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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 670463, 26 pages
A Survey on Extremal Problems of Eigenvalues
1Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China
Received 3 July 2012; Accepted 7 August 2012
Academic Editor: Yeong-Cheng Liou
Copyright © 2012 Ping Yan and Meirong Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Given an integrable potential , the Dirichlet and the Neumann eigenvalues and 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 metric for q is given; . Note that the spheres and balls are nonsmooth, noncompact domains of the Lebesgue space . 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 , will be used. Then the problems will be solved by passing . 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.
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 norms. Based on such eigenvalue properties and some topological facts on spaces, several interesting extremal problems for eigenvalues with 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 , all eigenvalues of the Sturm-Liouville operator associated with the Dirichlet boundary conditions are given by a sequence with . Let us consider such extremal problems as for any , where is a ball of radius in . These problems are interesting and of difficulty, because are implicit functionals of , while, topologically, is not compact or sequentially compact even in the weak topology of [24, 25], and, geometrically, is also nonsmooth in the space . Therefore, they cannot be solved directly by using the standard variational method.
Firstly, for any , we consider the counterparts of and with potentials confined to (balls in of radius ), denoted by and , respectively. Since the functional is continuous (in weak topology) and continuously differentiable (in the usual norm) in potential (and hence in , ), and the balls , , are compact in weak topology and smooth in the usual norm , 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 , the critical eigenfunction, denoted by , and the extremal value or are all involved.
Secondly, we will employ the limiting approach to obtain and . This step is based on some topological facts that balls and spheres in space can be approximated by balls and spheres in spaces as . Then the strong continuity of eigenvalue in potentials ensures that and . In this step, besides critical equations in spaces, properties of critical potentials and critical eigen-functions should also be sufficiently utilized to get a final solution to and .
Several extremal problems on (Dirichlet, Neumann, and periodic) eigenvalues (of the Sturm-Liouville operator and the -Laplace operator) have been studied recently, where the potentials are confined to different sets such as balls or spheres in . 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 , , are listed in Section 2. The essential of this section is that those “bad” balls or spheres in (neither smooth nor weak compact) can be approximated by “good” balls or spheres in , (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 , 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 , . In Section 4 we introduce how variational method is applied to get critical equations for the extremal problems for cases . 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 spaces via limiting approaches. Final results of extremal values in case are stated in this section. For the Sturm-Liouville operator, these extremal values can be expressed by elementary functions of the radius . 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 Spaces
Definition 2.1. Let . We say that is weakly convergent to , written as in , or in , if Here is the conjugate exponent of : .
For , and , let us take the following notations: where is the mean value of .
2.1. Approximation to Balls/Spheres in Strong Topology
Lemma 2.2 (see [21, Lemma 2.1]). Given that and , there exists such that .
Lemma 2.3 (see [19, Lemma 2.3]). Let and . For any , there exists , , such that .
2.2. Approximation to Balls/Spheres in Weak Topology
Lemma 2.5 (see [19, Lemma 2.5]). Suppose that . Then, for any and , is compact and sequentially compact in the space .
The balls and 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 norm, by “good” balls and , which are compact in weak topology and smooth in geometry. In fact, there hold the following stronger topological facts.
Lemma 2.6 (see [19, Lemma 2.6]). (i) Let , and . For any , there exists a sequence such that in . In other words, the closure of in the space is .
(ii) Consequently, for any , , there exists a sequence such that in .
3. Properties of Eigenvalues on Potentials/Weights
Denote by the scalar -Laplacian, that is, for any and . For any integrable potential , it is well known that all eigenvalues of associated with the Dirichlet boundary condition consist of a sequence and all eigenvalues associated with the Neumann boundary condition consist of a sequence as follows: see [28, 29]. To emphasize the dependence on the interval and the boundary conditions, we also write the eigenvalues as , where the superscript can be chosen as (for the Dirichlet eigenvalues) or (for the Neumann eigenvalues).
3.1. Scaling Results on Eigenvalues
Let be a finite interval of length . Given that and , define a potential by the following:
By this lemma, we need only to consider eigenvalues for those potentials defined on the interval , that is, for .
3.2. Relationship between the First- and Higher-Order Eigenvalues
Let us identify with , where . For , define the mapping by
In fact, (3.8) can be proved to be equalities. Therefore, we need only to consider extremal values of the first eigenvalues. For cases , the converse inequalities can be proved by using critical equations (4.16). See Remark 4.1. Moreover, results for case can be obtained by limiting approaches . Case is trivial.
3.3. Continuous Differentiability in Strong Topology
Theorem 3.3 (see [14, Theorem 1.2]). Given and an admissible . The functional is continuously Fréchet differentiable. The Fréchet derivative is the following bounded linear functional: or written simply as Here is a normalized eigenfunction associated with so that the norm .
Remark 3.4. Since (3.11) is always a negative functional of , , eigenvalues possess the following monotonicity: Moreover, if in addition, holds on a subset of 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 and an admissible , the following functional is continuous:
4. Extremal Problems in Balls,
In this section, we always assume that , , and or .
4.1. Preliminary Results for Minimization/Maximization Problems
For any admissible integer , let us take some notations as follows If , then can only be chosen as . For any positive integers , these notations are still reasonable, and the extremal values are independent of the choice of or due to the relationship between Dirichlet and Neumann eigenvalues [29, Theorem 4.3]. Furthermore, these extremal values on balls with radius are exactly the same as those on spheres with radius , that is, Equation (4.2) follows immediately from (3.12), the monotonicity of eigenvalues. Equation (4.3) follows from the facts that the closure of in the space is (See Lemma 2.6(i)), and is strongly continuous in .
Note that and are compact and sequentially compact in the space (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 is a case apart, unlike any other. It is well known that [31, Lemma 3.3] Therefore one has Moreover, cannot be attained by any potential on . Consequently, All other minimizers and maximizers are located exactly on the boundaries of the balls, that is, Again, (4.7) and (4.8) are immediate consequences from monotonicity of eigenvalues. Especially, the minimizers/maximizers on the sphere 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 , it is not easy to obtain monotonicity of eigenvalues on . 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 .
4.2. Variational Construction for Minimizer/Maximizer in
The extremal value problems (4.7) and (4.8) can be considered uniformly. The only constraint wherein is . As , the norm is continuously Fréchet differentiable in , with the Fréchet derivative as follows: where is . The eigenvalues are also Fréchet differentiable in , and the Fréchet derivative is as in (3.11). By the Lagrangian Multiplier Method, the critical potential satisfies the following equation: for some constant . Here is a normalized eigen-function associated with . Since the minimizer is nonnegative and the maximizer is nonpositive, for the minimization problem (4.7) and for the maximization problem (4.8). Let and Then is also an eigenfunction associated with . Specifically, we identify it by for Dirichlet case and for Neumann case. This identified is called the critical eigenfunction, and it satisfies the original ODE (3.1), that is, and corresponding boundary conditions. By (4.12), the critical potential can also be written as Substituting (4.15) into (4.14), one has 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 of eigenvalues . By (4.15), the constraint is transformed to
Remark 4.1. Due to the autonomy and the symmetry of critical equation (4.16), is a periodic solution of (4.16) with minimal period . In fact, one has (we refer the proof of this to Lemma 3.2 in ). By (4.15), is periodic with minimal period . Therefore, the maximizer is in the range of . See (3.6). This tells us that the converse inequalities of (3.8) are also true. More precisely, there hold for any integer and any . This is why we need only to consider extremal values of and .
The phase portraits of (4.16) are distinguished in three cases. See Figures 1–3. In Figure 1, . This figure corresponds to the maximization problem , and the eigen-function is certain nonconstant periodic orbit surrounding the equilibrium . In Figure 2, and . The minimization problem is illustrated in this figure, and the positive eigen-function corresponds to some non-constant periodic orbit surrounding the rightmost equilibrium. In Figure 3, and . When the minimization problem is considered, both Figures 2 and 3 should be taken into account. In fact, the bigger is, the smaller is. For large enough, is negative and should be some sign-changing periodic orbit outside the homoclinic orbits in Figure 2. For small enough, is nonnegative, and should be some non-constant periodic orbit surrounding the equilibrium in Figure 3.
Choose in (4.22). Then , and are implicitly determined by
4.3. Variational Construction for Minimizer/Maximizer in
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 for maximization problems and for minimization problems. The existence of the new constant is caused by the constraint . The critical potential is The constraint is The constraint is A first integral of the critical equation is
5. Extremal Problems in Balls
In this section we will solve the following extremal problems:
By the topological facts in Section 2 and the strong continuity of eigenvalues in weak topology (see Theorem 3.5), the extremal values in balls are limits of those extremal values in balls, that is, for any , and any admissible integer . By relationship (4.18), we need only consider for cases and in (5.1). Similar arguments hold for (5.2).
Let us use a uniform notation to denote the extremal values , , or , respectively, in different extremal values problems, that is, Note that (see [18, Lemma 2.3]) and due to the asymptotical distribution of large eigenvalues [29, 32]. Thus in each case, is finite and is bounded.
Two different limiting approaches to this limit 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 -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 ()
By (5.4), to obtain the extremal values in balls, it is natural to compute firstly the extremal values in balls and then let . Since 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 -Laplacian.
Extremal values and for the Sturm-Liouville operator are studied in . As stated in the previous section, (or ) and are implicitly determined by (4.23), where the exponent . After some transformation, (4.23) is analyzed in , and these extremal values in balls are finally expressed by some singular integrals. More precisely, for any , let and for any , let and we define Then there hold Note that and cannot be written as elementary functions of the radius . However, after some delicate analysis on the singular integrals , , , , and , the limits of and as are proved to be elementary functions of , that is, where is a decreasing diffeomorphism mapping onto .
Extremal values for the Sturm-Liouville operator are studied in , again by singular integral method in the limiting approach. Now , , and are implicitly determined by (4.24) with . One can imagine that difficulty increases due to the presence of the additional new parameter . We only state the final results here and refer the readers to  for details. Let which is a decreasing diffeomorphism from onto . Then there holds
5.2. Limiting Approach from the Viewpoint of Conservation Laws ()
Define It follows from (4.17) that as . Passing to a subsequence if necessary,
Motivated by (5.17), one attempt to compute in (5.4) is to consider the limit equation of the conservation law (5.14). Intuitively, if we assume that and 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 and the restriction that the critical potentials will give more information on . All these conditions on and the solution to (5.29) will finally lead to the answer to the extremal value . Compared with the previous singular integral method, such a limiting approach to from the viewpoint of conservation laws cannot only greatly simplify the analysis but also deal with the -Laplace operator.
In the limiting approach from the viewpoint of conservation laws, it is natural to study the convergence of critical eigen-functions as . To this end, the boundedness of and as should be taken into consideration.
For both Neumann and Dirichlet eigenvalues, multiplying (4.16) by , integrating over and taking use of the restriction (4.17), that is, , one has Integrating (5.14) over , one has Eliminating from the above two equalities, one has For the principal Neumann eigenvalues, there hold an additional equality as follows: This equality can be deduced from (4.14), (5.14), and (4.17).
Note that is bounded as . The boundedness of and the restriction (4.17) guarantees (passing to a subsequence if necessary) the convergence of and Here . For detailed proof, see Lemmas 2.1 and 4.1 in . Consequently, is bounded as , and by (5.20), By (4.17), there holds , and hence It followed from (5.23) and (5.24) that
By the monotonicity and the symmetry of the eigen-function , (4.21) can be written as where . For the Dirichlet eigen-function, , and . For the Neumann eigen-function, , and . 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 and As the limit of Dirichlet eigen-functions , satisfies the boundary conditions as follows: As the limit of Neumann eigen-functions , satisfies the boundary conditions as follows: The limiting equality of (5.20) is By (5.27), the limiting equality of (5.21) is
Either or is determined by ODE (5.29), the boundary conditions (5.30) and the restriction (5.32). Meanwhile, is determined by (5.29), (5.31), (5.32) and (5.33). To solve (5.29) and compute , we need more information about , , and .
Case 1 (maximization problem ). In this case, there holds . In fact, we have
The rightmost equilibrium in the phase portrait Figure 1 is . Thus we have
Consequently, by (5.25).
Finally, can be proved to be the unique root of where We refer to  for details. Note that if , result (5.36) is consistent with (5.9).
Case 2 (Minimization problem ). In this case, there hold and . In fact, for the minimization problems. It follows from (5.14) that Set and in (5.38), respectively, and let . We see that and . Furthermore, cases and can be excluded by checking (5.32) after solving (5.29)-(5.30) and taking into account that
In this case, till now there is no such restrictions as positivity or negativity on . This is consistent to the phase portrait analysis in Section 4.2.
