A Survey on Extremal Problems of Eigenvalues

Abstract

Given an integrable potential 𝑞∈𝐿¹([0,1],ℝ), the Dirichlet and the Neumann eigenvalues 𝜆^𝐷_𝑛(𝑞) and 𝜆^𝑁_𝑛(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 𝐿^𝛼([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.

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 𝐿^𝛼 norms. Based on such eigenvalue properties and some topological facts on 𝐿^𝛼 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 potential 𝑞∈ℒ¹∶=𝐿¹([0,1],ℝ), all eigenvalues of the Sturm-Liouville operator 𝑥^′′+𝜆+𝑞(𝑡)𝑥=0,𝑡∈[0,1], (1.1) associated with the Dirichlet boundary conditions 𝑥(0)=𝑥(1)=0 are given by a sequence 𝜆₁𝑞<𝜆₂𝑞<⋯<𝜆_𝑛𝑞<⋯, (1.2) with lim_𝑛→∞𝜆_𝑛(𝑞)=+∞. Let us consider such extremal problems as 𝐼_𝑛(𝑟)∶=inf_{𝑞∈𝐵1[𝑟]}𝜆_𝑛𝑞,𝑀_𝑛(𝑟)∶=sup_{𝑞∈𝐵1[𝑟]}𝜆_𝑛𝑞 (1.3) for any 𝑛∈ℕ, where 𝐵₁[𝑟] is a ball of radius 𝑟 in ℒ¹. These problems are interesting and of difficulty, because 𝜆_𝑛(⋅) 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 any 𝛼>1, we consider the counterparts of 𝐼_𝑛 and 𝑀_𝑛 with potentials confined to 𝐵_𝛼[𝑟] (balls in ℒ^𝛼∶=𝐿^𝛼([0,1],ℝ) of radius 𝑟), denoted by 𝐼_𝑛,𝛼 and 𝑀_𝑛,𝛼, respectively. Since the functional 𝜆_𝑛(⋅) is continuous (in weak topology) and continuously differentiable (in the usual 𝐿^𝛼 norm) in potential 𝑞∈ℒ¹ (and hence in 𝑞∈ℒ^𝛼, 𝛼>1), and the balls 𝐵_𝛼[𝑟], 𝛼>1, are compact in weak topology and smooth in the usual ℒ^𝛼 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 𝑞_𝛼, the critical eigenfunction, denoted by 𝑦_𝛼, and the extremal value 𝐼_𝑛,𝛼 or 𝑀_𝑛,𝛼 are all involved.

Secondly, we will employ the limiting approach 𝛼↓1 to obtain 𝐼_𝑛 and 𝑀_𝑛. This step is based on some topological facts that balls and spheres in ℒ¹ space can be approximated by balls and spheres in ℒ^𝛼 spaces as 𝛼↓1. Then the strong continuity of eigenvalue 𝜆_𝑛(⋅) in potentials ensures that lim_𝛼↓1𝐼_𝑛,𝛼=𝐼_𝑛 and lim_𝛼↓1𝑀_𝑛,𝛼=𝑀_𝑛. In this step, besides critical equations in ℒ^𝛼 spaces, properties of critical potentials and critical eigen-functions should also be sufficiently utilized to get a final solution to 𝐼_𝑛 and 𝑀_𝑛.

The final solution to problems in (1.3) can yield optimal lower and upper bounds of eigenvalues. Actually, from (1.3), we have 𝐼_𝑛𝑞₁≤𝜆_𝑛𝑞≤𝑀_𝑛𝑞₁∀𝑞∈ℒ¹, (1.4) which are optimal.

Several extremal problems on (Dirichlet, Neumann, and periodic) eigenvalues (of the Sturm-Liouville operator and the 𝑝-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 ℒ^𝛼, 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 ℒ^𝛼, 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 ℒ^𝛼, 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 on ℒ^𝛼 Spaces

In the Lebesgue space ℒ^𝛼, 1≤𝛼≤∞, the usual ℒ^𝛼 topology is induced by ℒ^𝛼 norm ‖⋅‖_𝛼. Besides this strong topology, one has also the weak topology 𝑤_𝛼 which is defined as follows [24, 25].

Definition 2.1. Let 𝑞_𝑛,𝑞∈ℒ^𝛼. We say that 𝑞_𝑛 is weakly convergent to 𝑞, written as 𝑞−−→_𝑛^𝑤𝛼𝑞 in ℒ^𝛼, or 𝑞_𝑛→𝑞 in (ℒ^𝛼,𝑤_𝛼), if lim_𝑛→∞¹₀𝑞_𝑛(𝑡)𝑢(𝑡)d𝑡=¹₀𝑞(𝑡)𝑢(𝑡)d𝑡∀𝑢∈ℒ^𝛼∗. (2.1) Here 𝛼^∗ is the conjugate exponent of 𝛼: 𝛼^∗=𝛼/(𝛼−1)∈[1,∞].

For 𝛼∈[1,∞), 𝑟∈[0,∞) and ℎ∈[−𝑟,𝑟], let us take the following notations: 𝑆_𝛼[𝑟]=𝑞∈ℒ^𝛼∶𝑞_𝛼,𝐵=𝑟_𝛼[𝑟]=𝑞∈ℒ^𝛼∶𝑞_𝛼,𝑆≤𝑟_𝛼[𝑟,ℎ]=𝑞∈ℒ^𝛼∶𝑞_𝛼=𝑟,,𝐵𝑞=ℎ_𝛼[𝑟,ℎ]=𝑞∈ℒ^𝛼∶𝑞_𝛼≤𝑟,𝑞=ℎ, (2.2) where 𝑞=∫¹₀𝑞(𝑡)d𝑡 is the mean value of 𝑞.

2.1. Approximation to ℒ¹ Balls/Spheres in Strong Topology

Lemma 2.2 (see [21, Lemma 2.1]). Given that 𝑟>0 and 𝑞∈𝑆₁[𝑟], there exists 𝑞_𝛼∈𝑆_𝛼[𝑟] such that lim_𝛼↓1‖𝑞_𝛼−𝑞‖₁=0.

Lemma 2.3 (see [19, Lemma 2.3]). Let 𝑟>0 and ℎ∈(−𝑟,𝑟). For any 𝑞∈𝑆₁[𝑟,ℎ], there exists {𝑞_𝛼}∈𝑆_𝛼[𝑟,ℎ], 1<𝛼<∞, such that lim_𝛼↓1‖𝑞_𝛼−𝑞‖₁=0.

Lemmas 2.2 and 2.3 give very nice topological approximation to spheres in ℒ¹ space because the spheres in ℒ^𝛼 (1<𝛼<∞) space have nicer topological and geometric properties.

Remark 2.4. A direct consequence from Lemmas 2.2 and 2.3 is that, ℒ¹ balls 𝐵₁[𝑟] and 𝐵₁[𝑟,ℎ] can be approximated, in the sense of ℒ¹ norm, by ℒ^𝛼 balls 𝐵_𝛼[𝑟] and 𝐵_𝛼[𝑟,ℎ], respectively.

