ON THE DIRICHLET PROBLEM FOR DEGENERATE MONGE-AMPÈRE EQUATIONS

Pengfei Guan Neil S. Trudinger Xu-Jia Wang

Abstract. We prove global second derivative estimates for the Dirichlet problem for degenerate Monge-Ampère equations which yield corresponding existence and regularity results. Our conditions are essentially optimal and our techniques, while drawing on previous investigations, are substantially new.

1. Introduction

In this paper, we resolve an outstanding problem concerning the global regularity of solutions of the Dirichlet problem for the degenerate Monge-Ampère equation in Euclidean n-space, IR^{n}, n ≥ 2.

Theorem 1.1. Let Ω be a uniformly convex domain in IR^{n} with boundary ∂Ω ∈ C^{3,1}, φ ∈ C^{3,1}(\overline{Ω}) and let f be a non-negative function in Ω such that f^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}). Then there exists a unique convex solution u ∈ C^{1,1}(\overline{Ω}) of the Dirichlet problem,

detD^{2}u = f in Ω,  (1.1)
u = φ on ∂Ω.

Consequently, any generalized solution of (1.1) in C^{0}(\overline{Ω}) must belong to C^{1,1}(\overline{Ω}). Moreover, the solution u satisfies an estimate

∥u∥_{C^{1,1}(\overline{Ω})} ≤ C,  (1.2)

where C is a constant depending on Ω and the norms of the functions φ and f̃ = f^{1/(n−1)} in the spaces C^{3,1}(\overline{Ω}) and C^{1,1}(\overline{Ω}) respectively.

A function u ∈ C^{0}(\overline{Ω}) is a generalized solution of equation (1), in the sense of Aleksandrov, if u is convex in Ω with subgradient (or normal) mapping χ having density f with respect to Lebesgue measure in IR^{n}. The regularity assertion in Theorem 1.1 is an immediate consequence of the existence assertion and the uniqueness of the generalized solution, (see, for example, [19]). In turn, the existence assertion follows from the a priori estimate (1.2) applied to smooth solutions of approximating non-degenerate problems, whose solvability is guaranteed by the fundamental second derivative Hölder estimates of Calabi [4], Caffarelli, Nirenberg, and Spruck [1], and Krylov [14,15]. The Monge-Ampère equation (1.1) is referred to as non-degenerate when the inhomogeneous term f is positive, (or equivalently when it is uniformly elliptic with respect to a convex classical solution). For non-degenerate equations, the global second derivative estimate (1.2) was established by Ivochkina [11], and again with a different proof in [1], with constant C depending also on inf_{Ω} f > 0.

1991 Mathematics Subject Classification. 35J25, 35J60.

Key words and phrases. Monge-Ampère equation, regularity.

The global regularity problem for degenerate Monge-Ampère equations, (that is when the non-negative function f is allowed to vanish somewhere in \overline{Ω}), has been studied by various authors, (see [2,7,10,15-18, 20, 23, 24]), with the strongest result to date due to Krylov [16-18] who established Theorem 1.1 in the case when f^{1/n} ∈ C^{1,1}(\overline{Ω}), with constant C in (1.2) depending on the norm of f^{1/n} (rather than f^{1/(n−1)}). We remark that the techniques of this paper are completely different to those of [16-18]. When the equation is completely degenerate, that is the function f is identically zero, Theorem 1.1 was proved by Caffarelli, Nirenberg, and Spruck [2], following the corresponding interior regularity result of Trudinger and Urbas [24].

The assumption that f^{1/n} ∈ C^{1,1}(\overline{Ω}) appears natural as then the estimation of second derivatives is readily reduced to boundary estimation, by applying the Aleksandrov-Bakelman maximum principle [5] to the twice differentiated equation as in [3, 23]. Recently, Guan [7] observed that by following an argument analogous to that of Pogorelov [5,19], this reduction to boundary estimation can be achieved by only assuming f^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}), as in Theorem 1.1. However, he has to suppose that either φ ≡ 0 or f is positive near the boundary, to derive an a priori estimate for second derivatives on the boundary by the usual methods.

The significance of Theorem 1.1 is not so much that it is an improvement of earlier results but that it is optimal. Indeed an example of Wang [25] shows that if the function f̃ = f^{1/(n−1)} ∉ C^{1,1}(\overline{Ω}), then C^{1,1} regularity is false in general. The assumption φ ∈ C^{3,1}(\overline{Ω}) is also optimal for global regularity [2]. In the non-degenerate case, Wang [26] shows that φ ∈ C^{3}(\overline{Ω}) suffices for global C^{1,1} regularity while if only φ ∈ C^{2,1}(\overline{Ω}), the solution may even fail to lie in the Sobolev space W^{2,p}(Ω) for sufficiently large p. We also note here that for degenerate Monge-Ampère equations, C^{1,1} regularity is the best that can be expected. This is readily seen by letting B = B_{1}(0) be the unit ball in IR^{2}, and defining u on B by

u(x, y) = [ max{(x^{2} − 1/2)^{+}, (y^{2} − 1/2)^{+}}]^{2}.  (1.3)

We then have detD^{2}u = 0 in B, with u analytic on ∂B. In the two dimensional case, better regularity is possible if the Hessian matrix D^{2}u has at least one positive eigenvalue [8].

