The asymptotic behavior of viscosity solutions of Monge–Ampère equations in half space☆
Introduction
As we all know, Jgens [13] (), Calabi [7] () and Pogorelov [18] () proved that any classical convex solution of is a quadratic polynomial. Later, Cheng and Yau [9] gave an easier proof via investigating the completeness of affine metric. Caffarelli [5] extended the Jgens–Calabi–Pogorelov theorem to viscosity solutions. Furthermore, Caffarelli and Li [6] obtained the asymptotic behavior at infinity of viscosity solution of outside a bounded domain of . They showed that for , for some symmetric positive definite matrix with , vector and constant , and for , for some symmetric positive definite matrix with , vector and constants , . For , Ferrer et al. [10], [11] proved the same result using complex variable methods.
Motivated by above results, many authors investigated the exterior Dirichlet problems for other fully nonlinear equations such as k-Hessian equations [3], [15], special Lagrangian equations [14], [16], [17], parabolic Monge–Ampère equations [20], [21], [22], and the references therein.
Recently, Jia et al. [12] investigated the asymptotic behavior at infinity of convex viscosity solution of where and is a quadratic polynomial of with . It was shown that if satisfies the quadratical growth condition for some , then there exist some symmetric positive definite matrix with , vector and constant such that where , and satisfy the compatibility condition
It is clear that the boundary condition of (1.1) leads the difference between the whole space case and the half space one as . As , the asymptotic term is a polynomial plus a logarithm function. It is natural to ask about the asymptotic behavior at infinity when the boundary condition is a polynomial plus a logarithm function.
In this paper, we consider the asymptotic behavior at infinity of convex viscosity solution of the Monge–Ampère equation where is a symmetric positive definite matrix and for some , and is a non-zero constant and satisfies the convexity condition: without which is not convex. It is also assumed that satisfies (1.2).
The main result of this paper is the following.
Theorem 1.1 Let be a convex viscosity solution of (1.4) under hypotheses (1.2), (1.5), (1.6). Then and there exist some invertible upper-triangular matrix with , constant and some function satisfying as such that (i) if , where and depend only on , and . Furthermore, for any , where also depends on . (ii) if , for any , where and depend only on , , , and . Furthermore, for any , where also depends on .
Remark 1.2 We give several remarks on Theorem 1.1 as follows. (1) When and , by Lemma 3.8 and Remark 3.9, we have that . (2) If the boundary condition of (1.4) is replaced by where is a quadratic polynomial of with , and is some symmetric positive definite matrix, after subtracting a linear function and making a proper transformation, similar conclusion also holds. (3) From (1.4), (1.2), we know that for some bounded vector .
The proof of Theorem 1.1 partially follows the idea of [6] and [12]. In addition, in the paper, a new method is required to overcome the difficulties caused by the complex boundary value. Furthermore, it is also needed to establish the asymptotic results at infinity on solutions elliptic equations with non-homogeneous terms in half spaces.
The paper is organized as follows. In Section 2, the purpose is to study the asymptotic behavior at infinity of solutions elliptic equations with non-homogeneous terms in half spaces, which will be used repeatedly in the proof of Theorem 1.1. In Section 3, the proof of Theorem 1.1 is given and divided into two steps: nonlinear approach and linear approach. In the nonlinear approach, using the idea of [6], it is shown that for some small, there exist some invertible upper-triangular matrix with such that In the linear approach, by the asymptotic results obtained in Section 2 and introducing a new method, it can be deduced that for some constant . More exact asymptotic behaviors are also obtained.
Throughout this paper, the following notations are always used.
For any , , .
; .
For any and , and . and .
Section snippets
Asymptotic behavior of elliptic equation in the upper half space
In this section we investigate the asymptotic behavior at infinity of viscosity solutions of linear elliptic equations with non-homogeneous terms outside bounded domains of half spaces, which will play a key role in obtaining the linear part and asymptotic rate at infinity in Theorem 1.1. For related results of solutions of homogeneous linear elliptic equations in the upper half space, one can refer to [12].
We first give two useful lemmas as the following.
Lemma 2.1 Let , ,
Proof of Theorem 1.1
We begin this section with the proof of the smoothness of in .
Lemma 3.1 Let be as in Theorem 1.1. Then .
Proof For any , let . In view of the boundary value of and the convexity condition (1.6), we have that separates quadratically on from its tangent plane at . Then by the boundary estimates (cf. [19]), for some small and any . Linearizing the equation and applying the Schauder estimates, we have
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