Existence and regularity of the solutions of some singular Monge–Ampère equations

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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Existence and regularity of the solutions of some singular Monge–Ampère equations Haodi Chen a , Genggeng Huang b,a,∗ a Mathematical Science Institute, The Australian National University, Canberra, ACT 2601, Australia b School of Mathematical Sciences, Fudan University, Shanghai, China

Received 21 August 2018; revised 30 December 2018

Abstract In this paper, we investigate the following singular Monge–Ampère equation ⎧ ⎪ ⎨det D 2 u = ⎪ ⎩u = 0,

1 (H u)n+k+2 u∗ k

on

in  ⊂⊂ Rn ,

(0.1)

∂

where k ≥ 0, H < 0 are constants and u∗ = x · ∇u(x) − u(x) is the Legendre transformation of u. Equation (0.1) is related to proper affine hyperspheres. We will show the existence of solutions of (0.1) ¯ via regularization method. Using the technique in [10,12], we also obtain the optimal u ∈ C ∞ () ∩ C() graph regularity of the solution of (0.1). © 2019 Elsevier Inc. All rights reserved.

1. Introduction Affine hypersphere is an important class of hypersurfaces in affine geometry. The study of affine hyperspheres dates back to the beginning of 20th century due to the work of G. Tzitzèica [17,18] and Blaschke [1]. There are two typical types of affine hyperspheres which are called proper or improper affine hyperspheres. Suppose F : M → Rn+1 is an immersed hypersurface * Corresponding author.

E-mail addresses: [email protected] (H. Chen), [email protected] (G. Huang). https://doi.org/10.1016/j.jde.2019.01.030 0022-0396/© 2019 Elsevier Inc. All rights reserved.

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and can be locally represented as a graph: xn+1 = f (x1 , · · · , xn ) = f (x). From the view-point of analysis, one can characterize affine hyperspheres via the following Monge–Ampère equations: I. F is a proper affine hypersphere with center at the origin and affine mean curvature H , if and only if f satisfies   2    ∂ f det  = (H (x · ∇f − f ))n+2 .  ∂x ∂x  i

(1.1)

j

II. F is an improper affine hypersphere with center at the origin, if and only if f satisfies   2    ∂ f  = c, det  ∂x ∂x  i

for some constant c > 0.

(1.2)

j

The derivation of (1.1) and (1.2) for convex affine hyperspheres can be found in [5]. In fact, following similar arguments, one can prove (1.1) and (1.2) also hold for general affine hyperspheres. In the case of locally uniformly convex hypersurface F (M), both proper and improper affine hyperspheres are well studied. Also, in the locally uniformly convex case, we can always choose the orientation of F (M) such that the affine metric g is positive definite. So there are two types of convex proper affine hyperspheres – elliptic and hyperbolic, depending on the affine mean curvature positive or negative. For these results, we refer to [4–7,16,19]. Inspired by these studies, it is natural to consider the non-convex affine hyperspheres. In this paper, we are only interested in the non-convex proper affine hyperspheres. It seems difficult for us to study (1.1) as it is a fully nonlinear hyperbolic PDE. We turn to study a simpler case. Throughout this paper, we call a hypersurface V as a cone if and only if x ∈ V ⇔ tx ∈ V , ∀t > 0. We define the following special cone. Definition 1.1. 2 2 V,k = {(x1 , · · · , xn+k+1 ) ∈ Rn+k+1 | xn+1 + · · · + xn+k+1 = h (x1 , · · · , xn )} where  is a bounded convex domain in Rn and h is a 1-order homogeneous convex function with ∇h(Rn ) = . In order to find the proper affine hypersphere which is asymptotic to cone V,k in Rn+k+1 , we need to investigate the following singular Monge–Ampère equation ⎧ 1 ⎨det D 2 u = (H u)n+k+2 u∗ k ⎩ u = 0, on ∂

in  ⊂⊂ Rn ,

(1.3)

where k ≥ 0, H < 0 are constants and u∗ = x · ∇u(x) − u(x) is the Legendre transformation of u. The derivation of the above Monge–Ampère equation will be represented in Section 2. Our first theorem is the existence of such proper affine hyperspheres. Theorem 1.1. Let  ⊂ Rn be a bounded convex domain which contains the origin 0. For any ¯ ∩ C ∞ (). As a consequence, k ≥ 0 and H < 0, (1.3) admits a unique convex solution u ∈ C()

