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Qualitative analysis of eigenvalues and eigenfunctions of one boundary valuetransmission problem
Boundary Value Problems volume 2016, Article number: 82 (2016)
Abstract
The aim of this study is to investigate various qualitative properties of eigenvalues and corresponding eigenfunctions of one SturmLiouville 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 RayleighRitz formula, the minimax principle, and the monotonicity of eigenvalues for the considered boundary valuetransmission problem (BVTP).
Introduction
SturmLiouville 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 SturmLiouville 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 SturmLiouville 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 valuetransmission 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.
In this paper we shall investigate some qualitative properties of the eigenvalues and the corresponding eigenfunctions of one boundary value problem which consists of the SturmLiouville equation,
together with endpoint conditions given by
and with transmission conditions at the interior singular point \(x=0\) given by
where \(q(x)\) is a realvalued function; \(\delta_{i}\), \(\gamma_{i}\) (\(i=1,2\)) are real numbers; \(\alpha,\beta\in[0,\pi)\); λ is a complex spectral parameter. Throughout we shall assume that \(q(x)\) is continuous in \(\Omega_{1}:=[\pi,0)\) and \(\Omega_{2}:=(0,\pi]\) with finite onehand limits \(q(0^{\pm})\); \(\gamma_{1}\gamma_{2}>0\), and \(\delta_{1}\delta_{2}>0\).
It is the aim of this study to investigate such important spectral properties as the eigenfunction expansion, Parseval’s equality, the RayleighRitz formula (minimization principle), the minimax principle, and monotonicity of the eigenvalues for the SturmLiouville 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.
Preliminary results about eigenvalues and eigenfunctions
In the direct sum of the Lebesgue spaces \(H:=L_{2}(\Omega_{1})\oplus L_{2}(\Omega_{2})\) we shall define a new inner product in terms of the coefficients of the considered transmission conditions as follows:
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
Let u and υ be eigenfunctions of BVTP (1)(5) corresponding to distinct eigenvalues λ and μ, respectively. If \(\lambda\neq\overline{\mu}\) then u and υ are orthogonal in the Hilbert space H, i.e.
Proof
Since \(\tau(u)=\lambda u\) and \(\tau(\upsilon)=\mu\upsilon\),
By using the Lagrange identity we have
where \(W(u,\upsilon;x)\) denotes the Wronskians of u and υ. The boundary conditions (2) and (3) implies \(W(u,\upsilon;\pi)=W(u,\upsilon;\pi)=0\). Further the transmission conditions (4) and (5) imply
By using these equations we get \((\lambda\overline{\mu})\langle u,\upsilon\rangle _{H}\). Thus, \(\lambda\neq\overline{\mu}\) implies \(\langle u,\upsilon\rangle _{H}=0\), which completes the proof. □
Theorem 2.3
All eigenvalues of the BVTP (1)(5) are real.
Proof
Let \((\lambda_{0},u_{0}(x))\) be any eigenpair of the problem (1)(5). Taking the complexconjugate of the BVTP (1)(5) we see that the pair \((\overline{\lambda}_{0},\overline{u_{0}(x)})\) is also an eigenpair of this problem. From the boundarytransmission conditions (2)(5) it follows easily that
and
Putting these equalities in the equality (7) we have \((\lambda\overline{\lambda}_{0})\u_{0}\^{2}=0\). This implies that \(\lambda_{0}\overline{\lambda}_{0}=0\), i.e. \(\lambda_{0}\) is real. □
Remark 2.4
Let \(\lambda_{0}\) be an eigenvalue of (1)(5) with corresponding eigenfunction \(u_{0}(x)=\upsilon_{0}(x)+i\omega_{0}(x)\), where \(\upsilon_{0}(x)\) and \(\omega_{0}(x)\) are realvalued. Then both \(\upsilon_{0}(x)\) and \(\omega_{0}(x)\) are also eigenfunctions corresponding to the same eigenvalue \(\lambda_{0}\). Indeed, putting \(u=u_{0}=\upsilon_{0}+i\omega_{0}\) and \(\lambda=\lambda_{0}\) in (1)(5) and in view of \(\lambda_{0}\) being real, we have
from which it follows that both \(\upsilon_{0}(x)\) and \(\omega_{0}(x)\) are eigenfunctions corresponding to the same eigenvalue \(\lambda_{0}\).
