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

Authors: Juan Wang; Jinlin Yang; Xinzhi Liu
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class nonlinear parabolic equation of Monge-Ampère type. We show that such solution exists for all times and is unique. It converges eventually to a solution that satisfies a Neumann type problem for nonlinear elliptic equation of Monge-Ampère type.


---

## Body

## 1. 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.Many scholars have studied this kind of equations (see, e.g., [1–5] and the references given therein). Their main work is directed at the first or the third boundary value problem. But concerning Neumann boundary value problem, there is lack of research. In this paper, we consider the existence, uniqueness, and asymptotic behavior of a classical solution to the initial and Neumann boundary value problem for a class parabolic equation of Monge-Ampère type as follows:
(1)u˙=det1/n(Dx2u)-gσ(x,u)inΩ×(0,T],uν=φ(x,u)on∂Ω×[0,T],u|t=0=u0inΩ,
where u˙=∂u/∂t and Ω is a bounded, uniformly convex domain in Rn with the boundary ∂Ω∈C4+α. ν denotes the unit inner normal on ∂Ω which has been extended on QT¯ to become a properly smooth vector field independent of t. For some T0, T0∈(0,T), when t∈(0,T0], gσ(x,u)=g1(x,u), and when t∈(T0,T], gσ(x,u)=g2(x,u). The function gσ∈C2+α,2+α(Ω¯×R), σ=1,2. For each x∈Ω,  limt→T0+g2(x,u(x,t))=g1(x,u(x,T0)). Here φ∈C3+α,3+α(Ω¯×R) and  u0∈C4+α(Ω¯). The initial value u0 is a strictly convex function on Ω¯. In the sequel we assume for simplicity that 0∈Ω.To guarantee the existence of the classical solutions for (1) and convergence to a solution with prescribed curvature, we have to assume several structure conditions analogous to [6]. These are
(2)φz≡∂φ(x,z)∂z≥cφ>0,(3)gσ>0,(gσ)z≡∂gσ(x,z)∂z≥0,forσ=1,2,(4)det1/n(Dx2u0)-g(x,u0)≥0.Moreover, we will always assume the following compatibility conditions to be fulfilled on∂Ω×{t=0}:
(5)(u0)ν=φ(x,u0),(det1/n(Dx2u0)-g1(x,u0))ν=φz(x,u0)(det1/n(Dx2u0)-g1(x,u0)).Elliptic  equations of Monge-Ampère type have been explored in [7–10] 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 -∂u/∂t+det1/n(Dx2u)=f(x,t). 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 Section2, 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 C0-estimates. C1-estimates then follow from [7]. In Section 6, we shall derive C2-estimates and the C2+β,1+β/2-estimates. In Section 7, we will give the proof of Theorem 1.Our main result is as follows.Theorem 1 (the main theorem).
Assume thatΩ is a bounded, uniformly convex domain in Rn with the boundary ∂Ω∈C4+α. ν denotes the unit inner normal on ∂Ω which has been extended on QT¯ to become a properly smooth vector field independent of t. Let gσ∈C2+α,2+α(Ω¯×R), σ=1,2, and φ∈C3+α,3+α(Ω¯×R) that satisfy (2)-(3). Let u0∈C4+α(Ω¯) be a strictly convex function that satisfies (4). Moreover, the compatibility conditions (5) are fulfilled. Then there exists a unique strictly convex solution of (1) in K4+α for some α∈(0,1), where
(6)K4+α:={v(x,t)∣v(x,t)∈C2,1(QT)∩C1,0(QT¯)andv(·,t)isstrictlyconvexforeverytimet∈[0,T]C1,0(QT¯)}∩C4+α,2+α/2(QT¯),QT=Ω×(0,T].
As t→∞, the functions u|t converge to a limit function u∞ such that u∞ is of class C4(Ω¯) and satisfies the Neumann boundary value problem
(7)det1/n(Dx2u∞)=g2(x,u∞)inΩuν∞(x)=φ(x,u∞)on∂Ω,
where ν is the inward ponting unit normal  of ∂Ω.Proof.
Uniqueness of the strictly convex classical solution is given by Theorem5. From the estimates obtained in Sections 4–6, we get the existence and the asymptotic behavior of the classical solution in Section 7.


