The first initial-boundary value problem of parabolic Monge–Ampère equations outside a bowl-shaped domain

In this paper, we study the parabolic Monge–Ampère equations −utdet(D2u)=g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-u_{t}\det (D^{2}u)=g$\end{document} outside a bowl-shaped domain with g being the perturbation of g0(|x|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{0}(|x|)$\end{document} at infinity. Under the weaker conditions compared with the problem outside a cylinder, we obtain the existence and uniqueness of viscosity solutions with asymptotic behavior for the first initial-boundary value problem by using the Perron method.


Introduction
Monge-Ampère equation is a class of fully nonlinear partial differential equations. The Dirichlet problem of elliptic Monge-Ampère equations on exterior domains is closely related to a celebrated result of Jörgens (n = 2 [1]), Calabi (n ≤ 5 [2]), and Pogorelov (n ≥ 2 [3]). It asserts that any classical convex solution of elliptic Monge-Ampère equation det D 2 u = 1 in R n must be a quadratic polynomial. A simpler and more analytical proof was given by Cheng and Yau [4]. Caffarelli [5] proved that this result holds true for viscosity solutions. Then the result was extended to the Dirichlet problem of elliptic Monge-Ampère equation on exterior domains by Caffarelli and Li in [6] where the existence and uniqueness of the viscosity solutions were proved by the Perron method. Other results for elliptic Monge-Ampère equations on exterior domains can be referred to [7][8][9][10][11] and the references therein. The blow-up solutions to the Monge-Ampère equation and convex solutions of the Monge-Ampère systems can be referred to [12,13].
The Jörgens-Calabi-Pogorelov theorem for parabolic Monge-Ampère equation -u t det D 2 u = 1 in R n × (-∞, 0] (1.1) -d 1 ≤ u t (x, t) ≤ -d 2 , (x, t) ∈ R n × (-∞, 0], then u must be the form u(x, t) = Ct + P(x) with C < 0 and P being a convex quadratic polynomial. Then the Jörgens-Calabi-Pogorelov parabolic theorem was generalized to the equation u t = ρ(logdet D 2 u) with ρ = ρ(z) ∈ C 2 (R) by Xiong and Bao [15], the equation u t -logdet D 2 u = f by Wang and Bao [16], and the equation -u t det D 2 u = f by Zhang, Bao, and Wang [17]. In [18], the author, using the Perron method, studied the first initial-boundary value problem for parabolic Monge-Ampère equation outside a cylinder -u t det D 2 u = g in R n \ × (0,T], (1.2) (1.4) where u = u(x, t), x ∈ R n , t ∈ R, u t = ∂u/∂t, D 2 u is the Hessian matrix of u with respect to the spatial variables x,T > 0 and is a smooth, bounded, and strictly convex open subset in R n , g = g(x, t) = 1 + O(|x| -α ), |x| → ∞ with α > 2, φ(x, t) and ψ(x) are given continuous functions satisfying the compatibility condition. The existence and uniqueness of viscosity solutions with asymptotic behavior at infinity to (1.2)-(1.4) were obtained. The first initialboundary value problems of parabolic Monge-Ampère equations u t = ρ(logdet D 2 u) and u t -logdet D 2 u = f on exterior domains were also studied in [19][20][21]. Recently, the author and Bao [22] obtained the existence of entire solutions of the Cauchy problem for parabolic Monge-Ampère equations -u t det D 2 u = g with g = g 0 (|x|) + O(|x| -α ) at infinity. This kind of first initial-boundary value problem (1.2)-(1.4) on exterior domains is motivated by the interior problem of parabolic Monge-Ampère equations [23,24] In this paper, we study the parabolic Monge-Ampère equations -u t det D 2 u = g(x, t) with g = g 0 (|x|) + O(|x| -α ) (see the following details for g 0 and α) outside a bowl-shaped domain.
Let D ⊂ R n+1 be a bounded domain and t ∈ R, define Set t 0 = inf{t : D(t) = ∅}. The parabolic boundary of D is defined by where D denotes the closure of D and ∂D(t) denotes the boundary of D(t). The side boundary of D is defined by SD = t∈R (∂D(t) × {t}). The set D ⊂ R n+1 is called a bowl-shaped domain if for each t, D(t) is convex and for t 1 ≤ t 2 , D(t 1 ) ⊂ D(t 2 ). One can also refer to [14]. Let D be a bowl-shaped domain and T = sup{t : In the following, we shall abuse the notations SD and ∂D(t) × [t 0 , T].
We shall consider the first initial-boundary value problem of parabolic Monge-Ampère equations LetD ⊂ R n+1 , if for (x, t) ∈D a function u is 2kth continuous differentiable with spatial variables x ∈ R n and kth continuous differentiable with time variable t, we say that u ∈ C 2k,k (D). Let USC(D) and LSC(D) be the sets of upper and lower semicontinuous realvalued functions onD, respectively. We say that a function u ∈ USC(D) (or LSC(D)) is parabolically convex if u is convex in x and nonincreasing in t. The following definition of viscosity solutions is referred to [25].
is locally parabolically convex. We say that u is a viscosity subsolution (supersolution) of (1.5) if for any func- For the supersolution, we also need that D 2 ϕ(x, t) > 0 in the matrix sense.
u ∈ C 0 (R n+1 T \D) is a viscosity solution of (1.5) if it is both a viscosity subsolution and supersolution of (1.5). Definition 1. 2 We say that u ∈ USC(R n+1 T \D) (LSC(R n+1 T \D)) is a viscosity subsolution (supersolution) of problem (1.5)-(1.7) if u is a viscosity subsolution (supersolution) of We assume that g and ψ satisfy the following assumptions: and for the constant α > 0, is the solution of elliptic Monge-Ampère equations Our main result is as follows.

