A Survey on Extremal Problems of Eigenvalues

and Applied Analysis 3 The final solution to problems in 1.3 can yield optimal lower and upper bounds of eigenvalues. Actually, from 1.3 , we have In ∥q ∥ ∥ 1 ) ≤ λn ( q ) ≤ Mn ∥q ∥ ∥ 1 ) ∀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 ≥ 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 > 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 > 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 Spaces In the Lebesgue space L, 1 ≤ α ≤ ∞, the usual L topology is induced by L norm ‖ · ‖α. Besides this strong topology, one has also the weak topology wα which is defined as follows 24, 25 . Definition 2.1. Let qn, q ∈ L. We say that qn is weakly convergent to q, written as qn wα −−→ q in L, or qn → q in L,wα , if


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 L 1 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 ∈ L 1 : L 1 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 λ 1 q < λ 2 q < · · · < λ n q < · · · , 1.2 with lim n → ∞ λ n q ∞. Let us consider such extremal problems as I n r : inf q∈B 1 r λ n q , M n r : sup q∈B 1 r λ n q 1.3 for any n ∈ N, where B 1 r is a ball of radius r in L 1 . These problems are interesting and of difficulty, because λ n · are implicit functionals of q ∈ L 1 , while, topologically, B 1 r is not compact or sequentially compact even in the weak topology w 1 of L 1 24, 25 , and, geometrically, B 1 r is also nonsmooth in the space L 1 , · 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 ∈ L 1 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 L 1 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↓1 I n,α I n and lim α↓1 M 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 .

Abstract and Applied Analysis 3
The final solution to problems in 1.3 can yield optimal lower and upper bounds of eigenvalues. Actually, from 1.3 , we have 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 L 1 . 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 L 1 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 L 1 spaces via limiting approaches. Final results of extremal values in L 1 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.

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 .

Abstract and Applied Analysis
For α ∈ 1, ∞ , r ∈ 0, ∞ and h ∈ −r, r , let us take the following notations: where q 1 0 q t dt is the mean value of q.
Lemmas 2.2 and 2.3 give very nice topological approximation to spheres in L 1 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, L 1 balls B 1 r and B 1 r, h can be approximated, in the sense of L 1 norm, by L α balls B α r and B α r, h , respectively.

Approximation to L 1 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 B 1 r and B 1 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 L 1 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 } ⊂ S 1 r, h such that q n → q in L 1 , w 1 .
Abstract and Applied Analysis 5

Properties of Eigenvalues on Potentials/Weights
Denote by φ p · the scalar p-Laplacian, that is, φ p x |x| p−2 x for any x / 0 and φ p 0 0. For any integrable potential q ∈ L 1 a, b , R , it is well known that all eigenvalues of associated with the Dirichlet boundary condition x a x b 0 consist of a sequence λ D 1 q < λ D 2 q < · · · < λ D n q < · · · , 3.2 and all eigenvalues associated with the Neumann boundary condition x a x b 0 consist of a sequence as follows: 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 .

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: 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 .

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 n q q n , q n t : n p q nt , t ∈ R.

6
Abstract and Applied Analysis 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 p q, q n α n p q α , λ σ n q n n p λ σ 1 q , 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∈B α r λ σ n q ≥ n p · sup 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.

Continuous Differentiability in Strong Topology
is continuously Fréchet differentiable. The Fréchet derivative ∂ q λ σ n q ∈ L α , · α * is the following bounded linear functional: or written simply as 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.

3.12
Abstract and Applied Analysis 7 Moreover, if in addition, q 1 t < q 2 t holds on a subset of 0, 1 of positive measure, the conclusion inequality in 3.12 is strict.

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: 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 .

