THE NEUMANN PROBLEM FOR MONGE-AMPÈRE TYPE EQUATIONS REVISITED

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Ampère type equations in bounded domains in n dimensional Euclidean space Rn. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.


Introduction
In this paper, we revisit the following Neumann problem for Monge-Ampère type equations, considered in [7]: det[D 2 u − A(x, u, Du)] = B(x, u, Du), in Ω, (1.1) in Ω if A is co-dimension one convex, (strictly co-dimension one convex), with respect to p, in the sense that n i,j,k,l=1 A ij,kl (x, z, p)ξ i ξ j η k η l ≥ 0, (≥ c 0 > 0), (1.3) for all (x, z, p) ∈ Ω × R × R n , ξ, η ∈ R n and ξ · η = 0, where A ij,kl = D 2 p k p l A ij and c 0 is a positive constant. A domain Ω is called uniformly A-convex with respect to u ∈ C 1 (Ω), if ∂Ω ∈ C 2 and n i,j,k=1 on ∂Ω, for all vectors τ = τ (x) tangent to ∂Ω, and some positive constant δ 0 . For more information about the definition of uniformly A-convex domain, one can refer to the introduction of [7]. (1.7) We will assume the existence of a nonnegative barrier function Φ with respect to the linearized operator L in (1.7), in the sense that there exists a barrier function Φ ≥ 0, ∈ C 2 (Ω) satisfying in Ω, (1.8) for some constant C 0 .
In order to derive an estimate (1.10), which is independent of the positive lower bound of B, we let C i , (i = 1, 2, 3), denote positive constants satisfying and 13) in U = (·, u, Du)(Ω), where I denotes the identity matrix.
We can then formulate a global second order derivative estimate, (independent of inf B), for problem (1.1)-(1.2) under the conditions (1.11), (1.12), (1.13), which extends the corresponding result in [14] to general regular matrix functions A satisfying the same hypotheses as in Theorem 1.1. We also include the corresponding estimate for strictly regular matrix functions A, without the barrier conditions (1.8), (1.9) and condition (1.13). (1.14) holds for constant C depending on n, C i (i = 0, 1, 2, 3), A, sup B, Ω, ϕ, δ 0 , |Φ| 1;Ω and |u| 1;Ω . If A is strictly regular in Ω, then the barrier conditions (1.8), (1.9) and condition (1.13) are not necessary and C depends on c 0 instead of C 0 , C 3 and Note that in Theorems 1.1 and 1.2, we do not assume monotonicity conditions of A, B and ϕ with respect to u, although these are critical for the construction of barriers Φ in general domains as in [7]. Some historical results for the Neumann problem for equations of Monge-Ampère type can be found in [4,7,8,10,11,13,14]. The results of this paper, along with their application to full second derivative estimates and existence theorems as in [7], may also be extended to parabolic Monge-Ampère type equations, extending [13], as well as the corresponding problems on Riemannian manifolds, extending [4]. The derivation of second derivative estimates here and in all these papers adapts the approach for the Monge-Ampère equation originating in [10]. This paper is organized as follows. In Section 2, we prove the second order derivative estimates, Theorem 1.1, on uniformly A-convex domains. In Section 3, the second order derivative estimates in Theorem 1.2 are treated by combining the techniques in Section 2 and [7,14].
ACKNOWLEDGEMENT. This paper is dedicated to the memory of Vaughan Jones. The second author remembers meeting Vaughan for the first time at the Australian National University in 1989, during Vaughan's participation in a special year programme in mathematical physics sponsored by the Centre for Mathematical Analysis, (CMA) and recalls in particular his CMA research report entitled "Baxterization"; (CMA R23-89). They met several times after that in various parts of the world and their conversations were probably more about Vaughan's windsurfing passion than his formidable mathematics achievements. He also recalls only having limited interactions with Vaughan during his time at Berkeley in 2005, because of a windsurfing mishap on the San Francisco Bay, and just a couple of years ago hearing about Vaughan's love of windsurfing at Sanya, while he was attending a meeting there. Another very memorable experience of the second author was giving lectures at the 2008 NZMRI Summer Workshop at Nelson with Vaughan in attendance, (at least when he wasn't windsurfing).

