On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures

In this paper, we will first establish the necessary and sufficient conditions for the existence of the principal eigenvalues of second-order measure differential equations with indefinite weighted measures subject to the Neumann boundary condition. Then we will show the principal eigenvalues are continuously dependent on the weighted measures when the weak$^*$ topology is considered for measures. As applications, we will finally solve several optimization problems on principal eigenvalues, including some isospectral problems.

Here w(x) ∈ L 1 (I) are nonzero indefinite (integrable) weights. By a principal eigenvalue λ of problem (1)-(N ), it means that the problem admits eigenfunctions ϕ(x) so that ϕ(x) > 0 on I. For any weight w = 0, λ = 0 is a trivial principal eigenvalue with an eigenfunction ϕ(x) ≡ 1. Henceforth, our concern is on nontrivial principal eigenvalues. For simplicity, the principal eigenvalues in this paper are always meant the positive principal eigenvalues. For the ODE case (1), the existence of principal eigenvalues is clear [2,5]. In fact, the necessary and sufficient conditions for (1)-(N ) to admit a principal eigenvalue λ prin > 0 are Here w ± (x) = max{±w(x), 0} are the positive and negative parts of w(x). See, for example, [22,Section 4]. The proofs in [22] are based on the Prüfer transformation and detailed analysis for the resulted argument functions. When w satisfies (2), the principal eigenvalue λ prin = λ prin (w) is unique. On higher dimensional domains, the existence of principal eigenvalues of partial differential equations can be found from [4,28]. Principal eigenvalues λ prin = λ prin (w) are important in ecology systems and population dynamics [5,17,18]. When the environment is spatially heterogeneous, w(x) is the (indefinite) density of the distribution of resource on I and the principal eigenvalue λ prin (w) stands for the surviving threshold for a single species. This leads to many interesting optimization problems on principal eigenvalues. Some have been solved [7,15,18], while many are still open [17]. In this paper, we consider the following related optimization problem. Given 0 < A < B, we denote W A,B := w ∈ L 1 (I) : I w + (x) dx > 0, I w(x) dx ≤ −A, and I |w(x)| dx ≤ B .
(3) It is not difficult to see that sup w∈W A,B λ prin (w) = +∞.
The optimization problem we are interested in is which is well defined. Due to the non-compactness of W A,B (even in the weak topology of L 1 (I)), we will see that the solution of problem (5) will lead to more general distributions of resources which have no densities (with respect to the Lebesgue measure). Moreover, principal eigenvalues and eigenfunctions have to be explained using the so-called measure differential equations (MDEs or MDE), which are a special class of generalized ordinary differential equations (GODEs) studied extensively in [24,27]. Let us use M 0 (I) to denote the space of (normalized, real, Radon) measures on I. The norm · of M 0 (I) is the total variation of measures. The space M 0 (I) is isometrically identical as the dual space of C(I), the space of real continuous functions on I with supremum norm · ∞ . For more details on M 0 (I), see §2.
The second-order linear MDE with a measure µ ∈ M 0 (I) is written as following the notations from [23]. (Strong) solutions u(x) of initial value problems of (6) are explained using the equivalent system of integral equations. Here u • (x), x ∈ I stands for the generalized right derivative of the solution u(x). It is known that u • (x) is a function of bounded variation, or a non-normalized Radon measure on I. With a measure ν ∈ M 0 (I) as a potential, eigenvalues of MDE with the Dirichlet boundary condition or, with the Neumann boundary condition have been established in [23]. The structure of these eigenvalues are the same as those of the ODEs with integrable potentials. For example, problem (7)-(N ) possesses a sequence of eigenvalues τ 0 (ν) < τ 1 (ν) < · · · → +∞, where τ 0 (ν) is called the zeroth Neumann eigenvalue. Spectral theory of MDEs and some types of GODEs has been extensively studied in recent works like [11,12,13,16,26,37]. In [37], by considering a nonzero nonnegative measure ρ ∈ M 0 (I) as a weight, for the (weighted) eigenvalue problem with (D) or (N ), some completely different features have been revealed. For example, the numbers of eigenvalues of problems (8)-(D) and (8)-(N ) depend on measures ρ and may be finite. These results on second-order MDEs, together with the so-called completely continuous dependence of eigenvalues on measures, have been extended in [16] to some class of third-order