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

Preliminary Results for Minimization/Maximization Problems
For any admissible integer n, let us take some notations as follows

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, Therefore one has Moreover, M 0,α r, h cannot be attained by any potential on S α r, h . Consequently, All other minimizers and maximizers are located exactly on the boundaries of the balls, that is,

Variational Construction for Minimizer/Maximizer in B α r
The arguments in this subsection follow in part those of 17, 18, 20, 21 . Abstract and Applied Analysis 9 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: 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: 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 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 sign c α ±1 and Then y α y σ n,r,α is also an eigenfunction associated with 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 p y α α φ p y α q α t φ p y α 0, 4.14 and corresponding boundary conditions. By 4.12 , the critical potential can also be written as Substituting 4.15 into 4.14 , one has φ p y α α φ p y α − s α φ 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 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 p I 1,α r n p , M n,α r n p M 1,α r n p 4.18 for any integer n ≥ 2 and any r > 0. This is why we need only to consider extremal values of λ N 0 q and λ σ 1 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 eigenfunction y α is certain nonconstant periodic orbit surrounding the equilibrium 0, 0 . In Figure 2, s α −1 and 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 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 signchanging 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.
To study the parameter α in 4.16 , introduce an additional parameter as follows: and besides, for case n 0 one more parameter as follows: A first integral of 4.16 is

4.22
Note that the minimal period of y α is 2, and y α satisfies the constraint 4.17 . Theoretically, α M 1,α or α I 1,α and b α are implicitly determined by singular integrals as follows:

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 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 The constraint q α α r is 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.

Extremal Problems in L 1 Balls
In this section we will solve the following extremal problems: Let us use a uniform notation 0 to denote the extremal values I 0 r , I 1 r , or M 1 r , respectively, in different extremal values problems, that is, 0 lim α↓1 α .

5.4
Note that I 0 r > −∞ see 18, Lemma 2.3 and M 1 r < ∞ due to the asymptotical distribution of large eigenvalues 29, 32 . Thus in each case, 0 is finite and { α } α>1 is bounded. Two different limiting approaches to this limit 0 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 eigenfunctions 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.

Limiting Approach from the Viewpoint of Singular Integrals (p 2)
By 5.4 , to obtain the extremal values 0 in L 1 balls, it is natural to compute firstly the extremal values α in L α balls and then let α ↓ 1. Since 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 I 1 r and M 1 r for the Sturm-Liouville operator are studied in 20 . As stated in the previous section, α M 1,α or 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

5.6
and we define

5.7
Then there hold 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 andǓ 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, is a decreasing diffeomorphism mapping −∞, π 2 onto 0, ∞ . Extremal values I 0 r for the Sturm-Liouville operator are studied in 21 , again by singular integral method in the limiting approach. Now 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 which is a decreasing diffeomorphism from −∞, 0 onto 0, ∞ . Then there holds

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 is a constant. We call 5.14 a conservation law of 4.16 . Define

5.17
Motivated by 5.17 , one attempt to compute 0 in 5.4 is to consider the limit equation of the conservation law 5.14 . Intuitively, if we assume that e α → e 0 and y α → y 0 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 y 0 . All these conditions on y 0 and the solution to 5.29 will finally lead to the answer to the extremal value 0 . Compared with the previous singular integral method, such a limiting approach to 0 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 α y α p p − s α r α 0.

5.19
Eliminating y α p p from the above two equalities, one has p α y α p p e α s α p − 1 α r α .

5.20
Abstract and Applied Analysis

5.25
Note that q α 1 ≤ q α α r. On the other hand, it follows from 4.15 and 4.17 that as α ↓ 1. Thus one has slim α↓1 q α 1 r.

5.27
By the monotonicity and the symmetry of the eigen-function y α , 4.21 can be written as where π p 2π p − 1 1/p p sin π/p .

5.37
We refer to 17 for details. Note that if p 2, result 5.36 is consistent with 5.9 .

5.38
Set t 0 and 1/2 in 5.38 , respectively, and let α ↓ 1. We see that e 0 ≥ 0 and e 0 ≥ 0 . Furthermore, cases e 0 0 and e 0 0 can be excluded by checking 5.32 after solving 5.29 -5.30 and taking into account that 0 I 1 r ≤ λ 1 r π p p − r.

5.39
In this case, till now there is no such restrictions as positivity or negativity on 0 . This is consistent to the phase portrait analysis in Section 4.2.

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 . 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 .

