Abstract and Applied Analysis

Volume 2013 (2013), Article ID 535629, 8 pages

http://dx.doi.org/10.1155/2013/535629

Research Article

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

Juan Wang,¹ Jinlin Yang,¹ and Xinzhi Liu²

¹School of Mathematics, Physics and Biological Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China

²Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

Received 26 March 2013; Accepted 11 June 2013

Academic Editor: Sergey Piskarev

Copyright © 2013 Juan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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:

where and is a bounded, uniformly convex domain in with the boundary . denotes the unit inner normal on which has been extended on to become a properly smooth vector field independent of . For some , , when , , and when , . The function , . For each , . Here and . The initial value is a strictly convex function on . In the sequel we assume for simplicity that .

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

Moreover, we will always assume the following compatibility conditions to be fulfilled on :

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 . 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 . This will be used in Section 5 to derive -estimates. -estimates then follow from [7]. In Section 6, we shall derive -estimates and the -estimates. In Section 7, we will give the proof of Theorem 1.

Our main result is as follows.

Theorem 1 (the main theorem). Assume that is a bounded, uniformly convex domain in with the boundary . denotes the unit inner normal on which has been extended on to become a properly smooth vector field independent of t. Let , , and that satisfy (2)-(3). Let 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 for some , where

As , the functions converge to a limit function such that is of class and satisfies the Neumann boundary value problem

where is the inward ponting unit normal of .

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.

2. Review of Some Notations, Definitions, and Results

We first review some notations and definitions as follows:

  is the -dimensional Euclidean space with ;  is a bounded, uniformly convex domain in , and denotes the boundary of ; , and denotes the parabolic boundary of , ; , ; , ; , ;  denotes the inverse of ;  denotes the trace of the Hessian matrix ;  denotes the determinant of the Hessian matrix ;  and are all continuous in ; , for all and that satisfy with the norm

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, means . 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 normal of and -the dimension of . We point out that the inequalities remain valid when is enlarged.

Now, we state existence results.

Lemma 2 (see [11]). Let , ; then is a concave function, is a positive matrix, and .

Lemma 3 (see [12]). If , then there exists a constant which is independent of , such that

where .

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 and , are all convex for every time . For some , , when , , and when , . Let , , and . Moreover, assume that

(1) in ,(2), then on ,(3) on ,

where is the inward pointing unit normal of ; then in .

Proof. Consider

where , .

From the assumptions and Lemma 2, we obtain that is a positive matrix and .

Combining (10) with condition (1), we infer that

an application of the weak parabolic maximum principle gives . In addition, from condition (2), cannot admit a positive maximum on . And from condition (3), on . So we obtain that in .

Theorem 5. Under the assumptions of Theorem 1, there exists a unique classical solution of (1).

Proof. Assume that are two classical solutions of (1). Then we have

meanwhile,

Thus,

It follows that conditions (1) and (3) in Lemma 4 are satisfied.

From on and the structure condition (2), we obtain that condition (2) in Lemma 4 is satisfied.

Since and the structure condition (3) is satisfied, we obtained from Lemma 4 that for all .

4. -Estimates

The proof of the -estimates can be carried out as in [6]. For a constant we define the function ; 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 admits a positive local maximum in for a positive time, then we differentiate the Neumann boundary condition and obtain from (2) that

which contradicts the maximality of at .

Now we choose in (16) and get

Since , , we obtain from the parabolic maximum principle that

From the aforementioned a positive local maximum of cannot occur at a point of for a positive time, so

From the fact that the solution is smooth up to the initial time , we get

By (21) and (22), there exists a controllable constant such that . Here we have used the fact that .

Lemma 7. If for , then a solution of (1) satisfies or equivalently for and .

Proof. We use the methods known from [6]. Differentiating the equation

yields

From (24) and parabolic maximum principle, we see that

where .

If admits a negative local minimum in for a positive time, then we differentiate the Neumann boundary condition and get from (2) that

which contradicts the minimum of at . Since , it follows that . That is,

So or equivalently for and .

From (24) and the strong parabolic maximum principle [13], we obtain that has to vanish identically if it vanishes in , contradicting for . If for , the Neumann boundary condition implies that

but this is impossible in view of the Hopf lemma applied to (24).

Consequently, if for , then a solution of (1) satisfies or equivalently for and .

5. - and -Estimates

In this section we derive the - and -estimates of the solution to problem (1).