The rest of this paper is devoted to the proof of the second derivative estimates (1.2). In the following section, we reduce that estimation to estimation on the boundary, as in [7], (Lemma 2.1). The estimation of the second derivatives of the solution of (1.1) on the boundary ∂Ω is carried out in the ensuing sections. As in the non-degenerate case [1, 11], the estimation of the double normal derivatives can be achieved, from that of the other second derivatives, through the equation itself. However, in the degenerate case, certain precise forms of these preliminary estimates are absolutely vital. In Section 3, we provide the necessary estimate from below for the cofactor of the double normal derivative, (Lemma 3.1), while in Secions 4 and 5, crucial linear estimates for the mixed tangentialnormal second derivatives with respect to the square roots of the corresponding tangential derivatives. Our techniques rest strongly on the behaviour of the equation (1.1) with respect to affine transformations and on the convexity of the solutions.

Finally, in Section 6, we complete the proof of estimate (1.2) and remark on the extension to more general inhomogeneous terms. The notation in this paper is standard with Lipschitz spaces and their norms being defined, as for example, in [5]. Almost always our constants C will depend on the same quantities as in the statement of Theorem 1.1, but to avoid too much repetition, this is not always indicated.

2. Global second derivative bounds

As mentioned in the introduction, it suffices to prove the estimate (1.2) for smooth solutions of non-degenerate problems. The estimation of the solution itself and its gradient is well known from standard barrier considerations, [5]. In this section we follow [7] to reduce the estimation of second derivatives of solutions of (1.1) to their estimation on the boundary.

Lemma 2.1. Let u ∈ C^{4}(Ω) ∩ C^{2}(\overline{Ω}) satisfy equation (1.1) in Ω with f > 0 in Ω. Then

sup_{Ω} |D^{2}u| ≤ C + sup_{∂Ω} |D^{2}u|,  (2.1)

where C is a constant depending on |f̃|_{1,1} and Ω.

Proof. Since the estimate (2.1) is already proved in [7], we present here a proof for completeness.

Let w(x) = Δu(x) + \frac{1}{2}M|x|^{2}. Suppose w attains its maximum at x_{0}. If x_{0} ∈ ∂Ω, we are through. So to prove (2.1) we need only to consider the case x_{0} ∈ Ω. By rotating the coordinates we may suppose D^{2}u is diagonal at x_{0}. Then at x_{0} we have

0 = w_{k}(x_{0}) = ∑_{i=1}^{n} u_{iik} + Mx_{k},

0 ≥ w_{kk}(x_{0}) = ∑_{i=1}^{n} u_{iikk} + M.

Differentiating the equation log det(D^{2}u) = log f with respect to x_{k} gives

u^{ij}u_{ijkk} = \frac{f_{kk}}{f} − \frac{f_{k}^{2}}{f^{2}} + ∑ u^{ir}u^{js}u_{ijk}u_{rsk}

≥ \{\frac{f_{kk}}{f} − \frac{n − 2}{n − 1} \frac{f_{k}^{2}}{f^{2}} \} − \frac{1}{n − 1} \frac{f_{k}^{2}}{f^{2}} =: I + II

where (u^{ij}) is the inverse of (u_{ij}), which is also diagonal at x_{0}. Since f̃ = f^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}), we have

I = \frac{f_{kk}}{f} − \frac{n − 2}{n − 1} \frac{f_{k}^{2}}{f^{2}} ≥ −Cf^{−1/(n−1)},

|∇f̃| ≤ C\sqrt{f̃},

and hence also

II ≥ −Cf^{−1/(n−1)},

where C depends on |f̃|_{1,1} and Ω. Therefore we obtain

u^{ii}u_{iikk} ≥ −Cf^{−1/(n−1)}.

Consequently

0 ≥u^{ii}w_{ii}(x_{0}) = u^{ii}u_{iikk} + Mu^{ii}
≥M ∑ \frac{1}{u_{ii}} − Cf^{−1/(n−1)}.

Observing that

∑ \frac{1}{u_{ii}} ≥ [\frac{1}{u_{22} ⋯ u_{nn}} ]^{1/(n−1)} ≥ f^{−1/(n−1)}

if u_{11} = max u_{ii} ≥ 1, we reach a contradiction if M is large enough. □

3. Tangential derivatives

In this section we establish upper and lower bounds for the tangential second order derivatives of solutions on the boundary. For any point x_{0} ∈ ∂Ω, without loss of generality we may suppose x_{0} is the origin and the x_{n}-axis is the inner normal there. By transforming the coordinates x′ = (x_{1}, ⋯, x_{n−1}) we may suppose that in a neighbourhood N of the origin, ∂Ω is represented by

x_{n} = ρ(x′) = \frac{1}{2}|x′|^{2} + cubic of x′ + O(|x′|^{4})  (3.1)

as x′ → 0, where x′ = (x_{1}, ⋯, x_{n−1}). Subtracting a linear function we may suppose