2.2. Approximation to ℒ¹ Balls/Spheres in Weak Topology

For any 𝛼∈(1,∞), it is well known that 𝐵_𝛼[𝑟] is compact and sequentially compact in the space (ℒ^𝛼,𝑤_𝛼) [24, 25].

Lemma 2.5 (see [19, Lemma 2.5]). Suppose that 𝛼∈(1,∞). Then, for any 𝑟>0 and ℎ∈[−𝑟,𝑟], 𝐵_𝛼[𝑟,ℎ] is compact and sequentially compact in the space (ℒ^𝛼,𝑤_𝛼).

The balls 𝐵₁[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 𝐵_𝛼[𝑟] and 𝐵_𝛼[𝑟,ℎ], which are compact in weak topology 𝑤_𝛼 and smooth in geometry. In fact, there hold the following stronger topological facts.

Lemma 2.6 (see [19, Lemma 2.6]). (i) Let 𝛼∈[1,∞), 𝑟∈(0,∞) and ℎ∈(−𝑟,𝑟). For any 𝑞∈𝐵_𝛼[𝑟,ℎ], there exists a sequence {𝑞_𝑛}⊂𝑆_𝛼[𝑟,ℎ] such that 𝑞_𝑛→𝑞 in (ℒ^𝛼,𝑤_𝛼). In other words, the closure of 𝑆_𝛼[𝑟,ℎ] in the space (ℒ^𝛼,𝑤_𝛼) is 𝐵_𝛼[𝑟,ℎ].
(ii) Consequently, for any 𝑞∈𝑆_𝛼[𝑟,ℎ], 1≤𝛼<∞, there exists a sequence {𝑞_𝑛}⊂𝑆₁[𝑟,ℎ] such that 𝑞_𝑛→𝑞 in (ℒ¹,𝑤₁).

3. Properties of Eigenvalues on Potentials/Weights

Denote by 𝜙_𝑝(⋅) the scalar 𝑝-Laplacian, that is, 𝜙_𝑝(𝑥)=|𝑥|^𝑝−2𝑥 for any 𝑥≠0 and 𝜙_𝑝(0)=0. For any integrable potential 𝑞∈𝐿¹([𝑎,𝑏],ℝ), it is well known that all eigenvalues of 𝜙_𝑝𝑥^′′+𝜆+𝑞(𝑡)𝜙_𝑝(𝑥)=0,𝑡∈[𝑎,𝑏] (3.1) associated with the Dirichlet boundary condition 𝑥(𝑎)=𝑥(𝑏)=0 consist of a sequence 𝜆^𝐷₁𝑞<𝜆^𝐷₂𝑞<⋯<𝜆^𝐷_𝑛𝑞<⋯, (3.2) and all eigenvalues associated with the Neumann boundary condition 𝑥′(𝑎)=𝑥′(𝑏)=0 consist of a sequence as follows: 𝜆^𝑁₀𝑞<𝜆^𝑁₁𝑞<⋯<𝜆^𝑁_𝑛𝑞<⋯, (3.3) see [28, 29]. To emphasize the dependence on the interval [𝑎,𝑏] and the boundary conditions, we also write the eigenvalues as 𝜆^𝜎_𝑛(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

Let 𝐼=[𝑎,𝑏] be a finite interval of length |𝐼|=𝑏−𝑎. Given that 𝛼∈[1,∞] and 𝑞∈ℒ^𝛼∶=𝐿^𝛼([0,1],ℝ), define a potential 𝑞_𝐼∈𝐿^𝛼(𝐼,ℝ) by the following: 𝑞_𝐼(𝑡)∶=|𝐼|^−𝑝𝑞(𝑡−𝑎)|𝐼|,𝑡∈𝐼. (3.4)

Lemma 3.1 (see [17, Lemma 2.1]). Let 𝑞 and 𝑞_𝐼 be as in (3.4). Then for any admissible 𝑛, there hold 𝑞_{𝐼𝐿𝛼(𝐼)}=|𝐼|^{−𝑝+1/𝛼}𝑞_𝛼,𝜆^𝜎_𝑛𝑞_𝐼,𝐼=|𝐼|^−𝑝𝜆^𝜎_𝑛𝑞,[0,1], (3.5) 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∈ℒ^𝛼.

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

Let us identify ℒ^𝛼=𝐿^𝛼([0,1],ℝ) with 𝐿^𝛼(ℝ/ℤ,ℝ), where 𝛼∈[1,∞]. For 𝑛≥2, define the mapping 𝒯_𝑛∶ℒ^𝛼→ℒ^𝛼 by 𝒯_𝑛𝑞=𝑞_𝑛,𝑞_𝑛(𝑡)∶=𝑛^𝑝𝑞(𝑛𝑡),𝑡∈ℝ. (3.6)

Lemma 3.2 (see [17, Lemma 2.2]). For any integer 𝑛≥2, let 𝑞 and 𝑞_𝑛 be as in (3.6). Then there hold 𝑞_𝑛=𝑛^𝑝𝑞,𝑞_𝑛𝛼=𝑛^𝑝𝑞_𝛼,𝜆^𝜎_𝑛𝑞_𝑛=𝑛^𝑝𝜆^𝜎₁𝑞, (3.7) where the superscript 𝜎 can be chosen to be 𝐷 or 𝑁.

When 𝑛≥2, 𝒯_𝑛 of (3.6) is only injective but not surjective from 𝐵_𝛼[𝑟/𝑛^𝑝] to 𝐵_𝛼[𝑟]. It followed from the last two equalities in (3.7) that sup_{𝑞∈𝐵𝛼[𝑟]}𝜆^𝜎_𝑛𝑞≥𝑛^𝑝⋅sup_{𝑞∈𝐵𝛼[𝑟/𝑛𝑝]}𝜆^𝜎₁𝑞,inf_{𝑞∈𝐵𝛼[𝑟]}𝜆^𝜎_𝑛𝑞≤𝑛^𝑝⋅inf_{𝑞∈𝐵𝛼[𝑟/𝑛𝑝]}𝜆^𝜎₁𝑞, (3.8) 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]). Given 𝛼∈[1,∞] and an admissible 𝑛. The functional ℒ^𝛼,‖⋅‖_𝛼→ℝ,𝑞↦𝜆^𝜎_𝑛𝑞 (3.9) is continuously Fréchet differentiable. The Fréchet derivative 𝜕_𝑞𝜆^𝜎_𝑛(𝑞)∈(ℒ^𝛼,‖⋅‖_𝛼)^∗ is the following bounded linear functional: ℒ^𝛼∋ℎ↦−¹₀𝐸^𝜎_𝑛(𝑡)^𝑝ℎ(𝑡)d𝑡∈ℝ, (3.10) or written simply as 𝜕_𝑞𝜆^𝜎_𝑛𝑞=−𝐸^𝜎_𝑛^𝑝. (3.11) Here 𝐸^𝜎_𝑛(𝑡)=𝐸^𝜎_𝑛(𝑡;𝑞) is a normalized eigenfunction associated with 𝜆^𝜎_𝑛(𝑞) so that the 𝐿^𝑝 norm ‖𝐸^𝜎_𝑛‖_𝑝=1.