symmetric MDEs. Motivated by the study on the Camassa-Holm equation [8,9] and general strings, some spectral and inverse spectral problems on second-order GODEs with indefinite distributions as weights have been developed in [11,12,13] and are still under developing. In order to solve problems (4) and (5) and for the theoretical purpose, in this paper, by taking indefinite measures ρ ∈ M 0 (I), we will give a relatively complete study on the (positive) principal eigenvalues of MDEs (8) with respect to the Neumann boundary condition (N ). According to the Jordan's decomposition theorem [31,Theorem 3.3], any ρ ∈ M 0 (I) can be uniquely decomposed as ρ = ρ + − ρ − . Here ρ + and ρ − are the non-negative and non-positive part of ρ, respectively. As functions of bounded variation, both ρ ± (x) are non-decreasing in x ∈ I. Corresponding to (2) for indefinite integrable weights, let us introduce a subset M prin := ρ ∈ M 0 (I) : ρ + > 0 and I dρ < 0 of measures on I. As before, by a principal eigenvalue λ of (8)-(N ), we always refer λ to a positive principal eigenvalue. The first part of this paper contains two main results. This first one gives the necessary and sufficient conditions for the existence of positive principal eigenvalues. (ii) Let ρ ∈ M prin be given. Then problem (8)-(N ) has a unique principal eigenvalue, denoted by λ prin = λ prin (ρ) > 0. Moreover, λ prin can be characterized by and the minimum is attained and only attained by eigenfunctions associated with λ prin . Here U ρ := u ∈ W 1,2 (I) : I u 2 dρ > 0 ⊂ W 1,2 (I).
Because we will establish in Proposition 1 the equivalence of weak and strong solutions of (8), so different from the approach by the Prüfer transformation in [22,23,37], the proof of Theorem 1.1 is variational.
In the measure space M 0 (I), besides the topology induced by norm · , it also possesses the weak * topology w * induced by the weak * convergence of measures. The next result asserts that λ prin (ρ) is continuous in ρ ∈ M prin even when the weak * topology w * is considered for M prin . Theorem 1.2. Assume that ρ ∈ M prin and ρ n ∈ M prin for all n ≥ 1. If ρ n ρ with respect to the weak * topology w * , then λ prin (ρ n ) → λ prin (ρ).
Since the weak * topology w * is considered for M prin , Theorem 1.2 asserts that λ prin (ρ) is continuously dependent on ρ in a very strong way. Such a continuous dependence can be called the complete continuity or the strong continuity. This type of continuity has been first proved in [23,37] for eigenvalues of (7) in potential measures ν and of (8) in non-negative weight measures ρ. Different from the proofs in [23,37], in this paper, Theorem 1.2 is obtained from the minimization characterization (9) for principal eigenvalues.
In the second part of this paper, we will apply Theorems 1.1 and 1.
where δ 0 and δ 1 are the Dirac measures at 0 and 1, respectively. We will obtain the following results.  By using the minimization characterization (9) for principal eigenvalues, problems (12) and (13) can be reduced to finitely dimensional extremal problems, whose solutions are elementary. After giving the proofs in §4.1, we will give more comments on the method used in this paper.
Next, let 0 < A < B be given. It is known from Theorem 1.
The solution of minimization problem (14) is shown by Moreover, the minimal value T(γ) is attained and only attained by eitherρ γ orρ γ defined in (16).
The rest of this paper is organized as follows. In §2, we record some preliminaries and prove some basic facts regarding measure theory and second-order MDEs. §3 is devoted to prove Theorem 1.1 and Theorem 1.2. In §4.1 and §4.2, Theorem 1.3 and Theorem 1.4 are proved. Finally, in §4.3, by using the complete continuity of λ prin (ρ) in Theorem 1.2 and a smooth approximation theorem for general measures in [19], one can find that the solution of problem (5)  The Riesz representation theorem asserts that (M 0 (I), · ) is isometrically identical as the dual space of (C(I), · ∞ ), the Banach space of continuous functions on I endowed with the supremum norm · ∞ . In fact, by using the Riemann-Stieltjes integrals, any ρ ∈ M 0 (I) defines a linear functional ρ * ∈ (C(I), · ∞ ) * by For this reason, M(I) and M 0 (I) are respectively called the space of non-normalized Radon measures on I and the space of (normalized Radon) measures on I. We use δ a , a ∈ I to denote the (unit) Dirac measure at a. Clearly, δ a = 1 for any a ∈ I and δ a − δ b = 2 for all a, b ∈ I with a = b. We say that A measure ρ is non-negative if and only if ρ(x) is non-decreasing on I; see [31]. For example, the Dirac measures δ a are non-negative. The Jordan's decomposition theorem for measures is as follows.
