QUANTIZATION OF THE ENERGY FOR THE INHOMOGENEOUS ALLEN–CAHN MEAN CURVATURE

. We consider the varifold associated to the Allen–Cahn phase transition problem in R n +1 (or n + 1-dimensional Riemannian manifolds with bounded curvature) with integral L q 0 bounds on the Allen–Cahn mean curvature (ﬁrst variation of the Allen–Cahn energy) in this paper. It is shown here that there is an equidistribution of energy between the Dirichlet and Potential energy in the phase ﬁeld limit and that the associated varifold to the total energy converges to an integer rectiﬁable varifold with mean curvature in L q 0 , q 0 > n . The latter is a diﬀused version of Allard’s convergence theorem for integer rectiﬁable varifolds.


Introduction
Let Ω ⊂ (M n+1 , g) be an open subset in a Riemannian manifold with bounded curvature.Consider u ∈ W 2,p (Ω) satisfying the following equation where W (t) = (1−t 2 ) 2 2 is a double-well potential.The equation (1.1) can be viewed as a prescribed first variation problem to the Allen-Cahn energy For any compactly supported test vector field η ∈ C ∞ c (Ω, R n+1 ), we have a variation u s (x) = u (x + sη(x)) and the first variation formula at u 0 = u ε is given by where ν = ∇uε |∇uε| is a unit normal to the level sets at non-critical points of u.By [MM77], [Mod87], [Ste88] using the framework of [DG79], the sequence of functionals E ε Γ-converges to the n-dimensional area functional as ε → 0. This shows that minimizing solutions to (1.1) with f ε = 0 converge as ε → 0 to area minimizing hypersurfaces.For general critical points (f ε = 0) a deep theorem of Hutchinson-Tonegawa [HT00, Theorem 1] shows the diffuse varifold obtained by smearing out the level sets of u converges to limit which is a stationary varifold with a.e.integer density.The main result of this paper is to prove Hutchinson-Tonegawa's Theorem [HT00, Theorem 1] in the context of natural integrability conditions on the first variation of E ε .Under suitable controls on the first variation of the energy functional E ε (the diffuse mean curvature) we can show comparable behaviour for the limit.In the case where n = 2, 3 Röger-Schätzle [RS06] have shown under the assumption lim inf that the limit is an integer rectifiable varifold with L 2 generalised mean curvature.
The main focus of this paper is to generalise this result to higher dimensions.Before we state our main theorem, we give a choice of the diffused analogue of "mean curvature" in the Allen-Cahn setting, which will be used to state our bounded L q 0 Allen-Cahn mean curvature condition in the theorem.
Recall that for an embedded hypersurface Σ n ⊂ Ω ⊂ R n+1 restricted to a bounded domain Ω and a compactly supported variation Σ s with Σ 0 = Σ, we have the first variation area at s = 0 given by where H is the mean curvature scalar, H = −Hν is the mean curvature vector, ν is a unit normal vector field, η is the variation vector field, and dµ Σ is the hypersurface measure.By comparing the first variation formula (1.2) for Allen-Cahn energy and the first variation formula (1.3) for area , we can see that fε ε|∇u| roughly plays the role of the mean curvature scalar in the Allen-Cahn setting.In [All72], a result of Allard implies that if a sequence of integral varifolds has L q 0 integrable mean curvature scalar with q 0 > n, then after passing to a subsequence, there is a limit varifold which is also integer rectifiable.
This theorem shows we can prove a result analagous to Hutchinson-Tonegawa [HT00], Tonegawa [Ton03] and show as ε → 0, the diffuse varifold associated to the Allen-Cahn functional converges to an integer rectifiable varifold.This has some similarities with Allard's compactness theorem for rectifiable varifolds and for integral varifolds but here the sequence consists of diffuse varifolds and hence we require stronger conditions on the proposed mean curvature.As we shall see in a later paper, these conditions are exactly what is required to prove a version of Allard's regularity theorem for Allen-Cahn varifolds.As an application, we have the following Corollary Corollary 1.2.If u ε satisfies (1.1) and of one of the following conditions holds: (1) ≤ C 2 , for some t > n − 2 s − 2 s > max{s, n − 2}; (2) ≤ C, for some p > n + 1 2 , (c.f.[TT20]); (3) ) ≤ C 2 , if the ambient dimension n + 1 ≥ 3; then after passing to a subsequence as ε → 0, the associated varifolds V ε converge to an integral n-rectifiable varifold with generalized mean curvature in L q 0 for some q 0 > n. Proof.
