ON EIGENVALUE PROBLEMS ARISING FROM ...

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Volume 37, Number 2, February 2017

doi:10.3934/dcds.2017036 pp. 879–903

ON EIGENVALUE PROBLEMS ARISING FROM NONLOCAL DIFFUSION MODELS

Fang Li Center for PDE, East China Normal University 500 Dongchuan Road, Minhang 200241, Shanghai, China

Jerome Coville Biostatistique et Processus Spatiaux INRA 84000, Avignon, France

Xuefeng Wang Center for PDE, East China Normal University 500 Dongchuan Road, Minhang 200241, Shanghai, China and Department of Mathematics, Southern University of Science and Technology 1088 Xueyuan Road, Nanshan 518055, Shenzhen, China

Dedicated to the memory of Professor Paul Fife–a man of wisdom and kindness Abstract. We aim at saying as much as possible about the spectra of three classes of linear diffusion operators involving nonlocal terms. In all but one cases, we characterize the minimum λp of the real part of the spectrum in two max-min fashions, and prove that in most cases λp is an eigenvalue with a corresponding positive eigenfunction, and is algebraically simple and isolated; we also prove that the maximum principle holds if and only if λp > 0 (in most cases) or ≥ 0 (in one case). We prove these results by an elementary method based on the strong maximum principle, rather than resorting to Krein-Rutman theory as did in the previous papers. In one case when it is impossible to characterize λp in the max-min fashion, we supply a complete description of the whole spectrum.

1. Introduction. Of concern are three classes of linear diffusion operators involving nonlocal (integral) terms; these operators may not be symmetric. We want to say as much as possible about the spectrum of each operator, and in particular the minimum λp of the real part of the spectrum. λp plays an important role in the dynamics of the time-dependent diffusion equation, for example, any time-dependent solution is of order O(e−λp t ) as t → ∞. In all but one cases, we characterize λp 2010 Mathematics Subject Classification. Primary: 45C05, 45K05; Secondary: 35P15. Key words and phrases. Principal eigenvalue, nonlocal operators, max-min characterization, maximum principle. FL is supported by NSF of China (No. 11431005), NSF of Shanghai (No. 16ZR1409600). JC is supported by the French ANR through the ANR JCJC project MODEVOL: ANR-13JS01-0009 and the ANR project NONLOCAL: ANR-13-JS01-0009. XFW is supported by NSF of China (No. 11671190). ∗ Corresponding author: Xuefeng Wang.

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in two max-min fashions; and in that exceptional case when λp can not be characterized in this way, we provide a complete description of the whole spectrum. In most of the cases, λp is an eigenvalue of the diffusion operator with a continuous and positive eigenfunction; we call it principal eigenvalue. In this case, we prove the algebraic simplicity and isolatedness of λp , and that any eigenfunction φp corresponding to λp has one sign. We also show that the maximum principle holds for the diffusion operator if and only if λp > 0 (except in one case where λp > 0 is replaced by λp ≥ 0). The first class of nonlocal diffusion operators is Z L1 u(x) = − κ(x, y)u(y)dy + a(x)u(x) x ∈ Ω, Ω n

where Ω is a domain in R , and ¯ κ ∈ C(Ω ¯ × Ω) ¯ is nonnegative, and κ(x, x) > 0 for x ∈ Ω, ¯ which (A1) a ∈ C(Ω), is, in the case of bounded Ω, equivalent to ∃ constants ¯, δ > 0 such that ¯ and |x − y| < δ. κ(x, y) > ¯ if x, y ∈ Ω Using L1 to model nonlocal diffusion was used early in population genetics models, see for example, the work of Crow and Kimura [11]. In Ecology, to our knowledge, Othmer, Dunbar and Alt in [17] were the firsts to introduce a simple jump process to model the dispersion of individuals, which later, has been generalised by Huston, Martinez, Mischaikow and Vickers in [13]. The basic idea behind L1 is explained as follows. Let u(x, t) be the density function of dispersing particles. For any points x, y ∈ Rn , let dx and dy be the volume elements containing x and y. Let dt be a small change in time t. The total population change in dx from time t to time t + dt is (u(x, t + dt) − u(x, t)) dx, which should equal total population that moves into dx from all y sites in [t, t + dt] minus total population that moves out of dx to all y sites in [t, t + dt]. The former should be proportional to the size dx of the xsite, to time duration dt, and to the population size u(y, t)dy at ysite. Let the proportionality constant be κ(x, y). Then the total population that moves into dx from all y−sites in [t, t + dt] is given by Z κ(x, y)u(y, t)dydxdt. Rn

Similarly, the total population that moves out of dx to all ysites in [t, t + dt] is given by Z κ(y, x)dyu(x, t)dxdt. Rn

Thus we have Z (u(x, t + dt) − u(x, t)) dx =



Z κ(x, y)u(y, t)dy −

Rn

κ(y, x)dyu(x, t) dxdt, Rn

which implies that ∂u(x, t) = ∂t

Z

Z κ(x, y)u(y, t)dy −

Rn

There are two practical cases to consider.

κ(y, x)dyu(x, t). Rn

(1.1)

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• Hostile exterior Ωc . This is the case when particles land in Ωc , they die immediately and so u(x, t) ≡ 0, x ∈ Ωc . Now (1.1) becomes Z Z ∂u(x, t) = κ(x, y)u(y, t)dy − κ(y, x)dyu(x, t), x ∈ Ω. (1.2) ∂t Ω Rn This corresponds to the classical Dirichlet boundary condition; but we do not prescribe a boundary value. Indeed, (1.2) with an initial condition is already a well-posed problem. • Well-insulated ∂Ω. This is the case where particles can jump neither from Ω to Ωc nor from Ωc to Ω. Thus κ(x, y) = 0 if x ∈ Ω and y ∈ Ωc or x ∈ Ωc and y ∈ Ω. Now (1.1) becomes Z Z ∂u(x, t) = κ(x, y)u(y, t)dy − κ(y, x)dyu(x, t), x ∈ Ω. (1.3) ∂t Ω Ω This corresponds to Neumann Rboundary condition. Indeed it follows from (1.3) that the total population Ω u(x, t)dx is conserved in t. The second class of nonlocal diffusion operators is motivated by the so called shadow equation of the reaction-diffusion system   in Ω, d1 ∆u + f (u, v) = 0 d2 ∆v − v + g(u) = 0 in Ω,   ∂u/∂ν = ∂v/∂ν = 0 on ∂Ω, where ν is the unit outer normal vector field on ∂Ω. If we divide the v−equation by d2 and send d2 → ∞, then assuming u and v remain bounded in Ω, formally we have ∆v = 0 in Ω. This and the Neumann boundary condition imply that v is constant (in the limit of d2 → ∞). Integrating both sides of the v−equation, we have Z 1 v= g(u(x))dx. |Ω| Ω Substituting this into the u−equation, we have ( d1 ∆u + f (u, I(u)) = 0 in Ω, ∂u/∂ν = 0 on ∂Ω, where I(u) =

1 |Ω|

Z g(u(x))dx. Ω

The linearized problem is ( R 0 1 g (u(x))w(x)dx = 0 d1 ∆w + fu (u, I(u))w + fI (u, I(u)) |Ω| Ω ∂w/∂ν = 0

in Ω, on ∂Ω.

