Acta Math Vietnam https://doi.org/10.1007/s40306-018-0270-3

Elliptic Solutions to Nonsymmetric Monge-Amp`ere Type Equations II. A Priori Estimates and the Dirichlet Problem Ha Tien Ngoan1 · Thai Thi Kim Chung2

Received: 21 October 2017 / Revised: 1 February 2018 / Accepted: 27 February 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Abstract In this paper, we consider the Dirichlet problem for nonsymmetric MongeAmp`ere type equations in which a skew-symmetric matrix is introduced. We establish uniform with respect to a class of skew-symmetric matrix bounds for δ-elliptic C 2,α ()solutions to the Dirichlet problem. Then, we prove the classical solvability of the Dirichlet problem, provided those skew-symmetric matrices are sufficiently small in some sense. Keywords Monge-Amp`ere type equations · δ-elliptic solutions · Second derivative estimates · The method of continuity Mathematics Subject Classification (2010) 35J66

1 Introduction This paper is a continuation of our previous one [14]. We study the solvability of the following Dirichlet problem for the nonsymmetric Monge-Amp`ere type equations   det D 2 u − A(x, u, Du) − B(x, u, Du) = f (x, u, Du) in , (1) u(x) = ϕ(x) on ∂,

 Ha Tien Ngoan

[email protected] Thai Thi Kim Chung [email protected] 1

Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, 10307 Hanoi, Vietnam

2

University of Transport Technology, 54 Trieu Khuc, Thanh Xuan, Hanoi, Vietnam

(2)

H. T. Ngoan, T. T. K. Chung

where  is a bounded domain in n-dimensional Euclidean space Rn with smooth boundary ∂; Du and D 2 u are the gradient vector and the Hessian matrix of the function u :  → R, respectively; A(x, z, p) = [Aij (x, z, p)]n×n , B(x, z, p) = [Bij (x, z, p)]n×n and f (x, z, p) are respectively symmetric matrix function, skew-symmetric matrix function, and positive scalar valued function defined on , where  :=  × R × Rn ; ϕ is a scalar valued function defined on ∂. As usual, we use x, z, p, r to denote points in , R, Rn , Rn×n , respectively. When B ≡ 0, we have the symmetric Monge-Amp`ere type equation associated to (1) as follows det[D 2 u − A(x, u, Du)] = f (x, u, Du) in .

(3)

Our terminology and notations are as in [14]. Here and in what follows, we denote by n×n λmin (P ) and λmax (P ) the smallest and largest  eigenvalues of a symmetric matrix P ∈ R , 2 respectively. For any function u ∈ C  , we set ω(x, u) ≡ D 2 u(x) − A(x, u(x), Du(x)) = [ωij (x, u)], R(x, u) ≡ ω(x, u) − B(x, u(x), Du(x)) = [Rij (x, u)], and, unless otherwise specified, we denote λu := minλmin (ω(x, u)) = min

min

(Dij u(x) − Aij (x, u(x), Du(x)))ξi ξj .

x∈ ξ ∈Rn ,|ξ |=1

x∈

Here and in what follows in this paper, the repeated indices denote summation from 1 to n, if there is no other indication. We also the inverse matrices of ω = [ωij ] and    denote R = [Rij ] by ω−1 = ωij and R −1 = R ij , respectively. We recall that (1) or (3) is elliptic with respect to u(x) ∈ C 2 () whenever λu > 0. C 2 ()-solution

(C 2 ()-subsolution)

to (1) or (3) and these equations are elliptic If u(x) is with respect to it, then we say that u(x) is elliptic solution (elliptic subsolution) to (1) or (3). It is well-known that the solvability of the Dirichlet problem (3)–(2) for symmetric Monge-Amp`ere type equations in the class of elliptic solutions was proved in [1–3, 5–8, 10, 11] and [16], where by the method of continuity, the solvability was reduced to getting a priori C 2,α -estimates for their elliptic solutions. In turn, these a priori estimates have been obtained on the basis of the two conditions: (1) the regularity property, introduced by N. S. Trudinger for the symmetric matrix A(x, z, p); (2) the existence of elliptic subsolution u(x) to this problem, and on the basis of two important facts: (a) the concavity of the function F (ω) = log det ω on the set of positive-definite matrices ω; and (b) the comparison principle for elliptic solutions and subsolutions of (3). As mentioned in the paper [14], the question on the solvability of Dirichlet problem for nonsymmetric equations (1) is an open one for dimensions n ≥ 3 [4, 16]. To study the Dirichlet problem for nonsymmetric Monge-Amp`ere type equations (1), we have introduced in [14] a new notion of δ-elliptic solution for δ ∈ [0, 1) as follows: u(x) is called δ-elliptic solution (subsolution) of (1) if u(x) is elliptic solution (subsolution) and μ(B) ≤ δλu ,

  where and in what follows, μ(B) is defined for B = B(x, z, p) ∈ BC ; Rn×n , B T = −B as follows μ(B) :=

sup B(x, z, p) = (x,z,p)∈

sup

sup

|Bij (x, z, p)ξi ξ j |,

(x,z,p)∈ ξ ∈Cn ,|ξ |=1

(4)

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

where B is the operator norm of B. In this paper, we will use also the method of continuity to study the Dirichlet problem (1)–(2). We try to establish a priori C 2,α -estimates for δ-elliptic solutions of the problem. To do this, in the previous paper [14], we have introduced the notion of the d-concavity for the function F (R) = F (ω + β) = log det(ω + β) on the subset Dδ,μ ⊂ Rn×n , defined for 0 ≤ δ < 1, μ ≥ 0 as follows  Dδ,μ ≡ R | R = ω + β, ωT = ω, β T = −β, λmin (ω) > 0, μ ≤ δλmin (ω), β ≤ μ . (5) We have proved that the function F (R) is d-concave on the set Dδ,μ , where d ≥ 0 depends only on n and δ [14, Theorem 3]. In the same paper, we have also prepared another tool by proving the comparison principle for δ-elliptic solutions and subsolutions of (1) [14, Theorem 4]. By using the d-concavity as well as some other properties of the function F (R) = log det R, where R ∈ Dδ,μ(B) and the comparison principle, obtained in [14] and by adapting proof the techniques of estimation for elliptic solutions in the symmetric case where B(x, z, p) ≡ 0 of N.S. Trudinger and his associates [5, 8–10, 12, 13, 15–17], we shall establish for δ-elliptic solutions a priori C 2,α -estimates, that are uniform with respect to a class of skew-symmetric matrices B(x, z, p). Then, by using the method of continuity, we will prove the existence and uniqueness theorem of C 2,α ()-solutions to the Dirichlet problem (1)–(2), provided the matrices B(x, z, p) are sufficiently small in some sense. In this paper, as in the papers of N.S. Trudinger mentioned above, we assume that the matrix function A(x, z, p) is strictly regular in  [10, 16, 17], that is A is co-dimension one convex with respect to p, in the sense that, there exists a constant a0 > 0 such that Aij,k (x, z, p)ξi ξj ηk η ≥ a0 |ξ |2 |η|2 ,

(6)

for all (x, z, p) ∈ , ξ, η ∈ Rn , ξ ⊥ η, where Aij,k = Dpk p Aij . The paper is organized as follows: In Section 2, we establish estimates for the traces of the product of two matrices (Lemma 2) and then use them to prove the global second derivative estimates (Theorem 1) for δ-elliptic solutions to (1), which reduce the global second derivative bounds to the boundary ones. In Section 3, we obtain second derivative bounds on the boundary for δ-elliptic solutions to the Dirichlet problem (1)–(2) (Theorem 2). In Section 4, we get the uniform with respect to B(x, z, p) ∈ W (described in Definition 1) global bounds in C 2,α () (Theorem 3) for δ-elliptic solutions to the Dirichlet problem (1)–(2). In the last section, we apply the method of continuity to prove the existence and uniqueness theorem of δ-elliptic solution to the Dirichlet problem (1)–(2), provided the matrices B(x, z, p) are sufficiently small in some sense (Theorem 4). Finally, we give a simple example of the Dirichlet problem for nonsymmetric Monge-Amp`ere type equations in which we can verify the assumptions of Theorems 3 and 4.

2 Global Second Derivative Estimates via the Boundary Ones In this section, by using the d-concavity of the function F (R) = log det R and the strict regularity of the symmetric matrix A(x, z, p), we derive global bounds for second derivatives of δ-elliptic solutions to (1) in the case that B(x, z, p) ≡ 0. The method is almost the same as in the case B(x, z, p) ≡ 0 [8, 10, 16, 17]. Our main theorem in this section is the following.

H. T. Ngoan, T. T. K. Chung

  Theorem 1 Suppose u(x) ∈ C 4 () is an elliptic solution of (1), where A ∈ C 2 ; Rn×n ,   B ∈ BC 2 ; Rn×n and f ∈ C 2 (; R). Suppose the following conditions are satisfied: (i) sup|u| ≤ M0 , sup|Du| ≤ M1 ; 



(ii) A(x, z, p) is strictly

regular in  satisfying (6); (iii) Bij (x, z, p)ξi ξ j ≤ δλu |ξ |2 ;



(iv) Dz Bij (x, z, p)ξi ξ j ≤ β1 λu |ξ |2 ;





(v) Dxk Bij (x, z, p)ξi ξ j ηk , Dpk Bij (x, z, p)ξi ξ j ηk ≤ β2 λu |ξ |2 |η|;





(vi) Dxk x Bij (x, z, p)ξi ξ j ηk η , Dxk p Bij (x, z, p)ξi ξ j ηk η ≤ β3 |ξ |2 |η|2 ,





D B (x, z, p)ξi ξ j ηk , Dzp Bij (x, z, p)ξi ξ j ηk ≤ β3 |ξ |2 |η|, k

xk z ij

Dzz Bij (x, z, p)ξi ξ j ≤ β3 |ξ |2 ;



(vii) Dpk p Bij (x, z, p)ξi ξ j ηk η ≤ b0 |ξ |2 |η|2 ; (viii) f0 := inf f (x, z, p) > 0, 

where M0 , M1 , β2 , β3 are positive constants and δ, β1 , b0 are nonnegative constants, 0 ≤ δ < 1, 0 ≤ b0 ≤ a0 , a0 is the positive constant given in (6), the conditions (iii)–(vii) hold for all (x, z, p) ∈  and ξ ∈ Cn , η ∈ Rn . Then, we have the following estimates  (7) sup λmax (ω(x, u)) ≤ C 1 + sup λmax (ω(x, u)) , x∈ x∈∂ 





sup |D 2 u(x)| ≤ C 1 + sup D 2 u(x) , (8) x∈

x∈∂

where the constant C depends only on M0 , M1 , n, a0 , δ, β1 , β2 , β3 , b0 , f0 , A, f and . Remark 1 (1) We can easily check that the form of (1) and all assumptions of Theorem 1 are invariant under a rotation of coordinates with the quantities M0 , M1 , a0 , δ, β1 , β2 , β3 , b0 and f0 unchanged. (2) From the conditions (iii)–(vii), we get the following estimates for all (x, z, p) ∈  and k, = 1, . . . , n, B(x, z, p) ≤ δλu ;

(9)

Dz B(x, z, p) ≤ β1 λu ;

(10)

Dxk B(x, z, p), Dpk B(x, z, p) ≤ β2 λu ;

(11)

Dxk x B(x, z, p), Dxk p B(x, z, p), Dxk z B(x, z, p) ≤ β3 , Dzpk B(x, z, p), Dzz B(x, z, p) ≤ β3 ;

(12)

Dpk pk B(x, z, p) ≤ b0 , Dpk p B(x, z, p) ≤ 2b0 (k = ).

