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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

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/220


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

© 2013 Meng; licensee Springer.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2">View MathML</a> 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; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2">View MathML</a> norm of integrable potentials is given.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M4">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M5">View MathML</a> denote the Lebesgue space of real functions with the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M6">View MathML</a> norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M7">View MathML</a>. Given a potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M8">View MathML</a>, we consider Sturm-Liouville equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M9">View MathML</a>

(1.1)

with the Dirichlet boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M10">View MathML</a>

(1.2)

It is known that problem (1.1)-(1.2) has countably many eigenvalues (see [11,12]). They are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M12">View MathML</a> , and ordered in such a way that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M13">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M14">View MathML</a>, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M15">View MathML</a>

In this paper, we study the following minimization problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M16">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M17">View MathML</a>

Given constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M18">View MathML</a>, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M19">View MathML</a>

The problem is to find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M20">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2">View MathML</a> balls, which are not compact even in the weak topology <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M22">View MathML</a>. In order to overcome this difficulty in topology, we first solve the following approximating minimization problem of eigenvalues.

Theorem 1.1Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M23">View MathML</a>

(1.5)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M24">View MathML</a>. We have that

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M26">View MathML</a>

(1.6)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M27">View MathML</a>is attained for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M28">View MathML</a>

(1.7)

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M29">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M30">View MathML</a>is the unique solution of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M31">View MathML</a>. Here, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M32">View MathML</a>is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M33">View MathML</a>

(1.8)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M27">View MathML</a>is attained for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M35">View MathML</a>

(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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M36">View MathML</a>

(1.10)

Here, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M37">View MathML</a>is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M38">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M6">View MathML</a> balls, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M40">View MathML</a>. Then minimization problem (1.3) can be solved by complicated limiting analysis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M41">View MathML</a>. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M43">View MathML</a>, besides the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M6">View MathML</a> norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M45">View MathML</a>, one has the following weak topologies [19]. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M46">View MathML</a>, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M47">View MathML</a> to indicate the topology of weak convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M42">View MathML</a>, and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M49">View MathML</a>, by considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M50">View MathML</a> as the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M51">View MathML</a>, we have the topology <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M52">View MathML</a> of weak convergence. In a unified way, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M53">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M54">View MathML</a> iff

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M55">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M56">View MathML</a> 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, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M57">View MathML</a>are continuous in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M59">View MathML</a>.

By the continuity result above, we show that extremal problem (1.5) can be attained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M60">View MathML</a>.

Lemma 2.2There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M61">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M62">View MathML</a>.

Proof Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M63">View MathML</a> is compact and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M64">View MathML</a> is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M65">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M60">View MathML</a> is compact in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M65">View MathML</a>. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M68">View MathML</a>

(2.1)

Moreover, if, in addition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M69">View MathML</a>holds on a subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M70">View MathML</a>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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M70">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72">View MathML</a> (resp., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M73">View MathML</a>) the symmetrically increasing (resp., decreasing) rearrangement of f. We recall that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72">View MathML</a> is uniquely defined by the following conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72">View MathML</a> and f are equimeasurable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M70">View MathML</a>. That is, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M77">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M78">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72">View MathML</a> is symmetric about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M80">View MathML</a>.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M72">View MathML</a> is decreasing in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M82">View MathML</a>.

Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M73">View MathML</a> is (uniquely) defined by (i), (ii) and (iii)’: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M73">View MathML</a> is increasing in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M82">View MathML</a>. 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M86">View MathML</a>, we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M87">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M88">View MathML</a> be a positive eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M89">View MathML</a>. By [[22], Theorem 378] and [[6], Section 7], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M90">View MathML</a>

 □

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M91">View MathML</a>.

Proof of Theorem 1.1 (i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M25">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M93">View MathML</a>. From the monotonicity property (2.1) of eigenvalues, we have that the minimizer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M94">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M95">View MathML</a> by computing directly.

(ii) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M96">View MathML</a>. By Lemma 2.2, Lemma 2.3 and Lemma 2.4, there exists a minimizer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M97">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M99">View MathML</a> for a.e. t. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M88">View MathML</a> be a positive eigenfunction corresponding to the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M101">View MathML</a>. We have from the proof of Lemma 2.4 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M102">View MathML</a> is also an eigenfunction corresponding to the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M101">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M104">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M105">View MathML</a>.

Then, we have that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M106">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M107">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M108">View MathML</a>

(3.1)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M110">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M111">View MathML</a> is symmetric about <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M80">View MathML</a> and increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M113">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M114">View MathML</a>

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M115">View MathML</a>

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M116">View MathML</a>

(3.2)

In fact,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M117">View MathML</a>

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M118">View MathML</a>. Comparing (3.1) and (3.2), we know that the equality holds in (3.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M88">View MathML</a> is an eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M120">View MathML</a>. Since (3.2) is an equality, we have from the proof that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M121">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M122">View MathML</a> for a.e. t.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M123">View MathML</a>. Now, (1.1) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M124">View MathML</a>

(3.3)

We can find that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M111">View MathML</a> of (3.3) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M126">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M127">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M128">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M129">View MathML</a>, we get from (3.4) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M130">View MathML</a> is the unique solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M131">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M132">View MathML</a> is as in (1.8). □

Finally, we use the limiting approach to obtain a complete solution for the minimization problem of eigenvalues when the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M2">View MathML</a> norm of integrable potentials is given.

Proof of Theorem 1.2 By the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M134">View MathML</a> in (1.8) and Z in (1.11), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M135">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M136">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M137">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M138">View MathML</a> and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M139">View MathML</a>

(3.5)

On the other hand, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M89">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M89">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M143">View MathML</a>. Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M144">View MathML</a>, where the closure is taken in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M145">View MathML</a> space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M51">View MathML</a>. We have that for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M148">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M150">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M151">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M152">View MathML</a>

Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M148">View MathML</a> is arbitrary, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M154">View MathML</a>

Therefore, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M155">View MathML</a>

(3.6)

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M147">View MathML</a> is arbitrary.

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

Remark 3.1 Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M157">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M158">View MathML</a> is as in (1.7) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M25">View MathML</a> and as in (1.9) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M160">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M161">View MathML</a>. Moreover, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M162">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M163">View MathML</a>. In fact, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M164">View MathML</a>, it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M165">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M166">View MathML</a>.

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M167">View MathML</a> is not in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/220/mathml/M145">View MathML</a> 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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