PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS

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1 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS OVIDIU SAVIN Abstract. We consider local minimizers of the Ginzburg-Landau energy functional 2 u ( u2 ) 2 dx and prove that, if the level set is included in a flat cylinder then, in the interior, it is included in a flatter cylinder. As a consequence we prove a conjecture of De Giorgi which states that level sets of global solutions of u = u 3 u such that u, n u >, lim x n ± u(x, x n ) = ± are hyperplanes in dimension n 8.. Introduction In this paper we establish further properties of phase transitions that are similar to the properties of sets with minimal perimeter. The Ginzburg-Landau model of phase transitions leads to considerations of local minimizers for the energy functional () J(u, Ω) = If u is a local minimizer then (2) u = u 3 u. Ω 2 u ( u2 ) 2 dx, u. We explain below some analogies between the theory of phase transitions and the theory of minimal surfaces. The rescalings u ε (x) = u(ε x) are local minimizers for the ε-energy functional ε J ε (u ε ) = 2 u ε 2 + 4ε ( u2 ε) 2 dx. In [6] Modica proved that as ε, u ε has a subsequence (3) u εk χ E χ E c in L loc

2 2 OVIDIU SAVIN where E is a set with minimal perimeter. In [8] Caffarelli and Cordoba proved a uniform density estimate for the level sets of local minimizers u ε of J ε. Suppose u ε () =, then {u ε > } B δ B δ C for ε δ, C > universal. In particular, this implies that in (3), the level sets {u εk = λ} converge uniformly on compact sets to E. In [9] Modica proved monotonicity of the energy functional, i.e. J(u, B R )R n increases with R. Let us recall some facts about minimal surfaces (see for example Giusti [4]). Suppose that E is a set with minimal perimeter in Ω and E. Then ) Flatness implies regularity, i.e if Ω = { x < } { x n < }, E { x n < ε}, and ε ε, ε small universal, then E is analytic in { x < /2}. The proof uses an improvement of flatness lemma due to De Giorgi (see chapters 6, 7, 8 from Giusti [4]). More precisely, one can show that, possibly in a different system of coordinates, E can be trapped in a flatter cylinder { y η 2 } E { y n εη }, with < η < η 2 universal. This implies E is C,α, and therefore analytic by the elliptic regularity theory. 2) If Ω = R n, and n 7 then E is a hyperplane. 3) If Ω = R n and n = 8 then there exists nonhyperplane minimal sets, for example Simons cone x 2 + x2 2 + x2 3 + x2 4 < x2 5 + x2 6 + x2 7 + x2 8. If, in addition, we assume that E is a graph in some direction, then E is a hyperplane. 4) If Ω = R n and n 9 then there exists nonhyperplane minimal graphs (see [6]). 5) If Ω = R n and E is a graph in the e n direction that has at most linear growth at then E is a hyperplane. It is natural to ask if some of these properties hold for level sets of local minimizers of (), or solutions of (2). In connection to 3) above De Giorgi made the following conjecture in []: Let u C 2 (R n ) be a solution of u = u 3 u,

3 such that PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 3 u, n u > in whole R n. Is it true that all the level sets of u are hyperplanes, at least if n 8? The conjecture was proved for n = 2 by Ghoussoub and Gui in [2] and for n = 3 by Ambrosio and Cabre in [2]. Barlow, Bass and Gui proved in [3] that monotonic solutions in R n with Lipschitz level sets are planar in all dimensions. The main result of this paper is an improvement of flatness theorem for level sets of local minimizers (Theorem 2.). More precisely, if u is a local minimizer of () and {u = } is included in a flat cylinder { x < l} { x n < θ} with θ, l large and θl small then, {u = } is included, possibly in a different system of coordinates, in a flatter cylinder { x < η 2 l} { x n < η θ} with η 2 > η > universal. If {u εk = } converges uniformly on compact sets to a hyperplane then, one can apply Theorem 2. and conclude that {u = } is included in flatter and flatter cylinders, therefore it is a hyperplane. This fact allows us to extend some of the minimal surfaces properties listed above to level sets of local minimizers of (). In particular, we prove the weak form of De Giorgi s conjecture, i.e we also assume that (4) lim x n ± u(x, x n ) = ±. The approach of Modica to study local minimizers of J ε uses variational techniques and the notion of Γ convergence. More precisely, by coarea formula, one has J ε (u ε, Ω) ( u 2 ε ) u ε dx = 2 Ω = ( s 2 )H n ({u ε = s} Ω)ds. 2 Heuristically, we minimize J ε (u ε, Ω) if, in the interior of Ω, we take the level sets {u ε = s} to be (almost) minimal and (5) u ε = 2ε ( u 2 ε ). Notice that, if Γ is a smooth surface then (6) u ε (x) = tanh d Γ(x) 2ε satisfies (5), where d Γ represents the signed distance to the surface Γ.

4 4 OVIDIU SAVIN In this paper we use the method of subsolutions and supersolutions together with the sliding method. Suppose that Γ is a smooth surface and consider the function given by (6). Then ε ( u 2 ε ) 2 ε 2 u ε = u 3 ε u ε d Γ κ i where κ i represent the principal curvatures of Γ at the point where the distance is realized. Heuristically, if Γ has positive (negative) mean curvature then we can find a supersolution (subsolution) whose level set is Γ. In a forthcoming paper we use the same techniques to prove similar results for solutions of F(D 2 u) = f(u), u C 2 (R n ), u, n u > where F is uniformly elliptic, and F, f are such that there exists a one dimensional solution g which solves the equation in all directions, i.e. F(D 2 g(x ν) = f(g(x ν)), ν R n, ν =. 2. Main results Consider the more general energy functional (7) J(u, Ω) = 2 u 2 + h (u)dx, u with Ω h C 2 [, ], h ( ) = h () =, h > on (, ) h ( ) = h () =, h ( ) >, h () >. We say that u is a local minimizer in Ω if, for every open set A Ω relatively compact in Ω. J(u, A) J(u + v, A), A local minimizer of (7) satisfies (8) u = h (u), u. κ i v H (A). Our goal is to prove the following theorem for flat level sets of u. Theorem 2.. Improvement of flatness Let u be a local minimizer of (7) in { x < l} { x n < l}, and assume that the level set {u = } stays in the flat cylinder and contains the point. { x < l} { x n < θ}.

5 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 5 Then there exist small constants < η < η 2 < depending only on n such that: Given θ > there exists ε (θ ) > depending on n, h and θ such that if θ l ε (θ ), θ θ then {u = } ({ π ξ x < η 2 l} { x ξ < η 2 l}) is included in a cylinder { π ξ x < η 2 l} { x ξ < η θ} for some unit vector ξ (π ξ denotes the projection along ξ). We prove Theorem 2. by compactness from the following Harnack inequality for flat level sets of minimizers Theorem 2.2. Let u be a local minimizer of J in the cylinder { x < l} { x n < l} and assume that {u = } { x n < θ}, u() =. There exists a small universal constant η > depending on n and h such that: Given θ > there exists ε (θ ) > depending on n, h and θ, such that if θl ε (θ ), θ θ, then {u = } { x < η l} { x n < ( η )θ}. As a consequence of Theorem 2. we prove the following theorems. Theorem 2.3. Suppose that u is a local minimizer of J in R n, and n 7. Then the level sets of u are hyperplanes. It is known (see [5]) that monotone solutions of (8) satisfying (4) are local minimizers. Theorem 2.4. Let u C 2 (R n ) be a solution of (9) u = h (u), such that () u, n u >, lim x n ± u(x, x n ) = ±. a) If n 8 then the level sets of u are hyperplanes. b) If the level set has at most linear growth at then the level sets of u are hyperplanes.