Proof of Theorem 1.1
In this section, we consider the global second derivative estimate of the problem (1.1)-(1.2) in uniformly A-convex domains for regular matrices A, and prove Theorem 1.1.
For the arguments below, we assume the functions ϕ, ν have been smoothly extended toΩ × R andΩ respectively. We also assume that near the boundary, ν is extended to be constant in the normal directions. For convenience in later discussion, we denote D ξη u D ij uξ i η j , w ξη w ij ξ i η j = D ij uξ i η j − A ij ξ i η j for any vectors ξ and η. As usual, C denotes a constant depending on the known data and may change from line to line in the context. By differentiation of the equation (1.5) in the direction ξ, we have and a further differentiation in the direction of ξ yields, Here and below, we use the standard summation convention that repeated indices indicate summation from 1 to n unless otherwise specified.
Applying the tangential operator δ = D − (ν · D)ν to the boundary condition (1.2), we can get the mixed tangential normal derivative estimate on ∂Ω, namely for any unit tangential vector field τ , where the constant C depends on supΩ |Du| and Ω, see [7,11,14]. We shall deduce the estimate for D νν u on ∂Ω. First, the ellipticity of u gives a lower bound of D νν u inΩ, namely Next, from the uniform A-convexity of Ω with respect to u, we can employ a barrier argument in a boundary strip to obtain an upper bound of D νν u on ∂Ω, in accordance with the following lemma, which is a refinement of the corresponding estimates in [7,11,14]. Here it is convenient to cover simultaneously the degenerate case as it is embraced by the same proof, which is a modification of that in [11].
where the constant C depends on n, A, Ω, ϕ, δ 0 , |u| 1;Ω and |B Proof. Considering the function where L is the first operator in (1.7). By using the arithmetic-geometric mean inequality, from (1.1) and the positivity of B, we have where Λ denotes the maximum eigenvalue of the augmented Hessian w = D 2 u − A. with constants C and C depending on the same quantities as in the lemma statement, (except that C is independent of B).
Since Ω is uniformly A-convex with respect to u and A is regular with respect to p, there exists a defining function φ ∈ C 2 (Ω) satisfying φ = 0 on ∂Ω, Dφ = 0 on ∂Ω, and φ < 0 in Ω, together with in a neighbourhood N of ∂Ω, whenever D ν u ≥ ϕ(·, u), where δ 1 is a positive constant and I is the identity matrix, with N and δ 1 depending on δ 0 , A and for a positive constant d 0 depending on δ 0 , A and |u| 1;Ω . Since the exponent n−2 n−1 < 1 in (2.9), by taking −Kφ as a barrier function, for sufficiently large constant K and using (2.9) and (2.11), a standard barrier argument, as in the proof of the uniformly convex case in Corollary 14.5 in [2], (under the structure condition (14.32)), leads to the estimate (2.12) Thus, (2.5) follows from (2.12).
We introduce a perturbationΦ of the barrier function Φ, given bỹ where a is a small positive constant, φ ∈ C 2 (Ω) is a defining function of Ω satisfying For sufficiently small a > 0, the nonnegative functionΦ in (2.14) satisfies (2.16) With the above preparations, we give the proof for the global second order derivative estimate in Theorem 1.1. In order to embrace general regular matrix functions A, the argument is a refinement of that in [11] and [7], where the perturbed barrierΦ is used in its exponential form e κΦ for some positive constant κ.
Proof of Theorem 1.1. We employ an auxiliary function v, given by for x ∈Ω and ξ ∈ S n−1 , where α, κ are positive constants to be determined,Φ is the perturbed barrier function in (2.14), and v is defined by The term e κΦ in the auxiliary function v in (2.17) is inspired by that of [12] for the Pogorelov estimate. The strategy of our proof is to estimate v at a maximum point (x, ξ) ∈Ω × S n−1 . For this purpose, we divide the discussion into two cases.