(13)

(3) It follows from (9) that μ(B) ≤ δλu , so u is a δ-elliptic solution of (1) and R(x, u) ∈ Dδ,μ(B) , where Dδ,μ(B) is the set defined by (5) with μ = μ(B). By [14, Propositions 3 and 5], we have n

0 < det ω(x, u) ≤ det R(x, u) ≤ (1 + δ 2 )[ 2 ] det ω(x, u), ∀x ∈ .

(14)

2.1 Some Preliminary Lemmas In this subsection, we introduce some preliminary lemmas which are essential for our later estimates.

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

We begin by recalling some properties of the matrices R = ω + β ∈ Dδ,μ [14]. By [14, Proposition 4 and Corollary 1], T  R −1 + R −1 1 1 1 = ω− 2 C1 D2 C1−1 ω− 2 , ≤ (D2 )ii ≤ 1, i = 1, . . . , n, (15) 2 1 + δ2 T  R −1 − R −1 1 1 δ = ω− 2 C1 D3 C1−1 ω− 2 , |(D3 )ii | ≤ , i = 1, . . . , n, (16) 2 1 + δ2 where C1 ∈ Cn×n is an unitary matrix, D2 ∈ Rn×n and D3 ∈ Cn×n are diagonal matrices. Lemma 1 Suppose that R = ω + β ∈ Dδ,μ . Then, T  R −1 + R −1 1 −1 ≤ ω−1 . ω ≤ 2 1 + δ2 Consequently, 1 ωii ≤ R ii ≤ ωii , i = 1, . . . , n. 1 + δ2

(17)

(18)

Proof The proof follows easily from (15). The following lemma provides some estimates for the trace of the product of two matrices. Lemma 2 Suppose that R = ω + β ∈ Dδ,μ . Then, we have the following estimates





(a) Tr R −1 P ≤ P TrR −1 ≤ P Trω−1 , ∀P ∈ Rn×n , P T = P ; (19)





(b) Tr R −1 Q ≤ δQTrR −1 ≤ δQTrω−1 , ∀Q ∈ Rn×n , QT = −Q; (20)





(c) Tr R −1 M ≤ (1 + δ)MTrR −1 ≤ (1 + δ)MTrω−1 , ∀M ∈ Rn×n . (21) Proof (a) Since P T = P and R −1 > 0, we have  T  

 R −1 + R −1 −1 −1 ≤ λmax (P )TrR −1 . λmin (P )TrR ≤ Tr R P = Tr P 2 Therefore,





Tr R −1 P ≤ max{|λmin (P )|, |λmax (P )|}TrR −1 = P TrR −1 .

From this and (18), we obtain (19).

  (b) Since QT = −Q, we have Tr R −1 Q = Tr



T  R −1 − R −1 2

Q . From this and (16),

we have



 

 1 1 1 1 Tr R −1 Q = Tr D3 C1−1 ω− 2 Qω− 2 C1 = (D3 )jj C1−1 ω− 2 Qω− 2 C1 , j

and so that





Tr R −1 Q ≤



1 δ 

−1 − 1

. 2 Qω − 2 C1 ω C

1 jj

1 + δ2 j

jj

(22)

H. T. Ngoan, T. T. K. Chung

Since ωT = ω and ω > 0, we can write ω = CDC −1 , where C ∈ Rn×n is orthogonal, D = diag(λ1 , . . . , λn ), where λi > 0, i = 1, . . . , n are the eigenvalues of ω. Let C2 = ˜ − 12 C −1 . ˜ = C −1 QC, then C2 is unitary and C −1 ω− 12 Qω− 12 C1 = C2 D − 12 QD C1−1 C, Q 1 2 Therefore,









 

  1 1



−2 −2 −1

C −1 ω− 12 Qω− 12 C1 =



˜ Q (C ) λ λ C k 2 jk k 2

1



jj j

j j k,





    − 12 − 12



˜ k (C2 )j k λ Q (C2 )j λ = k





j k,  





 1 2

(C2 )j k λ− 2

˜ Q ≤ k



j

⎛

k

⎞   −1 ⎝ |(C2 )j k |2 ⎠ λ−1 = Q k = QTrω . k

j



 δ −1 . This together with (18) gives From this and (22), we get |Tr R −1 Q | ≤ 1+δ 2 QTrω (20). T T (c) By writing M = M+M + M−M = P + Q, then we can easily obtain (21) from 2 2 (19) and (20).

2.2 The Proof of Theorem 1 In this subsection, we will prove Theorem 1. Our arguments are modifications of those presented in [10] and [17]. First, we rewrite (1) as follows F (R(x, u)) := log det R(x, u) = fˆ(x, u, Du)

in ,

(23)

where fˆ = log f . By [14, Proposition 7], we have for i, j, k, = 1, . . . , n, F ij ≡

∂F (R(x, u)) = Rj i , ∂Rij

F ij,k ≡

∂ 2 F (R(x, u)) = −R i R j k . ∂Rij ∂Rk

(24)

We introduce the linearized operators of the operator F and of (23),   L = F ij Dij − Dpk Aij (x, u, Du)Dk − Dpk Bij (x, u, Du)Dk ,

L = L − Dpk fˆ(x, u, Du)Dk . (25)  −1 T  ij  . From this and since R > 0, we see that the From (24), we have F n×n = R operators L and L are elliptic in . We also note ∈ Dδ,μ(B) , so by Lemma 2   that R(x, u)  and (24), we can estimate the sums of the form i,j F ij Mij = i,j R j i Mij = Tr R −1 M    via i R ii = i F ii or i ωii , where M = [Mij ] ∈ Rn×n . This plays an important role in our later estimates. Proof Let v(x, ξ ) be an auxiliary function given by v(x, ξ ) = ωξ ξ (x),

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

where ωξ ξ (x) = ωij (x, u)ξi ξj = (Dij u(x) − Aij (x, u(x), Du(x)))ξi ξj , x ∈ , ξ ∈ S n , S nis the set of the unit vectors in Rn . Suppose that v(x, ξ ) attains its maximum  consisting 0 0 n at x , ξ ∈  × S . Then,

 v x 0 , ξ 0 = maxλmax (ω(x, u)). (26) x∈

We consider the following cases. Case 1 x 0 ∈ . Without loss of generality, we can choose an orthogonal coordinate system e1 , . . . , en at x 0 such that e1 (x 0 ) = ξ 0 , [ωij ] is diagonal at x 0 and ω11 ≥ · · · ≥ ωnn > 0. Then, ω−1 = [ωij ] is diagonal at x 0 and ω11 ≤ · · · ≤ ωnn at x 0 . Let v(x) ≡ v(x, e1 ) =  ω11 (x). Then,   the function v(x) also attains its maximum at the point x 0 . Hence, Dv x 0 = 0, D 2 v x 0 ≤ 0 and we get



 Lω11 x 0 = Lv x 0 ≤ 0. (27)   Next, we shall estimate Lω11 x 0 . We start with some identities. By differentiation of (23) in the direction ξ , we have F ij [Dij uξ − Dξ Aij − (Dz Aij )uξ − (Dpk Aij )Dk uξ − Dξ Bij − (Dz Bij )uξ

 

−(Dpk Bij )Dk uξ ] = Dξ fˆ + Dz fˆ uξ + Dpk fˆ Dk uξ ,

(28)

and a further differentiation in the direction of ξ yields F ij [Dij uξ ξ − Dξ ξ Aij − (Dzz Aij )(uξ )2 − (Dpk p Aij )Dk uξ D uξ − (Dz Aij )uξ ξ −(Dpk Aij )Dk uξ ξ − 2(Dξ z Aij )uξ − 2(Dξpk Aij )Dk uξ − 2(Dzpk Aij )(Dk uξ )uξ −Dξ ξ Bij − (Dzz Bij )(uξ )2 − (Dpk p Bij )Dk uξ D uξ − (Dz Bij )uξ ξ − (Dpk Bij )Dk uξ ξ −2(Dξ z Bij )uξ − 2(Dξpk Bij )Dk uξ − 2(Dzpk Bij )(Dk uξ )uξ ] + F ij,k Dξ Rij Dξ Rk

   

= Dξ ξ fˆ + Dzz fˆ (uξ )2 + Dpk p fˆ Dk uξ D uξ + Dz fˆ uξ ξ + Dpk fˆ Dk uξ ξ   

+2 Dξ z fˆ uξ + 2 Dξpk fˆ Dk uξ + 2 Dzpk fˆ (Dk uξ )uξ . (29) Note that

Lω11 = L(u11 − A11 ) = Lu11 − LA11 .

(30)

Taking ξ = e1 in (29), and noting that uk = ωk + Ak , where ωk = 0 if k = , then from Lemma 2, (12) we have

Lu11 ≥ F ij Aij,k u1k u1 + F ij Bij,k u1k u1 + F ij Dz Bij u11   2 −F ij,k D1 Rij D1 Rk − C ωii (1 + ω11 ) + ω11 , where C depends on M0 , M1 , n, δ, β3 , f0 , A, f and . In this proof, unless otherwise specified, we shall use C to denote a positive constant with such dependence. By applying [14, Theorem 1] for the upper estimate for second-order differentials of the function

H. T. Ngoan, T. T. K. Chung

F (R) = log(det R) with the matrices R = R(x 0 , u) ∈ Dδ,μ(B) , M = D1 R(x 0 , u) = D1 ω(x 0 , u) − D1 B(x 0 , u(x 0 ), Du(x 0 )) and η = 1, we have  

1 2 4n2 δ 2

− 1 ij,k 2 D1 Bω − 2

D1 Rij D1 Rk ≥ − 1 + −F

ω 1 − δ2    D1 B 2 4n2 δ 2 C 2 ≥ −n 1 + ≥ − 1 + ω 11 , λu 1−δ 1 − δ2 where the second inequality holds by [14, Proposition 12], the last inequality holds by (10), (11) and the equality D1 B = Dx1 B + Dz Bu1 + k Dpk Bu1k , the constant C depends additionally on β1 and β2 . By Lemma 2 and (10), we have

 Dz B F ij Dz Bij u11 ≥ − δDz Bωii |u11 | ≥ −nδ |u11 | ≥ −C(1 + ω11 ). λu By Lemma 2 and (13), we have 2 F ij Aij,k u1k u1 + F ij Bij,k u1k u1 ≥ F ij (Aij,11 + Bij,11 )ω11 − Cωii (1 + ω11 ),

where C depends additionally on b0 . From these estimates, we obtain  C  ii 2 2 Lu11 ≥ F ij (Aij,11 + Bij,11 )ω11 − . ω (1 + ω11 ) + ω11 1−δ

(31)

From (24) and note that AT = A, we can write n n    1  ij F ij Aij,11 = R 1i + R i1 A1i,11 − R 11 A11,11 + R + R j i Aij,11 . 2 i,j =2

i=1

Denote



T   ≡ R ij + R j i R −1 + R −1

2≤i,j ≤n

, Dp1 p1 A ≡ [Aij,11 ]2≤i,j ≤n .

