A LOCALIZATION PROPERTY AT THE BOUNDARY FOR MONGE-AMPERE EQUATION

Transcription

1 A LOCALIZATION PROPERTY AT THE BOUNDARY FOR MONGE-AMPERE EQUATION O. SAVIN. Introduction In this paper we study the geometry of the sections for solutions to the Monge- Ampere equation det D 2 u = f, u : Ω R convex, which are centered at a boundary point x 0 Ω. We show that under natural local assumptions on the boundary data the domain, the sections S h (x 0 ) = {x Ω u(x) < u(x 0 ) + u(x 0 ) (x x 0 ) + h} are equivalent to ellipsoids centered at x 0, that is, for each h > 0 there exists an ellipsoid E h such that ce h Ω S h (x 0 ) x 0 CE h Ω, with c, C constants independent of h. The situation in the interior is well understood. Caffarelli showed in [C] that if for some x Ω, 0 < λ f Λ in Ω, S h (x) Ω, then S h (x) is equivalent to an ellipsoid centered at x i.e. ke S h (x) x k E for some ellipsoid E of volume h n/2 for a constant k > 0 which depends only on λ, Λ, n. This property provides compactness of sections modulo affine transformations. This is particularly useful when dealing with interior C 2,α W 2,p estimates of strictly convex solutions of det D 2 u = f when f > 0 is continuous (see [C2]). Sections at the boundary were also considered by Trudinger Wang in [TW] for solutions of det D 2 u = f but under stronger assumptions on the boundary behavior of u Ω, with f C α (Ω). They proved C 2,α estimates up to the boundary by bounding the mixed derivatives obtained that the sections are equivalent to balls. The author was partially supported by NSF grant

2 2 O. SAVIN 2. Statement of the main Theorem. Let Ω be a bounded convex set in R n. We assume throughout this note that (2.) B ρ (ρe n ) Ω {x n 0} B ρ, for some small ρ > 0, that is Ω (R n ) + Ω contains an interior ball tangent to Ω at 0. Let u : Ω R be convex, continuous, satisfying (2.2) det D 2 u = f, λ f Λ in Ω. We extend u to be outside Ω. By subtracting a linear function we may assume that (2.3) x n+ = 0 is the tangent plane to u at 0, in the sense that u 0, u(0) = 0, any hyperplane x n+ = ɛx n, ɛ > 0 is not a supporting hyperplane for u. In this paper we investigate the geometry of the sections of u at 0 that we denote for simplicity of notation S h := {x Ω : u(x) < h}. We show that if the boundary data has quadratic growth near {x n = 0} then, as h 0, S h is equivalent to a half-ellipsoid centered at 0. Precisely, our main theorem reads as follows. Theorem 2.. Assume that Ω, u satisfy (2.)-(2.3) above for some µ > 0, (2.4) µ x 2 u(x) µ x 2 on Ω {x n ρ}. Then, for each h < c(ρ) there exists an ellipsoid E h of volume h n/2 such that ke h Ω S h k E h. Moreover, the ellipsoid E h is obtained from the ball of radius h /2 by a linear transformation A h (sliding along the x n = 0 plane) A h E h = h /2 B A h (x) = x νx n, ν = (ν, ν 2,..., ν n, 0), with ν k log h. The constant k above depends on µ, λ, Λ, n c(ρ) depends also on ρ. Theorem 2. is new even in the case when f =. The ellipsoid E h, or equivalently the linear map A h, provides information about the behavior of the second derivatives near the origin. Heuristically, the theorem states that in S h the tangential second derivatives are bounded from above below the mixed second derivatives are bounded by log h. This is interesting given that f is only bounded the boundary data Ω are only C, at the origin. Remark. Given only the boundary data ϕ of u on Ω, it is not always easy to check condition (2.4). Here we provide some examples when (2.4) is satisfied: ) If ϕ is constant the domain Ω is included in a ball included in {x n 0}.

