AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/671537 671537 Research Article Exterior Dirichlet Problem for Translating Solutions of Gauss Curvature Flow in Minkowski Space Ju Hongjie Zheng Sining School of Science, Beijing University of Posts and Telecommunications, Beijing 100876 China bupt.edu.cn 2014 12 8 2014 2014 20 06 2014 30 07 2014 12 8 2014 2014 Copyright © 2014 Hongjie Ju. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove the existence of solutions to a class of Monge-Ampère equations on exterior domains in n ( n 2 ) and the solutions are close to a cone. This problem comes from the study of the flow by powers of Gauss curvature in Minkowski space.

1. Introduction and Main Results

The Euclidean space R n + 1 endowed with the Lorentz metric d s 2 = i = 1 n d x i 2 - d x n + 1 2 is called Minkowski space. We denote it by R n , 1 . A space-like hypersurface in R n , 1 is a Riemanian n -manifold, having an everywhere lightlike normal field ν which we assume to be future directed and thus satisfy the condition ν , ν = - 1 . Locally, such surfaces can be expressed as graphs of functions x n + 1 = u ( x 1 , , x n ) : R n R satisfying the space-like condition | D u ( x ) | < 1 for all x R n .

If a family of space-like hypersurfaces X t = X ( · , t ) : R n R n , 1 satisfies the evolution equation (1) X t = - K α ν on some time interval, we say that the surfaces M t = X t ( M ) are evolved by K α -flow, where K ( · , t ) is the Gauss curvature of M t and α 0 is a constant. When the initial surface is a graph over a domain Ω R n , (1) is equivalent, up to a diffeomorphism in R n , to (2) V t = 1 - | D V | 2 [ det ( D 2 V ) ( 1 - | D V | 2 ) ( n + 2 ) / 2 ] α with | D V ( · , t ) | < 1 , where V is a function defined in Ω × [ 0 , T ) .

The flow (2) was studied in  for the special case α = 1 . In fact, the authors in  used the flow (2) to prove existence and stability of smooth entire strictly convex space-like hypersurfaces of prescribed Gauss curvature and give a new proof of Theorem 3.5 in .

A function u = u ( x ) is called a translating solution to the K α -flow if the function V ( x , t ) = u ( x ) + t solves (2). Equivalently, u ( x ) is an initial hypersurface satisfying (3) det ( D 2 u ) = ( 1 - | D u | 2 ) ( ( n + 2 - ( 1 / α ) ) / 2 ) . The space-like condition reads as (4) | D u ( x ) | < 1 .

The space-like hypersurfaces evolved by mean curvature flow in Minkowski space were studied in . The translating solutions were introduced in [3, 4] and studied in [7, 8].

In this paper, we consider strictly convex space-like hypersurfaces of translating solutions to K α -flow as graphs over R n D , where D R n is an open domain whose boundary D is a smooth submanifold of R n . We want to look for a function u C ( R n D ) , which solves the problem (3)-(4) with the boundary condition (5) u = ϕ on D , where ϕ C ( D ) is a given function.

There are similar problems for the equation of translating solution of Gauss curvature flow in Eucliden space , the equation of prescribed Gauss curvature in Eucliden space , and the equation of prescribed Gauss curvature in Minkowski space , respectively. It was shown that there are convex solutions to the Dirichlet problems for the three equations on exterior domains, and the solution is close to the rotationally symmetric one at infinity for the first equation and close to a cone for the second and third equation under the assumption that there exists a strictly convex subsolution which is close to a cone up to the third order (see (7) and (8)).

In this paper, we will show that the same results as in [10, 11] hold for the problem (3)–(5). We would like to point out that (3) is essentially different from the equations in . For example, the equation of prescribed Gauss curvature in Minkowski space, det ( D 2 u ) = ( 1 - | D u | 2 ) ( n + 2 ) / 2 , has an explicit solution u = 1 + | x | 2 , from which one can easily construct subsolution or supersolution for given Dirichlet problems. However, it is unknown if there is such a solution to (3). In particular, it has no solution in the form of u = ( 1 + | x | 2 ) γ .