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
By way of contradiction suppose that there exist two linearly independent eigenfunctions \(u_{0}(x)\) and \(\upsilon_{0}(x)\) corresponding to the same eigenvalue \(\lambda_{0}\). The boundary conditions (1)(3) imply that \(W(u_{0},\upsilon_{0};\pi)=0\) and consequently \(W(u_{0},\upsilon_{0};x)=0\) for all \(x\in[\pi,0)\). Since \(u_{0}(x)\) and \(\upsilon_{0}(x)\) satisfy equation (1), \(u_{0}(x)\) and \(\upsilon_{0}(x)\) are linearly dependent on \(\Omega_{1}\) by the wellknown theorem of ordinary differential equation theory, i.e. there exists a constant \(c_{1}\neq0\) such that \(u_{0}(x)=c_{1}\upsilon_{0}(x)\) for all \(x\in\Omega_{1}\). Similarly, from the second boundary condition it follows that there exists a constant \(c_{2}\neq0\) such that \(u_{0}(x)=c_{2}\upsilon_{0}(x)\) for all \(x\in\Omega_{2}\). Hence
Substituting the transmission conditions (4)(5) we have
and
From these equalities we get \(c_{1}c_{2}=0\). Consequently \(u_{0}(x)\) and \(\upsilon_{0}(x)\) are linearly dependent on the whole \(\Omega=\Omega_{1}\cup\Omega_{2}\). Hence we have obtained a contradiction, which completes the proof. □
Remark 2.6
By virtue of Theorem 2.5 the eigenfunctions of a BVTP (1)(5) can be chosen to be realvalued. 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 realvalued 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 realvalued.
Now from Lemma 2.2, Theorem 2.3, and Remark 2.6 we have the next corollary.
Corollary 2.7
Let \(u_{1}\) and \(u_{2}\) be eigenfunctions of BVTP (1)(5) corresponding to distinct eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\). Then \(u_{1}\) and \(u_{2}\) are orthogonal in the sense of the following equality:
Reduction of (1)(5) to the integral equation with the Green kernel
Let \(u_{1}(x,\lambda)\) be the solution of equation (1) on the left interval \(\Omega_{1}\) (the socalled lefthand solution) satisfying \(u_{1}(\pi)=\sin\alpha\), \(u_{1}'(\pi)=\cos\alpha\). Next we proceed from \(u_{1}(x,\lambda)\) to define the righthand solution \(u_{2}(x,\lambda)\) of equation (1) on the righthand interval \(\Omega_{2}\) by the initial conditions
Now, let \(v_{2}(x,\lambda)\) be the solution of equation (1) on the righthand interval \(\Omega_{2}\) satisfying the initial conditions \(v_{1}(\pi)=\sin\beta\), \(v_{1}'(\pi,\lambda)=\cos\beta\). Similarly we proceed from \(v_{2}(x,\lambda)\) to define the lefthand solution \(v_{1}(x,\lambda)\) of equation (1) on the lefthand interval \(\Omega_{2}\) by the initial conditions
The existence of the solutions \(u_{i}\) and \(v_{i}\) (\(i=1,2\)) is obvious. Moreover, by using totally similar arguments as in [24] we can prove that each of these solutions is an entire function of the parameter \(\lambda\in\mathbb{C}\) for each fixed x. Since the Wronskian \(W[u_{i}(x,\lambda),v_{i}(x,\lambda)]\) is independent of the variable \(x\in\Omega_{i}\) (\(i=1,2\)), we can denote \(\omega_{i}(\lambda):=W[\upsilon_{i}(\cdot,\lambda),\nu _{i}(\cdot,\lambda)]\) (\(i=1,2\)). Using the transmission conditions (4)(5) it is easy to see that \(\gamma_{1}\gamma_{2}\omega_{1}(\lambda)=\delta_{1}\delta_{2}\omega_{2}(\lambda)\). Both sides of this equality we shall denote by \(\omega(\lambda)\). Now consider the following nonhomogeneous BVTP:
Let us define a Banach space \(\oplus C^{k}(\Omega)\) as
(\(k=0,1,2,\ldots\)) with the norm \(\f\_{\oplus C^{k}(\Omega)}:=\max\{\f_{(1)}\_{C^{k}[\pi,0]}, \f_{(2)}\_{C^{k}[0,\pi]}\}\). Below instead of \(\oplus C^{0}(\Omega)\) we shall write \(\oplus C(\Omega)\).
