# A Survey on Extremal Problems of Eigenvalues

Authors: Ping Yan; Meirong Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Given an integrable potentialq∈L1([0,1],ℝ), the Dirichlet and the Neumann eigenvalues λnD(q) and λnN(q) of the Sturm-Liouville operator with the potential q are defined in an implicit way. In recent years, the authors and their collaborators have solved some basic extremal problems concerning these eigenvalues when the L1 metric for q is given; ∥q∥L1=r. Note that the L1 spheres and L1 balls are nonsmooth, noncompact domains of the Lebesgue space (L1([0,1],ℝ),∥·∥L1). To solve these extremal problems, we will reveal some deep results on the dependence of eigenvalues on potentials. Moreover, the variational method for the approximating extremal problems on the balls of the spaces Lα([0,1],ℝ), 1<α<∞ will be used. Then the L1 problems will be solved by passing α↓1. Corresponding extremal problems for eigenvalues of the one-dimensional p-Laplacian with integrable potentials have also been solved. The results can yield optimal lower and upper bounds for these eigenvalues. This paper will review the most important ideas and techniques in solving these difficult and interesting extremal problems. Some open problems will also be imposed.


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## Body

## 1. Introduction

Minimization and maximization problems for eigenvalues are important in applied sciences like optimal control theory, population dynamics [1–3], and propagation speeds of traveling waves [4, 5]. They are also interesting mathematical problems [6, 7] because the solutions to them involve many different branches of mathematics. In recent years, some fundamental properties of eigenvalues have been revealed [8–16], such as strong continuity of eigenvalues in potentials/weights in the sense of weak topologies and continuous differentiability of eigenvalues in potentials/weights in the sense of the usual Lα norms. Based on such eigenvalue properties and some topological facts on Lα spaces, several interesting extremal problems for eigenvalues with L1 potentials/weights have been solved via variational methods and limiting approaches [17–23]. This paper will give a brief survey of the papers mentioned above and outline the ideas wherein to solve the extremal problems of eigenvalues.To illustrate the problems and the ideas more explicitly, let us first focus on one typical model. It is well known that for any integrable potentialq∈ℒ1:=L1([0,1],ℝ), all eigenvalues of the Sturm-Liouville operator
(1.1)x′′+(λ+q(t))x=0,t∈[0,1],
associated with the Dirichlet boundary conditions x(0)=x(1)=0 are given by a sequence
(1.2)λ1(q)<λ2(q)<⋯<λn(q)<⋯,
with limn→∞λn(q)=+∞. Let us consider such extremal problems as
(1.3)In(r):=infq∈B1[r]λn(q),Mn(r):=supq∈B1[r]λn(q)
for any n∈ℕ, where B1[r] is a ball of radius r in ℒ1. These problems are interesting and of difficulty, because λn(·) are implicit functionals of q∈ℒ1, while, topologically, B1[r] is not compact or sequentially compact even in the weak topology w1 of ℒ1 [24, 25], and, geometrically, B1[r] is also nonsmooth in the space (ℒ1,∥·∥1). Therefore, they cannot be solved directly by using the standard variational method.The main ideas developed in recent papers [17–23] to solve such extremal problems for implicit functionals on noncompact nonsmooth sets are in two steps.Firstly, for anyα>1, we consider the counterparts of In and Mn with potentials confined to Bα[r] (balls in ℒα:=Lα([0,1],ℝ) of radius r), denoted by In,α and Mn,α, respectively. Since the functional λn(·) is continuous (in weak topology) and continuously differentiable (in the usual Lα norm) in potential q∈ℒ1 (and hence in q∈ℒα, α>1), and the balls Bα[r], α>1, are compact in weak topology and smooth in the usual ℒα norm ∥·∥α, both the minimum and the maximum can be obtained, and one can study the critical potentials via standard variational methods. In this step, one can find a critical equation, in which the critical potential, denoted by qα, the critical eigenfunction, denoted by yα, and the extremal value In,α or Mn,α are all involved.Secondly, we will employ the limiting approachα↓1 to obtain In and Mn. This step is based on some topological facts that balls and spheres in ℒ1 space can be approximated by balls and spheres in ℒα spaces as α↓1. Then the strong continuity of eigenvalue λn(·) in potentials ensures that limα↓1In,α=In and limα↓1Mn,α=Mn. In this step, besides critical equations in ℒα spaces, properties of critical potentials and critical eigen-functions should also be sufficiently utilized to get a final solution to In and Mn.The final solution to problems in   (1.3) can yield optimal lower and upper bounds of eigenvalues. Actually, from   (1.3), we have
(1.4)In(∥q∥1)≤λn(q)≤Mn(∥q∥1)∀q∈ℒ1,
which are optimal.Several extremal problems on (Dirichlet, Neumann, and periodic) eigenvalues (of the Sturm-Liouville operator and thep-Laplace operator) have been studied recently, where the potentials are confined to different sets such as balls or spheres in ℒ1. For definition of these problems, see   (4.1). In the following sections, we give a slightly detailed description on these results. Some topological facts about ℒα, α≥1, are listed in Section 2. The essential of this section is that those “bad” balls or spheres in ℒ1 (neither smooth nor weak compact) can be approximated by “good” balls or spheres in ℒα, α>1 (smooth and weak compact). Section 3 is devoted to some properties of eigenvalues, including the scaling results which enable us to consider only those integrable potentials defined on the interval [0,1], the relationship between the first and higher-order eigenvalues which plays a role to enable us to consider only the first order, and the strong continuity (in weak topology) together with the continuous differentiability (in strong topology) which enable us to apply the variational method to problems confined on those “good” balls or spheres in ℒα, α>1. In Section 4 we introduce how variational method is applied to get critical equations for the extremal problems for cases α>1. After analysis on the critical equations, these extremal values are determined by some singular integrals. However, they cannot be expressed by elementary functions. Section 5 deals with the extremal problems in L1 spaces via limiting approaches. Final results of extremal values in ℒ1 case are stated in this section. For the Sturm-Liouville operator, these extremal values can be expressed by elementary functions of the radius r. In Section 6, the corresponding extremal problems for eigenvalues of measure differential equations [26, 27] are discussed briefly. The minimizing measures for the problems in   (1.3) are explained. In Section 7, two open problems for further study are imposed. One is on the eigenvalue gaps and the other is on the corresponding extremal problems of eigenvalues of the beam equation with integrable potentials.


## 2. Some Topological Facts onℒα Spaces

In the Lebesgue spaceℒα, 1≤α≤∞, the usual ℒα topology is induced by ℒα norm ∥·∥α. Besides this strong topology, one has also the weak topology wα which is defined as follows [24, 25].Definition 2.1.
Letqn,q∈ℒα. We say that qn is weakly convergent to q, written as qn→wαq in ℒα, or qn→q in (ℒα,wα), if
(2.1)limn→∞∫01‍qn(t)u(t)dt=∫01‍q(t)u(t)dt∀u∈ℒα*.
Here α* is the conjugate exponent of α: α*=α/(α-1)∈[1,∞].Forα∈[1,∞), r∈[0,∞) and h∈[-r,r], let us take the following notations:
(2.2)Sα[r]={q∈ℒα:∥q∥α=r},Bα[r]={q∈ℒα:∥q∥α≤r},Sα[r,h]={q∈ℒα:∥q∥α=r,q-=h},Bα[r,h]={q∈ℒα:∥q∥α≤r,q-=h},
where q¯=∫01‍q(t)dt is the mean value of q.


### 2.1. Approximation toℒ1 Balls/Spheres in Strong Topology

Lemma 2.2 (see [21, Lemma 2.1]).
Given thatr>0 and  q∈S1[r], there exists qα∈Sα[r] such that limα↓1∥qα-q∥1=0.Lemma 2.3 (see [19, Lemma 2.3]).
Letr>0 and h∈(-r,r). For any q∈S1[r,h], there exists {qα}∈Sα[r,h], 1<α<∞, such that limα↓1∥qα-q∥1=0.Lemmas2.2 and 2.3 give very nice topological approximation to spheres in ℒ1 space because the spheres in ℒα (1<α<∞) space have nicer topological and geometric properties.Remark 2.4.
A direct consequence from Lemmas2.2 and 2.3 is that, ℒ1 balls B1[r] and B1[r,h] can be approximated, in the sense of ℒ1 norm, by ℒα balls Bα[r] and Bα[r,h], respectively.


### 2.2. Approximation toℒ1 Balls/Spheres in Weak Topology

For anyα∈(1,∞), it is well known that Bα[r] is compact and sequentially compact in the space (ℒα,wα) [24, 25].Lemma 2.5 (see [19, Lemma 2.5]).
Suppose thatα∈(1,∞). Then, for any r>0 and h∈[-r,r], Bα[r,h] is compact and sequentially compact in the space (ℒα,wα).The ballsB1[r] and B1[r,h] are not compact or sequentially compact even in weak topology. Remark 2.4 tells us that such “bad” balls can be approximated, in the sense of ℒ1 norm, by “good” balls Bα[r] and Bα[r,h], which are compact in weak topology wα and smooth in geometry. In fact, there hold the following stronger topological facts.Lemma 2.6 (see [19, Lemma 2.6]).
(i) Letα∈[1,∞), r∈(0,∞) and h∈(-r,r). For any q∈Bα[r,h], there exists a sequence {qn}⊂Sα[r,h] such that qn→q in (ℒα,wα). In other words, the closure of Sα[r,h] in the space (ℒα,wα) is Bα[r,h].
(ii) Consequently, for anyq∈Sα[r,h], 1≤α<∞, there exists a sequence {qn}⊂S1[r,h] such that qn→q in (ℒ1,w1).


