Unique Continuation on Convex Domains

In this paper, we obtain estimates on the quantitative strata of the critical set of non-trivial harmonic functions $u$ which vanish continuously on $V \subset \partial \Omega$, a relatively open subset of the boundary of a convex domain $\Omega \subset \mathbb{R}^n$. In particular, these estimates improve dimensional estimates on $\{|\nabla u| =0\}$ both in $V \subset \partial \Omega$ and as it \textit{approaches} $V \cap \overline{\Omega}.$ These estimates are not obtainable by naively combining interior and boundary estimates and represent a significant improvement upon existing results for boundary analytic continuation in the convex case.


Introduction
Unique continuation is a fundamental property for functions which solve the Laplace and related linear equations.A closely related problem is that of boundary unique continuation: given a domain Ω ⊂ R n and a function u which is harmonic in Ω and vanishes continuously on V ⊂ ∂Ω, how large can the set {Q ∈ V : |∇u| = 0} be if u ≡ 0? Boundary unique continuation is closely tied to the Cauchy problem and questions of well-posedness and stability of solutions to boundary value problems (see, for instance, [Tat03] and [ARRV09]).In this paper, we address two questions.First, we address the question of boundary unique continuation for harmonic functions on convex domains.Second, we also address the question of how the critical set {|∇u| = 0} ∩ Ω approaches V ⊂ ∂Ω.We follow the approach of Garofalo and Lin [GL87] insofar as we make essential use of the Almgren frequency function.And, because we want to obtain results on the full critical set {|∇u| = 0}, we use packing estimates inspired by Cheeger, Naber, and Valtorta [CNV15].These tools allow us to obtain results the strata of the critical set {p ∈ Ω : |∇u(p)| = 0} as it approaches V ⊂ ∂Ω.
1.1.Background on Boundary Unique Continuation for Harmonic Functions.For dimensions n ≥ 3, Bourgain and Wolff [BW90] have constructed an example of a function, u : R n + → R, which is harmonic in R n + , C 1 up to the boundary R n−1 ⊂ R n , and for which both u and ∇u vanish on a set of positive surface measure.This result has been generalized by Wang [Wan95] to C 1,α domains, Ω ⊂ R n , for n ≥ 3.However, the sets of positive measure for which these functions vanish are not open.
In general, the following question posed by Lin in [Lin91] is still open.
Question 1.1.Let n ≥ 2 and Ω ⊂ R n be an open, connected Lipschitz domain.If u is a harmonic function which vanishes continuously on a relatively open set V ⊂ ∂Ω, does imply that u is the zero function?
If u is non-negative, the techniques of PDEs on non-tangentially accessible (NTA) domains give a comparison principle [Dah77] which allows us to say that the norm of the normal derivative is point-wise comparable to the density of the harmonic measure with respect to the surface measure dσ.Additionally, for Lipschitz domains it is well-known that the harmonic measure is mutually absolutely continuous with respect to dσ.These two facts then imply that if the normal derivative vanishes on a set of positive (surface) measure, then u must be identically 0.
The challenge is for harmonic functions u which change sign.For such functions, the aforementioned techniques fail completely because we cannot apply the Harnack principle.Authors have therefore approached this problem by asking for additional regularity.In [Lin91], Lin proves that for C 1,1 domains, Ω ⊂ R n , for n ≥ 2, if u is a non-constant harmonic function which vanishes on an open set V ∩ ∂Ω, then dim H ({x ∈ V ∩ ∂Ω : |∇u| = 0}) ≤ n − 2. Similar results were later shown by Adolfsson and Escauriaza for domains with locally C 1,α boundary [AE97].Relatedly, Kukavica and Nystöm showed that H n−1 ({x ∈ V : |∇u| = 0}) > 0 implies that u ≡ 0 if ∂Ω is C 1 Dini [KN98].Recently, this result has been greatly improved.Kenig and Zhou [KZ21] employed powerful technicues from [NV17] and [dLMSV16], have shown that for C 1 Dini domains, the (n − 2)-generalized singular set {u = 0 = |∇u|} has finite (n − 2)-dimensional upper Minkowski content.