If we allow more general elliptic operators and general boundary conditions, we are led to the following operators Z L2 u = −aij (x)uxi xj + bi (x)uxi + c(x)u + a(x) b(y)u(y)dy, x ∈ Ω Ω

associated with the boundary operator Bu = u or Bu = αi (x)

∂u + β(x)u, x ∈ ∂Ω. ∂xi

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(So Dirichlet, Neumann and Robin boundary conditions are all covered.) Lastly, in this paper we propose a model that incorporates the effect of advection in the nonlocal diffusion equation (1.1). The example we have in mind is fresh water organisms living in a river, which is represented by the interval [0, L]. Let c(x) be the velocity function of water with c(x) > 0 on [0, L] (so the river flows from the left to the right). Since on land (−∞, 0) and in the ocean (L, ∞), the fresh water organisms do not survive, their population density function can be modeled by the following  ∂u(x,t) RL R∞  = 0 κ(x, y)u(y, t)dy − −∞ κ(y, x)dyu(x, t) − ∂(c(x)u(x,t)) ,  ∂t ∂x 0 < x < L, t > 0,   u(0, t) = 0, t > 0. The reason why we have a boundary condition is that the initial value problem would not be well-posed without the boundary condition (unlike that for (1.2)). We are thus led to the third class of operators considered in this paper. ( RL L3 u(x) = − 0 κ(x, y)u(y)dy + a(x)u(x) + (c(x)u(x))0 0 < x < L, u(0) = 0, ¯ replaced by [0, L]. where a and κ satisfy (A1) with Ω It is known that for local models the spectrum of linearized problem, especially properties related to the principal eigenvalue, is crucial in the studies of qualitative properties of solutions and the application of many developed approaches like linearized analysis, upper/lower solution method, bifurcation theory and so on. Likewise when nonlocal terms are incorporated into models, studying the spectrum of the linearized problem is also an important step and the corresponding results have wide applications. For example, Turing patterns driven by nonlocal diffusion are studied in [4], single species model and competition systems with nonlocal dispersal are investigated in [1, 2, 5, 9, 16, 21, 22], and in [20, 23], logistic equations with nonlocal interaction are considered. 2. The spectrum of L1 . The first result on the existence of the principal eigenpair of L1 was established in [13], where κ(x, y) is assumed to be symmetric: κ(x, y) = κ(y, x), Ω is a finite interval and a is Lipschitz continuous. Under the same symmetry assumption on κ, but in higher spatial dimension, Bates and Zhao [3] observed that if L1 has a nonnegative and continuous eigenfunction, then the corresponding eigenvalue is the infimum of the spectrum of L1 , and is isolated with the eigenspace being one dimensional. Coville [7] was the first to study the general case without assuming the symmetry of κ. Inspired by the work of Berestycki, Nirenberg and Varadhan [6], he defined λp =

sup ¯ u>0 on u∈C(Ω),

L1 u(x) , ¯ x∈Ω u(x) Ω inf

(2.1)

which is no larger than m = minΩ¯ a(x) (as can be easily checked). (This definition can be traced back to the old result of Joseph Barta in 1937 for the principal Dirichlet Laplace eigenvalue.) He proved that if Z 1 dx = ∞, (2.2) Ω a(x) − m

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then λp is an eigenvalue of L1 with a positive continuous eigenfunction. Consequently, λp is such an eigenvalue if one of the following holds: ¯ • n = 1, a(x) ∈ Lip(Ω); ¯ • n = 2, Da(x) ∈ Lip(Ω); n−1 ¯ and ∃ x0 ∈ Ω such that a(x0 ) = m, Dk−1 a(x0 ) = 0 • n ≥ 3, D a(x) ∈ Lip(Ω), for any 1 ≤ k ≤ n − 1. In [10], for the case κ(x, y) = κ(y −x) it was observed that a positive and continuous eigenfunction φp exists if and only if λp < m. In the case of λp = m, Coville [8] proved that φp becomes a positive measure, which is, in some cases, a sum of a L1 −function and a δ−function with its pole at a global minimum point x0 of a(x). Shen and Xie [18] considered the special case κ(x, y) = κ(y − x) and proved that if Z max a(x) − min a(x) < inf ¯ Ω

¯ Ω

¯ x∈Ω

κ(y − x)dy, Ω

then a positive and continuous eigenfunction exists. Smith [19] considered the case of one dimensional unbounded domain Ω, proving, among other things, that if a(x) converges to its infimum on Ω at either +∞ or −∞, and if κ(x, y) >  for any x, y ∈ Ω with |x − y| < δ for some constants , δ > 0, then a positive and continuous eigenfunction exists. The method used in [13] is variational and that in other papers mentioned above is to use Krein-Rutman Theorem and its variants. In this paper, we generalize these results and prove them by an elementary method (without resorting to Krein-Rutman Theorem), which is centered at (2.1) and is based on the maximum principle. This method applies to other classes of operators such as L2 and L3 with equal ease, providing a unified approach and reducing the amount of tricks special to each class of operators. We mention that when λp defined by (2.1) is not associated to a continuous eigenfunction, some of our results are new. ¯ (so an We will always think of L1 as an operator from X to X, where X = C(Ω) ¯ eigenfunction of L1 is always continuous in Ω). We define the essential spectrum σess (L1 ) of L1 as the set of λ ∈ C such that λI − L1 is not Fredholm with index zero. We denote the spectrum of L1 by σ(L1 ). Theorem 2.1 (Existence of principal eigenvalue). Suppose that Ω is bounded and assumption (A1) holds. If either there exists a measurable subset A of Ω such that Z κ(x, y) 1 < inf dx ≤ ∞, (2.3) y∈A A a(x) − m where m = minΩ¯ a(x), or if Z M − m < inf

x∈Ω

κ(x, y)dy,

(2.4)

Ω

where M = maxΩ¯ a(x), then we have (i) λp < m, where λp is defined by (2.1). ¯ (ii) λp is an eigenvalue of L1 , with a corresponding eigenfunction φp > 0 on Ω. (iii) λp is isolated in the spectrum σ(L1 ), and has algebraic multiplicity equal to one. ¯ then λ = λp and φ (iv) If there exists an eigenpair (λ, φ) of L1 with φ > 0 in Ω, is a constant multiple of φp .

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Remark 1. The sufficient conditions (2.3), (2.4) are milder than all the previous sufficient conditions in literature (thus this theorem covers all the previous existence theorems). To see this, we check that (2.2) implies (2.3): Let ¯, δ be given in ¯ by finitely many open balls of radii = δ/2. From (2.2) it Assumption (A1); cover Ω follows that for at least one ball B, Z 1 dx = ∞. a(x) −m Ω∩B Let A = Ω ∩ B. Note κ(x, y) > ¯ ∀x, y ∈ A. Now we see Z κ(x, y) inf dx = ∞. y∈A A a(x) − m In the next theorem, we do not need λp to be an eigenvalue of L1 . Theorem 2.2 (Structure of the spectrum). Suppose that Ω is bounded and assumption (A1) holds. Then (i) The essential spectrum of L1 is σess (L1 ) = [m, M ]. (ii) σ(L1 ) \ σess (L1 ) consists of isolated eigenvalues of finite algebraic multiplicity. In the special case κ ≡ constant ρ, ( R ρ dx > 1, {λp } ∪ [m, M ] if Ω a(x)−m R σ(L1 ) = ρ [m, M ] if Ω a(x)−m dx ≤ 1. (iii) λp , defined by (2.1), satisfies λp ∈ σ(L1 ) and λp = inf{Re λ | λ ∈ σ(L1 )}. Remark 2. Part (i) was first mentioned in [18], where it was also proved that inf{Re λ | λ ∈ σ(L1 )} ∈ σ(L1 ). Part (iii) says that, whether or not λp is an eigenvalue, inf{Re λ | λ ∈ σ(L1 )} can be characterized by (2.1) as well as (2.5) below, which may yield good estimates by choosing appropriate test functions. This is new and is expected to be useful in applications. ¯ implies We say L1 satisfies the maximum principle if φ ∈ X and L1 φ ≥ 0 in Ω ¯ that φ ≥ 0 in Ω. Theorem 2.3 (Maximum principle and sign of λp ). Suppose that Ω is bounded and assumption (A1) is satisfied. (i) If λp < m, where λp is defined by (2.1), then the maximum principle holds for L1 if and only if λp > 0. (ii) If λp = m, then the maximum principle holds for L1 if and only if λp ≥ 0. Remark 3. [7] covers the case of λp being an eigenvalue, which is equivalent to λp < m. In defining the maximum principle, extra boundary condition φ ∂Ω ≥ 0 is added. Hence the necessary and sufficient condition in [7] is different: λp ≥ 0. It seems that the case of λp = m was never covered in this regard in literature. The following theorem provides another min-max characterization of λp with the order of min and max reversed. While the definition of λp can be used to obtain a lower bound, this theorem can be used to obtain an upper bound. Theorem 2.4. Suppose that Ω is bounded and (A1) holds. Then λp =