T  > 0 and thus (R −1 + (R −1 )T ) > 0. On the other hand, by Note that R −1 + R −1 choosing orthogonal vectors η = e1 = (1, 0, . . . , 0) and ξ = (0, ξ2 , . . . , ξn ) ∈ Rn , from (6), we can easily infer that λmin (Dp1 p1 A (x, z, p)) ≥ a0 , for all (x, z, p) ∈ . Thus,   n

 T     R ij + R j i Aij,11 = Tr Dp1 p1 A R −1 + R −1 i,j =2



T    ≥ 2a0 R ii . ≥ λmin Dp1 p1 A Tr R −1 + R −1 

i>1

 −1 T

> 0, so any 2×2 diagonal minor has a positive determinant. Moreover, since R −1 + R Thus, by applying Cauchy’s inequality, we have for any positive constant θ ,



 1 11

1i R , i = 1, . . . , n.

R + R i1 ≤ 2 R 11 R ii ≤ 2θR ii + 2θ Consequently, we get   1 11 ij ii R 2θR + |A1i,11 | − R 11 |A11,11 | + a0 R ii . (32) F Aij,11 ≥ − 2θ i

i>1

It follows from Lemma 2 and (24) that F ij Bij,11 ≥ −δDp1 p1 BR ii .

(33)

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

From (13), (33) and note that 0 ≤ b0 ≤ a0 , we have F ij Bij,11 ≥ −δa0 R ii . This together with (32) gives   (1 − δ)a0 − 2θ|A1i,11 | R ii F ij (Aij,11 + Bij,11 ) ≥ i>1

 1  − (1 + 2θ)|A11,11 | + |A1i,11 | + δa0 R 11 . 2θ 

i

From (6), we have max max|Aij,k (x, z, p)| > 0, where V = {(x, z, p) ∈  : |z| ≤ i,j,k, V

M0 , |p| ≤ M1 }. Thus, by taking θ =

(1−δ)a0 4 max max |Aij,k (x,z,p)| ,

we then get

i,j,k, V

F ij (Aij,11 + Bij,11 ) ≥

(1 − δ)a0  ii C R 11 R − 2 1−δ i>1

(1 − δ)a0  ii C (1 − δ)a0  ii C   ≥  ω − ω − ω11 ≥  ω11 , 2 1−δ 1−δ 2 1 + δ i>1 4 1 + δ2 i where C depends additionally on a0 , the  follows from (18) and the last second inequality inequality follows from the inequality i>1 ωii ≥ 12 i ωii , which holds by the fact that ω11 ≤ · · · ≤ ωnn . From this and (31), we obtain the estimate for Lu11 ,  (1 − δ)a0 ii 2 C  ii 2  ω ω11 − Lu11 ≥  (1 + ω ) + ω . (34) ω 11 11 1−δ 4 1 + δ2 It remains to estimate LA11 . From Lemma 2 and (11), we have

LA11 = F ij [Dxi xj A11 + Dxi z A11 uj + Dxi pk A11 uj k + Dxj z A11 ui + Dzz A11 ui uj +Dzpk A11 ui uj k + Dz A11 uij + Dxj pk A11 uik + Dzpk A11 uj uik + A11,k uik uj +Dpk A11 uij k − Dpk Aij (Dxk A11 + Dz A11 uk + Dp A11 uk ) −Dpk Bij (Dxk A11 + Dz A11 uk + Dp A11 uk )] −Dpk fˆ(Dxk A11 + Dz A11 uk + Dp A11 uk ) ≤ F ij (A11,k uik uj + Dpk A11 uij k ) + C[ωii (1 + ωjj ) + ωii ]. Taking ξ = ek in (28), where ek is the kth unit vector, then from Lemma 2, (10) and (11), we have Dpk A11 F ij uij k = Dpk A11 [F ij (Dxk Aij + Dz Aij uk + Dp Aij uk +Dxk Bij + Dz Bij uk + Dp Bij uk ) + Dxk fˆ + Dz fˆuk + Dp fˆuk ] ≤ C[ωii (1 + ωjj ) + ωii ]. By Lemma 2, (9) and (24), we have F ij A11,k uik uj = R j i (Rik + Aik + Bik )(R j + A j + B j )A11,k = R j A11,j + (A j + B j )A11,j + (Aik + Bik )A11,ki +R j i (Aik + Bik )(A j + B j )A11,k ≤ C(1 + ωii + ωii ). From these estimates and note that ωii ≤ ω11 , i = 1, . . . , n, we get 2 LA11 ≤ C[ωii (1 + ω11 ) + ω11 ].

(35)

H. T. Ngoan, T. T. K. Chung

Combining (30), (34) and (35), we obtain the estimate for Lω11 at x 0 , (1 − δ)a0 ii 2 C 2 [ωii (1 + ω11 ) + ω11 ]. ω ω11 − 1−δ 4(1 + δ 2 ) There are two possibilities:

Lω11 ≥

–

If

(1−δ)a0 ii ω 8(1+δ 2 )

≤

C 1−δ ,

that is ωii ≤

8(1+δ 2 )C . (1−δ)2 a0

(36)

Then from (1) and (14), we can easily

obtain ω11 ≤ (1−δ)C2(n−1) , for a further constant C. C 0 ii – If (1−δ)a ω ≥ 1−δ , then from (36), we have 8(1+δ 2 ) 

Lω11 ≥

 (1 − δ)a0 2 C C ωii . ω − − ω 11 1−δ 8(1 + δ 2 ) 11 1 − δ

This together with (27) gives   (1 − δ)a0 2 C C ω −  ωii . 0≥ ω11 − 1−δ 8 1 + δ 2 11 1 − δ From this, we obtain the estimate ω11 ≤

C , (1−δ)2

for a further constant C.

Consequently, from (36), we have shown that ω11 (x 0 ) ≤ C, where C is a positive constant depending only on M0 , M1 , n, a0 , δ, β1 , β2 , β3 , b0 , f0 , A, f and . From this and (26), we get that sup λmax (ω(x, u)) ≤ C. (37) x∈

Case 2 x 0 ∈ ∂. Then from (26), we get that sup λmax (ω(x, u)) ≤ sup λmax (ω(x, u)). x∈

(38)

x∈∂

As a consequence, from (37) and (38), we obtain (7). From (7), condition (i) and the equality ω(x, u) = D 2 u − A(x, u, Du), we obtain (8). The proof is complete.

3 Boundary Estimates for Second Derivatives In Section 2, we have shown that the second derivatives of δ-elliptic solutions of (1) may be estimated in terms of their boundary values. In this section, we shall establish the boundary estimate |D 2 u| ≤ C on ∂ for δ-elliptic solutions of the Dirichlet problem (1)–(2). Our main theorem in this section is the following. Theorem 2 Suppose u ∈ C 4 () is an elliptic solution of the Dirichlet problem (1)–(2), where A ∈ C 2 (; Rn×n ), B ∈ BC 2 (; Rn×n ), f ∈ C 2 (; R), ϕ ∈ C 4 (∂) and ∂ ∈ C 4 . Suppose also there exists an elliptic subsolution u(x) ∈ C 2 () to the Dirichlet problem (3)–(2)  in which B ≡ 0and u = ϕ on ∂. Suppose the following conditions are satisfied:  (i) max sup|u|, sup|u| ≤ M0 , max sup|Du|, sup|Du| ≤ M1 ; 







(ii) Dz A(x, z, p) = [Dz Aij (x, z, p)] ≥ 0 in ; (iii) A(x, z, p) is strictly

regular in  satisfying (6); (iv) Bij (x, z, p)ξi ξ j ≤ δ min{λu , λu }|ξ |2 ;

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...



(v) Dz Bij (x, z, p)ξi ξ j ≤ β1 min{λu , λu }|ξ |2 ;





(vi) Dxk Bij (x, z, p)ξi ξ j ηk , Dpk Bij (x, z, p)ξi ξ j ηk ≤ β2 min{λu , λu }|ξ |2 |η|;



(vii) Dpk p Bij (x, z, p)ξi ξ j ηk η ≤ b0 |ξ |2 |η|2 ; (viii) f0 := inf f (x, z, p) > 0;   nδ Dz f (x, z, p) ≥ β1 , (ix) inf  f (x, z, p) 1 + δ2 where M0 , M1 , β2 are positive constants and δ, β1 , b0 are nonnegative constants, 0 ≤ δ < 1, 0 ≤ b0 ≤ a0 , a0 is the positive constant given in (6), the conditions (iv)–(vii) hold for all (x, z, p) ∈  and ξ ∈ Cn , η ∈ Rn . Then, we have the boundary estimates sup |D 2 u(x)| ≤ C,

(39)

x∈∂

sup λmax (ω(x, u)) ≤ C,

(40)

x∈∂

where the constant C depends only on M0 , M1 , n, a0 , δ, β1 , β2 , b0 , f0 , A, f, u, ϕ and . Remark 2 Since u is an elliptic subsolution of (3), then by [14, Proposition 3] we infer that u is also an elliptic subsolution of (1). Hence, it follows from condition (iv) that u and u are respectively δ-elliptic solution and subsolution of (1). From this and the conditions (ii), (v) and (ix), we see that all hypotheses of the comparison principle [14, Theorem 4] for (1) hold when applyied to the compared functions u and u. Thus, we get ∂u ∂u ≥ , on ∂ (ν is the unit inner normal to ∂), ∂ν ∂ν where the second inequality follows from the first inequality and the fact that u = u on ∂. u ≥ u, in  and

3.1 Boundary Transformation We first note that the form of (1) and the hypotheses of Theorem 2 are invariant under translation and rotation of coordinates. Therefore, for any given boundary point x 0 ∈ ∂, without loss of generality, we may take x 0 as the origin and take the inner normal to ∂ at this point as the positive direction of xn -axis. In a neighbourhood N of the origin, ∂ can be represented as the graph xn = h(x ), such that h(0) = 0, Dh(0) = 0, where x = (x1 , . . . , xn−1 ). Next, we flatten the boundary ∂ in N , that is we consider the following diffeomorphism ψ defined by ˜ := ψ(). y = ψ(x) = (x , xn − h(x )), x = (x , xn ) ∈ , y ∈ 