## 2.1. Approximation toℒ1 Balls/Spheres in Strong Topology

Lemma 2.2 (see [21, Lemma 2.1]).
Given thatr>0 and  q∈S1[r], there exists qα∈Sα[r] such that limα↓1∥qα-q∥1=0.Lemma 2.3 (see [19, Lemma 2.3]).
Letr>0 and h∈(-r,r). For any q∈S1[r,h], there exists {qα}∈Sα[r,h], 1<α<∞, such that limα↓1∥qα-q∥1=0.Lemmas2.2 and 2.3 give very nice topological approximation to spheres in ℒ1 space because the spheres in ℒα (1<α<∞) space have nicer topological and geometric properties.Remark 2.4.
A direct consequence from Lemmas2.2 and 2.3 is that, ℒ1 balls B1[r] and B1[r,h] can be approximated, in the sense of ℒ1 norm, by ℒα balls Bα[r] and Bα[r,h], respectively.


## 2.2. Approximation toℒ1 Balls/Spheres in Weak Topology

For anyα∈(1,∞), it is well known that Bα[r] is compact and sequentially compact in the space (ℒα,wα) [24, 25].Lemma 2.5 (see [19, Lemma 2.5]).
Suppose thatα∈(1,∞). Then, for any r>0 and h∈[-r,r], Bα[r,h] is compact and sequentially compact in the space (ℒα,wα).The ballsB1[r] and B1[r,h] are not compact or sequentially compact even in weak topology. Remark 2.4 tells us that such “bad” balls can be approximated, in the sense of ℒ1 norm, by “good” balls Bα[r] and Bα[r,h], which are compact in weak topology wα and smooth in geometry. In fact, there hold the following stronger topological facts.Lemma 2.6 (see [19, Lemma 2.6]).
(i) Letα∈[1,∞), r∈(0,∞) and h∈(-r,r). For any q∈Bα[r,h], there exists a sequence {qn}⊂Sα[r,h] such that qn→q in (ℒα,wα). In other words, the closure of Sα[r,h] in the space (ℒα,wα) is Bα[r,h].
(ii) Consequently, for anyq∈Sα[r,h], 1≤α<∞, there exists a sequence {qn}⊂S1[r,h] such that qn→q in (ℒ1,w1).


## 3. Properties of Eigenvalues on Potentials/Weights

Denote byϕp(·) the scalar p-Laplacian, that is, ϕp(x)=|x|p-2x for any x≠0 and ϕp(0)=0. For any integrable potential q∈L1([a,b],ℝ), it is well known that all eigenvalues of
(3.1)(ϕp(x′))′+(λ+q(t))ϕp(x)=0,t∈[a,b]
associated with the Dirichlet boundary condition x(a)=x(b)=0 consist of a sequence
(3.2)λ1D(q)<λ2D(q)<⋯<λnD(q)<⋯,
and all eigenvalues associated with the Neumann boundary condition x'(a)=x'(b)=0 consist of a sequence as follows:
(3.3)λ0N(q)<λ1N(q)<⋯<λnN(q)<⋯,
see [28, 29]. To emphasize the dependence on the interval [a,b] and the boundary conditions, we also write the eigenvalues as λnσ(q,[a,b]), where the superscript σ can be chosen as D (for the Dirichlet eigenvalues) or N (for the Neumann eigenvalues).


### 3.1. Scaling Results on Eigenvalues

LetI=[a,b] be a finite interval of length |I|=b-a. Given that α∈[1,∞] and q∈ℒα:=Lα([0,1],ℝ), define a potential qI∈Lα(I,ℝ) by the following:
(3.4)qI(t):=|I|-pq((t-a)|I|),t∈I.Lemma 3.1 (see [17, Lemma 2.1]).
Letq and qI be as in   (3.4). Then for any admissible n, there hold
(3.5)∥qI∥Lα(I)=|I|-p+1/α∥q∥α,λnσ(qI,I)=|I|-pλnσ(q,[0,1]),
where the superscript σ can be chosen to be D or N.By this lemma, we need only to consider eigenvalues for those potentials defined on the interval[0,1], that is, for q∈ℒα.


### 3.2. Relationship between the First- and Higher-Order Eigenvalues

Let us identifyℒα=Lα([0,1],ℝ) with Lα(ℝ/ℤ,ℝ), where α∈[1,∞]. For n≥2, define the mapping 𝒯n:ℒα→ℒα by
(3.6)𝒯n(q)=qn,qn(t):=npq(nt),t∈ℝ.Lemma 3.2 (see [17, Lemma 2.2]).
For any integern≥2, let q and qn be as in   (3.6). Then there hold
(3.7)q¯n=npq¯,∥qn∥α=np∥q∥α,λnσ(qn)=npλ1σ(q),
where the superscript σ can be chosen to be D or N.Whenn≥2, 𝒯n of   (3.6) is only injective but not surjective from Bα[r/np] to Bα[r]. It followed from the last two equalities in   (3.7) that
(3.8)supq∈Bα[r]λnσ(q)≥np·supq∈Bα[r/np]λ1σ(q),infq∈Bα[r]λnσ(q)≤np·infq∈Bα[r/np]λ1σ(q),
for any α∈[1,∞], n≥2, and σ chosen as D or N.In fact,   (3.8) can be proved to be equalities. Therefore, we need only to consider extremal values of the first eigenvalues. For cases α>1, the converse inequalities can be proved by using critical equations   (4.16). See Remark 4.1. Moreover, results for case α=1 can be obtained by limiting approaches α↓1. Case α=∞ is trivial.


### 3.3. Continuous Differentiability in Strong Topology

Theorem 3.3 (see [14, Theorem 1.2]).
Givenα∈[1,∞] and an admissible n. The functional
(3.9)(ℒα,∥·∥α)→ℝ,q↦λnσ(q)
is continuously Fréchet differentiable. The Fréchet derivative ∂qλnσ(q)∈(ℒα,∥·∥α)* is the following bounded linear functional:
(3.10)ℒα∋h↦-∫01‍|Enσ(t)|ph(t)dt∈ℝ,
or written simply as
(3.11)∂qλnσ(q)=-|Enσ|p.
Here Enσ(t)=Enσ(t;q) is a normalized eigenfunction associated with λnσ(q) so that the Lp norm ∥Enσ∥p=1.Remark 3.4.
Since   (3.11) is always a negative functional of ℒα, α∈[1,∞], eigenvalues possess the following monotonicity:
(3.12)q1,q2∈ℒ1,q1≤q2⇒λnσ(q1)≥λnσ(q2).
Moreover, if in addition, q1(t)<q2(t) holds on a subset of [0,1] of positive measure, the conclusion inequality in   (3.12) is strict.


### 3.4. Strong Continuity in Weak Topology

Theorem 3.5 (see [14, Theorem 1.1]).
Givenα∈[1,∞] and an admissible n, the following functional is continuous:
(3.13)(ℒα,wα)→ℝ,q↦λnσ(q).This theorem shows that eigenvaluesλnσ(q) have very strong continuous dependence on potentials. For other differential operators, similar results can be found in [9–11, 13, 16, 26, 30].


## 3.1. Scaling Results on Eigenvalues

LetI=[a,b] be a finite interval of length |I|=b-a. Given that α∈[1,∞] and q∈ℒα:=Lα([0,1],ℝ), define a potential qI∈Lα(I,ℝ) by the following:
(3.4)qI(t):=|I|-pq((t-a)|I|),t∈I.Lemma 3.1 (see [17, Lemma 2.1]).
Letq and qI be as in   (3.4). Then for any admissible n, there hold
(3.5)∥qI∥Lα(I)=|I|-p+1/α∥q∥α,λnσ(qI,I)=|I|-pλnσ(q,[0,1]),
where the superscript σ can be chosen to be D or N.By this lemma, we need only to consider eigenvalues for those potentials defined on the interval[0,1], that is, for q∈ℒα.


## 3.2. Relationship between the First- and Higher-Order Eigenvalues

Let us identifyℒα=Lα([0,1],ℝ) with Lα(ℝ/ℤ,ℝ), where α∈[1,∞]. For n≥2, define the mapping 𝒯n:ℒα→ℒα by
(3.6)𝒯n(q)=qn,qn(t):=npq(nt),t∈ℝ.Lemma 3.2 (see [17, Lemma 2.2]).
For any integern≥2, let q and qn be as in   (3.6). Then there hold
(3.7)q¯n=npq¯,∥qn∥α=np∥q∥α,λnσ(qn)=npλ1σ(q),
where the superscript σ can be chosen to be D or N.Whenn≥2, 𝒯n of   (3.6) is only injective but not surjective from Bα[r/np] to Bα[r]. It followed from the last two equalities in   (3.7) that
(3.8)supq∈Bα[r]λnσ(q)≥np·supq∈Bα[r/np]λ1σ(q),infq∈Bα[r]λnσ(q)≤np·infq∈Bα[r/np]λ1σ(q),
for any α∈[1,∞], n≥2, and σ chosen as D or N.In fact,   (3.8) can be proved to be equalities. Therefore, we need only to consider extremal values of the first eigenvalues. For cases α>1, the converse inequalities can be proved by using critical equations   (4.16). See Remark 4.1. Moreover, results for case α=1 can be obtained by limiting approaches α↓1. Case α=∞ is trivial.


