- Research
- Open Access
Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem
- Kadriye Aydemir^{1}Email author and
- Oktay S Mukhtarov^{2, 3}
https://doi.org/10.1186/s13661-016-0589-4
© Aydemir and Mukhtarov 2016
- Received: 1 January 2016
- Accepted: 8 April 2016
- Published: 15 April 2016
Abstract
The aim of this study is to investigate various qualitative properties of eigenvalues and corresponding eigenfunctions of one Sturm-Liouville problem with an interior singular point. We introduce a new Hilbert space and integral operator in it such a way that the problem under consideration can be interpreted as a spectral problem of this operator. By using our own approaches we investigate such properties as uniform convergence of the eigenfunction expansions, the Parseval equality, the Rayleigh-Ritz formula, the minimax principle, and the monotonicity of eigenvalues for the considered boundary value-transmission problem (BVTP).
Keywords
- Sturm-Liouville problems
- boundary-transmission conditions
- eigenvalues
- Fourier series of eigenfunctions
- minimax principle
1 Introduction
Sturm-Liouville eigenvalue problems appear frequently in solving several classes of partial differential equations, particularly in solving the heat equation or a wave equation by separation of variables. Other examples of Sturm-Liouville boundary value problems are Hermite equations, Airy equations, Legendre equations etc. Also, many physical processes, such as the vibration of strings, the interaction of atomic particles, electrodynamics of complex medium, aerodynamics, polymer rheology or the earth’s free oscillations, yield Sturm-Liouville eigenvalue problems (see, for example, [1–6] and references therein).
In different areas of applied mathematics and physics many problems arise in the form of boundary value problems involving transmission conditions at the interior singular points. Such problems are called boundary value-transmission problems (BVTPs). For example, in electrostatics and magnetostatics the model problem which describes the heat transfer through an infinitely conductive layer is a transmission problem (see [7] and references therein). Another completely different field is that of ‘hydraulic fracturing’ (see [8]) used in order to increase the flow of oil from a reservoir into a producing oil well. Some problems with transmission conditions arise in thermal conduction problems for a thin laminated plate (i.e. a plate composed by materials with different characteristics piled in the thickness; see [9]). Some aspects of spectral problems for differential equations having singularities with classical boundary conditions at the endpoints were studied among others in [10–23] and references therein.
It is the aim of this study to investigate such important spectral properties as the eigenfunction expansion, Parseval’s equality, the Rayleigh-Ritz formula (minimization principle), the minimax principle, and monotonicity of the eigenvalues for the Sturm-Liouville problem (1)-(5). The ‘Rayleigh quotient’ is the basis of an important approximation method that is used in solid state physics as well as in quantum mechanics. In the latter, it is used in the estimation of energy eigenvalues of nonsolvable quantum systems.
Often in physical problems, the sign of the eigenvalue λ is quite important. For example, the equation \(\frac{dh}{dt}+\lambda h=0\) occurs in certain heat flow problems. Here, positive λ corresponds to exponential decay in time, while negative λ corresponds to exponential growth. In the vibration problems \(\frac{d^{2}h}{dt^{2}}+\lambda h=0\) only positive λ corresponds to the ‘usual’ expected oscillations.
The Rayleigh quotient cannot be used to explicitly determine the eigenvalue since the eigenfunction is unknown. However, interesting and significant results can be obtained from the Rayleigh quotient without solving the differential equation. Particularly, it can be quite useful in estimating the eigenvalues.
2 Preliminary results about eigenvalues and eigenfunctions
Remark 2.1
It is easy to see that the space H is also a Hilbert space with respect to the modified inner product (6).
Lemma 2.2
Proof
Proof
Remark 2.4
Theorem 2.5
There exists only one independent eigenfunction corresponding to each eigenvalue of the BVTP (1)-(5), i.e. each of eigenvalues of this problem is geometrically simple.
Proof
Remark 2.6
By virtue of Theorem 2.5 the eigenfunctions of a BVTP (1)-(5) can be chosen to be real-valued. Indeed, let \(\lambda_{0}\) be an eigenvalue with the eigenfunction \(u_{0}(x)=\upsilon_{0}(x)+i\omega_{0}(x)\). By Remark 2.4 both \(\upsilon_{0}(x)\) and \(\omega_{0}(x)\) are also eigenfunctions corresponding to the same eigenvalue. By Theorem 2.5 there is a complex number \(C_{0}\neq0\) such that \(\omega_{0}(x)=C_{0}\upsilon_{0}(x)\). Hence \(u_{0}(x)=\upsilon_{0}(x)+i\omega_{0}(x)=(1+iC_{0})\upsilon_{0}(x)\), i.e. here is only one real-valued eigenfunction, except for a constant factor, corresponding to each eigenvalue. In view of this fact, from now on we can assume that all eigenfunctions of the BVTP (1)-(3) are real-valued.
Now from Lemma 2.2, Theorem 2.3, and Remark 2.6 we have the next corollary.