(41)

We will investigate how this boundary transformation affects the form of the MongeAmp`ere type (1) and the hypotheses of Theorem 2. ∂ψ Denote Jij = ∂xij, , i, j = 1, . . . , n, where ψ = (ψ1 , . . . , ψn ). Then      E   E(n−1)×(n−1) −Dh (n−1)×(n−1) Dh , J −1 = J ij = , (42) J = Jij = 0 1 0 1 and furthermore,

   0 D(D h)  xk Dxk J −1 = Dxk J ij = , k = 1, . . . , n. 0 0

(43)

H. T. Ngoan, T. T. K. Chung

Under the transformation (41), we set v(y) = u(x). Then (1) becomes   ˜ ˜ ˜ det D 2 v − A(y, v, Dv) − B(y, v, Dv) = f˜(y, v, Dv), in ,

(44)

⎧   T  ˜ ⎪ z, p) = J −1 A(ψ −1 (y), z, Jp) − k pk D 2 ψk J −1 , ⎨ A(y, T   ˜ (45) B ψ −1 (y), z, Jp J −1 , B(y, z, p) = J −1 ⎪  ⎩ ˜ −1 f (y, z, p) = f ψ (y), z, Jp . So, it is easy to see that the form of (1) remains invariant.  s1 (y), . .. , sn (y) denote the eigenvalues of the symmetric matrix S(y) =  Let ˜ where s1 ≥ · · · ≥ sn > 0. From (42), by calculation, we have J J T ψ −1 (y) in , sj = 1, j = 2, . . . , n − 1, and  " 2  " 2   2 + |Dh|2 + 2 + |Dh|2 − 4 2 + |Dh|2 − 2 + |Dh|2 − 4 s1 = , sn = . 2 2

where

Obviously, s1 , . . . , sn are also the eigenvalues of S −1 = (J −1 )T J −1 and so are those of J −1 (J −1 )T . Let B(0; ρ0 ) denotes the ball in the y-space of radius ρ0 centered at 0 such that B(0; ρ0 ) ⊂ N˜ := ψ(N ) and set s0 := max s1 (y) = max (sn (y))−1 . B(0;ρ0 )

(46)

B(0;ρ0 )

Note that s1 (0) = sn (0) = 1, so we have s0 ≥ 1. Further, since δ ∈ [0, 1), we may take ρ0 small enough such that (47) s06 δ < 1. Now we assume that all of the assumptions of Theorem 2 are satisfied. With the transformation (41), we set (48) v(y) = u(x), v(y) = u(x). From (45), we then have

T ˜ v, Dv) = J −1 ω(x, u) J −1 , ω(y, ˜ v) ≡ D 2 v − A(y,

T ˜ v, Dv) = J −1 ω(x, u) J −1 . ω(y, ˜ v) ≡ D 2 v − A(y,

From this and (46), we can easily get min{λu , λu } ≤ s0 min{λv , λv },

(49)

˜ ρ0 :=  ˜ ∩ B(0; ρ0 ). It is easy where λv = min λmin (ω(y, ˜ v)), λv = min λmin (ω(y, ˜ v)),  ˜ρ y∈ 0

˜ρ y∈ 0

˜ ρ0 and v(y) is an elliptic to see that v(y) is an elliptic solution of the transformed (44) in  ˜ subsolution of the corresponding equation to (44) in which B ≡ 0. Further, we have the following proposition. Proposition 1 Under the transformation (41), the forms of the conditions (i)–(viii) of ˜ ρ0 with the following new corresponding constants: Theorem 2 are invariant in  √ M˜ 0 = M0 , M˜ 1 = s0 M1 , a˜ 0 = s0−2 a0 , δ˜ = s02 δ, β˜1 = s02 β1 , ⎛ ⎞ # #

√ 2 # # 5/2 β˜2 = 2 ns0 δ ⎝max sup #Dxk J −1 ψ −1 (y) #⎠ + s0 β2 , b˜0 = s02 b0 , f˜0 ≥ f0 , (50) k

˜ρ y∈ 0

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

where Dxk J −1 , k = 1, . . . , n are the matrices given in (43). Proof The proof follows easily from the conditions (i)–(viii), (45), (46), and (49). ˜ ρ0 Remark 3 (1) From Remark 2, (48) and the fact that u = u on ∂, we get that v ≥ v in  ˜ and Dn v ≥ Dn v on ∂  ∩ B(0; ρ0 ). (2) From (47), (50) and note that s0 ≥ 1, we have δ˜ ∈ [0, 1). (3) From (50) we note that the inequality b˜0 ≤ a˜ 0 does not generally hold. Despite this fact, from Proposition 1, (47) and b0 ≤ a0 , we have







δ˜ Dpk p B˜ ij (y, z, p)ξi ξ j ηk η ≤ s02 δ s02 b0 |ξ |2 |η|2 ≤ s04 δa0 |ξ |2 |η|2 ≤ a˜ 0 |ξ |2 |η|2 , ˜ ρ0 × R × Rn and ξ ∈ Cn , η ∈ Rn . Hence we have for all (y, z, p) ∈ for all (y, z, p) ∈  n ˜ ρ0 × R × R and k, = 1, . . . , n, # # # # # # # # ˜ ˜ z, p)# ≤ a˜ 0 , δ˜ #Dpk p B(y, z, p)# ≤ 2a˜ 0 (k = ). δ˜ #Dpk pk B(y,

3.2 The Proof of Theorem 2 Take any point x 0 ∈ ∂. Suppose a part of ∂ near x 0 had been flattened by the transformation (41). Then, instead of the variables y = (y1 , . . . , yn ), we still use the variables x = (x1 , . . . , xn ) with the corresponding modifications of the conditions (i)–(viii) of Theorem 2, as indicated in Proposition 1. Specifically, we may assume that x 0 = 0 and ∂ is flat in a neighbourhood of this point, that is, for some constant ρ0 > 0 small enough, ρ0 :=  ∩ B(0; ρ0 ) ⊂ Rn+ = {x = (x1 , . . . , xn ) ∈ Rn : xn > 0}, Tρ0 := ∂ ∩ B(0; ρ0 ) ⊂ ∂Rn+ = {x = (x1 , . . . , xn ) ∈ Rn : xn = 0}. Furthermore, we may assume that the functions u and u are respectively δ-elliptic solution and subsolution of (1) in ρ0 , and we have the following inequalities $ % $ % max sup|u|, sup|u| ≤ M0 , max sup|Du|, sup|Du| ≤ M1 ; ρ0

ρ0

ρ0

(51)

ρ0

Aij,k (x, z, p)ξi ξj ηk η ≥ a0 |ξ |2 |η|2 , ∀ξ, η ∈ Rn , ξ ⊥ η;

(52)

B(x, z, p) ≤ δ min{λu , λu }, 0 ≤ δ < 1;

(53)

Dz B(x, z, p) ≤ β1 min{λu , λu };

(54)

Dxk B(x, z, p), Dpk B(x, z, p) ≤ β2 min{λu , λu }, k = 1, . . . , n;

(55)

δDpk pk B(x, z, p) ≤ a0 , δDpk p B(x, z, p) ≤ 2a0 (k = ), k, = 1, . . . , n; (56) inf

ρ0 ×R×Rn

f (x, z, p) ≥ f0 > 0,

(57)

where the inequalities (52)–(56) are satisfied for all (x, z, p) ∈ ρ0 × R × Rn . Here and in what follows in this proof, λu = min λmin (ω(x, u)) and λu = min λmin (ω(x, u)). x∈ρ0

x∈ρ0

Furthermore, we have u ≥ u, in ρ0 and Dn u ≥ Dn u, on Tρ0 .

(58)

H. T. Ngoan, T. T. K. Chung

Now, we will adapt the techniques in [8, 10] and [15] to prove Theorem 2. First, we will construct suitable barriers near the boundary. Let d = d(x) be the distance function from ∂. As in [8] and [10], the key ingredient is the following lemma. Lemma 3 Under the above assumptions, there exist positive constants K0 , N0 sufficiently large and μ0 , ρ sufficiently small such that the function ψ0 = 1 − e satisfies

Lψ0 ≤ −ε0



  2 K0 (u−u)−μ0 d+N0 d

F ii − ε0 , in ρ and ψ0 ≥ 0, on ∂ρ ,

(59)

i

for some positive constant ε0 , where K0 , N0 , μ0 , ρ, ε0 depend only on M0 , M1 , n, a0 , δ, β1 , β2 , f0 , ρ0 , A, f, u and ρ0 , L is the linear elliptic operator defined by (25) and ρ =  ∩ B(0; ρ). Proof Let uε = u − 2ε |x|2 , for some positive constant ε determined later. Then we may choose ε ∈ (0, 1) small enough such that ω(x, uε ) > 0 in ρ0 and λuε ≥ k0 λu > 0, for some positive constant k0 satisfying k0 > δ, where λuε

(60)    = min λmin ω x, uε . Set x∈ρ0

2

v = (u − u) − μ0 d + N0 d . 2

Then v = 2ε |x|2 + uε − u − μ0 d + N0 d . From this and note that L d

 2

(61)  = 2 d Ld + 

2F ij Di dDj d, we have Lv = εF ii − εF ij (Dpk Aij + Dpk Bij )(x, u, Du)xk   +F ij Dij uε − (Aij + Bij )(x, uε , Duε ) + 2N0 Di dDj d   −F ij Dij u − (Aij + Bij )(x, u, Du)   +F ij (Aij + Bij )(x, uε , Duε ) − (Aij + Bij )(x, u, Duε ) +F ij [(Aij + Bij )(x, u, Duε ) − (Aij + Bij )(x, u, Du)     −(Dpk Aij + Dpk Bij )(x, u, Du)Dk (uε − u)] − μ0 Ld + 2N0 d Ld . (62) First, we assume that ρ < 1 small enough such that B + (0; 2ρ) = {x ∈ Rn+ : |x| < 2ρ} ⊂ ρ0 . Then ρ ⊂ B + (0; 2ρ) ⊂  ρ0 . From Lemma 2, (55) and note that d(x) = xn for all

x = (x1 , . . . , xn ) ∈ ρ0 , we have Ld ≤ CF ii , and thus     (63) − μ0 Ld + 2N0 d Ld ≥ −C(μ0 + 2N0 ρ)F ii , in ρ , where C depends on M0 , M1 , n, δ, β2 , A, u and ρ0 . In this proof, unless otherwise specified, we shall use C to denote a positive constant with such dependence. Also by Lemma 2 and (55), we have



  √

ij

(64)

F (Dpk Aij + Dpk Bij )xk ≤ |x| |Dp A| + nδβ2 λu F ii , where |Dp A| =



k |Dpk A|

 2 1/2 .