Remark 3.4. Since (3.11) is always a negative functional of ℒ^𝛼, 𝛼∈[1,∞], eigenvalues possess the following monotonicity: 𝑞₁,𝑞₂∈ℒ¹,𝑞₁≤𝑞₂⇒𝜆^𝜎_𝑛𝑞₁≥𝜆^𝜎_𝑛𝑞₂. (3.12) 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]). Given 𝛼∈[1,∞] and an admissible 𝑛, the following functional is continuous: ℒ^𝛼,𝑤_𝛼→ℝ,𝑞↦𝜆^𝜎_𝑛𝑞. (3.13)

This theorem shows that eigenvalues 𝜆^𝜎_𝑛(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 in ℒ^𝛼 Balls, 1<𝛼<∞

In this section, we always assume that 1<𝛼<∞, 𝑟>0, |ℎ|<𝑟 and 𝜎=𝐷 or 𝑁.

4.1. Preliminary Results for Minimization/Maximization Problems

For any admissible integer 𝑛, let us take some notations as follows 𝐈_𝑛,𝛼(𝑟)∶=inf_{𝑞∈𝐵𝛼[𝑟]}𝜆^𝜎_𝑛𝑞,𝐌_𝑛,𝛼(𝑟)∶=sup_{𝑞∈𝐵𝛼[𝑟]}𝜆^𝜎_𝑛𝑞,ℑ_𝑛,𝛼(𝑟,ℎ)∶=inf_{𝑞∈𝐵𝛼[]𝑟,ℎ}𝜆^𝜎_𝑛𝑞,𝔐_𝑛,𝛼(𝑟,ℎ)∶=sup_{𝑞∈𝐵𝛼[𝑟,ℎ]}𝜆^𝜎_𝑛𝑞. (4.1) 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, 𝐈_𝑛,𝛼(𝑟)=inf_{𝑞∈𝑆𝛼[𝑟]}𝜆^𝜎_𝑛𝑞,𝐌_𝑛,𝛼(𝑟)=sup_{𝑞∈𝑆𝛼[𝑟]}𝜆^𝜎_𝑛𝑞,ℑ_𝑛,𝛼(𝑟,ℎ)=inf_{𝑞∈𝑆𝛼[]𝑟,ℎ}𝜆^𝜎_𝑛𝑞,𝔐_𝑛,𝛼(𝑟,ℎ)=sup_{𝑞∈𝑆𝛼[𝑟,ℎ]}𝜆^𝜎_𝑛𝑞. (4.2)(4.3) Equation (4.2) follows immediately from (3.12), the monotonicity of eigenvalues. Equation (4.3) follows from the facts that the closure of 𝑆_𝛼[𝑟,ℎ] in the space (ℒ^𝛼,𝑤_𝛼) is 𝐵_𝛼[𝑟,ℎ] (See Lemma 2.6(i)), and 𝜆_𝑛(𝑞) is strongly continuous in 𝑞∈(ℒ^𝛼,𝑤_𝛼).

Note that 𝐵_𝛼[𝑟] and 𝐵_𝛼[𝑟,ℎ] are compact and sequentially compact in the space (ℒ^𝛼,𝑤_𝛼) (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 𝑆_𝛼[𝑟,ℎ] is a case apart, unlike any other. It is well known that [31, Lemma 3.3] 𝜆^𝑁₀𝑞≤−𝑞∀𝑞,𝜆^𝑁₀𝑞=−𝑞⟺𝑞isconstant. (4.4) Therefore one has 𝔐_0,𝛼(𝑟,ℎ)=max_{𝑞∈𝐵𝛼[𝑟,ℎ]}𝜆^𝑁₀𝑞=sup_{𝑞∈𝑆𝛼[𝑟,ℎ]}𝜆^𝑁₀𝑞=𝜆^𝑁₀(ℎ)=−ℎ. (4.5) Moreover, 𝔐_0,𝛼(𝑟,ℎ) cannot be attained by any potential on 𝑆_𝛼[𝑟,ℎ]. Consequently, 𝐌_0,𝛼(𝑟)=max_{𝑞∈𝐵𝛼[𝑟]}𝜆^𝑁₀𝑞=𝜆^𝑁₀(−𝑟)=𝑟=max_{𝑞∈𝑆𝛼[𝑟]}𝜆^𝑁₀𝑞. (4.6) All other minimizers and maximizers are located exactly on the boundaries of the balls, that is, 𝐈_𝑛,𝛼(𝑟)=min_{𝑞∈𝑆𝛼[𝑟]}𝜆^𝜎_𝑛𝑞𝐌,𝑛=0,1,2,…_𝑛,𝛼(𝑟)=max_{𝑞∈𝑆𝛼[𝑟]}𝜆^𝜎_𝑛𝑞ℑ,𝑛=1,2,3,…_𝑛,𝛼(𝑟,ℎ)=min_{𝑞∈𝑆𝛼[]𝑟,ℎ}𝜆^𝜎_𝑛𝑞𝔐,𝑛=0,1,2,…_𝑛,𝛼(𝑟,ℎ)=max_{𝑞∈𝑆𝛼[]𝑟,ℎ}𝜆^𝜎_𝑛𝑞,𝑛=1,2,3,…. (4.7)(4.8)(4.9)(4.10) Again, (4.7) and (4.8) are immediate consequences from monotonicity of eigenvalues. Especially, the minimizers/maximizers on the sphere 𝑆_𝛼[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∈𝐵_𝛼[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 𝐵_𝛼[𝑟,ℎ].