∇u(0) = 0 and inf_{Ω} u(x) = u(0) = 0,  (3.2)

while, by rotation of the coordinates x′, we may further suppose u_{ij}(0) = 0 for i ≠ j, i, j = 1, ⋯, n − 1. By Taylor expansion,

φ(x′, ρ(x′)) = \frac{1}{2} ∑_{i=1}^{n−1} b_{i}x_{i}^{2} + R(x′) + O(|x′|^{4}),  (3.3)

where R(x′) denotes the cubic term. Then b_{i} = u_{ii}(0) > 0 for i = 1, ⋯, n − 1. We may further suppose

0 < b_{1} ≤ b_{2} ≤ ⋯ ≤ b_{n−1}.

Differentiating the equality u(x′, ρ(x′)) = φ(x′, ρ(x′)), we get

b_{i} = φ_{ii}(0) + φ_{n}(0), i = 1, ⋯, n − 1,  (3.4)

which implies an upper bound for b_{i}, (in terms of ∇u). To proceed further, we first observe the following obvious fact. Suppose n = 2, u is nonnegative, and

u = φ = αx_{1}^{2} + βx_{1}^{3} + R_{4} on ∂Ω,

where α ≥ 0, β ∈ IR and |R_{4}| ≤ A|x_{1}|^{4}. Then

|β| ≤ (1 + A)\sqrt{α},  (3.5)

since u ≥ 0 at the points x_{1} = ±\sqrt{α}. A lower bound for the tangential derivatives b_{i} is given by the following lemma.

Lamma 3.1. There exists a positive constant γ_{0} depending on Ω, |φ|_{3,1} and |f̃|_{1,1} such that

∏_{i=1}^{n−1} b_{i} ≥ γ_{0}f(0).  (3.6)

Proof. To prove (3.6), we make a dilation x → y = T (x) defined by

y_{i} = M_{i}x_{i}, i = 1, ⋯, n,

where

\{ \begin{matrix}M_{i} = b_{i}^{1/2}M, i = 1, ⋯, n − 1, \\ M_{n} = M = 1/b_{1}. \end{matrix}

Let

v(y) = M^{2}u(x).

Then

det(D_{y}^{2}v) = g(y) =: f(\frac{y_{i}}{M_{i}})M^{2n}/ ∏_{i=1}^{n} M_{i}^{2} in Ω̃ = T (Ω).  (3.7)

By (3.1), the boundary ∂Ω̃ can be represented in a neighbourhood Ñ, (= T (N)), of the origin, by

y_{n} = ρ̃(y′) = \frac{1}{2} ∑_{i=1}^{n−1} d_{i}y_{i}^{2} + O(|y′|^{3})  (3.8)

with

d_{i} = \frac{M_{n}}{M_{i}^{2}} = \frac{b_{1}}{b_{i}} ≤ 1, i = 1, ⋯, n − 1.  (3.9)

Moreover,

|D_{y′}^{k} ρ̃| ≤ |D_{x′}^{k}ρ| ≤ C

for k = 3, 4, where C depends on ∂Ω. On Ñ ∩ ∂Ω̃ we have,

v = φ̃(y′) = \frac{1}{2} ∑_{i=1}^{n−1} b̃_{i}y_{i}^{2} + R(y′) + O(|y′|^{4}), (y′, ρ̃(y′)) ∈ ∂Ω̃,  (3.10)

where

b̃_{i} = \frac{M^{2}}{M_{i}^{2}} b_{i} = 1,  (3.11)

and R(y′) is the cubic term. Noticing that

|D_{y_{i}}^{4} φ̃(y)| = \frac{M^{2}}{M_{i}^{4}} |D_{x_{i}}^{4}φ(x)| ≤ C,

we see that all fourth order derivatives of φ̃ are bounded. Hence in (3.10), the third term O(|y′|^{4}) ≤ C|y′|^{4} for some constant C depending only on ∂Ω and |φ|_{3,1}. Applying (3.5) to every direction in the tangent plane {y_{n} = 0} of ∂Ω̃, we conclude that the coefficients of R are uniformly bounded. Hence near the origin both ∂Ω̃ and φ̃ are C^{3,1} smooth and their C^{3,1}-norms are independent of the upper bound of M. Without loss of generality we may assume Ω̃ ∩ {y_{n} < 1} ⊂ Ñ. Let

ω = {y ∈ Ω̃ | y_{n} < 1, |y_{i}| < 1, i = 2, ⋯, n − 1}.

