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The exterior Dirichlet problems of Monge–Ampère equations in dimension two
Boundary Value Problems volume 2020, Article number: 179 (2020)
Abstract
In this paper, we study the Monge–Ampère equations \(\det D^{2}u=f\) in dimension two with f being a perturbation of \(f_{0}\) at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.
1 Introduction
A classical theorem for Monge–Ampère equation states that any classical convex solution of
must be a quadratic polynomial which was obtained by Jörgens [1] for \(n=2\), Calabi [2] for \(n\leq 5\), and Pogorelov [3] for \(n\geq 2\). For \(n=2\), the classical solution is either convex or concave, so the result is true without the convex hypothesis.
Cheng and Yau [4] later gave a simpler and more analytical proof of the Jörgens–Calabi–Pogorelov theorem along the lines of affine geometry. Caffarelli [5] extended this result to viscosity solutions. Jost and Xin [6] gave another proof of the theorem. On the other hand, it was proved by Trudinger and Wang [7] that if D is an open convex subset in \(\mathbb{R}^{n}\) and u is a convex \(C^{2}\) solution to \(\det D^{2}u=1\) in D with \(\lim_{x\to \partial D}u(x)=\infty \), then \(D=\mathbb{R}^{n}\).
In 2003, the Jörgens–Calabi–Pogorelov theorem was extended to exterior domains for \(n\geq 2\) by Caffarelli and Li [8]. They proved that any convex viscosity solution to \(\det D^{2}u=1\) in an exterior domain must be a quadratic polynomial at infinity for \(n\geq 3\) and be the sum of a quadratic polynomial and a logarithmic term at infinity for \(n=2\). That is, for \(n\geq 3\), there exists a symmetric positive definite matrix \(A\in \mathbb{R}^{n\times n}\), \(\hat{b}\in \mathbb{R}^{n}\), \(\hat{c}\in \mathbb{R}\) such that
and for \(n=2\), there exists a symmetric positive definite matrix \(A\in \mathbb{R}^{2\times 2}\), \(\hat{b}\in \mathbb{R}^{2}\), \(\hat{c},\hat{d}\in \mathbb{R}\) such that
For \(n=2\), similar problems were studied by Ferrer, Martínez, and Milán [9, 10] applying the complex variable method. We can also refer to Delanoë [11]. Later, in [12], Caffarelli and Li also extended the Jörgens–Calabi–Pogorelov theorem to \(\det D^{2}u=f\) in \(\mathbb{R}^{n}\) with f being a periodic positive function and obtained that any classical convex solution must be the sum of a quadratic polynomial and a periodic function. In 2015, Bao, Li, and Zhang [13] extended the Jörgens–Calabi–Pogorelov theorem to \(\det D^{2}u=f\) in the exterior domain for \(n\geq 2\) with f being a perturbation of 1 at infinity.