Definition 1.

A function u _ C ( R n D ) is called a subsolution of (3)–(5), if u _ is strictly convex and satisfies (6) det ( D 2 u _ ) ( 1 - | D u _ | 2 ) ( n + 2 - β ) / 2 , in R n D ¯ , | D u _ | < 1 , in R n D ¯ , u _ = ϕ , on D . Here and below, we set β = 1 / α .

The main result of this paper is the following theorem.

Theorem 2.

Let D R n    ( n 2 ) be an open set whose boundary D is a smooth submanifold of R n and ϕ C ( D ) . Suppose that β < ( 3 / 2 ) - 2 n and u _ C ( R n D ) is a subsolution of (3)–(5) which is close to a cone, that is, (7) sup R n D | u _ - | x | | < and satisfies the following decay conditions at infinity: (8) | D ( u _ - | x | ) | = O ( 1 | x | ) , | D 2 ( u _ - | x | ) | + | D 3 u _ | = O ( 1 | x | 2 ) . Then there exists a smooth, strictly convex hypersurface of the exterior Dirichlet problem (3)–(5) and the solution u is close to a cone in the sense that (9) sup R n D | u - | x | | < .

Although the above theorem has an obvious disadvantage that it assumes the existence of a locally strictly convex subsolution, this assumption is reasonable and necessary in some case for the Dirichlet problems on nonconvex domains; see  for the details. However, in the special case when D = B ρ 0 ( 0 ) is a ball and the boundary values are zero, we can construct an explicit subsolution.

Theorem 3.

Let D = B ρ 0 ( 0 ) with ρ 0 > 0 and ϕ 0 . If β 0 , then there is a strictly convex subsolution u _ of (3)–(5) such that (7) and (8) are satisfied.

We consider the local problem (10) det ( D 2 u R ) = ( 1 - | D u R | 2 ) ( n + 2 - β ) / 2 , in B R D ¯ , sup B R D ¯ | D u R | < 1 , u R = u _ , on D B R , where R > 4 R 0 and D B R 0 for some constant R 0 > 1 . It is well known from the standard continuity method as in  that the Dirichlet problem (10) has a locally strict convex solution in C ( B ¯ R D ) . Our main task is to show that the C 2 -norms of u R are uniformly bounded in R . Once this is established, by the standard Krylov/Shafanov theory, Schauder regularity theory, and a diagonal sequence argument, we can obtain a smooth locally strictly convex solution u to (3)–(5) on exterior domain R n D .

The paper is organized as follows. In Section 2, we prove the C 0 and C 1 a priori estimates for u R . The C 2 -estimates are given in Section 3. Finally, we prove Theorem 3 in the last section.

2. <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M78"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi></mml:mrow> <mml:mrow> <mml:mn>0</mml:mn></mml:mrow> </mml:msup></mml:mrow> </mml:math></inline-formula> and <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M79"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi></mml:mrow> <mml:mrow> <mml:mn>1</mml:mn></mml:mrow> </mml:msup></mml:mrow> </mml:math></inline-formula> A Priori Estimates

From now on, we assume D and u _ as in Theorem 2 and u R as in (10); lower indices denote partial derivatives in R n , for example, u i = u / x i . The inverse of the Hessian of u is denoted by ( u i j ) = ( u i j ) - 1 . We use the Einstein summation convention. The letter c denotes a constant independent of R which may change its value from line to line throughout the text.

Without loss of generality we can assume that 0 D . It is easy to check that u ¯ = 1 + | x | 2 + L is a supersolution to (3) for α < 0 , where the constant L > max D ( ϕ - 1 + | x | 2 ) .

Owing to the maximum principle, we can obtain the following lemma as Lemma 2.2 in .

Lemma 4.