inf

sup

¯ u>0 on Ω x∈Ω u∈C(Ω),

L1 u(x) u(x)

In the proofs of these theorems, we will repeatedly use the following

(2.5)

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Lemma 2.1 (Touching Lemma). Suppose that Ω is bounded and assumption (A1) ¯ such that u > 0 in Ω ¯ and L1 u ≥ 0 in Ω. ¯ holds. Assume that there exists u ∈ C(Ω) ¯ satisfies L1 v ≥ 0 in Ω, ¯ then v does not change sign in Ω. ¯ Moreover, if If v ∈ C(Ω) ¯ then L1 u = L1 v ≡ 0 and v is a constant multiple of u. (Of v < 0 somewhere in Ω, course, the same conclusion holds for the operator L1 − λ, ∀ real λ.) ¯ we have nothing to prove. If v < 0 somewhere in Ω, ¯ then Proof. If v ≥ 0 in Ω, ¯ ¯ u + δv < 0 somewhere in Ω if constant δ > 0 is large, and u + δv ≥ 0 in Ω if δ > 0 ¯ ≥ 0 in Ω ¯ and equals zero at is small. Thus there exists δ¯ ∈ (0, ∞) such that u + δv ¯ ¯ > 0} and hence ¯ ¯ some x0 ∈ Ω. If u + δv 6≡ 0, then we can take x0 ∈ ∂{x ∈ Ω | u + δv Z ¯ ¯ 0 ≤ L1 (u + δv)(x κ(x0 , y)(u + δv)(y)dy 0 ¯ if and only if λp < m. on Ω Remark 4. This theorem was first proved in [10] for the special case κ(x, y) = κ(x − y) by using a variant of Krein-Rutman theorem. Here we present an elementary proof, which was originally inspired by the proof of Evans [12] for the similar property of non-symmetric second order elliptic operators.(Our proof is more elementary and direct: unlike [12], we do not use fixed point theorem; and we prove the existence of principal eigenvalue and its characterization (2.1) in one stroke, while such a characterization in [12] appears in an exercise by using the dual operator.) Proof of Theorem 2.5. Suppose that L1 has an eigenpair (λp , φp ) with φp ∈ X, ¯ Take x0 ∈ Ω ¯ such that a(x0 ) = m = minΩ¯ a(x). Now we have φp > 0 on Ω. Z λp φp (x0 ) = − κ(x0 , y)φp (y)dy + a(x0 )φp (x0 ) < a(x0 )φp (x0 ), Ω

which implies that λp < m. Now assume that λp < m and will show the existence of such φp . Claim 1. For any λ < λp , the operator L1 − λ : φ ∈ X → L1 φ − λφ ∈ X is one-to-one and onto. To see this, observe that the operator φ ∈ X → (a(x) − λ)φ ∈ X is an isomorphism, and the operator L1 − a(x) : X → X is compact. Thus L1 − λ is a Fredholm operator with index zero. So it suffices to show that L1 −λ is one-to-one. Suppose that there exists φ ∈ X such that ¯ L1 φ − λφ = 0 on Ω. If φ 6≡ 0, then w.l.o.g. assume that φ < 0 somewhere. By the definition of λp in ¯ and (2.1), there exists φλ ∈ X such that φλ > 0 on Ω λ < inf

x∈Ω

L1 φλ (x) , φλ (x)

which implies that (L1 − λ) φλ (x) > 0

¯ on Ω.

(2.6)

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FANG LI, JEROME COVILLE AND XUEFENG WANG

Now applying Touching Lemma (Lemma 2.1) with L1 replaced by L1 − λ, u = φλ and v = φ, we have (L1 − λ) φλ ≡ 0, which is a contradiction to (2.6). Hence Claim 1 is proved. By Claim 1, for any constants  ∈ (0, 1) and λ ∈ [λp −1, λp ), there exists a unique φλ ∈ X such that ¯ (L1 − λ) φλ =  on Ω. (2.7) ¯ Claim 2. φλ > 0 on Ω. ¯ Suppose this is not true, then φ < 0 somewhere We first show that φλ ≥ 0 on Ω. λ ¯ Applying Touching Lemma (Lemma 2.1) with L1 replaced by L1 − λ, u = φλ on Ω. ¯ If there exists and v = φλ , we reach a contradiction to (2.6). Thus φλ ≥ 0 on Ω. ¯ such that φ (x1 ) = 0, then choose x1 ∈ ∂{x ∈ Ω ¯ | φ > 0}. Due to (2.7), we x1 ∈ Ω λ λ have Z − κ(x1 , y)φλ (y)dy = , Ω

which is clearly impossible. Hence Claim 2 is proved. There are two cases to consider: R Case 1. There exists  ∈ (0, 1) such that for any λ ∈ [λp − 1, λp ),R Ω φλ (x)dx < 1. Case 2. For any  ∈ (0, 1), there exists λ ∈ [λp − 1, λp ) such that Ω φλ (x)dx ≥ 1. In Case 1, the family of functions {(L1 − a(x)) φλ | λ ∈ [λp − 1, λp )} is, as can ¯ By Arzela-Ascoli be easily checked, uniformly bounded and equicontinuous on Ω. Theorem, there exists an sequence of λ % λp and some f ∈ X such that (L1 − a(x)) φλ → f

in X.