Theorem 3.1
Let \(f\in\oplus C(\Omega)\). Then for λ not an eigenvalue, the nonhomogeneous BVTP (14) has a unique solution \(u_{f}\) for which the following formula holds:
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. □
Let us introduce to the consideration the function \(G(x,\xi,\lambda)\) given by
Then equation (15) can be written in the following form:
i.e. \(u_{f}(x,\lambda)=\langle G(x,\cdot,\lambda),f(\cdot)\rangle_{H}\). Consequently the function \(G(x,\xi,\lambda)\) given by (16) is the Green’s function for the considered BVTP. Now suppose that \(\lambda=0\) is not an eigenvalue and let \(f\in\oplus C(\Omega)\) be an arbitrary function. Denoting \(G(x,\xi)=G(x,\xi;0)\) we have
has an unique solution \(u=u(x)\) given by
Putting \(f=\lambda u\) in equation (19) we have the following integral equation with Green’s kernel:
Uniform and meansquare convergence of the eigenfunction expansions
Let us define the integral operator \(\mathfrak{F}\) by
Then the BVTP (1)(5) converts to the spectral problem for the integral operator \(\mathfrak{F}\) given by
where I is the identity operator. Since the kernel \(G(x,\xi)\) of the integral operator \(\mathfrak{F}\) is symmetric and continuous we can apply the wellknown extremal principle (see, for example, [25]). Let \(\{\lambda_{n}\}\) be a sequence of eigenvalues of the integral operator \(\mathfrak{F}\) determined by the extremal principles and \(\{\phi_{n}(x)\}\) be the corresponding sequence of orthonormal eigenfunctions.
Lemma 4.1
Let \(g\in\oplus C(\Omega)\). Then
where \(c_{i}(\mathfrak{F}g)=\langle\mathfrak{F}g,\phi_{i}\rangle_{H}\) denote the Fourier coefficients of \(\mathfrak{F}g\) with respect to the orthonormal set \((\phi_{i})\).
Proof
Denote \(g_{m}(x)=g(x)\sum_{i=1}^{m}\langle g,\phi_{i}\rangle_{H}\phi_{i}\). Since \(\{\phi_{n}\}\) is the orthonormal system in H, \(\langle g_{m},\phi_{i}\rangle_{H}=0\) for \(i=1,\ldots,m\). From the fact that the eigenvalues \(\lambda_{n}\) are determined by the extremal principle with the corresponding sequence of orthonormal eigenfunctions \(\{\phi_{n}\}\) we have \(\\mathfrak{F}g_{m}\_{H}\leq\lambda_{m+1}\g_{m}\_{H}\). Since \(\lambda_{m+1}\rightarrow0\), \(\\mathfrak{F}g_{m}\_{H} \rightarrow0\). Then we have
for arbitrary \(m=1,2,\ldots \) . Letting \(m\rightarrow\infty\) we get
where the convergence is in the Hilbert space H, i.e. the equality (22) holds. □
Corollary 4.2
If \(g\in\oplus C(\Omega)\) then the Parseval equality
holds.
Corollary 4.3
The set of orthonormal eigenfunction of the integral operator \(\mathfrak{F}\) is complete in the range of the integral operator \(\mathfrak{F}\) given by
Theorem 4.4
Let the hypotheses and notation of Lemma 4.1 hold. Then, for any \(h\in R(\mathfrak{F})\),
where the series converges with respect to the norm \(\oplus C(\Omega)\), i.e. uniformly on \(\Omega=\Omega_{1}\cup\Omega_{2}\).