Then ω is bounded in terms of ∂Ω and |φ|_{3,1} since d_{1} = 1 in (3.8). By the convexity of v and noticing that v = φ̃ ≤ C on ∂ω ∩ ∂Ω̃, we have

v ≤ C in ω,  (3.12)

where C depends on Ω and |φ|_{3,1}. We claim

sup{g(y); y ∈ ω} ≤ C  (3.13)

for some constant C depending on |f̃|_{1,1}, |φ|_{3,1}, and Ω, where by (3.7),

g(y) = f(\frac{y_{i}}{M_{i}})/ ∏_{i=1}^{n−1} b_{i}.  (3.14)

¿From (3.13) and (3.14) we obtain (3.6) with γ_{0} = 1/C. To prove (3.13), we compute

\frac{∂}{∂y_{i}} g^{1/(n−1)}(y) = \frac{∂}{∂y_{i}} f^{1/(n−1)}/[ ∏_{i=1}^{n−1} b_{i}]^{1/(n−1)}

= \frac{∂}{∂x_{i}} f^{1/(n−1)}/M_{i}[ ∏_{i=1}^{n−1} b_{i}]^{1/(n−1)}.

Since f̃ = f^{1/(n−1)} ∈ C^{1,1} and f is nonnegative, we have, as in the proof of Lemma 2.1,

|∇f̃| ≤ C\sqrt{f̃},

where C depends on Ω and |f̃|_{1,1}. We then obtain

|\frac{∂}{∂y_{i}} g^{1/(n−1)}(y)| ≤ Cf^{1/2(n−1)}/M_{i}[ ∏_{i=1}^{n−1} b_{i}]^{1/(n−1)}

= Cg^{1/2(n−1)}/M_{i}[ ∏_{i=1}^{n−1} b_{i}]^{1/2(n−1)}.

Observing that

M_{i}^{2(n−1)} ∏_{i=1}^{n−1} b_{i} ≥ M_{1}^{2(n−1)} ∏_{i=1}^{n−1} b_{i}
≥ M_{1}^{2(n−1)}b_{1}^{n−1} = 1,

we obtain

|\frac{∂}{∂y_{i}} g^{1/2(n−1)}(y)| ≤ C.  (3.15)

Hence if sup_{ω} g is large enough, so is inf_{ω} g, which implies by (3.12) and the comparison principle that inf_{ω} v < 0. On the other hand, by (3.2) we have v ≥ 0. This contradiction shows that g is uniformly bounded in ω, in accordance with (3.13). Hence (3.6) holds. □

4. Mixed tangential-normal derivatives

In this and the following section we prove an estimate for the mixed tangential-normal second derivatives of solutions of the Dirichlet problem (1.1) in terms of the corresponding tangential second derivatives. We first prove a preliminary estimate in terms of the largest tangential second derivative. Using the coordinate system (3.1), introduced in the preceding section, we formulate this estimate as follows.

Lemma 4.1. Letting γ and τ denote respectively the unit inner normal to ∂Ω and any unit tangential vector to ∂Ω, we have for any x ∈ N ∩∂Ω, with |x| ≤ \frac{1}{2} \sqrt{b_{n−1}}, the estimate

|u_{γτ} (x)| ≤ C\sqrt{b_{n−1}},  (4.1)

where C is a positive constant depending on Ω, |φ|_{3,1} and |f̃|_{1,1}, and b_{n−1} is defined by (3.3).

Proof. Let M = 1/b_{n−1}. We make a dilation x → y = T (x) by defining

\{ \begin{matrix}y′ = \sqrt{M}x′, \\ y_{n} = Mx_{n}. \end{matrix}

Let v(y) = M^{2}u(x). Then v satisfies

det(D_{y}^{2}v) = g(y) =: M^{n−1}f(T^{−1}(y)) in Ω̃ = T (Ω).  (4.2)

In the neighbourhood Ñ = T (N), ∂Ω̃ is represented by

y_{n} = \frac{1}{2}|y′|^{2} + O(|y′|^{3}),  (4.3)

and on the boundary ∂Ω̃ ∩ Ñ,

v(y) = φ̃(y′) = \frac{1}{2} ∑_{i=1}^{n−1} b̃_{i}y_{i}^{2} + R(y′) + O(|y|^{4}), y = (y′, ρ̃(y′)) ∈ ∂Ω̃,  (4.4)

where

b̃_{i} = \frac{b_{i}}{b_{n−1}} ≤ 1.

As in the proceding section, we see that near the origin, both ∂Ω̃ and φ̃ are C^{3,1} smooth with C^{3,1} norms independent of M. Similar to (3.13) we also infer that the function g is uniformly bounded on the domain

ω = {y ∈ Ω̃ | y_{n} < 1},

which is also bounded independently of M, by virtue of (4.3). By (4.2) and the uniform convexity of Ñ ∩ ∂Ω̃ implied by (4.3), we then obtain by the usual barrier considerations for the Monge-Ampère equation ([5], Theorem 17.21), a normal derivative bound

|vγ̃(y)| ≤ C  (4.5)

for y ∈ ∂Ω̃ satisfying y_{n} ≤ 7/8, where γ̃ is the unit inner normal to ∂Ω̃ and C depends as usual on Ω, |ψ|_{3,1} and |f̃|_{1,1}. ¿From (4.5) we have

|Dv| ≤ C  (4.6)

on ∂Ω̃∩{y_{n} ≤ 3/4}, which extends to Ω̃∩{y_{n} ≤ 3/4} by the convexity of v. By differentiating equation (4.2) with respect to a tangential vector field as in [5,24], or with respect to a linear vector field with skew symmetric Jacobian matrix as in [1,12], we obtain, again from the uniform convexity of Ñ ∩∂Ω̃, the mixed tangential-normal second derivative estimates,

|v_{γ̃τ̃}| ≤ C  (4.7)

on ∂Ω̃ ∩ \{y_{n} < \frac{1}{2}\}, where τ̃ is a unit tangent vector to ∂Ω̃ and C depends on Ω, |φ|_{3,1} and |f̃|_{1,1}. Since a more complicated version of this argument will be needed later, in the absence of uniform convexity of ∂Ω̃, we omit the details here. Pulling back to our original coordinates, we obtain (4.1). □

