Dependence of eigenvalues of 2nth order boundary value transmission problems
- Kun Li^{1}Email authorView ORCID ID profile,
- Jiong Sun^{1} and
- Xiaoling Hao^{1}
Received: 18 May 2017
Accepted: 15 September 2017
Published: 29 September 2017
Abstract
The present paper deals with the dependence of eigenvalues of 2nth order boundary value transmission problems on the problem. The eigenvalues depend not only continuously but also smoothly on the problem. Some new differential expressions of eigenvalues with respect to an endpoint, a coefficient, the weight function, boundary conditions, and transmission conditions, are given.
Keywords
MSC
1 Introduction
It is well known that boundary value transmission problems are of great importance for their wide applications in physics and engineering. These problems, such as heat, mass transfer (see [1]), and diffraction problems, relate to discontinuous material properties, and their miscellaneous physical applications connected with these problems are found in many literature works, see, e.g., [1–17] and the corresponding references cited therein. To deal with interior discontinuities, some conditions are imposed on the discontinuous points, which are often called transmission conditions (see [2, 3, 7, 10, 18, 19]), interface conditions (see [14, 17]), or point interactions (see [5]).
In this paper, we study the dependence of eigenvalues of 2nth order boundary value transmission problems on the problem. Using the ideas of Mukhtarov and Yakubov [9] and Wang et al. [10], a new Hilbert space is constructed, in which the considered problems are put. We prove that if λ is an eigenvalue of the considered problem, then λ can be embedded in a continuous eigenvalue branch. We also give some new differential expressions of the eigenvalues, which generalize the previous results obtained by Kong et al. (see [24]).
This paper is composed as follows. We give some notations and preliminaries in Section 2. The continuity results of eigenvalues and eigenfunctions are obtained in Section 3. Section 4 presents differential expressions of the eigenvalues with respect to all the data.
2 Notations and preliminaries
3 Continuity of eigenvalues and eigenfunctions
In this section, we prove the continuity of eigenvalues and normalized eigenfunctions for the 2nth order boundary value transmission problems. Moreover, the characterization of the eigenvalues as zeros of an entire function is established.
Lemma 2
Proof
Theorem 1
Proof
From Lemma 2, we get that for \(\omega \in \Omega\), \(\lambda (\omega) \) is an eigenvalue of the operator L if and only if \(\Delta (\omega,\lambda)=0\). For any \(\omega \in \Omega \), \(\Delta (\omega,\lambda) \) is an entire function of λ and is continuous in ω (see [32], Theorems 2.7, 2.8), and \(\Delta (\omega_{0},\mu)=0 \). Since the operator L is self-adjoint, we know that μ is an isolated eigenvalue, and then \(\Delta (\omega_{0},\lambda) \) is not constant in λ. Hence there exists \(\rho_{0}>0 \) such that \(\Delta (\omega_{0},\lambda) \neq 0 \) for \(\lambda \in S_{\rho_{0}}:=\{\lambda \in \mathbb{C}:\vert \lambda -\mu \vert =\rho_{0}\}\). By the well known theorem on continuity of the roots of an equation as a function of parameters (see [33], 9.17.4), the proof for Theorem 1 is completed. □
In what follows we will always assume that each eigenvalue \(\lambda (\omega) \) is embedded in a continuous eigenvalue branch.
Lemma 3
Proof
For \(t_{0}=c- \) and \(x=c+ \), by transmission conditions (2.2) and \(\det C=\det C_{0}=\rho^{n}>0\), the result holds for \(x=c+ \). By the extension of continuity of \(y(x,\lambda) \) on \(J_{1} \) or \(J_{2} \), respectively, the statement can be seen from Lemma 3.1 in [27] when \(x\in J\). As \(t_{0}=c+ \), utilizing the same method, the result follows. For \(x\in J \), using Lemma 3.1 in [27] and the above method, the statement follows. □
Lemma 4
Let \(\omega_{0}=(a_{0},b_{0},A_{0},B_{0},C_{0},\frac{1}{p_{0_{0}}},p _{{1}_{0}},\ldots,p_{{n}_{0}},w_{0}) \). Let \(\lambda =\lambda ( \omega) \) be an eigenvalue of the operator L. If the multiplicity of \(\lambda (\omega_{0}) \) is 1, then there exists a neighborhood N of \(\omega_{0} \) belonging to Ω such that the multiplicity of \(\lambda (\omega) \) is 1 for every ω in N.
Proof
If \(\lambda (\omega_{0}) \) is simple, then \(\Delta '(\lambda (\omega _{0}))\neq 0 \). Since \(\Delta (\lambda) \) is an entire function of λ, then the conclusion follows from Theorem 1. □
Theorem 2
Particularly, if \(\lambda (\omega_{0}) \) is simple for some \(\omega_{0}\in \Omega \), then there exists a normalized eigenfunction \(u_{1}=u_{1}(\cdot,\omega) \) such that (3.8) holds for \(k=1 \).
Proof
Again by Lemma 3, \(u(x,\omega)\rightarrow u(x,\omega_{0}), u^{[1]}(x,\omega)\rightarrow u^{[1]}(x,\omega_{0}),\ldots, u^{[2n-1]}(x, \omega)\rightarrow u^{[2n-1]}(x,\omega_{0}) \) as \(\omega \rightarrow \omega_{0}\), and \(x\in J\). The conclusion follows.
