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Minimization of eigenvalues for some differential equations with integrable potentials
Boundary Value Problems volume 2013, Article number: 220 (2013)
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.
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
with the Dirichlet boundary condition
For , denote
In this paper, we study the following minimization problem:
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  the minimization problem of weighted Dirichlet eigenvalues of
Given constants , denote
The problem is to find
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 1.1 Let
where . We have that
If , then(1.6)
Moreover, is attained for
If , then is the unique solution of . Here, the function is defined as(1.8)
Moreover, is attained for
Theorem 1.2 The following holds:
Here, the function is defined as
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 . 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.
Lemma 2.1 As nonlinear functionals, are continuous in , .
By the continuity result above, we show that extremal problem (1.5) can be attained in .
Lemma 2.2 There exists such 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 .
Moreover, if, in addition, holds on a subset of of 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:
and f are equimeasurable on . That is, for all ,
is symmetric about .
is decreasing in the interval .
We can compare the first eigenvalue of q with the first eigenvalue of its rearrangement.
Lemma 2.4 For any , 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.
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 ,
Let and . Since is symmetric about and increasing in , we have that
We claim that
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
We can find that the solution of (3.3) is given by
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
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
Because is arbitrary, it holds that
Therefore, we have that
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
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 .
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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.
The author declares that he has no competing interests.
The author completed the paper himself. The author read and approved the final manuscript.