6 6 OVIDIU SAVIN The paper is organized as follows. In section 3 we prove Theorems 2.3 and 2.4 assuming Theorem 2.. In section 4 we introduce some notation and preliminaries. In section 5 we show that the level set of u satisfies in some weak viscosity sense a mean curvature equation at large scale. In section 6 we show that Theorem 2.2 implies Theorem 2. by a compactness argument. In the remaining part of the paper we prove Theorem 2.2. The proof uses some ideas of Caffarelli and Cordoba from a paper about regularity of minimal surfaces (see [9]). Next we explain the strategy of its proof. Let g denote the one dimensional solution of (8), g () =, and suppose that at one point {u = } is close to x n = θ. Then, using a family of sliding surfaces (see section 7), we prove that the graph of u is close in the e n direction to the graph of g (x n + θ) at points that project along e n in sets of positive measure (section 8). Using an iteration lemma we show that these sets almost fill in measure the strip {(x,, x n+ ) x n+ /2} (section 9). From this we obtain a contradiction with the fact that u is a local minimizer and u() = (section ). 3. Proof of Theorems 2.3 and 2.4 In this section we use Theorem 2. to prove Theorems 2.3 and 2.4. Let Ω R n be an open set and E be a measurable set. The perimeter of E in Ω is defined as P Ω (E) = sup div g dx, where the supremum is taken over all vector fields g C (Ω) with g. We say that E is a set with minimal perimeter in Ω if, for every open set A Ω, relatively compact in Ω E P A (E) P A (F), whenever E and F coincide outside a compact set included in A. We introduce the rescaled energies, ε () J ε (v, Ω) := 2 v 2 + ε h (v)dx. If u is a local minimizer of J(u, Ω), then the rescalings Ω u ε (x) = u( x ε ), are local minimizers for J ε (, εω), and J ε (u ε, εω) = ε n J(u, Ω).

7 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 7 Now we state two theorems that we mentioned in the introduction. The first theorem was proved by Modica in [6]. Theorem 3.. Let u k be a sequence of local minimizers for the energies J εk (, Ω) with ε k. There exists a subsequence u km such that u km χ E χ E c in L loc (Ω) where E is a set with minimal perimeter in Ω. Moreover, if A is an open set, relatively compact in Ω, such that Dχ E =, then A (2) lim m J ε km (u km, A) = P A (E) 2h (s)ds. The second theorem was proved by Caffarelli and Cordoba in [8]. Theorem 3.2. Given α >, β <, if u is a minimizer of J in B R and u() α, then {u > β} B R CR n for R R (α, β), where C is a constant depending on n and h. Next we use Theorem 2. to prove the following lemma. Lemma 3.3. Let u be a local minimizer of J in R n with u() =. Suppose that there exist sequences of positive numbers θ k, l k and unit vectors ξ k with l k, θ k l k such that {u = } ({ π ξk x < l k } { x ξ k < l k }) { x ξ k < θ k }. Then the level set is a hyperplane. Proof: Fix θ >, and choose k large such that θ k l k ε ε (θ ). If θ k θ then we apply Theorem 2. and obtain {u = } is trapped in a flatter cylinder. We apply Theorem 2. repeatedly till the height of the cylinder becomes less than θ. In some system of coordinates we obtain {u = } ({ y < l k} { y n < l k}) { y n θ k}, with θ θ k η θ and θ k l k θ k l k ε, hence l k ε η θ. We let ε and obtain {u = } is included in an infinite strip of width θ. The lemma is proved since θ is arbitrary. Proof of Theorem 2.3

8 8 OVIDIU SAVIN The rescalings u ε (x) = u(ε x) are local minimizers for J ε in R n. From Theorem 3., there exists a sequence ε k such that (3) u εk χ E χ E c in L loc (Rn ) with E a set with minimal perimeter. Claim: {u εk = } converges uniformly on compact sets to E. Assume not, then there exist δ >, z R n, and points x k {u εk = } B(z, δ) with, say B(z, 2δ) E. By Theorem 3.2, the set {u εk < } has uniform density in B(z, 2δ) for ε k small, which contradicts (3). Since E is a minimal surface in R n, n 7, and E, we conclude that E is a hyperplane going through the origin. This implies {u εk = } B { x n δ k }, with δ k. Rescaling back we find that u satisfies the hypothesis of lemma 3.3 and the theorem is proved. Proof of Theorem 2.4 First we prove that a function u satisfying (9), () is a local minimizer in R n. For this, it suffices to show that in B R, u is the unique solution of Since v = h (v), v <, v = u on B R. lim x u(x, x n ) =, n we conclude that the graph of u(x, x n + t) is above the graph of v for large t. We slide this graph in the e n direction till we touch v for the first time. From the Strong Maximum Principle we find that the first touching point occurs on B R. Since u is strictly increasing in the e n direction, we can slide the graph of u(x, x n + t) till it coincides with the graph of u, hence u v. Similarly we obtain u v which proves that u is a local minimizer in R n. Assume u() = and define u ε (x) = u(ε x). Again we find that (3) holds for some sequence ε k. Moreover, u n > implies E c is a subgraph, hence E is a quasi-solution in the e n direction (see chapters 6, 7 in Giusti [4]). In both cases a) and b) one has E is a hyperplane and the theorem follows from lemma 3.3.

9 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 9 4. Preliminaries First we introduce some notation. Notation (e,.., e n, e n+ ) the Euclidean orthonormal basis in R n+ X = (x, x n+ ) = (x, x n, x n+ ) = (x, x 2,.., x n, x n, x n+ ) R n+ X R n+, x R n, x R n, x n+ < B(x, r) the ball of center x and radius r in R n B(X, r) the ball of center X and radius r in R n+ graph u = {(x, u(x)), x R n } d Γ the signed distance to the surface Γ ν a vector in R n+, ξ a vector in R n (ν, ν 2 ) the angle between the vectors ν and ν 2 π ν X = X (X ν)ν the projection along ν π i the projection along e i P ν the hyperplane perpendicular to ν going through the origin P i the hyperplane perpendicular to e i going through the origin Constants depending on n, h are called universal and we denote them by C i, c i, C i, c i ( C i, c i are constants that we use throughout the paper). Preliminaries In the proof we find many times inequalities involving a strictly increasing function g, and its derivatives g, g. In this cases we consider s = g as the new variable and we define a new function h(s) = 2 (dg dt )2. We obtain g = dg dt = 2h, g = d2 g dt = d dh 2h = 2 dt ds = h and now the inequality involves only h and h. We can reconstruct g from h (up to a translation) since H(s) := In particular we define H (s) := s s 2h(ζ) dζ = g (s) g (). 2h (ζ) dζ, and we find g (t) = h (g (t)), thus, g is a one dimensional solution of (8). g (t) := H (t)