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

Next, we shall prove the following estimate for sufficiently small ρ,   ε (65) F ij (Aij + Bij )(x, uε , Duε ) − (Aij + Bij )(x, u, Duε ) ≥ − F ii , in ρ . 4 Indeed, by applying the mean value theorem to the function h1 (t) := F ij (Aij + Bij )(x, tuε + (1 − t)u, Duε ) and using Lemma 2, (51) and (54), we have   F ij (Aij + Bij )(x, uε , Duε ) − (Aij + Bij )(x, u, Duε ) = h1 (1) − h1 (0)   = h 1 (t1 ) = (uε − u) F ij (Dz Aij + Dz Bij )(x, z, Duε )

  ε ≥ −|uε − u| |Dz A(x, z, Duε )| + δβ1 λu F ii ≥ −h0 2M1 + |x|F ii , 2 where z = t1 uε + (1 − t1 )u, for some t1 ∈ (0, 1), h0 = max|Dz A(x, z, p)| + δβ1 λu ≥ 0, V

here and in what follows in this proof, ' & V := (x, z, p) ∈ ρ0 × R × Rn : |z| ≤ M0 + 1/2, |p| ≤ M1 + 1 . Here, we have used the estimate

ε ε |(uε − u)(x)| ≤ |u(x) − u(0)| + |u(x) − u(0)| + |x|2 ≤ 2M1 + |x|, 2 2 which holds by (51) and the facts that u(0) = u(0), ρ ∈ (0, 1). Hence, if h0 = 0 then (65) is straightforward, and if h0 > 0 then (65) is followed by taking ρ is small enough such that  ρ < min 16Mε1 h0 , 4h1 0 . Thus (65) is proved. Now, we assume that (66) μ0 + 2N0 ρ ≤ 1. By applying the Taylor expansion to the function h2 (t) := F ij (Aij + Bij )(x, u, tDuε + (1 − t)Du) and using Lemma 2, (51), (56), (61) and (66), we have F ij [(Aij + Bij )(x, u, Duε ) − (Aij + Bij )(x, u, Du)

−(Dpk Aij + Dpk Bij )(x, u, Du)Dk (uε − u)] = h2 (1) − h2 (0) − h 2 (0) 1 1 = h 2 (t2 ) = F ij (Aij,k + Bij,k )(x, u, p)Dk vD v 2 2   ε ij + F (Aij,k + Bij,k )(x, u, p) εxk − 2Dk v − 2(μ0 − 2N0 d)Dk d x 2   1 + (μ0 − 2N0 d)F ij (Aij,k + Bij,k )(x, u, p) 2Dk v + (μ0 − 2N0 d)Dk d D d 2 1 ij ε ≥ F (Aij,k + Bij,k )(x, u, p)Dk vD v − (4M1 + 1)(|Dpp A| + 2na0 )|x|F ii 2 2 −C(μ0 + 2N0 ρ)F ii , (67) 1/2  2 where p = t2 Duε + (1 − t2 )Du, for some t2 ∈ (0, 1), |Dpp A| = , the k, |Dpk p A| constant C depends additionally on a0 but does not depend on μ0 and N0 . Here, we have used the following estimate |εx − 2Dv − 2(μ0 − 2N0 d)Dd| = |εx − 2(Du − Du)| ≤ 4M1 + 1, which holds by (51) and (61). Next, we note that the inequality (53) implies that μ(B) ≤ δ min{λu , λu }, here and in what follows in this proof, the quantity μ(B) is defined by (4) with  replaced by ρ0 × R × Rn . Hence from (60), we have μ(B) ≤ δ λuε , where δ = kδ0 ∈ [0, 1). Set δ ∗ = max{δ, δ } ∈ [0, 1), then R(x, u), R(x, uε ) ∈ Dδ ∗ ,μ(B) , where Dδ ∗ ,μ(B) is the set defined

H. T. Ngoan, T. T. K. Chung

as in (5). From this and the fact that Dd ⊗ Dd ≥ 0, we infer that R (0) ≡ R(x, u), R (1) ≡ R(x, uε )+2N0 Dd ⊗Dd ∈ Dδ ∗ ,μ(B) . Hence we can use [14, Theorem 3] on the d-concavity of the function F (R) = log det R applied to R (0) , R (1) and obtain   F ij Dij uε − (Aij + Bij )(x, uε , Duε ) + 2N0 Di dDj d   −F ij Dij u − (Aij + Bij )(x, u, Du) ≥ log det R (1) − log det R (0) − d(n, δ ∗ )   ≥ log det ω(x, uε ) + 2N0 Dd ⊗ Dd − fˆ(x, u, Du) − d(n, δ ∗ ), (68)

 2 δ ∗2 where d(n, δ ∗ ) = 2nδ ∗ 2 1 + 4n and the last inequality holds by [14, Proposition 3] 1−δ ∗2 and (23). We also note that the eigenvalues of Dd ⊗ Dd are 0, . . . , 0, 1. From this and (60), we infer that   n−1     n−1   λuε + 2N0 ≥ k0 λu k0 λu + 2N0 . det ω(x, uε ) + 2N0 Dd ⊗ Dd ≥ λuε So, the quantity log det[ω(x, uε ) + 2N0 Dd ⊗ Dd] can be made as large as we want by choosing N0 large enough. Assume N0 is large enough such that ˜ log det[ω(x, uε ) + 2N0 Dd ⊗ Dd] ≥ max|fˆ(x, z, p)| + d(n, δ ∗ ) + M, V

(69)

where M˜ is a large positive constant to be chosen later. Set   √ 4M1 + 1 |Dpp A(x, z, p)| + nδβ2 λu + na0 (4M1 + 1). m0 = max |Dp A(x, z, p)| + V 2 Then by taking ρ small enough satisfying ρ < Lv ≥

1 4m0 , from (62)–(65) and (67)–(69), we obtain

ε ii 1 ij ˜ (70) F + F (Aij,k + Bij,k )(x, u, p)Dk vD v − C(μ0 + 2N0 ρ)F ii + M. 2 2

Let us choose μ0 =

ε ε and ρ < small enough. 8C 16N0 C

(71)

ε Then μ0 + 2N0 ρ < 4C . Thus we can always choose C large enough such that the inequality (66) is fulfilled. Furthermore, from (70) and (71), we get

Lv ≥

ε ii 1 ij ˜ F + F (Aij,k + Bij,k )(x, u, p)Dk vD v + M. 4 2

Let φ0 = eK0 v with positive constant K0 to be determined later. Then Lφ0 = K0 eK0 v Lv + K02 eK0 v F ij Di vDj v  1 K0 v ε ii ≥ K0 e F + F ij (Aij,k + Bij,k )(x, u, p)Dk vD v 4 2  +K0 F ij Di vDj v + M˜ , in ρ . Without loss of generality, we may assume that Dv = (D1 v, 0, . . . , 0) at a given point in ρ , so we get   ε ii 1 ij F + F (Aij,11 + Bij,11 )(x, u, p)(D1 v)2 + K0 F 11 (D1 v)2 + M˜ . Lφ0 ≥ K0 eK0 v 4 2 (72)

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

It follows from (24), (32), (33), and (56) that F (Aij,11 + Bij,11 ) ≥ −2θ ij



|A1i,11 |F − ii

i



 1  |A1i,11 | + |A11,11 | + a0 F 11 . 2θ i

This together with (72) gives   ε Lφ0 ≥ K0 eK0 v − θ |A1i,11 (x, u, p)|(D1 v)2 F ii + M˜ 4    1  1 1 2 11 . |A1i,11 (x, u, p)| − |A11,11 (x, u, p)| − a0 (D1 v) F + K0 − 4θ 2 2 i

(73) By choosing θ = and then choosing K0 large enough such i,j,k, V 

 1 1 |A (x, z, p)| + |A (x, z, p)| + 12 a0 , we obtain from (51), that K0 ≥ max 4θ 1i,11 11,11 i 2 ε , 8(2M1 +1)2 max max|Aij,k (x,z,p)|

V

(61), (66), and (73) that Lφ0 ≥ K0 eK0 v

ε

 F ii + M˜ , in ρ .

8 From this, (51), (61), and (66), we obtain in ρ ,

Lφ0 = Lφ0 − Dpk fˆ(x, u, Du)Dk φ0



ε



F ii + M˜ − |Dv| Dp fˆ(x, u, Du)

≥ K 0 e K0 v 8



ε



F ii + M˜ − (2M1 + 1) Dp fˆ(x, u, Du) . ≥ K0 e−K0 (2M0 +μ0 ) (74) 8





Let us choose M˜ = 8ε + (2M1 + 1)max Dp fˆ(x, z, p) . Since the constant M˜ is now fixed, V

the large enough constant N0 can be fixed now to guarantee the inequality (69). Further, from (74), we then have 

Lφ0 ≥ ε0 F ii + 1 , in ρ . where ε0 = 8ε K0 e−K0 (2M0 +μ0 ) . It follows that



Lψ0 = L(1 − φ0 ) ≤ −ε0 F ii + 1 , in ρ .

(75)

Obviously, ψ0 = 0 on ∂ ∩ B(0; ρ). Further, we infer from (58), (61) and (71) that   ε v ≤ −μ0 + N0 d d ≤ (−μ0 + N0 ρ)d ≤ − d ≤ 0, on  ∩ ∂B(0; ρ). 16C Thus φ0 = eK0 v ≤ 1, that is ψ0 = 1 − φ0 ≥ 0 on  ∩ ∂B(0; ρ). Consequently, we have ψ0 ≥ 0 on ∂ρ . This together with (75) gives the conclusion of Lemma 3. Continuing of the Proof of Theorem 2 Our arguments are almost the same as in [8] and [10]. We have the following claims. Claim 1 We have the double tangential derivative estimates |Dαβ u(0)| ≤ C, α, β = 1, . . . , n − 1.

(76)

H. T. Ngoan, T. T. K. Chung

Indeed, since u − u = 0 on Tρ0 , we have Dαβ u(0) = Dαβ u(0), α, β = 1, . . . , n − 1. This yields the desired estimate (76). Claim 2 We have the mixed tangential-normal derivative estimates |Dαn u(0)| ≤ C, α = 1, . . . , n − 1.

(77)

Indeed, we first recall that (1) is equivalent to the following   F [u] := log det D 2 u − A(x, u, Du) − B(x, u, Du) = fˆ(x, u, Du), fˆ = log f. Differentiating this equation with respect to xα , α = 1, . . . , n, we obtain

L(Dα u) = F ij (Dxα Aij + Dz Aij Dα u + Dxα Bij + Dz Bij Dα u) + Dxα fˆ + Dz fˆDα u. (78) By Lemma 2, (54), (55) and (78), we have for α = 1, . . . , n − 1,





L(Dα (u − u)) ≤ C F ii + 1 , in ρ ,

(79)

where ρ is the positive constant defined in Lemma 3. On the other hand, since u = u on Tρ0 , we have Dα (u − u) = 0 on Tρ0 , for α = 1, . . . , n − 1 and thus, |Dα (u − u)| ≤ C|x|2 , on ∂ρ .

