) satisfies μ(B2rb (xb )) < νrk b. (c) A ball Brc (xc ) is such that βk (xc , 4rc ) < δ2 and V(xc , rc ) is a (k, 2)linearly independent set. (d) A ball Brd (xd ) is such that V(xd , rd ) is not a (k, 2)-linearly independent set. (e) A ball Bre (xe ) is such that βk (xe , 4re ) > δ2 . (s) A ball Brs (xs ) is such that we have Dk (y, 2rs ) < D − δ6 whenever we have y ∈ Brs (xs )24 . (f) A ball Brf (xf ) is one for which we know nothing about. Before continuing let us discuss a little the role of each of these ball types. The simplest two types are the (a) and (b) balls, as of course these are what we are wanting to construct in the theorem and there will be nothing left to do with them. Part of the proof will involve an induction on Dk (x, r). Therefore the (s)-balls will represent balls for which the distortion has strictly dropped, and therefore we will apply our inductive hypothesis to handle them. Thus in practice we are also done on s-balls as well. The (f)-balls will also require starting over on, however in practice when we label a ball an (f)-ball we will make sure it is only on a set of very small content, therefore starting over will be okay as the errors will become a geometric series, see Section 14.6. The next easiest ball types to deal with are the (d) and (e) balls. For a (d)-ball we will be able to cover all of Brd (xd ) by (b)-balls away from a set of very small context of (f)-balls. For an (e)-ball we will be able to entirely cover Bre (xe ) by (s)-balls for which the distortion has strictly dropped, and thus we will be able to apply our inductive hypotheses to these new balls. The most complicated ball type to deal with in the construction is therefore a (c)-ball. The goal will be to build a neck region so that Brc (xc ) ⊆ N ∪ Brx (C). In order to proceed with the next step, this will have to be done in a maximal fashion so that each ball Brx (x) with x ∈ C is either a (b),(d), (e) or (s) ball. Then using the Neck Structure Theorem 11.0.11 in combination with the (d) and (e) coverings just discussed we can estimate the content of those balls we need to start over as being small. 14.3. Collapsing and d-Ball Covering set from Definition 14.0.3. 24 In

Recall the definition of the noncollapsing

¯ = sup Dk (x, R) will be the distortion of a potentially much larger ball. practice D BR

Aaron Naber

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Proposition 14.3.1 (d-Ball Covering). Let Brd (xd ) be such that V(xd , rd ) is not a (k, 2)-linearly independent set, then we can cover % % Brb (xb ) ∪ Brf (xf ) , Brd (xd ) ⊆ b

such that μ(Brb (xb )) <  k and f rk f < C(n) rd .

νrk b

f

for each (b)-ball with the estimates



k b rb

< C(n, )rk d

Proof. Since V(xd , rd ) is not a (k, 2)-linearly independent set we can find a subspace Lk−1 such that V(xd , rd ) ⊆ B2rd (L). In particular, let {xf } ∈ Brd ∩ L be rd -dense with rf ≡ 4rd . We see the (f)-balls satisfy the required property. Now for each x ∈ V(xd , rd ) let rx  12 rd be such that μ(B2rx (x)) < νrk b . Let {Brb (xb )} be any maximal subset such that B10−1 rb (xb ) are disjoint. Then a simple volume  estimate25 shows that {Brb (xb )} satisfies the conditions of the proposition. 14.4. Symmetry and e-Ball Covering Proposition 14.4.1 (e-Ball Covering). Let Bre (xe ) be such that βk (xe , 4re ) > δ2 . Then we can cover % Brs (xs ) , Bre (xe ) ⊆ 

b

rk s

such that y ∈ Brs (xs ).



C(n)rk e

and Dk (y, rs ) < Dk (y, 10rs ) − C(n)δ4 whenever we have

Proof. Let rs ≡ re with {xs } ∈ Bre (xe ) a maximal subset such that B10−1 rs (xs )  k rs  C(n)rk are disjoint. In particular, {Brs (xs )} is a covering of Bre (xe ) and e. Finally, let y ∈ B2re (xe ) and consider  10re  10re dr 2 dr βk (y, r) βk (y, r)2 Dk (y, 10rs ) − Dk (y, rs ) =  r r re 6re  C(n)βk (xe , 4re )2 > C(n)δ4 , where the last line uses (1) of Exercise 9.0.5.



14.5. Neck Regions and c-Ball Covering Proposition 14.5.1 (c-Ball Covering). Let Brc (xc ) be such that βk (xc , 4rc )  δ2 and V(xc , rc ) is a (k, 2)-linearly independent set. Then for δ < δ(n, , ν) there exists a (k, δ, , ν)-neck region % N = B2rc (xc ) \ Brx (C) , x

such that for each x ∈ C with rx > 0 we have that Brx (x) satisfies one of the following: (d) Brx (x) = Brd (xd ) is such that V(x, rx ) is not a (k, 2)-linearly independent set. (e) Brx (x) = Bre (xe ) is such that βk (x, 4rx ) > δ2 . 25 See

Exercise 2.3.2 and Lemma 6.0.1.