Proof
Let \(h=\mathfrak{F}g\). Then for any n, p we have
In view of the fact that the integral operator \(\mathfrak{F}\) is a bounded linear operator in the Banach space \(\oplus C(\Omega)\) we get from (25)
for some constant C independent of n. By Bessel’s inequality, the righthand side of this inequality tends to zero as \(n\rightarrow\infty\) uniformly. Thus the series
converges in the Banach space \(\oplus C(\Omega)\). Let \(\tilde{h}(x)\) be the sum of the last series. Consequently \(\tilde{h}\in\oplus C(\Omega)\) and
From (23) and (28) it follows that \(\\mathfrak{F}g\tilde{h}\_{H}=0\), i.e. \(h(x)=\tilde{h}(x)\) almost everywhere. Since h is also continuous in Ω we have \(h(x)=\tilde{h}(x)\) for all \(x\in\Omega\). Thus
where the series converges with respect to the norm of \(\oplus C(\Omega)\), i.e. uniformly on Ω. □
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
By way of contradiction, suppose there is a nonzero eigenvalue \(\lambda^{*}\) distinct from all eigenvalues \((\lambda_{n})\). Let \(u^{*}\) be the eigenfunction corresponding to the eigenvalue \(\lambda^{*}\). Then from Theorem 4.4 we get
since \(\langle u^{*},\phi_{i}\rangle_{H}=0\) for all \(i=1,2,\ldots \) by Theorem 2.7. Thus we get a contradiction. □
Theorem 4.6
Let \(f\in\oplus C^{2}(\Omega)\) and satisfy the boundarytransmission conditions (2)(5). Then the Fourier series of f with respect to \(\{\phi_{i}\}\) converges uniformly on \(\Omega_{1} \cup\Omega_{2}\), i.e.
Proof
Let \(f\in\oplus C^{2}(\Omega)\) satisfy the boundarytransmission conditions (2)(5) and denote \(g=\tau(f)\). Then \(g\in\oplus C(\Omega)\). By virtue of (17) and (21) we have \(f=\mathfrak{F}g\). From Lemma 4.1,
where the series is convergent in the Banach space \(\oplus C(\Omega)\). □
Theorem 4.7
The set of eigenfunctions \(\{\phi_{i}(x)\}\) is a complete orthonormal set in the Hilbert space H.
Proof
Denote by \(\oplus C_{0}^{k}(\Omega)\) the set of all functions \(f\in C^{k}(\Omega)\) which vanishes at some neighborhoods of the points \(x=\pi\), \(x=0\), and \(x=\pi\). Let \(f\in H\) and \(\epsilon>0\) be given. Then there exists a function \(g\in\oplus C_{0}^{2}(\Omega)\) such that \(\fg\_{\mathcal{H}}<\frac{\epsilon}{3}\) since the set \(\oplus C_{0}^{2}(\Omega)\) is dense in the Hilbert space H, i.e. \(\overline{\oplus C_{0}^{2}(\Omega)}=H \) (see, for example, [26]). It is clear that
for arbitrary m. By Bessel’s inequality we have
and, by Theorem 4.6, there exists an integer \(n_{0}=n_{0}(\epsilon)\) such that, for \(m>n_{0}\),
Finally, from (32) and (33) we get
for \(m>n_{0}\). The proof is complete. □
Now we are ready to prove the next important result.
Theorem 4.8
The set of eigenfunctions \((\phi_{i}(x))\) of the problem (1)(5) form an orthonormal basis in the Hilbert space H and for any \(f\in H\) the Parseval equality
holds.
Proof
Without loss of generality we shall assume that \(\lambda=0\) is not an eigenvalue. Otherwise, we can select a real \(\lambda_{0}\neq0\) such that the problem \(\tau u=\lambda_{0}u\), \(\ell_{i}(u)=t_{i}(u)=0\), \(i=1,2\) has no nontrivial solutions. Then denoting \(\tilde{\lambda}=\lambda\lambda_{0}\) and \(\tilde{q}(x)=q(x)\lambda_{0}\) we see that the problem
has the same properties for the eigenfunctions and eigenvalues as the considered problem (1)(5). Namely, the pair \((\tilde{\lambda},u(x))\) is the eigenpair of the problem (34) if and only if the pair \((\lambda,u(x))\) is an eigenpair of (1)(5). Clearly, \(\tilde{\lambda}=0\) is not an eigenvalue of the problem (34). Hence, without loss of generality we can assume that \(\lambda=0\) is not an eigenvalue of the considered BVTP (1)(5). Moreover, if \(\lambda\neq0\), then the pair \((\lambda,u(x))\) is the eigenpair of the BVTP (1)(5) if and only if the pair \((\frac{1}{\lambda},u(x))\) is the eigenpair of the integral operator \(\mathfrak{F}\). Consequently the set \({\phi_{i}}\) form an orthonormal set of eigenfunctions either for \(\mathfrak{F}\) and (1)(5). Moreover, this set is complete by Theorem 4.7. It is well known that any complete orthonormal set in a Hilbert space forms an orthonormal basis. Consequently, every function \(f\in H\) may be expanded in a Fourier series with respect to the orthonormal set of eigenfunctions \((\phi_{i})\), i.e. the equality
