Abstract
The main goal of this paper is to study the general Schrödinger equations with a superlinear Neumann boundary value problem in domains with conical points on the boundary of the bases. First the formulation and the complex form of the problem for the equations are given, and then the existence result of solutions for the above problem is proved by the complex analytic method and the fixed point index theory, where we absorb the advantages of the methods in recent works and give some improvement and development. Finally, we are also interested in the asymptotic behavior of solutions of the mentioned equation. These results generalize some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrödinger equations or of delay Schrödinger equations.
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1 Introduction
This article deals with solutions of the general Schrödinger equation with a superlinear Neumann boundary value problem. To clarify our aim, we will introduce a class of Schrödinger equations (see [7, 22])
where \(\epsilon> 0\) is a small parameter, S is a compact metric space in \(\mathbb{R}^{n} (n\ge2)\), \(\omega_{\varepsilon}(x)\) is a positive weight. Assume that the domain is divided by the hyperplane \(\varSigma=\{x:x_{n}=0\}\) into two parts \(S^{(1)}=S\cap\{x:x_{n}>0\}\), \(S^{(2)}=S\cap\{x:x_{n}<0\}\), and that
The general theory of PDEs like (1) with variable exponent has gained the interest of many mathematicians in recent years. We refer to the surveys [1, 8, 14, 15, 23, 27].
From a physical point of view, such Schrödinger equations with a superlinear Neumann boundary value problem have gained a lot of interest in recent years, in particular in the context of systems for the mean field dynamics of Bose–Einstein condensates [2, 5] and in applications to fields like nonlinear and fibers optics [25].
To define the solution of (1), we introduce a class of functions related to the exponent \(\varrho(x)\) (see [30])
This set is a Sobolev space of functions, locally summable on S together with their first order generalized derivatives. It follows that there exists a good approximation of \(g_{\varrho}\) based on a set of independent and identically distributed random samples \(\mathbf{w}=\{w_{i}\}_{i=1}^{m}=\{(s_{i}, t_{i})\}_{i=1}^{m} \in Z^{m}\) drawn according to the measure ϱ.
To the best of our knowledge, this notion of indirect observability was introduced for the first time in the context of coupled elliptic equations in [7], to obtain an exact indirect controllability result, in which one wants to drive back the fully coupled system to equilibrium by controlling only one component of the system. In 2017, Lai, Sun and Li (see [17]) used a two level energy method to estimate the solution of (1). In the case when \(\omega_{\varepsilon}(x)\) and \(\varrho(x)\) are fixed constants, there have been many results about the existence, uniqueness, blowing-up and so on; we refer to the bibliography (see [19, 29]). It follows that the hypothesis space is a Hilbert space \(\mathfrak{H}_{E}\) induced by a Mercer kernel K which is a continuous, symmetric, and positive semi-definite function on \(S\times S\) (see [24]). Space \(\mathfrak {H}_{E}\) is the completion of the linear span of the set of functions \(\{E_{s} :=E(s,\cdot) : s \in S \}\) with respect to the inner product
The reproducing property in \(\mathfrak{H}_{E}\) is (see [3])
where \(g\in \mathfrak{H}_{E}\) and \(s\in S\).
Then by (2), we have (see [4])
for any \(g\in\mathfrak{H}_{E}\), where
It implies that \(\mathfrak{H}_{E}\subseteq C(S)\).
We define the approximation \(g_{\mathbf{w},\chi}\) of \(g_{\varrho}\) by (see [16])
where \(\chi=\chi(m)>0\) is a regularization parameter, \(\zeta =\varsigma(m)>0\) is a window width, and Φ is defined as follows (see [12]):
where q, \(c_{q}\) and \(c_{\varPsi}>0\) are positive constants and \(q>n+1\).
Scheme (3) shows that regularization not only ensures computational stability but also preserves localization property for the algorithm. In this paper, we further study the asymptotic behaviors of solutions of (1).
We adopt the coefficient-based regularization and the data-dependent hypothesis space (see [6, 20, 26])
where \(1\leq q\leq2\), and
Compared with scheme (3), the first advantage of (4) is the efficacy of computations without any optimization processes. Another advantage is that we can choose a suitable parameter q according to the research interest, e.g., smoothness and sparsity.