Theorem 8. Let be a bounded, uniformly convex domain in . Also, is a strictly convex solution of (1). Then there exists a controllable constant , such that .

Proof. Since (4) is satisfied, we obtained from Lemma 7 that in . So . As , then there exists a controllable constant such that

Next we will prove that is uniformly a priori bounded from above.

At a maximum of , which necessarily occurs on since is convex, we have . Since on , then

From (2) we get that uniformly as . Then we can deduce that for all and , where is controllable constant. Combining (30) yields

This completes the proof of the theorem.

Theorem 9. Let be a bounded, uniformly convex domain in , and is a strictly convex solution of (1). Then one has

where is a controllable constant.

Proof. For any , is a continuous differentiable, convex function. From and the -estimates, we get

where is a controllable constant. Then using Theorem 2.2 in [7], we have

Since is arbitrary, we obtain that

where is a controllable constant. This completes the proof of the theorem.

6. - and -Estimates

This section is concerned with the -estimates and the -estimates of the solution to problem (1).

Theorem 10. Assume that is a bounded, uniformly convex domain in and is a strictly convex solution of (1). Let , , . Then one has

where is a controllable constant.

Proof. Let . First we observe that , since is strictly convex. So we only need to prove the fact that is a priori bounded from above.

We define for that

where is given by

Here is a smooth extension of the inner unit normal on 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

then

We compute that

Next, we estimate the right-hand side of (42), respectively.

Let . From Lemma 2, we have that is a concave function, is a positive matrix, and .

Differentiating the equation

twice in the direction , , we therefore obtain

Using the concavity of , we have

then

Differentiating the equation

in the th coordinate direction, we obtain

From , where is the inverse of , we have

Using the estimates of and , we obtain that is bounded. From

as well as - and -estimates, it follows that is bounded. Thus there exists a controllable constant such that

Since is positive definite, we can get that

Applying (52), -estimates, and the following equality:

we obtain that

where and are positive controllable constants.

From (51), (54), and the estimates like these, it follows that

where and 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 - and -estimates, there exist positive controllable constants and such that

Since , we fix and deduce that

Thus by the parabolic maximum principle, we have

As is known on , we need only to estimate on .

The estimation of on splits into four stages according to the direction . The first three stages: (i) the mixed tangential normal second derivatives of on , (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.

Set . By (48), a direct calculation yields

Thus, using , (52), and our a priori estimates, we have

where is a controllable constant.

Let , and is arbitrary. We observe that is a bounded, uniformly convex domain in , so there exists a uniformly closed ball such that

Meanwhile, we assume that for all .

We consider the auxiliary function in

where is a positive constant to be determined.

If we choose sufficiently large, it is easy to see that on . For sufficiently large , we have

where we have used the fact that is positive definite.

By (61) and (64), we get

thus we obtain on in view of the parabolic maximum principle. Since , it follows that

thus

Therefore,

Hence,

where is a controllable constant.

For , in a similar fashion we can obtain

where is a controllable constant.

Since is arbitrary, we obtain

where is a controllable constant.

Combining the estimates of the four stages, we obtain that there exists a controllable constant such that on .

Since is an arbitrary direction in , now let , where th standard coordinate vector. Thus we can get the required bounded for immediately. This completes the proof of the theorem.

From the uniform -estimates, -estimates, and the assumptions on , , we can conclude that has a priori positive bound from below. And using the uniform -estimates for , we obtain that (1) is uniformly parabolic. So we can apply the method of [14] to obtain the interior estimates and the estimates near the bottom. Using the estimates near the side in [15], we can get the Hölder seminorm estimates for and . Thus we have the -estimates.

7. 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 . Since there are sufficient a priori estimates, we can extend a solution of (1) on a time interval to for a small . In this way we obtain existence for all 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 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 . Moreover, in -norm.

Proof. We may assume that and proceed as in [17]. Integrating the equation

with respect to yields

The left-hand side is uniformly bounded in view of the -estimates. By applying Lemma 7, is nonnegative, and we can find that such that

On the other hand, is monotone, and therefore

exists and is of class in view of the a priori estimates.

From differential interpolation inequality in Lemma 3, we can obtain the interpolation inequality of the form

for , where denotes the sup-norm.

Dini’s theorem and interpolation inequalities of the form (77) yield in -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.

Acknowledgment

This project is supported by Inner Mongolia Natural Science Foundation of China under Grant 2011MS0107.

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