## 3.3. Continuous Differentiability in Strong Topology

Theorem 3.3 (see [14, Theorem 1.2]).
Givenα∈[1,∞] and an admissible n. The functional
(3.9)(ℒα,∥·∥α)→ℝ,q↦λnσ(q)
is continuously Fréchet differentiable. The Fréchet derivative ∂qλnσ(q)∈(ℒα,∥·∥α)* is the following bounded linear functional:
(3.10)ℒα∋h↦-∫01‍|Enσ(t)|ph(t)dt∈ℝ,
or written simply as
(3.11)∂qλnσ(q)=-|Enσ|p.
Here Enσ(t)=Enσ(t;q) is a normalized eigenfunction associated with λnσ(q) so that the Lp norm ∥Enσ∥p=1.Remark 3.4.
Since   (3.11) is always a negative functional of ℒα, α∈[1,∞], eigenvalues possess the following monotonicity:
(3.12)q1,q2∈ℒ1,q1≤q2⇒λnσ(q1)≥λnσ(q2).
Moreover, if in addition, q1(t)<q2(t) holds on a subset of [0,1] of positive measure, the conclusion inequality in   (3.12) is strict.


## 3.4. Strong Continuity in Weak Topology

Theorem 3.5 (see [14, Theorem 1.1]).
Givenα∈[1,∞] and an admissible n, the following functional is continuous:
(3.13)(ℒα,wα)→ℝ,q↦λnσ(q).This theorem shows that eigenvaluesλnσ(q) have very strong continuous dependence on potentials. For other differential operators, similar results can be found in [9–11, 13, 16, 26, 30].


## 4. Extremal Problems inℒα Balls, 1<α<∞

In this section, we always assume that1<α<∞, r>0, |h|<r and σ=D or N.


### 4.1. Preliminary Results for Minimization/Maximization Problems

For any admissible integern, let us take some notations as follows
(4.1)In,α(r):=infq∈Bα[r]λnσ(q),Mn,α(r):=supq∈Bα[r]λnσ(q),ℑn,α(r,h):=infq∈Bα[r,h]λnσ(q),𝔐n,α(r,h):=supq∈Bα[r,h]λnσ(q).
If n=0, then σ can only be chosen as N. For any positive integers n, these notations are still reasonable, and the extremal values are independent of the choice of σ=N or σ=D due to the relationship between Dirichlet and Neumann eigenvalues [29, Theorem 4.3]. Furthermore, these extremal values on balls with radius r are exactly the same as those on spheres with radius r, that is,
(4.2)In,α(r)=infq∈Sα[r]λnσ(q),Mn,α(r)=supq∈Sα[r]λnσ(q),(4.3)ℑn,α(r,h)=infq∈Sα[r,h]λnσ(q),𝔐n,α(r,h)=supq∈Sα[r,h]λnσ(q).
Equation (4.2) follows immediately from   (3.12), the monotonicity of eigenvalues. Equation (4.3) follows from the facts that the closure of Sα[r,h] in the space (ℒα,wα) is Bα[r,h] (See Lemma 2.6(i)), and λn(q) is strongly continuous in q∈(ℒα,wα).Note thatBα[r] and Bα[r,h] are compact and sequentially compact in the space (ℒα,wα) (see Lemma 2.5). Then all these extremal values in balls are actually minima or maxima. Among all these extremal values, the supremum of the principal Neumann eigenvalues with potentials on Sα[r,h] is a case apart, unlike any other. It is well known that [31, Lemma 3.3]
(4.4)λ0N(q)≤-q-∀q,λ0N(q)=-q-⟺qisconstant.
Therefore one has
(4.5)𝔐0,α(r,h)=maxq∈Bα[r,h]λ0N(q)=supq∈Sα[r,h]λ0N(q)=λ0N(h)=-h.
Moreover, 𝔐0,α(r,h) cannot be attained by any potential on Sα[r,h]. Consequently,
(4.6)M0,α(r)=maxq∈Bα[r]λ0N(q)=λ0N(-r)=r=maxq∈Sα[r]λ0N(q).
All other minimizers and maximizers are located exactly on the boundaries of the balls, that is,
(4.7)In,α(r)=minq∈Sα[r]λnσ(q),n=0,1,2,…(4.8)Mn,α(r)=maxq∈Sα[r]λnσ(q),n=1,2,3,…(4.9)ℑn,α(r,h)=minq∈Sα[r,h]λnσ(q),n=0,1,2,…(4.10)𝔐n,α(r,h)=maxq∈Sα[r,h]λnσ(q),n=1,2,3,….
Again,   (4.7) and   (4.8) are immediate consequences from monotonicity of eigenvalues. Especially, the minimizers/maximizers on the sphere Sα[r] are nonnegative/nonpositive. It is natural to use the Lagrangian Multiplier Method (LMM for short) to deal with extremal value problems   (4.7) and   (4.8). Due to the restriction q¯=h, it is not easy to obtain monotonicity of eigenvalues on q∈Bα[r,h]. We refer the proof of   (4.9) and   (4.10) to [19, Lemma 3.1], where LMM is applied to exclude any minimizer or maximizer in the interior of Bα[r,h].


### 4.2. Variational Construction for Minimizer/Maximizer inBα[r]

The arguments in this subsection follow in part those of [17, 18, 20, 21].The extremal value problems   (4.7) and   (4.8) can be considered uniformly. The only constraint wherein is ∥q∥α=r. As α∈(1,∞), the norm ∥q∥α is continuously Fréchet differentiable in q∈(ℒα,∥·∥α), with the Fréchet derivative as follows:
(4.11)∂q∥q∥α=∥q∥α1-αϕα(q),
where ϕα:ℝ→ℝ is ϕα(x):=|x|α-2x. The eigenvalues λnσ(q) are also Fréchet differentiable in q, and the Fréchet derivative is as in   (3.11). By the Lagrangian Multiplier Method, the critical potential qα=qn,r,α satisfies the following equation:
(4.12)-|Eα(t)|p=cαϕα(qα(t))a.e.t∈[0,1]
for some constant cα=cn,r,α. Here Eα(t)=En,r,ασ(t) is a normalized eigen-function associated with λnσ(qα)=ℓα. Since the minimizer is nonnegative and the maximizer is nonpositive, cα<0 for the minimization problem   (4.7) and cα>0 for the maximization problem   (4.8). Let 𝔰α=signcα=±1 and
(4.13)yα(t)=𝔰αEα(t)|cα|p.
Then yα=yn,r,ασ is also an eigenfunction associated with ℓα. Specifically, we identify it by yα′(0)>0 for Dirichlet case and yα(0)>0 for Neumann case. This identified yα is called the critical eigenfunction, and it satisfies the original ODE   (3.1), that is,
(4.14)(ϕp(yα′))′+ℓαϕp(yα)+qα(t)ϕp(yα)=0,
and corresponding boundary conditions. By   (4.12), the critical potential can also be written as
(4.15)qα(t)=-𝔰αϕα*(|yα(t)|p)=-𝔰α|yα(t)|p(α*-1).
Substituting   (4.15) into   (4.14), one has
(4.16)(ϕp(yα′))′+ℓαϕp(yα)-𝔰αϕpα*(yα)=0,
which is the critical equation to problem   (4.7) and problem   (4.8). Note that this is a stationary Schrödinger equation, and it is independent of the orders n of eigenvalues λn(q). By   (4.15), the constraint ∥qα∥α=r is transformed to
(4.17)∫01‍|yα(t)|pα*dt=rα.Remark 4.1.
Due to the autonomy and the symmetry of critical equation   (4.16), yn,r,α(t) is a periodic solution of (4.16) with minimal period 2/n. In fact, one has yn,r,α(t+1/n)≡-yn,r,α(t) (we refer the proof of this to Lemma 3.2 in [20]). By   (4.15), qn,r,α(t) is periodic with minimal period 1/n. Therefore, the maximizer qn,r,α is in the range of 𝒯n. See   (3.6). This tells us that the converse inequalities of   (3.8) are also true. More precisely, there hold
(4.18)In,α(r)=npI1,α(rnp),Mn,α(r)=npM1,α(rnp)
for any integer n≥2 and any r>0. This is why we need only to consider extremal values of λ0N(q) and λ1σ(q).The phase portraits of   (4.16) are distinguished in three cases. See Figures 1–3. In Figure 1, 𝔰α=1. This figure corresponds to the maximization problem M1,α, and the eigen-function yα is certain nonconstant periodic orbit surrounding the equilibrium (0,0). In Figure 2, 𝔰α=-1 and ℓα<0. The minimization problem I0,α is illustrated in this figure, and the positive eigen-function yα corresponds to some non-constant periodic orbit surrounding the rightmost equilibrium. In Figure 3, 𝔰α=-1 and ℓα≥0. When the minimization problem I1,α is considered, both Figures 2 and 3 should be taken into account. In fact, the bigger r is, the smaller I1,α(r) is. For r large enough, I1,α(r) is negative and yα should be some sign-changing periodic orbit outside the homoclinic orbits in Figure 2. For r>0 small enough, I1,α(r) is nonnegative, and yα should be some non-constant periodic orbit surrounding the equilibrium (0,0) in Figure 3.Figure 1
Phase portrait of critical equation (4.16), where 𝔰α=1.Figure 2
Phase portrait of critical equation (4.16), where 𝔰α=-1 and ℓα<0.Figure 3
Phase portrait of critical equation (4.16), where 𝔰α=-1 and ℓα≥0.To study the parameterℓα in   (4.16), introduce an additional parameter as follows:
(4.19)bα∶=maxt∈[0,1]yα(t)=∥yα∥∞,
and besides, for case n=0 one more parameter as follows:
(4.20)aα∶=mint∈[0,1]yα(t).
A first integral of   (4.16) is
(4.21)|yα′(t)|p+Fα(yα(t))=Fα(bα),t∈[0,1],
where
(4.22)Fα(x)=Fα(x,ℓα)∶=(p-1)-1(ℓα|x|p-𝔰α|x|pα*α*),x∈ℝ.Note that the minimal period ofyα is 2, and yα satisfies the constraint   (4.17). Theoretically, ℓα=M1,α (or ℓα=I1,α) and bα are implicitly determined by singular integrals as follows:
(4.23)∫0bα‍dx(Fα(bα)-Fα(x))1/p=12,∫0bα‍xpα*dx(Fα(bα)-Fα(x))1/p=rα2,
where 𝔰α=1 in   (4.22) for M1,α, and 𝔰α=-1 for I1,α.Choose𝔰α=-1 in   (4.22). Then ℓα=I0,α, aα and bα are implicitly determined by
(4.24)∫aαbα‍dx(Fα(bα)-Fα(x))1/p=1,∫aαbα‍xpα*dx(Fα(bα)-Fα(x))1/p=rα,Fα(aα)=Fα(bα).