To study the approximation quality of \(g_{\mathbf{w},\varsigma}\), we derive an upper bound of the error
with
and establish its convergence rate as \(m \to\infty\) (see [10]).
The remainder of this paper is organized as follows. In Sect. 2, we will provide the main results. In Sect. 3, some basic, but important estimates and properties are summarized. The proofs of main results will be given in Sect. 4. Section 5 contains the conclusions of the paper.
2 Main results
We first formulate some basic notations and assumptions.
Let \(\varrho_{S}\) be the marginal distribution of ϱ on S and \(L_{\varrho_{S}} ^{2}(S)\) be the Hilbert space of functions from S to T, which are square-integrable with respect to \(\varrho_{S}\) with the norm denoted by \(\|\cdot\| _{\varrho_{S}}\). The integral operator \(L_{E}:L_{\varrho_{S}} ^{2}(S)\rightarrow L_{\varrho_{S}} ^{2}(S)\) is defined by
where \(s\in S\).
Let \(\{\mu_{i}\} \) be the eigenvalues of \(L_{E}\) and \(\{e_{i}\}\) be the corresponding eigenfunctions. Then for \(g\in L_{\varrho_{S}} ^{2}(S)\),
see [9]. We assume that \(g_{\varrho}\) satisfies the regularity condition \(L_{E}^{-r}g_{\varrho}\in L^{2}_{\varrho_{S}}\), where \(r>0\).
We show the following useful feature of the capacity of \(\mathfrak {H}_{E, \mathbf{w}}\) when the \(l^{2}\)-empirical covering number is used (see [11]), namely
where \(\epsilon>0\), \(B_{1}= \{f\in\mathfrak{H}_{E, \mathbf{w}}: \| g\|_{E}\leq1 \}\), \(0< p<2\) and \(c_{p}>0\) (see [22]).
We use the projection operator to obtain a faster learning rate under the condition \(|y|\leq M\) almost surely (see [18, 21]).
Definition 2.1
Let \(A>0\). Then the projection operator \(\gamma_{A}\) on the space of solutions \(g:S\rightarrow\mathbb{R}\) is defined as
We assume all the constants are positive and independent of δ, m, χ, ς and ζ. Now we state our main results.
Theorem 1
Suppose \(L_{E}^{-r}g_{\varrho}\in L^{2}_{\varrho_{S}}\) with \(r>0\), and (5) holds with \(0< p<2\) and \(0<\delta<1\). Then we have
where
It follows that (see [13])
is a solution of (1). In order to estimate \(\|\gamma(g_{\mathbf{w},\varsigma})-g_{\varrho}\| ^{2}_{\varrho_{S}}\), we invoke the following proposition in [28].
Proposition 1
Let \(f\in\mathfrak{H}_{E}\cup\{{g_{\varrho}}\}\) satisfy the Lipschitz condition on S, that is,
where \(u, v\in S\) and \(c_{0}\) is a positive constant. Then
Then we need an upper bound of the integral in (9). In order to get it, we only need to give its decomposition by using \(g_{\mathbf{w},\chi}\) which provides a crucial connection between \(g_{\mathbf{w},\varsigma}\) and the regularization function \(g_{\chi}\), while different regularization parameters χ and ς are adopted.
Here \(g_{\chi}\) is given by
Define
Remark
\(\mathfrak{S}(\mathbf{w},\chi,\varsigma)\), \(\mathfrak{H}(\mathbf{w},\chi ,\varsigma)\) and \(\mathfrak{D}(\chi)\) are solutions of (1).
Theorem 2
Let \(g_{\mathbf{w},\zeta,\varsigma,s}\) be defined as in (4) and let
be a solution of (1). Then we have
3 Lemmas
Some basic, but important estimates and properties of solutions \(\gamma_{A}(g)\) are summarized in the following lemma.
Lemma 1
Under the assumptions of Theorem 1, we have
Proof
We will split the proof into four steps.
Step 1. Obtaining estimates of the terms:
We take the sum of the inner products with \(g_{\chi}(t)\) and \(-g_{\varrho}(t)\), respectively, and obtain
in \((\mathbb{R}^{N}, \Vert \cdot \Vert _{\mathbb{R}^{N}, g})\).
Hence, integrating the latter equation over \(t \in(0,\tau)\), we have
and
which yields
where
On the other hand,
for all \(\varepsilon_{1} > 0\).