(b) If the multiplicity of \(\lambda (\omega) \) is l (\(l=2, \ldots,2n \)) for all ω in some neighborhood N of \(\omega_{0} \) in Ω. Then we can choose eigenfunctions of \(\lambda (\omega) \) satisfying the same initial conditions at \(c_{0} \) for some \(c_{0}\in J \) since a linear combination of l linearly independent eigenfunctions can be chosen to satisfy arbitrary initial conditions.
The above discussion illustrates that for every self-adjoint boundary value transmission problem and every eigenvalue \(\lambda (\omega) \), the eigenfunction \(u(\cdot,\omega)\) and its quasi-derivatives \(u^{[1]}(\cdot,\omega),\ldots, u^{[2n-1]}(\cdot,\omega) \) are uniformly convergent in ω for \(x\in J \). Then we normalize the eigenfunctions to end the proof. □
4 Differential expressions of eigenvalues on the problem
In this section, we will obtain the differential expressions of eigenvalues with respect to the data. To this end, we will use Frechet derivative and list its definition as follows.
Definition 1
See [24]
Lemma 5
See [24]
Lemma 6
Proof
This follows directly from integration by parts. □
Theorem 3
- 1.Let all components of ω except \(p_{k} \) for some (\(k=1,2,\ldots,n\)) be fixed. Consider λ as a function of \(p_{k}\in L(J)\). Then λ is Frechet differentiable at \(p_{k} \) and$$ d\lambda_{p_{k}}(h)= \int^{c}_{a}(-1)^{n+k}\bigl\vert u^{(k)}\bigr\vert ^{2}h+\rho \int ^{b}_{c}(-1)^{n+k}\bigl\vert u^{(k)}\bigr\vert ^{2}h,\quad h\in L(J). $$(4.2)
- 2.Let all components of ω except \(1/p_{0} \) be fixed. Consider λ as a function of \(1/p_{0}\in L(J)\). Then λ is Frechet differentiable at \(1/p_{0} \) and$$ d\lambda_{1/p_{0}}(h)=- \biggl( \int^{c}_{a}\bigl\vert p_{0}u^{(n)} \bigr\vert ^{2}h+\rho \int ^{b}_{c}\bigl\vert p_{0}u^{(n)} \bigr\vert ^{2}h \biggr),\quad h\in L(J). $$(4.3)
- 3.Let all components of ω except w be fixed. Consider λ as a function of \(w\in L(J)\). Then λ is Frechet differentiable at w and$$ d\lambda_{w}(h)=-\lambda \biggl( \int^{c}_{a}\vert u\vert ^{2}h+\rho \int^{b}_{c}\vert u\vert ^{2}h \biggr),\quad h\in L(J). $$(4.4)
Proof
Theorem 4
Proof
Theorem 5
- 1.Let all components of ω except A be fixed. Denote the eigenvalue and the normalized eigenfunction with respect to A by \(\lambda =\lambda (A) \) and \(u=u(\cdot,A) \), respectively. For all E satisfying \(\rho (A+E)Q_{2n}(A+E)^{*}= BQ_{2n}B^{*} \) in the neighborhood of A, λ is differentiable at A and$$ d\lambda_{A}(E)=-R_{u}(a)Q_{2n}A^{-1}HC_{\bar{u}}(a). $$(4.6)
- 2.Let all components of ω except B be fixed. Denote the eigenvalue and the normalized eigenfunction with respect to B by \(\lambda =\lambda (B) \) and \(u=u(\cdot,B)\), respectively. For all E satisfying \(\rho AQ_{2n}A^{*}= (B+E)Q_{2n}(B+E)^{*} \) in the neighborhood of B, λ is differentiable at B and$$ d\lambda_{B}(E)=\rho R_{u}(b)Q_{2n}B^{-1}HC_{\bar{u}}(b). $$(4.7)
- 3.Let all components of ω except C be fixed. Denote the eigenvalue and the normalized eigenfunction with respect to C by \(\lambda =\lambda (C) \) and \(u=u(\cdot,C)\), respectively. For all E satisfying \(\det {(C+E)}=\rho^{n} \) and \((C+E)^{*}Q_{2n}(C+E)= \rho Q_{2n} \) in the neighborhood of C, λ is differentiable at C and$$ d\lambda_{C}(E)=-R_{u}(c+)C^{\ast }Q_{2n}HC_{\bar{u}}(c+). $$(4.8)
Proof
5 Conclusion
The dependence of eigenvalues with respect to the data plays an important role in the theory of differential operators. It gives theoretical support for the numerical computation of eigenvalues. Moreover, the properties of monotonicity of eigenvalues with respect to the parameters can be obtained by the derivatives of eigenvalues on the given parameter.
In this article, we obtained the continuity results of eigenvalues and eigenfunctions and presented some new differential expressions of the eigenvalues with respect to the data. Our results in this article generalize the previous results by Kong et al. [24] into a discontinuous version. It can be verified that it turns into the classical case when \(\rho =1\).
Declarations
Acknowledgements
The authors thank the referees for their comments and detailed suggestions. These have significantly improved the presentation of this paper. The work of the authors is supported by the National Nature Science Foundation of China (No. 11561050, No. 11401325).
Authors’ contributions
All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.
Competing interests
The authors declare that there are no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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