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for any cone V,k in Rn+k+1 , there exists a proper affine hypersphere with affine mean curvature H < 0 and asymptotic to V as x → ∞. For k = 0, it is well-known that Cheng–Yau [6] proved that there exists a unique convex proper affine hypersphere which is asymptotic to the boundary of a given convex cone in Rn+1 with affine mean curvature H < 0. Recently, Jian–Li [9] obtained the optimal boundary regularity of solution to (1.3) when k = 0. For general proper affine hyperspheres, to the best knowledge of the authors, there are few papers [14,15,20] concerning the proper affine hyperspheres with indefinite affine metrics. Our second main result is about the graph regularity of the solutions of (1.3). Theorem 1.2. Suppose  is a bounded, uniformly convex domain in Rn with C ∞ boundary. Let u be a convex solution to (1.3). Then the graph Mu is C n+k+2,α smooth up to its boundary, for any α ∈ (0, 1). Moreover, if n + k is even, the graph Mu is C ∞ smooth up to its boundary. For k = 0, Theorem 1.2 is already proved in [10,12]. Remark 1.1. The reason we consider such type of affine hyperspheres is due to the following examples: 2 • Hyperboloid of two sheets: x12 + · · · + xn2 − xn+1 = −1. This is a hyperbolic type affine hypersphere asymptotic to the cone {(x1 , · · · , xn+1 )|xn+1 = x12 + · · · + xn2 }. 2 • Hyperboloid of one sheet: x12 + · · · + xn2 − xn+1 = 1. This is an affine hypersphere which 2 embraces two cones {(x1 , · · · , xn+1 )|xn+1 = x12 + · · · + xn2 }. 2 2 − · · · − xn+k = 1. • General Hyperboloid: x12 + · · · + xn2 − xn+1

2. The proof of main theorems Suppose F is an (n + k)-dimensional affine hypersphere with affine mean curvature H < 0 and F is asymptotic to V,k near ∞. Due to the rotational symmetry of V,k in (xn+1 , · · · , xn+k+1 ), it is natural to find affine hypersphere F with rotational symmetry in (xn+1 , · · · , xn+k+1 ). We now derive the singular Monge–Ampère equation (1.3). The derivation of (1.3). Without loss of generality, locally, we may write F as ⎞

n+k 

F = (x1 , · · · , xn+k , f (x1 , · · · , xn+k )) = ⎝x1 , · · · , xn+k , g 2 (x1 , · · · , xn ) − xr2 ⎠ ⎛

r=n+1

for some f, g > 0. Since V = {(x1 , · · · , xn+1 )|xn+1 = h (x1 , · · · , xn )} is a convex cone, we n+k  can assume g is convex. Denote xr2 by P 2 , P > 0. By a direct computation, one has fi =  ggi , g 2 −P 2

fr = − 

xr g 2 −P 2

r=n+1

and

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P 2 gi gj −  , g 2 − P 2 ( g 2 − P 2 )3 ggi xr fir =  , ( g 2 − P 2 )3 fij = 

ggij

frs = − 

δrs g2

− P2

xr x s −  , 2 ( g − P 2 )3

for 1 ≤ i, j ≤ n and n + 1 ≤ r, s ≤ n + k. Here δrs = 0, if r = s; δrs = 1, if r = s. Hence we have  n   g  | det D 2 f | =  det D 2 g · (−f )−3k · f 2k−2 (f 2 + P 2 ) f =

g n+2 f n+k+2

det D 2 g

and x · ∇f − f =

g ∗ ·g f

where g ∗ is the Legendre transformation of g. From (1.1), one has det D 2 g = g k (Hg ∗ )n+k+2 . Set u = g ∗ . Then u satisfies ⎧ ⎨det D 2 u = ⎩

u = 0,

This ends the derivation of (1.3).

1 (H u)n+k+2 u∗ k on ∂.

(2.1)

in ,

2

For simplicity, in the following, we always assume H = −1 in (1.3). We will use the ε-regularization method to prove Theorem 1.1. Proof of Theorem 1.1. In order to prove the first part of Theorem 1.1, we consider an ε-regularization problem, ⎧ 1 ⎨det D 2 uε = in , n+k+2 (−uε ) u∗ε k (2.2) ⎩ uε = −ε, on ∂. Denote the right-hand side of (2.2) by G(x, uε , ∇uε ). Then Gu (x, uε , ∇uε ) > 0 which implies the solution of (2.2) is unique. In the following, we establish the a priori estimates of uε . Consider two balls B2r (0) ⊂  ⊂ B R (0) for r, R > 0. Set 2

v = −a (r + ε)2 − |x|2

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with 0 < a < 1 to be determined. In Br (0), we have n

det D 2 v = a n (r + ε)2 ((r + ε)2 − |x|2 )− 2 −1 ≤ (−v)−n−k−2 (x · ∇v − v)−k which is equivalent to a 2n+2k+2 ≤ (r + ε)−2k−2 . Then we can choose a ∈ (0, 1) small independent of ε such that v(x) is a super-solution of (2.2). By the maximum principle, one has uε (0) ≤ −a(r + ε).

(2.3)

u∗ε = x · ∇uε − uε ≥ −uε (0) ≥ ar.

(2.4)

From this, by convexity one gets

Similarly, for  v˜ = −A (R + ε)2 − x 2 in BR (0), one can directly compute that v˜ is a sub-solution of (2.2) provided A > 1 is large and independent of ε. By maximum principle, one has uε L∞ () ≤ AR. Suppose p = (0 , −γ ) ∈ ∂ and {xn = −γ } is a supporting plane of  at p. Now let 1

w = (|x |2 − C)(xn + γ ) n+k+1 − ε for some constant C to be determined later. By a direct computation, one has ⎧ 1 ⎪ 2(xn + γ ) n+k+1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨0, ∂ 2w n+k 2 = xi (xn + γ )− n+k+1 , ∂xi ∂xj ⎪ ⎪ n + k + 1 ⎪ ⎪ ⎪ 2n+2k+1 n+k ⎪ ⎪ (C − |x |2 )(xn + γ )− n+k+1 , ⎩ 2 (n + k + 1)

i=j =  n, i = j and i, j = n, i = j = n, i = j = n,

from which we obtain   n+2k+2 n−2 det D 2 w = Cn,k (C − |x |2 )(xn + γ )− n+k+1 − |x |2 (xn + γ ) n+k+1 . Taking C sufficiently large, we have n+2k+2 1 det D 2 w ≥ Cn,k (C − |x |2 )(xn + γ )− n+k+1 ≥ (−w)−n−k−2 (x · ∇w − w)−k , 2

in .

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Then by maximum principle, one gets 1

uε (x) − uε (0 , −γ ) ≥ (|x |2 − C)(xn + γ ) n+k+1 ¯ For δ > 0, set near (0 , −γ ). By convexity, this implies uε ∈ C n+k+1 (). 1

δ = {x ∈ |d(x, ∂) > δ}. Claim: For any x0 ∈ δ , p0,ε ∈ ∇uε (x0 ), there exists t0 > 0 independent of ε such that {uε (x) < uε (x0 ) + p0,ε · (x − x0 ) + t0 } ⊂⊂ . ¯ we may assume uε → u as ε → 0 in C() ¯ and the Monge– Suppose not, by uε ∈ C n+k+1 (), Ampère measure μuε → μu weakly in measure. Also the contradiction assumption implies there exists x0 ∈ δ , p0 ∈ ∇u(x0 ) such that {u(x) = u(x0 ) + p0 · (x − x0 )} contains a segment. We know u∞ ≥ ar > 0 by (2.3) and u < −Cδ < 0 in δ , ∀δ > 0. This implies fixed δ > 0, for ε small enough, 1

0 < cδ ≤ μuε ≤ Cδ < +∞,

in δ .

By the weak convergence of the measure μuε , we also know 0 < cδ ≤ μu ≤ Cδ < +∞, in δ . By [Theorem 1, [2]],  = {u(x) = u(x0 ) + p0 · (x − x0 )} can’t have extreme point in , i.e. there exists a segment l ⊂  such that the extreme points of this segment lie on ∂. By convexity, this implies u ≡ 0 which yields a contradiction. By the above claim, for x0 ∈ δ , we know t0 = {uε (x) < uε (x0 ) + p0 · (x − x0 ) + t0 = l0 (x)} ⊂⊂ . Then by Pogorelov’s interior estimate [Theorem 17.19, [8]], we know (l0 − u)|D 2 uε |(x) ≤ C(n, |uε |C 0,1 (t ) , δ), 0

in t0 .

This implies |D 2 uε (x0 )| ≤ Cδ . From the arbitrariness of x0 , we know D 2 uε L∞ (δ ) ≤ Cδ and (2.2) is uniformly elliptic in δ . By Evans–Krylov’s estimates [Theorem 17.15, [8]] and standard bootstrap arguments, we know that uε C k,α (δ ) ≤ C(δ, k),

∀k ∈ Z+ ,

∀δ > 0.

(2.5)

From [3], we know the existence of a smooth solution uε of (2.2). Let ε → 0, by the locally ¯ of (2.2). uniform estimate (2.5), one obtains the desired solution u ∈ C ∞ () ∩ C() It remains to prove the second part of Theorem 1.1. We follow the idea in [13]. We only need to show ∇u() = Rn . Suppose e−1 = (−1, 0, · · · , 0) ∈ ∂,

xλ = (−1 + λ, 0, · · · , 0),

For any λ > 0 small, set uλ (x) =

u(λx) λ

n n+k+1

,

x∈

 . λ

d(xλ , ∂) = d(xλ , e−1 ) = λ.

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Then by a direct computation, one knows uλ satisfies (1.3) in  λ . We now estimate the value of  xλ uλ (yλ ), yλ = λ . In B1 (yλ ) ⊂ λ , set h(x) = −a 1 − |x − yλ |2 . By a direct computation, one has det D 2 h = −(n+k+2)

h

an (1 − |x − yλ |2 ) ∗ −k

(h )

=a

n+2 2

,

−(n+2k+2)

(2.6) 2 − n+2 2

(1 − |x − yλ | )

−k

(1 + yλ · (x − yλ ))

.

h is a super-solution of (1.3) in B1 (yλ ) is equivalent to a 2n+2k+2 ≤ (1 + yλ · (x − yλ ))−k . k

This implies a ∼ cλ 2n+2k+2 . By maximum principle, one has n

2n+k

n

2n+k

u(xλ ) = λ n+k+1 uλ (yλ ) ≤ λ n+k+1 h(yλ ) = −cλ 2n+2k+2 = −c(d(xλ , ∂)) 2n+2k+2 . From this, we see ∇u(x) blows up as x → ∂. This implies ∇u() = Rn .

(2.7)

2

By the proof of Theorem 1.1, we know |∇u| blows up on the boundary ∂. However as a graph Mu near ∂, Mu earns much more regularity. Suppose p = (−1, 0, · · · , 0) ∈ ∂ and e1 = (1, 0, · · · , 0) is the inner normal of ∂ at p. In order to study the graph regularity near ∂, we make the rotation of the coordinates: y1 = −xn+1 ,

yl = x l ,

l = 2, · · · , n,

yn+1 = x1 .

In the new coordinates, the graph of u near p can be represented as yn+1 = v(y). Noticing that u(v(y), y2 , · · · , yn ) + y1 = 0, by a direct differentiation, one obtains, 1 , ux1 ux vyi = − i , ux1 vy1 = −

x · ∇u − u =

i = 2, · · · , n,

(2.8)

y · ∇v − v . vy1

By ux1 < 0 near p, we see that vy1 > 0. Since Gaussian curvature is invariant under this transformation, we see that det D 2 v = K(1 + |∇v|2 )

n+2 2

 = det D 2 u

1 + |∇v|2 1 + |∇u|2

 n+2 2 .

After a computation, one gets ⎧   ⎨det D 2 v = vy1 n+k+2 (v ∗ )−k , ⎩

y1

v(0, y) ˜ = φ(y), ˜

in {y1 > 0} ∩ B1 (0),

on {y1 = 0} ∩ B1 (0)

(2.9)

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where y˜ = (y2 , · · · , yn ) and φ(y) ˜ is a uniformly convex function depending only on ∂ with φ(0) = −1. The optimal regularity of (2.9) was studied by Jian–Wang [10] for k = 0. Later, Jian–Wang [11] consider the following problem: F (D 2 u, Du, u, u(0, y) ˜ = φ(y), ˜

uy1 , y) = f (y), y1

in

{y1 > 0} ∩ B1 (0),

(2.10)

on {y1 = 0} ∩ B1 (0).

Denote F = F (M, p, z, r, y), where M ∈ S n×n is a symmetric matrix, p ∈ Rn , z, r ∈ R1 , y ∈ {y1 > 0} ∩ B1 (0). F satisfies the following structural conditions: (F1). F is uniformly elliptic, i.e. there exist constants λ,  > 0 such that ∀ξ ∈ Rn λ|ξ |2 ≤

∂F (M, p, z, r, y)ξi ξj ≤ |ξ |2 ∂Mij

in S n×n × Rn+2 × {{y1 > 0} ∩ B1 (0)}. (F2). F is concave with respect to the variables M ∈ S n×n . (F3). F satisfies the growth condition: (i) |F (0, p, z, 0, x)| ≤ C(1 + |p|2 ). (ii) |Fx | + |Fz | + (1 + |p|)(|Fp | + |Fr |) ≤ C(1 + |p|2 + |M|). (F4). Fz ≤ 0. Set μ(y) = −

uy Fr (D 2 u, Du, u, 1 , y) + 1. FM11 y1

Then in Jian–Wang [11], they proved the following theorem. Theorem A. [11] If f ∈ C μ(0)−2 ({y1 > 0} ∩ B1 (0)), μ(0) > 2 and F satisfies conditions (F1)–(F4), then for any ε > 0 small, there exists r > 0 such that the solution u of (2.10) belongs to C μ(0)−ε ({y1 > 0} ∩ Br (0)). We will apply this result to obtain the first part of Theorem 1.2. Proof of the first part of Theorem 1.2. Set F (D 2 v, Dv, v,

  n+k+2 n 1 k vy1 vy1 , y) = (det D 2 v) n − (v ∗ )− n = 0. y1 y1

Then (F 2), (F 4) is obviously true. Suppose e−1 = (−1, 0, · · · , 0) ∈ ∂ and its inner normal is e1 = (1, 0, · · · , 0). By the uniform convexityof ∂, we may find two tangential balls Br ((1 − r)e−1 ) ⊂  ⊂ BR+1 (Re1 ). Set h1 (x) = −A (R + 1)2 − |x − Re1 |2 . Then det D 2 h1 =

An (R + 1)2 ((R + 1)2 − |x − Re1 |2 )

n+2 2

,

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and (−h1 )−n−k−2 (h∗1 )−k =A−(n+2k+2) ((R + 1)2 − |x − Re1 |2 )−

n+2 2

((R + 1)2 − (R − x1 )R)−k .

Since (R + 1)2 − (R − x1 )R has positive lower bound, for A large enough, h1 is a super-solution of (1.3). Similarly, we may construct sub-solution h2 (x) = −a r 2 − |x − (1 − r)e−1 |2 for a small enough. By comparison principle, one gets h1 ≤ u ≤ h2 . Since yn+1 = v(y) represents the same graph as xn+1 = u(x), this comparison property still holds after the rotation of coordinates, i.e. 

 2 y y2 − (R + 1)2 − 12 − |y| ˜ 2 + R ≤ v ≤ − r 2 − 12 − |y| ˜ 2+r −1 A a for |y| ≤ r0 small and y1 > 0. This also proves vy1 ≡ 0 on {y1 = 0}. By Taylor’s expansion, one sees that C1 |y|2 ≤ v(y) + 1 ≤ C2 |y|2 ,

in

Br0

for some constants C1 , C2 > 0. For any λ > 0 small, we define vλ (y) = vλ (y) ≤ C2 |y|2 in B r0 ∩ {y1 > 0} and

v(λy)+1 . λ2

Then C1 |y|2 ≤

λ

 det D 2 vλ =

vλ,y1 y1

n+k+2

(1 + λ2 (y · ∇vλ − vλ ))−k .

(2.11)

Set lλ = qλ · (y − p) ˜ + vλ (p) ˜ where p˜ = (1, 0, · · · , 0), qλ = ∇vλ (p). ˜ By the convexity and local boundedness of vλ , we know that 1 ≤ 1 + λ2 (y · ∇vλ − vλ ) ≤ CR 2 in any ball BR provided λ is small enough. Claim: there exists ε0 independent of λ such that {vλ < lλ + ε0 } ⊂⊂ Rn+ . Suppose not, we may find a sequence of vλ → v¯ such that {v¯ = l0 } contains at least a segment where l0 is the supporting plane of v¯ at p. ˜ Also v¯ solves  det D 2 v¯ =

v¯y1 y1

n+k+2 ,

in Rn+

(2.12)

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with C1 y 2 ≤ v(y) ¯ ≤ C2 y 2 . By convexity, v¯y1 (y1 , y) ˜ ≥

˜ − v(0, ¯ y) ˜ v(y ¯ 1 , y) ≥ C1 y1 y1

and v¯y1 (y1 , y) ˜ ≤

˜ − v(y ¯ 1 , y) ˜ v(2y ¯ 1 , y) ≤ (4C2 − C1 )y1 . y1

Hence by [Theorem 1, [2]], one sees {v¯ = l0 } can’t have extreme points in Rn+ . We may assume t (1, 0, · · · , 0) ∈ {v¯ = l0 } for any t > 0. This yields a contradiction since v¯ has quadratic growth and l0 only admits linear growth. This proves the claim. Set  = {vλ < lλ + ε0 } and  1 = {vλ < 2

lλ + 12 ε0 }. Then one can apply Pogorelov’s interior estimates to get D 2 vλ L∞ ( 1 ) ≤ C. 2

Scaling back, one gets |D 2 v(λ, 0, · · · , 0)| ≤ C. This proves that v ∈ C 1,1 (Br (0) ∩ {y1 > 0}) for r > 0 small. From the above arguments, we vy see that C1 ≥ y11 ≥ C2 , C3 ≥ v ∗ ≥ C4 for some constants Ci > 0, i = 1, 2, 3, 4. This implies (F 1), (F 3) is true. An easy computation yields that μ(0) = −

vy Fr (D 2 v, Dv, v, 1 , 0) + 1 = n + k + 3. FM11 y1

Then from Theorem A, we complete the proof of the first part of Theorem 1.2.

2

In general, the regularity for the solutions of (2.10) is optimal. Recently, Jian–Wang–Zhao [12] proved the C ∞ graph regularity for hyperbolic affine hyperspheres, i.e. k = 0 in the even dimensional case. The second part of Theorem 1.2 is inspired by their work. Proof of the second part of Theorem 1.2. The main idea of the proof comes from [12]. For the readers’ convenience, we provide a proof here with a slightly different computation. From equation (2.9), one can easily deduce that  n+k+2 v11 =

v1 y1

(v ∗ )−k

det Dy2˜ v

n −

∂ 2 det D 2 v i,j =2 v1i v1j ∂v1i ∂v1j det Dy2˜ v

where y˜ = (y2 , · · · , yn ). We adopt the following notations

(2.13)

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f (i) =

∂if ∂y1i

11

,

d˜ = det Dy2˜ v, g˜ = −

n 

v1i v1j

i,j =1

∂ 2 det D 2 v . ∂v1i ∂v1j

Then by differentiation, we can get the following iteration formulas: fi hi + gi , y1 fi−1 (1) , fi = fi−1 − y1 (1)

fi

=

(2.14)

hi = hi−1 − 1, (1)

(1)

gi = gi−1 + hi−1

fi−1 y1

where f0 = v11 −

v1 , y1

  (n + k + 2)(v ∗ )−k v1 n+k+1 − 1, y1 d˜  (1)  n+k+2  ∗ −k (1) g˜ v1 (v ) + g0 = y1 det Dx2˜ v d˜

h0 =

Moreover, we have h0 (0, y) ˜ = n + k + 1. In order to show the smoothness, by [10], we only need to prove gn+k (0, y) ˜ = 0.

(2.15)

For k = 0, the above iteration formulas are proved in [10]. The arguments for general k are the same. The key point to show (2.15) is the following lemma. Lemma 2.1.

 v (2l−1) (0, y) ˜ = 0,

l = 1, · · · ,

 n+k+3 . 2

We will prove Lemma 2.1 via induction. l = 1, it is true. Suppose it is true for l ≤ m < Proof. n+k+3 . Then by (2.13), one has 2 ⎛  n+k+2 ⎜ v (2m+1) = ⎝

v1 y1

(v ∗ )−k

det Dy2˜ v

⎞(2m−1) ∂ 2 det D 2 v v v i,j =2 1i 1j ∂v1i ∂v1j ⎟ ⎠ det Dy2˜ v

n −

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=

Ci 1 i 2 i 3

i1 +i2 +i3 =2m−1

+

n 

v1 y1

n+k+2 (i1 )





Ci1 i2 i3 (v1i v1j )(i1 )

i,j =2 i1 +i2 +i3 =2m−1

Notice that

v1 y1

=

1 0

 (p) v1 y1

(i2 )

((det Dy2˜ v)−1 )(i3 ) .

v1 y1

(p)

1

=

λp v (p+2) (λy1 , y)dλ. ˜ 0

(0, y) ˜ = 0 if p ≤ 2m − 3 is odd. And

∗ (p)

(v )

∂ 2 det D 2 v ∂v1i v1j

(2.16)

v (2) (λy1 , y)dλ. ˜ One has 

This implies

((v ∗ )−k )(i2 ) ((det Dy2˜ v)−1 )(i3 )

n n   (p) (p−1) (p+1) (p) =( yi v1i ) = y1 v + (p − 1)v + yi v i . i=1

i=2

This implies (v ∗ )(p) (0, y) ˜ = 0 if p ≤ 2m − 1 is odd. Since one of i1 , i2 , i3 must be odd, by induction assumption, at (0, y), ˜ there is only one possible nonzero term on the right handside of (2.16), (n + k + 2)

 n+k+1  (2m−1) v1 y1

v1 y1

(v ∗ )k det Dy2˜ v

=

n + k + 2 (2m+1) (0, y). ˜ v 2m

In getting the above equality, we have used det Dy2˜ v(0, y) ˜ = (v ∗ )−k vyn+k+1 (0, y) ˜ 1 y1 which comes from vy1 yi (0, y) ˜ = 0, i = 2, · · · , n, vy1 y1 (0, y) ˜ = limy1 →0 (2m+1) assumption 2m < n + k + 2, one has v (0, y) ˜ = 0. 2

vy1 (y1 ,y) ˜ y1

and (2.9). By

Now from the iteration formulas, one easily derives that

(n+k) gn+k = g0

+

n+k−1   i=0

(1) fi hi y1

(n+k−i−1)

(n+k) = g0

+

n+k−1  i=0



 (1) h0

v1 y1

(i+1) (n+k−i−1) . (2.17)

Then by a similar argument as in Lemma 2.1, one has gn+k (0, y) ˜ = 0 provided n + k is even. This ends the proof of present theorem. 2

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Acknowledgments The authors would like to thank Professor Xu-Jia Wang for bringing us to this problem and for some helpful discussions. This work was supported by Australian Research Council FL130100118 and DP170100929. The work of the second author is supported by NSFC11871160 and the Scholarship of International Postdoctoral Exchange Fellowship Program. The authors also would like to thank referee(s) for his/her kindly and helpful comments. References [1] W. Blaschke, Ber. Leip. Akad. 69 (1917) 167, Ber. Leip. Akad. 70 (1918) 27. [2] L. Caffarelli, A localization property of viscosity solutions to the Monge–Ampère equation and their strict convexity, Ann. of Math. 131 (1) (1990) 129–134. [3] L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I. Monge– Ampère equation, Comm. Pure Appl. Math. 37 (3) (1984) 369–402. [4] E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens, Michigan Math. J. 5 (1958) 105–126. [5] E. Calabi, Complete affine hypersphere, I, in: Ist. Naz. Alta Math. Symp. Math., vol. X, 1972, pp. 19–38. [6] S.Y. Cheng, S.T. Yau, On the regularity of the Monge–Ampère equation det((∂ 2 u/∂x i ∂x j )) = F (x, u), Comm. Pure Appl. Math. 30 (1977) 41–68. [7] S.Y. Cheng, S.T. Yau, Complete affine hypersurfaces. Part I. The completeness of affine metrics, Comm. Pure Appl. Math. 39 (6) (1986) 839–866. [8] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition Classics in Mathematics, Springer-Verlag, Berlin, 2001, xiv+517 pp. [9] H. Jian, Y. Li, Optimal boundary regularity for a singular Monge–Ampère equation, J. Differential Equations 264 (12) (2018) 6873–6890. [10] H. Jian, X.-J. Wang, Bernstein theorem and regularity for a class of Monge–Ampère equations, J. Differential Geom. 93 (3) (2013) 431–469. [11] H. Jian, X.-J. Wang, Optimal boundary regularity for nonlinear singular elliptic equations, Adv. Math. 251 (2014) 111–126. [12] H. Jian, X.-J. Wang, Y. Zhao, Global smoothness for a singular Monge–Ampère equation, J. Differential Equations 263 (11) (2017) 7250–7262. [13] F. Lin, L. Wang, A class of fully nonlinear elliptic equations with singularity at the boundary, J. Geom. Anal. 8 (4) (1998) 583–598. [14] M.A. Magid, P.J. Ryan, Flat affine spheres in R3 , Geom. Dedicata 33 (3) (1990) 277–288. [15] M.A. Magid, P.J. Ryan, Affine 3-spheres with constant affine curvature, Trans. Amer. Math. Soc. 330 (2) (1992) 887–901. [16] A.V. Pogorelov, On the improper convex affine hyperspheres, Geom. Dedicata 1 (1972) 33–46. [17] G. Tzitzèica, Sur certaines surfaces réglées, C. R. Acad. Paris 145 (1907) 132–133, C. R. Acad. Paris 146 (1908) 165–166. [18] G. Tzitzèica, Sur une nouvelle class de surfaces, Rend. Circ. Mat. Palermo 25 (1908) 180–187, Rend. Circ. Mat. Palermo 28 (1909) 210–216. [19] T. Sasaki, Hyperbolic affine hyperspheres, Nagoya Math. J. 77 (1980) 107–123. [20] L. Vrancken, The Magid–Ryan conjecture for equiaffine hyperspheres with constant sectional curvature, J. Differential Geom. 54 (1) (2000) 99–138.