(80)

With the barrier function ψ0 constructed in Lemma 3, that is ψ0 = 1 − 2 eK0 [(u−u)−μ0 xn +N0 xn ] , we consider a new barrier function ψ1 = aψ0 + b|x|2 , where a, b are some positive constants. Then from (59), we have for a  b,  aε0 ii Lψ1 ≤ − F + 1 , in ρ and ψ1 ≥ b|x|2 , on ∂ρ . 2 From this, (79) and (80), we have for a  b  1, |L(Dα (u − u))| + Lψ1 ≤ 0, in ρ and |Dα (u − u)| ≤ ψ1 , on ∂ρ , where α = 1, . . . , n − 1. Thus, by the maximum principle, we infer that |Dα (u − u)| ≤ ψ1 in ρ . From this and note that Dα (u − u)(0) = ψ1 (0) = 0, we obtain |Dαn (u − u)(0)| ≤ Dn ψ1 (0). Here we note that Dn ψ1 (0) = aDn ψ0 (0) ≥ 0, which is true by (58). Hence we get the desired estimate (77). Claim 3 We have the double normal derivative estimate |Dnn u(0)| ≤ C.

(81)

Indeed, by the ellipticity of u, we have Trω(0, u) > 0. From this and (76), we get an estimate from below for Dnn u(0), Dnn u(0) ≥

n 

Aii (0, u(0), Du(0)) −

i=1

n−1 

Dii u(0) ≥ −C.

i=1

Thus, it remains to show that Dnn u(0) ≤ C.

(82)

Since B is skew-symmetric, (1) can be rewritten as (Dnn u − Ann (x, u, Du)) det R (x, u) + H = f (x, u, Du),

(83)

where R (x, u) = ω (x, u) − B , ω (x, u) = [Dαβ u − Aαβ (x, u, Du)]1≤α,β≤n−1 , B = [Bαβ (x, u, Du)]1≤α,β≤n−1 , and H denotes the remaining terms which do not involve Dnn u − Ann (x, u, Du). From (76) and (77), we infer that H is bounded at x = 0. Hence

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

from (83) and the inequality det R (0, u) ≥ det ω (0, u) > 0, which holds by [14, Proposition 3], we see that (82) will be proved if one can get a positive lower bound for det ω (0, u). Let S n−1 denote the subset of Rn−1 consisting of the unit vectors. We fix a vector ξ = (ξ1 , . . . , ξn−1 ) ∈ S n−1 and set,         Q x, ξ := Q[u] x, ξ = ω (x, u)ξ , ξ = Dαβ u − Aαβ (x, u, Du) ξα ξβ   = Dαβ ϕ − Aαβ (x, ϕ, D ϕ, Dn u) ξα ξβ , on Tρ0 , with Q > 0 by virtue of the ellipticity of u. Here D = (D1 , . . . , Dn−1 ) denotes the tangential gradient. We extend ϕ to ρ0 by defining ϕ(x , xn ) = ϕ(x , 0) for xn > 0, and denote the corresponding extension of  Q also  by Q. Note  that Q is bounded and so that, for sufficiently large K, the function Q˜ x, ξ := Q x, ξ + K|x|2 will take a minimum on

 Tρ0 × S n−1 at some point x, ξ , where x ∈ Tρ ⊂ Tρ0 , and

     Q˜ x, ξ ≥ Q˜ x, ξ , ∀ x, ξ ∈ ∂ρ × S n−1 , (84) where ρ is the constant defined in Lemma 3. By computation, we have



 LQ˜ x, ξ ≤ −L(Dn u)Dpn Aαβ ξ α ξ β − F ij Din uDj n uDpn pn Aαβ ξ α ξ β + C F ii + 1 . From the regularity of A and note that [F ij ]n×n > 0, we get

  −F ij Din uDj n uDpn pn Aαβ ξ α ξ β = − F ij Din uDj n u Dpn pn Aαβ ξ α ξ β ≤ 0. From these estimates, (78) for α = n, Lemma 2, (54), (55) and (57), we obtain 



LQ˜ x, ξ ≤ C F ii + 1 , in ρ .

(85)

Now, we continue to employ the barrier function ψ0 constructed in Lemma 3 as above. From (59), (84) and (85), we can a positive constant h sufficiently large such that

choose  the function ψ2 = −hψ0 + Q˜ x, ξ satisfying



 Lψ2 (x) ≥ LQ˜ x, ξ , in ρ and ψ2 (x) ≤ Q˜ x, ξ , on ∂ρ .

 By the maximum principle, we get ψ2 (x) ≤ Q˜ x, ξ in ρ . From this and note that



 ψ2 (x) = Q˜ x, ξ , we obtain Dn ψ2 (x) ≤ Dn Q˜ x, ξ . It follows that Dnn u(x)Dpn Aαβ (x, ϕ, D ϕ, Dn u)ξ α ξ β ≤ C. Since u = ϕ on Tρ0 , we have     Q[u] x, ξ ≥ λu > 0, ∀ x, ξ ∈ Tρ0 × S n−1 .

(86)

(87)

It follows from (51) and (58) that 0 ≤ Dn (u − u) ≤ 2M1 , on Tρ0 .

(88)

By using the regularity of A, we get

  Q[u] x, ξ − Q[u] x, ξ ≥ Dn (u − u)(x)Dpn Aαβ (x, ϕ, D ϕ, Dn u)ξ α ξ β . (89)

 Now we show that Q[u] x, ξ has a positive bound from below. From (87)–(89), this

 is obvious if Dn (u − u)(x) = 0 or Q[u] x, ξ ≥ τ20 , where τ0 = λu . It remains to

H. T. Ngoan, T. T. K. Chung

 consider the case that Dn (u − u)(x) > 0 and Q[u] x, ξ < Dpn Aαβ (x, ϕ, D ϕ, Dn u)ξ α ξ β

τ0 2.

Indeed, from (87)–(89),

τ0 4M1 .

> We infer from this and (86) that we then obtain = D u(x), i = 1, . . . , n − 1, we can Dnn u(x) ≤ C. From this, (14), (57) and Dii u(x) ii

 easily obtain an estimate from below for Q[u] x, ξ .

 We have shown that Q x, ξ has a positive bound from below. From this and note that



     Q 0, ξ = Q˜ 0, ξ ≥ Q˜ x, ξ ≥ Q x, ξ , ∀ ξ ∈ S n−1 ,   we then obtain an estimate from below for Q 0, ξ . Therefore, we get a positive lower bound for det ω (0, u), and by (83), we obtain Dnn u(0) ≤ C. Hence we get (82) and thus (81) is proved. By combining (76), (77) and (81), we get |D 2 u(0)| ≤ C. By using this estimate and a translation of coordinates, we can easily obtain |D 2 u(x)| ≤ C on Tρ . Consequently, for any x 0 ∈ ∂, we have |D 2 u(x)| ≤ C in a neighbourhood small enough of x 0 on ∂. Thus, by using usual covering argument, we obtain (39). From (39), we can easily obtain (40). The proof of Theorem 2 is complete.

4 The Uniform with Respect to a Class of Skew-Symmetric Matrices C 2,α ()-Estimates In this section, we will establish the C 2,α ()-estimates for δ-elliptic solutions to the Dirichlet problem (1)–(2), that are uniform with respect to a set of skew-symmetric matrices B(x, z, p). We will preserve the structure condition on the symmetric matrix A(x, z, p), introduced in [8], to get the estimates for |u| and |Du| in . We introduce now a class of skew-symmetric matrices B(x, z, p). Suppose δ, β1 , b0 are nonnegative constants, 0 ≤ δ < 1, 0 ≤ b0 ≤ a0 , where a0 is the positive constant given in (6), β2 , β3 are positive constants and there exists a function u(x), that is an elliptic subsolution to the Dirichlet problem (3)–(2) in which B ≡ 0 and u = ϕ on ∂. Definition 1 We denote by W = W (δ, β1 , β2 , β3 , b0 , λu ) the set of skew-symmetric matri  ces B(x, z, p) = [Bij (x, z, p)] ∈ BC 2 ; Rn×n satisfying the following conditions for all ξ ∈ Cn \ 0, η ∈ Rn \ 0,

(a) sup Bij (x, z, p)ξi ξ j < δλu |ξ |2 if δ > 0, and B ≡ 0 if δ = 0; (x,z,p)∈



(b) sup Dz Bij (x, z, p)ξi ξ j < β1 λu |ξ |2 if β1 > 0, and Dz B ≡ 0 if β1 = 0; (x,z,p)∈



 

Dx Bij (x, z, p)ξi ξ j ηk , Dp Bij (x, z, p)ξi ξ j ηk < β2 λu |ξ |2 |η|; (c) sup k k (x,z,p)∈



(d) Dxk x Bij (x, z, p)ξi ξ j ηk η , Dxk p Bij (x, z, p)ξi ξ j ηk η ≤ β3 |ξ |2 |η|2 ,





Dx z Bij (x, z, p)ξi ξ j ηk , Dzp Bij (x, z, p)ξi ξ j ηk ≤ β3 |ξ |2 |η|, k

k

Dzz Bij (x, z, p)ξi ξ j ≤ β3 |ξ |2 , ∀(x, z, p) ∈ ;



(e) Dpk p Bij (x, z, p)ξi ξ j ηk η ≤ b0 |ξ |2 |η|2 , ∀(x, z, p) ∈ . Definition 2 Suppose B = B(x, z, p) is any matrix   belonging to W . We denote by U = U (δ, β1 , β2 , B) the set of all functions u(x) ∈ C 2  satisfying the following conditions 1. λu := min min (Dij u(x) − Aij (x, u(x), Du(x)))ξi ξj > 0; n x∈ ξ ∈R ,|ξ |=1

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

2.

sup (x,z,p)∈

3.

sup (x,z,p)∈

4.

sup



Bij (x, z, p)ξi ξ j < δλu |ξ |2 , ∀ξ ∈ Cn if δ > 0;



Dz Bij (x, z, p)ξi ξ j < β1 λu |ξ |2 , ∀ξ ∈ Cn if β1 > 0;



 

Dx Bij (x, z, p)ξi ξ j ηk , Dp Bij (x, z, p)ξi ξ j ηk < β2 λu |ξ |2 |η|, ∀ξ ∈ Cn \ k k

(x,z,p)∈ 0, η ∈ Rn \ 0.