We call ρ + the non-negative part and ρ − the non-positive part of ρ. One has then See [6,31]. For ρ ∈ M 0 (I) and x ∈ I, let ρ [0,x] be the total variation of ρ on interval [0, x]. If we consider ρ(x) and ρ [0,x] as functions of x ∈ I, then for any Based on such an observation, we have the following conclusion. (10) is nonempty.
Proof. The condition ρ + > 0 implies that ρ(x) cannot be a decreasing function in x ∈ I. Thus there exist 0 ≤ ξ < η ≤ 1 such that ρ(ξ) < ρ(η). Without loss of generality, we assume that 0 < ξ < η < 1 and let h : is right continuous at ξ and η, there exists δ > 0 small enough such that Let us take a smooth function u ∈ C ∞ (I) such that and with the support being contained in [ξ, η + δ]. Consequently, ≥ h 2 > 0. This completes the proof.

2.2.
Second-order linear MDEs. Given a measure µ ∈ M 0 (I), the second-order linear MDE with the measure µ is written as See [23]. Here the notation u • (x) stands for the generalized right derivative of u(x).
With an initial value , is explained using the following equivalent system of integral equations It is known that both u( See [24] or [23, Theorem 3.1]. By (19), one sees that u(x) ∈ W 1,∞ (I) and Here u (x) is the classical right derivative of u(x) and is the Lebesgue measure of Remark 1. Due to the uniqueness of solutions of initial value problems, if a non- is only the generalized right-derivative, it is possible to use integral equations (19) and (20) to show that x 0 is an isolated zero of u(x). In fact, if x 0 ∈ (0, 1), u(x) must change signs at two sides of x 0 .
The next proposition shows that the Neumann eigenvalues and eigenfunctions can also be determined by weak solutions. Proposition 1. The following are equivalent: Proof. Assume first that u ∈ W 1,∞ (I) is a solution of (18)-(N ). Then u and v = u • fulfill (19) and (20) with v 0 = 0 and u 0 = u(0) ∈ R. Since u satisfies (N ), it follows from v 0 = 0 and (20) that According to (21), one has u(t) dµ(t) for -a.e. x ∈ I.
For any x ∈ I, by taking in (22) the following test function xu(t) dµ(t) (by using (24)) Meanwhile, let us define It then follows from the Fubini's theorem that From (25) and (27) Let ρ ∈ M prin be fixed. By Lemma 2.2, U ρ ⊂ W 1,2 (I) is non-empty. Since I dρ < 0, any u ∈ U ρ is nonconstant. Hence the Rayleigh form is well defined and positive. Define We give the proof of Theorem 1.1 in a series of claims.
Claim 1. We claim that λ prin > 0 and the infimum in (28) is attained by some ϕ ∈ U ρ . Let us take any minimizing sequence u n ∈ U ρ so that lim n→∞ R(u n ) = λ prin . Due to the homogeneity of R(u), we can assume that u n ∞ = 1 for all n ≥ 1. Then for any ε > 0, there exists a sufficient large integer n ε such that u n 2 2 ≤ (λ prin + ε) Here · 2 = · L 2 (I) . Hence the sequence {u n } n≥1 is bounded in W 1,2 (I). It then follows from the local weak compactness of W 1,2 (I) that there is a subsequence u nj j≥1 such that u nj ϕ weakly in W 1,2 (I) for some ϕ ∈ W 1,2 (I).
This gives the identity Applying Proposition 1 to (33) , we know that λ prin is an eigenvalue of problem (8)-(N ), while ϕ(x) is an associated eigenfunction.

Claim 5.The necessity part of the theorem holds.
To see this, let λ > 0 be a principle eigenvalue of (8)-(N ) and ϕ(x) a principal eigenfunction. Accordingly, ϕ(x) is strictly positive on I and verifies the identity (33) with λ = λ prin . We must prove that the measure ρ in MDE (8) satisfies ρ + > 0 and I dρ < 0. If this is false, we can distinguish two cases. Case 1. ρ + = 0 and I dρ < 0. Let us take φ ≡ 1 in identity (33). It then follows from Lemma 2.1 that which is impossible.
Combining Claims 1-5, the proof of Theorem 1.1 is complete.