We remark here, for purposes of illustration, that Lemmas 3.1 and 4.1 are already sufficient to ensure a second derivative estimate in the two dimensional case. To see this, we write equation (1.1) at the origin, in the form

u_{11}(0)u_{22}(0) = u_{12}^{2}(0) + f(0),  (4.8)

so that

0 ≤ u_{22}(0) = \frac{u_{12}^{2}(0)}{b_{1}} + \frac{f(0)}{b_{1}} ≤ C,

by Lemmas 3.1 and 4.1.

§5. Mixed tangential-normal derivatives continued

We proceed from Lemma 4.1, by means of induction, to obtain the following refinement.

Lemma 5.1. For i = 1, ⋯, n − 1, we have the estimate

|u_{in}(0)| ≤ C_{0}\sqrt{b_{i}},  (5.1)

where C_{0} is a constant depending on Ω, |φ|_{3,1} and |f̃|_{1,1}, and b_{i} = u_{ii}(0) is defined by (3.3).

Proof. For i = 1, ⋯, n − 1, let us denote by τ_{i} = τ_{i}(x) the tangential direction of ∂Ω at x ∈ N ∩ ∂Ω, which lies in the two dimensional plane parallel to the x_{i} and x_{n} axes and passes through the point x. Our induction hypothesis is that for some k = 1, ⋯, n − 2 and i = k + 1, ⋯, n − 1, there exists a constant θ_{i} > 0, depending on Ω, |φ|_{3,1} and |f̃|_{1,1}, such that for x ∈ N ∩ ∂Ω, with |x| ≤ θ_{i}\sqrt{b_{i}}, we have the estimates

|u_{γτ_{j}}(x)| ≤ C\sqrt{b_{i}}  (5.2)

for j = 1, ⋯, i, where C is a constant depending on Ω, |φ|_{3,1} and |f̃|_{1,1}. When k = n − 2, (5.2) is exactly (4.1) with θ_{n−1} = \frac{1}{2}. We shall prove that there exists a constant θ_{k}, also depending on Ω, |φ|_{3,1}, and |f̃|_{1,1}, such that for x ∈ N ∩ ∂Ω, with |x| ≤ θ_{k}\sqrt{b_{k}}, we have

|u_{γτ_{j}}(x)| ≤ C\sqrt{b_{k}},  (5.3)

for j = 1, ⋯, k, where C is a constant depending on Ω, |φ|_{3,1} and |f̃|_{1,1}. The estimate (5.1) then follows from (4.1) and (5.3) by induction.

To prove (5.3), we introduce the dilation, x → y = T (x), defined by

y_{i} = M_{i}x_{i}, i = 1, ⋯, n,

where

M_{i} = \{ \begin{matrix}\sqrt{M}, & i = 1, ⋯, k, \\ \sqrt{b_{i}}M, & i = k + 1, ⋯, n − 1, \\ M, & i = n, \end{matrix}
M = \frac{1}{b_{k}} .

We may suppose that b_{k} ≤ \frac{1}{2}b_{k+1}, otherwise (5.3) follows immediately from (5.2). Let v(y) = M^{2}u(x). Then v satisfies

det(D_{y}^{2}v) = g(y) =: f(\frac{y_{i}}{M_{i}})M^{2n}/ ∏_{i=1}^{n} M_{i}^{2} in Ω̃ = T (Ω).

Near the origin ∂Ω̃ is represented by

y_{n} = ρ̃(y′) = \frac{1}{2} ∑_{i=1}^{n−1} d_{i}y_{i}^{2} + O(|y′|^{3}),  (5.4)

where

d_{i} = \{ \begin{matrix}1, & i ≤ k, \\ \frac{b_{k}}{b_{i}} (≤ 1), & i ≥ k + 1. \end{matrix}

After the transformation we have, for Ñ = T (N),

v(y) = φ̃(y′) = \frac{1}{2} ∑_{i=1}^{n−1} b̃_{i}y_{i}^{2} + R(y′) + O(|y′|^{4}), y = (y′, ρ̃(y′)) ∈ Ñ ∩ ∂Ω̃,  (5.5)

where

b̃_{i} = \{ \begin{matrix}1, & i ≥ k, \\ \frac{b_{i}}{b_{k}} (≤ 1), & i < k. \end{matrix}

As above we see that near the origin, both φ̃ and ∂Ω̃ are C^{3,1} smooth and their C^{3,1}-norms are independent of M.