Theorem 1.1 Let D be a bowl-shaped domain in R n+1
, n ≥ 3, and SD be smooth and strictly convex. Assume that g and ψ satisfy (G) and ( ) respectively and φ ∈ C 2,1 (D), φ is decreasing in t. Then, for the b ∈ R n and the constant c in (1.9) and (1.10), there exists a unique viscosity solution u So we extend the previous results [18][19][20] from g ≡ 1 or g = 1 + O(|x| -α ) to g = g 0 (|x|) + O(|x| -α ). Moreover, in the Dirichlet problem of elliptic Monge-Ampère equations on exterior domains, an important lemma (Lemma 5.1 [6]) is used to construct the viscosity subsolutions with asymptotic behavior. Similarly, for the parabolic Monge-Ampère equations, a viscosity subsolution with asymptotic behavior is needed to be constructed by an important lemma (Lemma 2.1 [19]) on a cylinder Q = × (0,T] ⊂ R n+1 . To construct the viscosity subsolutions of parabolic Monge-Ampère equations applying the lemma, we added the strong condition φ x i ,t (x, t) = 0 for any x ∈ ∂ , 0 ≤ t ≤T [18][19][20], which is not natural. In this paper, we establish a lemma on a bowl-shaped domain and then we use this lemma to construct the viscosity subsolutions without the strong condition φ x i ,t (x, t) = 0. This paper is arranged as follows. In Sect. 2, we give the important lemma on a bowlshaped domain with which the viscosity subsolution is constructed. Theorem 1.1 is proved in Sect. 3.

An important lemma
Suppose that SD is smooth and strictly convex and (x, t) ∈ C 2,1 (D). Then there exists some constant C 0 , depending only on n, , D, such that, for any and c * is any bounded positive constant. In addition, for some positive constant c 0 and some bounded domain Proof Let (ξ , λ) ∈ SD, and locally has the expansion where D x is the gradient of in x, D 2 is the Hessian matrix of in x, (ξ , t) ∈ D, and where ν(ξ , λ) is the unit internal normal vector of SD at (ξ , λ) andĉ is sufficiently large but bounded positive constant to be determined. Then By a translation, without loss of generality, we can assume that ξ = 0, λ = 0. Then We again rotate the coordinates to have ν(0, 0) as one of the axes. That is, let M be an orthogonal matrix such that Me n+1 = ν(0, 0). Set M(y, γ ) = (x, t), then where C 1 , C 2 are bounded and depend on C, c * and M, C 3 is bounded and depends on c * , C 2,1 (D) , M and diamD. Since SD is strictly convex, then SD can be locally represented by (2.6) Thus, by (2.4), Again by the fact that SD is strictly convex, there exists a constant δ > 0 depending only on D such that Clearly, by (2.6), for sufficiently large but bounded constantĉ, whereĉ depends only on δ, C 2,1 (D) , c * , and M. On the other hand, by (2.7), we have Then, for any (y, γ ) ∈ SD\{(y, ρ(y)) : |y| < δ}, by (2.5), Choosingĉ large enough (depending only on c * , δ, diamD, Therefore, for some positive constant c 0 and some bounded domain D 1 ⊂ R n × [t 0 , T], similar to the above arguments, by translation and rotation of the coordinates, we can chooseĉ sufficiently large but bounded such that (2.2) holds. The lemma is proved.

Remark 2.1 By (2.2), it is easy to see that even if
Then we avoid the bad condition x i ,t (x, t) = 0 for any (x, t) ∈ SD in [18][19][20].

Proof of Theorem 1.1
For the reader's convenience, we first give the following lemmas whose proof can be found in [18,26].
Then w ∈ C 0 (Q 1 ) is parabolically convex and satisfies, in the viscosity sense,

Lemma 3.2 ([26]) Let 1 be an open strictly convex subsets with smooth boundary in
T], and f ∈ C 0 (Q 1 ) be nonnegative. Suppose that S 0 is a nonempty family of subsolutions to the equation
Proof of Theorem 1.1 Through an affine transformation in the x-space and by subtracting a linear function to u, we may assume that b = 0. The proof is divided into six steps.
Step 5. We prove that u is a viscosity solution of (1.5). As the proof of Theorem 1.4 in [21], we can prove that u is a viscosity solution of (1.5).
Step 6. We prove the uniqueness. By the comparison principle, u ≡ v, (x, t) ∈ R n+1 T \D. Theorem 1.1 is proved.