For merely convex domains Ω ⊂ R n , Adolfsson, Escauriaza, and Kenig showed that if u is a harmonic function in Ω which vanishes continuously on a relatively open set V ⊂ ∂Ω, then if {x ∈ V ∩ ∂Ω : |∇u| = 0} has positive surface measure, u must be a constant function [AEK95].The method of attack pursued in [AEK95] (and [Lin91], [AE97], [KN98]) was centered on showing that the harmonic function is "doubling" on the boundary in the following sense.If Ω ⊂ R n , then there exists an absolute constant M < ∞ such that for all This doubling property allows the authors to show that the normal derivative is an A 2 -Muckenhoupt weight with respect to surface measure, a kind of quantified version of mutual absolute continuity.It is well-known that if u vanishes in a surface ball ∆ r (Q) and the normal derivative of u is a 2-weight with respect to surface measure, then either {Q ′ ∈ ∆ r (Q) : |∇u| = 0} has measure zero or {Q ′ ∈ ∆ r (Q) : |∇u| > 0} has measure zero.The improvement from measure to dimension bounds in [AE97] and [Lin91] comes from applying an additional Federer dimension-reduction type argument.
Recently, Tolsa has answered Question 1.1 in the affirmative for C 1 domains and Lipschitz domains with small Lipschitz constant [Tol20].
Theorem 1.2.Let Ω be a convex domain and u ∈ C 0 (Ω) be a non-constant function such that ∆u = 0 in Ω.Let V ⊂ ∂Ω be a relatively open set.If u = 0 on V , then for any compact subset K ⊂ V , there exists a radius 0 < r(K) such that The content of Theorem 1.2 is two-fold.First, consider the results restricted to the boundary {|∇u| = 0} ∩ V ⊂ ∂Ω.Alberti [Alb94] proved (among other things) that the singular set of a convex function is a C 2 (n − 2)-rectifiable set, which implies that the geometric singular set of a convex body satisfies dim H (sing(∂Ω)) ≤ n − 2. Thus, Theorem 1.2 combined with [Alb94] implies which gives a strong improvement on the results of [AEK95], which proved that in this situation H n−1 (V ∩ {|∇u| = 0}) = 0.
Second, Theorem 1.2 provides new insight into how the critical set interacts with ∂Ω.Returning to [Alb94], Alberti also proved that the singular set of a convex function may be prescribed to be any C 2 (n − 2)-rectifiable set.Thus, it can happen that dim M (sing(∂Ω)) = n − 1.On the other hand, from the interior perspective, [NV14] proved finite (n − 2)dimensional upper Minkowski content bounds on {|∇u| = 0} in the interior.But, naive application of these estimates degenerate as one approaches the boundary because upper Minkowski dimension is not stable under countable unions.Considering {|∇u| = 0} ∩ Ω as it approaches ∂Ω, it was unknown whether or not {|∇u| = 0} ∩ Ω could oscillate wildly and have positive (n − 1)-upper Minkowski dimension like sing(∂Ω), would remain (n − 2)-upper Minkowski dimensional like the interior, or if something in between these two held.Theorem 1.2 proves that the set {|∇v| = 0} ∩ Ω cannot oscillate too wildly as it approaches ∂Ω and {|∇u| = 0} ∩ Ω ∩ K inherits its upper Minkowski dimension bounds from the interior rather than the boundary.
The author would like to thank Tatiana Toro, whose direction, advice, patience, and support can only be described as sine qua non.Additional thanks are due to Zihui Zhou for gently pointing out the errors in a previous version of this project and for a very thorough and patient reviewer, whose comments have greatly improved the presentation.

Definitions and Main Results
Theorem 1.2 is a corollary to the Technical Theorem (Theorem 2.10) and a containment result (Lemma 2.11).In order to state these results, we need the following definitions.
We define the following class of domains.