(1) To see the first condition implies the conditions in Theorem 1.1, we choose q 0 = t(s−2) s + 2 (q 0 > n is satisfied due to the choice of t and s above).Then we have where we used Hölder's inequality in the second line with exponent s s−2 .
(2) In the paper [TT20], assuming condition (2) above, the authors proved the same integer rectifiability and L q 0 mean curvature bound for the limit varifold.We show this conditions implies the integral bounds in the hypothesis of Theorem 1.1 for some q 0 > n.To see this, we compute and applying [Zie89, 5.12.4](c.f.[TT20, Theorem 3.7]) and [TT20, Theorem 3.8], and Hölder's inequality, with ϕ = φ .
(3) If n + 1 = 2 then this is proven in [RS06].For n + 1 ≥ 3 it can be directly verified that the condition (3) implies the the conditions in Theorem 1.1.
Here we give an overview of our proof.In Section 2, we gather together some standard notation on varifolds and the first variation.In section 3, we prove the main estimates required for the proof of the integrality and rectifiability.Specifically we will need a monotonicity formula.For the homogeneous Allen-Cahn equation and Allen-Cahn flow, a strict monotonicity formula can be proven due to Modica's estimate showing the discrepancy is negative.This estimate is not true without a homogeneous left hand side to equation (1.1).Instead we will use the integral bound (1.6) to derive a decay bound for L 1 norm of the discrepancy which we eventually show vanishes in the limit ε → 0. This estimate constitutes one of the main advances of this paper.In section 4 we show the limiting varifold we obtain as ε → 0 is a rectifiable set and in section 5 we show the limiting varifold is in addition integral.
Acknowledgements.The first author was supported by EPSRC grant EP/S012907/1.The second author was supported by EPSRC grants EP/S012907/1 and EP/T019824/1.

Preliminaries and notations
Throughout the paper, we will denote a constant by C if it only depend on the constants n, E 0 , c 0 , Λ 0 which appear in the conditions of Theorem 1.1.A certain points we may increase this constant in some steps of the argument, but we will not relabel the constant unless there is a risk of confusion from the context.We associate to each solution of (1.1) a varifold in the following way : let G(n + 1, n) denote the Grassmannian (the space of unoriented n-dimensional subspaces in R n+1 ).We regard S ∈ G(n + 1, n) as the (n + 1) × (n + 1) matrix representing orthogonal projection of R n+1 onto S, that is Varifold convergence means convergence of Radon measures or weak- * convergence.We let V ∈ V n (Ω) and let V denote the weight measure of V and we define the first variation of V by We let δV be the total variation of δV .If δV is absolutely continuous with respect to δV then the Radon-Nikodym derivative δV V exists as vector valued measure.We denote by H V = − δV V , the generalised mean curvature.
Let u be a function, we define the associated energy measure as a Radon measure given by where L n+1 is the (n + 1) dimensional Lebesgue measure.We also denote the the energy of the 1 dimensional solution by There is an associated varifold V ∈ V n (Ω) to the functions u given by where I is the (n + 1) × (n + 1) identity matrix and is orthogonal projection onto the space orthogonal to ∇u(x), that is {a ∈ R n+1 | a, ∇u(x) = 0}.By definition V = µ {|∇u| =0} and the first variation may be computed as

Discrepancy bounds and monotonicity formula
In this section, we deduce integral bounds on the discrepancy.There exists an almost monotonicity formula for the Allen-Cahn energy functional, we will give estimates on the terms appearing in the almost monotonicity formulas under the assumptions in Theorem 1.1 and obtain a monotonicity formula for the n-dimensional volume ratio.It will be used in the next section to deduce rectifiability and integrality of the limit varifold as ε → 0. Conditions (1)-(3) in Theorem 1.1 are assumed to hold throughout this section.
