Minimization of eigenvalues for some differential equations with integrable potentials
© Meng; licensee Springer. 2013
Received: 13 March 2013
Accepted: 28 August 2013
Published: 7 November 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.
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.
- (i)If , then(1.6)
- (ii)If , then is the unique solution of . Here, the function is defined as(1.8)
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
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.
- (i)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 .
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 .
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.
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.
since is arbitrary.
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 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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