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
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.
with the Dirichlet boundary condition
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
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 1.2The following holds:
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
Eigenvalues possess the following monotonicity property .
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:
We can compare the first eigenvalue of q with the first eigenvalue of its rearrangement.
3 Main results
(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 .
We claim that
which implies that
Therefore, we have that
By (3.5) and (3.6), the proof is complete. □
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 .
The author declares that he has no competing interests.
The author completed the paper himself. The author read and approved the final manuscript.
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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