Exterior Dirichlet Problem for Translating Solutions of Gauss Curvature Flow in Minkowski Space
We prove the existence of solutions to a class of Monge-Ampère equations on exterior domains in 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 endowed with the Lorentz metric is called Minkowski space. We denote it by . A space-like hypersurface in is a Riemanian -manifold, having an everywhere lightlike normal field which we assume to be future directed and thus satisfy the condition . Locally, such surfaces can be expressed as graphs of functions satisfying the space-like condition for all .
If a family of space-like hypersurfaces satisfies the evolution equation on some time interval, we say that the surfaces are evolved by -flow, where is the Gauss curvature of and is a constant. When the initial surface is a graph over a domain , (1) is equivalent, up to a diffeomorphism in , to with , where is a function defined in .
The flow (2) was studied in  for the special case . 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 is called a translating solution to the -flow if the function solves (2). Equivalently, is an initial hypersurface satisfying The space-like condition reads as
The space-like hypersurfaces evolved by mean curvature flow in Minkowski space were studied in [3–6]. 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 -flow as graphs over , where is an open domain whose boundary is a smooth submanifold of . We want to look for a function , which solves the problem (3)-(4) with the boundary condition where 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 [9–11]. For example, the equation of prescribed Gauss curvature in Minkowski space, , has an explicit solution , 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 .
Definition 1. A function is called a subsolution of (3)–(5), if is strictly convex and satisfies Here and below, we set .
The main result of this paper is the following theorem.
Theorem 2. Let be an open set whose boundary is a smooth submanifold of and . Suppose that and is a subsolution of (3)–(5) which is close to a cone, that is, and satisfies the following decay conditions at infinity: Then there exists a smooth, strictly convex hypersurface of the exterior Dirichlet problem (3)–(5) and the solution is close to a cone in the sense that
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 is a ball and the boundary values are zero, we can construct an explicit subsolution.
Theorem 3. Let with and . If , then there is a strictly convex subsolution of (3)–(5) such that (7) and (8) are satisfied.
We consider the local problem where and for some constant . It is well known from the standard continuity method as in  that the Dirichlet problem (10) has a locally strict convex solution in . Our main task is to show that the -norms of are uniformly bounded in . 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 to (3)–(5) on exterior domain .
The paper is organized as follows. In Section 2, we prove the and a priori estimates for . The -estimates are given in Section 3. Finally, we prove Theorem 3 in the last section.
2. and A Priori Estimates
From now on, we assume and as in Theorem 2 and as in (10); lower indices denote partial derivatives in , for example, . The inverse of the Hessian of is denoted by . We use the Einstein summation convention. The letter denotes a constant independent of which may change its value from line to line throughout the text.
Without loss of generality we can assume that . It is easy to check that is a supersolution to (3) for , where the constant .
Owing to the maximum principle, we can obtain the following lemma as Lemma 2.2 in .
Lemma 4. The functions converge locally uniformly to a continuous function as . Moreover, in .
Proof. From the maximum principle we obtain that for any and for . Again by the maximum principle, we have We conclude that are monotone in and converge locally uniformly to a continuous function according to Dini’s theorem.
To simplify the notation, we will omit the index and from now on assume that is a solution of (10) with fixed sufficiently large. The estimate for the first derivatives is stated in the following lemma.
Lemma 5. For , there is a constant independent of such that where and are unit vectors parallel and orthogonal to , respectively.
Proof. From the convexity of 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 is strictly convex, for , attains its maximum at . In view of (8), we may take Hence for , by (14) and (17) we have On the other hand, by the proof of Theorem 4.1 in , The lemma is completed.
3. A Priori Estimates
In this section, we prove the a priori estimates for solutions of (10) under the assumption of Theorem 2. As in , one obtains that the second derivatives on are bounded uniformly in . Furthermore, by considering the function for some constant and assuming its maximum over is attained at an interior, one can prove that Therefore, it suffices to bound on the outer boundary .
Next, we will give estimates for the tangential second derivatives, the mixed second derivatives, and the normal second derivatives on the outer boundary , respectively.
Theorem 6 (tangential second derivatives at the outer boundary). Let and , be tangential directions at . Then we have at ,
Proof. We may assume that . Then is represented locally as graph of , where Note that the Dirichlet boundary condition implies We differentiate twice with respect to to obtain that, at , According to the decay conditions at infinity (8), we have . Observing that Then, by Lemma 5 we have
Theorem 7 (mixed second derivatives at the outer boundary). For , let be unit vectors in tangential and normal directions, respectively. Then
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 and represent locally as graph of with . We take the logarithm of (3), and differentiate with respect to , where . We introduce the linear differential operator by and define the linear operator for : In the following we restrict attention to the domain with and . Notice that .
Lemma 8. The function satisfies the following estimates: where .
Proof. For (33), by the assumption (8) and estimates of Lemma 5, we get
where and is unit vector orthogonal to .
For the second inequality (34) we use that and note that , ,,. Then for , with , which implies (34).
To prove (35), by Lemma 5, we may take . In view of (8), (30), and , we obtain
In the next lemma, we introduce a function , which will be the main part of a barrier function to prove Theorem 7.
Lemma 9. There exists a positive constant independent of such that fulfills the estimates provided that and is sufficiently large. Here is the distance from .
Proof. In view of , for , , and , we have We fix and set . Let belong to an orthonormal basis for the orthogonal complement of which we choose such that the submatrix is diagonal. Assume that and correspond to the indices and , respectively. We use the Einstein summation convention for . The matrix is positive, and thus testing with the vectors gives In view of Direct computations using (17) give By (8), (16), and (42) we have Then, Thus, for large enough, we have Expanding the determinant and using that is a diagonal matrix give By the inequality for arithmetic and geometric means, Hence for large , Note that we have used the fact in the last inequality, which is from the assumption .
Lemma 10. There exists a positive constant independent of such that satisfies where and is as in Lemma 9.
Proof. According to Lemma 9, the fact on follows from which can be attained by choosing sufficiently large. The property now follows from the inequality which holds for large enough.
Proof of Theorem 7. The maximum principle applied to (52) yields that in . Since , it follows that with . Thus we get That is, which, together with (8) and (14), implies That is, (28) holds.
Theorem 11 (double normal -estimates at the outer boundary). Under the assumption of Theorem 2 and the notation of Theorem 7, we have
Proof. As the proof of Lemma 9, we fix and set . Let belong to an orthonormal basis for the orthogonal complement of which we choose such that the submatrix is diagonal. Assume that and correspond to the indices and , respectively. We expand the determinant, Then, for , we have
Proof of Theorem 2. It follows from Theorems 6, 7, and 11 that are uniformly bounded in . By the standard Krylov/Shafanov theory, Schauder regularity theory, and a diagonal sequence argument, we obtain a smooth locally strictly convex solution to (3)–(5) on exterior domain .
4. Proof of Theorem 3
In this section, we prove Theorem 3, which gives a simple example of a barrier construction.
Proof of Theorem 3. We introduce functions
where will be determined. We define by
Then, for ,
Obviously, on . Moreover,
where . Therefore, 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 as follows: Take . Using the assumption and the fact that we conclude that Thus, The theorem is completed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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).
P. Bayard and O. C. Schnürer, “Entire spacelike hypersurfaces of constant Gauß curvature in Minkowski space,” Journal für die reine und angewandte Mathematik, vol. 627, pp. 1–29, 2009.View at: Publisher Site | Google Scholar | MathSciNet
B. Guan, H. Jian, and R. M. Schoen, “Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space,” Journal für die Reine und Angewandte Mathematik, vol. 595, pp. 167–188, 2006.View at: Publisher Site | Google Scholar | MathSciNet
M. A. S. Aarons, “Mean curvature flow with a forcing term in Minkowski space,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 2, pp. 205–246, 2006.View at: Publisher Site | Google Scholar | MathSciNet
K. Ecker, “Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space,” Journal of Differential Geometry, vol. 46, no. 3, pp. 481–498, 1997.View at: Google Scholar | Zentralblatt MATH | MathSciNet
K. Ecker and G. Huisken, “Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes,” Communications in Mathematical Physics, vol. 135, no. 3, pp. 595–613, 1991.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Liu and H. Jian, “Evolution of spacelike hypersurfaces by mean curvature minus external force field in Minkowski space,” Advanced Nonlinear Studies, vol. 9, no. 3, pp. 513–522, 2009.View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. Jian, “Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space,” Journal of Differential Equations, vol. 220, no. 1, pp. 147–162, 2006.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Ju, J. Lu, and H. Jian, “Translating solutions to mean curvature flow with a forcing term in Minkowski space,” Communications on Pure and Applied Analysis, vol. 9, no. 4, pp. 963–973, 2010.View at: Publisher Site | Google Scholar | MathSciNet
H. Ju, J. Bao, and H. Jian, “Existence for translating solutions of Gauss curvature flow on exterior domains,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 8, pp. 3629–3640, 2012.View at: Publisher Site | Google Scholar | MathSciNet
F. Finster and O. C. Schnürer, “Hypersurfaces of prescribed Gauss curvature in exterior domains,” Calculus of Variations and Partial Differential Equations, vol. 15, no. 1, pp. 67–80, 2002.View at: Publisher Site | Google Scholar | MathSciNet
Y. Huang, H. Jian, and N. Su, “Spacelike hypersurfaces of prescribed Gauss-Kronecker curvature in exterior domains,” Acta Mathematica Sinica, vol. 25, no. 3, pp. 491–502, 2009.View at: Publisher Site | Google Scholar | MathSciNet
B. Guan, “The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4955–4971, 1998.View at: Publisher Site | Google Scholar | MathSciNet
D. Gilbarg and N. S. Trudinger, Second Order Elliptic Partial Differential Equations, Springer, Berlin, Germany, 2nd edition, 1983.View at: Publisher Site | MathSciNet