Global existence of timelike minimal surface of general co-dimension in Minkowski space time

In this paper, we prove that the global existence of solutions to timelike minimal surface equations having arbitrary co-dimension with slow decay initial data in two space dimensions and three space dimensions, provided that the initial value is suitably small. MSC:35L70.


Introduction
The theory of minimal surfaces has a long history, originating with the papers of Lagrange () and the famous Plateau problem; we refer to the classical papers by Calabi [] and by Cheng and Yau []. Timelike minimal submanifolds may be viewed as simple but nontrivial examples of D-branes, 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 [-] and []). Huang and Kong [] studied the motion of a relativistic torus in the Minkowski space R +n (n ≥ ). They derived the equations for the motion of relativistic torus in the Minkowski space R +n (n ≥ ). This kind of equation also describes the three dimensional timelike extremal submanifolds in the Minkowski space R +n . 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 [] and []). 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 [] and Lindblad [].
Paul et al. [] investigated timelike minimal submanifolds of dimension  + n, n ≥ , of Minkowski spacetimes of dimension +n+q, q ≥ . The authors considered an embedding of R +n into Minkowski spacetime R +n+q given by the graph of a map f : R +n − → R q . Let Greek indices α, β, . . . take values in , , . . . , n and let uppercase Latin indices I, J, . . . take values in , . . . , q. Introduce cartesian coordinates x α on R +n and x I on R q . The induced http://www.boundaryvalueproblems.com/content/2014/1/32 metric R +n is is the Minkowski metric. By variational principles (see []), they derived the Euler-Lagrange equations Moreover for a small initial value with compact support, they also proved the global existence of classical solutions for (.).
In this paper, we consider (.) with the initial data where A >  is a constant and ε >  is a small parameter. The aim of this paper is to prove that the Cauchy problem (.), (.) has a global classical solution, provided that the initial value f I . 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 (.), we note that (.) can be written in divergence form where = η μν ∂ μ ∂ ν is the Minkowski wave operator and F μν = η μν -√ det hh μν , as well as in the form We raise and lower Greek (intrinsic) indices using h μν and its inverse, while Latin (extrinsic) indices are raised and lowered using the identity δ IJ and its inverse. From (.), it follows that H μν IJ has the symmetries Due to the symmetries, an energy estimate and local well posedness holds for the system (.). http://www.boundaryvalueproblems.com/content/2014/1/32 The plan of this paper is as follows. In Section , 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  and Section , respectively.
We need the following lemma that is basically established in [] and []. For completeness, the proof will also be sketched here. http://www.boundaryvalueproblems.com/content/2014/1/32 is a solution to the following Cauchy problem: Then we have Tsutaya [] has showed that the solution of the Cauchy problem (.) satisfies Obviously, Lemma . improves the result in [].
where χ is the characteristic function of positive numbers.
We next continue to make an estimate for (.); we make an estimate for the right-hand side of (.) by dividing into two cases.
We subdivide into three cases again.
In other words, when t + |x| ≥ , we get For  < t + |x| < , by changing variables r = |x -y|, we obtain (  .   ) http://www.boundaryvalueproblems.com/content/2014/1/32 Thus (.) and (.) imply that (.) Assume that u is a solution to the following Cauchy problem: and assume that φ decays to  at infinity. If It follows for  ≤ t ≤ T that For the proof of Lemma ., see Klainerman []. Using Lemmas ., . and the L  -L ∞ estimate of the linear wave equation with zero initial data, it is not difficulty to prove the following.

Lemma . Suppose that n = , . Let φ = φ(t, x) be the solution to the Cauchy problem
By Lemmas . and ., we can prove the following lemma.

Global existence in three space dimensions
Theorem . Suppose that f I  (x), f I  (x) ∈ C ∞ (R  ) and satisfy ). In what follows, we will prove the global existence of the classical solutions by a continuous induction, or a bootstrap argument. Let N ≥ , we set and To set up the bootstrap argument, we assume that there is a positive constant K so that on [, T) we have the following estimates for the norms defined in (.): 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   K .
Since h μν = η μν + O(|∂f |  ), (.) may also be written as Using Lemma ., (.), and (.), we get if K is sufficiently large and ε  is suitably small. So if K is sufficiently large and ε  is suitably small. From (.), we know that the estimate for N  (t) implies the desired estimate for N  (t). We have completed the proof of Theorem ..

Global existence in two space dimensions
where A >  is a constant. Then there exists ε  such that for  < ε ≤ ε  the Cauchy problem (.), (.) has a global classical solution for all t ≥ .
Proof The local existence argument follows from the method of Picard iteration [] (see also [] and []). In what follows, we will prove global existence of classical solutions by http://www.boundaryvalueproblems.com/content/2014/1/32 a continuous induction, or bootstrap argument. Let N ≥ , we set To set up the bootstrap argument, we assume that there is a positive constant K so that on [, T) we have following estimates for the norms defined in (.), where  < ι <   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   K .
It follows from Lemma . and (.) for |α| ≤ N that if K is sufficiently large and ε  is suitably small. Applying Lemma . to (.), we obtain if K is sufficiently large and ε  is suitably small. http://www.boundaryvalueproblems.com/content/2014/1/32 In what follows, we make an estimate for N  (t). In order to make this estimate for N  (t), define the following null forms: Let Q symbolically stand for any of the full forms (.) and (.). Then for some constants a ij . Let Q be one of null form in (.)-(.), we have Note that the Lagrangian associated to the volume element of the induced metric is √ det h. For small |∂f |, we have and thus the Euler-Lagrange equations take the form For small |∂f |, we obtain  + δ KL Q  f K , f L - =  + O |∂f |  .