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Continuous minimizer of eigenvalues for eigenvalue problem with equimeasurable weights
 Zhiyuan Wen^{1}Email author and
 Lijuan Zhou^{1}
 Received: 1 February 2018
 Accepted: 13 April 2018
 Published: 9 May 2018
Abstract
The problem in this paper is motivated by physical problems concerned with the case when a class of continuous and equimeasurable densities of a string is given then how to find minimal frequencies among these given densities, that is, what kind of densities minimize the frequencies. By taking Dirichlet eigenvalues into account, given a certain weight function ω, we will show the minimizer of the mth eigenvalue is the mdegree continuous symmetrical decreasing rearrangement of ω. The main result of this paper can be viewed as complementary to Schwarz’s work (Schwarz in J. Math. Mech. 10:401–422, 1961).
Keywords
 Distribution function
 Equimeasurable
 Eigenvalue
 Rearrangement
1 Introduction
The physical explanation of this problem is as follows. Given a class of continuous densities with the same distribution, what kind of density minimizes the frequencies of a string with fixed ends? The solution to this problem will be called a minimizer in this paper. The problem is called a minimization problem; it dates back to Courant and Krein [3, 9].
The results have many generalizations; see for example [2, 4, 10, 11]. In recent papers [6, 11–15] the authors generalized Krein’s result for various kinds of differential operators from second order linear differential operators or nonlinear pLaplacian to measure differential equations. The admissible set of functions varies from integrable potentials or weights to measures with the same integral value or with fixed variations. All in all the admissible set of functions in these minimization problems are either discontinuous or integrable functions, while the results of these papers show that minimizers of eigenvalues are symmetric.
Since ω is assumed “piecewise continuous”, \(\omega^{+}_{m}\) may also be piecewise continuous. But, in the case of a physical application, to determine minimizers over continuous functions is more meaningful, just as Schwarz wrote in his paper “While for physical application this theorem is of interest only for positive continuous functions, our method of proof forces us to consider equimeasurable classes of piecewise continuous functions”.
It seems that to determine minimizers over an admissible set of “bad continuous” functions is easier than “good continuous” functions. In this paper we point out \(\omega^{+}_{m}\) is continuous, provided ω satisfies (3) and (4). Our main theorem is the following one.
Theorem 1.1
Given \(m\in \mathbb{N}\), suppose \(\omega(x)\) satisfies (3) and (4). Then \(\omega^{+}_{m}(x)\) is continuous and such that (5) holds.
The result is meaningful in practical science. For instance, in a remarkable paper [5], after having solved a minimization problem associated with (1) where an admissible set of functions is a certain class of integrable functions, the authors gave an explanation of its application to spatial biological cases. The weight function ω represents a bio density or a density of resources. To determine the minimizer of the eigenvalue will be helpful in optimizing resources and to maintain the existence of species.
2 Proof of main theorem
2.1 Preliminaries
Let us give some preliminaries on level sets, distributions and mdegree symmetrical decreasing rearrangement of functions.
The symmetric rearrangement, or the Schwarz rearrangement [16] is a method used by Hardy, Littlewood and Polya in [16] to study the Hardy–Littlewood–Polya inequality Also it is useful in many mathematical and physical problems. The following is standard; see for example,[1, 8, 16, 17].
 (\(\mathbf{p}_{1}\)):

For any \(t\ge0\), one has equimeasurability,$$\mu\bigl\{ x\in J: f(x)\ge t\bigr\} =\mu\bigl\{ x\in J: f^{+}_{m}(x) \ge t\bigr\} . $$
 (\(\mathbf{p}_{2}\)):

For each \(j=1,\ldots,m\),Moreover, \(f^{+}_{m}(x)\) is symmetrical decreasing on each periodic interval. In particular, if a nonnegative measurable function \(f(x)\) satisfies (7) and also symmetrical decreasing on each periodic interval, then \(f^{+}_{m}(x)=f(x)\) a.e. \(x\in[a,b]\).$$f^{+}_{m}(x)\equiv f^{+}_{m}\biggl(2a+ \frac{(2j1)(ba)}{m}x\biggr),\quad \forall x\in\biggl[a+\frac{(j1)(ba)}{m},a+ \frac{j(ba)}{m}\biggr]. $$
 (\(\mathbf{p}_{3}\)):

If \(0\le f(x)\le M\) a.e. \(x\in[a,b]\) for some \(M>0\), then \(0\le f^{+}_{m}(x)\le M\) a.e. \(x\in[a,b]\). Moreover,and$$f^{+}_{m}\biggl(a+\frac{(j1)(ba)}{m}\biggr)=f^{+}_{m} \biggl(a+\frac{j(ba)}{m}\biggr) = \inf_{J} f(x), $$$$f^{+}_{m}\biggl(a+\frac{(2j1)(ba)}{2m}\biggr)=\sup _{J} f(x). $$
2.2 Proof of Theorem 1.1
The proof of Theorem 1.1 is based on the following lemma.
Lemma 2.1
Proof
Proof of Theorem 1.1
Declarations
Acknowledgements
Both authors would like to thank professor Meng Gang and Meirong Zhang for helpful discussions.
Funding
Both authors are supported by scientific starting research foundation of Inner Mongolia University, No. 212005175108 and No. 201005165106.
Authors’ contributions
Both authors have equally contributed to this article and read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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