## 2. Review of Some Notations, Definitions, and Results

We first review some notations and definitions as follows:Rn is the n-dimensional Euclidean space with n≥2;Ω is a bounded, uniformly convex domain in Rn, and ∂Ω denotes the boundary of Ω;QT=Ω×(0,T],  and  ∂PQT denotes the parabolic boundary of QT, ∂PQT=QT¯-QT;u˙=∂u/∂t, u¨=∂2u/∂t2;ui=Diu=∂u/∂xi, Du=(D1u,…,Dnu);|Du|2:=∑i=1n|Diu|2, Dij:=∂2/∂xixj;(uij) denotes the inverse of (uij);tr(Dx2u) denotes the trace of the Hessian matrix (uij);det(Dx2u) denotes the determinant of the Hessian matrix (uij);Cl,k(QT):={u(x,t)∣Dxiu and Dtju are all continuous in QT(0≤i≤l,0≤j≤k)};C2l+α,l+α/2:={u;DβDtru∈Cα,α/2(QT¯), for all β and r that satisfy |β|+2r≤2l} with the norm
(8)|u|2l+α,l+α/2;QT=∑|β|+2r≤2l|DβDtru|α,α/2;QT.Indicesz and pi denote partial derivatives with respect to the argument used for the function u and for its gradient, respectively. This paper adopts the Einstein summation convention and sums over repeated Latin indices from 1 to n. For example, uivi  means  ∑i=1nuivi.  We will say “a constant C under control” or “a controllable constant C” if the constant C (independent of T) depends only on the known or estimated quantities, for example, the C4 normal of u0 and n-the dimension of Rn. We point out that the inequalities remain valid when C is enlarged.Now, we state existence results.Lemma 2 (see [11]).
LetF(Dx2u)=det1/n(Dx2u), Fij(Dx2u)=∂F(Dx2u)/∂uij; then F(Dx2u) is a concave function, (Fij(Dx2u)) is a positive matrix, and tr(Fij(Dx2u))=∑i=1nFii(Dx2u)≥1.Lemma 3 (see [12]).
Iff∈C2([0,1]), then there exists a constant M which is independent of f, such that
(9)∥f′∥≤M∥f∥(∥f∥+∥f′′∥),
where ||f||=max{|f(x)|:0≤x≤1}.


