The Initial and Neumann Boundary Value Problem for a Class Parabolic Monge-Ampère Equation

and Applied Analysis 3 3. Comparison Principle and Uniqueness This section is concerned with the uniqueness of the strictly convex classical solution for (1). First of all, we will prove a comparison principle as follows. Lemma 4. Assume that u, v ∈ C(Q T ) and u(⋅, t), v(⋅, t) are all convex for every time t ∈ (0, T]. For some T 0 , T 0 ∈ (0, T), when t ∈ (0, T 0 ], g σ (x, u) = g 1 (x, u), and when t ∈ (T 0 , T], g σ (x, u) = g 2 (x, u). Letg σ ∈ C 2,2 (Ω×R),σ = 1, 2, and (g σ ) z = ∂g σ (x, z)/∂z ≥ 0. Moreover, assume that (1) −?̇? + det1/n(D x u) − g σ (x, u) ≥ −V̇ + det1/n(D x V) − g σ (x, V) in Ω × (0, T], (2) if u > V, then ] > V] on ∂Ω × [0, T], (3) u ≤ V on Ω × {t = 0}, where ] is the inward pointing unit normal of ∂Ω; then u ≤ V in Q T . Proof. Consider − ?̇?+det1/n (D2 x u)−g σ (x, u)−(−?̇?+det1/n (D2 x v)−g σ (x, v))


Introduction
Historically, the study of Monge-Ampère is motivated by the following two problems: Minkowski and Weyl problems.One is of prescribing curvature type, and the other is of embedding type.The development of Monge-Ampère theory in PDE is closely related to that of fully nonlinear equations.Generally speaking, there are two ways to tackle the problems.One is via continuity method involving some appropriate a priori estimates, and the other is weak solution theory.Monge-Ampère equations have many applications.In recent years new applications have been found in affine geometry and optimal transportation problem.
Moreover, we will always assume the following compatibility conditions to be fulfilled on Ω × { = 0}: Elliptic equations of Monge-Ampère type have been explored in [7][8][9][10] by using the continuity method.Some of the techniques used there will be applied in our paper as well.For the parabolic case, Schnürer and Smoczyk [6] consider the flow of a strictly convex hypersurface driven by the Gauss curvature.For the Neumann boundary value problem and for the second boundary value problem, they show that such a flow exists for all times and converges eventually to a solution of the prescribed Gauss curvature equation.Zhou and Lian [11] proved the existence and uniqueness of classical solutions to the third initial and boundary value problem for equation of parabolic Monge-Ampère type of the form −/ + det 1/ ( 2  ) = (, ).In this paper we will consider more general case than [11] under the structure and compatibility conditions analogous to [6] and extend some results in [7] from elliptic case to parabolic case.
The organization of this paper is as follows.In Section 2, we will review some notations, definitions, and results.In Section 3, we will obtain the uniqueness of the strictly convex classical solutions by the comparison principle.In Section 4, we shall prove uniform estimates for | u |.This will be used in Section 5 to derive  0 -estimates. 1 -estimates then follow from [7].In Section 6, we shall derive  2 -estimates and the  2+,1+/2 -estimates.In Section 7, we will give the proof of Theorem 1.
Our main result is as follows.
Proof.Uniqueness of the strictly convex classical solution is given by Theorem 5. From the estimates obtained in Sections 4-6, we get the existence and the asymptotic behavior of the classical solution in Section 7.

Review of Some Notations, Definitions, and Results
We first review some notations and definitions as follows: is the -dimensional Euclidean space with  ≥ 2; Ω is a bounded, uniformly convex domain in   , and Ω denotes the boundary of Ω; Indices  and   denote partial derivatives with respect to the argument used for the function  and for its gradient, respectively.This paper adopts the Einstein summation convention and sums over repeated Latin indices from 1 to .For example,   V  means ∑  =1   V  .We will say "a constant  under control" or "a controllable constant " if the constant  (independent of ) depends only on the known or estimated quantities, for example, the  4 normal of  0 and -the dimension of   .We point out that the inequalities remain valid when  is enlarged.Now, we state existence results.

Comparison Principle and Uniqueness
This section is concerned with the uniqueness of the strictly convex classical solution for (1).First of all, we will prove a comparison principle as follows.
Theorem 5.Under the assumptions of Theorem 1, there exists a unique classical solution of (1).

u -Estimates
The proof of the u -estimates can be carried out as in [6].For a constant  we define the function  =   ( u ) 2 ; thus So (1) implies the following evolution equation for : Theorem 6.As long as a strictly convex solution of (1) exists, one obtains the estimates where  is a controllable constant.
Proof.If ( u ) 2 admits a positive local maximum in  ∈ Ω for a positive time, then we differentiate the Neumann boundary condition and obtain from (2) that which contradicts the maximality of ( u ) 2 at .Now we choose  = 0 in ( 16) and get Since det  2   > 0, (  )  ≥ 0, we obtain from the parabolic maximum principle that max From the aforementioned a positive local maximum of ( u ) 2 cannot occur at a point of Ω for a positive time, so From the fact that the solution is smooth up to the initial time  = 0, we get By ( 21) and ( 22), there exists a controllable constant  such that | u | 0,  ≤ .Here we have used the fact that  0 ∈  4 (Ω).
Proof.We use the methods known from [6].Differentiating the equation yields From ( 24) and parabolic maximum principle, we see that inf where ( u ) − = min{ u , 0}.
If u admits a negative local minimum in  ∈ Ω for a positive time, then we differentiate the Neumann boundary condition and get from (2) that which contradicts the minimum of ( u ) at .Since 0 ≤ u (, 0), it follows that inf So u ≥ 0 or equivalently det 1/ ( 2  ) −   (, ) ≥ 0 for  = 1, 2 and  > 0.