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Lecture Notes on Rectifiable Reifenberg for Measures

(s) Brx (x) = Brs (xs ) is such that for y ∈ Brx (x) we have Dk (y, 2rx )
0 then Brx (x) satisfies one of the conditions (d), (e), or (s). However, by construction if rx > 0 then there exists α for which x ∈ Cα with (x) remained in Cα+1 only if it satisfied rx = rα x . In the inductive step the ball Brα x  (d), (e), or (s). This finishes the proof of Proposition 14.5.1. 14.6. Inductive Proof of the Neck Decomposition Theorem We now finish the proof of the Neck Decomposition Theorem. The idea will be simply to continuously apply the covering propositions of the previous subsections. We begin by applying each of the Propositions once in order to get a first covering: Proposition 14.6.1 (Induction Step 1). Let μ be a Borel measure and assume for each 4 x ∈ B2 that 0 βk (x, r)2 dr r  Γ . Then for each ν, , δ > 0 with  < (n, ν) and δ < δ(n, ν, ), there is a covering % % % B1 ⊆ N ∪ C0 ∪ Brb (xb ) ∪ Brs (xs ) ∪ Brf (xf ) b

s

f

such that (1) N = B2 \ Brx (C) is a (k, δ, , ν)-neck region. In particular, μ(N)  C(n)δ and C0 is k-rectifiable by Theorem 11.0.11. (2) Brb (xb ) satisfy the measure constraints μ(B2rb ) < ν rk b .

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Lecture Notes on Rectifiable Reifenberg for Measures

(3) For each y ∈ Brs (xs ) we have Dk (y, 2rs ) < Γ − δ6 .   k rs < C(n, ), (4) We have the content estimates Hk (C0 ) + rk b+  k (5) We have the content estimate rf  C(n). Remark 14.6.2. The Neck Decomposition will follow by repeated applications of this Proposition. The balls Brs (xs ) have a drop in the distortion, and thus we can handle them later by an induction argument. The balls Brf (xf ) we know nothing about, however this is a set with small k-content. Therefore in Inductive Step 2 we will start over on them and all errors become a geometric series which converge. Proof. We begin with the ball B1 and observe that it is either a (c), (d), (e) or (s) ball, as in Subsection 14.2. If B1 is a (d), (e) or (s) ball then the Proposition is immediately proved simply by applying either Proposition 14.3.1 or Proposition 14.4.1. Thus we will assume B1 is a (c)-ball. Now applying Proposition 14.5.1, we can build a Neck Region N = B2 \ Brx (C) such that each ball Brx (x) with rx > 0 is either a (d), (e) or (s)-ball. This gives us the covering % % % B1 ⊆ N ∪ C0 ∪ Brd (xd ) ∪ Bre (xe ) ∪ Brs (xs ) . e

d

s

By applying the Neck Structure Theorem 11.0.11 we get the estimates    k rk + r + rk Hk (C0 ) + e s  C(n) . d e

d

s

Now we wish to remove the (d) and (e) balls from this covering and control what is left. If we apply Proposition 14.4.1 we can choose coverings of each e#  k ball Bre (xe ) ⊆ s Bres (xes ) such that s rk es  C(n)re . In particular the new collection of s balls satisfies the estimate    rk rk rk es + s  C(n) + C(n) e  C(n) . e

Combining these (s)-balls together together gives the covering % % B1 ⊆ N ∪ C0 ∪ Brd (xd ) ∪ Brs (xs ) , s

d

with the estimates Hk (C0 ) +



rk d+



rk s  C(n) .

s

d

We can now apply Proposition 14.3.1 to construct coverings of each d-ball # #  k Brd (xd ) ⊆ b Brdb (xdb ) ∪ f Brdf (xdf ) such that we have b rk eb  C(n, )rd  k k and s rdf  C(n) rd . Combining all of these together we get the covering % % % B1 ⊆ N ∪ C0 ∪ Brb (xb ) ∪ Brs (xs ) ∪ Brf (xf ) , with



k b rb

+



b k s rs

 C(n, ) and



k f rf

s

 C(n)



f

rk d

 C(n) as claimed.