Definition 2.1.(A normalized class of convex domains) Let D(n) be the collection of connected, open domains Ω ⊂ R n which satisfy the following conditions: (1) 0 ∈ ∂Ω. ( One of the key tools of this paper will be an Almgren frequency function, introduced by Almgren in [Alm79]. )and p ∈ Ω.We define the following quantities: We now define the class of functions in which we will work in this paper.
Definition 2.4.(A class of functions) Let A(n, Λ) be the set of functions, u : R n → R, which have the following properties: (1) u : R n → R is harmonic in a convex domain, Ω ∈ D(n).
We shall use rescalings which are adapted to the quantitative stratification methods introduced by Cheeger and Naber in [CN13] for studying the regularity of stationary harmonic maps and minimal currents.Definition 2.5.(Rescalings) Let u ∈ A(n, Λ), and let Ω be its associated domiain.We define the rescaled function, T x,r u of u at a point x ∈ B 1 (0) at scale 0 < r < 1 by ´∂B 1 (0)∩Ω (u(x + ry) − u(x)) 2 dσ(y) 1/2 .
In the case that the denominator is zero, we define T x,r u = ∞.We shall break with established convention and denote the rescalings of Ω in analogy with the rescaling of functions.Let T p,r Ω := Ω−p r and T p,r ∂Ω := ∂Ω−p r .
The geometry we wish to capture with the rescalings T x,r f are encoded in their translational symmetries.
Definition 2.6.(The class of blow-up profiles) Let u ∈ C(R n ).We say u is 0-symmetric if u satisfies one of the following conditions.
(1) u is a homogeneous harmonic polynomial.
(2) u(x) = φ(x − p) + c for some function φ which is homogeneous and harmonic in a convex cone, Ω ′ ∈ D(n), some point p ∈ R n ∩ Ω ′ , and some c ∈ R. We will say that u is k-symmetric if u is 0-symmetric and there exists a k-dimensional subspace V such that u(x + y) = u(x) for all x ∈ R n and all y ∈ V .
We now define the quantitative version of symmetry which describes how close to being k-symmetric a function is in a ball, B r (x) ⊂ R n .Definition 2.7.(Quantitative symmetry) For any u ∈ A(n, Λ) with associated domain Ω, u will be called (k, ǫ, r, p)-symmetric if there exists a k-symmetric function P such that We shall say that u is (k, δ 0 , r, p)-symmetric with respect to a k-dimensional subspace V if there is a k-symmetric function P which verifies that u is (k, δ 0 , r, p)-symmetric such that P (x + y) = P (x) for all y ∈ V .Definition 2.8.(Quantitative Generalized Critical Strata) Let u ∈ A(n, Λ) with Ω ∈ D(n) its associated domain.For 0 < ǫ, 0 < r ≤ 1, and integer 0 ≤ k ≤ n − 1 we denote the (k, ǫ, r)-generalized critical strata of u by C k ǫ,r (u), and we define it by C k ǫ,r (u) := {x ∈ Ω : u is not (k + 1, ǫ, s, x)-symmetric for all r ≤ s ≤ 1}.We shall also use the notation C k ǫ (u) for C k ǫ,0 (u).The quantitative strata of the generalized critical set behave very well under L 2 -convergence of functions.That is, if u i ∈ A(n, Λ) and In turn, we define the strata of the generalized critical set as follows C k (u) := ∪ η ∩ r C k η,r (u).We shall use the convention that for any A ⊂ R n , B r (A) = {x ∈ R n : d(A, x) < r}.Recall that we can define upper Minkowski s-content by and upper Minkowski dimension as dim M (A) = inf{s : M * s (A) = 0} = sup{s : M * s (A) > 0}.Now, we can state the main technical results.