The functions u R converge locally uniformly to a continuous function u as R . Moreover, u _ u u ¯ in B R D .

Proof.

From the maximum principle we obtain that (11) u _ u R u ¯ in B R D for any R > 4 R 0 and (12) u R 1 = u _ u R 2 on B R 1 D for 4 R 0 < R 1 < R 2 . Again by the maximum principle, we have (13) u R 1 u R 2 in B R 1 D . We conclude that u R are monotone in R and converge locally uniformly to a continuous function u according to Dini’s theorem.

To simplify the notation, we will omit the index R and from now on assume that u is a solution of (10) with R fixed sufficiently large. The estimate for the first derivatives is stated in the following lemma.

Lemma 5.

For R / 2 | x | R , there is a constant c independent of R such that (14) | ν ( u - u _ ) ( x ) | c R , (15) | τ u ( x ) | c R , (16) 1 - c R | D u ( x ) | < 1 , where ν = x / | x | and τ are unit vectors parallel and orthogonal to x , respectively.

Proof.

From the convexity of u and Lemma 4, we can prove (14) and (15) by using the similar proof techniques of (2.2) and (2.3) in . Then, we need only to prove (16). Since u is strictly convex, for R / 2 | x | R , | D u | attains its maximum at B R . In view of (8), we may take (17) | D u _ | 2 = O ( 1 - c R ) . Hence for x B R , by (14) and (17) we have (18) | D u ( x ) | 2 = | τ u ( x ) | 2 + | ν u ( x ) | 2 = | τ u _ ( x ) | 2 + | ν u _ ( x ) + ν ( u - u _ ) ( x ) | 2 | D u _ ( x ) | 2 - 2 | ν ( u - u _ ) ( x ) | 1 - c R . On the other hand, by the proof of Theorem 4.1 in , (19) max B R | D u | max B R | D u _ | < 1 . The lemma is completed.

3. <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M125"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi></mml:mrow> <mml:mrow> <mml:mn>2</mml:mn></mml:mrow> </mml:msup></mml:mrow> </mml:math></inline-formula> A Priori Estimates

In this section, we prove the C 2 a priori estimates for solutions of (10) under the assumption of Theorem 2. As in , one obtains that the second derivatives on D are bounded uniformly in R . Furthermore, by considering the function (20) w = a 2 | D u | 2 + log u ξ ξ for some constant a > 0 and assuming its maximum over ( x , ξ ) B R D ¯ × S n - 1 is attained at an interior, one can prove that (21) max B R D ¯ | D u | 2 c + max B R D | D 2 u | . Therefore, it suffices to bound | D 2 u | on the outer boundary B R .

Next, we will give estimates for the tangential second derivatives, the mixed second derivatives, and the normal second derivatives on the outer boundary B R , respectively.

Theorem 6 (tangential second derivatives at the outer boundary).

Let x 0 B R and τ 1 , τ 2 be tangential directions at x 0 . Then we have at x 0 , (22) | u τ 1 τ 2 - | x | τ 1 τ 2 | c R 2 .

Proof.

We may assume that x 0 = R · e n R · ( 0 , , 0,1 ) . Then B R is represented locally as graph of ω , where (23) ω ( x ^ ) = R 2 - | x ^ | 2 , x ^ = ( x 1 , x 2 , , x n - 1 ) R n - 1 . Note that the Dirichlet boundary condition implies (24) ( u - u _ ) ( x ^ , ω ( x ^ ) ) = 0 . We differentiate twice with respect to x ^ i , x ^ j , 1 i , j n - 1 to obtain that, at x 0 , (25) ( u - u _ ) i j + ( u - u _ ) n ω i j + 2 ( u - u _ ) n j ω i = 0 . According to the decay conditions at infinity (8), we have | u _ i j - | x | i j | = O ( 1 / R 2 ) . Observing that (26) w i ( x 0 ) = 0 , ω i j ( x 0 ) = - δ i j R . Then, by Lemma 5 we have (27) | u i j - | x | i j | = | - ( u - u _ ) n ω i j + ( u _ - | x | ) i j | c R 2 .