The n-dimensional volume ratio of the energy measure satisfies the following almost monotonicity formula.
ε is the discrepancy measure (difference between the Dirichlet and potential energy) in B r .
Proof.Multiplying equation (1.1) by x, ∇u ε and integrating by parts on B r , we get The conclusion then follows by dividing both sides by r n+1 and noticing Integrating the almost monotonicity formula (3.1) from ε to r 0 for 0 < ε < r 0 < 1, we have where ω n+1 denotes the volume of unit ball in R n+1 .
We need to estimate the first and third term on the right hand side to obtain a monotonicity formula.In order to estimate the third term, we derive an a priori gradient bound for u.Condition (3) of Theorem 1.1 states a combined integrability for the inhomogeneity f ε and |∇u|.The following theorem allows us to obtain separate integrability and regularity for each quantity.Theorem 3.2.There exists C, ε 0 > 0 depending on E 0 , c 0 , Λ 0 as defined in Theorem 1.1 such that if u ε satisfies (1.1) in B 1 ⊂ R n+1 with ε < ε 0 and if q 0 > n + 1, then sup Furthermore, there exists a δ 0 > 0 so that f has the following improved integrability Proof.We first consider the case q 0 > n + 1: Define the rescaled solution ũ(x) := u(εx) and f (x) = εf ε (εx) which satisfies the equation By condition (3) in Theorem 1.1, we have by rescaling Proof of Claim.By the hypothesis B1 (x ) and |∇φ| ≤ 4. By integration by parts and Young's inequality, we obtain (3.9) We write q 0 and use Young's inequality with exponent q 0 to get Here we used (3.8) to bound and the fact that ε 1− n q 0 < 1 in the second inequality, Hölder's inequality with exponent 2(q 0 −1) q 0 −2 in the third inequality.And in the fourth inequality we used φ 4q 0 q 0 −2 ≤ φ 2 , and chose δ to be q 0 4Cn(q 0 −1) .We insert the above inequality into (3.9) and get We assume B 2 (x 0 ) φ 2 |∇ũ| 2 ≥ 1, otherwise the desired bound holds trivially.Then by moving the first term 1 2 B 2 (x 0 ) |∇ũ| 2 φ 2 and the fifth term B 2 (x 0 ) φ 2 |∇ũ| 2 on the right to the left, we prove the claim.Now suppose ∇ũ L p 0 (B 1 (x 0 )) ≤ C(c 0 , Λ 0 , q 0 , n) (independent of ε) for some p 0 > 1 (p 0 can be chosen to be 2 by the claim above).For any B 2 (x 0 ) ∈ B 1 ε (0), we have by Hölder's inequality Remark 3.3.Here q 0 > n will make the scaling subcritical and ensures a uniform bound of f Thus f is uniformly bounded in L p 0 q 0 p 0 +q 0 −2 (B 1 (x 0 )) independent of ε.By applying the Sobolev inequality to (3.7), standard Calderon-Zygmund estimates and finally using the L ∞ bound of u in condition (2) of Theorem 1.1, we have We remark that q 0 > n ensures the coefficient ε stays uniformly bounded as ε → 0. In the case p 0 q 0 p 0 +q 0 −2 > n + 1, by Calderon-Zygmund estimates we have The Sobolev inequality then gives ∇ũ L ∞ ≤ C. In the case p 0 q 0 p 0 +q 0 −2 ≤ n + 1, using q 0 > n + 1, we have p 0 < p 0 Consider now the case n < q 0 ≤ n + 1.For any

And thus
. And we can iterate until (3.13) fails, namely for any x 0 ∈ B 1 ε −2 (so that the condition in the claim above is satisfied).By Sobolev inequalities, we then have for any ≤ C(c 0 , Λ, 0, q 0 , n).

Rescaling back gives
which is (3.5).By (3.10) we improve the integrability of f in (3.10) up to f 2 + δ 0 for some δ 0 > 0. On the other hand, if q 0 > n + 1, using (3.8) and the uniform gradient bound of u in Theorem 3.2, we have f L q 0 (B 1 (x 0 )) ≤ Cε q 0 −n q 0 , where q 0 > n + 1 > n+1 2 + δ 0 .Combining both cases, for any Rescaling back gives the bound on f , Since in the case q 0 ∈ (n, n+1], we lack gradient bounds of u as in the case q 0 > n+1.In order to get better estimates of the discrepancy terms in the almost monotonicity formula, we use some ideas from [RS06].We will apply the following Lemma to (3.7) for ε sufficiently small such that Cε And for τ = δ p 3 , where Proof.Let us consider the auxiliary function ψ which solves the Dirichlet problem The auxiliary function will allows us to control the inhomogeneous part of the equation.
Claim.The function ψ defined in (3.21) satisfies the bounds Here we prove (3.22): by the Sobolev inequality since due to the choice of ω, where we used (n+1)(n+1+2δ 0 ) n+1−2δ 0 > n + 1.Here we prove the gradient bound (3.23): We define ũ0 := ũ + ψ ∈ W 2, n+1 2 +δ 0 (B R ).By (3.21), (3.22), ũ0 satisfies We compute for any β > 0, for some C > 0. Thus by (3.22) and (3.23), we have By choosing β = ω ≤ δ p 2 and using our hypothesis on ω : R 2 +δ 0 .By our choice of p 1 = 2, p 2 = 15, we ensure To prove (3.19), it suffices to show Here we estimate ũ.Define ũR (x) = ũ(Rx) then by the Calderon-Zygmund estimates we have By the Sobolev embedding ∇ũ since the derivatives of order 5 or higher of the potential Since we have |ũ 0 | ≤ c 0 , we apply Calderon-Zygmund to (3.25), for any B 1 (x) ⊂ B R and 1 < r < ∞ and we get Hence by the Morrey embedding We define a modified discrepancy for some function G ∈ C ∞ (R) and ϕ ∈ W 2,2 (B R ) that we choose as in the following claims Claim.If we make the following choice of G, ds dt (3.33) then we have the properties (3.34) Furthermore we have and Proof of Claim.The first three equations of (3.34) follow from the direct computations.For (3.35), since G δ ≥ δ, we obtain where we used W is an even function, increasing in [−1, 0] and decreasing in Claim.If we choose ϕ to satisfy the Dirichlet problem and hence we obtain (3.39).
We choose ϕ according to (3.37).Notice if ξ G > 0, then we have ∇ũ 0 = 0 and The case ξ G ≤ 0 immediately gives us our desired estimate since we are seeking an upper bound.
Claim.For the choice of G as in (3.33) and ϕ as in (3.37) we have the differential inequality Proof.We compute the Laplacian of the modified discrepancy (3.42)By differentiating (3.32), we have and thus The last term in (3.42) is Substituting these into (3.42) and rearranging, we have in We choose G to be (3.33)which allows us to apply the estimates (3.34) and (3.35) so that From (3.34), the bounds on G δ and its derivatives, we get Substituting in (3.37), (3.39), and (3.44) into (3.43) and using the fact that G ′′ < 0, we have Thus applying equation (3.36 and consider two cases : by (3.34) and (3.39) this implies Moreover, Since x 0 is a critical point, ∇(λξ G )(x 0 ) = 0, and we get At a maximum point x 0 , the Laplacian of the function λξ G satisfies and thus since δ ≪ 1. Combining (3.41) and (3.47) we have Thus the last term above is bounded by By our choice of p 1 = 2, p 2 = 15, we have R Namely, assuming (3.46) or not, we have and thus by (3.39) which proves (3.20).
Next we derive energy estimates away from transition regions. and (Notice the power of f ε in the above inequality will still be 2 instead of n+1 2 + δ 0 .)Proof.Define a continuous function where t 0 = 1 √ 3 is chosen to be the number in (0, 1) such that W ′′ (t 0 ) = 0. Clearly |g| ≤ |W ′ |.For η ∈ C 1 0 (Ω) satisfying 0 ≤ η ≤ 1, η ≡ 1 on Ω ′ and |∇η| ≤ Cr −1 , we get by integration by parts (3.48) The left hand side of (3.48) can be bounded by By the definition of g above, we have for |t| ≤ 1 − δ.Applying these estimates to the second term on the right hand side of (3.48) we get the bound Choosing τ = C W 2 , and using which completes the proof.
The following proposition shows for all ε sufficiently small, if u ε satisfies the inhomogeneous Allen-Cahn equation then we can control the last term {|uε|≥1}∩Ω ′ W ′ (u ε ) 2 in Proposition 3.5 by applying the proposition inductively.
for all k ∈ N 0 .
Proof.For any k ∈ N + we choose a sequence of open sets This sequence satisfies Applying Proposition (3.5) with δ = 0, we have for i = 1, ..., k.The conclusion is obtained by applying the above inequality inductively k times.
We conclude with the following integral bound for positive part of discrepancy measure.
Lemma 3.7 ([RS06, Lemma 3.1] for all n).Let n ≥ 2, 0 < δ ≤ δ 1 (where δ 1 given as in Lemma 3.4), 0 < ε ≤ ρ, ρ 0 := max{2, 1 + δ −M ε}ρ for some large universal constant then the positive part of the discrepancy measure satisfies Proof.We prove the case 0 < ε ≤ ρ = 1.The case for other ρ > 0 follows by rescaling to ρ = 1.For 0 < δ ≤ δ 1 we choose R(δ) = 1 δ p 1 and ω(δ) = C ω δ p 2 as in Lemma 3.4.Let {x i } i∈I ⊂ B 1 , I ⊂ N be a maximal collection of points satisfying min i =j Since ε ≤ 1, we have For i ∈ I and x ∈ B 2R , we define the rescaled and translated functions as ũi (x) := u ε (x i + εx), fi (x) := εf ε (x i + εx), which satisfy the rescaled equation For ũi , fi to be well-defined, we choose M ≥ 5n + 6 and δ 1 ≤ 1 2 so that We decompose the index set I into By the condition u L ∞ ≤ c 0 in the condition of Theorem 1.1, and choosing C ω sufficiently small, we have Applying Lemma 3.4 to ũi gives (with p 3 from Lemma 3.4) Rescaling back, we get Summing over i ∈ I 1 and noticing B ε 2 (x i ) are disjoint, we get where we used Proposition 3.5 in the last line.Since for n ≥ 3 (the n = 2 case requires δ 0 ≥ 1 2 , but has already been addressed in [RS06]) Thus for i ∈ I 2 (at least one of the bounds in I 1 does not hold), we have By elliptic estimates applied to the rescaled equation (3.51), we get where we used ũi L ∞ ≤ c 0 .Rescaling back gives Then summing over i ∈ I 2 we get for large enough M since both R = δ −p 1 and ω = δ p 2 are fixed powers of δ.Combining (3.52) and (3.53) we get This completes the proof for ρ = 1 and rescaling gives the cases for other ρ > 0.
As a result of these, we have the L 1 convergence of the positive part of the discrepancy measure as ε → 0. Lemma 3.8.If we consider ξ ε = ξ ε,+ − ξ ε,− the decomposition of ξ ε into positive and negative variations then Furthermore this shows ξ ≤ 0.

Rectifiability
We will proceed by proving upper and lower density bounds for the energy measure.Combining the estimates obtained in the previous section, we get an upper bound on the density ratio of the limit energy measure.
Proof.For the sake of simplicity we set x 0 = 0 and set B ρ (0) = B ρ .By the almost monotonicity formula (3.1), Lemma 3.8 and Holder's inequality We estimate the last term above as follows where we used the inequality a 1− 1 q 0 ≤ 1 + a which holds for all a ≥ 0. Inserting this inequality into (4.2) and discarding the positive second term on the right had side, we get for µ-a.e. in Ω.
