On the regularity of solutions of the boundary value problem without initial condition for Schrödinger systems in domain with conical points
© Lien and Hung; licensee Springer 2014
Received: 25 March 2014
Accepted: 4 July 2014
Published: 24 September 2014
In this paper, we deal with the boundary value problems without initial condition for Schrödinger systems in cylinders. We establish several results on the regularity of the solutions.
MSC: 35Q41, 35B65, 35D30.
The theory of initial boundary value problems for partial differential equations and systems in nonsmooth domains has attracted the attention of many researchers. These problems for Schrödinger systems in domain with conical points are considered in , , in which authors consider the existence, the uniqueness and the regularity of solutions of the mentioned problems. While problems without initial conditions arise when describing different nonstationary processes in nature under the hypothesis that the initial condition practically has no influence at the present time. These problems are investigated in many works, see – for example, with results only about the well-posedness. In , we studied the boundary value problem without initial condition for Schrödinger systems in domain with conical points. By studying the corresponding problem with initial condition , then passing to the limits , we obtained the existence and uniqueness of the solution. In the present paper, we continue considering the mentioned problem and our main aim here is to study the regularity of the solution.
There are four sections in this paper. In Section 1, we set the problem and recall an obvious result about the unique existence of solution. Afterwards, in Sections 2 and 3, by a similar method to , , we give results on the regularity of problems with initial condition for Schrödinger systems in domains with conical points. Then, by letting , the smoothness of the generalized solution of our problem is obtained in Section 4.
2 Setting problem and obvious result
Let be a bounded domain in () with boundary . Assume that is an infinitely differentiable surface everywhere, except for the coordinate origin, in a neighborhood satisfying and coincides with the cone , where is a smooth domain on the unit sphere . For , set , . If , we use to refer to and to refer to . For each multi-index , set and .
Denote , , and , .
Let us introduce some functional spaces (see ) used in this paper.
Especially, we set .
Denote by the completion of infinitely differentiable vector functions vanishing near with respect to norm.
for all and a.e. , where is the closure in of infinitely differentiable complex -dimensional vector functions with compact support in .
where is the unit vector of outer normal to the surrounding surface .
holds for all , with .
for all , for all .
We recall the results about the solvability of problem (2.2)-(2.3) proved in .
, for all , ;
3 The regularity with respect to time variable of the problem with initial condition
The regularity of problem (2.2)-(2.3) is implied by analogous property for problem (2.5)-(2.7). In , the regularity with respect to time variable of problem (2.5)-(2.7) is studied in the case and . Now, by a similar method, we consider the problem in the case and . Moreover, we also show that the constants in all prior estimates do not depend on .
, , , for ;
, for , for all , .
where the constantdoes not depend on, , .
From the assumptions it follows that the coefficients , defined uniquely by (3.2), have derivatives up to order l + 1 and . It is very easy to check that , , .
Step 1: Prove that for all , , .
Because of the fact that for all , , using induction with respect to from (3.4) we obtain for all , , .
Step 2: A priori estimate for .
where , .
where is a positive constant.
The inequality (3.11) implies that is uniformly bounded in . By a standard weakly convergence argument, we can conclude that the sequence possesses a subsequence convergent to a vector function in . Moreover, it follows from (3.11) that (3.1) holds. □
4 Further results on the regularity of solution of problem with initial condition
Proceeding similarly to Lemma 2.1 in , the following lemma holds.
where the constantis independent of, , .
where the constantis independent of, , .
where does not depend on , .
for a.e. , where .
That means (4.5) is proved for .
From the inductive hypothesis and repeating the arguments of the proof in the case , the inequality (4.5) holds for .
from (4.5) and Theorem 4.1, (4.4) is true. The lemma is proved. □
Firstly, we study the case outside .
We use the induction by . For , this theorem is proved by Lemma 4.2 with noting that . Assume that the theorem’s assertion holds up to , we need to prove that this holds up to .
for all , where the constant is independent of .
Since , . So using similar arguments in the proof of Theorem 4.1 we get . By Lemma 4.2, one obtains . This means that (4.7)holds for .
Assume that (4.7) holds for .
where is a constant independent of .
It means that (4.7) is proved and our theorem is completed by fixing in (4.7).
The theorem is proved. □
5 The regularity of solution of problem (2.2)-(2.3)
The generalized solution of problem (2.2)-(2.3) can be approximated by a sequence of solutions of problems with initial condition (2.5)-(2.7).
It is well known that there is a smooth function which is equal to 1 on , is equal to 0 on , and assumes value in on (see , Th. 5.5] for more details). Moreover, we can suppose that all derivatives of are bounded.
So it is very easy to verify that is a generalized solution of problem (2.2)-(2.3); see . We obtain the following main results.
, , , for ;
, for all , ;
, for all , .
where the constantdoes not depend on, .
We would like to thank the reviewers for a careful reading and valuable comments on the original manuscript.
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