## 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 thatu,v∈C2,1(QT¯) and u(·,t), v(·,t) are all convex for every time t∈(0,T]. For some T0, T0∈(0,T), when t∈(0,T0], gσ(x,u)=g1(x,u), and when t∈(T0,T], gσ(x,u)=g2(x,u). Let gσ∈C2,2(Ω¯×R), σ=1,2, and (gσ)z=∂gσ(x,z)/∂z≥0. Moreover, assume that(1)
-u˙+det1/n(Dx2u)-gσ(x,u)≥-v˙+det1/n(Dx2v)-gσ(x,v)  in  Ω×(0,T],(2)
ifu>v,  then  uν>vν  on  ∂Ω×[0,T],(3)
u≤v on Ω×{t=0},where ν is the inward pointing unit normal of ∂Ω; then u≤v in QT¯.Proof.
Consider(10)-u˙+det1/n(Dx2u)-gσ(x,u)-(-v˙+det1/n(Dx2v)-gσ(x,v))=-(u˙-v˙)+∫01∂det1/n[sDx2u+(1-s)Dx2v]∂(suij+(1-s)vij)ds(u-v)ij-∫01∂gσ(x,su+(1-s)v)∂(su+(1-s)v)ds(u-v)=-(u˙-v˙)+aij(u-v)ij-b(u-v),
whereaij=∫01(∂det1/n[sDx2u+(1-s)Dx2v]/∂(suij+(1-s)vij))ds, b=∫01(∂gσ(x,su+(1-s)v)/∂(su+(1-s)v))ds.
From the assumptions and Lemma2, we obtain that (aij) is a positive matrix and b≥0.
Combining (10) with condition (1), we infer that
(11)-(u˙-v˙)+aij(u-v)ij-b(u-v)≥0inΩ×(0,T];
an application of the weak parabolic maximum principle gives maxQT¯(u-v)≤max∂PQT(u-v)+. In addition, from condition (2), u-v cannot admit a positive maximum on ∂Ω×[0,T]. And from condition (3), u≤v on Ω×{t=0}. So we obtain that u≤v in QT¯.Theorem 5.
Under the assumptions of Theorem1, there exists a unique classical solution of (1).Proof.
Assume thatu,v∈C2,1(QT¯) are two classical solutions of (1). Then we have
(12)u˙=det1/n(Dx2u)-gσ(x,u)inΩ×(0,T],u=u0(x)onΩ×{t=0};
meanwhile,
(13)v˙=det1/n(Dx2v)-gσ(x,v)inΩ×(0,T],v=u0(x)onΩ×{t=0}.
Thus,
(14)-u˙+det1/n(Dx2u)-gσ(x,u)=-v˙+det1/n(Dx2v)-gσ(x,v)inΩ×(0,T],u=vonΩ×{t=0}.
It follows that conditions (1) and (3) in Lemma 4 are satisfied.
Fromuν=φ(x,u)  on ∂Ω×[0,T] and the structure condition (2), we obtain that condition (2) in Lemma 4 is satisfied.
Sincegσ∈C2+α,2+α(Ω¯×R) and the structure condition (3) is satisfied, we obtained from Lemma 4 that u=v for all (x,t)∈QT¯.


## 4.u˙-Estimates

The proof of theu˙-estimates can be carried out as in [6]. For a constant λ we define the function r=eλt(u˙)2; thus
(15)r˙=λeλt(u˙)2+2eλtu˙u¨=λr+2eλtu˙ddt(det1/n(Dx2u)-gσ(x,u))=λr+2neλtu˙u˙ijuij·det1/n(Dx2u)-2r(gσ)z=λr+1n(rij-2eλtu˙iu˙j)uij·det1/n(Dx2u)-2r(gσ)z=1ndet1/n(Dx2u)uijrij-2neλtdet1/n(Dx2u)uiju˙iu˙j+(λ-2(gσ)z)r.So (1) implies the following evolution equation for r:
(16)r˙=1ndet1/n(Dx2u)uijrij-2neλtdet1/n(Dx2u)uiju˙iu˙j+(λ-2(gσ)z)r.Theorem 6.
As long as a strictly convex solution of (1) exists, one obtains the estimates
(17)|u˙|0,Q¯T≤M¯,
where M¯ is a controllable constant.Proof.
If(u˙)2 admits a positive local maximum in x∈∂Ω for a positive time, then we differentiate the Neumann boundary condition and obtain from (2) that
(18)((u˙)2)ν=2u˙(u˙)ν=2u˙(u˙ν)=2(u˙)2φz>0
which contradicts the maximality of (u˙)2 at x.
Now we chooseλ=0 in (16) and get
(19)d(u˙)2dt=1nuijdet1/n(Dx2u)((u˙)2)ij-2nuijdet1/n(Dx2u)u˙iu˙i-2(gσ)z(u˙)2≤1ndet1/n(Dx2u)uij((u˙)2)ij-2(gσ)z(u˙)2.
Since detDx2u>0, (gσ)z≥0, we obtain from the parabolic maximum principle that
(20)maxQT¯(u˙)2≤max∂PQT((u˙)2)+.
From the aforementioned a positive local maximum of (u˙)2 cannot occur at a point of ∂Ω for a positive time, so
(21)(u˙)2≤maxt=0(u˙)2⟹|u˙|≤maxt=0|u˙|.
From the fact that the solution is smooth up to the initial time t=0, we get
(22)u˙=det1/n(Dx2u0)-g1(x,u0)onΩ¯×{t=0}.
By (21) and (22), there exists a controllable constant M¯ such that |u˙|0,Q¯T≤M¯. Here we have used the fact that u0∈C4(Ω¯).Lemma 7.
If0≤u˙(x,0)≢0 for t=0, then a solution of (1) satisfies u˙>0 or equivalently det1/n(Dx2u)-gσ(x,u)>0 for σ=1,2 and t>0.Proof.
We use the methods known from [6]. Differentiating the equation
(23)u˙=det1/n(Dx2u)-gσ(x,u)
yields
(24)u¨=1ndet1/n(Dx2u)uiju˙ij-(gσ)zu˙.
From (24) and parabolic maximum principle, we see that
(25)infQT(u˙)≥inf∂PQT(u˙)-,
where (u˙)-=min{u˙,0}.
Ifu˙ admits a negative local minimum in x∈∂Ω for a positive time, then we differentiate the Neumann boundary condition and get from (2) that
(26)(u˙)ν=(u˙ν)=φzu˙<0
which contradicts the minimum of (u˙) at x. Since 0≤u˙(x,0), it follows that inf∂PQT(u˙)-=0. That is,
(27)infQT(u˙)≥inf∂PQT(u˙)-=0.
So u˙≥0 or equivalently det1/n(Dx2u)-gσ(x,u)≥0 for σ=1,2 and t>0.
From (24) and the strong parabolic maximum principle [13], we obtain that u˙ has to vanish identically if it vanishes in (x0,t0)∈Ω×(0,T], contradicting u˙≢0 for t=0. If u˙=0 for x0∈∂Ω, the Neumann boundary condition implies that
(28)(u˙)ν=(u˙)φz=0,
but this is impossible in view of the Hopf lemma applied to (24).
Consequently, if0≤u˙(x,0)≢0 for t=0, then a solution of (1) satisfies u˙>0 or equivalently det1/n(Dx2u)-gσ(x,u)>0 for σ=1,2 and t>0.