Case 3 (minimization problem ). In this case, there hold and . In fact, it follows from (4.4) that
By (5.33), . Note that (5.38) still holds for this minimization problem. Let in (5.38) and . One has . One can exclude the case by checking (5.33) after solving (5.29)–(5.31).
Finally, () can be uniquely determined by For details we refer to . Note that if , 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 on , it means that it is an element of the dual space of the Banach space . Then, for any and any subinterval of , the Riemann-Stieltjes integral and the Lebesgue-Stieltjes integral  are defined. Given a measure , following the notations in , the second-order linear MDE with the measure is written as The solution of MDE (6.2) with the initial value is explained using the system of integral equations as follows: Here is the generalized right-hand derivative of or the velocity of . By (6.3), it is well known that solutions of initial value problems of (6.2) are uniquely defined on [33, 34]. See also [26, 27]. From (6.3), one sees that the solutions themselves are continuous, that is, . However, may have discontinuity at those such that the density does not exist. In case is absolutely continuous with respect to the Lebesgue measure on , that is, , solutions of MDE (6.2) reduce to that for ODE
Let be a (real) measure on . The corresponding eigenvalue problem has been studied in [26, 27]. Like problem (1.1), eigenvalues of (6.5) with the Dirichlet boundary condition are a real, increasing sequence , accumulating at . With the Neumann boundary condition , eigenvalues of (6.5) are a real, increasing sequence , again accumulating at .
By the Riesz representation theorem , the measure space can be characterized using real functions of of bounded variations. For , we use to denote the total variation of on . Then is a Banach space. Since is the dual space of , one has also in the wea topology defined as follows: in if and only if It is well known that bounded subsets of are relatively compact and relatively sequentially compact in [24, 25].
Some deep results on the dependence of eigenvalues of MDE on measures are as follows.
Theorem 6.1 (see [26, 27]). (i) In the norm topology , is continuously Fréchet differentiable in . Moreover, the formula for the Fréchet derivative is similar to (3.11).
(ii) In the weak topology , are continuous in .
Let us introduce spheres and balls of measures as follows: where . Because of Theorem 6.1 and the compactness of in , the following extremal problems where are well posed. Moreover, both the minimum and the maximum can be realized by some measures from . Here, when , the values and are the same for the Dirichlet and the Neumann eigenvalues. Since the Fréchet derivatives are nonzero, one sees that problems (6.8) are the same as That is, the minimizing and maximizing measures of (6.8) must be on the sphere .
Recall from Theorem 6.1 that the functionals we are minimizing/maximizing are continuously differentiable. However, is not differentiable in the space . The solution of problems (6.9) appeals for the LMM using the sub-differentials [36, 37].
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 induces an absolutely continuous measure defined by Since , one sees that can be isometrically embedded into . Hence one has On the other hand, the minimizing (singular) measures and in (6.9) can be approximated by sequences in the wea topology where are smooth potentials from . Hence we have Thus one has (5.13).
Let us return to the minimizing/maximizing problems of eigenvalues of (1.1) and (3.1) for potentials in . Though has no compactness in or , it has been shown that all maximizing problems can be realized by potentials from . In fact, these maximizing potentials are step potentials. For detailed construction, we refer to  for general .
On the other hand, because of the noncompactness of , , the corresponding minimizing problems cannot be realized by any potential from , but by some singular measures from . The following results hold for eigenvalues of (1.1), that is, in (3.1). For example, from (6.10) and (5.13), one has For the first Dirichlet eigenvalues , one has from [20, 26] that The main result of  states that where . Results (6.16)–(6.18) can yield a natural explanation to what kinds of integrable potentials in 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 , , of problem (1.1) with integrable potentials . In applied sciences, it is important to study eigenvalue gaps like , . 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 for in bounded subsets of , the following extremal problems for eigenvalue gaps: 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 , consider the eigenvalue problem of the beam equation, or the Euler-Bernoulli equation [43–45], with the Lidstone boundary condition Eigenvalues of problem (7.2)-(7.3) are still denoted by . Like the second-order Sturm-Liouville operator (1.1), it is desirable to develop analogous ideas so that the following extremal problems for eigenvalues: and the extremal problems for eigenvalue gaps like can be solved in a complete way.
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).
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