Our next step in our inductive proof to get rid of the (f)-balls in the covering. As the (f)-balls have only a small content, the trick is to count the errors which

Aaron Naber

343

appear and see that they form a geometric series. As not everything goes away in the limit, we may be left with a Hk null set: Proposition 14.6.3 (Induction Step 2.). Let μ be a Borel measure and assume for each 4 x ∈ B2 that 0 βk (x, r)2 dr r  Γ . Then for each ν, , δ > 0 with  < (n, ν) and δ < δ(n, ν, ), there is a covering % % %  %  Na ∩ Bra ∪ (C0,a ∩ Bra ∪ S− ∪ Brb (xb ) ∪ Brs (xs ) B1 ⊆ a

a

s

b

such that (1) Na = B2ra \ Brx (Ca ) are (k, δ, , ν)-neck regions so μ(Na )  C(n)δrk a and the C0,a are k-rectifiable by Theorem 11.0.11. (2) Brb (xb ) satisfy the measure constraints μ(B2rb ) < ν rk b . 6 (3) For each y ∈ Brs (xs ) we have Dk (y, 2rs ) < Γ − δ . (4) We have the measure estimate Hk (S− ) = 0.   k  k rb + rs < C(n, ), (5) We have the content estimates Hk (C0 ) + rk a+ Proof. To get from Proposition 14.6.1 to Proposition 14.6.3 we will simply continually recover the f-balls. Indeed, let us begin by applying Proposition 14.6.1 to B1 see that it is covered by % % % N ∪ C0 ∪ Br0 (xb ) ∪ Br0s (x0s ) ∪ Br0 (x0f ) , b

b

s

f

f

   where Hk (C0 ) + (r0b )k + (r0s )k < C(n, ) and (r0f )k  C(n). Let us now apply Proposition 14.6.1 to each f-ball Br0 (x0f ) in order to get the new covering f % % %  %  % 1 1 Na ∩ Bra (xa ) ∪ (C0,a ∩ Br1a (xa ) ∪ Br1 (x1b ) ∪ Br1s (x1s ) ∪ Br1 (x1f ) , a

a

b

b

s

f

f

with     (r1a )k + (r1b )k + (r1s )k  C(n, ) + C(n, ) (r0f )k  C(n, )(1 + C(n)) a

s

b

and



(r1f )k

f

2  C(n) .

f

Now we can again apply Proposition 14.6.1 to each f-ball Br1 (x1f ). Indeed, if we f continue this N-times then we get a covering %  %  Na ∩ BrN (xN (C0,a ∩ BrN (xN a) ∪ a) a a a

∪

% b

BrN (xN b )∪ b

%

a

BrN (xN s )∪ s

s

% f

BrN (xN f ), f

with  a

k (rN a) +

N     j k N k (rN ) + (r )  C(n, ) C(n) , s b b

s

 f

j=0

N k (rN . f )  C(n)

344

References

Now for each N < M let us observe by construction that (xN (xM {BrN a )} ⊆ {BrM a )} , a a M {BrN (xN b )} ⊆ {BrM (xb )} , b

and

b

{BrN (xN (xM s )} ⊆ {BrM s )} , s s % % BrM (xM B2rN (xN f )⊆ f ). f

f

In particular, we can take limits of the (a), (b) and (s) covers and form a Hausdorff limit S− ≡ lim{BrN (xN f )} to get the covering f % %  %  % Na ∩ Bra (xa ) ∪ (C0,a ∩ BrN ∪ B1 ⊆ B (x ) ∪ Brs (xs ) ∪ S− , r b (x ) b a a a

a

s

b

with the estimates ∞      j k k C(n)  C(n, ) , rk + r + r  C(n, ) a s b a

s

b

j=0

Hk (S− ) = 0 . The last estimate follows because for each N we have S− ⊆  N rf < (C(n))N → 0 if   (n).

#

(xN f B2rN f ) f

with 

Let us now finish the proof of the Neck Decomposition: Proof of the Neck Decomposition Theorem 11.0.15. Our goal then is to remove the sballs from Proposition 14.6.3 by iteratively applying the Proposition 14.6.3 some finite number of times. To begin, if Γ < δ6 then Theorem 11.0.15 follows immediately from Proposition 14.6.3, as in this case there cannot be any s-balls. Now assume we have proved Theorem 11.0.15 for Γ  > 0, let us now prove it holds for Γ = Γ  + δ6 . So apply Proposition 14.6.3 for Γ = Γ  + δ6 in order to get the covering % %  %  % Na ∩ Bra ∪ (C0,a ∩ Bra ∪ Brb (xb ) ∪ S− ∪ Brs (xs ) , B1 ⊆ a

a

s

b

Γ .

In particular, by our where for each Brs (xs ) we now have that Dk (y, 2rs ) < inductive hypothesis we can now apply Theorem 11.0.15 to these balls in order to then conclude Theorem 11.0.15 for Γ = Γ  + δ6 , as desired. As the amount we increased was some definite δ6 independent of Γ  , we can for any Γ > 0 simply apply this procedure δ−6 Γ times in order to then conclude Theorem 11.0.15 holds for all Γ . 

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The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today’s harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.

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