𝐶 0 -and 𝐶 1 -Estimates
In this section we derive the  0 -and  1 -estimates of the solution to problem (1).
Then we can deduce that (, ) > 0 for all  ∈ Ω and  >  2 , where  2 is controllable constant.Combining (30) yields This completes the proof of the theorem.
Since  0 is arbitrary, we obtain that where  * is a controllable constant.This completes the proof of the theorem.

𝐶
where   is a controllable constant.
Proof.Let  ∈  −1 .First we observe that    > 0, since  is strictly convex.So we only need to prove the fact that    is a priori bounded from above.We define for (, , ) ∈ Ω × [0, ] ×  −1 that  (, , ) =    −  (, , ) + || 2 , (37 where (, , ) is given by Here ] is a smooth extension of the inner unit normal on Ω that is independent of .  is given by is a constant to be chosen, and  indicates that the chain rule has not yet been applied to the respective terms.Let in the th coordinate direction, we obtain From (  ) = ( 2  )/  = (( 2  )/)  , where (  ) is the inverse of (  ), we have Using the estimates of u and , we obtain that ( 2  ) = u +  is bounded.From as well as  0 -and  1 -estimates, it follows that |    | is bounded.Thus there exists a controllable constant  such that Since (  ) is positive definite, we can get that Applying (52),  1 -estimates, and the following equality: where  1 and  2 are positive controllable constants.From (51), (54), and the estimates like these, it follows that where  1 and  2 are positive controllable constants.Then using ( 46) and (48), we can obtain where we have used the structure condition (3) and the convexity of .Using  0 -and  1 -estimates, there exist positive controllable constants  3 and  4 such that Since tr(  ) ≥ 1, we fix  ≥ (1/2)( 3 +  4 + 1) and deduce that Thus by the parabolic maximum principle, we have As  is known on Ω × { = 0} ×  −1 , we need only to estimate  on Ω × [0, ] ×  −1 .
The estimation of  on Ω × [0, ] ×  −1 splits into four stages according to the direction .The first three stages: (i) the mixed tangential normal second derivatives of  on Ω × [0, ] ×  −1 , (ii)  tangential, and (iii)  nontangential, can be carried out as in [7].The details of this procedure could be seen in [7].Stage (i) is readily estimated.Stages (ii) and (iii) are reduced to the purely normal case.So we only give the proof of the fourth stage: (iv)  normal.We extend the argument given in [2] and modified for the parabolic case.
We consider the auxiliary function in where  1 is a positive constant to be determined.If we choose  1 sufficiently large, it is easy to see that (, ) ≥ ℎ(, ) on     .For sufficiently large  1 , we have where we have used the fact that (  ) is positive definite.
Combining the estimates of the four stages, we obtain that there exists a controllable constant  such that    ≤  on   .
From the uniform  0 -estimates, u -estimates, and the assumptions on   ,  = 0, 1, we can conclude that ( 2 ) has a priori positive bound from below.And using the uniform  2 -estimates for , we obtain that (1) is uniformly parabolic.So we can apply the method of [14] to obtain the  2+,1+/2 interior estimates and the estimates near the bottom.Using the estimates near the side in [15], we can get the Hölder seminorm estimates for u and  2  .Thus we have the  2+,1+/2 -estimates.

The Proof of Theorem 1
In Section 3 we proved the uniqueness of the strictly convex solution for (1).The existence of the strictly convex solution for (1) is obtained by using the continuity method.Applying Theorem 5.3 in [16], the implicit function theorem, and the Arzela-Ascoli theorem, we can get the desired result.Then the standard regularity of parabolic equation implies that  ∈  4+,2+/2 (  ).Since there are sufficient a priori estimates, we can extend a solution of (1) on a time interval [0, ] to [0,  + ) for a small  > 0. In this way we obtain existence for all  ≥ 0 from the a priori estimates.We then need the following lemma to prove the asymptotic behavior of a classical solution of (1).
Lemma 11.If a solution of (1) exists for all  ≥ 0 and (4) is satisfied, then as  → ∞, the functions |  converge to a limit function  ∞ () such that  ∞ () satisfies the Neumann boundary value problem where ] is the unit inner normal on Ω.Moreover, (, ) →  ∞ () in  3 -norm.
Proof.We may assume that u (⋅, 0) ̸ ≡ 0 and proceed as in [17].Integrating the equation (74) The left-hand side is uniformly bounded in view of the  0 -estimates.By applying Lemma 7, det for ũ =  −  ∞ , where ‖ ⋅ ‖ denotes the sup-norm.Dini's theorem and interpolation inequalities of the form (77) yield (, ) →  ∞ () in  3 -norm.We finally, obtain in view of (75) that  ∞ is a solution of the problem (72).This complete, the proof of the lemma.Now we completed the proof of Theorem 1.