10 OVIDIU SAVIN 5. The limiting equation In this section we prove the following Proposition 5.. The limiting equation Let u be a local minimizer of J and assume that u() =. For some δ > small, we consider the surfaces Γ = {x n = P(x ) := 2 x T Mx }, (4) M M (n ) (n ), P = tr M > δ M, M δ. There exists σ (δ ) > small, such that if ε σ (δ ) then Γ cannot touch from below {u ε = } at in a δ ( P) 2ε 2 neighborhood. By Γ touches from below {u ε = } at in a δ ( P) 2ε 2 neighborhood we understand {u ε = } {x n < P(x )} { x < δ ( P) 2 ε 2 } =. Proposition 5. says that {u ε = } satisfies a mean curvature equation in some weak viscosity sense in which we have to specify the size of the neighborhood around the touching point. The size of the neighborhood depends on the polynomial P and ε. If P is fixed and ε then the radius of the neighborhood converges to. In particular, if {u ε = } converges uniformly to a surface, then this surface satisfies in the viscosity sense a mean curvature equation. One way to interpret the above proposition is the following: Suppose that P has positive mean curvature and let δ be small such that (4) holds. Consider a spherical neighborhood around such that P separates at just one point at a distance δ ε from x n =. If r denotes the radius of this neighborhood then, M r 2 2δ ε r 2 δ M ε δ 2 ( P) ε. Hence, if ε < σ (δ ) then P cannot touch from below {u ε = } at in the r neighborhood. We are going to prove the following version of Proposition 5.. Lemma 5.2. Let u be a local minimizer of J in { x < l} { x n < l} and assume that u() = and u < below the surface Γ := {x n = P (x ) = θ l 2 2 x T M x + θ l ξ x }, M < δ, ξ < δ for some small δ >. There exists σ(δ) > small, such that if θl σ(δ), θ δ,

11 then PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS tr M δ. Next we show that Proposition 5. follows from lemma 5.2. Assume by contradiction that for some ε σ (δ ) := σ 2 ( δ2 4 ) Γ touches {u ε = } from below at in a δ ( P) 2ε 2 neighborhood. By rescaling we find that {x n = ε 2 x T Mx } touches from below {u = } at in a δ ( P) 2ε 2 neighborhood. We apply lemma 5.2 with l = δ (tr M) 2 ε δ 2, δ = θ = 2 2 4, M = (tr M) M, ξ = thus, Γ = {x n = ε 2 x T Mx }. Since M = (tr M) M < δ < δ, θl = δ 2 (tr δ 2 M) 2 ε 2 < ε 2 σ( 4 ) = σ(δ) we conclude δ tr M = which is a contradiction. Before we prove lemma 5.2 we need to introduce a comparison function. Using this function and the fact that {u < } below Γ, we are able to bound u by above. Lemma 5.3. (Comparison function) For l > large, there exists an increasing function g l supported inside (, l/2), g l () =, g l (s) is constant for s l/2, such that the rotation surface Ψ(y, l) := {x n+ = g l ( x y l)} is, in the viscosity sense, a strict supersolution of (8) everywhere except on the sphere { x y = l}. Moreover, if H l = g l there exists universal constants c small, C large, such that H l is defined on ( + e c l, ), and H (s) H l (s) C l log( s ) if s < e c l/2.

12 2 OVIDIU SAVIN Proof. Define h l, H l (the corresponding functions for g l ) as (5) { h (s) h h l (s) = (s l ) C 2 l (( + s) 2 s 2 l ) if s l < s < h (s) + h ( s l ) + C 2 l (( s) 2 + s l ( s)) if s < (6) H l (s) = s We choose C 2 large, universal, such that 2hl (ζ) dζ (7) 4(n ) h (s) < C 2 ( s ). and s l = e c l with c small. For s l < s and l large we have 2 (h (s) h (s l )) h l (s), hence H l (s l ) (h (ζ) h (s l )) 2 dζ s l C ( + ζ s l ) 2 ( + ζ) 2 dζ C2 log s l l/2, s l if c is small enough. Moreover, for > g l (t) > s l we have g l (t) + 2(n )l g l (t) = h l (s) + 2(n )l 2h l (s) < < h (s) 2 C 2 l ( + s) + 4(n )l h (s) < h (s). On the domain where g l is constant, i.e g l = s l one has Ψ(, l) = < h (s l ). We remark that g l is a C, function on (, ). Its second derivative has a small jump at H l (s l ) from to h l (s l ). From the above inequalities we can conclude that g l ( x l) is, in the viscosity sense, a strict supersolution for x < l. If e c l/2 < s then hence h (s) h l (s) h (s l ) + C 2 l ( + s) 2 C l ( + s) 2 H (s) H l (s) = s 2hl (ζ) 2h (ζ) dζ h (ζ) h l (ζ) C 2 dζ s ( + ζ s l ) 2( C + ζ) 5 l log( + s). 2

13 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 3 and For < s < we have Also, H l () h l (s) 2 (h (s) + h ( s l )), (h (ζ) + h ( s l )) 2 dζ ( C ( ζ) 2 + s ) 2 2 l dζ C 2 log s l l/2. g l (t) + (n )l g l (t) = h l (s) + (n )l 2h l (s) < < h (s) C 2 l ( s + s l ) + 4(n )l h (s) + h ( s l ) < h (s), thus, g l ( x l) is a strict supersolution for x > l. We also remark that g l (H()) >. If s < e c l/2 then hence, h l (s) h (s) h ( s l ) + 2 C 2 l ( s) 2 C l ( s) 2 H (s) H l (s) C 2 s s With this the lemma is proved. 2h (ζ) 2hl (ζ) dζ h l (ζ) h (ζ) ( ζ) 3 dζ C l log( s). Next we construct a strict supersolution which is on a surface Γ with positive mean curvature. Lemma 5.4. Let Γ be such that Γ = {x n = P(x ) = ε 2 x T Mx + σξ x } { x < σε }, tr M δ, M δ, ξ δ for some small δ >. There exists σ (δ) >, such that if ε σ σ (δ), then we can find a function g Γ for which g Γ (d Γ ) is, in the viscosity sense, a strict supersolution, where d Γ represents the signed distance to Γ, d Γ > above Γ. (We consider only the set where the distance d Γ is realized at a point in the interior of Γ.)

14 4 OVIDIU SAVIN Proof. We define (8) h Γ (s) = max{, h (s) + c δερ(s)}, where c = /2 min h (s), /2 s /2 if s /2 ρ(s) = 2s if /2 s /2 if /2 s. Let s δ,ε be the point near for which h (s δ,ε ) = c δε, hence +s δ,ε (δε) 2. We have H Γ (s δ,ε ) = H Γ () = sδ,ε 2hΓ (ζ) dζ C s δ,ε (ζ sδ,ε )( + ζ) C 2(δ) log ε, 2hΓ (ζ) dζ C ( ζ)2 + δε dζ C 2(δ) log ε, H Γ (s) H (s). Thus, g Γ (d) = H Γ (d) is defined for d H Γ() and it is constant for d H Γ (s δ,ε ). Let d be the signed distance function to Γ. In an appropriate system of coordinates ( D 2 κ d = diag,.., dκ ) κ n, dκ n where κ i are the principal curvatures of Γ at the point where the distance is realized. Notice that κ i C 3 (δ)ε, hence, for d C 2 (δ) log ε one has n i= κ i n dκ i i= κ i + C(δ)ε 2 log ε P + C P 2 D 2 P + C(δ)ε 3 2 εδ + C(δ)(εσ 2 + ε 3 2 ) εδ + C4 (δ)εσ 2. Thus, for g Γ (d) > s δ,ε, g Γ (d) g Γ (d) ε(δ C 4(δ)σ 2 )g Γ (d) = h Γ (s) ε(δ C 4(δ)σ 2 ) 2hΓ (s) h (2c (s) + ε ) δχ [ /2,/2] (δ C 4 (δ)σ 2 ) 2hΓ (s).