Let

ω = {y ∈ Ω̃ | y_{n} < β^{2}, |y_{i}| < β, i = k + 1, ⋯, n − 1},  (5.6)

where β will be chosen small such that the third and high order terms in (5.4) and (5.5) do no harm to the following estimation. As before we may assume ω ⊂ Ñ and, by (5.4), ω is bounded independently of M. By the convexity of v we have

v ≤ C in ω.

Similar to (3.13) we have

sup{g(y); y ∈ ω} ≤ C.  (5.7)

To prove (5.3) it is crucial to establish a bound for the normal derivative of v near the origin. The main difficulty is that we cannot control the convexity of ∂Ω̃ near the origin. We construct a lower barrier \underline{v} by setting

\underline{v}(y) = \frac{σ}{2} |y′|^{2} + \frac{1}{2}Ky_{n}^{2} − K^{2}y_{n},  (5.8)

where σ > 0 small and K > 1 large will be chosen so that

det(D^{2}\underline{v}) = σ^{n−1}K ≥ sup_{ω} g(y).  (5.9)

We claim \underline{v} ≤ v on ∂ω (with appropriate choice of β, σ and K). For later application we will prove the stronger inequality

\underline{v} ≤ \frac{1}{2}v on ∂ω.  (5.10)

To prove (5.10) we first consider the piece ∂_{1}ω := ∂ω ∩ ∂Ω̃. For y ∈ ∂_{1}ω we have, by (5.5),

v(y) = \frac{1}{2} ∑_{i−1}^{n−1} b̃_{i}y_{i}^{2} + O(|y′|^{3})

≥ \frac{1}{4}|ỹ|^{2} − C|ŷ|^{2}

provided β is small, where ŷ = (y_{1}, ⋯, y_{k}), and ỹ = (y_{k+1}, ⋯ y_{n−1}). By (5.4) we have

\underline{v}(y) ≤ \frac{σ}{2} |y′|^{2} − \frac{1}{2}K^{2}y_{n}

≤ \frac{σ}{2} |ỹ|^{2} − \frac{1}{4}K^{2}|ŷ|^{2}.

Hence (5.10) holds on ∂_{1}ω. On ∂_{2}ω := ∂ω ∩ {y_{n} = β^{2}} we have v ≥ 0. For σ > 0 small and K > 1 large, we have \underline{v} ≤ −\frac{1}{2}K^{2}β^{2}, so that (5.10) also holds on ∂_{2}ω.

Finally we consider the piece ∂_{3}ω := ∂ω ∩ {|y_{i}| = β for some i = k + 1, ⋯, n − 1}. We only consider the piece ∂′_{3}ω := ∂ω ∩ {y_{n−1} = β} since other pieces of ∂_{3}ω can be handled in the same way. First we prove that

v(y) ≥ \frac{1}{4}β^{2} on ∂′_{3}ω ∩ \{y_{n} < ε_{0}β^{2}\}  (5.11)

provided ε_{0} is small enough. If ∂′_{3}ω ⊂ {y_{n} ≥ ε_{0}β^{2}}, we have nothing to prove, so we may suppose ∂′_{3}ω ∩ {y_{n} < ε_{0}β^{2}} ≠ ∅. To prove (5.11) we first fix a point p = (p̂, p̃, p_{n}) ∈ ∂′_{3}ω, where p̂ = (p_{1}, ⋯, p_{k}) ≠ 0, p̃ = (p_{k+1}, ⋯, p_{n−1}), p_{n} < ε_{0}β^{2}. For δ ≥ 0, sufficiently small, we then fix a further point p^{∗} = (0, p̃, p_{n} + δ) so that the straight line through p and p^{∗} meets ∂′_{3}ω in a point \overline{p} satisfying

\frac{1}{2}|p − p^{∗}| ≤ |\overline{p} − p^{∗}| ≤ |\overline{p} − p|.  (5.12)

In view of the convexity of Ω̃ and the representation (5.4), we may accomplish (5.12) by taking

δ = |∇ŷρ̃(0, p̃)| |p̂| ≤ O(|β|^{3})

and β sufficiently small. Now let p^{0} = (0, p̃, ρ̃(0, p̃)) be the projection of p^{∗} on ∂Ω̃. We claim

|vγ̃(p^{0})| ≤ C,  (5.13)

where γ̃ is the unit inner normal at p^{0}. Indeed, for any y = (0, ỹ, ρ̃(0, ỹ)) ∈ ∂Ω̃ ∩ B_{θ_{k+1}}(0), with θ_{k+1} as given in (5.2), we have, (for x ∈ T^{−1}(y)),

|x_{i}| = |y_{i}/M_{i}| ≤ θ_{k+1}b_{k}/b_{i}^{1/2}, i = k + 1, ⋯, n − 1.

Hence by (5.2),

|u_{γ}(x)| ≤ u_{γ}(0) + C ∑_{i=k+1}^{n−1} sup{|x_{i}| · |u_{γτ_{i}}(x)|; |x_{i}| ≤ θ_{k+1}b_{k}/b_{i}^{1/2}} ≤ Cb_{k},

and since u_{γ}(0) = 0 by (3.2), we obtain

|∇u(x)| ≤ Cb_{k} at x = T^{−1}(y).