## 5.C0- and C1-Estimates

In this section we derive theC0- and C1-estimates of the solution to problem (1).Theorem 8.
LetΩ be a bounded, uniformly convex domain in Rn. Also, u∈C2,1(QT)∩C1,0(QT¯) is a strictly convex solution of (1). Then there exists a controllable constant M0, such that |u|0,QT¯≤M0.Proof.
Since (4) is satisfied, we obtained from Lemma 7 that u˙≥0 in QT¯. So u(x,t)≥u(x,0)=u0(x). As u0(x)∈C4+α(Ω¯), then there exists a controllable constant N1 such that
(29)u(x,t)≥N1inQT¯.
Next we will prove thatu is uniformly a priori bounded from above.
At a maximum ofu, which necessarily occurs on ∂Ω since u is convex, we have uν≤0. Since uν=φ(x,u) on ∂Ω×[0,T], then
(30)φ(x,u)≤0(x,t)∈∂Ω×[0,T].
From (2) we get that φ(·,z)→∞ uniformly as z→∞. Then we can deduce that φ(x,z)>0 for all x∈∂Ω and z>N2, where N2 is controllable constant. Combining (30) yields
(31)u≤N2.
This completes the proof of the theorem.Theorem 9.
LetΩ be a bounded, uniformly convex domain in Rn, and u∈C4,2(QT)∩C1,0(QT¯) is a strictly convex solution of (1). Then one has
(32)supQT|Du|≤M*,
where M* is a controllable constant.Proof.
For anyt0∈[0,T], u(x,t0) is a continuous differentiable, convex function. From uν=φ(x,u) and the C0-estimates, we get
(33)uν≥-M,
where M is a controllable constant. Then using Theorem 2.2 in [7], we have
(34)|Du(x,t0)|≤M1onΩ¯.
Sincet0 is arbitrary, we obtain that
(35)|Du|≤M*,
where M* is a controllable constant. This completes the proof of the theorem.


