- Research Article
- Open Access
Cauchy-Neumann Problem for Second-Order General Schrödinger Equations in Cylinders with Nonsmooth Bases
© Nguyen Manh Hung et al. 2009
- Received: 26 February 2009
- Accepted: 18 June 2009
- Published: 14 July 2009
The main goal of this paper is to obtain the regularity of weak solutions of Cauchy-Neumann problems for the second-order general Schrödinger equations in domains with conical points on the boundary of the bases.
- Weak Solution
- Nonnegative Integer
- Neumann Problem
- Conical Point
- Smooth Domain
Cauchy-Dirichlet problem for general Schrödinger systems in domains containing conical points has been investigated in [1, 2]. Cauchy-Neumann problems have been dealt with for hyperbolic systems in  and for parabolic equations in [4–6]. In this paper we consider the Cauchy-Neumann problem for the second-order general Schrödinger equations in infinite cylinders with nonsmooth bases. The solvability of this problem has been considered in . Our main purpose here is to study the regularity of weak solution of the mentioned problem.
The paper consists of six sections. In Section 1, we introduce some notations and functional spaces used throughout the text. A weak solution of the problem is defined in Section 2 together with some results of its unique existence and smoothness with the time variable. Our main result, the regularity with respect to both of time and spatial variables of the weak solution of the problem, is stated in Section 3. The proof of this result is given in Section 4 with some auxiliary lemmas. In Section 5 we specify that result for the classical Schrödinger equations in quantum mechanics. Finally, some conclusions of our results are given in Section 6.
Let be a bounded domain in and denote the closure and the boundary of in . We suppose that is an infinitely differentiable surface everywhere except the coordinate origin and coincides with the cone in a neighborhood of the origin point where is a smooth domain on the unit sphere in We begin by introducing some notations and functional spaces which are used fluently in the rest.
Denote , is the closure of , . For each multi-index , set , .
is the conormal derivative on is the unit exterior normal to is a given function.
where is a positive constant independent of and
for all test functions , for all .
Now we derive here some our obvious results of the unique existence and smoothness with respect to time variable of the weak solution of the problem (2.1)–(2.3) as lemmas of main results.
where the constant does not depend on , .
The constant depends only on the operator and the dimension of the space
where is a constant independent of , .
where is an arbitrary local coordinate system on , is a linear operator with smooth coefficients.
It is well known in  that for each the spectrum of this problem is an enumerable set of eigenvalues.
Recall that is the positive real number in Lemma 2.1. Now, let us give the main result of the present paper.
where is a constant independent of .
holds for all functions .
Now we surround the origin by a neighborhood with a sufficiently small diameter such that the intersection of and coincides with the cone We begin by proving some auxiliary lemmas.
Let be a weak solution in of the problem (2.1)–(2.3) such that outside . Moreover, we assume that . Then for almost all one has
(i)if then ,
(ii)if then where arbitrary.
where the positive constant is independent of
Case 1 ( ).
for all almost all From (4.7), (4.11) and we receive for almost all
Case 2 ( ).
for all almost all That is . The lemma is proved.
where or according to or , does not contain any point of the spectrum of the Neumann problem for the equation (3.2) for all . Then .
Then is a function in the space
Repeating the proof in the case we achieve , too.
where . From the hypotheses of the operator and Lemma 2.2 we have for almost all . Repeating arguments used for function we receive or .
In another way, it follows from Lemma 2.2 that
From (4.23) and the assertion that both and are in the space we have . This lemma is proved.
where the constant is independent of
for , where is a positive constant.
Since for , so for . In another way, for . Then from Lemma 2.2 we have for all Hence, by using similar arguments in the proof of Lemma 4.3 we get . This means that (4.26) holds for .
where is a constant independent of . It means that (4.26) is proved. Finally we only need to fix in (4.26) to complete the proof of this lemma.
Now let us prove Theorem 3.1.
Now, let us prove Theorem 3.1 by induction by When then functions satisfy the hypotheses of Lemma 4.3. So It follows that is in Assume that the theorem holds up to then we have By using analogous arguments in the proof of Lemma 4.4, with note that (from the hypothesis of induction), we can prove that . So The inequality in Theorem 3.1 can derive from inequality (4.25) (for ) and inequality (4.32). The theorem is proved completely.
In this section we apply the previous result to the Cauchy-Neumann problem for classical Schrödinger equations in quantum mechanics. It is shown that the smoothness of the weak solution of this problem depends on the structure of the boundary of the domain, the right hand side and the dimension of the space
where is the unit exterior normal to
The regularity of the weak solution of the problem (5.1)–(5.3) can be stayed as follows.
Let be a weak solution in the space of the Cauchy-Neumann problem (5.1)–(5.3) and if , . Then .
does not contain any eigenvalue of the Neumann problem for (5.5). By applying Theorem 3.1 we have . The theorem is proved.
The Schrödinger equation has received a great deal of attention from mathematicians, in particular because of its application to quantum mechanics and optics. It is therefore important to research boundary value problems for it. Such problems have been previously proposed and analyzed for Schrödinger equations whose coefficients are independent of the time variable and in finite cylinders (see, e.g., ). In infinite cylinder , the first initial boundary value problem for this kind of equation with coefficients depend on both of time and spatial variables has been considered (see [1, 2]). In this paper, for a general Schrödinger equation in infinite cylinder with conical points in the boundary of base, we proved regularity property of solution of second initial boundary value problem. As a special application of these new results, we received the regularity of solution of a classical Schrödinger equation in quantum mechanics when the dimension of space . The similar questions for the case can be answered after researching the asymptotic of solution in the case the strip contains eigenvalues of the associated spectral problem. This is also the aim of our future research.
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