By the definition of T,

|v_{y_{n}}(y)| = M|u_{x_{n}}(x)| ≤ CMb_{k} = C.

Noticing that v = φ̃ ∈ C^{3} on ∂Ω̃, we obtain (5.13). ¿From (5.13) and the convexity of v, we have

v(p^{∗}) ≥ v(p^{0}) − C|p_{n}^{∗} − p_{n}^{0}|,  (5.14)

while, from (5.5),

v(p^{0}) = \frac{1}{2} ∑_{i=k+1}^{n−1} |p_{i}^{0}|^{2} + O(|p^{0}|^{3}),

and

v(\overline{p}) = \frac{1}{2} ∑_{i=1}^{n−1} b̃_{i}|\overline{p}_{i}|^{2} + O(|\overline{p}|^{3})

= v(p^{0}) + \frac{1}{2} ∑_{i=1}^{k} b̃_{i}|\overline{p}_{i}|^{2} + O(|\overline{p}|^{3})

Noticing that p_{n−1}^{0} = β, we have v(p^{0}) ≥ \frac{1}{2}β^{2} − O(β^{3}) ≥ \frac{3}{8}β^{2} if β is small enough. Since d_{i} = 1 in (5.4) when i ≤ k, we see that

∑_{i=1}^{k} b̃_{i}|\overline{p}_{i}|^{2} ≤ C(|\overline{p}_{n} − p_{n}^{0}| + β^{3}) ≤ C(\overline{p}_{n} + β^{3}).

Hence we may first fix β small and then choose ε_{0} small so that \overline{p}_{n} ≤ 2ε_{0}β^{2} and

v(\overline{p}) ≤ \frac{9}{8}v(p^{∗}),

whence by (5.14), v(p^{∗}) ≥ \frac{1}{3}β^{2}. Hence by (5.12) and the convexity of v, we have

v(p) ≥ 3v(p^{∗}) − 2v(\overline{p}) ≥ \frac{3}{4}v(p^{∗}) ≥ \frac{1}{4}β^{2}

and (5.11) is proved.

¿From (5.11) we have

v(y) ≥ \frac{1}{8}β^{2} − Cy_{n} on ∂′_{3}ω,

where C depends on β and ε_{0}. On the other hand,

\underline{v}(y) ≤ Cσ − K^{2}y_{n}.

Hence \underline{v} < \frac{1}{2}v on ∂′_{3}ω if σ is small and K large enough. This completes the proof of inequality (5.10).

The next step in our proof is to adapt the barrier \underline{v} to obtain a normal derivative bound near the origin. For any point y_{0} ∈ ∂_{1}ω, let ξ = (ξ_{1}, ⋯, ξ_{n}) = A(y − y_{0}) be an orthogonal basis at y_{0} so that ξ_{n} coincides with the inner normal, where A is some orthogonal matrix. Instead of (5.8), we now set

\underline{v}(y) = \frac{σ}{2} |ξ′|^{2} + \frac{1}{2}Kξ_{n}^{2} − K^{2}ξ_{n} + ℓ(ξ′),

where σ > 0 small and K > 1 large will be chosen so that (5.9) holds, ξ′ = (ξ_{1}, ⋯, ξ_{n−1}), ℓ is a linear function such that |ℓ(ξ′) − v(y)| = o(|ξ′|) as ξ′ → 0. Since both φ̃ and ∂Ω̃ are C^{3,1} smooth near the origin, arguing as above and by virtue of (5.11)) we see that \underline{v} ≤ \frac{1}{2}v on ∂ω if |y_{0}| and σ (> 0) are sufficiently small and K > 1 is sufficiently large. By (5.9) and the comparison principle it follows that \underline{v}(y) ≤ v(y) on ω. We therefore obtain, by the convexity of v,

|vγ̃(y)| ≤ C for y ∈ ∂Ω̃ near the origin.  (5.15)

By choosing a new β we can then ensure that the estimate (5.15) holds for all y ∈ ∂_{1}ω = ∂ω ∩ ∂Ω̃, where ω is given by (5.6) and subsequently, from the convexity of v,

|Dv| ≤ C in ω.  (5.16)

We can now complete the proof of (5.1) by standard arguments, [1, 5]. For convenience we follow that in [1]. Let

Lw = v^{ij}w_{ij},

where {v^{ij}} denotes the inverse of the Hessian D^{2}v, and T = (T_{1}, ⋯, T_{k}), where

T_{i} = ∂_{i} + (y_{i}∂_{n} − y_{n}∂_{i}), i = 1, ⋯, k.

Applying the operator T to both sides of the equation

F (D^{2}v) := log det(D^{2}v) = log g,

we obtain

L(T v) = T (log g),

where L = v^{ij}∂_{i}∂_{j}. Setting

w(y) = ±T (v − φ̃)(y) + B|y|^{2},

we have, since d_{i} = 1 for i ≤ k in (5.4),

|w(y)| ≤ C_{B}(|y|^{2} + y_{n}) on ∂ω.  (5.17)

Similar to (3.15) we have

|∇g^{1/2(n−1)}(y)| ≤ C,

so that

|∇g| ≤ Cg^{(2n−3)/2(n−1)}(y).