## 6.C2- and C2+β,1+β/2-Estimates

This section is concerned with theC2-estimates and the C2+β,1+β/2-estimates of the solution to problem (1).Theorem 10.
Assume thatΩ is a C4 bounded, uniformly convex domain in Rn and u∈C4,2(QT¯) is a strictly convex solution of (1). Let gσ∈C2,2(Ω¯×R), σ=1,2, φ∈C3,3(Ω¯×R). Then one has
(36)supQT|Dx2u|≤M′′,
where M′′ is a controllable constant.Proof.
Letξ∈Sn-1. First we observe that Dξξu>0, since u is strictly convex. So we only need to prove the fact that  Dξξu is a priori bounded from above.
We define for(x,t,ξ)∈Ω¯×[0,T]×Sn-1 that
(37)ω(x,t,ξ)=Dξξu-V(x,t,ξ)+K|x|2,
where V(x,ξ,t) is given by
(38)V(x,ξ,t)=2〈ξ,ν〉ξi′(Diφ-DkuDiνk).
Here ν is a smooth extension of the inner unit normal on ∂Ω that is independent of t. ξ′ is given by
(39)ξ′=ξ-〈ξ,ν〉ν,K is a constant to be chosen, and D indicates that the chain rule has not yet been applied to the respective terms.
Let(40)ak=2〈ξ,ν〉(φzξk′-ξi′Diνk),b=2〈ξ,ν〉ξi′φi,
then
(41)V(x,ξ,t)=akuk+b,ω(x,t,ξ)=Dξξu-akuk-b+K|x|2.
We compute that
(42)-∂ω∂t+FijDijω=-Dξξtu+akDtku+Dtak·uk+Dtb+FijDijξξu-akFij·uijk-2FijDiak·ukj-Fij·ukDijak-FijDijb+2KFijδij.
Next, we estimate the right-hand side of (42), respectively.
LetF(Dx2u)=det1/n(Dx2u). From Lemma 2, we have that F(Dx2u) is a concave function, (Fij(Dx2u)) is a positive matrix, and tr(Fij(Dx2u))=∑i=1nFii(Dx2u)≥1.
Differentiating the equation(43)u˙=F(Dx2u)-gσ(x,u)
twice in the direction ξ,  ξ∈Sn-1, we therefore obtain
(44)-Dtξξu+FijDijξξu+Fij,klDijξuDklξu=Dξξgσ(x,u).
Using the concavity of F, we have
(45)Fij,klDijξuDklξu≤0;
then
(46)-Dtξξu+FijDijξξu≥Dξξgσ(x,u).
Differentiating the equation(47)u˙=F(Dx2u)-gσ(x,u)
in the kth coordinate direction, we obtain
(48)-Dtku+FijDijku=Dkgσ.
From (Fij)=∂F(Dx2u)/∂uij=(F(Dx2u)/n)uij, where (uij) is the inverse of (uij), we have
(49)FijDiak·ukj=F(Dx2u)nuijukj·Diak=F(Dx2u)nDiai.
Using the estimates of u˙ and u, we obtain that F(Dx2u)=u˙+gσ is bounded. From
(50)Djaj=2ξlDjνl(φzξj′-ξi′Diνj)+2〈ξ,ν〉[(φzj+φzzDju)ξj′-ξi′Djiνj],
as well as C0- and C1-estimates, it follows that |Djaj| is bounded. Thus there exists a controllable constant C such that
(51)|FijDiak·ukj|≤C.
Since(Fij) is positive definite, we can get that
(52)|Fij|≤12(Fii+Fjj).
Applying (52), C1-estimates, and the following equality:
(53)Djhak=2ξlDjhνl(φzξk′-ξi′Diνk)+2ξlDjνl[(φzh+φzzuh)ξk′-ξi′Dihνk]+2ξsDhνs[(φzj+φzzuj)ξk′-ξi′Dijνk]+2〈ξ,ν〉{+φzzujh]ξk′-ξi′Dijhνk[(φzjh+φzjzuh)+(φzzh+φzzzuh)uj+φzzujh]ξk′-ξi′Dijhνk},
we obtain that
(54)|Fhj·ukDhjak|≤|ukFhj2〈ξ,ν〉φzzujhξk′|+C|Fhj|=|2uk〈ξ,ν〉φzzξk′F(Dx2u)nuhjujh|+C|Fhj|≤C1+C2tr(Fij),
where C1 and C2 are positive controllable constants.
From (51), (54), and the estimates like these, it follows that
(55)|Dtak·uk+Dtb-2FijDiak·ukj-Fij·ukDijak-FijDijb|≤c1tr(Fij)+c2,
where c1 and c2 are positive controllable constants. Then using (46) and (48), we can obtain
(56)-∂ω∂t+FijDijω≥Dξξgσ-akDkgσ-(c1tr(Fij)+c2)+2KFijδij=(gσ)ξξ+2(gσ)ξzuξ+(gσ)zzuξuξ+(gσ)zuξξ-ak((gσ)k+(gσ)zuk)-(c1tr(Fij)+c2)+2KFijδij≥(gσ)ξξ+2(gσ)ξzuξ+(gσ)zzuξuξ-ak((gσ)k+(gσ)zuk)-(c1tr(Fij)+c2)+2KFijδij,