3 A LOCALIZATION PROPERTY 3 2) If the domain Ω is tangent of order 2 to {x n = 0} the boundary data ϕ has quadratic behavior in a neighborhood of 0. 3) ϕ, Ω C 3 at the origin, Ω is uniformly convex at the origin. We obtain compactness of sections modulo affine transformations. Corollary 2.2. Under the assumptions of Theorem 2., assume that lim f(x) = f(0) x 0 u(x) = P (x) + o( x 2 ) on Ω with P a quadratic polynomial. Then we can find a sequence of rescalings ũ h (x) := h u(h/2 A h x) which converges to a limiting continuous solution ū 0 : Ω 0 R with such that kb + Ω 0 k B + det D 2 ū 0 = f(0) ū 0 = P on Ω 0 {x n = 0}, ū 0 = on Ω 0 {x n > 0}. In a future work we intend to use the results above obtain C 2,α W 2,p boundary estimates under appropriate conditions on the domain boundary data. 3. Preliminaries Next proposition was proved by Trudinger Wang in [TW]. Since our setting is slightly different we provide its proof. Proposition 3.. Under the assumptions of Theorem 2., for all h c(ρ), there exists a linear transformation (sliding along x n = 0) A h (x) = x νx n, with ν n = 0, ν C(ρ)h n 2(n+) such that the rescaled function ũ(a h x) = u(x), satisfies in S h := A h S h = {ũ < h} the following: (i) the center of mass of S h lies on the x n -axis; (ii) k 0 h n/2 S h = S h k0 hn/2 ;

4 4 O. SAVIN (iii) the part of S h where {ũ < h} is a graph, denoted by G h = S h {ũ < h} = {(x, g h (x ))} that satisfies g h C(ρ) x 2 µ 2 x 2 ũ 2µ x 2 on G h. The constant k 0 above depends on µ, λ, Λ, n the constants C(ρ), c(ρ) depend also on ρ. In this section we denote by c, C positive constants that depend on n, µ, λ, Λ. For simplicity of notation, their values may change from line to line whenever there is no possibility of confusion. Constants that depend also on ρ are denote by c(ρ), C(ρ). Proof. The function v := µ x 2 + Λ µ n x2 n C(ρ)x n is a lower barrier for u in Ω {x n ρ} if C(ρ) is chosen large. Indeed, then v u on Ω {x n ρ}, In conclusion, hence v 0 u v u on Ω {x n = ρ}, det D 2 v > Λ. in Ω {x n ρ}, (3.) S h {x n ρ} {v < h} {x n > c(ρ)(µ x 2 h)}. Let x h be the center of mass of S h. We claim that (3.2) x h e n c 0 (ρ)h α, α = n n +, for some small c 0 (ρ) > 0. Otherwise, from (3.) John s lemma we obtain S h {x n C(n)c 0 h α h α } { x C h α/2 }, for some large C = C (ρ). Then the function w = ɛx n + h ( x 2 C h α/2 ) 2 + ΛC 2(n ) h is a lower barrier for u in S h if c 0 is sufficiently small. Indeed, for all small h, ( xn h α ) 2 w h 4 + h 2 + ΛC2(n ) (C(n)c 0 ) 2 h < h in S h, w ɛx n + h α C 2 x 2 x n + C(ρ)hc 0 h α µ x 2 u on Ω,

5 A LOCALIZATION PROPERTY 5 Hence det D 2 w = 2Λ. w u in S h, we contradict that 0 is the tangent plane at 0. Thus claim (3.2) is proved. Now, define The center of mass of S h = A h S h is A h x = x νx n, ν = x h x h e, n ũ(a h x) = u(x). x h = A h x h lies on the x n -axis from the definition of A h. Moreover, since x h S h, we see from (3.)-(3.2) that ν C(ρ) (x h e n) /2 (x h e n) C(ρ)h α/2, this proves (i). If we restrict the map A h on the set on Ω where {u < h}, i.e. on we have S h Ω {x n x 2 ρ } { x < Ch /2 } A h x x = ν x n C(ρ)h α/2 x 2 C(ρ)h α 2 x, part (iii) easily follows. Next we prove (ii). From John s lemma, we know that after relabeling the x coordinates if necessary, (3.3) D h B S h x h C(n)D h B where Since d d 2 0 D h = d n ũ 2µ x 2 on G h = {(x, g h (x ))}, we see that the domain of definition of g h contains a ball of radius (µh/2) /2. This implies that d i c h /2, i =,, n, for some c depending only on n µ. Also from (3.2) we see that which gives x h e n = x h e n c 0 (ρ)h α d n c(n) x h e n c(ρ)h α.