4.2. Variational Construction for Minimizer/Maximizer in 𝐵_𝛼[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∈(ℒ^𝛼,‖⋅‖_𝛼), with the Fréchet derivative as follows: 𝜕_𝑞𝑞_𝛼=𝑞_𝛼^1−𝛼𝜙_𝛼𝑞, (4.11) where 𝜙_𝛼∶ℝ→ℝ is 𝜙_𝛼(𝑥)∶=|𝑥|^𝛼−2𝑥. The eigenvalues 𝜆^𝜎_𝑛(𝑞) are also Fréchet differentiable in 𝑞, and the Fréchet derivative is as in (3.11). By the Lagrangian Multiplier Method, the critical potential 𝑞_𝛼=𝑞_{𝑛,𝑟,𝛼} satisfies the following equation: −𝐸_𝛼(𝑡)^𝑝=𝑐_𝛼𝜙_𝛼𝑞_𝛼(𝑡)a.e.𝑡∈[0,1] (4.12) for some constant 𝑐_𝛼=𝑐_{𝑛,𝑟,𝛼}. Here 𝐸_𝛼(𝑡)=𝐸^𝜎_{𝑛,𝑟,𝛼}(𝑡) is a normalized eigen-function associated with 𝜆^𝜎_𝑛(𝑞_𝛼)=ℓ_𝛼. Since the minimizer is nonnegative and the maximizer is nonpositive, 𝑐_𝛼<0 for the minimization problem (4.7) and 𝑐_𝛼>0 for the maximization problem (4.8). Let 𝔰_𝛼=sign𝑐_𝛼=±1 and 𝑦_𝛼(𝑡)=𝔰_𝛼𝐸_𝛼(𝑡)_𝑝𝑐_𝛼. (4.13) Then 𝑦_𝛼=𝑦^𝜎_{𝑛,𝑟,𝛼} is also an eigenfunction associated with ℓ_𝛼. Specifically, we identify it by 𝑦^′_𝛼(0)>0 for Dirichlet case and 𝑦_𝛼(0)>0 for Neumann case. This identified 𝑦_𝛼 is called the critical eigenfunction, and it satisfies the original ODE (3.1), that is, 𝜙_𝑝𝑦^′_𝛼^′+ℓ_𝛼𝜙_𝑝𝑦_𝛼+𝑞_𝛼(𝑡)𝜙_𝑝𝑦_𝛼=0, (4.14) and corresponding boundary conditions. By (4.12), the critical potential can also be written as 𝑞_𝛼(𝑡)=−𝔰_𝛼𝜙_𝛼∗𝑦_𝛼(𝑡)^𝑝=−𝔰_𝛼𝑦_𝛼(𝑡)^{𝑝(𝛼∗−1)}. (4.15) Substituting (4.15) into (4.14), one has 𝜙_𝑝𝑦^′_𝛼^′+ℓ_𝛼𝜙_𝑝𝑦_𝛼−𝔰_𝛼𝜙_𝑝𝛼∗𝑦_𝛼=0, (4.16) 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 𝜆_𝑛(𝑞). By (4.15), the constraint ‖𝑞_𝛼‖_𝛼=𝑟 is transformed to ¹₀𝑦_𝛼(𝑡)^𝑝𝛼∗d𝑡=𝑟^𝛼. (4.17)

Remark 4.1. Due to the autonomy and the symmetry of critical equation (4.16), 𝑦_{𝑛,𝑟,𝛼}(𝑡) is a periodic solution of (4.16) with minimal period 2/𝑛. In fact, one has 𝑦_{𝑛,𝑟,𝛼}(𝑡+1/𝑛)≡−𝑦_{𝑛,𝑟,𝛼}(𝑡) (we refer the proof of this to Lemma 3.2 in [20]). By (4.15), 𝑞_{𝑛,𝑟,𝛼}(𝑡) is periodic with minimal period 1/𝑛. Therefore, the maximizer 𝑞_{𝑛,𝑟,𝛼} is in the range of 𝒯_𝑛. See (3.6). This tells us that the converse inequalities of (3.8) are also true. More precisely, there hold 𝐈_𝑛,𝛼(𝑟)=𝑛^𝑝𝐈_1,𝛼𝑟𝑛^𝑝,𝐌_𝑛,𝛼(𝑟)=𝑛^𝑝𝐌_1,𝛼𝑟𝑛^𝑝 (4.18) for any integer 𝑛≥2 and any 𝑟>0. This is why we need only to consider extremal values of 𝜆^𝑁₀(𝑞) and 𝜆^𝜎₁(𝑞).

The phase portraits of (4.16) are distinguished in three cases. See Figures 1–3. In Figure 1, 𝔰_𝛼=1. This figure corresponds to the maximization problem 𝐌_1,𝛼, and the eigen-function 𝑦_𝛼 is certain nonconstant periodic orbit surrounding the equilibrium (0,0). In Figure 2, 𝔰_𝛼=−1 and ℓ_𝛼<0. The minimization problem 𝐈_0,𝛼 is illustrated in this figure, and the positive eigen-function 𝑦_𝛼 corresponds to some non-constant periodic orbit surrounding the rightmost equilibrium. In Figure 3, 𝔰_𝛼=−1 and ℓ_𝛼≥0. When the minimization problem 𝐈_1,𝛼 is considered, both Figures 2 and 3 should be taken into account. In fact, the bigger 𝑟 is, the smaller 𝐈_1,𝛼(𝑟) is. For 𝑟 large enough, 𝐈_1,𝛼(𝑟) is negative and 𝑦_𝛼 should be some sign-changing periodic orbit outside the homoclinic orbits in Figure 2. For 𝑟>0 small enough, 𝐈_1,𝛼(𝑟) is nonnegative, and 𝑦_𝛼 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 parameter ℓ_𝛼 in (4.16), introduce an additional parameter as follows: 𝑏_𝛼∶=max_[]𝑡∈0,1𝑦_𝛼(𝑡)=𝑦_𝛼∞, (4.19) and besides, for case 𝑛=0 one more parameter as follows: 𝑎_𝛼∶=min_𝑡∈[0,1]𝑦_𝛼(𝑡). (4.20) A first integral of (4.16) is 𝑦^′_𝛼(𝑡)^𝑝+𝐹_𝛼𝑦_𝛼(𝑡)=𝐹_𝛼𝑏_𝛼,𝑡∈[0,1], (4.21) where 𝐹_𝛼(𝑥)=𝐹_𝛼𝑥,ℓ_𝛼∶=𝑝−1⁻¹ℓ_𝛼|𝑥|^𝑝−𝔰_𝛼|𝑥|^𝑝𝛼∗𝛼^∗,𝑥∈ℝ. (4.22)

Note that the minimal period of 𝑦_𝛼 is 2, and 𝑦_𝛼 satisfies the constraint (4.17). Theoretically, ℓ_𝛼=𝐌_1,𝛼 (or ℓ_𝛼=𝐈_1,𝛼) and 𝑏_𝛼 are implicitly determined by singular integrals as follows: ^𝑏𝛼₀d𝑥𝐹_𝛼𝑏_𝛼−𝐹_𝛼(𝑥)^1/𝑝=12_𝑏𝛼0,𝑥^𝑝𝛼∗d𝑥𝐹_𝛼𝑏_𝛼−𝐹_𝛼(𝑥)^1/𝑝=𝑟^𝛼2, (4.23) where 𝔰_𝛼=1 in (4.22) for 𝐌_1,𝛼, and 𝔰_𝛼=−1 for 𝐈_1,𝛼.