The converse problem is whether the exterior Dirichlet problem has a unique solution with the prescribed asymptotic behavior. This problem was resolved, and the existence of solutions to the exterior Dirichlet problem of \(\det D^{2}u=1\) was established in [8] for \(n\geq 3\) through the Perron method. More specifically, assume that D is a strictly convex and bounded domain with smooth boundary in \(\mathbb{R}^{n}\) and \(\phi \in C^{2}(\partial D)\), then for any symmetric positive definite matrix \(A\in \mathbb{R}^{n\times n}\), \(\hat{b}\in \mathbb{R}^{n}\), there exists a constant \(c_{1}=c_{1}(n,D,\phi ,\hat{b},A)\) such that, for any \(\hat{c}>c_{1}\), the exterior Dirichlet problem
has a unique solution \(u\in C^{\infty }(\mathbb{R}^{n}\backslash \overline {D})\cap C^{0}(\overline { \mathbb{R}^{n}\backslash D})\) satisfying the asymptotic behavior (1.1). For \(n=2\), the existence of solutions to the exterior Dirichlet problem was established by Bao and Li [14]. More precisely, for any symmetric positive definite matrix \(A\in \mathbb{R}^{2\times 2}\), \(\hat{b}\in \mathbb{R}^{2}\), there exists a constant \(d^{*}=d^{*}(D,\phi ,\hat{b},A)\) such that, for any \(\hat{d}>d^{*}\), the exterior Dirichlet problem (1.3) has a unique solution \(u\in C^{\infty }(\mathbb{R}^{2}\backslash \overline {D})\cap C^{0}(\overline { \mathbb{R}^{2}\backslash D})\) satisfying
with \(M_{\hat{d}}\), \(c(\hat{d})\) being functions of d̂. Bao, Li, and Zhang [13] established the existence of exterior solutions to \(\det D^{2}u=f\) with f being a perturbation of 1 at infinity for \(n\geq 2\). More existence results for \(n=2\) can be found in [15, 16]. In [15], global solutions and exterior solutions for Monge–Ampère equation were obtained through the new ideas in contrast to [13]. And in [16], Bao, Xiong, and Zhou proved the existence of entire solutions of Monge–Ampère equations with asymptotic behaviors employing the different method from the constructions of sub- and super- solutions. However, the existence of solutions to \(\det D^{2}u=f\) in the exterior domain with f being a perturbation of \(f_{0}(|x|)\) at infinity for \(n\geq 3\) was obtained by Ju and Bao [17].
The constants ĉ and d̂ in (1.1) and (1.2) play important roles in the existence and nonexistence of solutions to the exterior Dirichlet problem. Wang and Bao [18] studied the constants ĉ and d̂ among the radially symmetric solutions. They proved that, for \(n\geq 3\), there exists a unique convex radial solution \(u\in C^{2}(\mathbb{R}^{n}\backslash \overline {B_{1}(0)})\cap C^{1}( \mathbb{R}^{n}\backslash B_{1}(0))\) satisfying
if and only if \(\hat{c}\geq C^{*}=\hat{a}-\frac{1}{2}+\int _{1}^{\infty }s((1-s^{-n})^{ \frac{1}{n}}-1)\,ds\); for \(n=2\), there exists a unique radially symmetric solution \(u\in C^{2}(\mathbb{R}^{2}\backslash \overline {B_{1}(0)})\cap C^{1}( \mathbb{R}^{2}\backslash B_{1}(0))\) satisfying (1.4), (1.5), and
if and only if \(\hat{d}\geq -1\) and \(\hat{c}=\hat{a}+\frac{\hat{d}}{4}+\frac{\hat{d}}{2}\ln 2-\frac{1}{2}[(1+ \hat{d})^{1/2}+\hat{d}\ln (1+(1+\hat{d})^{1/2})]\). Recently, Li and Lu [19] characterized the existence and nonexistence of solutions in terms of the asymptotic behavior to the exterior Dirichlet problem with the right-hand side being 1 or the perturbation of 1 at infinity for \(n\geq 3\).
In this paper, we study the Monge–Ampère equation with the right-hand side being \(f_{0}(|x|)\) at infinity for \(n=2\). Because of the appearance of the logarithmic term, it seems more difficult than the case of \(n=3\). Assume that \(f\in C^{0}(\mathbb{R}^{2})\). Let \(\beta >2\) be some constant and
where \(f_{0}\in C^{0}([0,+\infty ))\) is positive and radially symmetric in x, and for some constant α,
Suppose that
Consider the Dirichlet problem outside a unit ball
Our first main result is the following.
Theorem 1.1
Suppose that f satisfies (1.6) and f is radially symmetric, that is, \(f(x)=f(|x|)\). Let (1.8) hold. Then the exterior Dirichlet problem (1.9) and (1.10) has a unique convex radial solution \(u\in C^{1}(\mathbb{R}^{2}\backslash B_{1})\cap C^{2}(\mathbb{R}^{2} \backslash \overline {B_{1}})\) satisfying, as \(|x|\to \infty \),
if and only if \(\tilde{\tau }\geq \tilde{\tau }_{0}\) and \(d=h(\tilde{\tau })\), where
and
Remark 1.2
Let \(F_{0}(x)\) denote the second term in (1.11), that is,
then \(F_{0}\) satisfies
Remark 1.3
For \(n\geq 3\), the necessary and sufficient conditions of the existence of solutions with prescribed asymptotic behavior at infinity to (1.9) and (1.10) are obtained in the author’s another manuscript which is submitted.
If \(f\equiv 1\), the solution can be integrated. But if f is general, the solution cannot be integrated directly. By extracting the corresponding quadratic polynomial and logarithmic term, we obtain the asymptotic behavior at infinity. Moreover, our approach is enough to establish the existence of solutions of the exterior Dirichlet problem. So, in the following, by constructing the sub- and super-solutions, we apply the Perron method to get the existence of solutions with prescribed asymptotic behavior at infinity to the exterior Dirichlet problem. First let us recall the definition of viscosity solutions, we can also refer to [8]. Let \(O\subset \mathbb{R}^{2}\) be a domain, \(g\in C^{0}(O)\) be positive, and \(\Phi \in C^{0}(\partial O)\).
Definition 1.4
([8])
Suppose that \(u\in C^{0}(O)\) is locally convex. We say that u is a viscosity subsolution (supersolution) of
if, for any point \(\hat{x}\in O\) and any convex function \(\psi \in {C}^{2}(O)\), whenever
we must have
\(u\in {C}^{0}(O)\) is a viscosity solution of (1.13) if it is both a viscosity subsolution and supersolution of (1.13).
Definition 1.5
([8])
We say that \(u\in C^{0}(O)\) is a viscosity subsolution (supersolution) of (1.13) with the boundary condition \(u=\Phi\) on ∂O if u is a viscosity subsolution (supersolution) of (1.13), and \(u\leq (\geq )\ \Phi \) on ∂O.
Then \(u\in {C}^{0}(\overline {O})\) is a viscosity solution of (1.13) with the boundary condition \(u=\Phi\) on ∂O if it is a viscosity solution of (1.13) and \(u=\Phi \) on ∂O.
Then we consider the exterior Dirichlet problem
where Ω is a bounded domain in \(\mathbb{R}^{2}\), \(f\in C^{0}(\mathbb{R}^{2})\) and \(\varphi \in C^{2}(\partial \Omega )\). Note that here f is not necessarily radially symmetric.
Let \(\bar{f}(|x|)\), \(\underline {f}(|x|)\) be two positive continuous functions in \(\mathbb{R}^{2}\) satisfying
and
where \(f_{0}\) is the same as (1.7). The following is the second main result.
Theorem 1.6
Let Ω be a strictly convex and bounded domain in \(\mathbb{R}^{2}\), \(0\in \Omega \), \(\partial \Omega \in C^{2}\), and \(\varphi \in C^{2}(\partial \Omega )\). Suppose that f satisfies (1.6) and (1.8) holds. Then, for any \(\tilde{b}\in \mathbb{R}^{2}\), there exists a constant \(\lambda ^{*}=\lambda ^{*}(\tilde{b},\varphi ,f,\Omega ,\alpha , \beta )\) such that, for any \(\lambda >\lambda ^{*}\), there exists a unique viscosity solution \(u\in C^{0}(\mathbb{R}^{2}\backslash \Omega )\) to the Dirichlet problem (1.14), (1.15) which satisfies
where
and \(a_{1}:=\min \{a:\Omega \subset B_{a}\}\), \(B_{a}=B_{a}(0)=\{x\in \mathbb{R}^{2}:|x|< a\}\), d̃ is a function of λ, and
Remark 1.7
If Ω is a ball, \(\varphi =\mbox{constant,}\) and \(\overline {f}\equiv \underline {f}\), then \(\Lambda (\lambda )=0\).
Finally, we would like to mention the blow-up solutions and asymptotic behavior of fully nonlinear equations and Monge–Ampère equations. In [20], the existence, asymptotic boundary estimates, and uniqueness of large solutions to fully nonlinear equations were studied, and in [21], the sharp conditions and asymptotic behavior of boundary blow-up solutions to the Monge–Ampère equation were studied.
This paper is organized as follows. In Sect. 2, we prove Theorem 1.1. Theorem 1.6 is proved in Sect. 3.
2 Proof of Theorem 1.1
Proof of Theorem 1.1
Let \(u(x)=u(r)=u(|x|)\) be the radial solution to (1.9). By a direct calculation, we have
Hence
Then integrating the above equality from 1 to r on both sides twice, we have
Let \(\tau =(u^{\prime }(1))^{2}\), and
Then the exterior problem (1.9) and (1.10) has the radial solution
Then \(\tilde{\tau }\geq \tilde{\tau }_{0}\), where \(\tilde{\tau }_{0}\) is the same as (1.12).
From (1.8), we know that \(\alpha +2>0\). Due to (1.6) and (1.7), we have
where \(\delta >0\) is the same as (1.8).
On the other hand, if \(\tilde{\tau }\geq \tilde{\tau }_{0}\), then
It is clear that (2.1) is a radial solution of (1.9) and (1.10) which satisfies (1.11).
If \(u_{1}\) and \(u_{2}\) all satisfy (1.9), (1.10), and (1.11), by the comparison principle, we know that \(u_{1}\equiv u_{2}\) and obtain the uniqueness. Then we complete the proof of Theorem 1.1. □
Let
Set
Then
Since \(\tilde{\tau }\geq \tilde{\tau }_{0}\), by (1.6) and (1.7), the infinite integral \(\int _{1}^{\infty } (\int _{0}^{s}2tf_{0}(t)\,dt )^{ \frac{1}{2}}\frac{\partial g}{\partial \tilde{\tau }}\,ds\) is convergent uniformly for τ̃. So, for \(h(\tilde{\tau })\) in Theorem 1.1,
Let
By (2.4), then \(\tilde{\tau }^{*}\geq \tilde{\tau }_{0}\). So \(h(\tilde{\tau })\) has the maximum in \(\tilde{\tau }=\tilde{\tau }^{*}\) and \(d=h(\tilde{\tau })\leq h(\tilde{\tau }^{*})\), where
Hence we have the following Corollary 2.1.
Corollary 2.1
Let (2.4) hold. Then the exterior Dirichlet problem (1.9) and (1.10) has a convex radial solution \(u\in C^{1}(\mathbb{R}^{2}\backslash B_{1})\cap C^{2}(\mathbb{R}^{2} \backslash \overline {B_{1}})\) satisfying (1.11) if and only if \(\tilde{\tau }\geq \tilde{\tau }_{0}\) or \(d\leq h(\tilde{\tau }^{*})\).
If \(f=f_{0}\equiv 1\), Theorem 1.1 corresponds to the results in [18]. In fact, by (2.2), we know that
Since
and
then, as \(|x|\to \infty \),
where \(\tilde{\tau }=\tau -1\). Then (2.5) corresponds to the results in [18]. In fact, in [18], \(f\equiv 1\), (2.1) can be integrated, and then the asymptotic behavior is obtained. But here we cannot integrate (2.1), and so the asymptotic behavior is more complicated.
3 Proof of Theorem 1.6
Proof of Theorem 1.6
By a translation, without loss of generality, we assume that \(\tilde{b}=0\). Let \(a_{1}:=\min \{a:\Omega \subset B_{a}\}\). Choose \(a_{2}>a_{1}\). Let
By Lemma 3.1 in [13], for any \(\xi \in \partial \Omega \), there exists a function \(w_{\xi }(x)\) such that
and \(w_{\xi }(\xi )=\varphi (\xi )\), \(w_{\xi }(x)<\varphi (x)\) on \(\partial \Omega \backslash \{\xi \}\).
Set
Then, in the viscosity sense,
and
For some constant \(\lambda _{1}\geq 0\), let
where f̄ is the same as (1.16). Then
Furthermore, similar to (2.2), we have
where
In addition, similar to (2.3),
where δ is the same as (1.8). Let
As a result,
Clearly,
We can choose \(\lambda _{1}^{*}>0\) fixed such that, for \(\lambda _{1}>\lambda _{1}^{*}\),
For \(\lambda _{2}\geq 0\), let
where \(\underline {f}\) is the same as (1.16). Then
Similar to (2.2), we get that
where
and
where δ is the same as (1.8). Let
As a result,
In addition,
We can choose \(\lambda _{2}^{*}>0\) such that, for \(\lambda _{2}>\lambda _{2}^{*}\),
So, by the comparison principle, we can know that
Choose \(\lambda ^{*}>\max \{\tilde{\lambda }_{1}(\lambda _{1}^{*}), \tilde{\lambda }_{2}(\lambda _{2}^{*})\}\) large. Then, for any \(\lambda >\lambda ^{*}\), we can choose appropriate constants \(\lambda _{1}\), \(\lambda _{2}\) such that
And thus
Therefore
Since \(\bar{f}(t)-f_{0}(t)\geq \underline {f}(t)-f_{0}(t)\), then by (3.2) and (3.5), we can deduce
From the comparison principle, we have that
Let \(\tilde{d}=\nu _{1}(\lambda )\), then
where \(\Lambda (\lambda )\) is the same as (1.17). Define
Then \(\underline {u}\in C^{0}(\mathbb{R}^{2}\backslash \Omega )\) satisfies
Let \(\mathbb{S}_{\tilde{d}}\) denote the set of subsolutions v to the Dirichlet problem (1.14), (1.15) satisfying
Then \(\underline {u}\in \mathbb{S}_{\tilde{d}}\). Define
So
Thus, by (3.9) and (3.10), we know that
Moreover, u is a viscosity subsolution of (1.14). By (3.11), for any \(x_{0}\in \partial \Omega \),
On the other hand,
Indeed, for some \(R>0\), let \(\Omega \subset B_{R}(0)\), we consider
Then by the comparison principle for any \(v\in \mathbb{S}_{\tilde{d}}\), \(v\leq v^{+}\) in \(B_{R}(0)\backslash \Omega \), so \(u\leq v^{+}\) in \(B_{R}(0)\backslash \Omega \), and then
Combining with (3.12), we deduce that
In the following, we prove that u is a viscosity solution of (1.14).
For any \(\hat{x}\in \mathbb{R}^{2}\), choose \(B_{\epsilon }(\hat{x}):=\{x:|x-\hat{x}|\leq \epsilon \}\subset \mathbb{R}^{2}\backslash \overline {\Omega }\). Consider the Dirichlet problem
By virtue of the comparison principle, we know
Let
Then \(\det D^{2}\hat{w}\geq f\) in \(\mathbb{R}^{2}\backslash \overline {\Omega }\) and \(\hat{w}(x)\leq \overline {u}_{\lambda _{2}}(x)\), \(x\in \mathbb{R}^{2} \backslash \Omega \). Therefore \(\hat{w}\in \mathbb{S}_{\tilde{d}}\). And then \(u(x)\geq \hat{w}(x)\), \(x\in \mathbb{R}^{2}\backslash \Omega \). Especially,
So, by (3.13), we can get
Since x̂ is arbitrary, we know that u is a solution of (1.14).
The uniqueness can be obtained by the comparison principle for viscosity solutions. Theorem 1.6 is proved. □
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The author is very grateful to the referees for their very helpful comments and suggestions.
Funding
The research was supported by the National Natural Science Foundation of China (No. 11201343) and the Shandong Provincial Natural Science Foundation (ZR2018LA006).
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Dai, L. The exterior Dirichlet problems of Monge–Ampère equations in dimension two. Bound Value Probl 2020, 179 (2020). https://doi.org/10.1186/s13661-020-01476-4
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DOI: https://doi.org/10.1186/s13661-020-01476-4