holds, where the series converges with respect to the norm of the Hilbert space H. Further, the Parseval equality follows immediately from the last equality. □
The RayleighRitz 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
Let \((\lambda,u(x))\) be any eigenpair of the problem (1)(5). Multiplying (1) by \(u(x)\) and integrating by parts from \(x=\pi\) to \(x=0\), and from \(x=0\) to \(x=\pi\), we have
By using the equalities (4)(5), we get \(\gamma_{1}\gamma_{2} uu'_{\pi}^{0^{}}+\delta_{1}\delta_{2} uu'_{0^{+}}^{\pi}=0\). Hence
Consequently
since \(q\geq0\) on \(\Omega_{1} \cup\Omega_{2}\) by assumption. □
Theorem 5.2
Let \(q(x)\geq0\) for all \(x\in\Omega\). Then, all the eigenvalues of the problem (1)(5) are positive if any one of the following conditions holds:

(1)
\(q\not\equiv0\), i.e. there exists at least one \(x_{0}\in\Omega\) such that \(q(x_{0})>0\);

(2)
\(\cos^{2}\alpha+\cos^{2}\beta\neq0\).
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
Suppose that any one of the following conditions holds:

(1)
\(q\not\equiv0\) and \(q(x)\geq0\);

(2)
\(q(x)\geq0\) and \(\cos^{2}\alpha+\cos^{2}\beta\neq0\).
Let \(\lambda_{1}<\lambda_{2}<\cdots\) be the sequence of eigenvalues with the corresponding normalized eigenfunctions \(\phi_{1}(x),\phi_{2}(x),\ldots \) and let
(naturally by \(S_{1}\) we mean \(S_{1}=\{u(x)u\in\oplus C^{2}(\Omega);u\not\equiv0;\ell_{i}(u)=t_{i}(u)=0 \textit{ for } i=1,2 \}\)). Then for all \(n=1,2,\ldots \) we have
where the functional \(I(u)\) is given by
Moreover, the minimizing function is \(\phi_{n}\), i.e. \(\lambda_{n}=I(\phi_{n})\).
Proof
Let \(\varphi(\cdot)\in\oplus C^{2}(\Omega)\) with \(\ell_{i}\varphi=t_{i}\varphi=0\), \(i=1,2\). Then by Theorem 4.6 we have
where the convergence is uniform on Ω. Then, by integration by parts, we get
Since \(\{\phi_{n}(x)\}\) is a complete orthonormal set, by Parseval’s equation
where \(c_{n}(\varphi)=\langle\phi_{n}(\cdot), \varphi(\cdot)\rangle_{H}\). By using (35) we get
Consequently
Putting \(\varphi=\phi_{1}\) in equation (41) we have \(c_{n}(\varphi)=\langle \phi_{1},\phi_{n}\rangle =0\) for \(n=2,3,\ldots \) and
From (42)(43) it follows immediately that
and the minimizing function is \(\varphi=\phi_{1}(x)\), i.e. \(\lambda_{1}=I(\phi_{1})\). Next, let \(\varphi(x)\in\oplus C^{2}(\Omega)\) with \(\ell_{i}\varphi=t_{i}\varphi=0\), \(i=1,2\), and \(\langle\varphi,\phi_{n}\rangle=0\) for \(n=1,\ldots,k\). Then
Hence, by the same arguments as before, we have
for \(k=1,2,\ldots \) and the minimizing function is \(\phi_{k+1}\), i.e. \(\lambda_{k+1}=I(\phi_{k+1})\). □
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.
The minimax property of eigenvalues
According to the minimization principle which is given by the preceding Theorem 5.3 we can find the nth eigenpair \((\lambda_{n},\phi_{n})\) only after the previous eigenfunctions \(\phi_{1}(x),\phi_{2}(x),\ldots, \phi_{n1}(x)\) are known. But in many applications it is important to have a characterization of any eigenpair \((\lambda_{k},\phi_{k})\) that makes no reference to other eigenpairs. By applying the following theorem we can determine the nth eigenpair \((\lambda_{n},\phi_{n})\) without using the preceding eigenfunctions \(\phi_{1}(x),\phi_{2}(x),\ldots,\phi_{n1}(x)\).