Choose 𝔰_𝛼=−1 in (4.22). Then ℓ_𝛼=𝐈_0,𝛼, 𝑎_𝛼 and 𝑏_𝛼 are implicitly determined by ^𝑏𝛼_𝑎𝛼d𝑥𝐹_𝛼𝑏_𝛼−𝐹_𝛼(𝑥)^1/𝑝=1,_{𝑏𝛼𝑎𝛼}𝑥^𝑝𝛼∗d𝑥𝐹_𝛼𝑏_𝛼−𝐹_𝛼(𝑥)^1/𝑝=𝑟^𝛼,𝐹_𝛼𝑎_𝛼=𝐹_𝛼𝑏_𝛼. (4.24)

4.3. Variational Construction for Minimizer/Maximizer in 𝐵_𝛼[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 𝜙_𝑝𝑦^′_𝛼^′+ℓ_𝛼𝜙_𝑝𝑦_𝛼−𝔰_𝛼𝜙_𝛼∗𝑦^𝑝_𝛼−𝔪_𝛼𝑦_𝛼=0, (4.25) where 𝔰_𝛼=1 for maximization problems and 𝔰_𝛼=−1 for minimization problems. The existence of the new constant 𝔪_𝛼 is caused by the constraint 𝑞=ℎ. The critical potential is 𝑞_𝛼(𝑡)≡−𝔰_𝛼𝜙_𝛼∗𝑦^𝑝_𝛼(𝑡)−𝔪_𝛼. (4.26) The constraint ‖𝑞_𝛼‖_𝛼=𝑟 is ¹₀𝑦^𝑝_𝛼(𝑡)−𝔪_𝛼^𝛼∗d𝑡=𝑟^𝛼. (4.27) The constraint 𝑞_𝛼=ℎ is ¹₀𝜙_𝛼∗𝑦^𝑝_𝛼(𝑡)−𝔪_𝛼d𝑡=−𝔰_𝛼ℎ. (4.28) A first integral of the critical equation is 𝑦^′_𝛼^𝑝+ℓ_𝛼𝑦𝑝−1^𝛼𝑝−𝔰_𝛼𝑦^𝑝_𝛼−𝔪_𝛼^𝛼∗𝑝−1𝛼^∗=const. (4.29)

The existence of 𝔪_𝛼 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 in ℒ¹ Balls

In this section we will solve the following extremal problems: 𝐈_𝑛(𝑟)=inf_{𝑞∈𝐵1[𝑟]}𝜆^𝜎_𝑛𝑞,𝐌_𝑛(𝑟)=sup_{𝑞∈𝐵1[𝑟]}𝜆^𝜎_𝑛𝑞,ℑ_𝑛(𝑟,ℎ)∶=inf_{𝑞∈𝐵1[]𝑟,ℎ}𝜆^𝜎_𝑛𝑞,𝔐_𝑛(𝑟,ℎ)∶=sup_{𝑞∈𝐵1[𝑟,ℎ]}𝜆^𝜎_𝑛𝑞. (5.1)(5.2)

By the topological facts in Section 2 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 ℒ^𝛼 balls, that is, 𝐈_𝑛(𝑟)=lim_𝛼↓1𝐈_𝑛,𝛼(𝑟),𝐌_𝑛(𝑟)=lim_𝛼↓1𝐌_𝑛,𝛼(𝑟),ℑ_𝑛(𝑟,ℎ)=lim_𝛼↓1ℑ_𝑛,𝛼(𝑟,ℎ),𝔐_𝑛(𝑟,ℎ)=lim_𝛼↓1𝔐_𝑛,𝛼(𝑟,ℎ), (5.3) 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 notation ℓ₀ to denote the extremal values 𝐈₀(𝑟), 𝐈₁(𝑟), or 𝐌₁(𝑟), respectively, in different extremal values problems, that is, ℓ₀=lim_𝛼↓1ℓ_𝛼. (5.4) 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 {ℓ_𝛼}_𝛼>1 is bounded.

Two different limiting approaches to this limit ℓ₀ 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 ℓ_𝛼 in 𝐿^𝛼 balls and then let 𝛼↓1. Since ℓ_𝛼 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 values 𝐈₁(𝑟) and 𝐌₁(𝑟) for the Sturm-Liouville operator are studied in [20]. As stated in the previous section, ℓ_𝛼=𝐌_1,𝛼 (or ℓ_𝛼=𝐈_1,𝛼) and 𝑏_𝛼 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 ℒ^𝛼 balls are finally expressed by some singular integrals. More precisely, for any 𝑑∈(0,∞), let 𝐴=𝐴_𝛼(𝑑)=𝑑^{2(𝛼∗−1)}𝛼^∗,𝐓_𝛼(𝑑)=4¹₀d𝑥1−𝑥²+𝐴1−𝑥^2𝛼∗,𝐔_𝛼(𝑑)=4𝛼^∗𝐴^𝛼1₀𝑥^2𝛼∗d𝑥1−𝑥²+𝐴1−𝑥^2𝛼∗,𝐓_𝛼(𝑑)=4¹₀d𝑥𝑥²−1+𝐴1−𝑥^2𝛼∗,𝐔_𝛼(𝑑)=4𝛼^∗𝐴^𝛼1₀𝑥^2𝛼∗d𝑥𝑥²−1+𝐴1−𝑥^2𝛼∗, (5.5) and for any 𝑑∈(0,1), let 𝐵=𝐵_𝛼(𝑑)=𝑑^{2(𝛼∗−1)}=𝑑^2/(𝛼−1),𝐓̌_𝛼(𝑑)=4¹₀d𝑥1−𝑥²−𝐵1−𝑥^2𝛼∗/𝛼^∗,𝐔̌_𝛼(𝑑)=4𝐵^𝛼1₀𝑥^2𝛼∗d𝑥1−𝑥²−𝐵1−𝑥^2𝛼∗/𝛼^∗, (5.6) and we define 𝐙_1,𝛼(𝑥)=0for𝑥=𝜋²,142√𝑥^2−1/𝛼𝐔_𝛼(𝐓_𝛼⁻¹(2√𝑥))^1/𝛼for𝑥∈0,𝜋²,𝐑_1,𝛼=𝐊2𝛼^∗1for𝑥=0,42√−𝑥^2−1/𝛼𝐔_𝛼(𝐓_𝛼⁻¹(2√−𝑥))^1/𝛼(),𝐘̌for𝑥∈−∞,0_1,𝛼=142√𝑥^2−1/𝛼𝐔̌_𝛼𝐔̌_𝛼⁻¹𝐓̆_𝛼⁻¹2√𝑥^1/𝛼for𝑥∈𝜋²,∞. (5.7) Then there hold 𝐈_1,𝛼(𝑟)=𝐙⁻¹_1,𝛼(𝑟),𝐌_1,𝛼(𝑟)=𝐘̌⁻¹_1,𝛼(𝑟). (5.8) Note that 𝐈_1,𝛼(𝑟) and 𝐌_1,𝛼(𝑟) cannot be written as elementary functions of the radius 𝑟. However, after some delicate analysis on the singular integrals 𝐓_𝛼, 𝐔_𝛼, 𝐓_𝛼, 𝐔_𝛼, 𝐓̌_𝛼 and 𝐔̌_𝛼, the limits of 𝐈_1,𝛼(𝑟) and 𝐌_1,𝛼(𝑟) as 𝛼↓1 are proved to be elementary functions of 𝑟, that is, 𝐌₁(𝑟)=14𝜋+√𝜋²+4𝑟²,𝐈₁(𝑟)=𝐙₁⁻¹(𝑟), (5.9)(5.10) where 𝐙₁(𝑥)=2√√−𝑥coth−𝑥2(),2√for𝑥∈−∞,04for𝑥=0,√𝑥cot𝑥2for𝑥∈0,𝜋² (5.11) is a decreasing diffeomorphism mapping (−∞,𝜋²] onto [0,∞).

Extremal values 𝐈₀(𝑟) for the Sturm-Liouville operator are studied in [21], again by singular integral method in the limiting approach. Now ℓ_𝛼=𝐈_0,𝛼, 𝑎_𝛼, and 𝑏_𝛼 are implicitly determined by (4.24) with 𝑝=2. One can imagine that difficulty increases due to the presence of the additional new parameter 𝑎_𝛼. We only state the final results here and refer the readers to [21] for details. Let 𝐙₀(𝑥)=2√√−𝑥tanh−𝑥2,𝑥≤0, (5.12) which is a decreasing diffeomorphism from (−∞,0] onto [0,∞). Then there holds 𝐈₀(𝑟)=𝐙₀⁻¹(𝑟). (5.13)

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 𝑦^′_𝛼(𝑡)^𝑝+ℓ_𝛼𝑦𝑝−1^𝛼(𝑡)^𝑝−𝔰_𝛼𝑝−1𝛼^∗𝑦_𝛼(𝑡)^𝑝𝛼∗=𝑒_𝛼[],𝑝−1,𝑡∈0,1 (5.14) where 𝑒_𝛼∶=𝑝−1𝐹_𝛼𝑏_𝛼=ℓ_𝛼𝑏_𝛼^𝑝−𝔰_𝛼𝛼^∗𝑏_𝛼^𝑝𝛼∗ (5.15) is a constant. We call (5.14) a conservation law of (4.16).

Define 𝐺_𝛼(𝑡)𝑦∶=^𝛼(𝑡)^𝑝𝛼∗𝛼^∗,𝑡∈[0,1]. (5.16) It follows from (4.17) that ‖𝐺_𝛼‖₁=𝑟^𝛼/𝛼^∗→0 as 𝛼↓1. Passing to a subsequence if necessary, lim_𝛼↓1𝐺_𝛼(𝑡)=0a.e.𝑡∈[0,1]. (5.17)

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 𝑒_𝛼→𝑒₀ and 𝑦_𝛼→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 𝑦_𝛼 and the restriction that the critical potentials 𝑞_𝛼∈𝑆_𝛼[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-functions 𝑦_𝛼 as 𝛼↓1. To this end, the boundedness of ‖𝑦_𝛼‖_𝑝 and ‖𝑦^′_𝛼‖_𝑝 as 𝛼↓1 should be taken into consideration.

For both Neumann and Dirichlet eigenvalues, multiplying (4.16) by 𝑦_𝛼(𝑡), integrating over [0,1] and taking use of the restriction (4.17), that is, ‖𝑦_𝛼‖^𝑝𝛼∗_𝑝𝛼∗=𝑟^𝛼, one has −𝑦^′_𝛼^𝑝_𝑝+ℓ_𝛼𝑦_𝛼^𝑝_𝑝−𝔰_𝛼𝑟^𝛼=0. (5.18) Integrating (5.14) over [0,1], one has 𝑦^′_𝛼^𝑝_𝑝+ℓ_𝛼𝑦𝑝−1^𝛼𝑝𝑝−𝔰_𝛼𝑝−1𝛼^∗𝑟^𝛼=𝑒_𝛼.𝑝−1 (5.19) Eliminating ‖𝑦^′_𝛼‖^𝑝_𝑝 from the above two equalities, one has 𝑝ℓ_𝛼𝑦_𝛼^𝑝_𝑝=𝑒_𝛼+𝔰_𝛼1𝑝−𝛼𝑟^𝛼. (5.20) For the principal Neumann eigenvalues, there hold an additional equality as follows: 𝑒_𝛼1𝑦_𝛼^𝑝_𝑝𝑞=−^𝛼1𝛼. (5.21) This equality can be deduced from (4.14), (5.14), and (4.17).

Note that ‖ℓ_𝛼+𝑞_𝛼‖₁≤|ℓ_𝛼|+‖𝑞_𝛼‖₁≤|ℓ_𝛼|+‖𝑞_𝛼‖_𝛼=|ℓ_𝛼|+𝑟 is bounded as 𝛼↓1. The boundedness of ‖ℓ_𝛼+𝑞_𝛼‖₁ and the restriction (4.17) guarantees (passing to a subsequence if necessary) the convergence of lim_𝛼↓1𝑏_𝛼∈(0,∞) and 𝑦_𝛼→𝑦₀≠0in𝒞,‖⋅‖_∞as𝛼↓1. (5.22) Here 𝒞=𝐶([0,1],ℝ). For detailed proof, see Lemmas 2.1 and 4.1 in [18]. Consequently, ‖𝑦_𝛼‖_𝑝 is bounded as 𝛼↓1, and by (5.20), 𝑒_𝛼=𝑝−1𝐹_𝛼𝑏_𝛼=ℓ_𝛼𝑏_𝛼^𝑝−𝔰_𝛼𝑏_𝛼^𝑝𝛼∗𝛼^∗→𝑒₀as𝛼↓1. (5.23) By (4.17), there holds 𝑟^𝛼≤𝑏^𝑝𝛼∗_𝛼, and hence lim_𝛼↓1𝑏^𝑝𝛼∗_𝛼≥1. (5.24) It followed from (5.23) and (5.24) that lim_𝛼↓1𝑏_𝛼=1,𝑒₀=ℓ₀−lim_𝛼↓1𝔰_𝛼𝑏_𝛼^𝑝𝛼∗𝛼^∗. (5.25)

Note that ‖𝑞_𝛼‖₁≤‖𝑞_𝛼‖_𝛼=𝑟. On the other hand, it follows from (4.15) and (4.17) that 𝑞_𝛼1=¹₀𝑦_𝛼(𝑡)^𝑝𝛼∗𝑦_𝛼(𝑡)^𝑝1d𝑡≥𝑦_𝛼^𝑝_∞¹⁰𝑦_𝛼(𝑡)^𝑝𝛼∗𝑟d𝑡=^𝛼𝑏^𝑝_𝛼→𝑟 (5.26) as 𝛼↓1. Thus one has 𝑠-lim_𝛼↓1𝑞_𝛼1=𝑟. (5.27)

By the monotonicity and the symmetry of the eigen-function 𝑦_𝛼, (4.21) can be written as 𝑦^′_𝛼(𝑡)=𝜅_𝑝⁻¹𝑒_𝛼−ℓ_𝛼𝑦^𝑝_𝛼(𝑡)−𝔰_𝛼𝐺_𝛼(𝑡)^1/𝑝,𝑡∈𝐼_𝜎, (5.28) where 𝜅_𝑝∶=(𝑝−1)^1/𝑝. For the Dirichlet eigen-function, 𝑦_𝛼(0)=0, 𝑦_𝛼(1/2)=𝑏_𝛼 and 𝐼_𝜎=𝐼_𝐷=[0,1/2]. For the Neumann eigen-function, 𝑦_𝛼(0)=𝑎_𝛼=min_𝑡∈[0,1]𝑦_𝛼(𝑡), 𝑦_𝛼(1)=𝑏_𝛼 and 𝐼_𝜎=𝐼_𝑁=[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=lim_𝛼↓1𝑦_𝛼∈𝐶¹(𝐼_𝜎) and 𝑦^′₀=𝜅_𝑝⁻¹𝑒₀−ℓ₀𝑦^𝑝₀^1/𝑝,𝑡∈𝐼_𝜎. (5.29) As the limit of Dirichlet eigen-functions 𝑦^𝐷_𝛼, 𝑦₀ satisfies the boundary conditions as follows: 𝑦₀(0)=0,𝑦₀12=1. (5.30) As the limit of Neumann eigen-functions 𝑦^𝑁_𝛼, 𝑦₀ satisfies the boundary conditions as follows: 𝑦₀(0)=𝑎₀=min_𝑡∈[0,1]𝑦₀(𝑡)∈(0,1),𝑦₀(1)=1. (5.31) The limiting equality of (5.20) is 𝑝ℓ₀𝑦₀^𝑝_𝑝=𝑒₀+𝑝−1𝑟. (5.32) By (5.27), the limiting equality of (5.21) is 𝑒₀1𝑦₀^𝑝_𝑝=−𝑟. (5.33)

Either ℓ₀=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 problem 𝐌₁(𝑟)). In this case, there holds 𝑒₀=ℓ₀>0. In fact, we have ℓ₀=𝐌₁(𝑟)≥𝜆₁(−𝑟)=𝑟>0. (5.34) The rightmost equilibrium in the phase portrait Figure 1 is (ℓ_𝛼^{(𝛼−1)/𝑝},0). Thus we have 0<𝑏_𝛼<ℓ_𝛼^{(𝛼−1)/𝑝},lim_𝛼↓1𝑏^𝑝𝛼∗_𝛼𝛼^∗≤lim_𝛼↓1ℓ^𝛼_𝛼𝛼^∗=0. (5.35) Consequently, 𝑒₀=ℓ₀ by (5.25).
Finally, ℓ₀=𝐌₁(𝑟) can be proved to be the unique root of 𝑥−𝜋_𝑝𝑥^1/𝑝∗=𝑟,𝑥∈[0,∞), (5.36) where 𝜋_𝑝=2𝜋𝑝−1^1/𝑝.𝑝sin𝜋/𝑝 (5.37) We refer to [17] for details. Note that if 𝑝=2, result (5.36) is consistent with (5.9).

Case 2 (Minimization problem 𝐈₁(𝑟)). In this case, there hold 𝑒₀>0 and 𝑒₀>ℓ₀. In fact, 𝔰_𝛼=−1 for the minimization problems. It follows from (5.14) that 𝑒_𝛼=𝑝−1𝐹_𝛼𝑏_𝛼≥ℓ_𝛼𝑦_𝛼(𝑡)^𝑝,∀𝑡∈[0,1]. (5.38) 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 ℓ₀=𝐈₁(𝑟)≤𝜆₁(𝑟)=𝜋^𝑝_𝑝−𝑟. (5.39)

In this case, till now there is no such restrictions as positivity or negativity on ℓ₀. This is consistent to the phase portrait analysis in Section 4.2.

Finally, (𝑒₀,ℓ₀) (𝑒₀>0,𝑒₀>ℓ₀) can be uniquely determined by ¹₀𝜅_𝑝d𝑢𝑒₀−ℓ₀𝑢^𝑝1/𝑝=12,𝑟=2𝑝−1^−1/𝑝∗𝑒₀−ℓ₀^1/𝑝∗. (5.40) 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 problem 𝐈₀(𝑟)). In this case, there hold ℓ₀<0 and 𝑒₀=ℓ₀𝑎^𝑝₀<0. In fact, it follows from (4.4) that ℓ₀=𝐈₀(𝑟)<−𝑟<0. (5.41) 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≥ℓ₀𝑎^𝑝₀. One can exclude the case 𝑒₀>ℓ₀𝑎^𝑝₀ by checking (5.33) after solving (5.29)–(5.31).
Finally, (𝑎₀,ℓ₀) (0<𝑎₀<1,ℓ₀<0) can be uniquely determined by 𝜅_𝑝¹_𝑎0d𝑢ℓ₀𝑎^𝑝₀−ℓ₀𝑢^𝑝1/𝑝=1,𝑝−1^−1/𝑝∗ℓ₀𝑎^𝑝₀−ℓ₀^1/𝑝∗=𝑟. (5.42) 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 Differential Equations

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 measure 𝜇 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] _[0,1]𝑢d𝜇_𝐽𝑢d𝜇 (6.1) are defined. Given a measure 𝜇∈ℳ₀, following the notations in [27], the second-order linear MDE with the measure 𝜇 is written as d𝑥^•+𝑥d𝜇(𝑡)=0,𝑡∈[0,1]. (6.2) 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: 𝑥(𝑡)=𝑥₀+_[0,𝑡]𝑣(𝑠)[],𝑣(𝑡)=𝑣d𝑠,𝑡∈0,1₀𝑣,𝑡=0,₀−∫_[0,𝑡]𝑥(𝑠)d𝜇(𝑠),𝑡∈(0,1]. (6.3) 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 𝑥^′′+𝑞(𝑡)𝑥=0. (6.4)

Let 𝜇∈ℳ₀ be a (real) measure on [0,1]. The corresponding eigenvalue problem d𝑥^•+𝜆𝑥d𝑡+𝑥d𝜇(𝑡)=0,𝑡∈[0,1] (6.5) 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 {𝜆^𝐷_𝑛(𝜇)}_𝑛∈ℕ, accumulating at +∞. With the Neumann boundary condition 𝑥^•(0)=𝑥^•(1)=0, eigenvalues of (6.5) are a real, increasing sequence {𝜆^𝑁_𝑛(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: 𝜇_𝑛→𝜇 in (ℳ₀,𝑤_∗) if and only if lim_{𝑛→∞[0,1]}𝑢d𝜇_𝑛=_[0,1]𝑢d𝜇∀𝑢∈𝒞. (6.6) 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 eigenvalues 𝜆^𝜎_𝑛(𝜇) of MDE on measures 𝜇 are as follows.

Theorem 6.1 (see [26, 27]). (i) In the norm topology ‖⋅‖_𝐕, 𝜆^𝜎_𝑛(𝜇) is continuously Fréchet differentiable in 𝜇∈(ℳ₀,‖⋅‖_𝐕). Moreover, the formula for the Fréchet derivative 𝜕_𝜇𝜆^𝜎_𝑛(𝜇) is similar to (3.11).
(ii) In the weak∗ topology 𝑤_∗, 𝜆^𝜎_𝑛(𝜇) are continuous in 𝜇∈(ℳ₀,𝑤_∗).