Corollary 2.7
3 Reduction of (1)-(5) to the integral equation with the Green kernel
Theorem 3.1
Proof
By differentiating equation (15) twice we can easily see that \(\tau(u)=\lambda u+f\), \(\ell_{i}(u_{f})=t_{i}(u_{f})=0\) (\(i=1,2\)) so the function \(u_{f}\) given by (15) is the solution of the problem. We shall prove the uniqueness by way of contradiction. Suppose that there are two different solutions \(u_{0}\) and \(\upsilon_{0}\) of the system (14) corresponding to the same \(\lambda_{0}\), which is not an eigenvalue. Denoting \(\omega_{0}:=u_{0}-\upsilon_{0}\) we get \(\tau(\omega_{0})=\lambda_{0}\omega_{0}\), \(\ell_{i}(\omega_{0})=t_{i}(\omega_{0})=0 \) for \(i=1,2\), i.e. \(\lambda _{0}\) is an eigenvalue with the corresponding eigenfunction \(\omega_{0}\). So we get a contradiction, which completes the proof. □
4 Uniform and mean-square convergence of the eigenfunction expansions
Lemma 4.1
Proof
Corollary 4.2
Corollary 4.3
Theorem 4.4
Proof
Theorem 4.5
The set of all nonzero eigenvalues of the integral operator \(\mathfrak{F}\) coincide with the set of the eigenvalues \((\lambda_{n})\) which are obtained from the extremal principle.
Proof
Theorem 4.6
Proof
Theorem 4.7
The set of eigenfunctions \(\{\phi_{i}(x)\}\) is a complete orthonormal set in the Hilbert space H.
Proof
Now we are ready to prove the next important result.
Theorem 4.8
Proof
5 The Rayleigh-Ritz principle for the BVTP (1)-(5)
In the last sections of this study we will investigate some extremal properties of the eigenvalues and corresponding eigenfunctions of the considered BVTP (1)-(5) by using some variational methods.
Lemma 5.1
Let \(q(x)\geq0\) for all \(x\in\Omega\). Then all eigenvalues of the problem (1)-(5) are nonnegative.
Proof
Theorem 5.2
Proof
Let \(\lambda_{1}\) be the first eigenvalue with the corresponding eigenfunction \(u_{1}(x)\). Show that \(\lambda_{1}>0\).
(1) Since \(q(x)\) is continuous in Ω there are \(\delta>0\) and \(q_{0}>0\) such that \(q(x)\geq q_{0}\) for all \(x\in[x_{0}-\delta,x_{0}+\delta]\subset\Omega\). Then from (37) it follows immediately that \(\lambda_{1}>0\).
(2) Suppose that it possible that \(\lambda_{1}=0\). Then from (37), \(u_{1}'(x)=0\) for all \(x\in\Omega\), i.e. \(u_{1}(x)\) is a constant function in each of \(\Omega_{1}\) and \(\Omega_{2}\). Putting in (2) and (3) we have \(\cos\alpha u_{1}(-\pi)=\cos\beta u_{1}(\pi)=0\). Consequently at least one of \(u_{1}(-\pi)\) and \(u_{1}(\pi)\) is equal to zero and therefore \(u_{1}\) is identically zero \(\Omega_{1}\) or \(\Omega_{2}\). Then by applying the transmission conditions (3) and (4) we see that \(u_{1}\) is identically zero on the whole \(\Omega=\Omega_{1}\cup\Omega_{2}\). We have a contradiction, which completes the proof. □
Theorem 5.3
- (1)
\(q\not\equiv0\) and \(q(x)\geq0\);
- (2)
\(q(x)\geq0\) and \(\cos^{2}\alpha+\cos^{2}\beta\neq0\).
Proof
Remark 5.4
By applying the minimization principle directly, it is not possible to determine explicitly the eigenvalues and corresponding eigenfunctions, since we do not know how to minimize over all ‘admissible’ functions. Nevertheless, using the Rayleigh functional (38) with appropriate test functions one can obtain useful approximations for the eigenvalues.
6 The minimax property of eigenvalues
According to the minimization principle which is given by the preceding Theorem 5.3 we can find the nth eigen-pair \((\lambda_{n},\phi_{n})\) only after the previous eigenfunctions \(\phi_{1}(x),\phi_{2}(x),\ldots, \phi_{n-1}(x)\) are known. But in many applications it is important to have a characterization of any eigen-pair \((\lambda_{k},\phi_{k})\) that makes no reference to other eigen-pairs. By applying the following theorem we can determine the nth eigen-pair \((\lambda_{n},\phi_{n})\) without using the preceding eigenfunctions \(\phi_{1}(x),\phi_{2}(x),\ldots,\phi_{n-1}(x)\).
Theorem 6.1
Proof
Remark 6.2
7 Dependence of eigenvalues on the potential
The minimax principle of the eigenvalues, i.e. equation (45) for the eigenvalues makes it possible to study the dependence of the eigenvalues on the coefficients of the differential equation. In this section we shall establish the monotonicity of the eigenvalues with respect to the potential \(q(x)\) for fixed boundary-transmission conditions.
Theorem 7.1
Let \(\lambda_{n}(q)\) be the nth eigenvalue of the BVTP (1)-(5). Then \(\lambda_{n}(q)\) is a monotonically increasing function with respect to the variable \(q=q(x)\), i.e. if \(q_{1}(x)\leq q_{2}(x)\) for all \(x\in\Omega\) then \(\lambda_{n}(q_{1})\leq\lambda_{n}(q_{2})\).
Proof
Since \(0\leq q_{1}(x)\leq q_{2}(x)\) for all \(x\in\Omega\) it is obvious that \(I_{1}(u)\leq I_{2}(u)\) for all \(u\in D_{n-1}(u_{1},u_{2},\ldots,u_{n-1})\). Then by virtue of Theorem 6.1, we find the required inequality \(\lambda_{n}(q_{1})\leq\lambda_{n}(q_{2})\). The proof is complete. □
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable comments.
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Authors’ Affiliations
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