In view of the latter two inequalities, we have
for each \(\varepsilon_{1} >0\).
So
which yields
Integrating by parts, we have
where
However, for this term we have
Moreover,
Inserting the latter inequality into (17), we have
On the other hand,
So
Using (16), (18), (19) and (15), we have
Next, we estimate \(\widetilde{ E}_{g}(g_{\chi};\tau) + \widetilde{\gamma}_{A}(g_{\chi};0)\). For this purpose, we take the inner product with \((-\partial_{g}^{2} )^{-1}g_{\chi}'(t)\) in the space \((\mathbb{R}^{N}, \Vert \cdot \Vert _{\mathbb{R}^{N}, g})\) to obtain
It follows that
We now estimate the second term of the right-hand side of the above equation as
Moreover, by (20) and having in mind (21), we can write
So
which implies that
Step 2. Improving estimates (15) and (20).
Taking \(\varepsilon_{1} = 1\) in (15) yields
Inserting (22) into the latter inequality, we have
On the other hand, equation (20) implies that
and we have
from (22).
Step 3. Estimating \(\gamma_{A}(g_{\varrho};\tau ) + \gamma_{A}(g_{\varrho};0)\).
We have
from (22), (23) and (24), which gives
It follows that
for each \(\varepsilon_{2} > 0\), and we have
in view of (23).
Next we have
Inserting the latter inequality into equations (22)–(24), we obtain
Step 4. Estimating \(\int_{S} \gamma _{A}(g_{\varrho};t) \,dt\).
From (25), we have
It follows that
for all \(\varepsilon_{3}>0\).
Integrating the latter inequality between 0 and τ, we obtain
and having in mind equation (27), we can improve the last estimate as follows:
So
which yields
In other words,
Since \(\delta\leq\sqrt{\delta_{0}}/2\), it follows that (12) holds. This completes the proof. □
The following result provides a uniform observability inequality.
Lemma 2
where \(h>0\) and \(\tau>0\).
Proof
We first have the discrete identity
by Lemma 1, where
We now estimate separately A, \(X_{g}\) and B.
Estimate for A. We have
Estimate for \(X_{g}\). Notice that
So
Estimate for B. We have
Next we obtain
Moreover,
In other words,
where
So
It follows that
From (38)–(39) we deduce (37). Next, using (36) together with (37), we obtain the desired estimate (31). □
4 Proofs of main results
Now we derive the learning rates.
Proof of Theorem 1
Combining the three bounds of Step 1 in Lemma 1, we have
By substituting (40) into (9), we have
When \(0< r<1/2\),
Let \(\chi=m^{-\tau_{1}}\), \(\varsigma=m^{-\tau_{2}}\) and \(\zeta =m^{-\tau_{3}}\). Then
where
To maximize the learning rate, we take
Let
Then
Let
Then we have
When \(r\geq1/2\),
Similarly, we choose
to maximize the convergence rate.
We complete the proof of Theorem 1. □
Proof of Theorem 2
which yields
This completes the proof of Theorem 2. □
5 Conclusions
In this paper, we studied a class of Schrödinger equations with Neumann boundary condition \(L_{\varepsilon}g= \operatorname{div}(\omega_{\varepsilon}(x) |\nabla g|^{\varrho(x)-2}\nabla g)=0\) on a compact metric space \(S\subset\mathbb{R}^{n}\), \(n\ge2\), with a positive weight \(\omega_{\varepsilon}(x)\). We were interested in the asymptotic behavior of solutions of the mentioned equation. More precisely, we formulated conditions on a function g, which guarantee that the graph of at least one solution for the above-mentioned equation stays in the prescribed domain. These results generalized some previous results concerning the asymptotic behavior of solutions of non-delay systems of Schrödinger equations or of delay Schrödinger equations.
Change history
18 May 2021
A Correction to this paper has been published: https://doi.org/10.1186/s13661-021-01524-7
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The work was supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region of China (No. 2016D01A014).
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Liu, Z. RETRACTED ARTICLE: Existence results for the general Schrödinger equations with a superlinear Neumann boundary value problem. Bound Value Probl 2019, 61 (2019). https://doi.org/10.1186/s13661-019-1174-4
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DOI: https://doi.org/10.1186/s13661-019-1174-4