A new proof of Savin's theorem on Allen-Cahn equations

arXiv:1401.6480v1 [math.AP] 25 Jan 2014

A new proof of Savin’s theorem on Allen-Cahn equations Kelei Wang

∗

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China [email protected]

Abstract In this paper we establish an improvement of tilt-excess decay estimate for the Allen-Cahn equation, and use this to give a new proof of Savin’s theorem on the uniform C 1,α regularity of flat level sets, which then implies the one dimensional symmetry of minimizers in Rn for n ≤ 7. This generalizes Allard’s ε-regularity theorem for stationary varifolds to the setting of Allen-Cahn equations.

Keywords: Allen-Cahn equation, phase transition, improvement of tilt-excess decay, harmonic approximation, De Giorgi conjecture.

AMS Subject Classification (2000): 35B06, 35B08, 35B25, 35J91.

Contents 1 Introduction

2

2 Settings and notations

6

I

8

Tilt-excess decay

3 Statement

9

4 Compactness results ∗

11

Supported by NSFC No. 11301522.

1

5 Lipschitz approximation

14

6 Estimates on htε

21

7 The blow up limit

27

8 Proof of the tilt-excess decay ¯ 8.1 The case ∇h(0) =0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30 30 34

II

Uniform C 1,α regularity of intermediate layers

37

9 Statement

37

10 A Morrey type bound

38

11 A direct proof of Theorem 1.1

42

12 The Lipschitz nature of intermediate layers

45

Appendix A A distance type function

48

Appendix B Several technical results

52

References

55

1

Introduction

This paper is devoted to generalize the Allard’s regularity theory in Geometric Measure Theory to the setting of Allen-Cahn equations and discuss its application to the De Giorgi conjecture. The Allen-Cahn equation ∆u = u3 − u, (1.1) is a typical model of phase transition. By now, it has been studied from various aspects. One particular feature of this equation is its close relation with the minimal surface theory, through the singularly perturbed Allen-Cahn equation ∆uε =

1 3 u − u . ε ε2 ε 2

By this connection and in view of the Bernstein theorem for minimal hypersurfaces [23], De Giorgi made the following conjecture in [8]: Let u ∈ C 2 (Rn+1 ) be a solution of (1.1) such that |u| ≤ 1,

∂u > 0 in Rn+1 . ∂xn+1

Then u depends only on one variable, if n ≤ 7. This conjecture has been considered by many authors, including Ghoussoub and Gui [14], Ambrosio and Cabre [3] and Savin [19], and counterexamples are constructed by del Pino, Kowalczyk and Wei [10] when n ≥ 8. In particular, Savin proved an improvement of flatness type result for minimizing solutions (i.e. a minimizer of the energy functional). This result says, if u is a minimizer, and if in a ball Bl , the zero level set of u is trapped in a flat cylinder {|xn+1 | < θ} which is sufficiently narrow (i.e. θl−1 sufficiently small), then by shrinking the radius of ball, possibly after a rotation of coordinates, the zero level set of u is trapped in a flatter cylinder. By using this estimate, Savin proved Theorem 1.1. Let u be a minimizer defined on the entire space Rn+1 where n ≤ 6. Then u is one dimensional. For n > 6, if we add some assumptions on the behavior of level sets of u, it is still possible to prove the one dimensional symmetry of u. This theorem also implies the original De Giorgi conjecture, with an addition assumption that lim

xn+1 →±∞

u(x, xn+1 ) = ±1.

This type of improvement of flatness result is a common theme in partial regularity theory for many elliptic problems, although sometimes in rather different forms. One main ingredient to establish this improvement of flatness is the blow up (or harmonic approximation) technique, which was first introduced in De Girogi’s solution of the almost everywhere regularity problem for minimal hypersurfaces in [7]. Although the statement of Savin’s improvement of flatness result bears many similarities with De Giorgi’s theorem, the proof in [19] is rather different. In fact, it is based on Caffarelli-Cordoba’s proof of De Giorgi theorem in [5]. This approach uses the “viscosity” side of the problem, and relies heavily on a Krylov-Safanov’s type argument. In particular, Savin first obtains a Harnack inequality (hence some kind of uniform H¨older continuity) and then use this to prove that the blow up sequence converges uniformly to a harmonic function. Thus this argument can be understood as a kind of harmonic approximation in L∞ norm. 3

Savin’s approach can be applied to many other problems, even without variational structure, see for example [20, 21, 22]. However, it seems that the maximum principle and Harnack inequality is crucial in this approach. At present it is not clear how to get this kind of improvement of flatness result for elliptic systems, where Harnack inequality may fail. Thus, in view of the connection between Allen-Cahn equations and the minimal surface theory, in this paper we intend to explore the variational side of improvement of flatness and establish some results which parallel the classical regularity theory in Geometric Measure Theory. In particular, as in Allard’s regularity theory [1] (see also [16, Section 6.5] for an account), we use the following excess quantity Z 1 − (νε · e)2 ε|∇uε|2 ,

where νε = ∇uε /|∇uε | is the unit normal vector to level sets of uε . This quantity was first used by Hutchinson-Tonegawa [15] to derive the integer multiplicity of the limit varifold arising from general critical points in the Allen-Cahn problem. This quantity can be seen as a measurement of the flatness of the level sets of uε (see Lemma 4.6 below). Similar to Allard’s ε-regularity theorem, if the excess in a ball is small, then after shrinking the radius of ball and possibly rotating the vector e, the excess becomes smaller. This improvement of tilt-excess is the main step in the proof of Allard’s ε-regularity theorem, and also in our argument. In contrast to the quantity used in Savin’s improvement of flatness result, the excess is an energy type quantity. Indeed, if all the level sets {uε = t} can be represented by graphs along the (n + 1)-th direction, in the form {xn+1 = h(x, t)}, then Z Z 1 Z |∇h(x, t)|2 2 2 1 − (νε · e) ε|∇uε| = ε|∇uε(x, h(x, t))|dx dt, 1 + |∇h(x, t)|2 −1

which is almost a weighted Dirichlet energy. From this, we see the problem can be approximated by harmonic functions (corresponding to critical points of the Dirichlet energy) if |∇h(x, t)| is small. This is exactly the content of harmonic approximation technique. Using the excess allows us to work in the Sobolev spaces and apply standard compactness Sobolev embeddings to get the blow up limit. We also note that this type of tilt-excess decay result was known by Tonegawa, see [24], where he showed that this result implies the uniform C 1,α regularity of intermediate transition layers in dimension 2. However, in this tilt-excess decay we need to assume that Z 1 − (νε · e)2 ε|∇uε|2 ≫ ε2 . (1.2) 4

Compared to Allard’s ε-regularity theorem, this condition is rather unsatisfactory. It prevents us from applying this improvement of decay directly to deduce the uniform C 1,α regularity of intermediate transition layers, while Savin’s improvement of flatness does implies this directly. (This was also discussed by Tonegewa [24].) In this paper we show this tilt-excess decay still implies the uniform C 1,α regularity, by exploiting some more properties implied by the fact that uε is close to a one dimensional solution at O(1) scale. We can show that uε is close to a one dimensional solution at all scale up to O(ε). This fact, together with the improvement of tilt-excess decay, gives a Morrey type bound. Here, once again due to the obstruction (1.2), this Morrey type bound does not imply the C 1,α regularity of {uε = 0}, but only a Lipschitz one. However, under the condition that {uε = 0} is a Lipschitz graph, Caffarelli and Cordoba [6] has shown that transition layers are uniformly bounded in C 1,α for some α ∈ (0, 1). Thus we get a full analogue of Allard’s ε-regularity theorem in the Allen-Cahn setting. In this paper we do not fully avoid the use of maximum principle. For example, it seems that the Modica inequality is indispensable in our argument, because we need it to obtain a monotonicity formula. We also need to apply the moving plane (or sliding) method (as in Farina [12]) to deduce the one dimensional symmetry of some entire solutions. There we also need a function to control the behavior of u at the place far away from the transition part. This function behaves like a distance function and this fact follows from the Modica inequality. However, we do avoid the use of any Harnack inequality. It may be possible to remove the above mentioned deficiency by strengthening the tilt-excess decay estimate (see Theorem 3.3 below), but the current version of Theorem 3.3 is already sufficient for proving Theorem 1.1, as discussed above. This approach was used by the author in [25], where we consider a De Giorgi type conjecture for the elliptic system ∆u = uv 2 , ∆v = vu2 , u, v > 0 in Rn . For the corresponding singularly perturbed system ( ∆uκ = κuκ vκ2 , ∆vκ = κvκ u2κ , an improvement of flatness result was established by considering the quantity Z ∇ (uκ − vκ − e · x) 2 .

Both proofs are same in the spirit. In particular, to prove that the blow up limit is a harmonic function, we mainly use the stationary condition associated to the equation but not the equation itself. This is more apparent in the current setting, because the 5

stationary condition for the singularly perturbed Allen-Cahn equations is directly linked to the corresponding one in their limit problem, the stationary condition for varifolds (in the sense of Allard [1], see [15]). Furthermore, since the excess is kind of H 1 norm, to prove the strong convergence in H 1 space, we implicitly use a Caccioppoli type inequality, which is again deduced from the stationary condition (see Remark 4.7 below). Finally, although in our tilt-excess result (Theorem 3.3) and ε-regularity result (Theorem 9.1), we do not assume the solution to be a minimizer, we still need a kind of multiplicity one property of transition layers. (This is associated to the unit density property of the limit varifold.) Thus essentially this argument does not remove the restriction of no folding phenomena in Savin’s result. The organization of this paper can be seen from the table of contents.

2

Settings and notations

Throughout this paper, we shall work in the following settings. We consider the Allen-Cahn equation in the general form as ∆u = W ′ (u), (2.1) where W is a double well potential. That is, W ∈ C 3 (R), satisfying • W ≥ 0, W (±1) = 0 and W > 0 in (−1, 1); • for some γ ∈ (0, 1), W ′ < 0 on (γ, 1) and W ′ > 0 on (−1, −γ); • W ′′ ≥ κ > 0 for all |x| ≥ γ. A typical example is W (u) = (1 − u2 )2 /4. Through a scaling uε (X) := u(ε−1 X), we have the singularly perturbed version of the Allen-Cahn equation: 1 ε∆uε = W ′ (uε ). (2.2) ε Note that a solution uε is a critical point of the functional Z ε 1 Eε (uε ) = |∇uε |2 + W (uε ). (2.3) 2 ε We say uε is a minimizer (or a minimizing solution), if for any ball B in the definition domain of uε , Z Z ε 1 1 ε 2 |∇uε | + W (uε ) ≤ |∇v|2 + W (v), ε ε B 2 B 2 for any v ∈ H 1 (B) such that v = u on ∂B.

6

We will always assume that any solution uε of (2.2) satisfies |uε | ≤ 1. Then by standard elliptic estimates, there exists a universal constant C(n) such that ε|∇uε| + ε2 |∇2 uε | ≤ C(n).

(2.4)

Hence uε is a classical solution. For any smooth vector field Y with compact support, by considering the domain variation in the form utε (X) := uε (X + tY (X)), for |t| small, from the definition of critical points we get Z ε 1 d 2 |∇uε | + W (uε ) = 0. t=0 dt 2 ε

Then after some integration by parts, we obtain the stationary condition satisfied by uε : Z 1 ε 2 |∇uε | + W (uε ) divY − εDY (∇uε , ∇uε ) = 0. (2.5) 2 ε Finally, by the assumption on W , there exists a one dimensional solution g(t) defined on t ∈ (−∞, +∞), g ′′(t) = W ′ (g(t)), lim g(t) = ±1, (2.6) t→±∞

where the convergence rate is exponential. The first integral for g can be written as p g ′ (t) = 2W (g(t)) > 0.

(2.7)

For any ε > 0, we denote gε (t) = g(ε−1t), which satisfies gε′′ (t) =

1 ′ W (gε (t)). ε2

Throughout this paper, σ0 is the constant defined by Z +∞ σ0 := g ′ (t)2 dt.

(2.8)

(2.9)

−∞

In this paper we adopt the following notations. • A point in Rn+1 will be denoted by X = (x, xn+1 ) ∈ Rn × R. • Br (X) denotes an open ball in Rn+1 and Br (x) an open ball in Rn . If the center is the origin 0, we write it as Br (or Br ). 7

• Cr (x) = Br (x) × (−1, 1) ⊂ Rn+1 the finite cylinder over Br (x) ⊂ Rn . • ei , 1 ≤ i ≤ n + 1 denote the standard basis in Rn+1 . • P denotes a hyperplane in Rn+1 and ΠP (or simply P ) the orthogonal projection onto it. If P = Rn , we simply use Π. • G(n) denotes the Grassman manifold of unoriented n-dimensional planes in Rn+1 . • A varifold V is a Radon measure on Rn+1 ×G(n). We use kV k to denote the weighted measure of V , that is, for any measurable set A ⊂ Rn+1 , kV k(A) = V (A × G(n)). • For a measure µ, sptµ denotes its support. • νε (X) =

∇uε (X) |∇uε (X)|

if ∇uε (X) 6= 0, otherwise we take it to be an arbitrary unit vector.

• µε := ε|∇uε|2 dX. • Hs denotes the s-dimensional Hausdorff measure. • ωn denotes the volume of the unit ball B1 in Rn . R 1/2 • H 1 denote the Sobolev space with the norm |∇u|2 + |u|2 .

• distH is the Hausdorff distance between sets in Rn+1 .

• Universal constants C, Ci and Ki (which are large) and ci (which are small) depend only on the dimension n and the potential function W . Throughout this paper uε always denotes a solution of (2.2), satisfying the Modica inequality ε 1 |∇uε |2 ≤ W (uε ). (2.10) 2 ε This inequality may not be essential in the argument, but we prefer to assume it to make the arguments clear. We use ε (more precisely, it should be written as εi ) to denote a sequence of parameters converging to 0.

8

Part I

Tilt-excess decay 3

Statement

The following quantity will play an important role in our analysis. Definition 3.1 (Excess). Let P be an n-dimensional hyperplane in Rn+1 and e one of its unit normal vector, Br (x) ⊂ P an open ball and Cr (x) = Br (x) × (−1, 1) the cylinder over Br (x). The excess of uε in Cr (x) with respect to P is Z −n E(r; x, uε , P ) := r 1 − (νε · e)2 ε|∇uε |2 dX. (3.1) Cr (x)

If P = Rn and e = en+1 , then

E(r; x, uε ) = r

−n

Z

ε

Cr (x)

2 n X ∂uε i=1

∂xi

dX.

Remark 3.2. For any unit vector ν, |ν − e||ν + e| ≥

√

2 min{|ν − e|, |ν + e|}.

Thus 1 (1 − ν · e) (1 + ν · e) 4 1 |ν − e|2 |ν + e|2 = 4 1 ≥ min{|ν − e|2 , |ν + e|2 }. 2

1 − (ν · e)2 =

By projecting the unit sphere Sn to the real projective space RPn (both with standard metric), we get 1 − (ν · e)2 ≥ c(n)distRPn (ν, e)2 , for some universal constant c(n). Our main objective in Part I is to prove the following decay estimate.

9

Theorem 3.3 (Tilt-excess decay). There exist five universal constants δ0 , τ0 , ε0 > 0, θ ∈ (0, 1/4) and K0 large so that the following holds. Let uε be a solution of (2.2) with ε ≤ ε0 in B4 , satisfying the Modica inequality (2.10), |uε (0)| ≤ γ, and Z 1 ε −n |∇uε |2 + W (uε ) ≤ (1 + τ0 ) σ0 ωn . (3.2) 4 ε B4 2 Suppose the excess with respect to Rn δε2 := E(2; 0, uε , Rn ) ≤ δ02 ,

(3.3)

where δε ≥ K0 ε. Then there exists another plane P , such that θ θ−n E(θ; 0, uε, P ) ≤ E(2; 0, uε, Rn ). 2

(3.4)

Moreover, there exists a universal constant C such that ke − en+1 k ≤ CE(2; 0, uε, Rn )1/2 , where e is the unit normal vector of P pointing to the above. In the next section we shall see that (3.3) always holds, provided that we have (3.2) with τ0 sufficiently small (depending on δ0 ). However, the assumption that δε ≫ ε is crucial here, which is not so satisfactory compared to Allard’s and Savin’s version. We shall prove this theorem by contradiction. Assume it does not hold. Thus there exists a sequence of εi (for simplicity we shall drop the subscript i) and corresponding solutions uε satisfying all of the assumptions in Theorem 3.3, in particular: 1. We have a sequence of τε → 0 such that Z 1 ε −n |∇uε |2 + W (uε ) ≤ (1 + τε ) σ0 ωn . 4 ε B4 2

(3.5)

2. The excess δε2 := E(2; 0, uε, Rn ) → 0, where

δε → +∞, ε

as ε → 0.

(3.6) (3.7)

But for any unit vector e with 1

ke − en+1 k ≤ CE(2; 0, uε, Rn ) 2 , 10

(3.8)

where the constant C will be determined below (by the constant in (8.1)), we have θ (3.9) θ−n E(θ; 0, uε, P ) ≥ E(2; 0, uε, Rn ). 2 Here P is the hyperplane orthogonal to e and θ is a constant to be determined later, too. The remaining part will be devoted to derive a contradiction from these assumptions (3.5)-(3.9).

4

Compactness results

In this section, we study the convergence of various quantities associated to uε . Recall that we have assumed the Modica inequality (2.10) holds for uε . An important consequence of this inequality is the Proposition 4.1 (Monotonicity formula). For any X ∈ B3 , Z 1 ε −n |∇uε |2 + W (uε ) r ε Br 2 is non-decreasing in r ∈ (0, 1).

By combining the monotonicity formula (4.1) with (3.8), we see

Corollary 4.2. For any Br (X) ⊂ B3 , Z 1 ε |∇uε |2 + W (uε ) ≤ 8n σ0 ωn r n . ε Br (X) 2

(4.1)

We can use the main result in Hutchinson and Tonegawa [15] to study the convergence of uε . Define the varifold Vε by Z < Vε , Φ(X, S) >= Φ(x, I − νε ⊗ νε )ε|∇uε |2 dx, ∀ Φ ∈ C0∞ (C2 × G(n)).

Hutchinson and Tonegawa proved that, as ε → 0, Vε converges in the sense of varifolds to a stationary, rectifiable varifold V with integer density (modulo division by the constant σ0 ). From this, it can be directly checked that the measures µε converge to kV k weakly. Moreover, for any t ∈ (−1, 1) fixed, {uε = t} converges to sptkV k in the Hausdorff distance. Hence we must have 0 ∈ sptkV k because 0 ∈ {|uε | ≤ γ}. In [15], using (11.5) and Proposition 4.1, Hutchinson and Tonegawa also proved that the discrepancy quantity 1 ε 2 (4.2) W (uε ) − |∇uε | → 0, in L1loc (C3 ). ε 2 With the bound (3.5) we claim that 11

Proposition 4.3 (Limit varifold). The limit kV k = σ0 H n ⌊Rn . Proof. This is because 4−n kV k(B4 ) = σ0 ωn ,

which is obtained by taking the limit in (3.5) and using the integer multiplicity of V . Then by the monotonicity formula for stationary varifold (cf. [16, Theorem 6.3.2]), we deduce that V is a cone. To show that V is the standard varifold associated to this plane with unit density, we can apply Allard’s ε-regularity result to deduce that sptkV k ∩ B1/2 is a smooth hypersurface and combine this fact with the cone property of V . Now we show that away from Rn , uε is close to ±1. Proposition 4.4. For any h > 0, if ε sufficiently small, |xn+1 | 1 − u2ε + |∇uε | ≤ C(h)e− C(h)ε

in

C2 \ {|xn+1 | ≤ h}.

In particular, {uε = 0} ∩ C2 lies in a h-neighborhood of Rn ∩ B2 . Proof. By [15], u2ε converges to 1 uniformly on any compact set outside sptkV k = Rn . In particular, for all ε small, h u2ε ≥ γ in C3 \ {|xn+1 | ≥ }. 2 Then there exists a constant C depending only on the potential function W such that ∆(1 − u2ε ) ≥

h 1 (1 − u2ε ) in C3 \ {|xn+1 | ≥ }. 2 Cε 2

From this we can apply Lemma B.1 to get the exponential decay of 1 − u2ε . The estimate for |∇uε | follows from standard interior gradient estimate. Remark 4.5. We can show that uε converges to 1 locally uniformly in C2 ∩ {xn+1 > 0}, and to −1 locally uniformly in C2 ∩ {xn+1 < 0}. Together with the previous proposition, this implies that distH ({uε = 0} ∩ C1 , Rn ∩ C1 ) → 0. Indeed, if uε converges to 1 (or −1) on both sides of Rn , the multiplicity of V will be greater than 2 (see [15, Theorem 1, (4)]). This contradicts Proposition 4.3. The following lemma implies that if we have (3.5), then (3.6) always holds. Lemma 4.6. Let uε be a sequence of solutions satisfying (3.5) and the Modica inequality (2.10) in B4 . Then the excess with respect to Rn , E(2; 0, uε ) → 0, as ε → 0. 12

Proof. For any η ∈ C0∞ (C2 ), take the vector field Y = (0, · · · , 0, ηxn+1) and substitute it into the stationary condition. This leads to Z 1 ∂η ε 2 xn+1 (4.3) |∇uε | + W (uε ) η+ 0= 2 ε ∂xn+1 C2 n+1 X ∂η 2 2 νε,i νε,n ε|∇uε|2 . −ηνε,n+1 ε|∇uε | − xn+1 ∂x i i=1 By (4.2) and our assumption, the measures 1 ε 2 |∇uε | + W (uε ) dx, 2 ε

νε,i νε,n ε|∇uε |2 dx

all converge to some measures supported on Rn . Thus Z n+1 X ε ∂η 1 ∂η 2 lim |∇uε | + W (uε ) xn+1 − xn+1 νε,i νε,n ε|∇uε|2 = 0. ε→0 C 2 ε ∂x ∂x n+1 i 2 i=1 Substituting this into (4.3) and applying the Modica inequality (2.10), we can finish the proof. Remark 4.7. Although we shall not use the Caccioppoli type inequality, here we show how to use the stationary condition to derive it. Take a ψ ∈ C0∞ ((−1, 1)) satisfying 0 ≤ ψ ≤ 1, ψ ≡ 1 in (−1/2, 1/2), |ψ ′ | ≤ 3. For any φ ∈ C0∞ (B1 ), take η(x, xn+1 ) = φ(x)2 ψ(xn+1 )2 and replace xn+1 by xn+1 − λ in (4.3), where λ ∈ (−1, 1) is an arbitrary constant. By this choice we get Z ε 1 2 0= |∇uε | + W (uε ) φ2 ψ 2 + 2φ2ψψ ′ (xn+1 − λ) 2 ε C1 n X ∂φ 2 2 2 2 −φ ψ νε,n+1 ε|∇uε| − (xn+1 − λ) 2φψ 2 νε,i νε,n+1 ε|∇uε |2 (4.4) ∂xi i=1 2 − (xn+1 − λ) 2φ2 ψψ ′ νε,n+1 ε|∇uε|2 .

First consider those terms containing ψ ′ (xn+1 ). Since ψ ′ ≡ 0 in B1 × {|xn+1 | < 1/2}, with the help of Proposition 4.4, we get Z 1 ε 1 2 2 |∇uε | + W (uε ) 2φ2 ψψ ′ (xn+1 − λ) − (xn+1 − λ) 2φ2ψψ ′ νε,n+1 ε|∇uε |2 = O(e− Cε ). ε C1 2 Substituting this into (4.4) leads to Z 1 ε 2 2 2 |∇uε | − νε,n+1ε|∇uε | + W (uε ) φ2 ψ 2 2 ε C1 13

(4.5)

=

Z

C1

(xn+1 − λ)

n X

2φψ 2

i=1

∂φ νε,i νε,n+1 ε|∇uε|2 + O(e−c/ε ). ∂xi

By the Cauchy inequality, Z n X ∂φ (xn+1 − λ) 2φψ 2 νε,i νε,n+1 ε|∇uε |2 ∂x i C1 i=1 Z Z n X 1 2 2 2 2 φψ |∇φ|2 ψ 2 (xn+1 − λ)2 ε|∇uε|2 . νε,i ε|∇uε| + 64 ≤ 4 C1 C1 i=1 Substituting this into (4.5) and noting that n X i=1

2 νε,i = 1 − (νε · en+1 )2 ,

and by the Modica inequality (2.10), ε 1 2 |∇uε |2 − νε,n+1 ε|∇uε|2 + W (uε ) ≥ 1 − (νε · en+1 )2 ε|∇uε|2 , 2 ε

we obtain the following Caccioppoli type inequality Z Z 1 2 2 2 2 8 |∇φ|2 ψ 2 (xn+1 − λ)2 ε|∇uε|2 + Ce− Cε . φ ψ 1 − (νε · en+1 ) ε|∇uε| ≤ 2

(4.6)

C1

C1

Remark 4.8. Since we only have a control on 1 − (νε · en+1 )2 , in view of Remark 3.2, νε may be close to en+1 or −en+1 . To exclude one possibility, we need to use the unit density property. A subtle point here is that, without such assumptions, we cannot say that 1 − νε · en+1 (or 1 + νε · en+1 ) is small everywhere. This is related to the possible interface foliation (and consequently the higher multiplicity of the limit varifold V ), see the examples constructed by del Pino-Kowalczyk-Wei-Yang in [9].

5

Lipschitz approximation

Let fε (x) =

Z

1

−1

1 − (νε (x, xn+1 ) · en+1 )2 ε|∇uε (x, xn+1 )|2 dxn+1 .

By Lemma 4.6, fε → 0 in L1 (B1 ). Consider the Hardy-Littlewood maximal function Z −n fε (y)dy. Mfε (x) := sup r r∈(0,1)

14

Br (x)

Given an l > 0, by the weak L1 estimate, there exists a constant C(n) depending only on the dimension n such that Hn ({Mfε > l} ∩ B1 ) ≤

δ2 C(n) kfε kL1 (B1 ) = C(n) ε . λ l

(5.1)

Denote the set B1 \ {Mfε > l} by Wε . Note that since the integrand in (5.1) and hence fε (x) are continuous functions, Wε is an open set. Given b ∈ (0, 1) and l > 0, we say a point X ∈ {|uε | < 1 − b} ∩ C1 is good, if Z −n sup r 1 − (νε · en+1 )2 ε|∇uε|2 dX < l. 0 0, there exist three constants R0 large and τ1 , l2 small so that the following holds. Suppose that uε is a solution of (2.2) in BR0 , where ε ≤ 1, satisfying |uε (0)| ≤ γ, the Modica inequality (2.10) and Z ε 1 −n R0 |∇uε |2 + W (uε ) ≤ (1 + τ1 ) σ0 ωn , ε BR0 2 17

and R0−n

Z

BR′

then

1 − (νε · en+1 )2 ε|∇uε |2 ≤ l2 ,

{uε = uε (0)} ∩ B1 ⊂ δ − neighborhood of Rn ∩ B1 . Proof. Assume that we have a sequence of solutions uk satisfying the assumptions in this lemma, with ε replaced by εk ∈ (0, 1], and Z 1 εk −n |∇uk |2 + W (uεk ) ≤ (1 + τk ) σ0 ωn , (5.7) R0 εk BR0 2 where τk → 0, and R0−n

Z

εk

BR0

2 n X ∂uk ∂xi

i=1

dX → 0,

(5.8)

but there exists Xk = (xk , xn+1,k ) ∈ {uε = 0} ∩ B1 with |xn+1,k | ≥ δ. The constant R0 will be determined below. Without loss of generality, assume that Xk converges to some point X∞ = (x∞ , xn+1,∞ ) ∈ B1 with |xn+1,∞ | ≥ δ. We divide the proof into two cases. Case 1. εk converges to some ε0 > 0 (after subtracting a subsequence). By standard elliptic estimates and Arzel`a-Ascoli theorem, uk converges to a function u∞ in C 2 (BR0 −1 ). Note that we always have |uk | < 1 and hence |u∞ | ≤ 1 in BR0 −1 . u∞ is a solution of (2.2) with ε replaced by ε0 . Since |u∞ (0)| ≤ γ < 1, by the strong maximum principle, |u∞ | < 1 strictly in BR0 −1 . It can be directly checked that u∞ is a solution of (2.2) with ε replaced by ε0 . We still have Z ε0 1 |∇u∞ |2 + W (u∞ ) ≤ σ0 ωn (R0 − 1)n . (5.9) ε0 BR0 −1 2 Since ε0 ≤ 1, we cannot have u∞ ≡ u∞ (0) because otherwise Z ε0 1 1 |∇u∞ |2 + W (u∞ ) ≥ W (u∞ (0))ωn+1 (R0 − 1)n+1 > σ0 ωn (R0 − 1)n , ε0 ε0 BR0 −1 2 if we choose R0 large to satisfy the last inequality. (It depends only the dimension n and W .) Passing to the limit in (5.8) we see R0−n

Z

BR0

ε0

2 n X ∂u∞ ∂xi

i=1

18

dX = 0.

Thus u∞ (X) ≡ u(xn+1 ). Moreover, if R0 is sufficiently large in (5.9), ∂u∞ (X) > 0, ∂xn+1

in B1 .

In particular, u∞ 6= u∞ (0) in B1 \ Rn . However, by the convergence of Xk and uniform convergence of uk , u∞ (X∞ ) = u∞ (0). Because X∞ ∈ B1 \ Rn , this is a a contradiction. Case 2. εk → 0. Let Vk be the varifold associated to uk as defined in Section 4. For any η ∈ C0∞ (BR0 ), let Φ(X, S) = η(X) < Sen+1 , en+1 >∈ C0∞ (BR0 × G(n)). By (5.8), < Vk , Φ >=

Z

η(X)εk

BR′

2 n X ∂uk i=1

∂xi

dX → 0.

Let V∞ be the limit varifold of Vk , which is stationary rectifiable with unit density. Then Z 0 =< V, Φ >= η(X) < T en+1 , en+1 > dkV∞ k, where T P is the weak tangent plane of V at X. This implies that T = Rn kV∞ k a.e.. Thus V∞ = σ0 j i(Tj ) in BR0 /2 × (−R0 /2, R0 /2), where Tj = Rn × {(0, tj )} for some j. By our assumptions, there are at least two components, say T0 and T1 , associated to 0 and X∞ respectively. However, by passing to the limit in (5.7), we obtain kV k(BR0 ) ≤ σ0 ωn R0n = σ0 kT0 k(BR0 ). Thus we cannot have any more components other than T0 , which is a contradiction. Remark 5.7. It will be useful to write the dependence of δ and l2 reversely as δ = c2 (l2 ). This function is a modulus of continuity, i.e. a non-decreasing function satisfying liml2 →0 c2 (l2 ) = 0. For any X0 = (x0 , x0,n+1 ) ∈ Aε and r ∈ (ε, 1/R0 ), by considering u˜ε,r (X) = uε (X0 + rX), the previous lemma implies that {uε = uε (X0 )} ∩ B1/2 (X0 ) \ Bε (X0 ) ⊂ {|xn+1 − x0,n+1 | ≤ c2 (l)|x − x0 |}. 19

(5.10)

Together with Lemma 5.4, this implies that for every t ∈ (−1 + b, 1 − b) and x ∈ Π(Aε ), there exists at most one point in Π−1 (x) ∩ {uε = t}. On the other hand, by Remark 4.5, for each x ∈ B1 , uε (x, 1) > 1 − b,

uε (x, −1) < −1 + b.

Thus there must exist one xn+1 ∈ (−1, 1) such that uε (x, xn+1 ) = t. In conclusion, for any x ∈ Π(Aε ), there exists a unique point (x, xn+1 ) ∈ Π−1 (x) ∩ {uε = t}. This completes the proof of Lemma 5.5. Note that we have assumed that uε > 0 in C1 ∩ {xn+1 > h} and uε < 0 in C1 ∩ {xn+1 < −h} for some h > 0. (This h can be made arbitrarily small as ε → 0.) Then by continuity, for any r ∈ (ε, 1/R0), (5.10) can be improved to {xn+1 − x0,n+1 > c2 (l)|x − x0 |} ∩ B1/2 (X0 ) \ Bε (X0 ) ⊂ {uε > uε (X0 )}. (5.11) When r = ε, we can combine this with Lemma 5.4 to obtain that,

∂uε c(b) (X) ≥ (1 − c1 (b, l))|∇uε (X)| ≥ , ∀ X ∈ Aε . ∂xn+1 ε

(5.12)

In the following we denote the Lipschitz functions by htε for t ∈ (−1 + b, 1 − b). Note that Lemma 5.4 implies that the definition domains of htε can be made to be a common one, Π(Aε ). By (5.12), htε is strictly increasing in t ∈ (−1 + b, 1 − b). In the above construction, htε is only defined on a subset of B1 , but we can extend it to B1 without increasing the Lipschitz constant by letting (see for example [16, Theoerm 3.1.3]) htε (x) := inf htε (y) + c3 (b, l)|y − x|, ∀x ∈ B1 . (5.13) y∈Π(Aε )

Here c3 (b, l) = max{c0 (b, l), c2 (l)}. This extension preserves the monotonicity of htε in t. In Section 7 and 8, b and hence l may be decreased further. Thus it is worthy to note the dependence of these Lipschitz functions on b and l. Remark 5.8. If we decrease l, the definition domain of htε also decreases. But on the common part, these two constructions coincide. If we define two families by choosing two 0 < b1 < b2 < 1, these two families also coincide on (−1 + b2 , 1 − b2 ). Notation: Dε = Π(Aε ). In the following it will be useful to keep in mind that, on {uε = t} ∩ Aε , ∂uε = ∂xn+1

∂htε ∂t

−1

,

∂uε = ∂xi 20

∂htε ∂t

−1

∂htε , 1 ≤ i ≤ n. ∂xi

(5.14)

6

Estimates on htε

First we give an H 1 bound. Lemma 6.1. There exists a constant C independent of ε such that, Z 1−b Z |∇htε |2 dxdt ≤ C(b)δε2 . −1+b

B1

Proof. First since Hn (B1 \ Dε ) ≤ Cδε2 , and |∇htε | ≤ c3 (b, l) in B1 for all t ∈ (−1 + b, 1 − b), we have Z Z 1−b

−1+b

B1 \Dε

|∇htε |2 dxdt ≤ Cδε2

(6.1)

Next, by noting that on Aε , ε|∇uε | ≥ c(b), we have Z 2 1 − (νε · en+1 )2 ε|∇uε |2 dX δε ≥ Aε Z 1−b Z 2 = 1 − (νε · en+1 ) ε|∇uε |dA dt −1+b

{uε =t} 1−b Z

p 1 t 2 1 + |∇hε | dx dt 1− ≥ c(b) 1 + |∇htε |2 −1+b Aε Z 1−b Z t 2 |∇hε | dx dt, ≥ c(b) Z

−1+b

Aε

if we have chosen l so small that the Lipschitz constants of htε , c3 (b, l) ≤ 1/2. With this lemma in hand, we first choose a tε ∈ (−1 + b, 1 − b) such that Z |∇htεε |2 dx ≤ C(b)δε2 , B1

and then take a λε so that the function defined by ¯ ε := 1 htε − λε , h δε ε satisfies

R

B1

(6.2)

¯ ε = 0. Then by the Poincare ienquality h Z

B1

¯ ε (x)2 dx ≤ C(n) h

Z

B1

21

|∇hε (x)|2 dx ≤ C(b).

(6.3)

¯ ε converges weakly Thus we can assume that (after passing to a subsequence of ε → 0), h 1 2 ¯ in H (B1 ) and strongly in L (B1 ). to an h Let ¯ t := 1 ht − λε . h ε δε ε In Dε , −1 ∂uε ∂htε = ≤ C(b)ε, (6.4) 0≤ ∂t ∂xn+1 with a constant C(b) depending only on b. Hence 0 ≤ h1−b − h−1+b ≤ C(b)ε, ε ε

in Dε .

This also holds for x ∈ B1 \ Dε by (5.13). Then for any −1 + b < t1 < t2 < 1 − b, Z 2 htε1 − htε2 ≤ C(b)ε2 .

(6.5)

(6.6)

B1

¯ in L2 (B1 ). ¯ t˜ε still converges to h Because δε ≫ ε, for any sequence of t˜ε ∈ (−1 + b, 1 − b), h ε −1 t 2 n Since δε ∇hε are uniformly bounded in L (B1 ×(−1 + b, 1 −b), R ), we can assume that it converges weakly to some limit in L2 (B1 × (−1 + b, 1 − b), Rn ). By the above discussion, ¯ this limit must be ∇h. ¯ is independent of the choice of b. Hence we have a universal constant By Remark 5.8, h C, which is independent of b and l, such that Z ¯ 2+h ¯ 2 ≤ C. |∇h| (6.7) B1

Concerning the size of λε , we have Lemma 6.2. limε→0 |λε δε | = 0. Proof. Note that λε δε = By Proposition 4.4, lim

Z

sup

B1

ε→0 C ∩{|u |≤1−b} ε 1

htεε .

(6.8)

|xn+1 | = 0.

Thus lim

sup

sup |htε (x)| = 0.

ε→0 t∈(−1+b,1−b) x∈Dε

22

(6.9)

For any x ∈ B1 \ Dε , by Lemma 5.2, 1

2

dist(x, Dε ) ≤ C(n)l− n δεn . Thus since the Lipschitz constant of htε is smaller than c3 (b, l) ≤ 1, sup

sup |htε (x)| ≤

t∈(−1+b,1−b) x∈B1 \Dε

2

sup |htε (x)| + C(n, l)δεn .

sup

t∈(−1+b,1−b) x∈Dε

Combining this with (6.9) we see lim

sup

sup |htε (x)| = 0.

ε→0 t∈(−1+b,1−b) x∈B1

Substituting this into (6.8) we can finish the proof. Next we establish an L2 bound. This can be viewed as a Poincare inequality on the varifold Vε (while the Caccioppoli type inequality is a reverse Poincare inequality). Lemma 6.3. There exists a universal constant C such that Z (xn+1 − λε δε )2 ε|∇uε |2 ≤ Cδε2 .

(6.10)

C3/4

Proof. We divide the proof into two steps. In the following we shall fix two numbers 0 < b2 < b1 < 1 so that W ′′ ≥ κ in (−1, −1 + b1 ) ∪ (1 − b1 , 1). Step 1. Here we give an estimate in the part {|uε | < 1 − b2 } ∩ C1 , i.e. Z (xn+1 − λε δε )2 ε|∇uε |2 ≤ Cδε2 . (6.11) {|uε | 1 − b2 } ∩ C3/4 , Z (xn+1 − λε δε )2 ε|∇uε |2 ≤ Cδε2 .

(6.15)

{|uε |>1−b2 }∩C3/4

Choose a function ζ ∈ C ∞ (R), satisfying    ζ(t) ≡ 1, in {|t| > 1 − b1 }, ζ(t) ≡ 0, in {|t| < 1 − b2 },   ′ |ζ | + |ζ ′′| ≤ C in {1 − b1 ≤ |t| ≤ 1 − b2 }.

We also fix an function η ∈ C0∞ (B1 × {|xn+1 | < 4/3}) so that 0 ≤ η ≤ 1, η ≡ 1 in C3/4 . It can be checked directly that κ (6.16) ∆ ε|∇uε|2 ≥ 2 ε|∇uε |2 , in {|uε | > 1 − b1 }. ε

Multiplying this equation by (xn+1 − λε δε )2 ηζ(uε) and integrating by parts, we obtain Z (xn+1 − λε δε )2 ηζ(uε)ε|∇uε |2 B2 Z ε2 ∆ (xn+1 − λε δε )2 η ζ(uε)ε|∇uε |2 ≤ κ B2 Z ε2 ∂uε + 4 (xn+1 − λε δε ) ηζ ′(uε )ε|∇uε |2 (6.17) κ B2 ∂xn+1 Z ε2 2 (xn+1 − λε δε )2 (∇η · ∇uε ) ζ ′(uε )ε|∇uε |2 + κ B2 Z ε2 ζ ′′(uε )|∇uε |2 + ζ ′(uε )∆uε (xn+1 − λε δε )2 ηε|∇uε|2 . + κ B2

In the right hand side, the first term is bounded by Z ε2 ∆ (xn+1 − λε δε )2 η ζ(uε)ε|∇uε |2 ≤ Cε2 , κ B2 24

(6.18)

because both ∆ (xn+1 − λε δε )2 η and ζ(uε) are bounded by a universal constant. Note that the supports of ζ ′(uε ) and ζ ′′(uε ) belong to {|uε | < 1 − b2 }. By the Cauchy inequality, the second term is bounded by Z ∂uε ε2 4 (xn+1 − λε δε ) ηζ ′(uε )ε|∇uε|2 (6.19) κ B2 ∂xn+1 # 21 2 21 "Z Z ∂uε ε ε|∇uε |2 (xn+1 − λε δε )2 η 2 ε|∇uε |2 ≤ Cε ∂xn+1 {|uε | δε }) ≤ Cδε → 0. For any Xε ∈ Π−1 (Eε ) ∩ Aε , consider vε (X) := uε (Xε + εX),

for X ∈ Bε−1 /2 .

vε is a solution of (2.1). By definition, vε (0) = uε (Xε ) = t ∈ (−1+b, 1−b) because Xε ∈ Aε . 2 As usual we can assume that vε converges to a v∞ in Cloc (Rn+1 ), which is a solution of (2.1) on Rn+1 . By the definition of Hardy-Littlewood maximal function and our choice of Eε , sup

r

0 0, there exists a K(δ) so that the following holds. For any X ∈ {|uε| ≤ γ} ∩ B1 and r ∈ (K(δ)ε, 1), there exists a hyperplane Pr (X) such that distH ({uε = uε (X)} ∩ Br (X), Pr (X) ∩ Br (X)) ≤ δr. By Lemma 4.6, if r ≥ K1 ε (K1 a constant determined by Lemma 4.6), the excess with respect to Pr (X) (with unit normal vector er (X)) E(r; X, uε, Pr (X)) ≤ δ02 .

(10.3)

Then Theorem 3.3 can be stated as Lemma 10.3. If E(r; X, uε, Pr (X)) ≥ K0 r −1 ε, we can find another hyperplane P˜r (X) such that θ θ−n E(θr; X, uε , P˜r (X)) ≤ E(r; X, uε , Pr (X)). 2 ˜ Here one of the unit normal vector of Pr (X), e˜r (X) satisfies 1

k˜ er (X) − er (X)k ≤ CE(r; X, uε , Pr (X)) 2 . 38

With this lemma in hand we can prove Lemma 10.1. Proof of Lemma 10.1. Assume X0 = 0 is the origin. For k ≥ 0, let rk = θk . Define Z −2 2−n 1 − (νε · e)2 ε|∇uε|2 . Ek := minn ε rk e∈S

Brk

Assume that e¯k is a unit vector attaining this minima. As in (10.3), for all rk ≥ K1 ε, Ek ≤ δ02 ε−2 rk2 .

(10.4)

Lemma 10.3 implies that, if Ek ≥ K0 , then Ek+1

θ2 ≤ Ek . 2

(10.5)

Moreover, by the definition of Ek , we always have Ek+1 ≤ θ2−n Ek .

(10.6)

Let k1 be the unique number satisfying θk1 ∈ [K1 ε, K1 θ−1 ε). Now we derive the claimed bound on Ek from (10.4)-(10.6), for k ≤ k1 . Let k0 be the smallest number such that, for all k > k0 , Ek ≤ K0 θ2−n .

(10.7)

If k0 = 0, then for any k > 0 and rk ≥ K1 ε, Z 1 − (νε · e¯k )2 ε|∇uε | ≤ K0 θ2−n ε2 rkn−2 . Brk

This can also be extended to those r ∈ [K1 ε, θ) by choosing a (unique) k so that r ∈ [rk+1 , rk ). Next we assume k0 > 0. By definition, Ek0 > K0 θ2−n . By (10.6), Ek0 −1 ≥ K0 . Then we can apply (10.5) to deduce that Ek0 −1 ≥

2 Ek . θ2 0

In particular, Ek0 −1 ≥ Ek0 ≥ K0 θ2−n .

With this estimate we can repeat the above procedure to obtain that, for all i ∈ [0, k0 ), Ei ≥

2 Ei+1 ≥ K0 θ2−n . θ2 39

This then implies that for all i ≤ k0 , Ei ≤ which implies that for all i ≤ k0 , Z Bri

θ2 2

i

E0 ,

1 − (νε · e¯i )2 ε|∇uε| ≤ δ02 rin+α .

(10.8)

This estimate can also be extended to those r ∈ [rk0 , θ) by choosing an i so that r ∈ [ri+1 , ri ). In conclusion, for r ∈ [rk0 , θ), we have the estimate (10.8), and for r ∈ (K1 ε, rk0 ) we can apply (10.7). These two give (10.1). Next we show that we can choose er (X0 ) independent of r. Lemma 10.4. For any σ > 0, there exist two universal constants K3 and K4 (K4 independent of σ) so that the following holds. For any X0 ∈ {|uε | ≤ γ} ∩ B1 and ball Br (X0 ) with r ∈ (K3 ε, θ), there exists a unit vector e(X0 ) such that Z −n r 1 − (νε · e(X0 ))2 ε|∇uε|2 ≤ σ 2 + K4 τA2 r α . (10.9) Br (X0 )

Here e(X0 ) is independent of r ∈ (K3 ε, θ).

Proof. We keep notations as in the proof of Lemma 10.1. For any r ∈ (K1 ε, θ), combining Remark 3.2 and Lemma B.5, we have Z Z 2 2 1 − (νε · er (X0 ))2 ε|∇uε |2 1 − (νε · e2r (X0 )) ε|∇uε | + Br B2r Z distRPn (νε , er (X0 ))2 + distRPn (νε , e2r (X0 ))2 ε|∇uε|2 ≥ c(n) Br Z 2 ε|∇uε |2 ≥ c(n)distRPn (e2r (X0 ), er (X0 )) Br 2 n

≥ c(n)distRPn (e2r (X0 ), er (X0 )) r .

For k < k0 , this gives α

α

distRPn (ek+1 (X0 ), ek (X0 )) ≤ CτA rk2 = CτA θ 2 k . Summing in i from k to k0 , we see α

α

distRPn (ek0 (X0 ), ek (X0 )) ≤ CτA θ 2 k = CτA rk2 , 40

∀k < k0 .

(10.10)

For k ∈ [k0 , k1 ), we have distRPn (ek+1 (X0 ), ek (X0 )) ≤ Cεrk−1 = Cεθ−k .

(10.11)

Let k2 ≤ k1 be the largest number satisfying ε

C θ−k2 −1 ≤ σ. −1

θ−1

We take K3 := C(σ −1 − 1)θ−1 σ −1 . Summing (10.11) from k to k2 , we get distRPn (ek2 (X0 ), ek (X0 )) ≤ ε

C θ−k2 −1 ≤ σ, −1 θ −1

∀k0 ≤ k ≤ k2 .

(10.12)

In particular, distRPn (ek2 (X0 ), ek0 (X0 )) ≤ σ.

(10.13)

Let e(X0 ) = ek2 (X0 ), by combining (10.10)-(10.13) we obtain, for any k ∈ (0, k2), α

distRPn (ek , e(X0 )) ≤ σ + CτA rk2 . Note that for any k ≥ 0, 1 − (ν · e(X0 ))2 ≤ 1 − (ν · ek )2 + C(n)distRPn (ek , e(X0 ))2 .

Together with (10.1), this gives Z 1 − (νε · e(X0 ))2 ε|∇uε|2 ≤ Cσ 2 + Cδ02 rkn+α . Brk

For any r ∈ (K3 ε, θ), by choosing a k so that r ∈ (rk , rk+1], we then get Z 1 − (νε · e(X0 ))2 ε|∇uε|2 ≤ C σ 2 + δ02 rkn+α ,

(10.14)

Br

for some universal constant C.

The following result will be used in the proof of Lipschitz regularity of {uε = 0}. Corollary 10.5. For any X0 ∈ {uε = 0} ∩ B1 , |e(X0 ) − en+1 | ≤ C (τA + σ) . 41

Proof. By taking r = θ in (10.9), we have Z 1 − (νε · e(X0 ))2 ε|∇uε |2 ≤ C σ 2 + τA2 . Bθ (X0 )

On the other hand, by (9.1) and Lemma 4.6, we also have Z 1 − (νε · en+1 )2 ε|∇uε |2 ≤ CτA2 . Bθ (X0 )

Similar to the proof of the previous lemma, combining these two and using Lemma B.5, we get distRPn (e(X0 ), en+1 )2 Z Z 2 −n 2 −n ≤ θ 1 − (νε · e(X0 )) ε|∇uε| + θ B1/4 (X0 )

2

≤ C σ +

τA2

B1/4 (X0 )

.

1 − (νε · en+1 )2 ε|∇uε |2

Finally, we can fix e(X0 ) so that it points above. Thus the estimate on the distance in RP can be lifted to an estimate in Sn . n

We shall use these results to show that the level sets of uε are Lipschitz graphs in the form of xn+1 = hε (x). However, before proving this we first sketch a direct proof of Theorem 1.1.

11

A direct proof of Theorem 1.1

This section is devoted to a direct proof of Theorem 1.1. In fact, we prove something more. Theorem 11.1. Suppose that u is a smooth solution of (2.1) on Rn+1 , satisfying Z 1 −n lim R |∇u|2 + W (u) ≤ (1 + τA ) ωn σ0 . R→+∞ BR 2 Then there exists a unit vector e such that u(X) ≡ u(e · X).

(11.1)

In the following we will show that if u is a minimizing solution of (2.1) on Rn+1 , where n ≤ 6, then (11.1) is satisfied. Thus Theorem 1.1 is a corollary of this theorem. Since u is an entire solution, by the main result of [17], u satisfies the Modica inequality and hence the monotonicity formula. This monotonicity ensures the existence of the limit in (11.1). It also implies that, for any ball BR (X), Z 1 −n R |∇u|2 + W (u) ≤ (1 + τA ) ωn σ0 . 2 BR (X) 42

With this bound, we can study the asymptotic behavior of u through the scalings uε (X) := u(ε−1 X). As before, by Hutchinson-Tonegawa’s theorem, the varifolds Vε associated to uε converge to a stationary varifold V with integer multiplicity. Furthermore, we claim that Proposition 11.2. V is a cone with respect to the origin 0. Proof. This is because for any R > 0, by the convergence of kVε k and (4.2), R−n kV k(BR ) = lim R−n kVε k(BR ) ε→0 Z 1 ε −n |∇uε |2 + W (uε) = lim R ε→0 ε BR 2 Z −n 1 = lim ε−1 R |∇u|2 + W (u). ε→0 2 Bε−1 R

(11.2)

However, by (11.5) and Proposition 4.1, there exists a constant Θ such that Z 1 n lim ε |∇u|2 + W (u) = ΘRn , ∀R > 0. ε→0 2 Bε−1 R Then by the monotonicity formula for stationary varifold (cf. [16, Theorem 6.3.2]), we deduce that V is a cone. By (11.1) and (11.2), kV k(B1 ) ≤ (1 + τA ) ωn σ0 .

Hence we can apply Allard’s ε-regularity result to deduce that sptkV k ∩ B1/2 is a smooth hypersurface. Then by the previous proposition, sptkV k must be a hyperplane and V is the standard varifold associated to this plane with unit density. This blowing down analysis also implies that, the distance function Φε := gε−1 ◦ uε (see Appendix A) converges to (up to a subsequence of ε → 0) a linear function in the form e · X in Cloc (Rn+1 ), where e is a unit vector. However, this argument does not show the uniqueness of this limit. Different subsequences of ε → 0 may lead to different limits. To obtain the uniqueness of the blowing down limit, we use the following lemma. Lemma 11.3. There exists a universal constant C(n), such that for any ball BR with R ≥ 1, we can find a unit vector eR to satisfy Z 1 − (ν · eR )2 |∇u|2 ≤ C(n)Rn−2 . (11.3) BR

43

The proof is similar to the one of Lemma 10.1, see also the proof of [25, Theorem 2.3]. Note that eR in this theorem may not be unique. In the following we assume that for each R > 1, we have fixed such a vector eR . If n = 1, as R → +∞, since eR are unit vectors, we can take a subsequence of Ri → +∞ so that eRi → e∞ ∈ S1 . Assume e∞ = e2 . Then by taking limit in (11.3), we get for any R > 0, 2 Z ∂u = 0. ∂x1 BR Thus u(x1 , x2 ) ≡ u(x2 ). Now consider the case n ≥ 2. Similar to Lemma 10.4, we also have

Lemma 11.4. There exists a unit vector e∞ and a universal constant C(n) such that Z 1 − (ν · e∞ )2 |∇u|2 ≤ C(n)Rn−2 , ∀R > 1. (11.4) BR

For the blowing down sequence uε , (11.4) implies that Z 1 − (νε · e∞ )2 ε|∇uε|2 ≤ C(n)ε2 . B1

For any η ∈ C0∞ (Rn+1 ), let Φ(X, S) = η(X)2 < Se∞ , e∞ >∈ C0∞ (Rn+1 × G(n)). Then we have 0 = lim < Vε , Φ >=< V, Φ > . ε→0

Thus for kV k a.a. X, the tangent plane of V at X is the hyperplane orthogonal to e∞ . It can be directly checked that V must be the standard varifold associated to this hyperplane. (This can also be seen by noting that we have proved that sptkV k is a hyperplane.) This also implies that, for all ε → 0, in Cloc (Rn+1 ).

Φε → e∞ · X, Without loss of generality, assume e∞ = en+1 . Then by Theorem A.4, for any δ > 0, ∇Φε → en+1 ,

uniformly on B1 ∩ {|xn+1 | > δ}.

By compactness, this still holds true if we replace the base point by any point X0 ∈ {u = 0}. Thus we arrive at Lemma 11.5. For any δ > 0, there exists an L such that, for any X ∈ {|Φ| ≥ L}, |∇Φ(X) − en+1 | ≤ δ. 44

In particular, in {|Φ| > L}, u is increasing along directions in the cone {e : e · en+1 ≥ δ}. Then we can proceed as in [12] to deduce that u is increasing along directions in this cone everywhere in Rn+1 . After letting L → +∞, we deduce that for any unit vector e orthogonal to en+1 , e · ∇u ≥ 0, −e · ∇u ≥ 0, in Rn+1 .

∂u Thus ∂x ≡ 0 in Rn+1 , for all 1 ≤ i ≤ n. This then implies that u depends only on xn+1 . i Finally, by using (11.1), it can be checked directly that we must have u(X) ≡ g(xn+1 +t) for some t ∈ R. Now assume that u is a minimizing solution of (2.1) on Rn+1 , where n ≤ 6. First we can use standard comparison functions to deduce an energy bound.

Lemma 11.6. There exists a constant M, which depends only on the dimension n, such that Z 1 |∇u|2 + W (u) ≤ MRn , (11.5) BR (X) 2 for any ball BR (X). As before we can consider the blowing down sequence uε and the associated varifold Vε . By [15, Theorem 2], its limit varifold V has unit density. In fact, in this case we can prove that sptkV k = ∂Ω, where Ω has minimizing perimeter, see [18]. Moreover, ∂Ω is still a cone. Now by the assumption that the dimension n ≤ 6, ∂Ω must be a hyperplane, see Simons [23]. Then (11.2) gives Z 1 −n lim R |∇u|2 + W (u) = kV k(B1 ) = ωn σ0 . R→+∞ BR 2 That is, u satisfies all of the assumptions in Theorem 11.1. Thus we get Theorem 1.1 by applying Theorem 11.1.

12

The Lipschitz nature of intermediate layers

Now we continue the proof of Theorem 9.1. In this section we first prove that {uε = t} can be represented by a Lipschitz graph in the xn+1 direction. This is the consequence of Corollary 10.5 and the following Lemma 12.2. Before coming to Lemma 12.2, we need the following

45

Lemma 12.1. Let v be a solution of (2.1) in Rn+1 . Assume there exists a constant σ small so that for all r large, Z 1 − (ν · en+1 )2 |∇v|2 ≤ σ 2 r n , (12.1) Br

and

lim r

−n

r→+∞

Z

Br

1 |∇v|2 + W (v) ≤ (1 + τA ) σ0 ωn . 2

(12.2)

Then there exists a unit vector e satisfying |e − en+1 | ≤ C(n)σ,

(12.3)

so that v(X) ≡ v(e · X). Proof. The only thing we need to check is that (12.1) implies (12.3). This can be directly verified by substituting u(X) ≡ g(e · X + t) (t a constant) into (12.1).

¯ > R so that the following Lemma 12.2. For any R > 1 and σ > 0 small, there exists R holds. Let v be a solution of (2.1) in BR¯ , satisfying |v(0)| ≤ γ, the Modica inequality (2.10) and Z 1 −n ¯ |∇v|2 + W (v) ≤ (1 + τA ) σ0 ωn . R BR¯ 2

¯ Suppose that for any r ∈ (R, R), Z Br

1 − (ν · en+1 )2 |∇v|2 ≤ σ 2 r n .

Then by denoting Φ := g −1 ◦ v and assuming that v > 0 when xn+1 ≫ 0, 1 sup |∇Φ − en+1 | ≤ . 4 BR Proof. Assume by the contrary, there exists an R > 0, a sequence of Ri → +∞ and a sequence of solutions vi to (2.1) defined on BRi , satisfying |vi (0)| ≤ γ, the Modica inequality (2.10) and Z 1 −n Ri |∇vi |2 + W (vi ) ≤ (1 + τA ) σ0 ωn , (12.4) BRi 2 and

But

Z

Br

1 − (νi · en+1 )2 |∇vi |2 ≤ σ 2 r n , 1 sup |∇Φi − en+1 | > . 4 BR 46

∀r ∈ (R, Ri ).

(12.5)

(12.6)

Then we can assume vi converges to a smooth solution v∞ on any compact set of Rn+1 . By the monotonicity formula and (12.4), for any r > 0, Z 1 −n |∇v∞ |2 + W (v∞ ) ≤ (1 + τA ) σ0 ωn . r 2 Br Passing to the limit in (12.5) we also have Z 1 − (ν∞ · en+1 )2 |∇v∞ |2 ≤ σ 2 r n , Br

∀r > R.

Then by the previous lemma (noting that v∞ (0) = limi→+∞ vi (0) and v∞ > γ above Rn ), v∞ (X) ≡ g(e · X + g −1(v∞ (0))) for some unit vector e satisfying 1 |e − en+1 | ≤ . 8 Then Φi (X) := g −1 ◦ vi (X) → e · X

In particular, for all i large,

in C 1 (BR ).

1 sup |∇Φi − e| ≤ . 4 BR

However, this contradicts (12.6). We can apply this lemma to v(X) := uε (X0 + εX), where uε is as in Theorem 9.1 and X0 ∈ {uε = uε (0)} ∩ B1 . Combined with Corollary 10.5 (provided ε and σ are sufficiently small), this results in Lemma 12.3. For any X0 ∈ {uε = uε (0)} ∩ B1 , ∇uε 6= 0 in BK3 ε (X0 ) and 1 |νε − en+1 | ≤ . 2 Here we only need to note that, at the beginning we have assumed that uε > uε (0) in {xn+1 > 1/2} ∩ B1 . Then by Lemma 10.2, uε < uε (0) in {xn+1 < −1/2} ∩ B1 . Next by combining Lemma 10.2 and Lemma 10.4, for any r ≥ K4 ε, r {(X − X0 ) · e(X0 ) ≥ } ∩ Br (X0 ) ⊂ {uε > uε (0)}, 2

(12.7)

thanks to the continuous dependence on r. (12.7) also implies that for any x ∈ B1 , there exists a unique xn+1 ∈ (−1, 1) so that (x, xn+1 ) ∈ {uε = uε (0)}. Combined with the previous lemma, this then implies that {uε = uε (0)} ∩ B1 = {xn+1 = hε (x)}, 47

x ∈ B1 .

Here hε is a function with its Lipschitz constant bounded by 4. (This constant can be made as small as possible by decreasing ε, τA and σ.) To complete the proof of Theorem 9.1, we directly apply the main result in [6]. Note that instead of the minimizing condition assumed in that paper, with our assumption (9.1) the argument still goes through.

A

A distance type function

In this appendix uε always denotes a solution of (2.3), satisfying the Modica inequality (2.10). Here we introduce a distance type function associated to uε and study its properties and convergence as ε → 0. This is perhaps well known (see for example [11] for the parabolic Allen-Cahn case). However we do not find an exact reference, so we include some details here. Recall that gε (t) := g(ε−1t) is a one dimensional solution of (2.2). Define Φε (X) = gε−1 (uε (X)) . It satisfies W ′ (g(t))

where f (t) := √

2W (g(t))

− ε∆Φε = f (ε−1Φε )(1 − |∇Φε |2 ),

(A.1)

. Note that f (t) < 0 is C 2 in a neighborhood of 1 (and f (t) > 0,

C 2 in a neighborhood of −1). The following result is a consequence of the Modica inequality. Proposition A.1. |∇Φε | ≤ 1. Proof. Since uε = gε (Φε ), Thus |∇Φε | ≤ 1 is equivalent to

|∇uε | = gε′ (Φε )|∇Φε |. |∇uε |2 ≤ gε′ (Φε )2 .

The first integral for gε is

ε ′ 2 1 (g ) = W (gε ). 2 ε ε Then the final equivalent statement is exactly the Modica inequality for uε . By using (A.1), the limit of Φε , Φ0 can be characterized as a viscosity solution of the eikonal equation: In {Φ0 > 0}, Φ0 is a viscosity solution of |∇Φ0 |2 − 1 = 0. 48

In {Φ0 < 0}, Φ0 is a viscosity solution of 1 − |∇Φ0 |2 = 0. . This is similar to the vanishing viscosity method (see for example Fleming-Souganidis [13]). However, here we would like to give a direct proof. Proposition A.2. For any δ > 0, there exists ε♯ , τ♯ > 0, if uε is a solution of (2.2) in B2 with ε ≤ ε♯ , satisfying |uε (0)| ≤ γ and (9.1) with τA replaced by τ♯ , then there exists a set Ω ⊂ B1 , with 0 ∈ ∂Ω and ∂Ω being a smooth minimal hypersurface, such that sup |Φε − d∂Ω | ≤ δ.

(A.2)

B1

Here d∂Ω is the signed distance function to ∂Ω, which is positive in Ω. Proof. We fix τ♯ so that it fulfills the assumption in Allard’s ε-regularity theorem. By Hutchinson-Tonegawa [15], if we have a sequence of uε with ε → 0, satisfying all of the assumptions in this proposition, then the associated varifold Vε converges to a stationary rectifiable varifold V with integer multiplicity. Moreover, (9.1) and the monotonicity formula implies that 2−n kV k(B2 ) ≤ (1 + τ♯ ) σ0 ωn .

Then Allard’s regularity theorem (cf. [1] and [16, Theorem 6.5.1]) implies that sptkV k∩B3/2 is a smooth hypersurface and V ∩B3/2 is the standard varifold associated to this hypersurface with density one. This hypersurface also divides B1 into two parts, say Ω and B1 \ Ω. Moreover, as in Remark 4.5, uε converges to 1 uniformly in any compact set of Ω, and to −1 uniformly in any compact set of Ωc . Thus if ε♯ is small enough, we can assume that there exists a set Ω with 0 ∈ ∂Ω and ∂Ω being a smooth minimal hypersurface, such that δ distH ({uε > 0} ∩ B1 , Ω ∩ B1 ) ≤ . 8 In particular,

δ sup dist{uε =0} − d∂Ω ≤ . 8 B1

By Proposition A.1, in {uε > 0} ∩ B1 ,

Φε (X) ≤ dist{uε =0} (X) ≤ dist∂Ω (X) + δ. Similarly, in {X : |dist∂Ω (X)| ≤ 4δ }, |Φε | ≤ δ/2. Thus in this part, Φε − d∂Ω ≤ Φε + d∂Ω ≤ δ. 49

Thus to prove (A.2), we only need to show that if X ∈ Ω ∩ {X : dist∂Ω (X) ≥ 4δ }, where dist{uε =0} ≥ δ/8, then δ Φε (X) ≥ dist{uε =0} (X) − . 16 However, this can be proved directly by constructing a comparison function in Bdist{u =0} (X)− δ (X), ε 16 by noting that uε is close to 1 in this ball, if we have chosen ε♯ sufficiently small (compared to δ). In the following we assume that as ε → 0, Φε converges to a distance function Φ0 uniformly. Now we present a fact about the C 1 convergence of Φε near a C 1 point of Φ0 . First we establish the uniform semi-concavity of Φε . Lemma A.3. Let Φε > 1/2, |∇Φε | ≤ 1, satisfy (A.1) in B1 . Then ∇2 Φε (0) ≤ C, where C is a constant depending only on the dimension n. The constant 1/2 is not essential here. It can be replaced by any positive constant. Proof. We shall work in the setting where ε = 1 and the ball is BR . Here ε = R−1 . For simplicity, all subscripts will be dropped. Take a unit vector ξ. By directly differentiating (A.1) in the direction ξ, we get ′