Let us introduce spheres and balls of measures as follows: 𝑆₀[𝑟]∶=𝜇∈ℳ₀∶𝜇_𝐕=𝑟,𝐵₀[𝑟]∶=𝜇∈ℳ₀∶𝜇_𝐕≤𝑟, (6.7) where 𝑟≥0. Because of Theorem 6.1 and the compactness of 𝐵₀[𝑟] in (ℳ₀,𝑤_∗), the following extremal problems 𝐼_𝑛(𝑟)∶=min_{𝜇∈𝐵0[𝑟]}𝜆^𝜎_𝑛𝜇,𝑀_𝑛(𝑟)∶=max_{𝜇∈𝐵0[𝑟]}𝜆^𝜎_𝑛𝜇, (6.8) 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 𝐼_𝑛(𝑟) and 𝑀_𝑛(𝑟) are the same for the Dirichlet and the Neumann eigenvalues. Since the Fréchet derivatives 𝜕_𝜇𝜆^𝜎_𝑛(𝜇) are nonzero, one sees that problems (6.8) are the same as 𝐼_𝑛(𝑟)≡min_{𝜇∈𝑆0[𝑟]}𝜆^𝜎_𝑛𝜇,𝑀_𝑛(𝑟)≡max_{𝜇∈𝑆0[𝑟]}𝜆^𝜎_𝑛𝜇. (6.9) That is, the minimizing and maximizing measures of (6.8) must be on the sphere 𝑆₀[𝑟].

Recall from Theorem 6.1 that the functionals 𝜆^𝜎_𝑛(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 eigenvalues 𝜆^𝑁₀(𝜇), problems (6.9) have been solved using this idea in [22]. The results are as follows.

Theorem 6.2 (see [22]). For any 𝑟≥0, one has 𝐼₀(𝑟)=min_{𝜇∈𝑆0[𝑟]}𝜆^𝑁₀𝜇=𝜆^𝑁₀𝑟𝛿₀=𝜆^𝑁₀𝑟𝛿₁=𝐙₀⁻¹(𝑟),𝑀₀(𝑟)=max_{𝜇∈𝑆0[𝑟]}𝜆^𝑁₀𝜇=𝜆^𝑁₀(−𝑟ℓ)=𝑟, (6.10) where 𝐙₀ is defined by (5.12), and for 𝑎∈[0,1], 𝛿_𝑎 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 𝜇_𝑞 defined by 𝜇_𝑞(𝑡)∶=_[0,𝑡]𝑞(𝑠)d𝑠for𝑡∈[0,1]. (6.11) Since ‖𝜇_𝑞‖_𝐕=‖𝑞‖₁, one sees that 𝐵₁[𝑟]⊂(ℒ¹,‖⋅‖₁) can be isometrically embedded into 𝐵₀[𝑟]⊂(ℳ₀,‖⋅‖_𝐕). Hence one has inf_{𝑞∈𝐵1[𝑟]}𝜆^𝑁₀𝑞≥min_{𝜇∈𝑆0[𝑟]}𝜆^𝑁₀𝜇. (6.12) On the other hand, the minimizing (singular) measures 𝑟𝛿₀ and 𝑟𝛿₁ in (6.9) can be approximated by sequences {𝜇_𝑞𝑛} in the weak^∗ topology 𝑤_∗ where 𝑞_𝑛 are smooth potentials from 𝐵₁[𝑟]. Hence we have min_{𝜇∈𝑆0[𝑟]}𝜆^𝑁₀𝜇=lim_𝑛→∞𝜆^𝑁₀𝜇_𝑞𝑛≥inf_{𝑞∈𝐵1[𝑟]}𝜆^𝑁₀𝑞. (6.13) 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 sup_{𝑞∈𝐵1[𝑟]}𝜆^𝜎_𝑛𝑞 (6.14) 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 of 𝐵₁[𝑟], 𝑟>0, the corresponding minimizing problems inf_{𝑞∈𝐵1[𝑟]}𝜆^𝜎_𝑛𝑞 (6.15) 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 inf_{𝑞∈𝐵1[𝑟]}𝜆^𝑁₀𝑞=𝜆^𝑁₀𝑟𝛿₀=𝜆^𝑁₀𝑟𝛿₁. (6.16) For the first Dirichlet eigenvalues 𝜆^𝐷₁(𝑞), one has from [20, 26] that inf_{𝑞∈𝐵1[𝑟]}𝜆^𝐷₁𝑞=𝜆^𝐷₁𝑟𝛿_1/2. (6.17) The main result of [23] states that inf_𝑞∈𝐵_1[𝑟]𝜆^𝑁₀𝑞=𝜆^𝑁₀±𝑟2𝛿₀−𝛿₁𝑟=−²4, (6.18) where 𝐵₁[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 eigenvalues 𝜆_𝑛(𝑞), 𝑛∈ℕ, of problem (1.1) with integrable potentials 𝑞∈ℒ¹. In applied sciences, it is important to study eigenvalue gaps like 𝜆_𝑛+1(𝑞)−𝜆_𝑛(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 𝜆_𝑛(𝑞) for 𝑞 in bounded subsets of (ℒ¹,‖⋅‖₁), the following extremal problems for eigenvalue gaps: inf_{𝑞∈𝑆1[𝑟]}𝜆_𝑛+1𝑞−𝜆_𝑛𝑞sup_{𝑞∈𝑆1[𝑟]}𝜆_𝑛+1𝑞−𝜆_𝑛𝑞,𝑛∈ℕ, (7.1) 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 potential 𝑞∈ℒ¹, consider the eigenvalue problem of the beam equation, or the Euler-Bernoulli equation [43–45], 𝑥⁽⁴⁾+𝑞(𝑡)𝑥=𝜆𝑥,𝑡∈[0,1], (7.2) with the Lidstone boundary condition 𝑥(0)=𝑥(1)=0,𝑥^′′(0)=𝑥^′′(1)=0. (7.3) Eigenvalues of problem (7.2)-(7.3) are still denoted by {𝜆_𝑛(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: inf_{𝑞∈𝑆1[𝑟]}𝜆_𝑛𝑞,sup_{𝑞∈𝑆1[𝑟]}𝜆_𝑛𝑞, (7.4) and the extremal problems for eigenvalue gaps like inf_{𝑞∈𝑆1[𝑟]}𝜆₂𝑞−𝜆₁𝑞,sup_{𝑞∈𝑆1[𝑟]}𝜆₂𝑞−𝜆₁𝑞, (7.5) can be solved in a complete way.

Acknowledgments

The first author is supported by the National Natural Science Foundation of China (Grant no. 10901089 and 11171090), and the second author is supported by the Doctoral Fund of Ministry of Education of China (Grant no. 20090002110079) and the 111 Project of China (2007).