### 4.3. Variational Construction for Minimizer/Maximizer inBα[r,h]

The extremal value problems   (4.9) and   (4.10) can be considered uniformly by LMM as in the previous subsection. The critical equation is
(4.25)(ϕp(yα′))′+ℓαϕp(yα)-𝔰αϕα*(yαp-𝔪α)yα=0,
where 𝔰α=1 for maximization problems and 𝔰α=-1 for minimization problems. The existence of the new constant 𝔪α is caused by the constraint q¯=h. The critical potential is
(4.26)qα(t)≡-𝔰αϕα*(yαp(t)-𝔪α).
The constraint ∥qα∥α=r is
(4.27)∫01‍|yαp(t)-𝔪α|α*dt=rα.
The constraint q¯α=h is
(4.28)∫01‍ϕα*(yαp(t)-𝔪α)dt=-𝔰αh.
A first integral of the critical equation is
(4.29)|yα′|p+ℓαp-1|yα|p-𝔰α|yαp-𝔪α|α*(p-1)α*=const.The existence of𝔪α brings more complexity and difficulties. For the case of the Sturm-Liouville operator, we refer the readers to [19, 23] for more details.


## 4.1. Preliminary Results for Minimization/Maximization Problems

For any admissible integern, let us take some notations as follows
(4.1)In,α(r):=infq∈Bα[r]λnσ(q),Mn,α(r):=supq∈Bα[r]λnσ(q),ℑn,α(r,h):=infq∈Bα[r,h]λnσ(q),𝔐n,α(r,h):=supq∈Bα[r,h]λnσ(q).
If n=0, then σ can only be chosen as N. For any positive integers n, these notations are still reasonable, and the extremal values are independent of the choice of σ=N or σ=D due to the relationship between Dirichlet and Neumann eigenvalues [29, Theorem 4.3]. Furthermore, these extremal values on balls with radius r are exactly the same as those on spheres with radius r, that is,
(4.2)In,α(r)=infq∈Sα[r]λnσ(q),Mn,α(r)=supq∈Sα[r]λnσ(q),(4.3)ℑn,α(r,h)=infq∈Sα[r,h]λnσ(q),𝔐n,α(r,h)=supq∈Sα[r,h]λnσ(q).
Equation (4.2) follows immediately from   (3.12), the monotonicity of eigenvalues. Equation (4.3) follows from the facts that the closure of Sα[r,h] in the space (ℒα,wα) is Bα[r,h] (See Lemma 2.6(i)), and λn(q) is strongly continuous in q∈(ℒα,wα).Note thatBα[r] and Bα[r,h] are compact and sequentially compact in the space (ℒα,wα) (see Lemma 2.5). Then all these extremal values in balls are actually minima or maxima. Among all these extremal values, the supremum of the principal Neumann eigenvalues with potentials on Sα[r,h] is a case apart, unlike any other. It is well known that [31, Lemma 3.3]
(4.4)λ0N(q)≤-q-∀q,λ0N(q)=-q-⟺qisconstant.
Therefore one has
(4.5)𝔐0,α(r,h)=maxq∈Bα[r,h]λ0N(q)=supq∈Sα[r,h]λ0N(q)=λ0N(h)=-h.
Moreover, 𝔐0,α(r,h) cannot be attained by any potential on Sα[r,h]. Consequently,
(4.6)M0,α(r)=maxq∈Bα[r]λ0N(q)=λ0N(-r)=r=maxq∈Sα[r]λ0N(q).
All other minimizers and maximizers are located exactly on the boundaries of the balls, that is,
(4.7)In,α(r)=minq∈Sα[r]λnσ(q),n=0,1,2,…(4.8)Mn,α(r)=maxq∈Sα[r]λnσ(q),n=1,2,3,…(4.9)ℑn,α(r,h)=minq∈Sα[r,h]λnσ(q),n=0,1,2,…(4.10)𝔐n,α(r,h)=maxq∈Sα[r,h]λnσ(q),n=1,2,3,….
Again,   (4.7) and   (4.8) are immediate consequences from monotonicity of eigenvalues. Especially, the minimizers/maximizers on the sphere Sα[r] are nonnegative/nonpositive. It is natural to use the Lagrangian Multiplier Method (LMM for short) to deal with extremal value problems   (4.7) and   (4.8). Due to the restriction q¯=h, it is not easy to obtain monotonicity of eigenvalues on q∈Bα[r,h]. We refer the proof of   (4.9) and   (4.10) to [19, Lemma 3.1], where LMM is applied to exclude any minimizer or maximizer in the interior of Bα[r,h].


## 4.2. Variational Construction for Minimizer/Maximizer inBα[r]

The arguments in this subsection follow in part those of [17, 18, 20, 21].The extremal value problems   (4.7) and   (4.8) can be considered uniformly. The only constraint wherein is ∥q∥α=r. As α∈(1,∞), the norm ∥q∥α is continuously Fréchet differentiable in q∈(ℒα,∥·∥α), with the Fréchet derivative as follows:
(4.11)∂q∥q∥α=∥q∥α1-αϕα(q),
where ϕα:ℝ→ℝ is ϕα(x):=|x|α-2x. The eigenvalues λnσ(q) are also Fréchet differentiable in q, and the Fréchet derivative is as in   (3.11). By the Lagrangian Multiplier Method, the critical potential qα=qn,r,α satisfies the following equation:
(4.12)-|Eα(t)|p=cαϕα(qα(t))a.e.t∈[0,1]
for some constant cα=cn,r,α. Here Eα(t)=En,r,ασ(t) is a normalized eigen-function associated with λnσ(qα)=ℓα. Since the minimizer is nonnegative and the maximizer is nonpositive, cα<0 for the minimization problem   (4.7) and cα>0 for the maximization problem   (4.8). Let 𝔰α=signcα=±1 and
(4.13)yα(t)=𝔰αEα(t)|cα|p.
Then yα=yn,r,ασ is also an eigenfunction associated with ℓα. Specifically, we identify it by yα′(0)>0 for Dirichlet case and yα(0)>0 for Neumann case. This identified yα is called the critical eigenfunction, and it satisfies the original ODE   (3.1), that is,
(4.14)(ϕp(yα′))′+ℓαϕp(yα)+qα(t)ϕp(yα)=0,
and corresponding boundary conditions. By   (4.12), the critical potential can also be written as
(4.15)qα(t)=-𝔰αϕα*(|yα(t)|p)=-𝔰α|yα(t)|p(α*-1).
Substituting   (4.15) into   (4.14), one has
(4.16)(ϕp(yα′))′+ℓαϕp(yα)-𝔰αϕpα*(yα)=0,
which is the critical equation to problem   (4.7) and problem   (4.8). Note that this is a stationary Schrödinger equation, and it is independent of the orders n of eigenvalues λn(q). By   (4.15), the constraint ∥qα∥α=r is transformed to
(4.17)∫01‍|yα(t)|pα*dt=rα.Remark 4.1.
Due to the autonomy and the symmetry of critical equation   (4.16), yn,r,α(t) is a periodic solution of (4.16) with minimal period 2/n. In fact, one has yn,r,α(t+1/n)≡-yn,r,α(t) (we refer the proof of this to Lemma 3.2 in [20]). By   (4.15), qn,r,α(t) is periodic with minimal period 1/n. Therefore, the maximizer qn,r,α is in the range of 𝒯n. See   (3.6). This tells us that the converse inequalities of   (3.8) are also true. More precisely, there hold
(4.18)In,α(r)=npI1,α(rnp),Mn,α(r)=npM1,α(rnp)
for any integer n≥2 and any r>0. This is why we need only to consider extremal values of λ0N(q) and λ1σ(q).The phase portraits of   (4.16) are distinguished in three cases. See Figures 1–3. In Figure 1, 𝔰α=1. This figure corresponds to the maximization problem M1,α, and the eigen-function yα is certain nonconstant periodic orbit surrounding the equilibrium (0,0). In Figure 2, 𝔰α=-1 and ℓα<0. The minimization problem I0,α is illustrated in this figure, and the positive eigen-function yα corresponds to some non-constant periodic orbit surrounding the rightmost equilibrium. In Figure 3, 𝔰α=-1 and ℓα≥0. When the minimization problem I1,α is considered, both Figures 2 and 3 should be taken into account. In fact, the bigger r is, the smaller I1,α(r) is. For r large enough, I1,α(r) is negative and yα should be some sign-changing periodic orbit outside the homoclinic orbits in Figure 2. For r>0 small enough, I1,α(r) is nonnegative, and yα should be some non-constant periodic orbit surrounding the equilibrium (0,0) in Figure 3.Figure 1
Phase portrait of critical equation (4.16), where 𝔰α=1.Figure 2
Phase portrait of critical equation (4.16), where 𝔰α=-1 and ℓα<0.Figure 3
Phase portrait of critical equation (4.16), where 𝔰α=-1 and ℓα≥0.To study the parameterℓα in   (4.16), introduce an additional parameter as follows:
(4.19)bα∶=maxt∈[0,1]yα(t)=∥yα∥∞,
and besides, for case n=0 one more parameter as follows:
(4.20)aα∶=mint∈[0,1]yα(t).
A first integral of   (4.16) is
(4.21)|yα′(t)|p+Fα(yα(t))=Fα(bα),t∈[0,1],
where
(4.22)Fα(x)=Fα(x,ℓα)∶=(p-1)-1(ℓα|x|p-𝔰α|x|pα*α*),x∈ℝ.Note that the minimal period ofyα is 2, and yα satisfies the constraint   (4.17). Theoretically, ℓα=M1,α (or ℓα=I1,α) and bα are implicitly determined by singular integrals as follows:
(4.23)∫0bα‍dx(Fα(bα)-Fα(x))1/p=12,∫0bα‍xpα*dx(Fα(bα)-Fα(x))1/p=rα2,
where 𝔰α=1 in   (4.22) for M1,α, and 𝔰α=-1 for I1,α.Choose𝔰α=-1 in   (4.22). Then ℓα=I0,α, aα and bα are implicitly determined by
(4.24)∫aαbα‍dx(Fα(bα)-Fα(x))1/p=1,∫aαbα‍xpα*dx(Fα(bα)-Fα(x))1/p=rα,Fα(aα)=Fα(bα).


## 4.3. Variational Construction for Minimizer/Maximizer inBα[r,h]

The extremal value problems   (4.9) and   (4.10) can be considered uniformly by LMM as in the previous subsection. The critical equation is
(4.25)(ϕp(yα′))′+ℓαϕp(yα)-𝔰αϕα*(yαp-𝔪α)yα=0,
where 𝔰α=1 for maximization problems and 𝔰α=-1 for minimization problems. The existence of the new constant 𝔪α is caused by the constraint q¯=h. The critical potential is
(4.26)qα(t)≡-𝔰αϕα*(yαp(t)-𝔪α).
The constraint ∥qα∥α=r is
(4.27)∫01‍|yαp(t)-𝔪α|α*dt=rα.
The constraint q¯α=h is
(4.28)∫01‍ϕα*(yαp(t)-𝔪α)dt=-𝔰αh.
A first integral of the critical equation is
(4.29)|yα′|p+ℓαp-1|yα|p-𝔰α|yαp-𝔪α|α*(p-1)α*=const.The existence of𝔪α brings more complexity and difficulties. For the case of the Sturm-Liouville operator, we refer the readers to [19, 23] for more details.


## 5. Extremal Problems inℒ1 Balls

In this section we will solve the following extremal problems:(5.1)In(r)=infq∈B1[r]λnσ(q),Mn(r)=supq∈B1[r]λnσ(q),(5.2)ℑn(r,h):=infq∈B1[r,h]λnσ(q),𝔐n(r,h):=supq∈B1[r,h]λnσ(q).By the topological facts in Section2 and the strong continuity of eigenvalues in weak topology (see Theorem 3.5), the extremal values in ℒ1 balls are limits of those extremal values in ℒα balls, that is,
(5.3)In(r)=limα↓1In,α(r),Mn(r)=limα↓1Mn,α(r),ℑn(r,h)=limα↓1ℑn,α(r,h),𝔐n(r,h)=limα↓1𝔐n,α(r,h),
for any r>0, |h|<r and any admissible integer n. By relationship   (4.18), we need only consider for cases n=0 and n=1 in   (5.1). Similar arguments hold for   (5.2).For simplicity, we only illustrate the extremal value problems   (5.1) with potentials varied in B1[r] and refer to [19, 23] for   (5.2) with potentials varied in B1[r,h].Let us use a uniform notationℓ0 to denote the extremal values I0(r), I1(r), or M1(r), respectively, in different extremal values problems, that is,
(5.4)ℓ0=limα↓1ℓα.
Note that I0(r)>-∞ (see [18, Lemma 2.3]) and M1(r)<∞ due to the asymptotical distribution of large eigenvalues [29, 32]. Thus in each case, ℓ0 is finite and {ℓα}α>1 is bounded.Two different limiting approaches to this limitℓ0 are reviewed in the following two subsections. One is from the viewpoint of singular integrals as in [20, 21], where singular integrals involving extremal values for the Sturm-Liouville operator are analyzed directly and delicately. To overcome the difficulties caused by the presence of the p-Laplace operator, the other limiting approach is from the viewpoint of conservation laws (involving eigen-functions and extremal values) as in [10, 17, 18, 23]. Compared with the singular integral method, the conservation law method can simplify and refine the limiting process because more information from eigen-functions can be used.


### 5.1.   Limiting Approach from the Viewpoint of Singular Integrals (p=2)

By   (5.4), to obtain the extremal values ℓ0 in L1 balls, it is natural to compute firstly the extremal values ℓα in Lα balls and then let α↓1. Since ℓα will finally be formulated by singular integrals, we say that such a limiting approach is from the viewpoint of singular integrals. We have only applied such a method to the Sturm-Liouville operator because of the complexity in analyzing singular integrals. More difficulties will be added to such a process by the presence of p-Laplacian.Extremal valuesI1(r) and M1(r) for the Sturm-Liouville operator are studied in [20]. As stated in the previous section, ℓα=M1,α (or ℓα=I1,α) and bα are implicitly determined by   (4.23), where the exponent p=2. After some transformation,   (4.23) is analyzed in [20], and these extremal values in ℒα balls are finally expressed by some singular integrals. More precisely, for any d∈(0,∞), let
(5.5)A=Aα(d)=d2(α*-1)α*,Tα(d)=4∫01‍dx1-x2+A(1-x2α*),Uα(d)=4(α*A)α∫01‍x2α*dx1-x2+A(1-x2α*),T^α(d)=4∫01‍dxx2-1+A(1-x2α*),U^α(d)=4(α*A)α∫01‍x2α*dxx2-1+A(1-x2α*),
and for any d∈(0,1), let
(5.6)B=Bα(d)=d2(α*-1)=d2/(α-1),Tˇα(d)=4∫01‍dx1-x2-B(1-x2α*)/α*,Uˇα(d)=4Bα∫01‍x2α*dx1-x2-B(1-x2α*)/α*,
and we define
(5.7)Z1,α(x)={0forx=π2,14(2x)2-1/α(Uα(Tα-1(2x)))1/αforx∈(0,π2),R1,α=K(2α*)forx=0,14(2-x)2-1/α(U^α(T^α-1(2-x)))1/αforx∈(-∞,0),Yˇ1,α=14(2x)2-1/α(Uˇα(Uˇα-1T˘α-1(2x)))1/αforx∈[π2,∞).
Then there hold
(5.8)I1,α(r)=Z1,α-1(r),M1,α(r)=Yˇ1,α-1(r).
Note that I1,α(r) and M1,α(r) cannot be written as elementary functions of the radius r. However, after some delicate analysis on the singular integrals Tα, Uα, T^α, U^α, Tˇα and Uˇα, the limits of I1,α(r) and M1,α(r) as α↓1 are proved to be elementary functions of r, that is,
(5.9)M1(r)=14(π+π2+4r)2,(5.10)I1(r)=Z1-1(r),
where
(5.11)Z1(x)={2-xcoth(-x2)forx∈(-∞,0),4forx=0,2xcot(x2)forx∈(0,π2]
is a decreasing diffeomorphism mapping (-∞,π2] onto [0,∞).Extremal valuesI0(r) for the Sturm-Liouville operator are studied in [21], again by singular integral method in the limiting approach. Now ℓα=I0,α, aα, and bα are implicitly determined by   (4.24) with p=2. One can imagine that difficulty increases due to the presence of the additional new parameter aα. We only state the final results here and refer the readers to [21] for details. Let
(5.12)Z0(x)=2-xtanh(-x2),x≤0,
which is a decreasing diffeomorphism from (-∞,0] onto [0,∞). Then there holds
(5.13)I0(r)=Z0-1(r).


### 5.2. Limiting Approach from the Viewpoint of Conservation Laws (p>1)

The first integral   (4.21) of the critical equation   (4.16) can also be written as
(5.14)|yα′(t)|p+ℓαp-1|yα(t)|p-𝔰α(p-1)α*|yα(t)|pα*=eαp-1,t∈[0,1],
where
(5.15)eα∶=(p-1)Fα(bα)=ℓα|bα|p-𝔰αα*|bα|pα*
is a constant. We call   (5.14) a conservation law of   (4.16).Define(5.16)Gα(t):=|yα(t)|pα*α*,t∈[0,1].
It follows from   (4.17) that ∥Gα∥1=rα/α*→0 as α↓1. Passing to a subsequence if necessary,
(5.17)limα↓1Gα(t)=0a.e.t∈[0,1].Motivated by   (5.17), one attempt to compute ℓ0 in   (5.4) is to consider the limit equation of the conservation law   (5.14). Intuitively, if we assume that eα→e0 and yα→y0 in appropriate sense, then it followed from   (5.4) and   (5.17) that the limit equation of   (5.14) should be of the form   (5.29), which is a first-order ODE simpler than   (5.14). Boundary conditions on the critical eigen-functions yα and the restriction that the critical potentials qα∈Sα[r] will give more information on y0. All these conditions on y0 and the solution to   (5.29) will finally lead to the answer to the extremal value ℓ0. Compared with the previous singular integral method, such a limiting approach to ℓ0 from the viewpoint of conservation laws cannot only greatly simplify the analysis but also deal with the p-Laplace operator.In the limiting approach from the viewpoint of conservation laws, it is natural to study the convergence of critical eigen-functionsyα as α↓1. To this end, the boundedness of ∥yα∥p and ∥yα′∥p as α↓1 should be taken into consideration.For both Neumann and Dirichlet eigenvalues, multiplying   (4.16) by yα(t), integrating over [0,1] and taking use of the restriction   (4.17), that is, ∥yα∥pα*pα*=rα, one has
(5.18)-∥yα′∥pp+ℓα∥yα∥pp-𝔰αrα=0.
Integrating   (5.14) over [0,1], one has
(5.19)∥yα′∥pp+ℓαp-1∥yα∥pp-𝔰α(p-1)α*rα=eαp-1.
Eliminating ∥yα′∥pp from the above two equalities, one has
(5.20)pℓα∥yα∥pp=eα+𝔰α(p-1α)rα.
For the principal Neumann eigenvalues, there hold an additional equality as follows:
(5.21)eα∥1yα∥pp=-∥qα∥1α.
This equality can be deduced from   (4.14),   (5.14), and   (4.17).Note that∥ℓα+qα∥1≤|ℓα|+∥qα∥1≤|ℓα|+∥qα∥α=|ℓα|+r is bounded as α↓1. The boundedness of ∥ℓα+qα∥1 and the restriction   (4.17) guarantees (passing to a subsequence if necessary) the convergence of limα↓1bα∈(0,∞) and
(5.22)yα→y0≠0in(𝒞,∥·∥∞)asα↓1.
Here 𝒞=C([0,1],ℝ). For detailed proof, see Lemmas 2.1 and 4.1 in [18]. Consequently, ∥yα∥p is bounded as α↓1, and by   (5.20),
(5.23)eα=(p-1)Fα(bα)=ℓα|bα|p-𝔰α|bα|pα*α*(→e0asα↓1).
By   (4.17), there holds rα≤bαpα*, and hence
(5.24)limα↓1bαpα*≥1.
It followed from   (5.23) and   (5.24) that
(5.25)limα↓1bα=1,e0=ℓ0-limα↓1𝔰α|bα|pα*α*.Note that∥qα∥1≤∥qα∥α=r. On the other hand, it follows from   (4.15) and   (4.17) that
(5.26)∥qα∥1=∫01‍|yα(t)|pα*|yα(t)|pdt≥1∥yα∥∞p∫01‍|yα(t)|pα*dt=rαbαp→r
as α↓1. Thus one has
(5.27)s-limα↓1∥qα∥1=r.By the monotonicity and the symmetry of the eigen-functionyα,   (4.21) can be written as
(5.28)yα′(t)=κp-1(eα-ℓαyαp(t)-𝔰αGα(t))1/p,t∈Iσ,
where κp:=(p-1)1/p. For the Dirichlet eigen-function, yα(0)=0, yα(1/2)=bα and Iσ=ID=[0,1/2]. For the Neumann eigen-function, yα(0)=aα=mint∈[0,1]yα(t), yα(1)=bα and Iσ=IN=[0,1]. Since there holds   (5.17), the Lebesgue-dominated convergence theorem can be applied to the integral equation equivalent to   (5.28) with corresponding boundary conditions. Then it can be proved that y0=limα↓1yα∈C1(Iσ) and
(5.29)y0′=κp-1(e0-ℓ0y0p)1/p,t∈Iσ.
As the limit of Dirichlet eigen-functions yαD, y0 satisfies the boundary conditions as follows:
(5.30)y0(0)=0,y0(12)=1.
As the limit of Neumann eigen-functions yαN, y0 satisfies the boundary conditions as follows:
(5.31)y0(0)=a0=mint∈[0,1]y0(t)∈(0,1),y0(1)=1.
The limiting equality of   (5.20) is
(5.32)pℓ0∥y0∥pp=e0+(p-1)r.
By   (5.27), the limiting equality of   (5.21) is
(5.33)e0∥1y0∥pp=-r.Eitherℓ0=M1(r) or ℓ0=I1(r) is determined by ODE   (5.29), the boundary conditions   (5.30) and the restriction   (5.32). Meanwhile, ℓ0=I0(r) is determined by   (5.29),   (5.31),   (5.32) and   (5.33). To solve   (5.29) and compute ℓ0, we need more information about e0, ℓ0, and a0.Case 1 (maximization problemM1(r)).
In this case, there holdse0=ℓ0>0. In fact, we have
(5.34)ℓ0=M1(r)≥λ1(-r)=r>0.
The rightmost equilibrium in the phase portrait Figure 1 is (ℓα(α-1)/p,0). Thus we have
(5.35)0<bα<ℓα(α-1)/p,limα↓1bαpα*α*≤limα↓1ℓααα*=0.
Consequently, e0=ℓ0 by   (5.25).
Finally,ℓ0=M1(r) can be proved to be the unique root of
(5.36)x-πpx1/p*=r,x∈[0,∞),
where
(5.37)πp=2π(p-1)1/ppsin(π/p).
We refer to [17] for details. Note that if p=2, result   (5.36) is consistent with   (5.9).Case 2 (Minimization problemI1(r)).
In this case, there holde0>0 and e0>ℓ0. In fact, 𝔰α=-1 for the minimization problems. It follows from   (5.14) that
(5.38)eα=(p-1)Fα(bα)≥ℓα|yα(t)|p,∀t∈[0,1].
Set t=0 and 1/2 in   (5.38), respectively, and let α↓1. We see that e0≥0 and e0≥ℓ0. Furthermore, cases e0=0 and e0=ℓ0 can be excluded by checking   (5.32) after solving   (5.29)-(5.30) and taking into account that
(5.39)ℓ0=I1(r)≤λ1(r)=πpp-r.In this case, till now there is no such restrictions as positivity or negativity onℓ0. This is consistent to the phase portrait analysis in Section 4.2.Finally,(e0,ℓ0) (e0>0,e0>ℓ0) can be uniquely determined by
(5.40)∫01‍κpdu(e0-ℓ0up)1/p=12,r=2(p-1)-1/p*(e0-ℓ0)1/p*.
For details we refer to [18]. Note that if p=2, the integral in   (5.40) can be evaluated explicitly, and one can get   (5.10).Case 3 (minimization problemI0(r)).
In this case, there holdℓ0<0 and e0=ℓ0a0p<0. In fact, it follows from   (4.4) that
(5.41)ℓ0=I0(r)<-r<0.
By   (5.33), e0<0. Note that   (5.38) still holds for this minimization problem. Let t=0 in   (5.38) and α↓1. One has e0≥ℓ0a0p. One can exclude the case e0>ℓ0a0p by checking   (5.33) after solving   (5.29)–(5.31).
Finally,(a0,ℓ0) (0<a0<1,ℓ0<0) can be uniquely determined by
(5.42)κp∫a01‍du(ℓ0a0p-ℓ0up)1/p=1,(p-1)-1/p*(ℓ0a0p-ℓ0)1/p*=r.
For details we refer to [18]. Note that if p=2, the integral in   (5.42) can be evaluated explicitly, and one can get   (5.13).


## 5.1.   Limiting Approach from the Viewpoint of Singular Integrals (p=2)

By   (5.4), to obtain the extremal values ℓ0 in L1 balls, it is natural to compute firstly the extremal values ℓα in Lα balls and then let α↓1. Since ℓα will finally be formulated by singular integrals, we say that such a limiting approach is from the viewpoint of singular integrals. We have only applied such a method to the Sturm-Liouville operator because of the complexity in analyzing singular integrals. More difficulties will be added to such a process by the presence of p-Laplacian.Extremal valuesI1(r) and M1(r) for the Sturm-Liouville operator are studied in [20]. As stated in the previous section, ℓα=M1,α (or ℓα=I1,α) and bα are implicitly determined by   (4.23), where the exponent p=2. After some transformation,   (4.23) is analyzed in [20], and these extremal values in ℒα balls are finally expressed by some singular integrals. More precisely, for any d∈(0,∞), let
(5.5)A=Aα(d)=d2(α*-1)α*,Tα(d)=4∫01‍dx1-x2+A(1-x2α*),Uα(d)=4(α*A)α∫01‍x2α*dx1-x2+A(1-x2α*),T^α(d)=4∫01‍dxx2-1+A(1-x2α*),U^α(d)=4(α*A)α∫01‍x2α*dxx2-1+A(1-x2α*),
and for any d∈(0,1), let
(5.6)B=Bα(d)=d2(α*-1)=d2/(α-1),Tˇα(d)=4∫01‍dx1-x2-B(1-x2α*)/α*,Uˇα(d)=4Bα∫01‍x2α*dx1-x2-B(1-x2α*)/α*,
and we define
(5.7)Z1,α(x)={0forx=π2,14(2x)2-1/α(Uα(Tα-1(2x)))1/αforx∈(0,π2),R1,α=K(2α*)forx=0,14(2-x)2-1/α(U^α(T^α-1(2-x)))1/αforx∈(-∞,0),Yˇ1,α=14(2x)2-1/α(Uˇα(Uˇα-1T˘α-1(2x)))1/αforx∈[π2,∞).
Then there hold
(5.8)I1,α(r)=Z1,α-1(r),M1,α(r)=Yˇ1,α-1(r).
Note that I1,α(r) and M1,α(r) cannot be written as elementary functions of the radius r. However, after some delicate analysis on the singular integrals Tα, Uα, T^α, U^α, Tˇα and Uˇα, the limits of I1,α(r) and M1,α(r) as α↓1 are proved to be elementary functions of r, that is,
(5.9)M1(r)=14(π+π2+4r)2,(5.10)I1(r)=Z1-1(r),
where
(5.11)Z1(x)={2-xcoth(-x2)forx∈(-∞,0),4forx=0,2xcot(x2)forx∈(0,π2]
is a decreasing diffeomorphism mapping (-∞,π2] onto [0,∞).Extremal valuesI0(r) for the Sturm-Liouville operator are studied in [21], again by singular integral method in the limiting approach. Now ℓα=I0,α, aα, and bα are implicitly determined by   (4.24) with p=2. One can imagine that difficulty increases due to the presence of the additional new parameter aα. We only state the final results here and refer the readers to [21] for details. Let
(5.12)Z0(x)=2-xtanh(-x2),x≤0,
which is a decreasing diffeomorphism from (-∞,0] onto [0,∞). Then there holds
(5.13)I0(r)=Z0-1(r).