Now as λ % λp , we have  − f (x)  − (L1 − a(x)) φλ (x) → in X. a(x) − λ a(x) − λp ¯ because f (x) ≤ 0 and λp < a(x) Denote the limiting function by φ . φ > 0 on Ω ¯ Then it follows that on Ω. L1 u(x) L1 φ (x)  λp = sup inf ≥ inf = λp + inf  > λp ,  (x) x∈Ω x∈Ω x∈Ω u(x) φ φ (x) ¯ u∈X,u>0 on Ω φλ =

which yields a contradiction. Therefore, Case 1 does not occur. In Case 2, let φ (x)  , c = R  . ψ  (x) = R λ  φ (x)dx φ (x)dx Ω λ Ω λ R  ¯ After passing Then Ω ψ (x)dx = 1, c → 0 as  → 0+ and (L1 − λ ) ψ  = c on Ω. + ¯ to a subsequence of  → 0 , we have λ → some λ ∈ [λp − 1, λp ], and (L1 − a(x)) ψ  converges in X. Because ψ =

c − (L1 − a(x)) ψ  (x) a(x) − λ

¯ ≤ λp < m, one sees that ψ  → some ψ 0 in X, and hence and λ − (L1 − a(x)) ψ 0 (x) ¯ > 0 on Ω. ¯ a(x) − λ ¯ ψ 0 ) is an eigenpair of L1 . λ ¯ < λp is impossible because of Claim 1; thus Thus (λ, ¯ = λp and we can take φp to be ψ 0 . λ ψ0 =

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Proof of Theorem 2.1. Recall that λp ≤ m. If λp = m, then for any λ < m, there exists φλ ∈ X with φλ > 0 such that (2.6) holds, which we rewrite as R κ(x, y)φλ (y)dy ¯ φλ > Ω on Ω. (2.8) a(x) − λ Let B be an arbitrary measurable subset of Ω. Integrating both sides in x−variable on B, we have  Z Z Z Z R κ(x, y)φλ (y)dy κ(x, y) Ω dx ≥ dx φλ (y)dy φλ (x)dx > a(x) − λ B B B a(x) − λ B Z Z κ(x, y) ≥ inf dx φλ (y)dy, y∈B B a(x) − λ B which gives that Z

κ(x, y) dx. (2.9) B a(x) − λ If the diameter of B is less than δ in Assumption (A1), then using Fatou’s lemma, we see from (2.9) that 1/(a(x) − m) ∈ L1 (B) and hence ∈ L1 (Ω). Now by the continuity of κ and by using Lebesgue dominated convergence theorem, we have that the function Z κ(x, y) ¯→ y∈Ω dx A a(x) − m ¯ Now (2.3) is continuous, and hence its infimum over A is the same as that over A. becomes Z κ(x, y) dx > 1. (2.10) inf ¯ A a(x) − m y∈A In (2.9) take B = A, pick yλ ∈ A such that Z κ(x, yλ ) 1> dx. a(x) −λ A 1 > inf

y∈B

Sending λ % m and using Lebesgue dominated convergence theorem again, we have ¯ that for some y¯ ∈ A, Z κ(x, y¯) 1≥ dx. a(x) −m A This contradicts to (2.10). ¯ such that φλ (xλ ) = minΩ¯ φλ . Now we On the other hand, in (2.8), take xλ ∈ Ω have R R κ(xλ , y)dy κ(xλ , y)φλ (y)dy Ω ≥ Ω min φλ . min φλ = φλ (xλ ) > ¯ ¯ a(xλ ) − λ a(xλ ) − λ Ω Ω R This shows that a(xλ ) − λ > Ω κ(xλ , y)dy and thus Z M −λ> κ(xλ , y)dy. Ω

¯ as λ % m, then we have After passing to a subsequence, assume xλ → x ¯∈Ω Z M −m≥ κ(¯ x, y)dy, Ω

contradicting (2.4). We have shown (i) under either condition (2.3) or (2.4). Moreover, (ii) follows from (i) and Theorem 2.5.

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We now prove (iii). By the proof of Claim 1 in the proof of Theorem 2.5, L1 − λp is a Fredholm operator with index zero and λ is in the resolvent set of L1 for any λ < λp . By [14, Theorem 5.31, Page 241], there exists  > 0 such that dim N (L1 − λ) = constant

for any λ ∈ C with 0 < |λ − λp | < .

But for such real λ < λp , the dimension of N (L1 −λ) is zero. Now dim N (L1 −λ) = 0 for any λ ∈ C with 0 < |λ − λp | < . Since the set of Fredholm operators with index zero is open in the operator topology, L1 − λ is Fredholm with index zero if |λ − λp | < . Now we see for 0 < |λ − λp | < , λ is in the resolvent S∞ set of L1 . Now we show the algebraic multiplicity λp is one, i.e. dim k=1 N (L1 −λp )k = 1. It suffices to show dim N (L1 − λp )2 = 1. We first show dim N (L1 − λp ) = 1. For any ¯ By using Touching Lemma φ ∈ N (L1 − λp ), w.l.o.g., assume φ < 0 somewhere in Ω. (Lemma 2.1) with L1 replaced by L1 − λp , u = φp , v = φ, we see φ is a constant multiple of φp , proving dim N (L1 − λp ) = 1. Now suppose ψ ∈ N (L1 − λp )2 . Then (L1 − λp )ψ = cφp for a constant c. If c = 0, then ψ ∈ N (L1 − λp ) and we are done. If c 6= 0, we divide both sides by c and redefine ψ to get (L1 − λp )ψ = φp . ¯ then by Touching Lemma (Lemma 2.1) again, we are led If ψ < 0 somewhere in Ω, ¯ then ψ > 0 in Ω, ¯ and using Touching Lemma to (L1 − λp )ψ = 0; if ψ ≥ 0 in Ω, (Lemma 2.1) with L1 replaced by L1 − λp , u = ψ, v = −φp , we reach the same (L1 − λp )ψ = 0. We have proved (iii). To prove (iv), observe that λ ≥ λp (recall λ is in the resolvent set of L1 for any λ < λp ), so (L1 − λp )φ ≥ 0. By Touching Lemma (Lemma 2.1) with L1 replaced by L1 − λp , u = φ, v = −φp , we have φ is a constant multiple of φp and (L1 − λp )φ = 0. Proof of Theorem 2.2. Note L1 is a compact perturbation of the operator A : φ ∈ X → a(x)φ ∈ X. By the well known fact, σess (L1 ) = σess (A). It is easy to see that the resolvent set of A is C \ [m, M ], and hence σ(A) = [m, M ]. For any λ ∈ [m, M ], let ¯ | a(x) = λ}. Sλ = {x ∈ Ω Case 1. Sλ has no interior points. Then for any φ ∈ N (A − λ), φ(x) = 0 for any ¯ We have shown N (A−λ) = {0}. x ∈ Ω\Sλ . Since Ω\Sλ is dense, φ ≡ 0 on Ω. Obviously 1 6∈ R(A − λ). Thus A − λ is not Fredholm of index zero, implying λ ∈ σess (A). Case 2. Sλ contains an open ball B. Let C0 (B) be the set of continuous functions, compactly supported on B. Then C0 (B) ⊂ N (A − λ). But C0 (B) is infinite dimensional, hence again A − λ is not Fredholm of index zero. We have shown in both cases, λ ∈ σess (A), hence (i) follows. (ii) for general κ(x, y) follows from the general theory, as discussed in [14, Page 243]. In the special case that k ≡ constant ρ > 0, since L1 is symmetric, σ(L1 ) consists of real numbers. For any λ ∈ σ(L1 ) \ σess (L1 ) = σ(L1 ) \ [m, M ], λ is an eigenvalue of L1 . Take a corresponding eigenfunction φ(x). Then R ρ Ω φ(y)dy ¯ , x∈Ω φ(x) = − λ − a(x)

ON EIGENVALUE PROBLEMS

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and hence φ does not change sign. Integrating both sides, we have Z ρ dx = 1. a(x) −λ Ω This, together with (iv) of Theorem 2.1, completes the proof of (ii). If λp < m, then by Theorem 2.5, λp is an eigenvalue of L1 ; if λp = m, then λp ∈ σess (L1 ). In both cases, λp ∈ σ(L1 ). For any λ ∈ σ(L1 ), write λ = a + bi, a, b ∈ R. We want to show a ≥ λp . Suppose that a < λp . We just need to consider the case λ ∈ σ(L1 ) \ σess (L1 ). Then λ is an eigenvalue of L1 with an eigenfunction φ(x) (maybe complex-valued). Obviously e−λt φ(x) is a solution of ∂w (x, t) = −L1 w(x, t), ∂t

¯ t ≥ 0. x ∈ Ω,

¯ ∈ (a, λp ). Since λ ¯ < λp , by the Let u(x, t) be the real part of e−λt φ(x). Take λ ¯ ¯ ¯ definition of λp , there exists φ ∈ X such that φ(x) > 0 on Ω and ¯ L1 φ(x) ¯ > λ, ¯ φ(x)

¯ x ∈ Ω.

¯

¯ Define v(x, t) = e−λt φ(x). Then ∂v ¯ ¯ ¯ ¯ −λt ¯ ¯ t ≥ 0. (x, t) = −λe φ(x) > −e−λt L1 φ(x) = −L1 v(x, t) x ∈ Ω, ∂t If necessary, replace φ¯ by a large multiple of itself, so that ¯ ¯ ¯ φ(x) > Re φ(x) = u(x, 0) > −φ(x), x ∈ Ω. It is easy to see that v(x, t) ≥ u(x, t) ≥ −v(x, t), ¯ This is impossible because a < λ.

¯ t ≥ 0. x ∈ Ω,

Proof of Theorem 2.3. Let S = {λ ∈ R | L1 − λI satisfies the maximum principle}. We will show that Case 1. S = (−∞, λp ) if λp < m, Case 2. S = (−∞, λp ] if λp = m. Then Theorem 2.3 follows. Claim 1. (−∞, λp ) ⊂ S. For any λ ∈ (−∞, λp ), by the definition of λp , there exists φλ ∈ X with φλ > 0 ¯ satisfying (L1 − λ) φλ (x) > 0 on Ω. ¯ Suppose that φ ∈ X satisfies on Ω, ¯ (L1 − λ) φ(x) ≥ 0 on Ω. By Touching Lemma (Lemma 2.1) with L1 replaced by L1 − λ, u = φλ and v = φ, ¯ The proof of Claim 1 is complete. we see φ ≥ 0 on Ω. Claim 2. S ∩ (λp , ∞) = ∅. Suppose otherwise, i.e., there exists λ ∈ S ∩ (λp , ∞). • Suppose λp < m. Then the principal eigenfunction φp > 0 exists by Theorem 2.5. Observe that ¯ (L1 − λ) (−φp ) = (λ − λp )φp > 0 on Ω.

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Then λ ∈ S implies that −φp ≥ 0, which is a contradiction. The same proof also says λp 6∈ S, so Case 1 is proved. • Suppose λp = m. There exists an open ball B ⊂ Ω such that a(x) − λ < 0 on B. Take φ ∈ C0∞ (B) such that φ < 0 on B. Then we have Z ¯ (L1 − λ) φ(x) = − κ(x, y)φ(y)dy + (a(x) − λ)φ(x) ≥ 0 on Ω, Ω

which yields that φ ≥ 0 on Ω because λ ∈ S. This is a contradiction. Claim 2 is proved and it remains to show m ∈ S in Case 2. ¯ satisfying Claim 3. If λp = m, then there exists z ∈ X with z > 0 on Ω Z z(y) ¯ dy ≤ z(x), x ∈ Ω. κ(x, y) a(y) − m Ω To prove this, we use the notation in the proof of Theorem 2.5. Recall that we proved that for any constants  ∈ (0, 1) and λ ∈ [m − 1, m), there exists a unique ¯ and φλ ∈ X such that φλ > 0 on Ω ¯ (L1 − λ) φλ =  on Ω. There are two possibilities to be considered. R (a) There exists  ∈ (0, 1) such that for any λ ∈ [m − 1, m), RΩ φλ (x)dx < 1. (b) For any  ∈ (0, 1), there exists λ ∈ [m − 1, m) such that Ω φλ (x)dx ≥ 1. R When (a) holds, after passing to a subsequence of λ % m, Ω κ(x, y)φλ (y)dy converges in X. Hence Z (a(x) − λ)φλ (x) =  + κ(x, y)φλ (y)dy Ω

also converges in X as λ % m. Let ¯ zλ (x) = (a(x) − λ)φλ (x) (> 0 on Ω),

z(x) = lim− zλ (x). λ→m

Then by Fatou’s lemma,   Z zλ (y) dy z(x) = lim−  + κ(x, y) a(y) − λ λ→m Ω Z Z zλ (y) z(y) =  + lim− κ(x, y) dy ≥  + κ(x, y) dy. a(y) − λ a(y) − m λ→m Ω Ω This proves Claim 3, when (a) holds. ¯ If λ ¯ < m, When (b) is true, by passing to a subsequence, assume lim→0+ λ = λ. 0 ¯ ¯ then (λ, ψ ) is an eigenpair of L1 and hence by Theorem 2.5, λ = λp < m, which is ¯ = m. Define impossible. Thus λ z (x) = (a(x) − λ )ψ  (x), then

Z z (x) = c +

κ(x, y)ψ  (y)dy

Ω

converges to some z(x) in X after passing to a subsequence of  → 0+ . Now we have   Z Z z(y) z (y) dy ≥ κ(x, y) dy. z(x) = lim+ c + κ(x, y) a(y) − λ a(y) − m →0 Ω Ω

ON EIGENVALUE PROBLEMS

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¯ To show the strict positivity of z, it suffices to show z 6≡ 0. Obviously, z ≥ 0 on Ω. This can be seen from Z Z Z z(x)dx = lim+ ( κ(x, y)ψ  (y)dy)dx →0 Ω ZΩ ZΩ = lim+ ( κ(x, y)dx)ψ  (y)dy →0

≥

Ω

Ω

C

R ¯ for a positive constant C, because as a continuous function of y ∈ Ω, κ(x, y)dx Ω is pointwise positive (by Assumption (A1)), and because the integral of ψ  is equal to 1. The proof of Claim 3 is complete. ¯ Suppose that m 6∈ S, then there exists φ ∈ X such that (L1 − m)φ(x) ≥ 0 on Ω ¯ ¯ and x0 ∈ Ω such that φ(x0 ) < 0. The set Γ = {x ∈ Ω | a(x) − m > 0} is dense in ¯ (otherwise (2.3) would be satisfied and then λp < m). Thus we can take x0 such Ω that we also have a(x0 ) − m > 0. Let ψ(x) = (a(x) − m)φ(x) ∈ X. Then ψ(x0 ) < 0 and Z ψ(y) ¯ dy on Ω. ψ(x) ≥ κ(x, y) a(y) − m Ω Let ¯ δ¯ = sup{δ > 0 | z(x) + δψ(x) > 0 on Ω}. ¯ ¯ and there exists x1 ∈ Ω ¯ such that Then 0 < δ¯ < ∞, z(x) + δψ(x) ≥ 0 on Ω ¯ z(x1 ) + δψ(x1 ) = 0. But then Z ¯ z(y) + δψ(y) ¯ 0 = z(x1 ) + δψ(x κ(x1 , y) dy 1) ≥ a(y) − m Ω ¯ ¯ | z(x) + δψ(x) This cannot hold if {x ∈ Ω > 0} is nonempty and we take x1 to be ¯ on the boundary of this set. Thus z(x) + δψ(x) ≡ 0, which cannot be true because ¯ and ψ = 0 at the minimum point of a(x). We have proved m ∈ S in z > 0 on Ω Case 2. ¯ p . We first show Proof of Theorem 2.4. Denote the right hand side of (2.5) by λ ¯ p . If this is not true, then there exists λ ¯ p < λ < λp and ψ, ¯ ψ ∈ X with that λp ≤ λ ¯ such that ψ¯ > 0 in Ω and ψ > 0 in Ω ¯ L1 ψ(x) L1 ψ(x) < λ < inf sup ¯ , x∈Ω ψ(x) ψ(x) x∈Ω i.e., (L1 − λ)ψ(x) > 0,