15 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 5 If σ (δ) is chosen small enough, then g Γ (d) < h (g Γ (d)). On the domain where g Γ is constant, i.e d H Γ (s δ,ε ) we have g Γ = < h (s δ,ε ). Since g Γ is a C, function we conclude that g Γ (d) is a strict supersolution in the viscosity sense and the lemma is proved. Proof of lemma 5.2 Assume by contradiction that tr M > δ. We apply lemma 5.4 to the surface Γ 2 = {x n = P (x ) εδ 2 x 2 } { x l} with ε = θl 2, σ = θl and we find that g Γ2 (d Γ2 ) is a strict supersolution if σ is small enough. On the other hand we claim that (9) u(x) g l/4 (d Γ ) if x l/2, x n l/2. In order to prove this we use Theorem 3.2. We choose α < small such that h is strictly increasing on [, α] and β =. Then, there exists C 2 universal such that if u(x) α then If l 8C 2, then B(x, C 2 ) {u > }. (2) u(x) < α for x B((, l/2), l/4) Since Ψ((, l/2), l/4) is a supersolution of (8) in B((, l/2), l/4) and it is supported inside B((, l/2), 3l/8) (see lemma 5.3), we conclude from the maximum principle that u is below Ψ((, l/2), l/4). We slide this surface continuously along vectors ν,with ν e n+ =, ν e n, till we touch the graph of u. Since Ψ((, l/2), l/4) is a strict supersolution everywhere except on the level set, we find that the touching points can occur only on the level set. The inequality (9) now follows from the fact that, if σ is small enough, at each point of Γ we have a tangent sphere of radius l/4 from below which can be obtained from the sphere x (, l/2) = l/4 by sliding it continuously inside the domain {u < }. Now it suffices to prove that for σ σ (δ) we have (2) g Γ2 (d Γ2 ) > g l/4 (d Γ ) on { x = l/2} { d Γ l/4}. Then we slide g Γ2 (d Γ2 ) from below in the e n direction in the cylinder { x l/2} { x n l/2} till we touch u. By (9), (2) this cannot happen on { x = l/2} therefore the contact point is an interior

16 6 OVIDIU SAVIN point which is a contradiction with the fact that g Γ2 (d Γ2 ) is a strict supersolution. We notice that on { x = l/2} { d Γ l/4} we have d Γ2 d Γ + c (δ), thus, in order to prove (2), it suffices to show that (22) H Γ2 (s) < H l/4 (s) + c (δ). From (5),(8) we find that for l = θσ C (δ) large h Γ2 (s) h l/4 (s) if s + c 2 (δ)l 2 h Γ2 (s) h l/4 (s) if s c 2 (δ)l 2 This implies that the maximum of H Γ2 (s) H l/4 (s) occurs for s c 2 (δ)l 2. For these values of s we have H Γ2 (s) H (s) H l/4 (s) + 4 C l log l 2 With this the lemma is proved. c 2 (δ) < H l/4(s) + c (δ). 6. Theorem 2.2 implies Theorem 2. The proof is by compactness. Assume by contradiction that there exists u k, θ k, l k, ξ k such that u k is a local minimizer of J, u k () =, the level set {u k = } stays in the flat cylinder { x < l k } { x n < θ k } θ θ, θ k l k as k for which the conclusion of Theorem 2. doesn t hold. Let A k be the rescaling of the level sets given by (x, x n ) {u k = } (y, y n ) A k y = x l k, y n = x n θ k. Claim : A k has a subsequence that converges uniformly on y /2 to a set A = {(y, w(y )), y /2} where w is a Holder continuous function. In other words, given ε, all but a finite number of the A k s from the subsequence are in an ε neighborhood of A. Proof: Fix y, y /2 and suppose (y, y k) A k. We apply Theorem 2.2 for the function u k in the cylinder { x l k y < l k /2} { x n θ k y k < 2θ k }

17 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 7 in which the set {u k = } is trapped. Thus, there exists a universal constant η > and an increasing function ε (θ) >, ε (θ) as θ, such that {u k = } is trapped in the cylinder { x l k y < η l k /2} { x n θ k y k < 2( η )θ k } provided that 4θ k l k ε (2θ k ). Rescaling back we find that A k { y y η /2} { y n y k 2( η )}. We apply the Harnack inequality repeatedly and we find that (23) A k { y y η m /2} { y n y k 2( η ) m } provided that 4θ k l k η m ε (2( η ) m θ k ). Since these inequalities are satisfied for all k large we conclude that (23) holds for all but a finite number of k s. There exist positive constants α, β depending only on η, such that if (23) holds for all m m then A k is above the graph y n = y k 2( η ) m α y y β in the cylinder y /2. Taking the supremum over these functions as y varies we obtain that A k is above the graph of a Holder function y n = a k (y ). Similarly we obtain that A k is below the graph of a Holder function y n = b k (y ). Notice that (24) b k a k 4( η ) m and that a k, b k have a modulus of continuity bounded by the Holder function αt β. From Arzela-Ascoli Theorem we find that there exists a subsequence a kp which converges uniformly to a function w. Using (24) we obtain that b kp, and therefore A kp, converge uniformly to w. Claim 2: The function w is harmonic (in the viscosity sense). Proof: The proof is by contradiction. Fix a quadratic polynomial y n = P(y ) = 2 y T My + ξ y, M < δ, ξ < δ such that P > δ, P(y ) + δ y 2 touches the graph of w, say, at for simplicity, and stays below w in y < 2δ. Thus, for all k large we find points (y k, y kn ) close to such that P(y )+const touches A k from

18 8 OVIDIU SAVIN below at (y k, y k n) and stays below it in y y k < δ. This implies that, after eventually a translation, there exists a surface { x n = θ k lk 2 2 x T Mx + θ } k ξ k x, ξ k < 2δ l k that touches {u k = } at the origin and stays below it in the cylinder x < δl k. We write the above surface in the form { x n = δ2 θ k 2(δl k ) 2x T Mx + δ2 θ k δl k δ ξ k x and we contradict lemma 5.2 since θ k θ, θ k l k and P > δ. Since w is harmonic, there exist < η < η 2 small (depending only on n) such that w ξ y < η /2 for y < 2η 2. Rescaling back and using the fact that A k converge uniformly to the graph of w we conclude that for k large enough {u k = } { x < 3l k η 2 /2} { x n θ k l k ξ x < 3θ k η /4}. This is a contradiction with the fact that u k doesn t satisfy the conclusion of the Theorem 2.. }, 7. Construction of the sliding surfaces S(Y, R) In this section we introduce a family of rotation surfaces in R n+ which we denote by S(Y, R). We say that the point Y is the center of S and R the radius. The surfaces S are defined for centers Y in the strip { y n+ /4} and for radius R large. They have the following property: Suppose that for fixed R, some surfaces S(Y, R) are tangent by above to the graph of u. Then the contact points project along e n into a set with measure comparable with the measure of the projection of the centers Y along e n (see Proposition 7.). We define S(Y, R) as (25) S(Y, R) := {x n+ = g yn+,r(h (y n+ ) + x y R)}, y n+ /4, where the function g s,r, respectively h s,r, H s,r associated with it, are constructed below for s /4 and large R. For simplicity of notation we denote them by g, h, H. Denote (26) C3 = + 8(n ) max h