Theorem 7 (mixed second derivatives at the outer boundary).

For x 0 B R , let τ , ν be unit vectors in tangential and normal directions, respectively. Then (28) | u τ ν ( x 0 ) | c R .

The proof is going to be put in three lemmas and will be finished below Lemma 10. Similar to Theorem 6, we may assume that x 0 = R · e n and represent B R locally as graph of ω with ω ( x ^ ) = R 2 - | x ^ | 2 . We take the logarithm of (3), (29) log det u i j - n + 2 - β 2 log ( 1 - | D u | 2 ) = 0 , and differentiate with respect to x k , (30) u i j u i j k + n + 2 - β 1 - | D u | 2 u i u i k = 0 , where ( u i j ) = ( u i j ) - 1 . We introduce the linear differential operator L by (31) L w = u i j w i j + n + 2 - β 1 - | D u | 2 u i w i and define the linear operator for t < n : (32) T = x t + γ = 1 n - 1 ω  tr ( 0 ) x γ x n x t - x t R x n . In the following we restrict attention to the domain Ω δ = B δ ( x 0 ) B R with x 0 B R and δ R / 2 . Notice that Ω δ B R D ¯ .

Lemma 8.

The function u - u _ satisfies the following estimates: (33) | T ( u - u _ ) | c R in Ω δ , (34) | T ( u - u _ ) | c R 2 | x - x 0 | 2 on B R , (35) | L T ( u - u _ ) | c R + c R 2 tr  ( u i j ) in Ω δ , where tr  ( u i j ) Σ i = 1 n u i i .

Proof.

For (33), by the assumption (8) and C 1 estimates of Lemma 5, we get (36) | T ( u - u _ ) | | ( u - u _ ) t | + | ( u - u _ ) n | 2 | D ( u - u _ ) | 2 | ( u - u _ ) ν | + 2 | ( u - u _ ) τ | 2 | ( u - | x | ) τ | + 2 | ( u _ - | x | ) τ | + c R 2 | u τ | + c R c R , where ν = x / | x | and τ is unit vector orthogonal to x .

For the second inequality (34) we use that ( u - u _ ) t + ( u - u _ ) n ω t = 0 and note that ω t ( 0 ) = 0 ,   | ω i | c , | ω i j | c / R , | ω i j k | c / R 2 . Then for x B R , (37) T ( u - u _ ) = - ( ω t - γ = 1 n - 1 ω t γ ( 0 ) x γ ) ( u - u _ ) n = - γ , s = 1 n - 1 x γ ω t γ s ( θ x ^ ) x s ( u - u _ ) n with 0 < θ < 1 , which implies (34).

To prove (35), by Lemma 5, we may take 1 / ( 1 - | D u | 2 ) = O ( R ) . In view of (8), (30), and u i j u j k = δ i k , we obtain (38) | L T ( u - u _ ) | = | L ( ( u - u _ ) t - x t R ( u - u _ ) n ) | = | u i j [ ( u - u _ ) t i j - ( x t R ( u - u _ ) n ) i j ] k + n + 2 - β 1 - | D u | 2 u i [ ( u - u _ ) t i - ( x t R ( u - u _ ) n ) i ] | | u i j u t i j + n + 2 - β 1 - | D u | 2 u i u t i | + | x t R ( u i j u n i j + n + 2 - β 1 - | D u | 2 u i u n i ) | + | u i j ( u _ t i j - x t R u _ n i j ) | + n + 2 - β 1 - | D u | 2 | u i ( u _ t i + x t R u _ n i ) | + | u i j ( δ i t R u n j + δ j t R u n i ) | + | u i j ( δ i t R u _ n j + δ j t R u _ n i ) | + n + 2 - β 1 - | D u | 2 | u i · δ i t R ( u - u _ ) n | c · tr ( u i j ) · ( | D 3 u _ | + 1 R | D 2 u _ | ) + c · n + 2 - β 1 - | D 2 u _ | 2 [ | D 2 u _ | + 1 R 2 ] + c c R 2 tr ( u i j ) + c .