6 6 O. SAVIN We claim that for all small h, n d i k 0 h n/2, i= with k 0 small depending only on µ, n, Λ, which gives the left inequality in (ii). To this aim we consider the barrier, n ( ) 2 xi w = ɛx n + ch. We choose c sufficiently small depending on µ, n, Λ so that for all h < c(ρ), i= d i w h on S h, on the part of the boundary G h, we have w ũ since w ɛx n + c c 2 x 2 + ch µ 4 x 2 + chc(n) x n d n ( ) 2 xn d n µ 4 x 2 + ch α C(ρ) x 2 µ 2 x 2. Moreover, if our claim does not hold, then det D 2 w = (2ch) n ( d i ) 2n > Λ, thus w ũ in S h. By definition, ũ is obtained from u by a sliding along x n = 0, hence 0 is still the tangent plane of ũ at 0. We reach again a contradiction since ũ w ɛx n the claim is proved. Finally we show that S h Ch n/2 for some C depending only on λ, n. Indeed, if then Since we obtain the desired conclusion. v = h on S h, det D 2 v = λ v u 0 in S h. h h min S h v c(n, λ) S h 2/n In the proof above we showed that for all h c(ρ), the entries of the diagonal matrix D h from (3.3) satisfy d i ch /2, i =,... n

7 A LOCALIZATION PROPERTY 7 d n c(ρ)h α, α = n n + ch n/2 d i Ch n/2. The main step in the proof of Theorem 2. is the following lemma that will be proved in the remaining sections. Lemma 3.2. There exist constants c, c(ρ) such that (3.4) d n ch /2, for all h c(ρ). Using Lemma 3.2 we can easily finish the proof of our theorem. Proof of Theorem 2.. Since all d i are bounded below by ch /2 their product is bounded above by Ch n/2 we see that Ch /2 d i ch /2 for all h c(ρ). Using (3.3) we obtain Moreover, since S h Ch /2 B. i =,, n x h e n d n ch /2, ( x h) = 0, the part G h of the boundary S h contains the graph of g h above x ch /2, we find that ch /2 B Ω S h, with Ω = A h Ω, S h = A h S h. In conclusion We define the ellipsoid E h as ch /2 B Ω A h S h Ch /2 B. E h := A h (h/2 B ), hence ce h Ω S h CE h. Comparing the sections at levels h h/2 we find we easily obtain the inclusion ce h/2 Ω CE h A h A h/2 B CB. If we denote A h x = x ν h x n then the inclusion above implies which gives the desired bound for all small h. ν h ν h/2 C, ν h C log h

8 8 O. SAVIN We introduce a new quantity b(h) which is proportional to d n h /2 which is appropriate when dealing with affine transformations. Notation. Given a convex function u we define b u (h) = h /2 sup S h x n. Whenever there is no possibility of confusion we drop the subindex u use the notation b(h). Below we list some basic properties of b(h). ) If h h 2 then 2) A rescaling ( h h 2 ) 2 b(h ) b(h 2 ) ũ(ax) = u(x) ( h2 h ) 2. given by a linear transformation A which leaves the x n coordinate invariant does not change the value of b, i.e bũ(h) = b u (h). 3) If A is a linear transformation which leaves the plane {x n = 0} invariant the values of b get multiplied by a constant. However the quotients b(h )/b(h 2 ) do not change values i.e bũ(h ) bũ(h 2 ) = b u(h ) b u (h 2 ). then 4) If we multiply u by a constant, i.e. From (3.3) property 2 above, ũ(x) = βu(x) bũ(βh) = β /2 b u (h), bũ(βh ) bũ(βh 2 ) = b u(h ) b u (h 2 ). c(n)d n b(h)h /2 C(n)d n, hence Lemma 3.2 will follow if we show that b(h) is bounded below. We achieve this by proving the following lemma. Lemma 3.3. There exist c 0, c(ρ) such that if h c(ρ) b(h) c 0 then (3.5) for some t [c 0, ]. b(th) b(h) > 2,