¯ (L1 − λ)(−ψ(x)) >0

in Ω,

which gives a contradiction by Touching Lemma (Lemma 2.1). We have shown ¯p. λp ≤ λ ¯ then taking ψ = φp , If λp corresponds to a principal eigenfunction φp > 0 on Ω, ¯ ¯ we are led to λp ≤ λp . Hence λp = λp . If λp is not a principal eigenvalue, i.e. λp = m, then for any  > 0, we modify ¯ | a(x) = m}, such that the resulting a(x) in a small neighborhood N of {x ∈ Ω function a (x) is equal to its own global minimum in N and ka − a kX < . By Theorem 2.1, L1 , which is L1 with a(x) replaced by a (x), has a principal eigen-pair ¯  in the obvious way. Then with the eigenfunction being positive. Define λp and λ p   + ¯ . Sending  → 0 , we complete the proof. λp = λ p

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Finally in this section, we discuss the existence of the principal eigenvalue in the case of unbounded domain Ω. In the remainder of this section, we assume that • Ω is unbounded; ¯ and is bounded on Ω; ¯ • a ∈ C(Ω) ¯ ¯ ¯ ¯ where c > 0 is a constant, and there exist • κ ∈ C(Ω × Ω), 0 ≤ κ ≤ c on Ω × Ω, , δ, µ > 0 such that ( ¯ >  if |x − y| < δ, x, y ∈ Ω, κ(x, y) ¯ = 0 if |x − y| > µ, x, y ∈ Ω We still define λp as in (2.1). To indicate the dependence of λp on Ω, we write λp (Ω). Then  Z  m = inf a(x) ≥ λp (Ω) ≥ inf − κ(x, y)dy + a(x) ≥ −c|Bµ (0)| + m, ¯ Ω

¯ Ω

Ω

where Bµ (0) is the ball centered at 0 with radius µ. By the definition of λp , it is easy to see the following domain monotonicity property: for any domains Ω1 ⊂ Ω2 ⊂ Ω, λp (Ω2 ) ≤ λp (Ω1 ). ¯ with φp > 0 on Ω ¯ correTheorem 2.6. There exists an eigenfunction φp ∈ C(Ω) sponding to λp (Ω), if one of the following holds (i) m = inf Ω¯ a(x) is assumed at some x∗ ∈ Ω and there exists a bounded domain Ω0 ⊂ Ω such that x∗ ∈ Ω0 and λp (Ω0 ) < m. (ii) lim|x|→∞, x∈Ω¯ a(x) = m, and there exist bounded and measurable subsets ¯ x ∈ Ak } → ∞ as k → ∞, |Ak | ≥ α > 0, {Ak }∞ k=1 of Ω such that inf{|x| κ(x, y) ≥ β > 0 for x, y ∈ Ak , where α, β are positive constants. Remark 5. For special conditions to satisfy λp (Ω0 ) < m, see (2.3) and (2.4) with ¯r (xk ) ⊂ Ω ¯ with r < δ/2 fixed Ω replaced by Ω0 . If there exists a sequence of balls B ¯ and xk → ∞ as k → ∞, then we can take Ak = Br (xk ) in (ii). Proof of Theorem 2.6. Assume that (i) is satisfied. Take k large enough such that Ω0 ⊂ Ω ∩ Bk (0). Let Ωk = Ω ∩ Bk (0). By the domain monotonicity property, λp (Ω) ≤ λp (Ωk ) ≤ λp (Ω0 ) < m ¯ such that limk→∞ λp (Ωk ) = λ. ¯ Define the operator on Ωk , and there exists λ Z ¯ k. Lk u(x) = − κ(x, y)u(y)dy + a(x)u(x), x ∈ Ω Ωk

¯ k ) and φk > 0 By Theorem 2.5, L has an eigenpair (λp (Ωk ), φk (x)) with φk ∈ C(Ω ∗ ¯ on Ωk . We normalize φk such that φk (x ) = 1. ¯ φk is uniformly bounded on K. We claim that for any fixed compact set K ⊂ Ω, To prove this, first observe that Z (a(x∗ ) − λp (Ωk )) φk (x∗ ) = κ(x∗ , y)φk (y)dy, k

Ωk

which gives that Z m − λp (Ωk ) ≥  φk (y)dy, Dk ∗ ¯ k |x − y| < δ}. Thus for large k, where Dk = {y ∈ Ω Z φk (y)dy < (m − λp (Ω)) / = O(1). Dk

(2.11)

ON EIGENVALUE PROBLEMS

893

Note Dk is independent of large k. We now show that the above inequality holds with Dk replaced by larger and larger sets, which eventually cover K. For this purpose, note that we have  Z Z Z (a(x) − λp (Ωk )) φk (x)dx = κ(x, y)φk (y)dy dx Dk Ωk  ZDk Z = κ(x, y)dx φk (y)dy  ZΩk ZDk ≥ κ(x, y)dx φk (y)dy, (2.12) 1 Dk

where

Dk1

Dk

¯ k | dist(x, Dk ) < δ/2}. For y ∈ D1 and large k, = {x ∈ Ω k Z Z κ(x, y)dx ≥ dx ≥ c1 

(2.13)

|x−y| 1. a(x) − m

Now for all large k such that ¯ k = Bk (0) ∩ Ω ⊃ Ak , Ω 0 we have Z inf

y∈Ak0

Ak0

κ(x, y) dx > 1. a(x) − inf Ωk a(x)

Now applying Theorem 2.1, for large k, Lk has a principal eigenpair (λp (Ωk ), φk (x)) ¯ k ) and φk > 0 on Ω ¯ k . Pick a point x∗ in Ak and normalize φk by with φk ∈ C(Ω 0 ∗ φk (x ) = 1. The proof of the claim in (i) can be repeated to obtain that Z φk (y)dy ≤ O(1) as k → ∞, ` Dk

where ` is large enough such that ¯ | dist(y, K) < µ} ⊂ Dk` {y ∈ Ω ¯ k . Then for k ≥ k1 , for large k. Take a large k1 such that K ⊂ Ω 1 λp (Ωk ) ≤ λp (Ωk1 ) < inf a(x) ≤ inf a(x) Ωk1

K

and for x ∈ K, 0 < φk (x)

= ≤ ≤ =

Z 1 κ(x, y)φk (y)dy a(x) − λp (Ωk ) Ωk Z 1 κ(x, y)φk (y)dy inf K a(x) − λp (Ωk1 ) Ωk Z 1 cφk (y)dy inf K a(x) − λp (Ωk1 ) |y−x|≤µ,y∈Ω O(1)

as k → ∞.

Now the proof of (i) can be repeated again. 3. The spectrum of L2 . First, assume that ¯ bi (x), c(x), a(x) ∈ L∞ (Ω), b(x) ∈ L1 (Ω); (aij )n×n is sym(A2) aij (x) ∈ C(Ω), ¯ for constant c0 > 0. metric and (aij )n×n ≥ c0 In×n , x ∈ Ω (A2)c a(x) ≤ 0 and b(x) ≥ 0 on Ω, or a(x) ≥ 0 and b(x) ≤ 0 on Ω.