Proof.Without loss of generality, we assume 0 ∈ spt µ ∩ Ω ′ and want to prove a density lower bound at 0. We first integrate (4.4) from s to r.
By (4.6) in Proposition 4.2, the discrepancy term By the ε-Upper Density Bound (4.1) we get The last term in (4.8) may be estimated as follows Using the bound and the ε-Upper Density Bound (4.1), we get Thus, plug all the above estimates of terms in (4.7), we get Next, we estimate the term Claim.There exists x ∈ B r 2 such that |f ε | 2 dρ, (4.10) for some universal constant θ0 > 0.
≤ 1(otherwise the conclusion automatically follows), and so From Theorem 3.2 we have inequality in the conclusion of the claim holds.Applying the error estimates Proposition 3.5 with the choice Ω ′ = B r 4 and Ω = B r 2 , for sufficiently small τ Notice by (3.55), the second term . So the last three terms are at most of order O(ε).Hence, as 0 ∈ spt µ, by passing to limit ε → 0 we have (This guarantees we can always choose such a point x ∈ B r 2 with |u ε (x)| ≤ 1 − τ if 0 ∈ spt µ.)To complete the proof, we define for 0 < ρ < r 1 the convolution Now we can estimate the term on the right hand side in the claim, by a change of variables t = 3ρ 1−β .Here β := β(r 1 ) is chosen small enough such that 3 We find is a decreasing function.Hence we get the bound (4.13) Choosing Mγ < 1 2 and β sufficiently small so that Mγ + nβ < 1 2 , and applying the weak L 1 inequality for the distribution function and (4.13), we get for some Cβ depending on β We can thus combine (4.12) with (4.14) to find an x ∈ B r 2 so that the upper bound and lower bound in the claim holds.
With this claim, we proceed with the proof of the density lower bound.For the θ0 obtained from the claim, we denote by s := sup{0 ≤ ρ ≤ r 4 : µε(Bρ(x)) ρ n ≥ 2 θ0 }.And it is obvious from (4.11) Proof.We first prove the lower n-dimensional density of the discrepancy measure vanishes.Namely If not, there exists 0 < ρ 0 , δ < 1 and Multiplying both sides of (4.2) by an integrating factor and integrating from r to ρ 0 as in the proof of Theorem 4.1 we get Using Lemma 3.8, that is ξ + = 0 and Theorem 4.1, we have when passing to the limit ε → 0 This gives a contradiction by letting r → 0. By the density lower bound Theorem 4.3 and differentiation theorem for measures, we have and this shows Proposition 4.5.We choose a Borel measurable function ν ε : Ω → ∂B 1 (0) extending ∇uε |∇uε| on ∇u ε = 0 and consider the varifold The first variation is given by Proof.By equation (2.1), we have The Stress-Energy tensor for the Allen-Cahn equation is given by Integrating by parts, we get Hence inserting this into our expression for the first variation we get Combining Theorem 4.1, Theorem 4.3 and Proposition 4.4, we obtain Theorem 4.6.After passing to a subsequence, the associated varifolds V ε → V where V is a rectifiable n-varifold with the weak mean curvature in L q 0 loc (µ V ).
Proof.We first compute the first variation of the associated varifolds V ε to the energy measure µ ε (c.f.[RS06, Proposition 4.10], [TT20, Equation 4.3]).For any η ∈ C 1 0 (Ω; R n+1 ), using Proposition 4.5 and Proposition 4.4 (4.17) So we see the limit varifold has locally bounded first variation, combining with the density lower bound Theorem 4.3 we conclude the limit varifold is rectifiable by Allard's rectifiability theorem.Moreover, the above calculation shows δV is a bounded linear functional on L q 0 q 0 −1 loc (µ V ) and thus itself is in L q 0 loc (µ V ).

Integrality
In this section, we prove the integrality of the limit varifold.
