Minimization of eigenvalues for some differential equations with integrable potentials

In this paper we use the limiting approach to solve the minimization problem of the Dirichlet eigenvalues of Sturm-Liouville equations when the L^1 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.

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 L^1 norm of integrable potentials is given.

For 1\le p\le \mathrm{\infty }, let {\mathcal{L}}^p:=L^p([0,1],\mathbb{R}) denote the Lebesgue space of real functions with the L^p norm {\parallel \cdot \parallel }_p={\parallel \cdot \parallel }_{L^p[0,1]}. Given a potential q\in {\mathcal{L}}^p, we consider Sturm-Liouville equations

with the Dirichlet boundary condition

It is known that problem (1.1)-(1.2) has countably many eigenvalues (see [11, 12]). They are denoted by {\lambda }_j, j=1,2,\dots , and ordered in such a way that

For r\in (0,\mathrm{\infty }), 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 [7] 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 L^1 balls, which are not compact even in the weak topology w_1. In order to overcome this difficulty in topology, we first solve the following approximating minimization problem of eigenvalues.

Theorem 1.1 Let

where r,H\in (0,\mathrm{\infty }). We have that

1. (i)

   If r\ge H, then

   \mathbf{L}(r,H)={\pi }^2-H.

   (1.6)

Moreover, \mathbf{L}(r,H) is attained for

1. (ii)

   If , then \mathbf{L}(r,H)\in (-\mathrm{\infty },{\pi }^2-r) is the unique solution of {\mathbf{Z}}_{r,H}(x)=0. Here, the function {\mathbf{Z}}_{r,H}:(-\mathrm{\infty },{\pi }^2-r)\to \mathbb{R} is defined as

   {\mathbf{Z}}_{r,H}(x)=\{\begin{array}{cc}1+\frac{\sqrt{-x-H}}{\sqrt{-x}}tanh\sqrt{-x}\frac{H-r}{2H}tanh\sqrt{-x-H}\frac{r}{2H}\hfill & \mathit{\text{for }}x\in (-\mathrm{\infty },-H),\hfill \\ 1-\frac{\sqrt{x+H}}{\sqrt{-x}}tanh\sqrt{-x}\frac{H-r}{2H}tan\sqrt{x+H}\frac{r}{2H}\hfill & \mathit{\text{for }}x\in [-H,0),\hfill \\ 1-\frac{H-r}{2\sqrt{H}}tan\frac{r}{2\sqrt{H}}\hfill & \mathit{\text{for }}x=0,\hfill \\ 1-\frac{\sqrt{x+H}}{\sqrt{x}}tan\sqrt{x}\frac{H-r}{2H}tan\sqrt{x+H}\frac{r}{2H}\hfill & \mathit{\text{for }}x\in (0,{\pi }^2-r).\hfill \end{array}

   (1.8)

Moreover, \mathbf{L}(r,H) is attained for

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 1.2 The following holds:

Here, the function \mathbf{Z}:(-\mathrm{\infty },{\pi }^2]\to [0,\mathrm{\infty }) 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 L^p balls, . Then minimization problem (1.3) can be solved by complicated limiting analysis of p\downarrow 1. 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.

In the Lebesgue spaces {\mathcal{L}}^p, p\in [1,\mathrm{\infty }], besides the L^p norms {\parallel \cdot \parallel }_p, one has the following weak topologies [19]. For p\in [1,\mathrm{\infty }), we use w_p to indicate the topology of weak convergence in {\mathcal{L}}^p, and for p=\mathrm{\infty }, by considering {\mathcal{L}}^{\mathrm{\infty }} as the dual space of ({\mathcal{L}}^1,{\parallel \cdot \parallel }_1), we have the topology w_{\mathrm{\infty }} of weak^∗ convergence. In a unified way, q_m\to q_0 in ({\mathcal{L}}^p,w_p) iff

Here, p^{\ast }:=p/(p-1)\in [1,\mathrm{\infty }] is the conjugate exponent of p.

To solve problem (1.5), let us quote from [14, 16] some important properties on eigenvalues.

Lemma 2.1 As nonlinear functionals, {\lambda }_n(q) are continuous in q\in ({\mathcal{L}}^p,w_p), p\in [1,\mathrm{\infty }].

By the continuity result above, we show that extremal problem (1.5) can be attained in B_1[r]\cap B_{\mathrm{\infty }}[H].

Lemma 2.2 There exists q_1\in B_1[r]\cap B_{\mathrm{\infty }}[H] such that \mathbf{L}(r,H)={\lambda }_1(q_1).

Proof Notice that B_{\mathrm{\infty }}[H] is compact and B_1[r]\cap {\mathcal{L}}^{\mathrm{\infty }} is closed in ({\mathcal{L}}^{\mathrm{\infty }},w_{\mathrm{\infty }}). Hence B_1[r]\cap B_{\mathrm{\infty }}[H] is compact in ({\mathcal{L}}^{\mathrm{\infty }},w_{\mathrm{\infty }}). 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

Moreover, if, in addition, holds on a subset of [0,1] 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 [0,1], we denote by f^{+} (resp., f^{-}) the symmetrically increasing (resp., decreasing) rearrangement of f. We recall that the function f^{+} is uniquely defined by the following conditions:

1. (i)

   f^{+} and f are equimeasurable on [0,1]. That is, for all s\ge 0,

   m\left(\{t:f^{+}(t)\ge s\}\right)=m\left(\{t:f(t)\ge s\}\right).
2. (ii)

   f^{+} is symmetric about t=1/2.

3. (iii)

   f^{+} is decreasing in the interval [0,1/2].

Similarly, f^{-} is (uniquely) defined by (i), (ii) and (iii)’: f^{-} is increasing in the interval [0,1/2]. 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 2.4 For any q\in {\mathcal{L}}^p, we have {\lambda }_1(q)\ge {\lambda }_1({|q|}^{-}).

Proof Let y_1(t) be a positive eigenfunction corresponding to {\lambda }_1(q). By [[22], Theorem 378] and [[6], Section 7], we have

 □

Now we are ready to prove the main results of this paper. First, we solve the minimization problem for eigenvalues when potentials q\in B_1[r]\cap B_{\mathrm{\infty }}[H].

Proof of Theorem 1.1 (i) If r\ge H, then B_1[r]\cap B_{\mathrm{\infty }}[H]=B_{\mathrm{\infty }}[H]. From the monotonicity property (2.1) of eigenvalues, we have that the minimizer q_1=H and then \mathbf{L}(r,H)={\pi }^2-H by computing directly.

1. (ii)

   When . By Lemma 2.2, Lemma 2.3 and Lemma 2.4, there exists a minimizer 0\le q_1\in B_1[r]\cap B_{\mathrm{\infty }}[H] such that \mathbf{L}(r,H)={\lambda }_1(q_1) and q_1(t)=q_1^{-}(t) for a.e. t. Let y_1(t) be a positive eigenfunction corresponding to the first eigenvalue {\lambda }_1(q_1). We have from the proof of Lemma 2.4 that y_1^{-}(t) is also an eigenfunction corresponding to the first eigenvalue {\lambda }_1(q_1), which implies that y_1=y_1^{-} because {\int }_0^1{y}_1\mathrm{d}t={\int }_0^1{y}_1^{-}\mathrm{d}t.

Then, we have that for each q\in B_1[r]\cap B_{\mathrm{\infty }}[H],

Hence,

Let \eta :=\frac{H-r}{2H}\in (0,1/2) and \xi :=y_1(\eta )>0. Since y_1 is symmetric about t=1/2 and increasing in [0,1/2], we have that

Define

We claim that

In fact,

Notice that q_{\eta }\in B_1[r]\cap B_{\mathrm{\infty }}[H]. Comparing (3.1) and (3.2), we know that the equality holds in (3.2) and y_1(t) is an eigenfunction corresponding to {\lambda }_1(q_{\eta }). Since (3.2) is an equality, we have from the proof that {\lambda }_1(q_1)={\lambda }_1(q_{\eta }) and q_1(t)=q_{\eta }(t) for a.e. t.

Let {\mu }_1:={\lambda }_1(q_1). Now, (1.1) becomes

We can find that the solution y_1 of (3.3) is given by

where a=\xi \frac{\sqrt{{\mu }_1+H}}{\sqrt{{\mu }_1}}tan\sqrt{{\mu }_1+H}\frac{r}{2H}. Since y_1(0)=y_1(1)=0 and {\mu }_1\le {\pi }^2-r, we get from (3.4) that {\mu }_1 is the unique solution of {\mathbf{Z}}_{r,H}(x)=0, where {\mathbf{Z}}_{r,H}(x) is as in (1.8). □

Finally, we use the limiting approach to obtain a complete solution for the minimization problem of eigenvalues when the L^1 norm of integrable potentials is given.

Proof of Theorem 1.2 By the definitions of {\mathbf{Z}}_{r,H} in (1.8) and Z in (1.11), we have that

which implies that

Since B_1[r]\supset B_1[r]\cap B_{\mathrm{\infty }}[H], we have that \mathbf{L}(r)\le \mathbf{L}(r,H) and then

On the other hand, since {\lambda }_1(q) is continuous in q\in ({\mathcal{L}}^1,w_1), {\lambda }_1(q) is continuous in q\in ({\mathcal{L}}^1,{\parallel \cdot \parallel }_1). Notice that B_1[r]\subset \overline{{\bigcup }_{H>0}(B_1[r]\cap B_{\mathrm{\infty }}[H])}, where the closure is taken in the {\mathcal{L}}^1 space ({\mathcal{L}}^1,{\parallel \cdot \parallel }_1). We have that for arbitrary ϵ>0 and q\in B_1[r], there exist H_0 and q_{H_0}\in B_1[r]\cap B_{\mathrm{\infty }}[H_0] such that

Hence,

Because q\in B_1[r] is arbitrary, it holds that

Therefore, we have that

since ϵ>0 is arbitrary.

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

Remark 3.1 Fix r>0. Assume that q_H\in B_1[r]\cap B_{\mathrm{\infty }}[H] is as in (1.7) when r\ge H and as in (1.9) when . Then \mathbf{L}(r,H)={\lambda }_1(q_H). Moreover, we have that

as H\to +\mathrm{\infty }. In fact, for any f\in C[0,1], it holds that

as H\to +\mathrm{\infty }.

Notice that r{\delta }_{1/2} is not in the {\mathcal{L}}^1 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].

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.

Affiliations

1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

   • Gang Meng

Corresponding author

Correspondence to Gang Meng.

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.

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