The main theorem in this section is the following one, in which we obtain C 2,α ()estimates for δ-elliptic solutions of the Dirichlet problem (1)–(2), that are uniform with respect to B(x, z, p) ∈ W . Theorem 3 Suppose that  is a bounded domain in Rn with boundary ∂ ∈ C 4 . Suppose the following conditions are satisfied: (i) The symmetric matrix A(x, z, p) = [Aij (x, z, p)] ∈ C 2 (; Rn×n ) satisfies the structure conditions

 A(x, z, p) ≥ −γ0 1 + |p|2 E, and λmax (A(x, z, 0)) ≥ 0,

for all x ∈ , z ∈ R, p ∈ Rn and for some positive constant γ0 ; (ii) Dz A(x, z, p) = [Dz Aij (x, z, p)] ≥ 0 in ; (iii) A(x, z, p) is strictly regular in  satisfying (6) with a0 > 0; (iv) The function f (x, z, p) ∈ C 2 (; R) satisfying f0 := inf f (x, z, p) > 0;   nδ Dz f (x, z, p) ≥ β1 , β1 ≥ 0; (v) inf  f (x, z, p) 1 + δ2 (vi) There exists an elliptic subsolution u(x) ∈ C 2 () to the Dirichlet problem (3)–(2) in which B ≡ 0 and u = ϕ on ∂, where ϕ ∈ C 4 ().   Suppose B(x, z, p) is any matrix from W and u(x) ∈ U ∩ C 4  is a δ-elliptic solution to the corresponding Dirichlet problem (1)–(2), where W = W (δ, β1 , β2 , β3 , b0 , λu ) and U = U (δ, β1 , β2 , B) had been introduced in the Definitions 1 and 2, respectively. Then there exist positive constants C0 , λ0 , α0 ∈ (0, 1), C1 , that depend only on n, γ0 , a0 , δ, β1 , β2 , β3 , b0 , f0 , A, f, u, ϕ and  such that max λmax (ω(x, u)) ≤ C0 ,

(90)

λu ≥ λ0 ,

(91)

|u|2,α0 ; ≤ C1 ,

(92)

x∈

where λ0 =

−[ n ]

(1+δ 2 ) 2 f0 , C0n−1

and these estimates are uniform with respect to B ∈ W .

Proof From the assumptions and Remark 2, we have u ≥ u, in  and

∂u ∂u ≥ , on ∂, ∂ν ∂ν

where ν is the unit inner normal to ∂. By using this and the same arguments as in [8], from condition (i), we can easily obtain the following estimates sup|u| ≤ M0 , sup|Du| ≤ M1 , 



where M0 depends on |u|0; and |ϕ|0; , M1 depends on n, γ0 , |u|1; , |ϕ|2; and .

(93)

H. T. Ngoan, T. T. K. Chung

From (93), we see that all of the assumptions of Theorem 1 and Theorem 2 are satisfied. Thus, we can obtain the estimates (7) and (40). From these estimates we can easily obtain the desired estimate (90). Let λi (x), i = 1, . . . , n be the eigenvalues of ω(x, u), where λ1 ≥ · · · ≥ λn > 0. It follows from (90) that λi (x) ≤ C0 , ∀x ∈ , i = 1, . . . , n. From this and the following inequality

−[ n ] −[ n ] −[ n ] 2 2 2 det ω(x, u) ≥ 1 + δ 2 det R(x, u) = 1 + δ 2 f (x, u, Du) ≥ 1 + δ 2 f0 , which is true by (14) and condition (iv), we get that −[ n ]  1 + δ 2 2 f0 det ω(x, u) , ∀x ∈ . λn (x) = ≥ λ0 := λ1 (x) · · · λn−1 (x) C0n−1 This gives the desired estimate (91). It remains to prove (92). We recall that (1) can be rewritten in the form F (R(x, u)) = fˆ(x, u, Du)

in ,

where F (R(x, u)) = log det R(x, u), R(x, u) = ω(x, u) − B(x, u, Du) and fˆ = log f. We have the following lemma. Lemma 4 Under the assumptions of Theorem 3, we have the following statements: (1) F is uniformly elliptic with respect to u(x), that is  1 ij λ|ξ |2 ≤ F + F j i ξi ξj ≤ |ξ |2 , ∀ξ ∈ Rn , 2

(94)

where λ,  are positive constants depending only on n, γ0 , a0 , δ, β1 , β2 , β3 , b0 , f0 , A, f, u, ϕ and , and the estimate (94) is uniform with respect to B ∈ W . (2) F is strong concave with respect to ω on the range of ω(x, u) in the sense that F ij,k Pij Pk ≤ −ε1 |P |2 , ∀P = [Pij ] ∈ Rn×n , P T = P ,

(95)

where ε1 is a positive constant depending only on n, γ0 , a0 , δ, β1 , β2 , β3 , b0 , f0 , A, f, u, ϕ and , and the estimate (95) is uniform with respect to B ∈ W . Proof The estimate (94) easily follows from (17), (24), (90) and (91). The estimate (95) easily follows from [14, Corollary 3] and (90). The proof is complete. Under the assumptions of Theorem 3, we can apply the Theorems 1 and 2 to obtain the estimates (8) and (39). From these two estimates, we obtain |u|2; ≤ C.

(96)

From this, we see that in order to prove (92), it remains to prove the global H¨older estimates for second derivatives of u, that is [D 2 u]α0 ; ≤ C. This estimate is inferred from the interior H¨older estimates and the H¨older estimates at any point x 0 ∈ ∂ for D 2 u. Indeed, by Lemma 4, we can use the arguments which are almost the same as the proof of [5, Theorem 17.26’]. Since the matrix R(x, u) is nonsymmetric,

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

we indicate some differences in our arguments. First, in the proof of the interior H¨older estimates for second derivatives, it suffices to prove the following estimate   F ij (R(y, u)) Dij u(y) − Dij u(x) ≤ D0 |x − y|, (97) for all x, y ∈  such that tx + (1 − t)y ∈  for all t ∈ [0, 1], where D0 is some positive constant under control. Indeed, since u is a δ-elliptic of (1) in , we can apply [14, Theorem 2] to the matrices R (0) ≡ R(y, u(y)) and R (1) ≡ R(x, u(x)) from the set Dδ,μ(B) and obtain F ij (R(y, u))(Dij u(y) − Dij u(x)) ≤ F ij (R(y, u))[Aij (y, u(y), Du(y)) +Bij (y, u(y), Du(y)) − Aij (x, u(x), Du(x)) − Bij (x, u(x), Du(x))] +fˆ(y, u(y), Du(y)) − fˆ(x, u(x), Du(x))   4n2 δ 2 1 2 1+ λ−2 + 0 |B(y, u(y), Du(y)) − B(x, u(x), Du(x))| , 2 1 − δ2 where λ0 is the constant given in (91). Hence, by the mean value theorem, Lemmas 1, 2, (90), (91), and (96), we can easily obtain the desired estimate (97), where D0 depends on n, γ0 , a0 , δ, β1 , β2 , β3 , b0 , f0 , A, f, u, ϕ and . Next, in the proof of H¨older estimates for D 2 u at any boundary point x 0 ∈ ∂, we will use the C 4 coordinate change ψ as in Section 3.1 to flatten ∂ in a neighborhood small enough of x 0 . The rest of the argument is the same as in [5]. The proof of Theorem 3 is complete.

5 Solvability of the Dirichlet Problem 5.1 A Necessary Condition for the Existence of δ-Elliptic Solutions We have the following necessary condition on the skew-symmetric matrix B(x, z, p) for the existence of δ-elliptic solutions to (1). Proposition 2 Let Uδ () be the set of all δ-elliptic solutions of (1) in  with δ > 0. If Uδ () is not empty, then it is necessary that the following inequality holds 

n  1 n [2] sup B(x, z, p) + 2 − 1 det B(x, z, p) n (x,z,p)∈ δ

[ n ] 2 ≤ 1 + δ2 inf inf f (x, u(x), Du(x)). (98) u∈Uδ () x∈

Proof Since u ∈ Uδ (), the matrices R(x, u) and R(x, u, y, z, p) ≡ ω(x, u) − B(y, z, p) are in the set Dδ,μ(B) , for all x, y ∈ , z ∈ R and p ∈ Rn . By [14, Proposition 5], we then have

n  1 n 2 − 1 det B(y, z, p) ≤ det R(x, u, y, z, p) B(y, z, p) + 2 δn

[ n ] [ n ] 2 2 ≤ 1 + δ2 det ω(x, u) ≤ 1 + δ 2 f (x, u(x), Du(x)), for all x, y ∈ , z ∈ R and p ∈ Rn , where the last inequality follows from the estimate det ω(x, u) ≤ det R(x, u) = f (x, u, Du), which is true by [14, Proposition 3] and (1). From this, we can easily obtain (98). The proof is complete.

H. T. Ngoan, T. T. K. Chung

5.2 Sufficient Conditions for the Existence of δ-Elliptic Solutions The necessary condition (98) says that, in order to get the solvability of the Dirichlet problem (1)–(2), we must impose on the skew-symmetric matrices B(x, z, p) some conditions. In this subsection, we will apply the method of continuity to prove the existence and uniqueness of the δ-elliptic solution to problem (1)–(2). The main theorem in this subsection is the following. Theorem 4 Suppose the assumptions (i)–(vi) of Theorem      3 are fulfilled, where A ∈ C 3 ; Rn×n , f ∈ C 3 (; R), ϕ ∈ C 5  , u ∈ C 5  and ∂ ∈ C 5 . Suppose W = W (δ, β1 , β2 , β3 , b0 , λu ) is the set given in Definition 1. Then there exist positive constants λ∗ and α∗ ∈ (0, 1) depending only on n, γ0 , a0 , δ, β1 ,β2 , β3 , b0, f0 , A, f , u, ϕ and  such that if B(x, z, p) is any matrix from the set W ∩ C 3 ; Rn×n and satisfies the following additional conditions for all ξ ∈ Cn \ 0, η ∈ Rn \ 0,



(a)’ sup Bij (x, z, p)ξi ξ j < δλ∗ |ξ |2 if δ > 0, and B ≡ 0 if δ = 0; (x,z,p)∈



(b)’ sup Dz Bij (x, z, p)ξi ξ j < β1 λ∗ |ξ |2 if β1 > 0, and Dz B ≡ 0 if β1 = 0; (x,z,p)∈



 

Dx Bij (x, z, p)ξi ξ j ηk , Dp Bij (x, z, p)ξi ξ j ηk < β2 λ∗ |ξ |2 |η|, (c)’ sup k k (x,z,p)∈   then the Dirichlet problem (1)–(2) has a unique δ-elliptic solution in C 2,α∗  . Proof In this proof, for any w ∈ C 2 (), we denote   F [w] := log det D 2 w − A(x, w, Dw) − B(x, w, Dw) , G[w] := F [w] − log f (x, w, Dw). Suppose B(x, z, p) is any fixed matrix from W . We consider for t ∈ [0, 1] the family of problems:       det D 2 u(t) − A x, u(t) , Du(t) − B x, u(t) , Du(t) = g (t) in , (99) u(x) = ϕ(x) on ∂,     where g (t) := g (t) x, u(t) , Du(t) = f x, u(t) , Du(t) e(1−t)G[u] . Lemma 5 SupposeB(x,  z, p) ∈ W . In addition to the assumptions of Theorem 4, suppose that u(t) ∈ U ∩ C 4  is a δ-elliptic solution of problem (99), where U = U (δ, β1 , β2 , B) is the set given in Definition 2. Then there exist positive constants C∗ , λ∗ ≤ λu , α∗ ∈ (0, 1), C∗∗ , that depend only on n, γ0 , a0 , δ, β1 , β2 , β3 , b0 , f0 , A, f, u, ϕ and  such that u(t) ∈ C 2,α∗ () and we have the following estimates

 (100) ≤ C∗ , max λmax ω x, u(t) x∈



(t)

u

λu(t) ≥ λ∗ ,

2,α∗ ;

where λ∗ = B ∈ W.