where we have used the structure condition (3) and the convexity of u. Using C0- and C1-estimates, there exist positive controllable constants c3 and c4 such that
(57)-∂ω∂t+FijDijω≥(2K-c3)tr(Fij)-c4.
Sincetr(Fij)≥1, we fix K≥(1/2)(c3+c4+1) and deduce that
(58)-∂ω∂t+FijDijω≥1.
Thus by the parabolic maximum principle, we have
(59)ω≤sup∂PQTω.
As ω is known on Ω¯×{t=0}×Sn-1, we need only to estimate ω on ∂Ω×[0,T]×Sn-1.
The estimation ofω on ∂Ω×[0,T]×Sn-1 splits into four stages according to the direction ξ. The first three stages: (i) the mixed tangential normal second derivatives of u on ∂Ω×[0,T]×Sn-1, (ii) ξ tangential, and (iii) ξ 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) ξ normal. We extend the argument given in [2] and modified for the parabolic case.
Seth(x,t)=νkDku-φ(x,u)=Dνu-φ(x,u). By (48), a direct calculation yields
(60)Lh=-Dth+FijDijh=-νkDktu+φzDtu+νkFijDijku+2FijDiνkDjku+FijDijνkDku-FijDijφ=νk((gσ)k+(gσ)zuk)+φzDtu+2FijDiνkDjku+FijDijνkDku-Fij(φij+2φizuj+φzzuiuj+φzuij).
Thus, using (Fij)=(F(Dx2u)/n)uij, (52), and our a priori estimates, we have
(61)|Lh|=|-Dth+FijDijh|≤C0(1+tr(Fij))≤Ctr(Fij),
where C is a controllable constant.
Let(x0,t0)∈∂Ω×[0,T], and (x0,t0) is arbitrary. We observe that Ω is a bounded, uniformly convex domain in Rn, so there exists a uniformly closed ball BR(x*) such that
(62)Ω⊂BR(x*)⊂Rn,∂BR(x*)∩∂Ω={x0}.
Meanwhile, we assume that |x-x*|>1 for all x∈Ω-.
We consider the auxiliary function inBR(x*)×[0,T](63)q(x,t)=eK1R2-eK1|x-x*|2,
where K1 is a positive constant to be determined.
If we chooseK1 sufficiently large, it is easy to see that q(x,t)≥h(x,t) on ∂PQT. For sufficiently large K1, we have
(64)-∂q∂t+FijDijq=-2K1tr(Fij)eK1|x-x*|2-4K12Fij(x-x*)i(x-x*)jeK1|x-x*|2≤-2K1tr(Fij)eK1|x-x*|2≤-Ctr(Fij),
where we have used the fact that (Fij) is positive definite.
By (61) and (64), we get
(65)-∂(q-h)∂t+FijDij(q-h)≤0;
thus we obtain q-h≥inf∂PQT(q-h)≥0 on QT¯ in view of the parabolic maximum principle. Since q(x0,t0)=h(x0,t0)=0, it follows that
(66)(q-h)(x0+ρν)-(q-h)(x0,t0)ρ≥0
thus
(67)limρ→0+(q-h)(x0+ρν)-(q-h)(x0,t0)ρ≥0.
Therefore,
(68)Dν(q-h)(x0,t0)≥0.
Hence,
(69)hν(x0,t0)≤qν(x0,t0)≤c1,
where c1 is a controllable constant.
For-q, in a similar fashion we can obtain
(70)hν(x0,t0)≥-c2,
where c2 is a controllable constant.
Since(x0,t0)∈∂Ω×[0,T] is arbitrary, we obtain
(71)sup∂Ω×[0,T]Dννu≤C,
where C is a controllable constant.
Combining the estimates of the four stages, we obtain that there exists a controllable constantC such that Dξξu≤C on QT¯.
Sinceξ is an arbitrary direction in Sn-1,  now let ξ=ei±ej/21/2,  where ei=(0,0…,1,…0)=ith  standard coordinate vector. Thus we can get the required bounded for Dx2u immediately.  This completes the proof of the theorem.From the uniformC0-estimates, u˙-estimates, and the assumptions on gσ, σ=0,1, we can conclude that F(D2u) has a priori positive bound from below. And using the uniform C2-estimates for u, we obtain that (1) is uniformly parabolic. So we can apply the method of [14] to obtain the C2+β,1+β/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 Dx2u. Thus we have the C2+β,1+β/2-estimates.