2

−∆Φξ = f (Φ)(1 − |∇Φ| )Φξ − 2f (Φ)

n+1 X

Φk Φξk ,

k=1

−∆Φξξ = f ′ (Φ)(1 − |∇Φ|2 )Φξξ + f ′′ (Φ)(1 − |∇Φ|2 )Φ2ξ ′

−4f (Φ)

n+1 X k=1

Φk Φξk Φξ − 2f (Φ)

n+1 X k=1

Φk Φξξk − 2f (Φ)

n+1 X

Φ2ξk .

k=1

Take an η ∈ C0∞ (BR/2 ) such that η ≡ 1 in BR/4 , 0 ≤ η ≤ 1, |∇η| ≤ 8R−1 and η −1 |∇η|2 + |∆η| ≤ 100R−2. Denote w := ηΦξξ . Since w = 0 on ∂BR/2 , it attains its maxima at an interior point X0 , where ∇w = η∇Φξξ + Φξξ ∇η = 0, 0 ≥ ∆w = ∆Φξξ η + 2∇Φξξ ∇η + Φξξ ∆η ≥ −f ′ (Φ)(1 − |∇Φ|2 )w − f ′′ (Φ)(1 − |∇Φ|2 )Φ2ξ η 50

(A.3)

′

+4f (Φ)

n+1 X

Φk Φξk Φξ η + 2f (Φ)

n+1 X

Φk Φξξk η + 2f (Φ)η

k=1

k=1

−1

+2∇Φξξ ∇η + wη ∆η.

n+1 X

Φ2ξk

k=1

Substituting (A.3) into this, and applying the Cauchy inequality to the third term, we obtain f ′ (Φ)2 |∇Φ|2 Φ2ξ η + f ′′ (Φ)(1 − |∇Φ|2 )Φ2ξ η f (Φ) ≥ −f ′ (Φ)(1 − |∇Φ|2 )w − 2f (Φ)∇Φ∇ηη −1 w + f (Φ)η −1 w 2 + wη −1∆η − 2wη −2|∇η|2. 4

This implies that w(X0) ≤

f ′ (Φ) 1 2 −1 (1 − |∇Φ|2 )η + 2|∇Φ||∇η| + |∆η| + η |∇η|2 f (Φ) f (Φ) f (Φ) ′ 2 21 1 f (Φ) 2 2 ′′ 2 2 + 4 |∇Φ| Φξ η + f (Φ)(1 − |∇Φ| )Φξ η f (Φ) f (Φ)

(A.4)

Since Φ ≥ R/2 in BR/2 , by the definition of f and some standard estimates on g(t), f (Φ) > c

in BR/2 ,

|f ′ (Φ)| + |f ′′ (Φ)| ≤ Ce−cR

in BR/2 .

Substituting these into (A.4) and noting that |∇Φ| ≤ 1 and the condition on η, we get sup Φξξ ≤ w(x0 ) ≤ CR−1 .

BR/4

Rescaling back we get the claimed estimate. Theorem A.4. Assume that Φε converges to Φ0 in C 0 (Ω), where Ω ⊂ Rn+1 is an open set 1 and Φ0 > 0 in Ω. If Φ0 ∈ C 1 (Ω), then Φε converges to Φ0 in Cloc (Ω). Proof. Fix an open set Ω0 ⊂⊂ Ω. Take an arbitrary sequence Xε ∈ Ω0 such that Xε → X0 ∈ Ω0 as ε → 0. By the uniform semi-concavity of Φε in Ω0 , there exists a constant C(Ω0 ) such that, for all ε > 0 ˜ ε (X) := Φε (X) − C(Ω0 )|X − Xε |2 Φ are concave in Ω0 . In particular, for any unit vector e and h < dist(Ω0 , ∂Ω), ˜ ε (Xε + he) ≤ Φ ˜ ε (Xε ) + h∇Φε (Xε )e. Φ 51

Because |∇Φε (Xε )| ≤ 1, we can assume that ∇Φε (Xε ) converge to a vector ξ. By the uniform convergence of Φε in Ω, in the limit we have Φ0 (X0 + he) ≤ Φ0 (X0 ) + h (ξ · e) +

C(Ω0 )h2 , ∀h > 0. 2

Since Φ0 is differentiable at X0 , letting h → 0, we see ξ = ∇Φ0 (X0 ).

B

Several technical results

Here we collect some technical results used in this paper. The first one is an exponential decay estimate. This has been used in many places and can be proved by various methods, so here we only state the result. Lemma B.1. If in the ball B2R (0), u ∈ C 2 satisfies ( ∆u ≥ Mu, 0 ≤ u ≤ 1, then

(B.1)

1

sup u ≤ C1 (n)e−C2 (n)RM 2 ,

BR (0)

where C1 (n) and C2 (n) depend on the dimension n only. The next one gives a control of the discrepancy using the excess. Lemma B.2. Given M > 0, for any τ > 0 there exists a δ > 0 so that the following holds. Suppose that u is a solution of (2.1) in B2 , satisfying Z 1 |∇u|2 + W (u) ≤ M, B2 2 Z 1 − (ν · en+1 )2 |∇u|2 ≤ δ. B2

Then

1 W (u) − |∇u|2 ≤ τ. 2 B1

Z

52

Proof. This can be proved by a contradiction argument. We need to use the fact that for any one dimensional solution, i.e. a solution u satisfying Z 1 − (ν · en+1 )2 |∇u|2 = 0, B2

we have the first integral (2.7), that is,

1 W (u) = |∇u|2 . 2 The following result says the energy ε|∇uε |2 is mostly concentrated on the transition part {|uε | ≤ 1 − b}.

Lemma B.3. Let uε be a solution of (2.2) defined in B2 , satisfying Z ε 1 |∇uε |2 + W (uε ) ≤ M. ε B2 2 For any δ > 0, there exists a b > 0 such that Z

{|uε |>1−b}∩B1

ε|∇uε |2 ≤ δ.

This is essentially [15, Proposition 5.1]. We just need to note that by the Modica inequality, we can bound ε|∇uε|2 by ε−1 W (uε). In fact we can give a different proof, by using the method in the proof of Corollary 6.4. More precisely, we have Lemma B.4. Let uε be as in the previous lemma. For any 1 ≤ i ≤ n + 1 and δ > 0, there exists a b > 0 such that, 2 2 Z Z ∂uε ∂uε ε ε ≤δ + Cε2 . ∂xi ∂xi {|uε |>1−b}∩B1 {|uε | 0, there exist two constants c(M) and R(M) so that the following holds. Assume u is a solution of (2.2) in BR where R ≥ 2R(M), satisfying |u(0)| ≤ γ, the Modica inequality and the energy bound Z 1 |∇u|2 + W (u) ≤ MRn . BR 2 Then for any r ∈ [1, R/2],

Z

Br

|∇u|2 ≥ c(M)r n . 53

(B.2)

Proof. We first prove that, under the assumptions of this lemma (with a different constant R1 (M)), Z |∇u|2 ≥ c(M)Rn , (B.3) BR/2

if R ≥ R1 (M). Assume this is not true, that is, the claimed R1 (M) does not exist. Then there exists an M > 0 and a sequence of ui , which are solutions of (2.2) in BRi where Ri → +∞, satisfying |ui (0)| ≤ γ, the Modica inequality and the energy bound Z 1 |∇ui |2 + W (ui ) ≤ MRin , BRi 2 but Ri−n

Z

BRi /2

|∇u|2 → 0.

Let εi = Ri−1 and uεi (X) := ui (Ri X). Then Z 1 εi |∇uεi |2 + W (uεi ) ≤ M, εi B1 2 Z εi |∇uεi |2 → 0.

(B.4)

(B.5)

B1/2

By the main result in [15], εi |∇ui |2 dX ⇀ µ as measures, where µ is a positive Radon measure (in fact, the weight measure associated to the limit varifold, as in Section 4). Moreover, uεi → ±1 locally uniformly outside sptµ. However, (B.5) obviously implies that µ(B1/2 ) = 0. Thus for all i large, |uεi | > γ in B1/2 . This contradicts our assumption that |uεi (0)| ≤ γ and proves (B.3). By the monotonicity formula (recall that we have assumed the validation of the Modica inequality), for any r ∈ (1, R) Z 1 |∇u|2 + W (u) ≤ Mr n . (B.6) 2 Br Thus the above discussion covers the case r ∈ [R1 (M), R] in (B.2), that is, (B.3) holds for every r ∈ [R1 (M), R]. For the remaining case, we only need to note that it is impossible to have u ≡ u(0), because otherwise Z 1 |∇u|2 + W (u) = ωn+1 W (u(0))r n+1 ≥ Mr n , 2 Br 54

provided r ≥

M . ωn+1 (inf s∈[−γ,γ] W (s))

Then it can be directly verified that Z

B1

by using (B.6) with r =

|∇u|2 ≥ c(M),

M . ωn+1 (inf s∈[−γ,γ] W (s))

Choosing R(M) = max{R1 (M), ω

M n+1

(inf s∈[−γ,γ] W (s))

} we can finish the proof.

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