## 5.2. Limiting Approach from the Viewpoint of Conservation Laws (p>1)

The first integral   (4.21) of the critical equation   (4.16) can also be written as
(5.14)|yα′(t)|p+ℓαp-1|yα(t)|p-𝔰α(p-1)α*|yα(t)|pα*=eαp-1,t∈[0,1],
where
(5.15)eα∶=(p-1)Fα(bα)=ℓα|bα|p-𝔰αα*|bα|pα*
is a constant. We call   (5.14) a conservation law of   (4.16).Define(5.16)Gα(t):=|yα(t)|pα*α*,t∈[0,1].
It follows from   (4.17) that ∥Gα∥1=rα/α*→0 as α↓1. Passing to a subsequence if necessary,
(5.17)limα↓1Gα(t)=0a.e.t∈[0,1].Motivated by   (5.17), one attempt to compute ℓ0 in   (5.4) is to consider the limit equation of the conservation law   (5.14). Intuitively, if we assume that eα→e0 and yα→y0 in appropriate sense, then it followed from   (5.4) and   (5.17) that the limit equation of   (5.14) should be of the form   (5.29), which is a first-order ODE simpler than   (5.14). Boundary conditions on the critical eigen-functions yα and the restriction that the critical potentials qα∈Sα[r] will give more information on y0. All these conditions on y0 and the solution to   (5.29) will finally lead to the answer to the extremal value ℓ0. Compared with the previous singular integral method, such a limiting approach to ℓ0 from the viewpoint of conservation laws cannot only greatly simplify the analysis but also deal with the p-Laplace operator.In the limiting approach from the viewpoint of conservation laws, it is natural to study the convergence of critical eigen-functionsyα as α↓1. To this end, the boundedness of ∥yα∥p and ∥yα′∥p as α↓1 should be taken into consideration.For both Neumann and Dirichlet eigenvalues, multiplying   (4.16) by yα(t), integrating over [0,1] and taking use of the restriction   (4.17), that is, ∥yα∥pα*pα*=rα, one has
(5.18)-∥yα′∥pp+ℓα∥yα∥pp-𝔰αrα=0.
Integrating   (5.14) over [0,1], one has
(5.19)∥yα′∥pp+ℓαp-1∥yα∥pp-𝔰α(p-1)α*rα=eαp-1.
Eliminating ∥yα′∥pp from the above two equalities, one has
(5.20)pℓα∥yα∥pp=eα+𝔰α(p-1α)rα.
For the principal Neumann eigenvalues, there hold an additional equality as follows:
(5.21)eα∥1yα∥pp=-∥qα∥1α.
This equality can be deduced from   (4.14),   (5.14), and   (4.17).Note that∥ℓα+qα∥1≤|ℓα|+∥qα∥1≤|ℓα|+∥qα∥α=|ℓα|+r is bounded as α↓1. The boundedness of ∥ℓα+qα∥1 and the restriction   (4.17) guarantees (passing to a subsequence if necessary) the convergence of limα↓1bα∈(0,∞) and
(5.22)yα→y0≠0in(𝒞,∥·∥∞)asα↓1.
Here 𝒞=C([0,1],ℝ). For detailed proof, see Lemmas 2.1 and 4.1 in [18]. Consequently, ∥yα∥p is bounded as α↓1, and by   (5.20),
(5.23)eα=(p-1)Fα(bα)=ℓα|bα|p-𝔰α|bα|pα*α*(→e0asα↓1).
By   (4.17), there holds rα≤bαpα*, and hence
(5.24)limα↓1bαpα*≥1.
It followed from   (5.23) and   (5.24) that
(5.25)limα↓1bα=1,e0=ℓ0-limα↓1𝔰α|bα|pα*α*.Note that∥qα∥1≤∥qα∥α=r. On the other hand, it follows from   (4.15) and   (4.17) that
(5.26)∥qα∥1=∫01‍|yα(t)|pα*|yα(t)|pdt≥1∥yα∥∞p∫01‍|yα(t)|pα*dt=rαbαp→r
as α↓1. Thus one has
(5.27)s-limα↓1∥qα∥1=r.By the monotonicity and the symmetry of the eigen-functionyα,   (4.21) can be written as
(5.28)yα′(t)=κp-1(eα-ℓαyαp(t)-𝔰αGα(t))1/p,t∈Iσ,
where κp:=(p-1)1/p. For the Dirichlet eigen-function, yα(0)=0, yα(1/2)=bα and Iσ=ID=[0,1/2]. For the Neumann eigen-function, yα(0)=aα=mint∈[0,1]yα(t), yα(1)=bα and Iσ=IN=[0,1]. Since there holds   (5.17), the Lebesgue-dominated convergence theorem can be applied to the integral equation equivalent to   (5.28) with corresponding boundary conditions. Then it can be proved that y0=limα↓1yα∈C1(Iσ) and
(5.29)y0′=κp-1(e0-ℓ0y0p)1/p,t∈Iσ.
As the limit of Dirichlet eigen-functions yαD, y0 satisfies the boundary conditions as follows:
(5.30)y0(0)=0,y0(12)=1.
As the limit of Neumann eigen-functions yαN, y0 satisfies the boundary conditions as follows:
(5.31)y0(0)=a0=mint∈[0,1]y0(t)∈(0,1),y0(1)=1.
The limiting equality of   (5.20) is
(5.32)pℓ0∥y0∥pp=e0+(p-1)r.
By   (5.27), the limiting equality of   (5.21) is
(5.33)e0∥1y0∥pp=-r.Eitherℓ0=M1(r) or ℓ0=I1(r) is determined by ODE   (5.29), the boundary conditions   (5.30) and the restriction   (5.32). Meanwhile, ℓ0=I0(r) is determined by   (5.29),   (5.31),   (5.32) and   (5.33). To solve   (5.29) and compute ℓ0, we need more information about e0, ℓ0, and a0.Case 1 (maximization problemM1(r)).
In this case, there holdse0=ℓ0>0. In fact, we have
(5.34)ℓ0=M1(r)≥λ1(-r)=r>0.
The rightmost equilibrium in the phase portrait Figure 1 is (ℓα(α-1)/p,0). Thus we have
(5.35)0<bα<ℓα(α-1)/p,limα↓1bαpα*α*≤limα↓1ℓααα*=0.
Consequently, e0=ℓ0 by   (5.25).
Finally,ℓ0=M1(r) can be proved to be the unique root of
(5.36)x-πpx1/p*=r,x∈[0,∞),
where
(5.37)πp=2π(p-1)1/ppsin(π/p).
We refer to [17] for details. Note that if p=2, result   (5.36) is consistent with   (5.9).Case 2 (Minimization problemI1(r)).
In this case, there holde0>0 and e0>ℓ0. In fact, 𝔰α=-1 for the minimization problems. It follows from   (5.14) that
(5.38)eα=(p-1)Fα(bα)≥ℓα|yα(t)|p,∀t∈[0,1].
Set t=0 and 1/2 in   (5.38), respectively, and let α↓1. We see that e0≥0 and e0≥ℓ0. Furthermore, cases e0=0 and e0=ℓ0 can be excluded by checking   (5.32) after solving   (5.29)-(5.30) and taking into account that
(5.39)ℓ0=I1(r)≤λ1(r)=πpp-r.In this case, till now there is no such restrictions as positivity or negativity onℓ0. This is consistent to the phase portrait analysis in Section 4.2.Finally,(e0,ℓ0) (e0>0,e0>ℓ0) can be uniquely determined by
(5.40)∫01‍κpdu(e0-ℓ0up)1/p=12,r=2(p-1)-1/p*(e0-ℓ0)1/p*.
For details we refer to [18]. Note that if p=2, the integral in   (5.40) can be evaluated explicitly, and one can get   (5.10).Case 3 (minimization problemI0(r)).
In this case, there holdℓ0<0 and e0=ℓ0a0p<0. In fact, it follows from   (4.4) that
(5.41)ℓ0=I0(r)<-r<0.
By   (5.33), e0<0. Note that   (5.38) still holds for this minimization problem. Let t=0 in   (5.38) and α↓1. One has e0≥ℓ0a0p. One can exclude the case e0>ℓ0a0p by checking   (5.33) after solving   (5.29)–(5.31).
Finally,(a0,ℓ0) (0<a0<1,ℓ0<0) can be uniquely determined by
(5.42)κp∫a01‍du(ℓ0a0p-ℓ0up)1/p=1,(p-1)-1/p*(ℓ0a0p-ℓ0)1/p*=r.
For details we refer to [18]. Note that if p=2, the integral in   (5.42) can be evaluated explicitly, and one can get   (5.13).


## 6. Extremal Problems for Eigenvalues of Measure DifferentialEquations

Generalized ordinary differential equations (GODEs for short) [33, 34] can describe the jumps of solutions caused by impulses, and so forth. In this paper we will only consider the so-called measure differential equations (MDEs for short) [26, 27], a special class of GODE.By a real measureμ on [0,1], it means that it is an element of the dual space ℳ0:=(𝒞,∥·∥∞)* of the Banach space (𝒞,∥·∥∞). Then, for any u∈𝒞 and any subinterval J of [0,1], the Riemann-Stieltjes integral and the Lebesgue-Stieltjes integral [35]
(6.1)∫[0,1]‍udμ∫J‍udμ
are defined. Given a measure μ∈ℳ0, following the notations in [27], the second-order linear MDE with the measure μ is written as
(6.2)dx•+xdμ(t)=0,t∈[0,1].
The solution x(t) of MDE   (6.2) with the initial value (x(0),x•(0))=(x0,v0)∈ℝ2 is explained using the system of integral equations as follows:
(6.3)x(t)=x0+∫[0,t]‍v(s)ds,t∈[0,1],v(t)={v0,t=0,v0-∫[0,t]‍x(s)dμ(s),t∈(0,1].
Here v(t):=x•(t) is the generalized right-hand derivative of y(t) or the velocity of y(t). By   (6.3), it is well known that solutions of initial value problems of   (6.2) are uniquely defined on [0,1] [33, 34]. See also [26, 27]. From   (6.3), one sees that the solutions themselves are continuous, that is, x∈𝒞. However, v(t)=x•(t) may have discontinuity at those t such that the density dμ(t)/dt does not exist. In case μ∈ℳ0 is absolutely continuous with respect to the Lebesgue measure ℓ on [0,1], that is, dμ(t)/dt=q(t)∈ℒ1, solutions of MDE   (6.2) reduce to that for ODE
(6.4)x′′+q(t)x=0.Letμ∈ℳ0 be a (real) measure on [0,1]. The corresponding eigenvalue problem
(6.5)dx•+λxdt+xdμ(t)=0,t∈[0,1]
has been studied in [26, 27]. Like problem   (1.1), eigenvalues of   (6.5) with the Dirichlet boundary condition x(0)=x(1)=0 are a real, increasing sequence {λnD(μ)}n∈ℕ, accumulating at +∞. With the Neumann boundary condition x•(0)=x•(1)=0, eigenvalues of   (6.5) are a real, increasing sequence {λnN(μ)}n∈ℤ+, again accumulating at +∞.By the Riesz representation theorem [25], the measure space ℳ0 can be characterized using real functions of [0,1] of bounded variations. For μ∈ℳ0, we use ∥μ∥V to denote the total variation of μ on [0,1]. Then (ℳ0,∥·∥V) is a Banach space. Since ℳ0 is the dual space of (𝒞,∥·∥∞), one has also in ℳ0 the weak* topology w* defined as follows: μn→μ in (ℳ0,w*) if and only if
(6.6)limn→∞∫[0,1]‍udμn=∫[0,1]‍udμ∀u∈𝒞.
It is well known that bounded subsets of (ℳ0,∥·∥V) are relatively compact and relatively sequentially compact in (ℳ0,w*) [24, 25].Some deep results on the dependence of eigenvaluesλnσ(μ) of MDE on measures μ are as follows.Theorem 6.1 (see [26, 27]).
(i) In the norm topology∥·∥V, λnσ(μ) is continuously Fréchet differentiable in μ∈(ℳ0,∥·∥V). Moreover, the formula for the Fréchet derivative ∂μλnσ(μ) is similar to   (3.11).
(ii) In the weak* topology w*, λnσ(μ) are continuous in μ∈(ℳ0,w*).Let us introduce spheres and balls of measures as follows:(6.7)S0[r]:={μ∈ℳ0:∥μ∥V=r},B0[r]:={μ∈ℳ0:∥μ∥V≤r},
where r≥0. Because of Theorem 6.1 and the compactness of B0[r] in (ℳ0,w*), the following extremal problems
(6.8)In(r):=minμ∈B0[r]λnσ(μ),Mn(r):=maxμ∈B0[r]λnσ(μ),
where n∈ℤ+ are well posed. Moreover, both the minimum and the maximum can be realized by some measures from B0[r]. Here, when n∈ℕ, the values In(r) and Mn(r) are the same for the Dirichlet and the Neumann eigenvalues. Since the Fréchet derivatives ∂μλnσ(μ) are nonzero, one sees that problems   (6.8) are the same as
(6.9)In(r)≡minμ∈S0[r]λnσ(μ),Mn(r)≡maxμ∈S0[r]λnσ(μ).
That is, the minimizing and maximizing measures of   (6.8) must be on the sphere S0[r].Recall from Theorem6.1 that the functionals λnσ(μ) we are minimizing/maximizing are continuously differentiable. However, S0[r] is not differentiable in the space (ℳ0,∥·∥V). The solution of problems   (6.9) appeals for the LMM using the sub-differentials [36, 37].For the zeroth Neumann eigenvaluesλ0N(μ), problems   (6.9) have been solved using this idea in [22]. The results are as follows.Theorem 6.2 (see [22]).
For anyr≥0, one has
(6.10)I0(r)=minμ∈S0[r]λ0N(μ)=λ0N(rδ0)=λ0N(rδ1)=Z0-1(r),M0(r)=maxμ∈S0[r]λ0N(μ)=λ0N(-rℓ)=r,
where Z0 is defined by   (5.12), and for a∈[0,1], δa denotes the unit Dirac measure located at a, and ℓ is the Lebesgue measure of [0,1].Remark 6.3.
Result   (6.10) can give another explanation to result   (5.13) in Section 5. To see this, one can notice that any integrable potential q∈ℒ1 induces an absolutely continuous measure μq defined by
(6.11)μq(t):=∫[0,t]‍q(s)dsfort∈[0,1].
Since ∥μq∥V=∥q∥1, one sees that B1[r]⊂(ℒ1,∥·∥1) can be isometrically embedded into B0[r]⊂(ℳ0,∥·∥V). Hence one has
(6.12)infq∈B1[r]λ0N(q)≥minμ∈S0[r]λ0N(μ).
On the other hand, the minimizing (singular) measures rδ0 and rδ1 in   (6.9) can be approximated by sequences {μqn} in the weak* topology w* where qn are smooth potentials from B1[r]. Hence we have
(6.13)minμ∈S0[r]λ0N(μ)=limn→∞λ0N(μqn)≥infq∈B1[r]λ0N(q).
Thus one has   (5.13).Let us return to the minimizing/maximizing problems of eigenvalues of   (1.1) and   (3.1) for potentials q in B1[r]. Though B1[r]⊂ℒ1 has no compactness in ∥·∥1 or w1, it has been shown that all maximizing problems
(6.14)supq∈B1[r]λnσ(q)
can be realized by potentials from S1[r]⊂B1[r]. In fact, these maximizing potentials are step potentials. For detailed construction, we refer to [17] for general p∈(1,∞).On the other hand, because of the noncompactness ofB1[r], r>0, the corresponding minimizing problems
(6.15)infq∈B1[r]λnσ(q)
cannot be realized by any potential from B1[r], but by some singular measures from S0[r]⊂B0[r]. The following results hold for eigenvalues of   (1.1), that is, p=2 in   (3.1). For example, from   (6.10) and   (5.13), one has
(6.16)infq∈B1[r]λ0N(q)=λ0N(rδ0)=λ0N(rδ1).
For the first Dirichlet eigenvalues λ1D(q), one has from [20, 26] that
(6.17)infq∈B1[r]λ1D(q)=λ1D(rδ1/2).
The main result of [23] states that
(6.18)infq∈B~1[r]λ0N(q)=λ0N(±r2(δ0-δ1))=-r24,
where B~1[r]={q∈ℒ1:∥q∥1=r,q¯=0}. Results   (6.16)–(6.18) can yield a natural explanation to what kinds of integrable potentials in B1[r] will decrease the eigenvalues.


## 7. Some Open Problems

We end this paper with two open problems.Problem 1 (eigenvalue gaps).
Let us consider, for example, the Dirichlet eigenvaluesλn(q), n∈ℕ, of problem   (1.1) with integrable potentials q∈ℒ1. In applied sciences, it is important to study eigenvalue gaps like λn+1(q)-λn(q), n∈ℕ. See [38–42]. In most literature, only single-well and symmetric potentials are considered, and lower and upper bounds for these gaps are obtained. Because of the boundedness of λn(q) for q in bounded subsets of (ℒ1,∥·∥1), the following extremal problems for eigenvalue gaps:
(7.1)infq∈S1[r](λn+1(q)-λn(q))supq∈S1[r](λn+1(q)-λn(q)),n∈ℕ,
are well posed. The problem is how to solve these explicitly, including the extremal values and the corresponding minimizers/maximizers.Problem 2 (eigenvalues of the beam equation with integrable potentials).
Given an integrable potentialq∈ℒ1, consider the eigenvalue problem of the beam equation, or the Euler-Bernoulli equation [43–45],
(7.2)x(4)+q(t)x=λx,t∈[0,1],
with the Lidstone boundary condition
(7.3)x(0)=x(1)=0,x′′(0)=x′′(1)=0.
Eigenvalues of problem   (7.2)-(7.3) are still denoted by {λn(q)}n∈ℕ. Like the second-order Sturm-Liouville operator   (1.1), it is desirable to develop analogous ideas so that the following extremal problems for eigenvalues:
(7.4)infq∈S1[r]λn(q),supq∈S1[r]λn(q),
and the extremal problems for eigenvalue gaps like
(7.5)infq∈S1[r](λ2(q)-λ1(q)),supq∈S1[r](λ2(q)-λ1(q)),
can be solved in a complete way.



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*Source: 670463-2012-09-25.xml*