# Minimization of eigenvalues for some differential equations with integrable potentials

Gang Meng

Author Affiliations

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

Boundary Value Problems 2013, 2013:220  doi:10.1186/1687-2770-2013-220

 Received: 13 March 2013 Accepted: 28 August 2013 Published: 7 November 2013

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper we use the limiting approach to solve the minimization problem of the Dirichlet eigenvalues of Sturm-Liouville equations when the norm of integrable potentials is given. The construction of an approximating problem in this paper can simplify the analysis in the limiting process.

MSC: 34L15, 34L40.

##### Keywords:
eigenvalue; Sturm-Liouville equations; minimization problem; integrable potential; ball

### 1 Introduction

Extremal problems for eigenvalues are important in applied sciences like optimal control theory, population dynamics [1-3] and propagation speeds of traveling waves [4,5]. These are also interesting mathematical problems [6-10], because the solutions are applied in many different branches of mathematics. The aim of this paper is to solve the minimization problem of the Dirichlet eigenvalues of Sturm-Liouville equations when the norm of integrable potentials is given.

For , let denote the Lebesgue space of real functions with the norm . Given a potential , we consider Sturm-Liouville equations

(1.1)

with the Dirichlet boundary condition

(1.2)

It is known that problem (1.1)-(1.2) has countably many eigenvalues (see [11,12]). They are denoted by ,  , and ordered in such a way that

For , denote

In this paper, we study the following minimization problem:

(1.3)

Note that the minimization problems in [1,3,7] are taking over order intervals of potentials/weights which are compact in the weak topologies, and therefore always have minimizers. For example, Krein studied in [7] the minimization problem of weighted Dirichlet eigenvalues of

Given constants , denote

The problem is to find

(1.4)

Using compactness of the class S and continuity of the eigenvalues in the weak topologies, problem (1.4) can be realized by some optimal weight w. However, our problem (1.3) is taking over balls, which are not compact even in the weak topology . In order to overcome this difficulty in topology, we first solve the following approximating minimization problem of eigenvalues.

Theorem 1.1Let

(1.5)

where. We have that

(i) If, then

(1.6)

Moreover, is attained for

(1.7)

(ii) If, thenis the unique solution of. Here, the functionis defined as

(1.8)

Moreover, is attained for

(1.9)

Then, using the continuous dependence of eigenvalues on potentials with respect to the weak topologies (see [13-16]), we obtain a complete solution for minimization problem (1.3).

Theorem 1.2The following holds:

(1.10)

Here, the functionis defined as

(1.11)

This paper is organized as follows. In Section 2, we give some preliminary results on eigenvalues. In Section 3, we first consider the approximating minimization for eigenvalues and obtain Theorem 1.1. Then, by the limiting analysis, we give the proof of Theorem 1.2.

We end the introduction with the following remark. In [17,18], the authors first considered the approximating minimization for eigenvalues on the corresponding balls, . Then minimization problem (1.3) can be solved by complicated limiting analysis of . In this paper, we study a different approximating problem, which also has a sense from mathematical point of view. Such a construction can simplify the analysis in the limiting process.

### 2 Auxiliary lemmas

In the Lebesgue spaces , , besides the norms , one has the following weak topologies [19]. For , we use to indicate the topology of weak convergence in , and for , by considering as the dual space of , we have the topology of weak convergence. In a unified way, in iff

Here, is the conjugate exponent of p.

To solve problem (1.5), let us quote from [14,16] some important properties on eigenvalues.

Lemma 2.1As nonlinear functionals, are continuous in, .

By the continuity result above, we show that extremal problem (1.5) can be attained in .

Lemma 2.2There existssuch that.

Proof Notice that is compact and is closed in . Hence is compact in . Consequently, the existence of minimizers of (1.5) can be deduced from Lemma 2.1 in a direct way. □

Eigenvalues possess the following monotonicity property [12].

Lemma 2.3

(2.1)

Moreover, if, in addition, holds on a subset ofof positive measure, the conclusion inequality in (2.1) is strict.

Next, we use the theory of Schwarz symmetrization as a tool. For a given nonnegative function f defined on the interval , we denote by (resp., ) the symmetrically increasing (resp., decreasing) rearrangement of f. We recall that the function is uniquely defined by the following conditions:

(i) and f are equimeasurable on . That is, for all ,

(iii) is decreasing in the interval .

Similarly, is (uniquely) defined by (i), (ii) and (iii)’: is increasing in the interval . For more information on rearrangements, see [20] and [21].

We can compare the first eigenvalue of q with the first eigenvalue of its rearrangement.

Lemma 2.4For any, we have.

Proof Let be a positive eigenfunction corresponding to . By [[22], Theorem 378] and [[6], Section 7], we have

□

### 3 Main results

Now we are ready to prove the main results of this paper. First, we solve the minimization problem for eigenvalues when potentials .

Proof of Theorem 1.1 (i) If , then . From the monotonicity property (2.1) of eigenvalues, we have that the minimizer and then by computing directly.

(ii) When . By Lemma 2.2, Lemma 2.3 and Lemma 2.4, there exists a minimizer such that and for a.e. t. Let be a positive eigenfunction corresponding to the first eigenvalue . We have from the proof of Lemma 2.4 that is also an eigenfunction corresponding to the first eigenvalue , which implies that because .

Then, we have that for each ,

Hence,

(3.1)

Let and . Since is symmetric about and increasing in , we have that

Define

We claim that

(3.2)

In fact,

Notice that . Comparing (3.1) and (3.2), we know that the equality holds in (3.2) and is an eigenfunction corresponding to . Since (3.2) is an equality, we have from the proof that and for a.e. t.

Let . Now, (1.1) becomes

(3.3)

We can find that the solution of (3.3) is given by

(3.4)

where . Since and , we get from (3.4) that is the unique solution of , where is as in (1.8). □

Finally, we use the limiting approach to obtain a complete solution for the minimization problem of eigenvalues when the norm of integrable potentials is given.

Proof of Theorem 1.2 By the definitions of in (1.8) and Z in (1.11), we have that

which implies that

Since , we have that and then

(3.5)

On the other hand, since is continuous in , is continuous in . Notice that , where the closure is taken in the space . We have that for arbitrary and , there exist and such that

Hence,

Because is arbitrary, it holds that

Therefore, we have that

(3.6)

since is arbitrary.

By (3.5) and (3.6), the proof is complete. □

Remark 3.1 Fix . Assume that is as in (1.7) when and as in (1.9) when . Then . Moreover, we have that

as . In fact, for any , it holds that

as .

Notice that is not in the space. So, we get the so-called measure differential equations. For some basic theory for eigenvalues of the second-order linear measure differential equations, see [23].

### Competing interests

The author declares that he has no competing interests.

### Author’s contributions

The author completed the paper himself. The author read and approved the final manuscript.

### Acknowledgements

The author is supported by the National Natural Science Foundation of China (Grant No. 11201471), the Marine Public Welfare Project of China (No. 201105032) and the President Fund of GUCAS.

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