In the space M 0 (I) of measures, ρ n ρ in (M 0 (I), w * ) if and only if, as n → ∞, We give the proof of Theorem 1.2 in a series of claims.
Claim 3.We claim that the eigenvalue λ of Claim 2 must be the principal eigenvalue λ prin (ρ).
Finally, it follows from Claim 2 and Claim 3 that, as n → ∞, {λ prin (ρ n )} n≥1 itself is convergent to λ prin (ρ). The proof of Theorem 1.2 is now complete.
4. Optimization problems on principal eigenvalues and on measures.
Moreover, for the case of a < b, is an eigenfunction associated with λ prin (αδ a − βδ b ). For the case of a > b, is an eigenfunction associated with λ prin (αδ a − βδ b ).
For the case of a > b, conclusions (37) and (39) can be verified via a similar argument.
Now we can give the proof of Theorem 1.3. At first we note that the left-hand side λ prin (ρ) of (42) is an infinitely dimensional functional on M A,B , while the right-hand side is a 4-dimensional elementary function F (a, b, α, β) defined on It is elementary to verify that F (a, b, α, β) takes its minimum Moreover, corresponding to (48), one has measuresρ A,B ,ρ A,B ∈ M A,B . Immediately, it follows from (40) and (50) that We keep the same notations as in the proof of Lemma 4.2, i.e. ϕ(x) = ϕ(x; ρ), α = ρ + , β = ρ − , and the maximal and minimal points a and b of ϕ(x) on I, respectively. In the present case, (42) and (46) are now identities. It then follows that L(A, B) = λ prin (αδ a − βδ b ) and ϕ(x) is an eigenfunction associated with λ prin (αδ a − βδ b ). It is evident from (47) that one has (48). This implies that ϕ(x) is an eigenfunction associated with either λ prin (ρ A,B ) or λ prin (ρ A,B ). For the case (a, b) = (0, 1), one has from (38) an eigenfunction associated with λ prin (ρ A,B ). Clearly, x = 0 is the unique maximal point of ϕ(x) and x = 1 is the unique minimal point of ϕ(x). Then the identity (43) implies that Similarly, for the case (a, b) = (1, 0), one can obtain Thus (51) and (52) show that either ρ =ρ A,B or ρ =ρ A,B .
Remark 3. Optimization problems like (12) and (13) for (principal) eigenvalues are infinitely dimensional variational problems. Some typical approaches to these problems include the Pontryagin's maximum principle [7,17], and the Lyapunovtype inequalities [25,34]. By using the Lagrangian multiplier method, a variational method has been developed in [35,36] to deal some optimization problems of eigenvalues. However, in the present proof of Theorem 1.1, we have adopted a direct method, as done in [20,21]. In fact, based on the minimization characterization (9) for λ prin (ρ), an MDE explanation (45) to the estimates of the Rayleigh form can reduce problems (12) and (13) to finitely dimensional optimization problems. Owing to this advantage, our approach is relatively elementary.

4.2.
Minimizing measures with given principal eigenvalue -Proof of Theorem 1.4. This subsection is devoted to study the minimization problem (14). Using the complete continuity in Theorem 1.2, we first show the minimal value T(γ) defined in (14) can be attained by some measures from M A,B . Proof. Take a minimizing sequence ρ n ∈ M γ such that I dρ n → T(γ) as n → ∞.
A crucial observation on ρ γ is that, when the measure ρ is taken as ρ γ , (42) will become an equality.
Proof. In view of Lemma 4.1 and Lemma 4.2, we have It remains to show this is in fact an equality. Otherwise, let us assume the strict inequality Then there exists some 0 < α < α γ such that Thus αδ aγ − β γ δ bγ ∈ M γ . However, This contradicts with the assumption that ρ γ is a minimizer.
Step 1. By keeping the notations as same as in Lemma 4.4, we claim that |a γ − b γ | = 1.
Step 3. To obtain the explicit expression of T(γ), we need to derive the explicit forms of α γ and β γ .

4.3.
A relation between optimization problems. Let us go back to the optimization problem (5) with integrable weights. Proof. For w(x) ∈ W A,B , it easy to verify that the absolutely continuous measure It is easy to verify that w n ∈ W A,B for all n ≥ 2, and, moreover, as measures, ρ wn ρ A,B with respect to the weak * topology w * . By the complete continuity of λ prin (ρ) in Theorem 1.2, we obtaiñ This gives another explanation to equality (56).