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On the regularity of solutions of the boundary value problem without initial condition for Schrödinger systems in domain with conical points
Boundary Value Problems volume 2014, Article number: 181 (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.
We use to denote the space of -dimensional vector functions defined in with the norm
Denote by the space consisting of all vector functions satisfying
and is the space of vector functions with norm
Especially, we set .
is the space of all functions which have generalized derivatives , , , , satisfying
is the space of all functions which have generalized derivatives , , , satisfying
Denote by the completion of infinitely differentiable vector functions vanishing near with respect to norm.
Now we introduce a differential operator of order
where are matrices with the bounded complex-valued components in , ( is the transposed conjugate matrix to ). Set
We assume further that the form satisfies the following condition: there exists a constant independent of and such that
for all and a.e. , where is the closure in of infinitely differentiable complex -dimensional vector functions with compact support in .
Now we consider the following problem in the cylinder :
where is the unit vector of outer normal to the surrounding surface .
Let , a complex vector-valued function is called a generalized solution of problem (2.2)-(2.3) if and only if, for any , the equality
holds for all , with .
For any , we also consider the following problem with initial condition corresponding to problem (2.2)-(2.3):
A function is a generalized solution of problem (2.5)-(2.7) if and only if, for any , we have
for all , for all .
We recall the results about the solvability of problem (2.2)-(2.3) proved in .
, for all , ;
where, are positive constants. Then for all, , there exists a uniquely generalized solutionof problem (2.2)-(2.3) satisfying
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 .
Assume that and there exists such that
, , , for ;
, for , for all , .
Then for all, the solutionof problem (2.5)-(2.7) has derivatives with respect toup to ordersatisfying, , such that
where the constantdoes not depend on, , .
Suppose that is an orthogonal basis of which is orthonormal in . For any , the approximating solutions can be found in the form where is the solution of the ordinary differential system
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 , , .
Taking the derivative times of both sides of (3.2) with respect to , then multiplying both sides by and taking the sum with respect to from 1 to , we arrive at
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 .
For , adding (3.4) to its complex conjugate, integrating with respect to from to , and then integrating by parts, we get
We denote by , , , , , and the terms from the first, second, third, fourth, fifth, sixth, and seventh, respectively, of the right-hand sides of (3.5). We will give estimations for these terms. Using Cauchy’s inequality, for any , we have
where , .
Denote for , then for all , using above estimations and (3.5), we obtain
where is a positive constant.
On the other hand, in  one already has the following estimate:
Multiplying both sides of (3.7) by , then integrating with respect to from to we obtain
where the constant does not depend on . Now using (3.8) and (3.6) for yields
So we can choose small enough such that . By (3.9) and using the Fubini theorem we obtain
Repeating the above arguments, from (3.6) one can prove for any , if , , that has derivative with respect to up to the th and for any
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
Assume that is a local coordinate system on . Moreover, assume that the principal part of the operator at origin 0 can be written in the form
where is a linear operator with smooth coefficients. Consider the following spectral problem:
Proceeding similarly to Lemma 2.1 in , the following lemma holds.
Let, and. Ifis a generalized solution of problem (2.5)-(2.7) in the spacesuch thatwhenever, , thenand the following estimate holds:
where the constantis independent of, , .
Letbe a generalized solution of problem (2.5)-(2.7), and letfor, for. In addition suppose that the strip
does not contain points of the spectrum of problem (4.1)-(4.2) for every. Thenand the following estimate holds:
where the constantis independent of, , .
First, we prove that
where does not depend on , .
Doing the same in , Proposition 2.1] we have
for a.e. , where .
Multiplying both sides of this inequality by , then integrating with respect to from to , we get
and by Lemma 4.1
from which it follows that
That means (4.5) is proved for .
Now assume that (4.5) is true for . By differentiating the systems (2.5) times with respect to and by putting , we obtain
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. □
Let l be a nonnegative integer. Assume thatis a weak solution in the spacewithof problem (2.5)-(2.7) and, for, for. In addition, suppose that in the strip
there is no point from the spectrum of (4.1)-(4.2). Then we haveand the following estimate holds:
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 .
Firstly we will prove the following inequality:
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 .
By the assumption of the induction of , put then . Differentiating (2.5) -times with respect to , we have
where . Following the assumption of the induction of and the hypothesis of the function one has , . It follows that . Because the strip does not contain spectral points of problem (4.1)-(4.2), then using results in , one gets , a.e. . So . Note that , then by using similar method to the one used in Lemma 2.2 in  we have and the following estimate holds:
where is a constant independent of .
So using the inductive assumption and (4.9), we obtain
It means that (4.7) is proved and our theorem is completed by fixing in (4.7).
2. The general case: Consider a function such that in some neighborhood of 0. Definite , which satisfies the system
where is a linear operator having order less than and the coefficients of this operator is equal to 0 outside . So the first case of this theorem implies that
Write where . The function is equal to 0 in , so we can apply the famous results on the smoothness of solutions of Schrödinger systems in a smooth domain to this function and obtain
From inequalities (4.11) and (4.12) it follows that
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.
Let be an integer. We set and we consider a generalized solution and of problems (2.5)-(2.7) in cylinders and with replaced by and , respectively. If , can be defined in with , , then we put , and is the generalized solution of problem (2.5)-(2.7) in which is replaced by . According to (4.6),
Since , when . So when . Repeating this argument, we discover when , for all . It follows that is a Cauchy sequence and is convergent to in . Moreover, is continuously embedded in , since
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.
Assume that and there exists such that
, , , for ;
, for all , ;
, for all , .
Then for all, the solutionof problem (2.2)-(2.3) has derivatives with respect toup to ordersatisfying, , such that
where the constantdoes not depend on, .
Let l be a nonnegative integer. Assume thatis a weak solution in the spacewithof the problem (2.2)-(2.3) and, for. In addition, suppose that in the strip
there is no point from the spectrum of (4.1)-(4.3). Then we haveand the following estimate holds:
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We would like to thank the reviewers for a careful reading and valuable comments on the original manuscript.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Lien, N.T., Hung, N.M. On the regularity of solutions of the boundary value problem without initial condition for Schrödinger systems in domain with conical points. Bound Value Probl 2014, 181 (2014). https://doi.org/10.1186/s13661-014-0181-8
- regularity of generalized solution
- problems without initial condition