Theorem 2.10.(Technical Theorem) Let u ∈ A(n, Λ), then for any r 0 > 0 and all 0 < r 0 < r In particular, letting r 0 → 0 Lemma 2.11.(Containment) There exists an 0 < ǫ = ǫ(n, Λ) such that Proof of Theorem 1.2 assuming Lemma 2.11 For each point x ∈ K ⊂ V, there is a radius 0 < r such that B 4r (x) ∩ ∂Ω ⊂ V. Since K is compact, we may find a finite subcover {B r i (x i )} i .Thus, Lemma 2.11 implies that in each Since upper Minkowski dimension is stable under finite unions, the first claim of Theorem 1.2 holds.The second follows from an identical argument using Lemma 2.11.

2.1.
Outline of the Paper.The structure of this paper is roughly in four parts.Section 3 and Section 4 use the geometric techniques of [GL87], [HL] (and many, many others) to establish that the Almgren frequency is monotonically non-decreasing and bounded on {u = 0}.Section 5 uses these results to establish compactness of {T p,r u} for u(p) = 0. Section 6 extends these results to p ∈ Ω such that u(p) = 0.
The second part of this paper is devoted to obtaining geometric control upon C k ǫ,r (u).The general idea is to employ the usual "frequency pinching" (Lemma 7.2) and cone-splitting results (Lemma 7.5).However, because we are considering N Ω (p, r, u) at points p such that u(p) = 0, the Almgren frequency is not monotonic.This is overcome by proving that if dist(p, {u = 0}) << r, then The third part of this paper is devoted to obtaining packing estimates to prove Theorem 2.10.To do so, we use the tools of [CNV15], which do not require restricting to a level set or the delicate machinery which powers the finer estimates of [dLMSV16].The fact that we do not control the tilt of approximating L k at different scales accounts for the (k+ǫ)-dimensional results.
The fourth part of this paper is devoted to proving the containment results which prove Lemma 2.11.
Throughout this paper, the constant C will by ubiquitous and represent different constants even within the same string of inequalities.A constant written C(n, Λ) will only depend upon n and Λ, but each instantiation may represent a distinct constant.

The Almgren frequency function
In this section, we develop crucial properties of the Almgren frequency function.The main results of this Section are the monotonicity of the Almgren frequency on p ∈ {u = 0} ∩ Ω.
We now note some of the elementary properties of H Ω (p, r, u), D Ω (p, r, u), N Ω (p, r, u), and their derivatives.
where η is the unit outer normal of the relevant domains.
Proof.In the interior setting, for B r (p) ⊂ Ω, these identities follow from straightforward computation.(5) follows from the change of variables, y → rx + p, and the divergence theorem.(6) relies upon the Rellich-Necas Identity, the divergence theorem, and the fact that u vanishes on the boundary.The last two equations follow immediately from 5. Without exception, the standard interior computations go through identically for radii for which B r (p) ∩ ∂Ω = ∅, where we also use the identity where η is the outer unit normal vector to Ω. See [HL] Theorem 2.2.3, Corollary 2.2.5 and [AEK95] (Proof of the Doubling Property) for details.
The following lemma records a useful identity which follows from the previous lemma by straightforward computation.
Lemma 3.2.For u ∈ A(n, Λ) and p ∈ Ω ∩ B 1 (0) and all 0 < r < 1, d dr N Ω (p, r, u) may be decomposed into four terms where and η is the unit outer normal.
Proof.Recall that by the Cauchy-Schwarz inequality, we have that for Choosing w = ∇u • (y − p) and v = u − u(p), we have where The Divergence theorem then implies that for u(p) = 0, λ(p, r, u) = N Ω (p, r, u).

The Zero set: Uniform Frequency Bounds
The main result in this section is Lemma 4.3, which gives a uniform bound on the Almgren frequency function for all p ∈ Ω ∩ B 1 4 (0) for which u(p) = 0 and all 0 < r ≤ 1 2 .We begin with a few basic results.
Proof.Recalling Equations ( 5) and ( 7) We bound N Ω (p, r, u) by N Ω (p, S, u) using Lemma 3.4.Plugging in these bounds, we have that for r ∈ [s, S] Evaluating and exponentiating gives the desired result.
We are now ready for the main result of this section.