Hence

Lw ≥ B ∑ v^{ii} − C(g^{−1/2(n−1)} + ∑ v^{ii})

≥ \frac{1}{2}B ∑ v^{ii} − Cg^{−1/2(n−1)}

≥ \frac{1}{2}Bg^{−1/n} − Cg^{−1/2(n−1)} ≥ 0.

provided B is large enough. Now set

w̃(y) = A(\underline{v} − v − y_{n}) + w(y),

where \underline{v} is given by (5.8), and A > 1 is a sufficient large constant to be chosen later. By (5.9) and the concavity of F, we have

L(\underline{v} − v) ≥ F (D^{2}\underline{v}) − F (D^{2}v) ≥ 0.

Consequently

Lw̃ ≥ 0,  (5.18)

which implies the function w̃ attains the maximum on the boundary of ω.

We claim w̃(y) ≤ 0 on ∂ω for sufficiently large A. This is because by (5.10) and (5.17),

w̃(y) ≤ C_{B}(|y′|^{2} + y_{n}) − \frac{1}{2}A(v(y) + y_{n}).

Using (5.4) and (5.5), we then choose A large enough so that w̃(y) ≤ 0 on ∂ω.

Noticing that w̃(0) = 0, we have therefore \frac{∂}{∂y_{n}} w̃(0) ≤ 0. Namely, |v_{in}(0)| ≤ C, i = 1, ⋯, k. Similarly we have |v_{γ̃τ̃_{i}}(y)| ≤ C, i = 1, ⋯, k, for y ∈ ∂ω ∩ ∂Ω̃ near the origin, where τ̃_{i} is a tangential direction of ∂Ω̃ at y which lies in the plane parallel to the y_{i} and y_{n} axes and passes through the point y. Pulling back to the x-coordinates we obtain (5.1). □

§6. Concluding remarks

To complete the proof of the second derivative estimate (1.2) in Theorem 1.1, we write, (as in the case n = 2 in (4.8)), equation (1.1) at the origin in the form

( ∏_{i=1}^{n−1} b_{i})u_{nn}(0) = ∑_{j=1}^{n−1} ( ∏_{i≠j} b_{i})u_{jn}^{2}(0) + f(0)  (6.1)

so that

0 ≤ u_{nn}(0) = ∑_{j=1}^{n−1} \frac{u_{jn}^{2}(0)}{b_{j}} + f(0)/ ∏_{i=1}^{n−1} b_{i} ≤ C  (6.2)

by Lemmas 3.1 and 5.1. Combining (3.4), (4.1) or (5.1) and (6.2), we thus obtain

|D^{2}u| ≤ C  (6.3)

on ∂Ω, and subsequently in Ω by (2.1), where in both (6.2) and (6.3) the constant C depends on Ω, |f̃|_{1,1} and |φ|_{3,1}. The estimate (1.1) is thus established for the non-degenerate case f > 0 on \overline{Ω}. To get the full generality of Theorem 1.1, we need to solve approximating Dirichlet problems, (using [1] or [5]), with f replaced by f + ε, for ε > 0, constant, and deduce the existence assertion of Theorem 1.1 by sending ε to zero.

The conditions on the function f can be weakened somewhat. In particular, in the derivation of the estimate (6.3) on the boundary ∂Ω, we have only used the condition, f^{1/2(n−1)} ∈ C^{0,1}(\overline{Ω}). For the full second derivative estimate in Ω, we only need to assume additionally that Δf^{1/(n−1)} is bounded from below (in the sense of distributions) in the proof of Lemma 2.1 [7]. Moreover by employing the Aleksandrov-Bakelman estimate [5] instead, we can replace the latter condition by Δf^{1/n} being bounded from below by a function in L^{n}(Ω). For application to problems in differential geometry, (as, for example in [8]), which provided the motivation for Guan’s approach in [7], it is desirable to impose no restriction on the non-negative function f apart from smoothness, as in the cases n = 2, 3 in [7].

Finally we remark that the results of this work carry over to more general Monge-Ampère equations of the form

detD^{2}u = f(x, u)ψ(Du),  (6.4)

where f is monotone increasing in u and ψ convex. In particular, we may consider the equation of prescribed Gauss curvature

detD^{2}u = f(x)(1 + |Du|^{2})^{(n+2)/2}  (6.5)

with again f^{1/(n−1)} ∈ C^{1,1}(\overline{Ω}), in the presence of barriers, so that Theorem 1.1 would continue to hold if

∫_{Ω} f < ω_{n}  (6.6)

and, for example, f = 0 on ∂Ω, [23].

Acknowledgement. This research was supported by an Australian Research Council grant at the Australian National University and a Canadian NSERC grant at McMaster University.

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Pengfei Guan Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1, Canada.

Neil S. Trudinger, and Xu-Jia Wang Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia.