Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation

In this paper, we use the Perron method to prove the existence and uniqueness of the exterior problem for a kind of parabolic Monge-Ampere equation \begin{document}$ -u_t+\log\det D^2u = f(x) $\end{document} with prescribed asymptotic behavior at infinity, where \begin{document}$ f $\end{document} is asymptotically close to a radial function at infinity. We generalize the results of both the elliptic exterior problems and the parabolic interior problems for the Monge-Ampere equations.


1.
Introduction. Elliptic Monge-Ampère equations, a hot research field in partial differential equation, have many valuable results over the years, which also promote the extensive study of parabolic Monge-Ampère equations. These two kinds of equations have some common research perspectives, such as the liouville property, asymptotic behavior and existence, while the existence and uniqueness is the focus of this paper.
We start with some known significant conclusions of elliptic Monge-Ampère equations and mainly prove an existence theorem of the exterior problem of a kind of parabolic Monge-Ampère equation. In the following statement, unless otherwise specified, we always assume that Ω is a smooth, bounded and strictly convex domain in R n .
In 1984, Caffarelli, Nirenberg and Spruck [4] obtained the existence and uniqueness of strictly convex smooth solutions for the interior Dirichlet problem where f ∈ C ∞ (Ω) is a positive function, and ϕ ∈ C ∞ (Ω). The result also holds for viscosity solutions, see [16], [3] and [11]. On this basis, many mathematicians have discussed the existence and uniqueness of exterior problems and global solutions with asymptotics, see [1,2,5,13]. The following theorem is a major representative of the exterior problems. Theorem 1.1 ( [5]). Let n ≥ 3, ϕ ∈ C 2 (∂Ω). Then for any given real n × n symmetric positive definite matrix A with det A = 1, b ∈ R n , there exists some constant c * depending only on n, Ω, ϕ, A and b, such that for every c > c * , there exists a unique function u ∈ C ∞ (R n \ Ω) ∩ C 0 (R n \ Ω) that satisfies The corresponding result for n = 2 was given by Ferrer, Mart1nez and Milán in [9,10] through the complex variable method. In regard to the conclusion of Theorem 1.1, Li and Lu took a further step in [14], discovering the definite relationship between c * and the existence and nonexistence of solutions for n ≥ 3.
In addition, under some asymptotic condition, Caffarelli and Li [5] obtained an important existence result about the global solutions of where n ≥ 3, and f is a continuous function which is constant outside a bounded set and satisfies In [1], Bao, Li and Zhang weakened the requirement of f in (1.2) to a perturbation of 1 and got a similar conclusion. Recently, Bao, Xiong and Zhou solved the twodimensional case, see [2]. On the other hand, there are many studies about the existence and uniqueness of solutions of different forms of parabolic Monge-Ampère equations, see [12,17,19].
Let Q T = Ω × (0, T ], where T is a positive constant, and ∂ p Q T denote the parabolic boundary of Q T . By means of approximation and nonlinear perturbation, Wang and Wang [19] proved the existence and uniqueness of viscosity solutions to the initial-boundary value problem where f is Lipschitz continuous in Q T and ϕ ∈ C 2,2 (Q T ) is strictly convex in x for any fixed t ∈ [0, T ]. Moreover, (1.3) has application in geometric problems such as surface deformation as well as Minkowski problems, see [6].
In this paper, we will prove that under certain conditions, there exists a unique viscosity solution to the exterior problem of the parabolic Monge-Ampère equation where n ≥ 2. When f ≡ −1, the result was proved by Dai in [8] with the condition ϕ xi,t (x, t) = 0, x ∈ ∂Ω, 0 ≤ t ≤ T. Now we consider the case where f is asymptotically close to a radial function at infinity and the restriction on ϕ is removed.
To begin with, let us recall the definition of viscosity solutions of parabolic equations [18,20]. Denote Q r (x,t) := {(x, t) |x −x| < r,t − r 2 < t ≤t}. A PARABOLIC MONGE-AMPÈRE EQUATION   4923 For simplicity, USC is short for upper-semicontinuous and LSC is short for lowersemicontinuous.
in the viscosity sense, where τ is a constant, and there exist b ∈ R n , c ∈ R such that is the solution of det D 2 u 0 = e f0(|x|)+τ and satisfies u 0 (0) = 0, u 0 (0) = 0. For a domain D ⊂ R n+1 , we say a function u ∈ C k,j (D) if u is k-th continuously differentiable with the spatial variable x ∈ R n and j-th continuously differentiable with the time variable t for (x, t) ∈ D.
Remark 1. If f 0 = 0, we can easily see that f (x) is a perturbation of 0 at infinity. More precisely, the condition (F ) is: (F ) f ∈ C 0 (R n ) satisfies that for the positive constant β > 2, The condition (Φ) is: where τ is a constant.
We draw the following conclusion at once.
The paper is arranged as follows. In Section 2, we give some useful lemmas. In Section 3, we prove Theorem 1.4 using the Perron method.
. Then for any positive constant c, there exists a positive constant C 0 , depending on n, ϕ, Ω, T,c, such that for any Proof. Let ξ ∈ ∂Q. Through a translation, scaling and rotation, without loss of generality, we may assume ξ = 0, and ∂Ω is locally represented by and ϕ has the local expansion Therefore for x ∈ N , where C 1 depends on ||ϕ|| C 2,0 (∂Ω×[0,T ]) andc. On account of the strict convexity of ∂Ω, there exists a positive constant δ depending only on Ω, such that for |x | < δ, Besides, from (2.2) and the strict convexity of ∂Ω, for any Choosingx n large (depending on δ, T, Ω, ||ϕ|| C 2,0 (∂Ω×[0,T ]) ), we also have We say D is an open set in the parabolic sense if D = D \ ∂ p D.

3)
and Then w ∈ U SC(D) is locally convex in x and satisfies .
Now we give the comparison principle below.
in the viscosity sense, respectively. Then we have Under the assumptions u, v ∈ C 0 (D) and D being a cylindrical domain, the lemma was proved by Wang and Wang in [17]. Based on their result, we can then directly obtain our comparison principle. Definition 2.4. Let g ∈ C 0 (Ω) be a positive function, and u ∈ C 0 (Ω) a locally convex function. We say that u is a viscosity subsolution of or a viscosity solution of det D 2 u ≥ g in Ω if for everyx ∈ Ω and every convex ϕ ∈ C 2 (Ω) satisfying ϕ ≥ u in Ω and ϕ(x) = u(x), we have det Similarly, u is a viscosity supersolution of (2.6) if for for everyx ∈ Ω and every convex ϕ ∈ C 2 (Ω) satisfying ϕ ≤ u in Ω and ϕ(x) = u(x), we have u is a viscosity solution of (2.6) if u is both a viscosity subsolution and a viscosity supersolution of (2.6).
[7] Let n ≥ 2, u ∈ U SC(Ω) and v ∈ LSC(Ω) be the viscosity subsolution and supersolution of respectively, where f (x) ∈ C 0 (Ω) is a positive function. Furthermore, Then sup To introduce the Perron method for parabolic equations, we first define viscosity solutions which do not satisfy (semi) continuous properties.
Definition 2.6. We say a function u is a weak viscosity subsolution of (1.4) if the USC envelope of u, namely, is finite and a viscosity subsolution, where Similarly, one uses LSC envelope u * = −(−u) * for supersolutions. If u is a weak viscosity sub-and supersolution, we call u a weak viscosity solution.
We can also define weak viscosity solutions of the problem (1.4), (1.5), (1.6) by giving the boundary condition like Definition 1.3.
Similar to the process by Y. Zhan [20], we have the two lemmas below. We give the proof here for completeness.
then u is a weak viscosity subsolution of (2.8).
Proof. By the definition of weak viscosity subsolutions, we need to prove that for all functions ϕ ∈ C 2,1 (D), if there exists (x,t) ∈ D such that for some Q r := Q r (x,t), then Without loss of generality, we can assume that (u * − ϕ)(x,t) = 0. Set ψ(x, t) = ϕ(x, t) + |x −x| 4 + |t −t| 2 , then u * − ψ attains its strict maximum in Q r at (x,t). So in Q r , Since v * k − ψ ∈ U SC(D), it attains its maximum at (y k , s k ) in some compact neighborhood B ⊂ Q r of (x,t). Noting that in Q r , Since v * k is a viscosity subsolution of (2.8), and v * k − ψ attains its local maximum at (y k , s k ), then

Lemma 2.8 ([7]
). Let F denote the nonempty set of weak viscosity subsolutions of (2.7). Set w(x) = sup{u(x)| u ∈ F} for x ∈ Ω. Suppose w * (x) < ∞ f or x ∈ Ω, then w is a weak viscosity subsolution of (2.7). Lemma 2.9. Let g be a weak viscosity supersolution of (2.8). Let S g := {v v is a weak viscosity subsolution of (2.8) and v ≤ g} and u(x, t) := sup{v(x, t) v ∈ S g }. If S g is not empty, then u is a weak viscosity solution of (2.8).
Clearly ϕ ≤ u * ≤ g * in Q r , so u * (x,t) = ϕ(x,t) < g * (x,t), otherwise it would contradict the fact that g is a weak viscosity supersolution of (2.8).
Since F and ϕ are continuous, for δ > 0 small enough, we have (2.9) indicates that the function ϕ(x, t) + δ 2 is a subsolution in B 2δ . Furthermore, we have Define w(x, t) by By Lemma 2.2, w is a weak viscosity subsolution of (2.8) over D. Since w ≤ g, w ∈ S g . By the definition of u, we have u ≥ w.
On the other hand, since there is a point (z, s) ∈ B δ such that u(z, s) − ϕ(z, s) < δ 2 and u(z, s) < w(z, s), which leads to a contradiction.
To ensure the definition of the upper bound meaningful, we need the set S c to be nonempty, that is we need to find an appropriate function satisfying all the requirements for the functions in S c at the same time. In fact, the function u we construct is a viscosity subsolution of (1.4), (1.5), (1.6) satisfying (1.11). Let us start doing it now. Choose R 1 > 2. Suppose B 2 (0) ⊂⊂ Ω ⊂⊂ B R1 (0), and then choose R 2 > 3R 1 . By Lemma 2.1, for any ξ ∈ ∂Ω, t ∈ [0, T ], there existsx(ξ, t) = − 1 c * D x ϕ(ξ, t)+x n ν ∈ R n satisfying |x(ξ, t)| < ∞, and w ξ ( and c * is sufficiently large to make sure the following two inequalities hold simultaneously then w is a locally Lipschitz function in R n+1 , and By Lemma 2.7 and Lemma 2.8, Let f (|x|), f (|x|) be two positive continuous functions in [0, +∞) satisfying Choose a 0 > 0 such that for a ≥ a 0 , the following four inequalities hold at the same time We can deduce α + n > 0 from −n(min{β,n}−2) n−1 We can also obtain that as |x| → +∞, For sufficiently large c in (1.9), there exist a 1 (c), a 2 (c) satisfying µ 1 (a 1 (c)) = µ 2 (a 2 (c)) = c.
From the discussions above, it only remains to prove that u c is a viscosity solution of (1.4). In fact, from the definition of u c and Lemma 2.9, it can be obtained directly that u c is a weak viscosity solution of (1.4). By Lemma 2.3 and the asymptotic behavior, u * c ≤ u c * . From the definition of u * c and u c * , we know that u * c ≥ u c * . Then we have u * c = u c * = u c . Thus, u c is continuous and a viscosity solution of (1.4).