− n

(1+δ 2 ) [ 2 ] f0 , C∗n−1

≤ C∗∗ ,

(101) (102)

and these estimates are uniform with respect to t ∈ [0, 1] and

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

Proof It is easy to see that problem (99) has the same form as problem (1)–(2) and satisfies all assumptions of Theorem 3. Hence, we can apply Theorem 3 and obtain the estimates (100)–(102). Note that u is a solution of problem (99) when t = 0, and thus from (101), we have λu ≥ λ∗ . The proof is complete. The solvability of problem (1)–(2) follows from the following lemma. defined in Lemma 5. Suppose Lemma 6 Let λ∗ and α∗ ∈ (0, 1) be the positive constants   B = B(x, z, p) is any matrix from the set W ∩ C 3 ; Rn×n satisfying the additional conditions (a)’, (b)’, and (c)’. We set   V = V (δ, β1 , β2 , α∗ , B) := U (δ, β1 , β2 , B) ∩ C 2,α∗  . Then, the Dirichlet problem (99) is solvable in V for any t ∈ [0, 1].   Proof Suppose u(t) ∈ V is a solution of problem (99), that is u(t) ∈ C 2,α∗  is a δ-elliptic solution of problem (99). From this and the smoothnessconditions on A, B, f, ϕ, u and ∂,  we can use the linear theory to infer that u(t) ∈ C 4,α∗  (see [5, Section 6.4 and Section 17.5]). Hence we can apply Lemma (101) and (102).   5 to obtain  the estimates Set v (t) = u(t) − u and T v (t) , t = G v (t) + u − (1 − t)G[u]. Then problem (99) is equivalent to   T v (t) , t = 0 in , v (t) = 0 on ∂.

Denote    X = v ∈ C 2,α∗  : v = 0 on ∂ ,

' & X˜ = v ∈ X : v + u ∈ V ,

(103)   Y = C 0,α∗  .

Then, X is a Banach space. Note that V is an open set in C 2,α∗ () and u ∈ V , so X˜ is an open set in X. Let us define    I = t ∈ [0, 1] : T v (t) , t = 0 for some v (t) ∈ X˜ ,    E = v (t) ∈ X˜ : T v (t) , t = 0 for some t ∈ [0, 1] . Solving problem (103) is equivalent to finding a function v (t) ∈ E . It is easy to see that v (0) = 0 ∈ E , and thus 0 ∈ I . The proof will be finished if we prove that 1 ∈ I . To see this, we prove that I is open and closed in [0, 1]. Claim 1 I is open.   First, we find the Fr´echet derivative of T with respect to the first variable at v (t) , t given by       Tv (t) : X → Y, Tv (t) h = Grij u(t) Dij h + Gpi u(t) Di h + Gz u(t) h, where u(t) = v (t) +u. Then, Tv (t) is a uniformly elliptic linear operator with C α∗ coefficients, and (see the proof of [14, Theorem 4]),

     D f  z (t) ij (t) (t) (t) x, u(t) , Du(t) ≤ 0. = −F u (Dz Aij +Dz Bij ) x, u , Du − Gz u f By the classical Schauder theory, Tv (t) is an invertible operator. Suppose t0 ∈ I , i.e.,  ˜ By the inverse function theorem, for any t ∈ [0, 1] T v (t0 ) , t0 = 0 for some v (t0 ) ∈ X.

H. T. Ngoan, T. T. K. Chung

  close to t0 , there is a function v (t) ∈ X˜ satisfying T v (t) , t = 0. Hence t ∈ I for all such t, and therefore I is open. Claim 2 I is closed. Let {tk }k∈N ⊂ I be converging to some t ∗ ∈ [0, 1]. Let {v (tk ) }k∈N ⊂ X˜ be such that T [v (tk ) , tk ] = 0, then u(tk ) = v (tk ) + u ∈ V and u(tk ) is the solution of (99). From (102), the sequence {u(tk ) } is uniformly bounded in C 2,α∗ () and therefore, by the Arzel`a-Ascoli theorem, there exists a subsequence {u(tk ) } converging in C 2 () to some function u ∈ C 2,α∗ (). From (101), we have λu ≥ λ∗ > 0. This together with the conditions (a)’, (b)’ and (c)’ gives u ∈ V . Consequently, the sequence {v (tk ) } = {u(tk ) − u} converges in C 2 () to the function v = u − u ∈ X˜ satisfying T [v, t ∗ ] = 0. This means that t ∗ ∈ I and I is closed. We have proved the existence of a δ-elliptic solution for the Dirichlet problem (1)–(2) in V . The uniqueness of the δ-elliptic solution follows directly from the comparison principle ([14, Theorem 4]). The proof of Theorem 4 is complete.

5.3 An Example In this subsection, we give a simple example in which we can verify the assumptions of Theorems 3 and 4. Specifically, we consider the following Dirichlet problem for the nonsymmetric Monge-Amp`ere type equations      1 1  2 2 det D u− |Du| E −Du ⊗ Du −B(x, Du) = n 1+eu 1+|Du|2 in , (104) 2 5 u = 0 on ∂, (105) where  = {x ∈ Rn : |x| < 1}. Here, A(x, z, p) = 12 |p|2 E − p ⊗ p, that appears in   conformal geometry ([16]), B T (x, p) = −B(x, p), f (x, z, p) = 51n (1 + ez ) 1 + |p|2 . In order to show that problem (104)–(105) satisfies the assumptions of Theorem 3 and Theorem 4, we need only show that all conditions (i)–(vi) 3 are satisfied. Indeed,  of Theorem  since 0 ≤ p ⊗ p ≤ |p|2 E, then A ≥ − 12 |p|2 E > − 12 1 + |p|2 E. So, it is easy to see that the matrix A satisfies the conditions (i) and (ii), with γ0 = 12 . Further, by computation, we have for all ξ, η ∈ Rn , ξ ⊥ η, Aij,k ξi ξj ηk η = (δij δk − δik δj − δj k δi )ξi ξj ηk η = |ξ |2 |η|2 − 2(ξ, η)2 = |ξ |2 |η|2 . This means that A is strictly regular in  satisfying (6) with a0 = 1, and therefore condition

 Dz f (x,z,p) 1 (iii) is satisfied. Next, it is easy to see that inff (x, z, p) = 5n > 0 and inf f (x,z,p) = 0. 



This means that the conditions (iv) and (v) are satisfied with f0 = 51n and β1 = 0. It remains   to check condition (vi). We choose u(x) = 12 |x|2 − 1 ; then u = 0 on ∂. Further, since Du(x) = x, D 2 u(x) = E, we have  1 2 1 ω(x, u) = D 2 u − A(x, u, Du) = E − |x| E − x ⊗ x ≥ E, ∀x ∈ . 2 2 This implies that λu = 12 . Thus, we get for n ≥ 2,  n   1 1 4 1 2 det ω(x, u) ≥ > n ≥ n 1 + e 2 (|x| −1) 1 + |x|2 = f (x, u, Du), ∀x ∈ . 2 5 5

Elliptic Solutions to Nonsymmetric Monge-Amp`ere...

Hence, u is an elliptic subsolution of (104) in which B ≡ 0. Suppose δ, β2 , β3 , b0 are any fixed numbers such that 0 < δ < 1, β2 > 0, β3 > 0, 0 ≤ b0 ≤ a0 = 1. Then Theorem 4 states that there exist positive constants λ∗ ≤ λu = 12 and α∗ ∈ (0, 1) depending on n,   δ, β2 , β3 , b0 such that if B(x, p) is any skew-symmetric matrix from W ∩ C 3 ; Rn×n , where W = W (δ, 0, β2 , β3 , b0 , 1/2) and satisfies the additional conditions (a)’ and (c)’, then the Dirichlet problem (104)–(105) has a unique δ-elliptic solution in C 2,α∗  .

References 1. Caffarelli, L.A., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations I. Monge-Amp`ere equation. Comm. Pure Appl. Math. 37(3), 369–402 (1984) 2. Caffarelli, L.A., Nirenberg, L., Spruck, J.: Correction to: the Dirichlet problem for nonlinear second order elliptic equations I. Monge-Amp`ere equation. Comm. Pure Appl. Math. 40(5), 659–662 (1987) 3. Caffarelli, L.A., Cabr´e, X.: Fully Nonlinear Elliptic Equations. Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995) 4. De Philippis, G., Figalli, A.: The Monge-Amp`ere equation and its link to optimal transportation. Bull. Am. Math. Soc. (N.S.) 51(4), 527–580 (2014) 5. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001) 6. Ivochkina, N.M.: A priori estimates of uc2, α () of convex solutions of the Dirichlet problem for the Monge-Amp`ere equation. J. Soviet Math. 21, 689–697 (1983) 7. Ivochkina, N.M.: Classical solvability of the Dirichlet problem for the Monge-Amp`ere equation. J. Soviet Math. 30, 2287–2292 (1985) 8. Jiang, F., Trudinger, N.S., Yang, X.-P.: On the Dirichlet problem for Monge-Amp`ere type equations. Calc. Var. Partial Differ. Equ. 49(3-4), 1223–1236 (2014) 9. Jiang, F., Trudinger, N.S.: On Pogorelov estimates in optimal transportation and geometric optics. Bull. Math. Sci. 4(3), 407–431 (2014) 10. Jiang, F., Trudinger, N.S., Yang, X.-P.: On the Dirichlet problem for a class of augmented Hessian equations. J. Differ. Equ. 258(5), 1548–1576 (2015) 11. Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Math. USSR-Izv 22, 67–98 (1984) 12. Liu, J., Trudinger, N.S.: On Pogorelov estimates for Monge-Amp`ere type equations. Discrete Contin. Dyn. Syst. Ser. A 28(3), 1121–1135 (2010) 13. Ma, X.-N., Trudinger, N.S., Wang, X.-J.: Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 177(2), 151–183 (2005) 14. Ngoan, H.T., Chung, T.T.K.: Elliptic solutions to nonsymmetric Monge-Amp`ere type equations I. The d-concavity and the comparison principle. Acta Math. Vietnam. (2017). https://doi.org/10.1007/s40306017-0231-2 15. Trudinger, N.S.: On the Dirichlet problem for Hessian equations. Acta Math. 175(2), 151–164 (1995) 16. Trudinger, N.S.: Recent developments in elliptic partial differential equations of Monge-Amp`ere type. International Congress of Mathematicians III, 291–301 (2006). Eur. Math. Soc., Z¨urich 17. Trudinger, N.S., Wang, X.-J.: On the second boundary value problem for Monge-Amp`ere type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(1), 143–174 (2009)