The optimal upper bound for the first eigenvalue of the fourth order equation

In this paper we will give the optimal upper bound for the first eigenvalue of the fourth order equation with integrable potentials when the L 1 norm of potentials is known. Combining with the results for the corresponding optimal lower bound problem in [ 12 ], we have completely obtained the optimal estimation for the first eigenvalue of the fourth order equation.

Minimization and maximization problems for eigenvalues are important in applied sciences like quantum mechanics [2], population dynamics [5] and propagation speeds of traveling waves [8]. These are also interesting mathematical problems [1,6,7,9,18,19,20,21], because the solutions are involved of many different branches of mathematics.
Since the L 1 ball B 1 [r] is not compact even in the weak topology of L 1 , we usually do not know if maximization problem (3) can be attained by some potentials from B 1 [r]. To overcome this, different from the approach in [13,17], here we will extend the problem to the measure case. More precisely, let µ : [0, 1] → R be a measure. Firstly, we will find the explicit optimal upper bound for the first eigenvalue λ 1 (µ) of the corresponding measure differential equation (MDE) with the corresponding Lidstone boundary condition when the total variation of measure µ is known. Secondly, Based on the relationship between maximization problem of ODE and of MDE, we can obtain the main result of this paper as follows.
and B 1 (≈ 500.56) be the unique root of the equation One has the following conclusions.
(i) When 0 < r ≤ H(B 1 ) (≈ 215.42), it holds that Here, the invertible elementary function H : (π 4 , A 1 ) → (0, +∞) is defined as (ii) When r > H(B 1 ), it holds that Here, t 0 ∈ (0, 1/2) is the unique root of polynomial We should point out that the methods utilized in this paper are original and constructive. We will illustrate in further work that the methods of this paper are more powerful to solve some types of extremal problems for eigenvalues. This paper is organized as follows. In Section 2, we will recall some basic facts on eigenvalue theory of MDE and introduce useful properties about some elementary functions. In Section 3, based on the minimization characterization of the first eigenvalue and the relationship between maximization problem of ODE and of MDE, we will prove Theorem 1.1.  |µ(t i+1 ) − µ(t i )| : 0 < t 0 < · · · < t n−1 < t n = t, n ∈ N .
Typical examples of measures are as follows.
• Let : I → R be (t) ≡ t. Then yields the Lebesgue measure of I and the Lebesgue integral. More generally, any q ∈ L 1 (I, R) induces an absolutely continuous measure defined by In this case, one has , for any f ∈ C(I, R) and subinterval I 0 ⊂ I.
• For a = 0, the unit Dirac measure at t = 0 is • For a ∈ (0, 1], the unit Dirac measure at t = a is By the Riesz representation theorem [10], Moreover, one has In the space M 0 (I, R) of measures, one has the usual topology induced by the norm · V . Due to duality relation (16), one has also the following weak * topology w * . Let µ 0 , µ n ∈ M 0 (I, R), n ∈ N. We say that µ n is weakly * convergent to µ 0 iff, for each f ∈ C(I, R), one has We remark that in some literature, this topology is just called the weak topology for measures.
In general, a measure cannot be a limit of smooth measures in the norm · V . However, in the w * topology, the following conclusion holds.
Moreover, if µ 0 is increasing (decreasing) on I, then the sequence {µ n } above can be chosen such that for each n ∈ N, µ n is increasing (deceasing) on I and µ n V = µ 0 V .
Considering q ∈ L 1 (I, R) as a density, one has the measure or distribution given by (14). Since µ q V = q 1 , Via (14), by the Hölder inequality and the isometrical embedding (17), the following inclusion is proper . As for the compactness of these balls in weak * topology, we have that B 0 [r] ⊂ (M 0 (I, R), w * ) is sequentially compact. See [10].
Given a measure µ ∈ M 0 := M 0 (I, R), we will write the fourth order linear MDE with the measure µ as • there exist (y 0 , y 1 , y 2 , y 3 ) ∈ R 4 and functions y (1) , y (2) , y (3) : [0, 1] → R such that the following are satisfied The initial condition of MDE (19) can be written as Since we have assumed that y ∈ C := C([0, 1], R), the right-hand sides of (20), be the Sobolev spaces with the norm · W n,p . For p = 2, W n,2 and W n,2 0 are denoted simply by H n and H n 0 , respectively, with the norm · H n . By the properties of Lebesgue integral and Lebesgue-Stieltjes integral, some regularity results for solutions y(t) are as follows.  We use y(t, y 0 , y 1 , y 2 , y 3 ) to denote the unique solution of (19) and (24). Let  (19). By the linearity of (19) and the uniqueness of solution, one has that, for t ∈ [0, 1], We consider eigenvalue problem of the fourth order equation (7) with the Lidstone boundary condition (2). Given µ ∈ M 0 , we say that λ ∈ R is an eigenvalue of the Lidstone problem (7) and (2), if MDE (7) with such a parameter λ has non-zero solutions y(t) satisfying (2). The corresponding solutions y(t) are called eigenfunctions associated with λ.
Besides the Sobolev spaces H 2 0 and H 3 0 , let us introduce H 3 00 := {y ∈ H 3 : y satisfies (2)} = {y ∈ H 3 : y(0) = y(1) = y (0) = y (1) = 0}. One has the proper inclusions H 3 00 ⊂ H 3 0 ⊂ H 2 0 . In [12], the authors have established the minimization characterization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. Here, Moreover, we have the following variational characterizations of the higher order eigenvalues, which is a limiting case of the minimax principle [4].
where S is defined to be the class of piecewise twice differentiable functions satisfying Let us introduce the following ordering for measures. We say that measures As a consequence of (25) in Lemma 2.5, we can obtain the following result.
Lemma 2.10. For each r > H(B 1 ), the following equation because B 0 [r] is sequentially compact in (M 0 , w * ) and Lemma 2.8. We need the following result about the problem (7) and (2).
We will find the explicit optimal upper bound of the first eigenvalue of MDE.
To this end, we need to solve the following equation From the explanation to solutions of MDE, one knows that solutions y(t) of (36) satisfies the classical ODE y (t) = λy(t) Denote From ODE (37) and the first two conditions of (2), we obtain By using this as the initial value at t = 1/2, we obtain from ODE (37) that for t ∈ (1/2, 1]. Now the last two conditions y(1) = y (1) = 0 of (2) are the following linear system for (c 1 , c 2 ) This can yield the relation In order that system (41) has non-zero solutions (c 1 , c 2 ), the corresponding determinant of (41) is necessarily zero. This yields the following equation Then, by the existence of the first eigenvalue, we conclude that Here A 1 and H are defined in (8) and (10), respectively.