19 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 9 and let ϕ be such that (27) 2ϕ(s) = 2h (s) C R (s s ). where C is large enough such that the following holds (28) ϕ(s) < h (s) 2 C 3 R, if s [ 3/4, /2] ϕ(s) > h (s) + 2 C 3 R, if s [/2, 3/4]. Let s R near be such that h (s R ) = R, hence + s R R 2. We define h s,r : [s R, ] R as h (s) h (s R ) C 3 R (s s R ) if s [s R, ] 2 (29) h(s) = ϕ(s) if s ( /2, /2) h (s) + R + C 3 R ( s) if s [, ]. 2 For R large, h(s) c ( + s)(s s R ) on [s R, ], thus h is positive on (s R, ]. Define (3) H s,r(s) = H (s ) + and for R large enough (3) H(s R ) H (s ) H() H (s ) + s s s R 2h(ζ) dζ s c ( + ζ)(ζ s R ) dζ C log R c2 ( ζ) 2 + R dζ C log R. s Finally we define g s,r as { sr if t < H(s (32) g s,r(t) = R ) H (t) if H(s R ) t H(). Next we list some properties of the surfaces S(Y, R): ) Notice that (33) h(s) > h (s) 2 C 3 R > ϕ(s) if s [ 3/4, /2] h(s) < h (s) + 2 C 3 R < ϕ(s) if s [/2, 3/4]. From (27), (29), (33) we have (34) H(s) = H (s) C 2R (s s ) 2, if s /2 H(s) > H (s) C 2R (s s ) 2 if /2 < s < 3/4.

20 2 OVIDIU SAVIN Let ρ s,r be the function whose graph is obtained from the graph of g by the transformation (t, s) (t C 2R (s s ) 2, s) for s < 3/4. From (34) we obtain that g = ρ for s /2, and g < ρ at all other points where ρ is defined. In other words, if S(Y, R) is the rotation surface (35) S(Y, R) := {x n+ = ρ yn+,r(h (y n+ ) + x y R)}, y n+ /4, then, S(Y, R) coincides with S(Y, R) in the set x n+ /2 and stays below it at all the other points where S is defined. Notice that S(Y, R) { x n+ 3/4} and it is defined only in a neighborhood of the sphere x y = R which is the y n+ level set of S(Y, R). 2) We remark that S(Y, R) is constant s R when x y R 2 R 3 R H (y n+ ) + H(s R ), and grows from s R to when 2 R 3 < x y R H (y n+ ) + H() < 2 R 3. 3) The function g is C, in (, H( /2)) (H(/2), H()) and g has a small jump from to h (s R ) at H(s R ). If s (s R, /2) (/2, ), then on the s level set we have (see (26)) (36) S h (s) + 2(n )R 2h(s) h (s) C 3 R + 4(n ) h(s)r < h (s). Moreover, from (29), (33) we have (37) lim H (s) < lim H (s), s /2 s /2 + lim H (s) < lim H (s), s /2 s /2 + lim H (s) < s which together with (36) implies that S(Y, R) is, in the viscosity sense, a strict supersolution for x n+ /2. In other words S(Y, R) cannot touch from above a C 2 subsolution at a point X with x n+ /2.

21 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 2. ( s H ( ), ) s g g s, R 4) If s < 3/4, then on the s level set of S(Y, R), defined in (35), one has (see (27)) (38) h (s) C 2R ϕ (s) S S ϕ 2(n ) (s) + 2ϕ(s) h R (s) + C 2 R. This shows that S(Y, R) is an approximate solution of equation (8) with a R error. 5) From (27), (29) we see that if R R 2, then thus, h s,r (s) h s,r 2 (s) if s R s s h s,r (s) h s,r 2 (s) if s s (39) H s,r (s) H s,r 2 (s) in the domain where H s,r is defined. Next proposition is the key tool in proving Theorem 2.2. Proposition 7.. (Measure estimate for contact points) Let u be a C 2 subsolution of (8), i.e u h (u), u. Let ξ be a vector perpendicular to e n+ and A be a closed set in P ξ { x n+ /4}. Assume that for each Y A the surface S(Y + tξ, R), R large, stays above the graph of u when t and, as t increases, it touches the graph from above for the first time at a point (contact point). If B denotes the projection of the contact points along ξ in P ξ, then, µ A B where µ > universal, small and A represents the n-dimensional Lebesgue measure.

22 22 OVIDIU SAVIN Proof. Assume that S(Y, R) touches u from above at the point X = (x, u(x)). From the discussion above we find u(x) < /2. Denote by ν the normal to the surface at X, i.e. ν = (ν, ν n+ ) = ( u, ). + u 2 The center Y is given by (4) Y (X) = (x + ν ν σ, x n+ + ω) = F(X, ν), where (4) ω = R C (ν n+ ν H (x n+)) σ = C 2R ω2 + H (x n+ ) H (x n+ + ω) + R. The function F is smooth defined on {X R n+ : x n+ < /2} {ν R n+ : ν =, c < ν n+ < c } The differential D X Y is a linear map defined on T X, the tangent plane at X, and (42) D X Y = F X (X, ν) + F ν (X, ν)d X ν = F X (X, ν) F ν (X, ν)ii u where II u represents the second fundamental form of u at X. Writing the above formula for the surface S(Y, R) at X, we find thus, (42) becomes = F X (X, ν) F ν (X, ν)ii S (43) D X Y = F ν (X, ν)(ii S II u ). From (4) and (4), it is easy to check that (44) F ν (X, ν) C R. Since S touches u by above at X, we find that D 2 S D 2 u. On the other hand, from (38), which implies or S h (x n+) + C 2 R u + C 2 R D 2 S D 2 u C 3 R (45) II S II u C 4 R. From (43), (44), (45) we conclude D X Y C 5.

23 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 23 The centers Z for which X S(Z, R) describe a rotation surface, around X. Note if S(, R) is above u, then its center is above this surface. The normal to the surface at Y (X), which we denote by τ, belongs to the plane spanned by ν and e n+, and c 2 < τ < c 2. Thus, if ξ is perpendicular to e n+, we have τ ξ C 6 ν ξ. (Notice that the tangent plane to the surface at Y (X) is the range of F ν (X, ν).) Let B be the set of contact points, à the set of the corresponding centers, B = π ξ B and A = πξ Ã. Remark that π ξ is injective on à and B by construction. From above, we know that à belongs to a Lipschitz surface. One has A = τ(y ) ξ dy τ(y ) ξ D X Y dx à B C 7 ν(x) ξ dx = C 7 B and the proposition is proved. B 8. Extension of the contact set In this section we prove that the contact set from Proposition 7. becomes larger and larger when possibly we decrease the radius R. Denote L = P n { x n+ /2} Q l = {(x,, x n+ )/ x l, x n+ /2}. Let D k, represent the set of points on the graph of u that have from above a tangent surface S(Y, RC k ), where C is a large universal constant. Suppose that we have some control on the e n coordinate of these sets and denote by D k their projections into L. Recall that S(Y, RC k ) is an approximate solution of equation (8) with a C k R error. If S(Y, RC k ) touches u from above at X then, from Harnack inequality, the two surfaces stay C k R close to each other in a neighborhood of X (see lemma 8.). Thus, if we denote E k = {Z L/ dist(z, D k ) C } then we control the e n coordinate of a set on the graph of u that projects along e n into E k. We want to prove that, in measure, E k almost covers Q l as k becomes larger and larger. At large scale the interface satisfies a mean curvature equation. In lemma 8.2 we prove that near (large scale) a point Z D k we can find