## 7. The Proof of Theorem1

In Section3 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 u∈C4+β,2+β/2(QT¯). Since there are sufficient a priori estimates, we can extend a solution of (1) on a time interval [0,T] to [0,T+ϵ) for a small ϵ>0. In this way we obtain existence for all t≥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 t≥0 and (4) is satisfied, then as t→∞, the functions u|t converge to a limit function u∞(x) such that u∞(x) satisfies the Neumann boundary value problem
(72)det1/n(Dx2u∞)=g2(x,u∞)x∈Ω,uν∞=φ(x,u∞)x∈∂Ω,
where ν is the unit inner normal on ∂Ω. Moreover, u(x,t)→u∞(x) in C3-norm.Proof.
We may assume thatu˙(·,0)≢0 and proceed as in [17]. Integrating the equation
(73)u˙=det1/n(Dx2u)-g2(x,u)
with respect to t yields
(74)u(x,t)-u(x,T0)=∫T0t(det1/n(Dx2u(x,τ))-g2(x,u(x,τ)))dτ.
The left-hand side is uniformly bounded in view of the C0-estimates. By applying Lemma 7, det1/n(Dx2u)-g2 is nonnegative, and we can find that tk=tk(x)→∞ such that
(75)(det1/n(Dx2u)-g2)|t=tk⟶0.
On the other hand,u(x,·) is monotone, and therefore
(76)limt→∞u(x,t)=:u∞(x)
exists and is of class C4(Ω-) in view of the a priori estimates.
From differential interpolation inequality in Lemma3, we can obtain the interpolation inequality of the form
(77)∥Du~∥≤C∥u~∥·(∥Dx2u~∥+∥u~∥)
for u~=u-u∞, where ∥·∥ denotes the sup-norm.
Dini’s theorem and interpolation inequalities of the form (77) yield u(x,t)→u∞(x) in C3-norm. We finally, obtain in view of (75) that u∞ is a solution of the problem (72). This complete, the proof of the lemma.
Now we completed the proof of Theorem1.



---

*Source: 535629-2013-07-21.xml*