Case 1. v takes its maximum at an interior point x 0 ∈ Ω and a unit vector ξ. Let then the function H also attains its maximum at the point x 0 ∈ Ω and the unit vector ξ. At the point x 0 , we have for i = 1 · · · n, and Next, we shall estimate each term on the right hand side of (2.22) at the maximum point x 0 . Using (2.2) and the regular condition of A, (see (3.9) in [17]), we have (2.23) By Cauchy's inequality, we have for any θ > 0, where C(θ) is a positive constant depending on θ. Since A ξξ + v is a function of x, z and p, by direct calculations and (2.1), we have for some constant C depending on A, B, ϕ and |u| 1;Ω . Using (2.1), we have (2.28) In order to treat the terms with third order derivatives, we need to derive a lower bound for the quantity From the inequality (3.48) in [11], we havẽ Using w ij = u ij − A ij (·, u, Du) and by direct calculations, we have (2.32) Inserting (2.31) and (2.32) into (2.29), we get where (2.30) and {w ij } > 0 are used in the inequality.
Without loss of generality, we assume that {w ij } is diagonal at x 0 with the maximum eigenvalue w 11 . We can always assume that w 11 > 1 and is as large as we want; otherwise we are done. Since v is bounded, w 11 and w ξξ are comparable in the sense that for any θ > 0, there exists a further constant C(θ) such that if w 11 > C(θ). Then (2.34) guarantees the following relationship between w 11 and We now return to the third order derivative terms in (2.28). Using (2.35), we where the inequality (2.33) for P is used in the second inequality. Recalling D i H = 0 in (2.20) and using (2.35), we have for i = 1, · · · , n. By a direct calculation, we have for i = 1, · · · , n, where C is a constant depending on A and |u| 1;Ω . Using (2.37) and (2.38), we get (2.39) and 2θ By Cauchy's inequality, we have By taking κ > C 2 and inserting (2.42) into (2.41), sinceΦ ≥ 0, we get (2.43) By successively choosing α > 2C, κ > max{4C(2+α), C 2 } and θ = 1/(α 2 + κ 2 e 2κ supΦ ), we thus obtain an estimate S ≤ C at x 0 , which implies a corresponding estimate for |D 2 u(x 0 )|. α 2 |Du| 2 +e κΦ [w ξξ − v (·, ξ)] attains its maximum overΩ at x 0 ∈ ∂Ω and a unit vector ξ. We then consider the following two subcases of different directions of ξ.
Combining (2.50) with (2.53), and using (2.13), we obtain We now conclude from the above two subcases (i) and (ii) that if v attains its maximum overΩ at a point x 0 ∈ ∂Ω, then v(x 0 , ξ) is bounded from above as in (2.56), which implies the second derivative D ξξ u(x 0 ) is also similarly bounded from above.
Combining Case 1 and Case 2, we obtain the desired estimate (1.10) and complete the proof of Theorem 1.1.
We conclude this section with some remarks.
Remark 2.1. In [7], inequality (2.24) is written incorrectly with ξ and i, j interchanged. When we use the correct version (2.30), we need to control the difference P −P and this is done through the adjusted auxiliary function (2.17). Rather than indicate all the consequent changes to our previous proof in [7], (and also taking account of Remark 2.3 below), we have for clarity written out the full proof.
Remark 2.2. If we assume monotonicity conditions as in [7], namely that A, B and ϕ are non-decreasing with respect to the solution variable u, the barrier in (1.8) and (1.9) can be constructed in the form, for some proper constants 1 and K, and an elliptic supersolutionū ∈ C 2 (Ω) satisfying the same boundary condition; see Lemma 2.1 in [7], Lemma 2.2 in [5] or Lemma 2.1 in [6] for a simpler proof. When A ≡ 0, such an elliptic supersolution u is readily constructed; see [11]. With the barrier in (2.57), by using the proof of this section, we thus recover Theorems 1.1 and 1.2 in [7] for general regular matrix functions A as stated. for a unit vector ξ in any direction.
We are now ready to prove Theorem 1.2. The proofs are the modifications of those in Section 2 for the regular A case and [7] for the strictly regular for some constantĈ depending on n, C 1 and C 2 . By (1.13), we have for some constant C 3 depending on C 3 , A and |u| 1;Ω .
For the regular A case, we will follow the lines of the proof of Theorem 1.1 and make necessary modifications. We still employ the same auxiliary function v in (2.17).
