Abstract
In this paper, we prove that the global existence of solutions to timelike minimal surface equations having arbitrary codimension with slow decay initial data in two space dimensions and three space dimensions, provided that the initial value is suitably small.
MSC: 35L70.
Keywords:
timelike minimal surface; global existence; slow decay initial value1 Introduction
The theory of minimal surfaces has a long history, originating with the papers of Lagrange (1760) and the famous Plateau problem; we refer to the classical papers by Calabi [1] and by Cheng and Yau [2]. Timelike minimal submanifolds may be viewed as simple but nontrivial examples of Dbranes, which play an important role in string theory, and the system under consideration here thus has natural generalizations motivated by string theory. The case of timelike surfaces has been investigated by several authors (see [35] and [6]). Huang and Kong [7] studied the motion of a relativistic torus in the Minkowski space (). They derived the equations for the motion of relativistic torus in the Minkowski space (). This kind of equation also describes the three dimensional timelike extremal submanifolds in the Minkowski space . They showed that these equations can be reduced to a dimensional quasilinear symmetric hyperbolic system and the system possesses some interesting properties, such as nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, and the strong null condition (see [8] and [9]). They also found and proved the interesting fact that all plane wave solutions to these equations are lightlike extremal submanifolds and vice versa, except for a type of special solution. For small initial data with compact support, the global existence problem for timelike minimal hypersurfaces has been considered by Brendle [10] and Lindblad [11].
Paul et al.[12] investigated timelike minimal submanifolds of dimension , , of Minkowski spacetimes of dimension , . The authors considered an embedding of into Minkowski spacetime given by the graph of a map . Let Greek indices take values in and let uppercase Latin indices take values in . Introduce cartesian coordinates on and on . The induced metric is
where , and is the Minkowski metric. By variational principles (see [13]), they derived the EulerLagrange equations
Moreover for a small initial value with compact support, they also proved the global existence of classical solutions for (1.2).
In this paper, we consider (1.2) with the initial data
where is a constant and is a small parameter. The aim of this paper is to prove that the Cauchy problem (1.2), (1.3) has a global classical solution, provided that the initial value is sufficiently small and satisfy , (, ). We reduce the restriction on compact support of the initial data to some decay. In other words, we show the global existence of solutions to timelike minimal surface in two space dimensions and three space dimensions, provided that the initial value is suitably small.
To study (1.2), we note that (1.2) can be written in divergence form
where is the Minkowski wave operator and , as well as in the form
where
We raise and lower Greek (intrinsic) indices using and its inverse, while Latin (extrinsic) indices are raised and lowered using the identity and its inverse. From (1.6), it follows that has the symmetries
Due to the symmetries, an energy estimate and local well posedness holds for the system (1.5).
The plan of this paper is as follows. In Section 2, we cite some estimates and prove some estimates on the solution of linear wave equations. The global existence of solutions to timelike minimal surface equations with slow decay initial value in two space dimensions and three space dimensions will be proved in Section 3 and Section 4, respectively.
2 Preliminaries
Following Klainerman [14], we introduce a set of partial differential operators
where
and
denotes a product of of the vector fields (2.2), (2.3), (2.4), and (2.5). is a multiindex, , σ is the number of partial differential operators in and
It is easy to prove Lemma 2.1 (see [15]).
Lemma 2.1For any multiindex, we have
and
wherestands for the Poisson bracket, are multiindices, □ is the wave operator, and, , andare constants.
We need the following lemma that is basically established in [16] and [17]. For completeness, the proof will also be sketched here.
Assume thatis a solution to the following Cauchy problem:
Then we have
Remark 2.1 Under the condition that
Tsutaya [18] has showed that the solution of the Cauchy problem (2.9) satisfies
Obviously, Lemma 2.2 improves the result in [18].
Proof The solution of (2.9) is given
First, we make an estimate for ; switching to polar coordinates, we have
where and , and χ is the characteristic function of positive numbers.
Let be a continuous function on and . Define
and
where, as before, φ is given by
We will use the following proposition, which is proved in Kovalyov [19].
Proposition 2.1 (I) Ifand, thensatisfies
whereχis the characteristic function of positive numbers.
We next continue to make an estimate for (2.12); we make an estimate for the righthand side of (2.12) by dividing into two cases.
By (2.14), we get
We subdivide into three cases again.
Note that
and
Thus,
From (2.15), we have
It follows from (2.15) that
Hence
In other words, if and , from (2.16)(2.18), we get
In what follows, we prove that (2.19) also holds if and . In this case, we also subdivide into two cases.
By changing variables, , we obtain
Note that and , thus we obtain . From Cases (i)(iii), we get
In other words, when , from (2.19)(2.21), we have
From (2.12), we get
where
and
In what follows, we make an estimate for I and II, respectively, when .
It follows from (2.14) that
By (2.13), we get
In what follows, we make estimate II by dividing into three cases.
For , by changing variables , we obtain
Combining (2.25)(2.27) gives
Thus (2.22) and (2.28) imply that
By Tsutaya [18], we obtain
and
Equation (2.10) follows from (2.29)(2.31), and (2.11) immediately. Then we have completed the proof of lemma. □
The following lemma plays a key role in our main results. It is basically established in [20] and [21].
Assume thatuis a solution to the following Cauchy problem:
Then
and assume thatϕdecays to 0 at infinity. If
For the proof of Lemma 2.4, see Klainerman [22].
Using Lemmas 2.2, 2.3 and the estimate of the linear wave equation with zero initial data, it is not difficulty to prove the following.
Lemma 2.5Suppose that. Letbe the solution to the Cauchy problem
Then
By Lemmas 2.2 and 2.3, we can prove the following lemma.
Lemma 2.6Assume thatandis the solution to the Cauchy problem
where the coefficients () are constants. Then we have
where
3 Global existence in three space dimensions
Theorem 3.1Suppose thatand satisfy
whereis a constant. Then there existssuch that forthe Cauchy problem (1.2), (1.3) has a global classical solution for all.
Proof The local existence argument follows from the method of Picard iteration [23] (see also [24] and [12]). In what follows, we will prove the global existence of the classical solutions by a continuous induction, or a bootstrap argument. Let , we set
and
To set up the bootstrap argument, we assume that there is a positive constant K so that on we have the following estimates for the norms defined in (3.1):
and
To close the bootstrap, we can prove that we can in fact choose K sufficiently large and ε suitably small so that the above inequalities hold independent of T with K replaced by .
From Lemma 2.1 and (1.5), we obtain
It follows from Lemma 2.4 and (3.2), (3.3) for that
if K is sufficiently large and is suitably small.
Note that ; from (1.4) and Lemma 2.1, we have
where again at most one of the can satisfy .
Applying Lemma 2.6 to (3.5), we obtain
if K is sufficiently large and is suitably small.
Since , (3.3) may also be written as
Using Lemma 2.5, (3.2), and (3.7), we get
if K is sufficiently large and is suitably small.
So
if K is sufficiently large and is suitably small.
From (3.1), we know that the estimate for implies the desired estimate for . We have completed the proof of Theorem 3.1. □
4 Global existence in two space dimensions
Theorem 4.1Suppose thatand satisfy
whereis a constant. Then there existssuch that forthe Cauchy problem (1.2), (1.3) has a global classical solution for all.
Proof The local existence argument follows from the method of Picard iteration [23] (see also [24] and [12]). In what follows, we will prove global existence of classical solutions by a continuous induction, or bootstrap argument. Let , we set
and
To set up the bootstrap argument, we assume that there is a positive constant K so that on we have following estimates for the norms defined in (4.1),
and
where is a fixed, arbitrary constant.
To close the bootstrap, we can prove that we can in fact choose K sufficiently large and ε suitably small so that the above inequalities hold independent of T with K replaced by .
It follows from Lemma 2.4 and (3.3) for that
if K is sufficiently large and is suitably small.
Applying Lemma 2.6 to (3.5), we obtain
if K is sufficiently large and is suitably small.
In what follows, we make an estimate for . In order to make this estimate for , define the following null forms:
and
Let Q symbolically stand for any of the full forms (4.5) and (4.6). Then
Let Q be one of null form in (4.5)(4.7), we have
Note that the Lagrangian associated to the volume element of the induced metric is . For small , we have
and thus the EulerLagrange equations take the form
So we have
By Lemma 2.1, (4.9), we have
From Lemma 2.5 and (4.1), (4.2), (4.8), (4.10) when , i.e., , we get
if K is sufficiently large and is suitably small and since .
From (4.1), we know that the estimate for implies the desired estimate for . We have completed the proof of the theorem. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally and read and approved the final manuscript.
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