Theorem 6.1
Let \(u_{1}(x),u_{2}(x),\ldots,u_{n1}(x)\in\oplus C(\Omega)\) be arbitrary functions. Denote
(naturally by \(D_{0}(u_{1},u_{2},\ldots,u_{n1})\) we mean the linear manifold
Then the nth eigenvalue of the BVTP (1)(5) is
Proof
Let \(0<\lambda_{1}<\lambda_{2}<\cdots\) be the sequence of eigenvalues determined by the extremal principles and \(\phi_{1}(x),\phi_{2}(x),\ldots \) be the corresponding sequence of orthonormal eigenfunctions and let \(u_{1},u_{2},\ldots,u_{n1}\in\oplus C(\Omega)\) be arbitrary functions. Define
Now let \(\psi_{1}(x),\psi_{2}(x),\ldots,\psi_{n1}(x)\in\oplus C^{1}(\Omega)\) be any given functions, such that \(\ell_{i}(\psi_{j})=t_{i}(\psi_{j})=0\) (\(i=1,2\); \(j=1,2,\ldots,n1\)). Denoting \(a_{ij}=\langle\phi_{i},\psi_{i}\rangle_{H}\) for \(i,j=1,2,\ldots,n1\), consider a system of \(n1\) homogeneous linear equations in n unknowns \(z_{1},z_{2},\ldots,z_{n}\) given by
Obviously this system of \(n1\) homogeneous linear equations in n unknowns has a nontrivial solution. Let \(\alpha_{1},\alpha_{2},\ldots,\alpha_{n1}\) be any nontrivial solution of system (46). Define a function \(\upsilon_{n}(x)\) by
It is easy to see that
Consequently \(\upsilon_{n}\in D_{n1}(\psi_{1},\psi_{2},\ldots,\psi_{n1}) \) and \(\\upsilon\_{H}^{2}=\alpha_{1}^{2}+\alpha_{2}^{2}+\cdots+\alpha_{n}^{2}\). Then we have
Integrating by parts we get
where \(\delta_{ij}\) is the Kronecker delta. Then we have
Consequently \(F_{n}(u_{1},u_{2},\ldots,u_{n1})\leq I(\upsilon_{n})\leq\lambda_{n}\) for all \(u_{1},u_{2},\ldots,u_{n1}\in\oplus C(\Omega)\). Furthermore, by virtue of the preceding theorem
Hence
which completes the proof. □
Remark 6.2
In many problems of mathematical physics, the smallest eigenvalue (the socalled principal eigenvalue) plays an important role. For example, the principal eigenvalue of the simple boundary value problem
where \(\rho(x)>0\) is the given function, can be interpreted as the square of the lowest frequency of vibration of a rod of nonuniform cross section given by \(\rho(x)\). Therefore it is significant to determine explicitly the principal eigenvalue, or at least a ‘good’ estimation of it. Note that useful approximation values for the principal eigenvalue can be drawn from the minimax property (45) by using certain principles of the theory of the calculus of variations. In particular we can find an upper bound for the lowest eigenvalue \(\lambda_{1}\). In fact let \(\oplus C(\Omega)\) be any nontrivial function satisfying the boundarytransmission conditions (2)(5) called the trial function. Then by virtue of the minimax principle we have the inequality
which gives us an upper bound for the principal eigenvalue. By taking the trial function \(\upsilon(x)\) as close as possible to the corresponding eigenfunction we can expect to get the ‘good’ estimation for principal eigenvalue \(\lambda_{1}\). In many special cases the useful trial function can be found by applying some principles of variational analysis.
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 boundarytransmission 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
Define
Let the notation of the preceding theorem hold and let \(u_{1}(x),u_{2}(x),\ldots,u_{n1}(x)\in\oplus C(\Omega)\) be any arbitrary functions.
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 \(u\in D_{n1}(u_{1},u_{2},\ldots,u_{n1})\). 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. □
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Keywords
 SturmLiouville problems
 boundarytransmission conditions
 eigenvalues
 Fourier series of eigenfunctions
 minimax principle