ON EIGENVALUE PROBLEMS

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The boundary operator is specified as follows ∂u Bu = u or Bu = αi (x) + β(x)u, x ∈ ∂Ω, ∂xi where ∂Ω ∈ C 2 , αi , β ∈ C 1 (∂Ω), (α1 , α2 , ..., αn ) · ν > 0 on ∂Ω, where ν is the unit outer normal vector on ∂Ω. Let Y = W 2,p (Ω), p > n. We think of L2 as an operator from {u ∈ Y | Bu = 0} to X = Lp (Ω). Define λp =

sup

inf

u∈Y,Bu≥0 on ∂Ω,u>0 on Ω x∈Ω

L2 u(x) . u(x)

(3.1)

Theorem 3.1. Suppose that (A2) and (A2)c hold. Then (i) λp defined by (3.1) is an eigenvalue of L2 , with a corresponding eigenfunction φp > 0 on Ω. (ii) λp is the only eigenvalue which corresponds to an eigenfunction positive on Ω. (iii) The spectrum of L2 consists of isolated eigenvalues of finite algebraic multiplicity; the algebraic multiplicity of λp is equal to 1. (iv) All eigenvalues λ of L2 satisfy Re λ ≥ λp . Theorem 3.2. Assume that (A2) and (A2)c hold. Then λp > 0 if and only if L2 satisfies the maximum principle, i.e., if u ∈ Y , Bu ≥ 0 on ∂Ω, and L2 u ≥ 0 on Ω, ¯ then u ≥ 0 on Ω. Theorem 3.3. Assume that (A2) and (A2)c hold. Then λp =

inf

sup

u∈Y,Bu≤0 on ∂Ω,u>0 on Ω x∈Ω

L2 u(x) . u(x)

The proofs of these theorems use the following property. Lemma 3.1 (Touching Lemma). Assume that (A2) and (A2)c hold. Suppose that there exists u ∈ Y such that Bu ≥ 0 on ∂Ω, u > 0 on Ω and L2 u ≥ 0 on Ω. If there exists v ∈ Y satisfying Bv ≥ 0 on ∂Ω, and L2 v ≥ 0 on Ω, then v does not change ¯ Moreover, if v < 0 somewhere in Ω, then L2 u = 0 = L2 v and v is a sign on Ω. constant multiple of u. The result holds for L2 − λ with any λ ∈ R. Proof. Because u > 0 in Ω , L2 u ≥ 0 and (A2)c , we have −aij (x)uxi xj + bi (x)uxi + c(x)u ≥ 0, and hence Hopf boundary point lemma applies to u. In the Dirichlet case Bu = u, if u = 0 at some point on ∂Ω, then by Hopf boundary point lemma, at that point ∂u ∂ν < 0; in the other case (Neumann/Robin case), by Hopf boundary point lemma again and the boundary condition Bu ≥ 0, it is not possible for u to be 0 anywhere ¯ on ∂Ω, and hence u > 0 on Ω. We only need to consider the case that v < 0 somewhere in Ω. Let δ¯ = sup{δ > 0 | u + δv > 0 on Ω}. By the discussion at the beginning of this proof about u, we have that 0 < δ¯ < ∞, ¯ ≥ 0 on Ω, ¯ ¯ ¯ either (u + δv)(x u + δv 0 ) = 0 for some x0 ∈ Ω or u + δv > 0 on Ω and ¯ (u + δv)(x1 ) = 0 for some x1 ∈ ∂Ω. Furthermore, in the case of Dirichlet boundary operator because both u and v are nonnegative on the boundary, u(x1 ) = 0 = v(x1 ). ¯ δv) ¯ Note that we have (x1 ) = 0 (otherwise we can increase δ). So ∂(u+ ∂ν ¯ ≥0 L2 (u + δv)

896

FANG LI, JEROME COVILLE AND XUEFENG WANG

which implies that ¯ x x + bi (x)(u + δv) ¯ x + c(x)(u + δv) ¯ ≥ 0. −aij (x)(u + δv) i j i ¯ By the strong maximum principle, in the case of (u + δv)(x 0 ) = 0 we have u + ¯ ¯ δv ≡ 0; by Hopf boundary point lemma, in the case of (u + δv)(x 1 ) = 0 we have ¯ ∂(u+δv) ¯ B(u+ δv)(x1 ) < 0 if the boundary operator is Neumann/Robin, and ∂ν (x1 ) < 0 ¯ ≥ 0 on ∂Ω, and in the Dirichlet if the boundary operator is Dirichlet. But B(u + δv) ¯ ∂(u+δv) (x1 ) = 0. Thus x1 does not exist and we must have case we already said ∂ν ¯ ¯ u + δv ≡ 0 on Ω, from which the desired conclusion follows. Proof of Theorem 3.1. It is well-known that for any λ ∈ C, L2 − λ without the integral term is Fredholm of index zero. So is L2 − λ and thus the spectrum of L2 consists of isolated eigenvalues with finite algebraic multiplicity by [14, Page. 243]. Next, we claim that for any λ < λp , λ 6∈ σ(L2 ). Suppose that this is not true, i.e., there exists λ < λp , 0 6≡ u ∈ Y and Bu = 0 such that (L2 − λ)u = 0

in Ω.

Since λ < λp , there exists uλ ∈ Y and uλ > 0 in Ω such that (L2 − λ)uλ > 0

in Ω,

which is impossible according to Touching Lemma (Lemma 3.1), because w.l.o.g., we can assume u < 0 somewhere in Ω. Thanks to this claim, for any λ < λp and  ∈ (0, 1), there exists unique uλ ∈ Y with Buλ = 0 such that (L2 − λ)uλ =  in Ω. Again Touching Lemma (Lemma 3.1) implies that uλ ≥ 0 in Ω. Case 1. There exists  ∈ (0, 1), such that for any λ ∈ [λp − 1, λp ), kuλ kL∞ ≤ 1. Recall that uλ satisfies Z −aij (x)(uλ )xi xj + bi (x)(uλ )xi + c(x)uλ + a(x) b(y)uλ (y)dy − λuλ = , x ∈ Ω. Ω

Using Lp estimates, we see that after passing to a subsequence of λ % λp , ¯ there exists u ∈ Y such that uλ → u weakly in Y and strongly in C 1,α (Ω)   for some 0 < α < 1. Then u ≥ 0 satisfies Bu = 0 and Z −aij (x)(u )xi xj + bi (x)(u )xi + c(x)u + a(x) b(y)u (y)dy − λp u = , x ∈ Ω. Ω

By strong maximum principle, we have u > 0 in Ω. Thus λp u (x) +  L2 u (x) = inf > λp , x∈Ω x∈Ω u (x) u (x) inf

which contradicts the definition of λp . Thus Case 1 does not occur. Case 2. For any  ∈ (0, 1), there exists λ ∈ [λp − 1, λp ) such that kuλ kL∞ > 1. Let u (x)  , c =  . w (x) = λ  kuλ kL∞ kuλ kL∞ Then kw kL∞ = 1 and Z −aij (x)(w )xi xj + bi (x)(w )xi + c(x)w + a(x)

b(y)w (y)dy − λ w = c . Ω

ON EIGENVALUE PROBLEMS

897

Just like in Case 1, we can show that after passing to a subsequence of  → 0+ , there exist λ0 ∈ [λp − 1, λp ], w0 ∈ Y such that λ → λ0 , w → w0 weakly in ¯ w0 > 0 in Ω, Bw0 = 0 and Y and strongly in C 1,α (Ω), Z −aij (x)(w0 )xi xj + bi (x)(w0 )xi + c(x)w0 + a(x) b(y)w0 (y)dy − λ0 w0 = 0. Ω