Lemma 4.3.(Uniform bound on frequency) Let u ∈ A(n, Λ), as above.There is a constant, Proof.Recall that 0 ∈ ∂Ω and that the Almgren frequency function is invariant under rescalings.Therefore, we normalize our function u by the rescaling v = T 0,1 u.
Therefore, applying Lemma 4.1 to Q = 0, letting r = cR, and integrating both sides with respect to R from 0 to S, we have that for any c ∈ (0, 1) Thus, letting S = 1 and c = 1 16 and dividing by ω n we obtain (16) |v| 2 dV.

The Zero Set: Compactness
The uniform bounds on the Almgren frequency function allow us to prove compactness results on the collection of rescaling {T p,r u}.The main results of this Section are weak compactness (Lemma 5.4), the geometric non-degeneracy of the domains Ω (Corollary 5.2).
We now state a sequence of preliminary corollaries to Lemma 4.3.We shall denote the C 0,γ (B 1 (0))-norm by We defer the proof of this statement to the Appendix A. The techniques are standard.
where we choose C(n, Λ) to be the same constant in Lemma 5.1 for which we have sup Weiner regular domain and the boundary data is piecewise continuous, so a unique solution φ must exist.By the maximum principle By [JK82] Theorem 5.1.,there is a constant C such that we have that for all x ∈ B 1 4 where C depends only upon the geometry of B 1 4 (Q) ∩ H Q .Since this geometry is always a half-ball, this constant is uniform.Therefore, we have that for all x ∈ B 1 4 Applying this argument to ±T Q 0 ,r u, we obtain the desired estimate.
Lemma 5.4.(Preliminary Compactness) Let u i ∈ A(n, Λ), Q i ∈ ∂Ω i ∩ B 1/4 (0), and 0 < r i ≤ 1 4 .Then there exists a subsequence (also indexed by i) such that (1) Hausdorff metric on compact subsets and Ω ∞ is a non-degenerate convex domain with 0 ∈ ∂Ω ∞ which satisfies the same non-degeneracy as Corollary 5.2. (3 Proof.By definition, T Q i ,r i u i (0) = 0. Therefore, Lemma 5.1 implies the first convergence result by Arzela-Ascoli.Note that since By taking a further subsequence, we may assume that lim i T Q i ,r i Ω i = Ω ′ exists in a set theoretic sense.The uniform convergence in (1) implies that for all 0 To see the converse containment, we observe that for any Now, let 0 < δ and suppose that for all sufficiently large i there exists . By passing to a subsequence, we may assume But this contradicts u ∞ being non-trivial.Thus, there must be a subsequence such that in the Hausdorff metric on compact subsets.Convexity and the conclusion of Corollary 5.2 are preserved under this mode of convergence, and so (2) is proved.Now that we know that Ω ′ = Ω ∞ , the previous argument proves (3), as well.
On the other hand, Lemma 6.1 implies that Therefore, we have N Ω (p, r, u) (0) and all radii, 0 < r ≤ 1 8 , T Q,r u ∈ Lip(B 1 (0)) with uniform Lipschitz constant Lip(T Q,r u) ≤ C(n, Λ).Proof.Since T Q,r u is continuous and constant outside of T Q,r Ω, we reduce to bounding ∇T Q,r u at interior points y ∈ T Q,r Ω ∩ B 1 (0).Note that by our definition of the rescalings (Definition 2.5) and Lemma 4.1 udV .Recall that |∇u| is subharmonic, and therefore by Lemma 6.2 Thus, we have that, We now prove that the Almgren frequency is a function of uniformly bounded variation.
Now, if we let Q 0 ∈ ∂Ω be a point such that |p − Q 0 | = δ we may calculate by Lemma 6.1, Lemma 4.1, Lemma 5.3, and Lemma 6.3 Thus, by Lemma 6.2, we may bound This proves the lemma.
(3) follows analogously as in the proof of Lemma 5.4(2).That is, if Thus, after possibly passing to a subsequence so that the lim i Q i exists, the argument of Lemma 5.4(2) applies.(3) follows immediately.