24 24 OVIDIU SAVIN a set of positive measure in D k+. Using a covering argument we show that the sets E k almost cover Q l as k increases. Next we state and prove two technical lemmas, lemma 8. and lemma 8.2. At the end of the section we prove a covering lemma which links the two scales. Lemma 8.. (Small scale extension) Suppose that the surface S(Y, R) touches a solution u from above at X = (x, u(x )) with ( u u (x ), e n ) π 8. Given a constant a > large, there exists C(a) > depending on universal constants and a such that for each point Z L B(π n X, a) there exists x with ) π n (x, u(x)) = Z, x x 2a 2) (x x ) u u (x ) H (u(x)) H (u(x )) + C(a)R. Lemma 8.2. (Large scale extension) Suppose that the surface S(Y, R) stays above a C 2 subsolution u in the cylinder { x < l} { x n < l}, l > 4R 3 and touches the graph of u at (x, u(x )) with u(x ) < /2, x n < l/4, x = q, q < l/4, ( u u (x ), e n ) π 8. There exist universal constants C 4, C5, large, c 2, small, such that if C 4 q, l R c 2 then the set of points (x, u(x)) that satisfy the following four properties ) x < q/5, u(x) < /2, x x < 2q 2) there exists a surface S(Y, R/ C 5 ) that stays above u and touches its graph at (x, u(x)) 3) ( u u (x), u u (x )) C 4 qr 4) (x x ) u u (x ) C 4 q 2 R + H (u(x)) H (u(x )) project along e n into a set of measure greater than c 2 q n.

25 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 25 Remark: The term H (u(x)) H (u(x )) that appears in property 2 of lemma 8. and property 4 of lemma 8.2 represents the distance between the u(x) level surface and the u(x ) level surface of a one dimensional solution. Now we state the iteration lemma that links lemmas 8.2 and 8.. Lemma 8.3. (Covering lemma) Let D k be closed sets, D k L, with the following properties: ) D Q l, D D D ) if Z D k Q 2l, Z L, Z Z = q and 2l q a then, D k+ B(Z, q/) µ B(Z, q) L where a > (large), µ (small) are given positive constants and l > 2a. Denote by E k the set E k := {Z L/ dist(z, D k ) a}. Then there exists µ > depending on n, µ such that Q l \ E k ( µ) k Q l. Proof of lemma 8. Let S(Y, R) be the surface defined in (35). Notice that S(Y, R) touches u from above at X. The restrictions π n S : S(Y, R) P n, π n+ S : S(Y, R) P n+ are diffeomorphisms in a 3a neighborhood of X for R large. Denote by T the map In the set T := π n+ S π n S : P n { x n+ < 3/4} P n+. O := T (P n { x n+ < 3/4} B(π n X, a + 2)) we have S u, = S(x ) u(x ). From (38) and the fact that h is Lipschitz we find The open set C (S u) + C R (S u). O 2 := T (P n { x n+ < 5/8} B(π n X, a + )) satisfies O 2 O, dist(o 2, O ) c, with c >, universal. From Harnack inequality, one obtains (46) sup x O 2 (S u) C (a)r.

26 26 OVIDIU SAVIN For each Z L B(π n X, a) we consider the line Z +te n and denote by X its intersection with S(Y, R). Notice that in O we have n S c 2, c 2 > universal. From this, (46), and the continuity of u we find that Z + te n intersects the graph of u at a point X 2 = (x 2, u(x 2 )) with Since (x x ) we conclude that (x 2 x ) and the lemma is proved. X 2 X C (a)r. u u (x ) H (z n+ ) H (u(x )) + C 2 R u u (x ) H (u(x 2 )) H (u(x )) + C(a)R Proof of lemma 8.2 The proof consists in 2 steps. In step we find a point that satisfies properties 2-4 and property with q/4 instead of q/5. In step 2 we use Proposition 7. to extend properties 2-4 from that point to a set of positive measure. Before we start, we introduce some notation. For a surface S(Y, R) we associate its level surface, the n dimensional sphere { Σ(y, r) = x y = r := R H (y n+ ) C } 2R y2 n+. We remark that the s level surface of S, s < /2, is a concentric sphere at a (signed) distance (47) H (s) + O() C R, O() < /2 from Σ(y, r). Also for a point X = (x, x n+ ) S(Y, R), x n+ < /2 we associate the point x = [y, x) Σ(y, r) where [y, x) represents the half line from y going through x. First we prove the lemma in the following situation (this is a rotation of the above configuration): The surface S(Y, R ) stays above the graph of u in the cylinder { x 2q} { x n l/2} and touches it at X = (x, u(x )), u(x ) < /2. Assume x { x = q} {x n = }, y = e n r 2 q 2,

27 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 27 q c large, and q/r c, c small, universal. Step We prove the existence of a surface S(Y, R ) that stays above u in the cylinder x 2q and touches it at (x, u(x )) such that Y = Y + t e n, R > R /C 3, x {x n < C 4 q 2 where C 3, C 4 are large universal constants. R } { x < q } From (26), (27), (29) we obtain the existence of C, C3 universal such that (48) (2h s,r(s)) /2 h s,r (s) h (s) C R if s < /2 h s,r (s) = h (s) C 3 R if s (s R, /2) (/2, ) C 3 = + 8(n ) max h. We consider the function ψ : R n R ψ(z ) = γ ( z γ ) z R n where γ is such that (49) γ = 4(C + 6(n 2)). Finally, we choose ω <, universal, such that ω γ 2 = 2. The graph (5) x n = q 2 r 2 q 2ψ(x q ) has by below the tangent sphere Σ(y, r ) when x = q, and a tangent sphere of radius r ω and center y ω when x = ωq, where r ω = ω γ+2 r 2 + q 2 (ω 2γ 2 ) r /2. Let Γ denote the graph of Σ(y, r ) for x > q below x n =, Γ 2 the graph of the above function for ωq x q and Γ 3 the graph of x y ω = r ω when x < ωq, x n >. We notice that Γ = Γ Γ 2 Γ 3 is a C, surface in R n. We define the following surface in R n+ Ψ = { x n+ = g yn+,r ( dγ + H yn+,r () )}, where d Γ represents the signed distance to the surface Γ (d Γ positive in the exterior of Γ). Note that Ψ coincides with S(Y, R ) if d Γ is realized on Γ.

