Monge–Ampère equation on exterior domains

Abstract

We consider the Monge–Ampère equation $\det (D^2u)=f$ where $f$ is a positive function in ${\mathbb {R}}^n$ and $f=1+O(|x|^{-\beta })$ for some $\beta >2$ at infinity. If the equation is globally defined on ${\mathbb {R}}^n$ we classify the asymptotic behavior of solutions at infinity. If the equation is defined outside a convex bounded set we solve the corresponding exterior Dirichlet problem. Finally we prove for $n\ge 3$ the existence of global solutions with prescribed asymptotic behavior at infinity. The assumption $\beta >2$ is sharp for all the results in this article.

Appendix: Interior estimate of Caffarelli and Jian–Wang

The following theorem is a combination of the interior estimate of Caffarelli [7] and an improvement by Jian and Wang [25].

Theorem 6.1

(Caffarelli, Jian and Wang) Let $u\in C^0(\Omega )$ be a convex viscosity solution of

$$\begin{aligned}&det(D^2u)=f, \quad \Omega , \\&u=0\quad \hbox {on }\quad \partial \Omega , \end{aligned}$$

where $\Omega$ is a convex bounded domain satisfying $B_1\subset \Omega \subset B_n$. Assume that $f$ is Dini continuous on $\Omega$ and

$$\begin{aligned} \frac{1}{c_0}\le f \le c_0, \quad \Omega . \end{aligned}$$

Then $u\in C^2(B_{1/2})$ and $\forall x,y\in B_{1/2}$

$$\begin{aligned} |D^2u(x)-D^2u(y)|\le C \left( d+\int _0^d\frac{\omega _f(r)}{r}+d\int _d^1\frac{\omega _f(r)}{r^2}\right) \end{aligned}$$

(6.1)

where $d=|x-y|, C>0$ depends only on $n$ and $c_0, \omega _f$ is the oscillation function of $f$ defined by

$$\begin{aligned} \omega _f(r):=\sup \{|f(x)-f(y)|:\quad |x-y|\le r \}. \end{aligned}$$

It follows that (i) If $f$ is Dini continuous, then $u\in C^2(B_{1/2})$, and the modulus of convexity of $D^2u$ can be estimated by (6.1). (ii) If $f\in C^{\alpha }(\Omega )$ and $\alpha \in (0,1)$, then

$$\begin{aligned} \Vert D^2u\Vert _{C^{\alpha }(B_{1/2})}\le C \left( 1+\frac{\Vert f\Vert _{C^{\alpha }(\Omega )}}{\alpha (1-\alpha )}\right) . \end{aligned}$$

(iii) If $f\in C^{0,1}(\Omega )$, then

$$\begin{aligned} |D^2u(x)-D^2u(y)|\le Cd (1+\Vert f\Vert _{C^{0,1}(\Omega )}|\log d| ). \end{aligned}$$

Here we recall that $f$ is Dini continuous if the oscillation function $\omega _f$ satisfies $\int _0^1\omega _f(r)/rdr<\infty$.

Remark 6.1

Note that in Caffarelli’s interior estimate $u=0$ is assumed on $\partial \Omega$. Since $\Omega$ is very close to a ball, by [5, 6] $u$ is strictly convex in $\Omega$. But there is no explicit formula that describes how the higher order derivatives of $u$ depend on $f$. In Jian-Wang’s theorem, this dependence is given as in (6.1) but instead of assume $u=0$ on $\partial \Omega$, they assumed $u$ is strictly convex and their constant depends on the strict convexity. We feel the way that Theorem 6.1 is stated is more convenience for application. We only used the $(ii)$ and $(iii)$ of Theorem 6.1 in this article.