By our choice of rescaling, T p j ,r j u, we have that N Ω (0, 1, T p j ,r j u j ) = ´B1 (0) |∇T p j ,r j u j | 2 dV.Therefore, Lemma 6.2 gives that ∇T p j ,r j u j are uniformly bounded in L 2 (B 1 (0); R n ).Therefore, Rellich compactness gives weak convergence.
The only thing remaining to show is that ∇T p j ,r j u j → ∇u ∞ .By (3), we may choose a subsequence such that, ∂Ω j have a convergent subsequence such that T p i ,r i ∂Ω i → ∂Ω ∞ locally in the Hausdorff metric to a non-degenerate convex domain.Since the boundary of a convex domain is locally the graph of a Lipschitz function dim M (∂Ω ∞ ∩B 1 (0)) = n−1.Thus, by continuity of measures and Lemma 6.3, for all ǫ > 0 we can find a τ (Λ, n, ǫ) independent of T p j ,r j u j , such that Therefore, using the notation ∂Ω j,τ = B τ (T p j ,r j ∂Ω j ) where the last equality follows from W 1,2 -convergence of harmonic functions in the region B 1 (0) \ B τ (∂Ω ∞ ).Since ǫ > 0 was arbitrary, we have that lim j→∞ D Ω i (1, 0, T p j ,r j u j ) ≤ D Ω∞ (1, 0, u ∞ ).The other inequality follows from the same trick or from lower semi-continuity.Thus, lim j→∞ D(1, 0, T Q j ,r j u j ) = D Ω∞ (1, 0, u ∞ ).This implies strong convergence.
Corollary 6.6.(Convergence of the Almgren frequency) For u j ∈ A(n, Λ), p j ∈ B 1 8 (0) ∩ Ω i , and r j ∈ (0, 1 8 ], there exists a subsequence and a limit function such that N Tp j ,r j Ω i (0, 1, T p j ,r j u j ) → N Ω∞ (0, 1, u ∞ ).(20) Proof.The continuous convergence of T p j ,2r j u j in B 1 (0) and the strong convergence ∇T p j ,2r j u j in B 1 (0) give the desired convergence of H T p j ,2r j Ω i (0, 1 2 , T p j ,2r j u j ) and D T p j ,2r j Ω j (0, 1 2 , T p j ,2r j u j ), respectively.Recall that by Definition 2.2 and Definition 2.5 Corollary 6.7.(Limit functions are harmonic in the limit domain) Let the sequence of functions T p j ,r j u converge to the function u ∞ in the senses of Lemma 6.5.Then, u ∞ is harmonic in Ω ∞ .

Geometric Control
The main results of this section are two "quantitative rigidity" results about homogeneous harmonic functions.Both are essentially consequences of the compactness obtained in Lemma 6.5.
Proof.The hypotheses imply that N Ω (p, s, u) is a constant for all r/10 ≤ s ≤ r.Furthermore, using the notation in Lemma 3.2, = 0 for all r/10 ≤ s ≤ r.Thus, by Lemma 3.3 we have that for all y ∈ ∂B s (p) Since N Ω (p, s, u) is a constant for all r/10 ≤ s ≤ r, this becomes a separable ODE in polar coordinates, and In particular, u is (0, 0, s, p)-symmetric for all 0 < s ≤ 1.

Standard quantitative rigidity results usually prove that if N
However, these results rely essentially upon the monotonicity of N Ω (p, r, u).If N Ω (p, r, u) is not monotonic, then N Ω (p, 1, u) = N Ω (p, r, u) does not imply that the Almgren frequency is constant.In fact, even if N Ω (p, r, u) is constant for 1/10 < r < 1, if u(p) = 0, it is not clear that u would be homogeneous.To overcome this technical issue we consider p which are merely very close to {u = 0}.
The case that B 1 (0) ⊂ T p i ,r i Ω i , repeat the argument to obtain the same contradiction.