In the next lemma, we introduce a function V , which will be the main part of a barrier function to prove Theorem 7.

Lemma 9.

There exists a positive constant ɛ independent of R such that (39) V = ( u - u _ ) + 1 R d - 1 2 R 5 / 4 d 2 fulfills the estimates (40) L V - ɛ R - ( 1 / n ) ( β - ( 7 / 4 ) ) - ɛ R - ( 5 / 4 ) tr  ( u i j ) , in Ω δ , V 0 , on Ω δ provided that δ = R 3 / 4 and R is sufficiently large. Here d = R - | x | is the distance from B R .

Proof.

In view of δ = R 3 / 4 , for x Ω δ , d = R - | x | δ = R 3 / 4 , and u u _ , we have (41) V = ( u - u _ ) + 1 R d - 1 2 R 5 / 4 d 2 1 R d - 1 2 R 5 / 4 d 2 0 on Ω δ . We fix x Ω δ and set ν = x / | x | . Let τ , τ belong to an orthonormal basis for the orthogonal complement of ν which we choose such that the submatrix ( u τ τ ) is diagonal. Assume that ν and τ , τ correspond to the indices n and 1 , , n - 1 , respectively. We use the Einstein summation convention for τ , τ . The matrix u i j is positive, and thus testing with the vectors ν ± τ gives (42) | u ν τ | 1 2 ( u ν ν + u τ τ ) . In view of (43) u ν ν = u i j x i | x | x j | x | , tr ( u τ τ ) = tr ( u i j ) - u ν ν = u i j ( δ i j - x i | x | x j | x | ) . Direct computations using (17) give (44) L u = u i j u i j + ( n + 2 - β ) | D u | 2 1 - | D u | 2 ( n + 2 - β ) | D u | 2 1 - | D u | 2 + c . By (8), (16), and (42) we have (45) L u _ = u i j u _ i j + n + 2 - β 1 - | D u | 2 u i u _ i = n + 2 - β 1 - | D u | 2 u i u _ i + u i j [ | x | i j + ( u _ - | x | ) i j ] = n + 2 - β 1 - | D u | 2 u i u _ i + 1 | x | u i j ( δ i j - x i | x | x j | x | ) + u i j ( u _ - | x | ) i j = n + 2 - β 1 - | D u | 2 u i u _ i + 1 | x | tr ( u τ τ ) + u τ τ ( u _ - | x | ) τ τ + 2 u τ ν ( u _ - | x | ) τ ν + u ν ν ( u _ - | x | ) ν ν n + 2 - β 1 - | D u | 2 u i u _ i + ( 1 | x | - c | x | 2 ) tr ( u τ τ ) - c | x | 2 u ν ν , L d = u i j ( R - | x | ) i j + n + 2 - β 1 - | D u | 2 u i ( R - | x | ) i = - 1 | x | tr ( u τ τ ) - n + 2 - β 1 - | D u | 2 u i x i | x | , L d 2 = u i j ( ( R - | x | ) 2 ) i j + n + 2 - β 1 - | D u | 2 u i ( ( R - | x | ) 2 ) i - 2 R | x | tr ( u τ τ ) + 2 tr ( u i j ) - n + 2 - β 1 - | D u | 2 u i ( 2 R x i | x | - 2 x i ) = - 2 d | x | tr ( u τ τ ) + 2 u ν ν - n + 2 - β 1 - | D u | 2 u i ( 2 R x i | x | - 2 x i ) . Then, (46) L V = L ( u - u _ ) + 1 R L d - 1 2 R 5 / 4 L d 2 c + n + 2 - β 1 - | D u | 2 u i ( u i - u _ i ) + n + 2 - β 1 - | D u | 2 u i x i R 5 / 4 | x | ( d - R 3 / 4 ) - [ 1 | x | - c | x | 2 + 1 | x | ( 1 R - d R 5 / 4 ) ] tr ( u τ τ ) - ( 1 R 5 / 4 - c | x | 2 ) u ν ν . Thus, for R large enough, we have (47) L V c R 1 / 2 - 1 2 R tr ( u τ τ ) - 1 2 R 5 / 4 u ν ν . Expanding the determinant and using that ( u τ τ ) is a diagonal matrix give (48) det ( u i j ) = det ( u 11 0 0 u 1 n 0 0 0 0 u n - 1 n - 1 u n - 1 n u 1 n u n - 1 n u n n ) = i u i i - τ | u n τ | 2 τ τ u τ τ i u i i . By the inequality for arithmetic and geometric means, (49) 1 R tr ( u τ τ ) + 1 R 5 / 4 u ν ν n [ ( 1 R ) n - 1 · 1 R 5 / 4 i u i i ] 1 / n n [ ( det ( u i j ) ) ] - ( 1 / n ) R - ( 1 / n ) ( n + ( 1 / 4 ) ) . Hence for large R , (50) L V c R 1 / 2 - 1 4 R tr ( u τ τ ) - 1 4 R 5 / 4 u ν ν - 1 4 R tr ( u τ τ ) - 1 4 R 5 / 4 u ν ν c R 1 / 2 - c [ det ( u i j ) ] - ( 1 / n ) R - ( 1 / n ) ( n + ( 1 / 4 ) ) - 1 4 R 5 / 4 tr ( u i j ) = c R 1 / 2 - c [ ( 1 - | D u | 2 ) ( n + 2 - β ) / 2 ] - ( 1 / n ) R - ( 1 / n ) ( n + ( 1 / 4 ) ) - 1 4 R 5 / 4 tr ( u i j ) c R 1 / 2 - c R ( 3 - 2 β - 2 n ) / 4 n - 1 4 R 5 / 4 tr ( u i j ) - c R ( 3 - 2 β - 2 n ) / 4 n - 1 4 R 5 / 4 tr ( u i j ) . Note that we have used the fact ( 3 - 2 β - 2 n ) / 4 n > 1 / 2 in the last inequality, which is from the assumption β < ( 3 / 2 ) - 2 n .

Lemma 10.

There exists a positive constant A independent of R such that (51) Θ = V + A · 1 R 2 · | x - x 0 | 2 ± T ( u - u _ ) satisfies (52) L Θ 0 , in Ω δ , Θ 0 , on Ω δ , where δ = R 3 / 4 and V is as in Lemma 9.

Proof.

According to Lemma 9, the fact Θ 0 on Ω δ follows from (53) A · 1 R 2 · | x - x 0 | 2 ± T ( u - u _ ) 0 on Ω δ , which can be attained by choosing A sufficiently large. The property L Θ 0 now follows from the inequality (54) - ɛ R ( 3 - 2 β - 2 n ) / 4 n - ɛ R - 5 / 4 tr ( u i j ) + c A R - ( 1 / 4 ) + c + c · 1 + A R 2 tr ( u i j ) 0 , which holds for R large enough.

Proof of Theorem <xref ref-type="statement" rid="thm3.2">7</xref>.

The maximum principle applied to (52) yields that Θ 0 in Ω δ . Since Θ ( x 0 ) = 0 , it follows that (55) Θ ν ( x 0 ) 0 with ν = - x 0 / | x 0 | . Thus we get (56) V ν ( x 0 ) | ( T ( u - u _ ) ) ν | ( x 0 ) . That is, (57) [ - ( u - u _ ) n - 1 R ( R - | x | ) n + 1 R 5 / 4 ( R - | x | ) n 2 ] ( x 0 ) | ( u - u _ ) t n + x t R ( u - u _ ) n n | ( x 0 ) = | ( u - u _ ) t n | ( x 0 ) , which, together with (8) and (14), implies (58) | u t n ( x 0 ) | | u _ t n ( x 0 ) | + | ( u - u _ ) n | + 1 R c R . That is, (28) holds.

Theorem 11 (double normal <inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M252"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi></mml:mrow> <mml:mrow> <mml:mn mathvariant="normal">2</mml:mn></mml:mrow> </mml:msup></mml:mrow> </mml:math></inline-formula>-estimates at the outer boundary).

Under the assumption of Theorem 2 and the notation of Theorem 7, we have (59) | u ν ν ( x 0 ) | c .

Proof.

As the proof of Lemma 9, we fix x 0 B R and set ν = x 0 / | x 0 | . Let τ , τ belong to an orthonormal basis for the orthogonal complement of ν which we choose such that the submatrix ( u τ τ ) is diagonal. Assume that ν and τ , τ correspond to the indices n and 1 , , n - 1 , respectively. We expand the determinant, (60) ( 1 - | D u | 2 ) ( n + 2 - β ) / 2 = det ( u i j ) = u n n · i < n u i i - k < n u k n 2 · k i < n u i i = u n n · i < n u i i - i < n u i i k < n u k n 2 1 u k k . Then, for β < ( 3 / 2 ) - 2 n , we have (61) u n n = ( 1 - | D u | 2 ) ( n + 2 - β ) / 2 i < n u i i + k < n u k n 2 u k k c R - ( ( n + 2 - β ) / 2 ) ( c / R ) n - 1 + k < n ( c / R ) 2 c / R c .

Proof of Theorem <xref ref-type="statement" rid="thm1.1">2</xref>.

It follows from Theorems 6, 7, and 11 that u R C 2 are uniformly bounded in R . By the standard Krylov/Shafanov theory, Schauder regularity theory, and a diagonal sequence argument, we obtain a smooth locally strictly convex solution u to (3)–(5) on exterior domain R n D .

4. Proof of Theorem <xref ref-type="statement" rid="thm1.2">3</xref>

In this section, we prove Theorem 3, which gives a simple example of a barrier construction.

Proof of Theorem <xref ref-type="statement" rid="thm1.2">3</xref>.

We introduce functions (62) φ ( τ ) = a ρ 0 2 τ - 3 , η ( r ) = - ρ 0 r ( ρ φ ( τ ) d τ ) d ρ , where 0 < a < 1 will be determined. We define u _ by (63) u _ : R n B ρ 0 R x | x | - ρ 0 + η ( | x | ) . Then, for r ρ 0 , (64) 0 < - η ( r ) = r φ ( τ ) d τ = a 2 ρ 0 2 r - 2 a 2 < 1 . Obviously, u _ = 0 on B ρ 0 . Moreover, (65) sup | u _ - r | ρ 0 + ρ 0 ( ρ φ ( τ ) d τ ) d ρ , | D ( u _ - r ) | = | η ( r ) | = O ( 1 r ) , | D 2 ( u _ - r ) | + | D 3 u _ | = O ( 1 r 2 ) , where r = | x | . Therefore, u _ is close to a cone in the sense of (7) and satisfies the regularity conditions (8) and (17).

We compute the Gauss curvature of graph u _ as follows: (66) u _ i = ( 1 + η ) x i r , | D u _ | = 1 + η , u _ i j = ( 1 + η ) 1 r ( δ i j - x i x j r 2 ) + η ′′ ( r ) x i x j r 2 det D 2 u _ = η ′′ ( r ) r n - 1 ( 1 + η ) n - 1 = φ r n - 1 ( 1 + η ) n - 1 . Take 0 < a < min { 1 , 1 / ( 2 ρ 0 ) 2 } . Using the assumption β < 0 and the fact that (67) | D u _ | 2 = ( 1 + η ) 2 = 1 - a ρ 0 2 r - 2 + a 2 4 ρ 0 4 r - 4 1 - a ρ 0 2 r - 2 , we conclude that (68) det D 2 u _ ( 1 - | D u _ | 2 ) ( n + 2 - β ) / 2 = φ ( r ) ( 1 + η ) n - 1 · 1 ( 1 - ( 1 + η ) 2 ) ( n + 2 - β ) / 2 · 1 r n - 1 a ρ 0 2 r - 3 2 1 - n ( a ρ 0 2 r - 2 ) - ( n + 2 - β ) / 2 · r 1 - n = 2 1 - n a - ( n + β ) / 2 ρ 0 β - n r - β ( 2 a ρ 0 ) - n 1 . Thus, (69) det D 2 u _ ( 1 - | D u _ | 2 ) ( n + 2 - β ) / 2 , | x | > ρ 0 , u _ = 0 , | x | = ρ 0 . The theorem is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (11301034, 11201011, and 11391240188) and the Fundamental Research Funds for the Central Universities (2013RC0901).

Bayard P. Schnürer O. C. Entire spacelike hypersurfaces of constant Gauß curvature in Minkowski space Journal für die reine und angewandte Mathematik 2009 627 1 29 10.1515/CRELLE.2009.009 MR2494911 2-s2.0-58449122844 Guan B. Jian H. Schoen R. M. Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space Journal für die Reine und Angewandte Mathematik 2006 595 167 188 10.1515/CRELLE.2006.047 MR2244801 2-s2.0-33745461829 Aarons M. A. S. Mean curvature flow with a forcing term in Minkowski space Calculus of Variations and Partial Differential Equations 2006 25 2 205 246 10.1007/s00526-005-0351-8 MR2188747 2-s2.0-33244485047 Ecker K. Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space Journal of Differential Geometry 1997 46 3 481 498 MR1484889 ZBL0909.53045 2-s2.0-0031184693 Ecker K. Huisken G. Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes Communications in Mathematical Physics 1991 135 3 595 613 10.1007/BF02104123 MR1091580 ZBL0721.53055 2-s2.0-0001556478 Liu Y. Jian H. Evolution of spacelike hypersurfaces by mean curvature minus external force field in Minkowski space Advanced Nonlinear Studies 2009 9 3 513 522 MR2536952 ZBL1180.53025 2-s2.0-77953362777 Jian H. Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space Journal of Differential Equations 2006 220 1 147 162 10.1016/j.jde.2005.08.005 MR2182083 ZBL1189.53065 2-s2.0-28344448247 Ju H. Lu J. Jian H. Translating solutions to mean curvature flow with a forcing term in Minkowski space Communications on Pure and Applied Analysis 2010 9 4 963 973 10.3934/cpaa.2010.9.963 MR2610255 2-s2.0-77957907352 Ju H. Bao J. Jian H. Existence for translating solutions of Gauss curvature flow on exterior domains Nonlinear Analysis: Theory, Methods & Applications 2012 75 8 3629 3640 10.1016/j.na.2012.01.020 MR2901343 2-s2.0-84857911898 Finster F. Schnürer O. C. Hypersurfaces of prescribed Gauss curvature in exterior domains Calculus of Variations and Partial Differential Equations 2002 15 1 67 80 10.1007/s005260100118 MR1920715 Huang Y. Jian H. Su N. Spacelike hypersurfaces of prescribed Gauss-Kronecker curvature in exterior domains Acta Mathematica Sinica 2009 25 3 491 502 10.1007/s10114-008-6010-1 MR2495530 2-s2.0-63849210732 Guan B. The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature Transactions of the American Mathematical Society 1998 350 12 4955 4971 10.1090/S0002-9947-98-02079-0 MR1451602 2-s2.0-22444453647 Gilbarg D. Trudinger N. S. Second Order Elliptic Partial Differential Equations 1983 2nd Berlin, Germany Springer 10.1007/978-3-642-61798-0 MR737190