28 28 OVIDIU SAVIN Claim : The surface Ψ is a supersolution of (8) everywhere except the set where x n+ < /2 and d Γ (x) is realized on Γ Γ 3. Proof: Let h yn+,r be the corresponding function for g yn+,r that we are going to denote by h and g for simplicity. At distance d from Γ we have in an appropriate system of coordinates D 2 g = diag [ κ κ d g,.., g ] = diag [ κ 2h(s),.., h (s) κ d where κ i represent the principal curvatures of Γ (upwards) at the point where d is realized. Case : If d is realized at a point on Γ, then the result follows from the construction of S(Y, R). Case 2: If d is realized at a point on Γ 2, then κ i r ω 3R i =,.., n 2 κ n γ + R 2 provided that q/r is small. Without loss of generality we assume d R 3 since otherwise, g is constant. On the /2, respectively /2, level sets g(d) is a supersolution from (37). On the other level sets one has h + n i= κ i κ i d n 2 2h h + ( 2 i= κ i κ n 2 ) 2h < h + C R γ + 2h + (6(n 2) )R 2h < h 4 (s) (we used (48) and (49)). Case 3: If d is realized at a point on Γ 3 and s > /2, then h + n i= κ i 2h h κ i d C 3 R + 4(n )R 2h < h (s), (we used (48)) and the claim is proved. ] Γ 3 Σ( y, r ) ω Γ 2 Σ( y, r ) Γ

29 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 29 We remark that Ψ and S(Y, R ) coincide outside the cylinder x < 2q. Next we consider S(Y ω, R ) with R = r ω + H (y n+) + 5 C R Y ω = (y ω, y n+). The sphere Σ(y ω, r ) stays at distance greater than 3 C R above Γ 3 and stays at distance greater than 3 C R below Γ if x > q(+ω)/2 > /2 ωq + 8 C. This implies (see (47)): ) the region of Ψ where x n+ < /2 and the distance to Γ is realized on Γ 3 is above S(Y ω, R ) 2) the region of S(Y ω, R ) where x n+ < /2 and the distance to Σ(y ω, r ) is realized at a point outside { x < q( + ω)/2} is above Ψ 3) S(Y ω, R ) is above Ψ outside { x < 2q}. We slide from below Ψ in the e n direction till we touch u for the first time. This cannot happen at (x, u(x )) since Ψ is a strict supersolution in the viscosity sense at x and u C 2 is a subsolution. We conclude that there exists β > such that the surface Ψ βe n = {X βe n, X Ψ} touches u at a point (z, u(z)) with u(z) < /2 and the distance from z + βe n to Γ is realized on Γ 3. Now we consider the surfaces S(Y + te n, R ) and increase t till we touch for the first time the graph of u. We notice that when Y +te n = Y ω βe n then the point (z, u(z)) is above the surface S(Y + te n, R ). Thus we can find < t < Y Y ω β such that S(Y, R ), Y = Y + t e n touches u from above at a point (x, u(x )), u(x ) < /2 in the cylinder x < 2q. Moreover from the above remarks x { x < q( + ω)/2} {x n < C 2 q 2 R } R > R /3. We apply the above argument with (x, u(x )) and S(Y, R ) instead of (x, u(x )) and S(Y, R ) and continue inductively at most a finite number of times till we find a point (x, u(x )) with the required properties. Step 2 Using the result from step, we prove that the set of contact points (x, u(x)) such that ) x < q/4, u(x) < /2, x x < 4q/3 2) in the cylinder { x < 2q}, u is touched by above at (x, u(x)) by S(Y, R /C 5 ), and S(Y, R /C 5 ) is above S(Y, R ) outside this cylinder 3) ( ) u u (x), u u (x q ) < C 9 R

30 3 OVIDIU SAVIN and the contact points belong in each level set to a Lipschitz graph with Lipschitz constant less than C 9 qr 4) u (x x ) u (x q 2 ) H (u(x)) H (u(x )) + C 9 R project along e n in a set of measure greater than c 2 q n, where C 5, C 9, c 2 are appropriate universal constants. We slide from below, in the e n direction, the surfaces S(Y, R) with (5) y x q 5, y n+ 4, R = R, C 5 = 4C 4 (4) 2 C 5 till they touch u. First we show that ( x, 2C 4q 2 R ) is in the exterior of Σ(y, r). Assume not, then Σ(y, r) is above x n = 3C 4 q 2 (2R ) in the cylinder x x q() 2. One has x = x + u u (x )(H (u(x )) + O() C C 3 R ), ( u u (x ), e n ) qc 3 R, hence x e n x e n + H (u(x )) + C 6 (q 2 R 2 + R ). Thus, if q is greater than a large universal constant, one has that x is at a signed distance less than H (u(x )) + C 6 (q 2 R 2 + R ) C 4 q 2 (2R ) < H (u(x )) C C 5 R from Σ(y, r). This implies that x is in the interior of the u(x ) level surface of S(Y, R) which is a contradiction. Since ( x, 2C 4q 2 R ) is in the exterior of Σ(y, r), we find from (5) that Σ(y, r) is below x n = 4C 4 q 2 R and below x n = outside x < q/5. Thus, Σ(y, r) is at a distance greater than q 2 (4R ) in the interior of Σ(y, r ) outside {x n > } { x < q/5}. The s level surface of S(Y, R ) is at distance greater than (see (39)) H yn+,r (s) H yn+,r () H yn+,r(s) H yn+,r () H yn+,r(s) C 2R from Σ(y, r ). The s level surface of S(Y, R) is at distance less than H yn+,r(s) H yn+,r() H y n+,r (s) + C C 5 R

31 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 3 from Σ(y, r). Hence, at the points x for which d Σ(y,r) (x) d Σ(y,r )(x) 2 C C 5 R S(Y, R) is above S(Y, R ). Since S(Y, R ) is constant outside a R 3 neighborhood of Σ(y, r ), we can conclude that, for q greater than a large universal constant, S(Y, R) is above S(Y, R ) outside x < q/4. This implies that the contact points (x, u(x)) have the properties u(x) < /2, ( u u (x), e n ) < C 7 q R, x n < 4C 4 q 2 R, x < q 4 and, from Proposition 7. they project along e n in a set of measure greater than c 2 q n. We notice that on each level set the contact points belong to a Lipschitz graph with Lipschitz constant less than 2C 7 qr. Also, one has thus, x x < 4 3 q, x = x + u u (x) ( H (u(x)) + O() C ) C 5 R x n = x n + H (u(x)) + C 8 (q 2 R 2 + R ) q 2 (x x ) e n 5C 4 + H (u(x)) H (u(x )) R u (x x ) u (x q 2 ) C 9 + H (u(x)) H (u(x )) R which proves step 2. End of proof of lemma 8.2 In the general case we denote by X S(Y, R ) the point such that π n X = and let ξ = x y x y. The cylinder { (x x ) ξ < l/2} { π ξ (x x ) < 2 π ξ ( x x ) } is included in { x < l} { x n < l}. Also, x /2 < π ξ ( x x ) < x 3/2, hence we are in the above situation. The contact points obtained in step 2 belong in each level set to a Lipschitz graph (in the e n direction) with Lipschitz constant less than. The result follows now by projecting these points along the e n direction.

32 32 OVIDIU SAVIN With this the lemma is proved. Proof of lemma 8.3 Denote by F k E k the closed set F k = {Z L/ dist(x, D k Q l+a ) a}. We prove that there exists µ(n, µ ) > small, such that (52) Q l \ F k ( µ) k Q l. Let Z Q l \ F k, Z F k be such that Z Z = dist(z, F k ) = r. We claim that for some µ 2 (n, µ ) > (53) F k+ Q l B(Z, r) µ 2 Q l B(Z, r). Let Z D k Q l+a be the point for which Z Z = r + a and Z belongs to the segment [Z, Z ]. If 2r a, let Z 2 be such that Z Z 2 = r 2, B(Z 2, r 2 ) L Q l. From property 2 and a + r/2 Z 2 Z 5r we obtain F k+ Q l B(Z, r) D k+ B(Z 2, r 2 ) D k+ B(Z 2, Z 2 Z /) µ B(Z 2, r 2 ) L µ 2 B(Z, r) Q l. If 2r < a then, from property 2, there exists a point thus, Z 3 D k+ B(Z, r + a ) Q l+a Q l B(Z, r) Q l B(Z 3, a) F k+, which proves (53). We take a finite overlapping cover of Q l \F k with balls B(Z, r). Using (53) we find a constant µ(µ 2, n) > such that F k+ (Q l \ F k ) µ Q l \ F k hence, and (52) is proved. Q l \ F k+ ( µ) Q l \ F k,

33 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS Estimate for the projection of the contact set In this section we use the results of the previous section and prove the following Lemma 9.. Let u be a local minimizer of J in { x < 32l} { x n < 32l}, and assume that u() =, u < if x n < θ. There exists universal constants C, µ, c 4 such that: Given θ >, there exists ε (θ ) > such that if then the set of points that satisfy θl = ε ε (θ ), θ θ, Ck ε c 4 (x, u(x)) { x l} { x n+ /2} x n C k θ + H (u(x)) project along e n into a set of measure greater than ( ( µ) k ) Q l. Before we prove lemma 9. we need another lemma that gives us a first surface S(Y, R) that touches u from above. Lemma 9.2. (The first touching surfaces) Let u be a local minimizer of J in { x < 32l} { x n < 32l}, and assume that u() =, u < if x n < θ. Given θ >, there exists ε (θ ) > such that if θl ε (θ ), θ θ, then the points (x, u(x)) with the following properties ) x < l, u(x) < /2 2) there exists a surface S(Y, R ) that stays above u in the cylinder { x < 6l} { x n < 6l} and touches its graph at (x, u(x)), where 3) 4) R = l2 32θ, l > R 3 ( ) u u (x), e n lr x n θ 4 + H (u(x)) project along e n into a set of measure greater than c 3 l n, where c 3 > is small, universal.

34 34 OVIDIU SAVIN Proof of lemma 9.2 We slide from below surfaces Ψ(y, l) and as in the proof of (9) we obtain (54) g l (x n + θ) u(x) if x < 6l, x n < 6l where g l ( respectively Ψ(y, l)) is the comparison function (surface) constructed in lemma 5.3. Let R = l 2 (32θ) and notice that lr is small and l > R 3 if ε (θ ) is small. Consider the surfaces S(Y, R ) that contain the point (, ) with y l/6, y n+ /4. Claim: The surfaces S(Y, R ) are above g l (x n +θ) (and therefore above u) in the region l < x < 6l. Proof: The level surface of S(Y, R ) is a sphere x y = r, which is below the hyperplane x n = θ/8. Let d, d 2, denote the signed distance to the sphere x y = r, respectively to the hyperplane x n = θ. If b R 3, the sphere x y = r + b is below x n = 2θ + b outside x < l/2, thus (55) d d 2 + θ, in { d R 3 } {l < x < 6l}. Now it suffices to show (56) H yn+,r (s) H yn+,r () H l (s) + θ which implies g yn+,r (d + H yn+,r ()) g l (d θ) hence, g yn+,r (d + H yn+,r ()) g l (d θ) g l (d 2 ), or S(Y, R ) is above g l (x n + θ) in the region l < x < 6l. The proof of (56) is similar to the proof of (22). Notice that (57) H yn+,r (s) H yn+,r () = s 2hyn+,R (ζ) dζ, (58) H yn+,r (s) H yn+,r () H (s) + C 2R H (s) + C θl 2 H (s) + θ/2 for l large. From (5), (29) we find that h yn+,r (s) h l (s), if s + c (θ )l 2

35 PHASE TRANSITIONS: REGULARITY OF FLAT LEVEL SETS 35 h yn+,r (s) h l (s), if s c (θ )l 2, provided that l θ ε (θ ) C (θ ) large. This implies that the maximum of H yn+,r (s) H l (s) occurs for s c (θ )l 2. For these values of s we have (see lemma 5.3) H (s) H l (s) + C l log which together with (58) proves (56). l 2 c (θ ) < H l(s) + θ /2 In conclusion, we slide from below surfaces S(Y, R ) in the cylinder { x 6l} { x n 6l} with y l/6, y n+ /4 and we touch u for the first time at points (x, u(x)) that satisfy properties, 2, 3 of the lemma and x n H yn+,r (u(x)) H yn+,r () + θ/8 H (u(x)) + 32 C θ + θ/8 H 2l 2 (u(x)) + θ/4. Now the lemma follows from Proposition 7.. Proof of lemma 9. Let R = l 2 (32θ) and define D k the set of points (x, u(x)) with the following properties ) x 6l, u(x) < /2 2) the graph of u is touched from above in { x 6l} { x n 6l} at (x, u(x)) by S(Y, R k ) with R k R C k 6 3) 4) where ( ) u u (x), e n C 6lR k x e n C k 6θ + H (u(x)) C 6 = max{ C 5, 2 C 4, C( C 4 )}. Also, we define D k = π n ( D k ). From lemma 9.2 we find that if ε ε (θ ), then D Q l. Claim: As long as 8 C 6 k lr min{ c 2, π} D k satisfies property 2 of lemma 8.3 with a = C 4.

36 36 OVIDIU SAVIN Proof: Let Z k = π n (x k, u(x k )) Q 2l D k and let Z L, x k z = q, 2l q C 4. We apply lemma 8.2 in the cylinder { x z 8l} { x n 8l} and obtain that the points (x, u(x)) with the following four properties project along e n in a set of measure greater than c 2 q n. ) x x q/5, u(x) < /2, x x k 4l 2) the graph of u is touched from above in { x 6l { x n 6l} at (x, u(x)) by S(Y, R k+ ) with 3) hence, 4) thus, R k+ R C k 5 R C k 6 ( ) u u (x), u u (x k) 2 C 4 Ck 6 lr ( ) u u (x), e n 2 C 4 Ck 6 lr + C 6 k lr (x x k ) k+ C 6 lr u u (x k) 4 C 4 l 2 Ck 6 R + H (u(x)) H (u(x k )) (x x k ) e n 4 C k 6l 2 R + 4 C 4 l 2 Ck 6 R + H (u(x)) H (u(x k )) k+ x n C 6 θ + H (u(x)). All these points are in D k+ which proves the claim. Let E k be the sets defined in lemma 8.3. From lemma 8. we know that each point in E k is the projection of a point (x, u(x)) with x x k 2 C 4 and (x x k ) u u (x k) H (u(x)) H (u(x k )) + C( C 4 )R k, for some point (x k, u(x k )) D k. Thus, (x x k ) e n C( C 4 ) C k 6R + 2 C 4 Ck 6 lr + H (u(x)) H (u(x k )) or k+ x n C 6 θ + H (u(x)). We apply lemma 8.3 and obtain that there exist positive universal constants c 4, µ, small, C := C 6 2 such that if C ε k c 4 := 3 min{ c 2, π}, ε ε (θ ),