Remark 7.3.By the scale invariance of the Almgren frequency, Lemma 7.2 implies that for all 0 < δ, if 0 < γ ≤ γ 0 (n, Λ, δ), then Next, we obtain a "cone-splitting" result.The prototypical example of a result like this is the following proposition.See [HL] Theorem 4.1.3for the proof of similar results.
Proposition 7.4.Let P : R n → R be a 0-symmetric function.Let k ≤ n − 2. If P is symmetric with respect to some k-dimensional subspace V and P is homogeneous with respect to some point x ∈ V , then P is (k + 1)-symmetric with respect to span{x, V }.
We now prove a similar result for our almost-symmetric functions u ∈ A(n, Λ).
Proof.Assume that there exists a δ, τ > 0 for which there exist a sequence of 0 < r i ≤ 1, function, u i ∈ A(n, Λ) and points {p i } for which u i is (k, i −1 , r i , p i )-symmetric with respect to some V i and (0, i −1 , r i , x i )-symmetric for some x i ∈ B r i (p i ) \ B τ (V i + p i ), but that all u i are not (k + 1, δ, r i , p i )-symmetric.
By considering T p i ,r i u i and applying Lemma 6.5 and Lemma 6.7 there exists a function u ∞ ∈ L 2 (B 1 (0)) such that a subsequence T p j ,r j u j → u ∞ in the senses of the lemma.Note that u ∞ is non-degenerate.Taking further subsequences, we may reduce to a sequence for which Note that u ∞ is (k, 0, 1, 0)-symmetric with respect to V , and harmonic.Therefore, u ∞ is (k + 1, 0, 1, 0)-symmetric.Since u i → u in L 2 (B 2 (0)), we have our contradiction.By taking the smallest 0 < δ 0 for 0 ≤ k ≤ n − 2, we eliminate the dependence upon k.

The Covering and its Properties
The lemmata in the previous section allow us to inductively define a covering with the right packing conditions.Quantitative rigidity allows us to prove a "Quantitative Differentiation" lemma that bounds the number of scales across which the frequency can change by more than some threshold γ > 0. Cone splitting on the other hand, will give us good geometric control of the singular set at scales for which v is close to a homogeneous harmonic polynomial.Together, these things will give us the necessary packing conditions.First, we describe the covering.
8.1.The General Construction.Let u ∈ A(n, Λ), and let ǫ, r > 0, k ≤ n − 2, and N ∈ N be given.We use the notation ρ i = 10 −i .In this section we describe a general procedure which will produce a cover of C k ǫ,r (u) ∩ B 1 10 (0) by balls of radius ρ N .
We begin by defining an auxiliary quantity.Let D(u, x, r) = inf{δ ′ > 0 : u is (0, δ ′ , r, x)-symmetric}.(22) Let 0 < δ 0 .We shall refer to δ 0 as the sorting threshold.For any i ∈ N we can assign to each x ∈ C k ǫ,r (u) ∩ B 1 8 (0) an i-tuple T i (x) according to the rule For any T i we shall use |T i | to denote the sum of the entries.Note that there is a partial ordering on the set of these i-tuples.That is, if k < i, we can say that T k < T i if (T k ) j = (T i ) j for all j ∈ {1, 2, ..., k}.Now, we partition our set according to these i-tuples.For any given i-tuple, T i ∈ {0, 1} i , we define It follows immediately from the definitions that E(T i ) ⊂ E(T k ) if and only if T k < T i .We now define our covering inductively.For i = 1, we let C k ǫ,r (T i ) = B 1 10 (0) for both 1-tuples T i ∈ {0, 1} 1 .Now, assume that i ∈ N, i < N, and C k ǫ,r (T i ) has been defined and consists of balls of radius ρ i .Within each ball B ρ i (y) ∈ C k ǫ,r (T i ) partition the set B ρ i (y) ∩ E(T i ) into the sets E(T i+1 ) for T i+1 such that T i < T i+1 .For either such T i+1 , take a minimal covering of B ρ i (y) ∩ E(T i+1 ) by balls of radius ρ i+1 centered at points in B ρ i (y) ∩ E(T i+1 ).The union of these balls is C k ǫ,r (T i+1 ).For some i-tuples, the set E(T i ) may be empty.In this case, we simply allow the corresponding collection of balls, C k ǫ,r (T i ), be empty.If i = N, we terminate the procedure.Note that for any sorting threshold 0 < δ 0 and N ∈ N this procedure defines a sequence of collections such that 8.2.Properties of the Construction.Now, we argue that there is a choice of sorting threshold 0 < δ 0 with the desired properties.