From the previous section, we have already shown the limiting varifold V is rectifiable.And thus for a.e.x 0 ∈ spt µ V , we have for any sequence ρ i → 0 where D ρ i (x) = ρ −1 i x and T x 0 (x) = x − x 0 represent dilations and translations in R n+1 and θ x 0 is the density of µ V at x 0 .By choosing a sequence of rescaling factors ρ i such that εi := ε i ρ i → 0, (5.1) the new sequence ũε i (x) := u ε i (ρ i x + x 0 ), fε i (x) := ρ i fi (ρ i x + x 0 ) satisfies εi ∆ũ εi − W ′ (ũ εi ) εi = fε i and the associated varifold Ṽi of this new sequence ũε i converges to θ x 0 P 0 .By (3.55), we also have as q 0 > n.Furthermore, by choosing more carefully so that Therefore we have reduced Theorem 5.1 to the following proposition Proposition 5.2.If the limit varifold is θ 0 H n ⌊P 0 for some P 0 ∈ G(n + 1, n) and θ 0 > 0, then α −1 θ 0 is a nonnegative integer, where α = ∞ −∞ (tanh ′ x) 2 dx is the total energy of the heteroclinic 1-d solution.
In order to prove Proposition 5.2, we need two lemmas.The first Lemma 5.5 is a multisheet monotonicity formula (c.f.[All72, Theorem 6.2] for the version for integral varifolds, which is used to prove the integrality of the limits of sequences of integral varifolds).The second Lemma 5.7 says at small scales, the energy of each layers are almost integer multiple of the 1-d solution.We first gather some apriori bounds on energy ratio for µ ε .
As a corollary, we have Corollary 5.4.If in addition to the conditions in Proposition 5.3, we assume then the following upper bound for the energy ratio for µ ε holds Proof.We have Thus by Proposition 5.3 and ε ≤ ρ, we have The conclusion then follows by substituting in (5.6) and applying Gronwall's inequality to the above differential inequality.
Lemma 5.7 (Lemma 5.5 of [RS06]).Suppose the conditions in Theorem 1.1 are satisfied.For any τ ∈ (0, 1),δ > 0 small, Λ > 0 large, there exists ω > 0 sufficiently small and L > 1 sufficiently large such that the following holds: Suppose u ε satisfies condition of Theorem 1.1 in B 4Lε (0) ⊂ R We want to apply Lemma 5.5 and Lemma 5.7.We choose 0 < ω = ω(N, δ, 1 2 , 1 2 , C) and ω(δ, τ, C) ≤ 1 where L = L(δ, τ ) which are the parameters that appear in Lemma 5.5 and Proposition 5.6 and C is the constant so that We define A ε to be the set where the hypotheses for our Propositions hold, that is We show the complement of the set A ε has small measure.By Besicovitch's covering theorem, we find a countable sub-covering ∪ i B ρ i (x i ), ρ i ∈ [ε, 3] of {|u ε | ≤ 1 − τ } \ A ε such that every point x ∈ {|u ε | ≤ 1 − τ } \ A ε belongs to at most B n balls in the covering, where B n depends only on the dimension n.For each i, either On the other hand, by (5.2), for sufficiently small ε, we have 1 By (5.7), for each i, we obtain µ ε B ρ i (x i ) ≤ Cρ n i .Since the overlap in the Besicovitch covering is finite and (5.34), we get On the other hand, since lim ε→0 µ ε (B 1 (0)) = θω n and δ > 0 is arbitrarily small, we obtain θ ≤ (N − 1)α.
And since by definition N is the smallest integer such that θ < Nα, we have θ = (N − 1)α.
Proof.Since we have ϕ ≥ 0 in ∂B R 2 by applying the maximum principle, we have ϕ ≥ 0 in B R 2 which gives us (3.38).The estimates (3.31), (3.29) and (3.30) bound the right hand side of (3.37), that is ∆ϕ This proves (3.26) and as a consequence (3.19).If |ũ| ≥ 1 − τ in B 1 2 , then (3.20) follows because the left hand side is less than the second term on the right.So we only need to consider the case there exists x 0 ∈ B 1 2 with ũ(x 0 ) ≤ 1 − τ .By the Sobolev inequality and Calderon-Zygmund estimates we bound ũ in the Hölder norm as follows ũ