In Case 1, v takes its maximum at an interior point x 0 ∈ Ω and a unit vector ξ. Plugging (3.3), (3.4) and (3.5) into (2.2), then in place of (2.23), we now have where the constant C depends also on C 3 , and the constant C depends onC andĈ.
Similarly, by using the first inequality in (3.3), there are also terms C(1 + S)B − 1 n−1 and CB − 1 n−1 on the right hand side of (2.25) and (2.26) respectively. In place of (2.27), by the second inequality in (3.3), we have for some positive constantC 0 depending onC andΦ. Consequently, in place of (2.28), we have at the maximum point x 0 ∈ Ω of v overΩ, for a further constant C.
Next, we need to treat the terms in (3.8) involving B − 1 n−1 and B − 1 2(n−1) . Without loss of generality, we assume that {w ij } is diagonal at x 0 with maximum eigenvalue w 11 . We can always assume that w 11 > 1 and is as large as we want; otherwise the proof is completed. From the arithmetic-geometric mean inequality as in (2.8), we then have By using Cauchy's inequality, we also have for any positive constant , where C( ) is a positive constant depending on , C 0 andC 0 . Taking = n−1 4 , from (3.9) and (3.10), we obtain Inserting (3.9) and (3.11) into (3.8), and following the same steps as in the proof of Theorem 1.1, we can get an estimate S ≤ C at x 0 , which implies an estimate for Since (3.2) still holds, the proof for Case 2 when x 0 ∈ ∂Ω is the same as that of Theorem 1.1. We omit the details.
For the strictly regular A case, we present a proof without the barrier conditions (1.8), (1.9) and condition (1.13) by adapting the proof of Theorem 4.1 in [7]. In order to fit the Neumann case here, we restrict the oblique vector field β on ∂Ω in [7] to be the unit inner normal vector field ν on ∂Ω. As in [7], we employ the auxiliary function v = w τ τ − K(1 + M 2 )φ, (3.12) where τ is the tangential vector field which is C 2 extended toΩ, M 2 = sup Ω |D 2 u|, φ ∈ C 2 (Ω) is a negative defining function for Ω, and K is a proper chosen constant such that the maximum of v onΩ must occur at an interior point of Ω. Using (3.3) and (3.4), in place of (4.6) in [7], we have at the interior maximum point x 0 of v, where the constant C depends also on C 1 and C 2 . Without loss of generality, we choose coordinates so that w is diagonalised at x 0 with the maximum eigenvalue w 11 . We can assume that with C being the constant in (3.13), otherwise we have already obtained an upper bound for u τ τ (x 0 ). Similarly to (3.9), using (3.14) we get which implies the estimate (4.9) in [7] for u τ τ (x 0 ). The rest of the proof can be completed in the same way as that of Theorem 4.1 in [7]. Note that we only use the strict regularity of A here. The barrier conditions (1.8), (1.9) and condition (1.13) are not used. Therefore, the constant C in the desired estimate (1.14) depends on c 0 instead of C 0 , C 3 and |Φ| 1;Ω .
Remark 3.1. Alternatively, in place of (3.7), we can also use the first inequality in  for the perturbed barrier functionΦ, which also suffices in the proof.
Remark 3.2. If B is separated in the sense that B(x, z, p) = f (x, z)g(p) with f > 0 and g ≥ g 0 > 0 for some positive constant g 0 , then condition (1.13) is clearly satisfied and constant C in (1.14) depends on g 0 instead of C 3 .
If B only depends on x, (1.11) holds locally in Ω if B 1 n−1 ∈ C 1,1 holds globally inΩ; see Lemma 3.1 in [1]. In this case, we only need to assume that (1.11) holds near ∂Ω, with B ϕ = ϕ(x) as in [14], by assuming B is strictly increasing in z, the barrier in (1.8) and (1.9) can be constructed as in (2.57) by replacingū with a subsolution u ∈ C 2 (Ω) satisfying the same boundary condition; see [14]. Then following the proof of this section, the second derivative estimate in Theorem 1.1 and the existence of globally C 1,1 solutions in Theorem 1.2 in [14] are valid for general regular matrix functions A, that is the alternative restrictions (a) and (b) in Theorems 1.1 and 1.2 in [14] can be dispensed with.