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 M 0 : C, · ∞ * of the Banach space C, · ∞ . Then, for any u ∈ C and any subinterval J of 0, 1 , The solution x t of MDE 6.2 with the initial value x 0 , x • 0 x 0 , v 0 ∈ R 2 is explained using the system of integral equations as follows: x t x 0

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 ∈ M 0 is absolutely continuous with respect to the Lebesgue measure on 0, 1 , that is, dμ t /dt q t ∈ L 1 , solutions of MDE 6.2 reduce to that for ODE x q t x 0. 6.4 Let μ ∈ M 0 be a real measure on 0, 1 . The corresponding eigenvalue problem dx • λx dt x dμ t 0, t ∈ 0, 1 6.5 Abstract and Applied Analysis 21 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 M 0 can be characterized using real functions of 0, 1 of bounded variations. For μ ∈ M 0 , we use μ V to denote the total variation of μ on 0, 1 . Then M 0 , · V is a Banach space. Since M 0 is the dual space of C, · ∞ , one has also in M 0 the weak * topology w * defined as follows: μ n → μ in M 0 , w * if and only if lim n → ∞ 0,1 u dμ n 0,1 u dμ ∀u ∈ C. 6.6 It is well known that bounded subsets of M 0 , · V are relatively compact and relatively sequentially compact in M 0 , w * 24, 25 . Some deep results on the dependence of eigenvalues λ σ n μ of MDE on measures μ are as follows.
Let us introduce spheres and balls of measures as follows: where r ≥ 0. Because of Theorem 6.1 and the compactness of B 0 r in M 0 , w * , the following extremal problems I n r : min μ∈B 0 r λ σ n μ , M n r : max μ∈B 0 r λ σ n μ , 6.8 where n ∈ Z are well posed. Moreover, both the minimum and the maximum can be realized by some measures from B 0 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 ∂ μ λ σ n μ are nonzero, one sees that problems 6.8 are the same as I n r ≡ min μ∈S 0 r λ σ n μ , M n r ≡ max μ∈S 0 r λ σ n μ . 6.9 That is, the minimizing and maximizing measures of 6.8 must be on the sphere S 0 r . Recall from Theorem 6.1 that the functionals λ σ n μ we are minimizing/maximizing are continuously differentiable. However, S 0 r is not differentiable in the space M 0 , · V . The solution of problems 6.9 appeals for the LMM using the sub-differentials 36, 37 . For the zeroth Neumann eigenvalues λ N 0 μ , problems 6.9 have been solved using this idea in 22 . The results are as follows.

22
Abstract and Applied Analysis Theorem 6.2 see 22 . For any r ≥ 0, one has

6.10
where Z 0 is defined by 5.12 , and for a ∈ 0, 1 , δ a denotes the unit Dirac measure located at a, and 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 ∈ L 1 induces an absolutely continuous measure μ q defined by μ q t : 0,t q s ds for t ∈ 0, 1 .

6.12
On the other hand, the minimizing singular measures rδ 0 and rδ 1 in 6.9 can be approximated by sequences {μ q n } in the weak * topology w * where q n are smooth potentials from B 1 r . Hence we have min μ∈S 0 r λ N 0 μ lim n → ∞ λ N 0 μ q n ≥ inf q∈B 1 r λ N 0 q . 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 B 1 r . Though B 1 r ⊂ L 1 has no compactness in · 1 or w 1 , it has been shown that all maximizing problems sup q∈B 1 r λ σ n q 6.14 can be realized by potentials from S 1 r ⊂ B 1 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 B 1 r , r > 0, the corresponding minimizing problems inf q∈B 1 r λ σ n q 6.15 Abstract and Applied Analysis 23 cannot be realized by any potential from B 1 r , but by some singular measures from S 0 r ⊂ B 0 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∈B 1 r λ N 0 q λ N 0 rδ 0 λ N 0 rδ 1 .

6.16
For the first Dirichlet eigenvalues λ D 1 q , one has from 20, 26 that inf q∈B 1 r λ D 1 q λ D 1 rδ 1/2 . 6.17 The main result of 23 states that where B 1 r {q ∈ L 1 : q 1 r, q 0}. Results 6.16 -6.18 can yield a natural explanation to what kinds of integrable potentials in B 1 r will decrease the eigenvalues.

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 ∈ L 1 . 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 L 1 , · 1 , the following extremal problems for eigenvalue gaps: