The asymptotic behavior of viscosity solutions of Monge–Ampère equations in half space

Abstract

In this paper we report the asymptotic behavior at infinity of convex viscosity solution of $det{D}^{2}u=1$ outside a bounded domain of the upper half space. It is shown that if the solution is a quadratic polynomial plus a logarithmic function at the flat boundary, then it tends to a quadratic polynomial plus a “$log$” term at infinity, where the “$log$” term means that it can be controlled by logarithmic function. Meanwhile, more accurate asymptotic behaviors at infinity are acquired.

Introduction

As we all know, J$\ddot{\text{o}}$gens [13] ($n=2$), Calabi [7] ($n\le 5$) and Pogorelov [18] ($n\ge 2$) proved that any classical convex solution of

$$det{D}^{2}u=1\text{in}{\mathbb{R}}^n$$

is a quadratic polynomial. Later, Cheng and Yau [9] gave an easier proof via investigating the completeness of affine metric. Caffarelli [5] extended the J$\ddot{\text{o}}$gens–Calabi–Pogorelov theorem to viscosity solutions. Furthermore, Caffarelli and Li [6] obtained the asymptotic behavior at infinity of viscosity solution of $det{D}^{2}u=1$ outside a bounded domain of ${\mathbb{R}}^n$. They showed that for $n\ge 3$,

$$|u\left(x\right)-\left(\frac{1}{2}{x}^{T}Ax+b^{T}x+c\right)|=O\left({\left|x\right|}^{2-n}\right)$$

for some symmetric positive definite matrix $A$ with $detA=1$, vector $b$ and constant $c$, and for $n=2$,

$$|u\left(x\right)-\left(\frac{1}{2}{x}^{T}Ax+b^{T}x+dlog\sqrt{x^{T}Ax}+c\right)|=O\left({\left|x\right|}^{-1}\right)$$

for some symmetric positive definite matrix $A$ with $detA=1$, vector $b$ and constants $c$, $d$. For $n=2$, 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

$$\left\{\begin{array}{cc}\hfill & det{D}^{2}u\left(x\right)=1\text{ in }{\mathbb{R}}_{+}^n\setminus \overline{B_1^{+}},\hfill \\ \hfill & u(x^{\prime },x_n)=p\left(x^{\prime }\right)\text{on }\{\left|x^{\prime }\right|>1,x_n=0\},\hfill \end{array}\right.$$

where $n\ge 2$ and $p\left(x^{\prime }\right)$ is a quadratic polynomial of $x^{\prime }$ with $\left[D_{x^{\prime }}^{2}p\right]>0$. It was shown that if $u$ satisfies the quadratical growth condition

$$\mu {\left|x\right|}^2\le u\left(x\right)\le {\mu }^{-1}{\left|x\right|}^2\text{in }\overline{{\mathbb{R}}_{+}^n\setminus B_1^{+}}$$

for some $0<\mu \le \frac{1}{2}$, then there exist some symmetric positive definite matrix $A$ with $detA=1$, vector $b$ and constant $c$ such that

$$|u\left(x\right)-\left(\frac{1}{2}{x}^{T}Ax+b^{T}x+c\right)|=O\left(\frac{x_n}{{\left|x\right|}^n}\right)\text{as }\left|x\right|\to \infty ,$$

where $A$, $b$ and $c$ satisfy the compatibility condition

$$p\left(x^{\prime }\right)=\frac{1}{2}{(x^{\prime },0)}^{T}A(x^{\prime },0)+b^T(x^{\prime },0)+c.$$

It is clear that the boundary condition of (1.1) leads the difference between the whole space case and the half space one as $n=2$. As $n=2$, 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

$$\left\{\begin{array}{cc}\hfill & det{D}^{2}u\left(x\right)=1\text{in }{\mathbb{R}}_{+}^n\setminus \overline{B_1^{+}},\hfill \\ \hfill & u\left(x\right)=\frac{1}{2}{\left|x\right|}^2+dlog\sqrt{x^{T}Qx}\text{on }\{\left|x^{\prime }\right|>1,x_n=0\},\hfill \end{array}\right.$$

where $Q$ is a $n\times n$ symmetric positive definite matrix and for some $\varrho \in (0,1]$,

$$\varrho {\left|x\right|}^2\le x^{T}Qx\le {\varrho }^{-1}{\left|x\right|}^2\forall x\in {\mathbb{R}}^n,$$

and $d$ is a non-zero constant and satisfies the convexity condition:

$$\frac{1}{2}{\left|x\right|}^2+dlog\sqrt{x^{T}Qx}\text{ is strictly convex for all }x\in \{\left|x^{\prime }\right|>1,x_n=0\},$$

without which $u$ is not convex. It is also assumed that $u$ satisfies (1.2).

The main result of this paper is the following.

Theorem 1.1

Let $u\in C^0\left(\overline{{\mathbb{R}}_{+}^n\setminus B_1^{+}}\right)$ be a convex viscosity solution of (1.4) under hypotheses (1.2), (1.5), (1.6). Then $u\in C^{\infty }(\overline{{\mathbb{R}}_{+}^n}\setminus \overline{B_1^{+}})$ and there exist some invertible upper-triangular matrix $T$ with $detT=1$, constant $b_n\in \mathbb{R}$ and some function $\Phi \left(x\right)\in C^{\infty }(\overline{{\mathbb{R}}_{+}^n}\setminus \overline{B_1^{+}})$ satisfying $\Phi \left(x\right)=O(log\left|x\right|)$ as $\left|x\right|\to \infty$ such that

(i) if $n=2$,

$$|u\left(x\right)-\left(\frac{1}{2}{x}^T{T}^{T}Tx+b_n{x}_n+\Phi \left(Tx\right)\right)|\le C\frac{x_2}{{\left|x\right|}^2}\text{in }\overline{{\mathbb{R}}_{+}^n\setminus B_R^{+}},$$

where $C>0$ and $R\ge 1$ depend only on $\mu$, $d$ and $\varrho$. Furthermore, for any $k\ge 1$,

$${\left|x\right|}^{k+1}\left|D^k\left(u\left(x\right)-\left(\frac{1}{2}{x}^T{T}^{T}Tx+b_2{x}_2+\Phi \left(Tx\right)\right)\right)\right|\le C\text{in }\overline{{\mathbb{R}}_{+}^n\setminus B_R^{+}},$$

where $C$ also depends on $k$.

(ii) if $n\ge 3$, for any $\delta \in \left(0,\frac{2}{n-1}\right)$,

$$|u\left(x\right)-\left(\frac{1}{2}{x}^T{T}^{T}Tx+b_n{x}_n+\Phi \left(Tx\right)\right)|\le C{\left(\frac{x_n}{{\left|x\right|}^n}\right)}^{\delta }\text{in }\overline{{\mathbb{R}}_{+}^n\setminus B_R^{+}},$$

where $C>0$ and $R\ge 1$ depend only on $n$, $\mu$, $d$, $\varrho$ and $\delta$. Furthermore, for any $k\ge 1$,

$${\left|x\right|}^{k+(n-1)\delta }\left|D^k\left(u\left(x\right)-\left(\frac{1}{2}{x}^T{T}^{T}Tx+b_n{x}_n+\Phi \left(Tx\right)\right)\right)\right|\le C\text{in }\overline{{\mathbb{R}}_{+}^n\setminus B_R^{+}},$$

where $C$ also depends on $k$.

Remark 1.2

We give several remarks on Theorem 1.1 as follows.

(1) When $n=2$ and $Q=I_2$, by Lemma 3.8 and Remark 3.9, we have that $\Phi \left(x\right)=log\left|x\right|$.

(2) If the boundary condition of (1.4) is replaced by

$$u(x^{\prime },x_n)=p\left(x\right)+dlog\sqrt{x^{T}Qx}\text{on }\{x_n=0,\left|x^{\prime }\right|>1\},$$

where $p\left(x\right)$ is a quadratic polynomial of $x$ with $\left[D^{2}p\right]>0$, and $Q$ is some $n\times n$ 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

$$T=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill {\nu }_1\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill {\nu }_{n-1}\hfill \\ \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$$

for some bounded vector $\nu =({\nu }_1,\dots ,{\nu }_{n-1},0)\in {\mathbb{R}}^n$.

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 $\epsilon >0$ small, there exist some invertible upper-triangular matrix $T$ with $detT=1$ such that

$$|u\left(T^{-1}x\right)-\frac{1}{2}{\left|x\right|}^2|=O\left({\left|x\right|}^{2-\epsilon }\right)\text{as }\left|x\right|\to \infty .$$

In the linear approach, by the asymptotic results obtained in Section 2 and introducing a new method, it can be deduced that

$$|u\left(T^{-1}x\right)-\frac{1}{2}{\left|x\right|}^2-b_n{x}_n|=O(log\left|x\right|)\text{as }\left|x\right|\to \infty$$

for some constant $b_n$. More exact asymptotic behaviors are also obtained.

Throughout this paper, the following notations are always used.

$\cdot$ For any $x\in {\mathbb{R}}^n$, $x=(x_1,x_2,\dots ,x_n)=(x^{\prime },x_n)$, $x^{\prime }\in {\mathbb{R}}^{n-1}$.

$\cdot$ ${\mathbb{R}}_{+}^n=\{x\in {\mathbb{R}}^n:x_n>0\}$; $\overline{{\mathbb{R}}_{+}^n}=\{x\in {\mathbb{R}}^n:x_n\ge 0\}$.

$\cdot$ For any $x\in {\mathbb{R}}^n$ and $r>0$, $B_r\left(x\right)=\{y\in {\mathbb{R}}^n:|y-x|<r\}$ and $B_r^{+}\left(0\right)=B_r\left(0\right)\cap \{x_n>0\}$. $B_r=B_r\left(0\right)$ and $B_r^{+}=B_r^{+}\left(0\right)$.