9 A LOCALIZATION PROPERTY 9 This lemma states that if the value of b(h) on a certain section is less than a critical value c 0, then we can find a lower section at height still comparable to h where the value of b doubled. Clearly Lemma 3.3 property above imply that b(h) remains bounded for all h small enough. The quotient in (3.5) is the same for ũ which is defined in Proposition 3.. We normalize the domain S h ũ by considering the rescaling v(x) = hũ(h/2 Ax) where A is a multiple of D h (see (3.3)), A = γd h such that det A =. Then ch /2 γ Ch /2, the diagonal entries of A satisfy The function v satisfies a i c, i =, 2,, n, cb u (h) a n Cb u (h). λ det D 2 v Λ, v 0, v(0) = 0, is continuous it is defined in Ω v with Then for some x, Ω v := {v < } = h /2 A Sh. x + cb Ω v CB +, ct n/2 S t (v) Ct n/2, t, where S t (v) denotes the section of v. Since ũ = h in S h {x n C(ρ)h}, then v = on Ω v {x n σ}, σ := C(ρ)h α. Also, from Proposition 3. on the part G of the boundary of Ω v where {v < } we have n (3.6) 2 µ n a 2 i x 2 i v 2µ a 2 i x 2 i. i= In order to prove Lemma 3.3 we need to show that if σ, a n are sufficiently small depending on n, µ, λ, Λ then the function v above satisfies (3.7) b v (t) 2b v () for some > t c 0. Since α <, the smallness condition on σ is satisfied by taking h < c(ρ) sufficiently small. Also a n being small is equivalent to one of the a i, i n being large since their product is a i are bounded below. i=

10 0 O. SAVIN In the next sections we prove property (3.7) above by compactness, by letting σ 0, a i for some i. First we consider the 2D case in the last section the general case. 4. The 2 dimensional case. In order to fix ideas, we consider first the 2 dimensional case. We study the following class of solutions to the Monge-Ampere equation. Fix µ > 0 small, λ, Λ. We denote by D σ the set of convex, continuous functions such that (4.) (4.2) (4.3) λ det D 2 u Λ; u : Ω R 0 Ω, B µ (x 0 ) Ω B + /µ for some x 0 ; µh n/2 S h µ h n/2 ; (4.4) u = on Ω \ G, 0 u on G, u(0) = 0, with G a closed subset of Ω included in B σ, G Ω B σ. Proposition 4.. Assume n = 2. For any M > 0 there exists c 0 small depending on M, µ, λ, Λ, such that if u D σ σ c 0, then for some h c 0. b(h) := (sup x)h /2 > M S h Property (3.7) easily follows from the proposition above. Indeed, by choosing M = 2µ > 2b() we prove the existence of a section h c 0 such that b(h) 2b(). Also, the function v of the previous section satisfies v D c0 (after renaming the constant µ) provided that σ is sufficiently small a sufficiently large. We prove Proposition 4. by compactness. First we discuss briefly the compactness of bounded solutions to Monge-Ampere equation. For this we need to introduce solutions with possibly discontinuous boundary data. Let u : Ω R be a convex function with Ω R n bounded convex. We denote by Γ u := {(x, x n+ ) Ω R x n+ u(x)} the upper graph of u. Definition 4.2. We define the values of u on Ω to be equal to ϕ i.e if the upper graph of ϕ : Ω R { } u Ω = ϕ, Φ := {(x, x n+ ) Ω R x n+ ϕ(x)}

11 A LOCALIZATION PROPERTY is given by the closure of Γ u restricted to Ω R, Φ := Γ u ( Ω R). From the definition we see that ϕ is always lower semicontinuous. The following comparison principle holds: if w : Ω R is continuous det D 2 w Λ det D 2 u, w Ω u Ω, then w u in Ω. Indeed, from the continuity of w we see that for any ε > 0, there exists a small neighborhood of Ω where w ε < u. This inequality holds in the interior from the stard comparison principle, hence w u in Ω. Since the convex functions are defined on different domains we use the following notion of convergence. Definition 4.3. We say that the convex functions u m : Ω m R converge to u : Ω R if the upper graphs converge Γ um Γ u in the Hausdorff distance. Similarly, we say that the lower semicontinuous functions ϕ m : Ω m R converge to ϕ : Ω R if the upper graphs converge Φ m Φ in the Hausdorff distance. Clearly if u m converges to u, then u m converges uniformly to u in any compact set of Ω, Ω m Ω in the Hausdorff distance. Remark: When we restrict the Hausdorff distance to the nonempty closed sets of a compact set we obtain a compact metric space. Thus, if Ω m, u m are uniformly bounded then we can always extract a subsequence m k such that u mk u u mk Ωmk ϕ. Next lemma gives the relation between the boundary data of the limit u ϕ. Lemma 4.4. Let u m : Ω m R be convex functions, uniformly bounded, such that λ det D 2 u m Λ u m u, u m Ωm ϕ. Then λ det D 2 u Λ, the boundary data of u is given by ϕ the convex envelope of ϕ on Ω. Proof. Clearly Φ Γ u, hence Φ Γ u. It remains to show that the convex set K generated by Φ contains Γ u ( Ω R). Indeed consider a hyperplane x n+ = l(x) which lies strictly below K. Then, for all large m {u m l 0} Ω m, by Alexrov estimate we have that u m l Cd /n m

12 2 O. SAVIN where d m (x) represents the distance from x to Ω m. By taking m we see that u l Cd /n thus no point on Ω below l belongs to Γ u. In view of the lemma above we introduce the following notation. Definition 4.5. Let ϕ : Ω R be a lower semicontinuous function. When we write that a convex function u satisfies u = ϕ on Ω we underst u Ω = ϕ where ϕ is the convex envelope of ϕ on Ω. Whenever ϕ ϕ do not coincide we can think of the graph of u as having a vertical part on Ω between ϕ ϕ. It follows easily from the definition above that the boundary values of u when we restrict to the domain Ω h := {u < h} are given by ϕ h = ϕ on Ω {ϕ h} Ω h ϕ h = h on the remaining part of Ω h. The comparison principle still holds. Precisely, if w : Ω R is continuous det D 2 w Λ det D 2 u, w Ω ϕ, then w u in Ω. The advantage of introducing the notation of Definition 4.5 is that the boundary data is preserved under limits. Proposition 4.6 (Compactness). Assume λ det D 2 u m Λ, u m = ϕ m on Ω m, Ω m, ϕ m uniformly bounded. Then there exists a subsequence m k such that with u mk u, ϕ mk ϕ λ det D 2 u Λ, u = ϕ on Ω. Indeed, we see that we can also choose m k such that ϕ m k ψ. Since ϕ mk ϕ we obtain ϕ ψ ϕ, the conclusion follows from Lemma 4.4. Now we are ready to prove Proposition 4.. Proof of Proposition 4.. If c 0 does not exist we can find a sequence of functions u m D /m such that b um (h) M, h m.

13 A LOCALIZATION PROPERTY 3 By Proposition 4.6 there is a subsequence which converges to a limiting function u satisfying (4.)-(4.2)-(4.3) (see Definition 4.5) u = ϕ on Ω with (4.5) ϕ = on Ω \ {0}, ϕ(0) = 0, moreover u has an obstacle by below in Ω (4.6) u M 2 x2 2. We consider the barrier w := δ( x + 2 x2 ) + Λ δ x2 2 Nx 2 with δ small depending on µ, N large so that Then Hence which gives Λ δ x2 2 Nx 2 0 in B + /µ. w ϕ on Ω, det D 2 w > Λ. w u in Ω u δ x Nx 2. Next we construct another explicit subsolution v such that whenever v is above the two obstacles δ x Nx 2, M 2 x2 2, we have det D 2 v > Λ v. Then we can conclude that u v, we show that this contradicts the lower bound on S h. We look for a function of the form v := rf(θ) + 2M 2 x2 2, where r, θ represent the polar coordinates in the x, x 2 plane. The domain of definition of v is the angle K := {θ 0 θ π θ 0 } with θ 0 small so that 2M 2 x2 2 2 (δ x Nx 2 ) on K B µ. In the set {v M 2 x2 2} i.e. where r sin2 θ 2M 2 f

14 4 O. SAVIN we have (4.7) det D 2 v = r (f + f) sin2 θ M 2 f (f + f) sin4 θ 0 2M 4. We let f(θ) = σe C0 π 2 θ, where C 0 is large depending on θ 0, M, Λ so that (see (4.7)) det D 2 v > Λ in the set where {v M 2 x2 2}. On the other h we can choose σ small so that v δ x Nx 2 on K B µ v on the set {v M 2 x2 2}. In conclusion u v ɛx 2, hence u max{ɛx 2, δ x Nx 2 }. This implies S h Ch 2 for all small h we contradict that S h µh, h [0, ]. 5. The higher dimensional case In higher dimensions it is more difficult to construct an explicit barrier as in Proposition 4. in the case when in (3.6) only one a i is large the others are bounded. We prove our result by induction depending on the number of large eigenvalues a i. Fix µ small λ, Λ. For each increasing sequence with we consider the family of solutions α α 2... α n α µ, D µ σ(α, α 2,..., α n ) of convex, continuous functions u : Ω R that satisfy (5.) λ det D 2 u Λ in Ω, u 0 in Ω; (5.2) 0 Ω, B µ (x 0 ) Ω B + /µ for some x 0 ; (5.3) µh n/2 S h µ h n/2 ; (5.4) u = on Ω \ G;

15 A LOCALIZATION PROPERTY 5 (5.5) n n µ αi 2 x 2 i u µ αi 2 x 2 i on G, where G is a closed subset of Ω which is a graph in the e n direction is included in boundary in {x n σ}. For convenience we would like to add the limiting solutions when α k+ σ 0. We denote by D µ 0 (α,..., α k,,,..., ) the class of functions u : Ω R that satisfy properties (5.)-(5.2)-(5.3) (see Definition 4.5) u = ϕ on Ω with (5.6) ϕ = on Ω \ G; (5.7) µ where G is a closed set k αi 2 x 2 i ϕ min{,, µ G Ω {x i = 0, k αi 2 x 2 i } on G, i > k}, if we restrict to the space generated by the first k coordinates then k k { µ αi 2 x 2 i } G { µ αi 2 x 2 i }. We extend the definition of D µ σ(α, α 2,..., α n ) to include also the pairs with µ α... α k <, α k+ = = α n = for which σ = 0 i.e. D µ 0 (α, α 2,..., α k,,..., ). Proposition 4.6 implies that if u m D µ σ m (a m,..., a m n ) is a sequence with σ m 0 a m k+ for some fixed 0 k n 2, then we can extract a convergent subsequence to a function u with u D µ 0 (a,.., a l,,.., ), for some l k a... a l. Proposition 5.. For any M > 0 k n there exists C k depending on M, µ, λ, Λ, n, k such that if u D µ σ(α, α 2,..., α n ) with α k C k, σ C k then b(h) = (sup x n )h /2 M S h for some h with C k h. As we remarked in the previous section, property (3.7) therefore Lemma 3.3 follow from Proposition 5. by taking k = n M = 2µ. We prove the proposition by induction on k.

16 6 O. SAVIN Lemma 5.2. Proposition 5. holds for k =. Proof. By compactness we need to show that there does not exist u D µ 0 (,..., ) with b(h) M for all h. The proof is almost identical to the 2 dimensional case. One can see as before that u max{δ x Nx n, M 2 x2 n} then construct a barrier of the form v = rf(θ) + 2M 2 x2 n, θ 0 θ π 2 where r = x θ represents the angle in [0, π/2] between the ray passing through x the {x n = 0} plane. Now, det D 2 v = f ( + f f cos θ f sin θ r r cos θ We have f r > sin2 θ 2M 2 on the set {v > M 2 x2 n} we choose a function of the form which is decreasing in θ. Then det D 2 v > f + f f if C 0 is chosen large. We obtain as before that which gives we reach a contradiction. f(θ) := νe C0( π 2 θ) ) n 2 sin 2 θ M 2. ( sin 2 ) n θ 0 2M 2 > Λ u max{δ x Nx n, ɛx n } S h Ch n Now we prove Proposition 5. by induction on k. Proof of Proposition 5.. In this proof we denote by c, C positive constants that depend on M, µ, λ, Λ, n k. We assume that the statement holds for k we prove it for k +. It suffices to show the existence of C k+ only in the case when α k < C k, otherwise we use the induction hypothesis. If no C k+ exists then we can find a limiting solution with u D µ 0 (,,...,,,..., ) (5.8) b(h) < Mh /2, h > 0 where µ depends on µ C k. We show that such a function u does not exist.

17 A LOCALIZATION PROPERTY 7 Denote x = (y, z, x n ), y = (x,..., x k ) R k, z = (x k+,..., x n ) R n k. On the Ω plane we have ϕ w := δ x 2 + δ z + Λ δ n x2 n Nx n for some small δ depending on µ, N large so that Since we obtain u w on Ω hence Λ δ n x2 n Nx n 0 on B + / µ. det D 2 w > Λ, (5.9) u(x) δ z Nx n. We look at the section S h of u. From (5.8)-(5.9) we see that (5.0) S h {x n > N (δ z h)} {x n Mh /2 }. with We notice that an affine transformation x T x, T x := x + ν z + ν 2 z ν n k z n k + ν n k x n ν, ν 2,..., ν n k span{e,..., e k } i.e a sliding along the y direction, leaves the z, x n coordinate invariant together with the subspace (y, 0, 0). The section S h := T S h of the rescaling satisfies (5.0) ũ = ϕ on S h with ũ(t x) = u(x) ϕ = ϕ on G := {ϕ h} G, ϕ = h on S h \ G. From John s lemma we know that S h is equivalent to an ellipsoid E h. We choose T an appropriate sliding along the y direction, so that T E h becomes symmetric with respect to the y (z, x n ) subspaces, thus x h + c(n) S h /n AB S h C(n) S h /n AB, det A = the matrix A leaves the y the (z, x n ) subspaces invariant. By choosing an appropriate system of coordinates in the y z variables we may assume A(y, z, x n ) = (A y, A 2 (z, x n )) with with 0 < β β k, β β 2 0 A = β k

18 8 O. SAVIN γ k+ 0 0 θ k+ 0 γ k+2 0 θ k+2 A 2 = γ n θ n θ n with γ j, θ n > 0. Next we use the induction hypothesis show that S h is equivalent to a ball. Lemma 5.3. There exists C 0 such that S h C 0 h n/2 B +. Proof. Using that we obtain We need to show that Since S h satisfies (5.0) we see that S h h n/2 x h + ch /2 AB S h Ch /2 AB. A C. S h { (z, x n ) Ch /2 }, which together with the inclusion above gives A 2 C hence Also S h contains the set which implies We define the rescaling {(y, 0, 0) γ j, θ n C, θ j C. y µ /2 h /2 } G, β i c > 0, i =,, k. w(x) = hũ(h/2 Ax) which is defined in a domain Ω w := h /2 A Sh such that w = ϕ w on Ω w with B c (x 0 ) Ω w B + C, 0 Ω w, ϕ w = on Ω w \ G w, µ β 2 i x 2 i ϕ w min{, µ β 2 i x 2 i } on G w, where G w := h /2 A G. This implies that w D µ 0 (β, β 2,..., β k,,..., ) for some value µ depending on µ, M, λ, Λ, n, k. We claim that b u (h) c. First we notice that b u (h) = bũ(h) θ n.

19 A LOCALIZATION PROPERTY 9 Since θ n βi γj = det A = γ j C, we see that if b u (h) ( therefore θ n ) becomes smaller than a critical value c then β k C k ( µ, M, λ, Λ, n), with M := 2 µ, by the induction hypothesis b w ( h) M 2b w () for some h > C k. This gives b u (h h) b u (h) = b w( h) b w () 2, which implies b u (h h) 2b u (h) our claim follows. Next we claim that γ j are bounded below by the same argument. Indeed, from the claim above θ n is bounded below if some γ j is smaller than a small value c then β k C k ( µ, M, λ, Λ, n) with M := 2M µc. By the induction hypothesis b w ( h) M 2M c b w (), hence b u (h h) b u (h) 2M c which gives b u (h h) 2M, contradiction. In conclusion θ n, γ j are bounded below which implies that β i are bounded above. This shows that A is bounded the lemma is proved. Next we use the lemma above show that the function u has the following property. Lemma 5.4. If for some p, q > 0, u p( z qx n ), q q 0 then u p ( z (q η)x n ) for some p p, with η > 0 depending on q 0 µ, M, λ, Λ, n, k. Proof. From Lemma 5.3 we see that after performing a linear transformation T (siding along the y direction) we may assume that Let S h C 0 h /2 B. w(x) := h u(h/2 x)

20 20 O. SAVIN for some small h p. Then our hypothesis becomes S (w) := Ω w = h /2 S h B + C 0 (5.) w p h /2 ( z qx n), Moreover the boundary values ϕ w of w on Ω w satisfy ϕ w = on Ω w \ G w µ y 2 ϕ w min{, µ y 2 } on G w, where G w := h /2 {ϕ h}. Next we show that ϕ w v on Ω w where v is defined as v := δ x 2 + Λ δ n (z qx n ) 2 + N(z qx n ) + δx n, δ is small depending on µ C 0, N is chosen large such that Λ δ n t2 + Nt is increasing in the interval t ( + q 0 )C 0. From the definition of v we see that det D 2 v > Λ. On the part of the boundary Ω w where z qx n we use that Ω w B C0 obtain v δ( x 2 + x n ) ϕ w. On the part of the boundary Ω w where z > qx n we use (5.) obtain = ϕ w C( z qx n ) C(z qx n ) with C arbitrarily large provided that h is small enough. We choose C such that the inequality above implies Then Λ δ n (z qx n ) 2 + N(z qx n ) < 2. ϕ w = > 2 + δ( x 2 + x n ) v. In conclusion ϕ w v on Ω w hence the function v is a lower barrier for w in Ω w. Then w N(z qx n ) + δx n, since this inequality holds for all directions in the z-plane, we obtain Scaling back we get w N( z (q η)x n ), η := δ N. u p ( z (q η)x n ) in S h. Since u is convex u(0) = 0, this inequality holds globally, the lemma is proved.

21 A LOCALIZATION PROPERTY 2 We remark that Lemma 5.4 can be used directly to prove Proposition 4. Lemma 5.2. End of the proof of Proposition 5.. From (5.9) we obtain an initial pair (p, q 0 ) which satisfies the hypothesis of Lemma 5.4. We apply this lemma a finite number of times obtain that u ɛ( z + x n ), we contradict that S h is equivalent to a ball of radius h /2. References [C] [C2] [TW] Caffarelli L., A localization property of viscosity solutions to the Monge-Ampere equation their strict convexity, Ann. of Math. 3 (990), Caffarelli L., Interior W 2,p estimates for solutions of Monge-Ampere equation, Ann. of Math. 3 (990), Trudinger N., Wang X.J, Boundary regularity for Monge-Ampere affine maximal surface equations, Ann. of Math. 67 (2008), Department of Mathematics, Columbia University, New York, NY address: savin@math.columbia.edu