Lemma 8.1.Let u ∈ A(n, Λ).Let 0 < δ ′ be the sorting threshold in the construction above.For any i ∈ N there are at most N D(n,Λ,δ ′ ) nonempty sets E(T i ) such that E(T i ) is non-empty.
Proof.First, we prove (3).Because for all integers k and all 0 < ǫ the set C k ǫ (u) is closed, we reduce to proving that there is an 0 < ǫ(n, Λ) such that C n−2 (u)∩B 1/8 (0) ⊂ C n−2 ǫ (u).Suppose that this containment is false.Then, there would exist a sequence of functions u i ∈ A(n, Λ), points p i ∈ C n−2 (u i ) ∩ B 1 8 (0) and scales 0 < r i ≤ 1 such that u i is (n − 1, 2 −i , p i , r i )symmetric.We rescale to the functions T p i ,r i u i .By Lemma 6.5, there exists a subsequence (also indexed by i) and a (n − 1)-symmetric function u ∞ such that and so by Lemma 3.4 for sufficiently large i N Ω i (Q i , r i , u i ) ≥ 1 + δ.Letting i → ∞ we obtain that by Lemma 6.6 that N Ω∞ (0, 1, u ∞ ) > 1 + δ.This is a contradiction.Thus, there exists an 0 For prove (4), we note that if Q ∈ C n−2 (u) ∩ ∂Ω \ sing(∂Ω), then letting r → 0, we may extract a subsequence such that T Q,r j u → u ∞ in the sense of Lemma 5.4 and Lemma 6.5.By the monotonicity of the Almgren frequency, Lemma 6.6, and by considering T Q,cr j u for any 0 < c < 1 we see that N Ω∞ (0, r, u ∞ ) ≡ lim r→0 + N Ω (Q, r, u).Thus, Lemma 7.1 implies that u ∞ is a homogeneous function which is harmonic in Ω ∞ .Since Q ∈ sing(∂Ω), Ω ∞ is a half-space and we may extend u ∞ to an entire, homogeneous harmonic function by reflection.Since Q ∈ C n−2 (u) this polynomial must be a non-linear homogeneous harmonic polynomial and lim r→0 + N Ω (Q, r, u) ≥ 2. By Lemma 3.4, then N Ω (Q, r, u) ≥ 2 for all 0 < r ≤ 1 and Lemma 9.2 gives the claim.

Appendix A: Hölder Continuity
In this section, we provide a proof of Lemma 5.1.First, some standard results.
Remark 2.3.(Invariances of the Almgren frequency function) This normalized version of the Almgren frequency function is invariant in the following senses.Let a, b, c ∈ R with a, r = 0.If w(x) = au(bx + p) + c and T p,b Ω = 1 b (Ω − p) then N Ω (p, r, u) = N T p,b Ω (0, b −1 r, w).

0 (∂Ω) \ B γ j+1 0 (
3. Now, decompose Ω = ∪∞ j=0 A j (Ω) where A j (Ω) = Ω ∩ B γ j ∂Ω).We shall argue that there is aD = D(n, Λ, δ ′ ) such that |T N (p)| ≤ D for all p ∈ Ω ∩ B 10 (0).If the claimis true, then if N ≤ D there are at most 2 N ≤ N D N-tuples with |T N | ≤ D. And, if N ≥ D there are at most N D many N-tuples with |T N | ≤ D. Since N D ≤ N D we have the desired claim: C k ǫ,r (u) ∩ B 1/10 (0) is contained in the union of at most N D nonempty sets E(T N ) and covered by at most N D collections C k ǫ,r (T N ).