Then (λ0 , w0 ) is the eigenpair sought in (i). Moreover, λ0 < λp is impossible due to the claim, thus λ0 = λp . The rest of the theorem can be proved similarly as for the case of L1 and we omit the details. The proofs of Theorems 3.2 and 3.3 are similar to those of Theorems 2.3 and 2.4 and we also omit the details. We include an example to demonstrate that when assumption (A2)c is violated, principal eigenvalue of L2 might not exist. Example 3.1. Consider ( R1 u00 (x) +  0 u(x)dx + λu(x) = 0 u0 (0) = u0 (1) = 0,

x ∈ (0, 1),

(3.2)

R1 Assume that (λ, u) is an eigenpair. If 0 u(x)dx = 0, then (λ, u) = some (λk , uk ) = (k 2 π 2 , cos kπx), k ≥ 1; conversely, all these (λk , uk ) are eigenpairs of (3.2). If R1 R1 u(x)dx 6= 0, w.l.o.g., assume 0 u(x)dx = 1, then it is easy to see that (λ, u) = 0 (λ0 , u0 ) = (−, 1). We see that when  > −π 2 , λ0 is the smallest eigenvalue, which also corresponds to a positive eigenfunction; when  = −π 2 , λ0 = λ1 is a double eigenvalue; when  < −π 2 , λ1 becomes the smallest eigenvalue, which is simple but associated eigenfunctions change sign. This example shatters our hope of using λp as a characterization of the minimum of the real part of all eigenvalues. We now present a result which helps us to find all eigenvalues; we do so for the case of the local part of the operator being symmetric: Z b(y)u(y)dy, x ∈ Ω, Ls2 u = − (aij (x)uxi )xj + c(x)u + a(x) Ω

where ¯ c(x), a(x) ∈ L∞ (Ω), b(x) ∈ L2 (Ω); (aij )n×n is symmetric (A2)s aij (x) ∈ C 1 (Ω), ¯ for constant c0 > 0. and (aij )n×n ≥ c0 In×n , x ∈ Ω Then we provide a characterization of all eigenvalues of Ls2 as follows. Theorem 3.4. Assume that (A2)s holds. Then the set of all eigenvalues of Ls2 , associated with boundary conditions either u = 0 on ∂Ω or (aij (x)uxi )·ν +β(x)u = 0 on ∂Ω, consists of the union of three sets: { µi | µi is a multiple eigenvalue of L } ∪ { µi | µi is a simple eigenvalue of L and < a, ψi >< b, ψi >= 0 } ∪ { λ 6= µi for all i |

∞ X < a, ψi >< b, ψi > = 1 }, λ − µi i=1

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FANG LI, JEROME COVILLE AND XUEFENG WANG

where Lu = − (aij (x)uxi )xj + c(x)u with either u = 0 on ∂Ω or (aij (x)uxi ) · ν + β(x)u = 0 on ∂Ω, µi , i ≥ 1 denote the eigenvalues with normalized eigenfunctions ψi , i.e., Z ψi ψj dx = δij , i, j ≥ 1, Ω

and < ·, · > is the standard L2 inner product. The proof of this theorem is similar to that of [15, Theorem 2.1] and we include the details for the convenience of readers. Proof of Theorem 3.4. For clarity, our arguments will be divided into several steps. Step 1. We claim that if λ 6= µi for all i, then λ is an eigenvalue of Ls2 if and only if ∞ X < a, ψi >< b, ψi > = 1. λ − µi i=1 Consider Ls2 u

Z = Lu + a(x)

b(y)u(y)dy = λu, Ω

then since λ 6= µi for all i, we have Z Z ∞ X < a, ψi > −1 ψi (x), u = (λ − L) a(x) b(y)u(y)dy = b(y)u(y)dy λ − µi Ω Ω i=1 which gives that ∞ X < a, ψi >< b, ψi > = 1. λ − µi i=1

since

R Ω

b(y)u(y)dy 6= 0. On the other hand, if the above equality holds, then u=

∞ X < a, ψi > ψi (x) λ − µi i=1

satisfies Ls2 u = λu. The claim is proved. Step 2. Claim that if λ = µi with µi being a a multiple eigenvalue of L, then λ = µi is an eigenvalue of Ls2 . Now there are at least two linearly independent eigenfunctions, denoted by ψi1 and ψi2 . Obviously, there exist constants c1 , c2 , where at least one of them is nonzero, such that Z
b(y) c1 ψi1 (y) + c2 ψi2 (y) dy = 0. Ω

Then clearly λ = µi is an eigenvalue of Ls2 with eigenfunction c1 ψi1 + c2 ψi2 . The claim is proved. Step 3. We will show that if µi is a simple eigenvalue of L and < a, ψi >< b, ψi >= 0, then µi is an eigenvalue of Ls2 . This is obvious if < b, ψi >= 0. Now assume < a, ψi >= 0. Then ψi belongs to the kernel of the adjoint operator of Ls2 − µi , and thus µi is an eigenvalue of Ls2 by Fredholm Alternatives. Step 4. It remains to prove that if λ = µi with µi being a simple eigenvalue of L is an eigenvalue of Ls2 , then < a, ψi >< b, ψi >= 0. Let φ denote the eigenfunction

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corresponding to µi , i.e., Ls2 φ = Lφ + a(x)

Z b(y)φ(y)dy = µi φ. Ω

This gives that Z

Z a(y)ψi (y)dy b(y)φ(y)dy = 0. Ω R Ω R If Ω a(y)ψi (y)dy 6= 0, then RΩ b(y)φ(y)dy = 0 and thus φ is a constant multiple of ψi since µi is simple. Hence Ω b(y)ψi (y)dy = 0. The proof is complete. 4. The spectrum of L3 . Throughout this section, we assume (A3) κ ∈ C([0, L] × [0, L]) is nonnegative, κ(x, x) > 0 for any x ∈ [0, L]; a ∈ C([0, L]), c ∈ C 1 ([0, L]) and c(x) > 0 in [0, L]. Let Y = {u ∈ C 1 ([0, L]) | u(0) = 0}, equipped with the standard C 1 −norm. We think of L3 as an operator from Y to X = C([0, L]). Define λp =

sup

inf

u∈Y,u>0 on (0,L] x∈(0,L]

L3 u(x) . u(x)

(4.1)

For convenience, let v(x) = u(x)c(x), Z L v(y) a(x) L¯3 v(x) = − κ(x, y) dy + v(x) + v 0 (x). c(y) c(x) 0 Obviously, λp =

sup v∈Y,v>0 on

L¯3 v(x) . (0,L] x∈(0,L] v(x)/c(x) inf

(4.2)

λp will be proved to be the principal eigenvalue. We now show, using (4.2), that λp is a finite number. Choose a function v ∈ Y such that for x ≈ 0, Z L v(y) dy. v 0 (x) > κ(x, y) c(y) 0 Then

L¯3 v(x) > −∞. x∈(0,L] v(x)/c(x) Suppose that λp = ∞. Then there exist {vn }∞ n=1 ⊂ Y such that vn > 0 on (0, L] and Z L vn (y) c(x)vn0 (x) + a(x)vn (x) − c(x) κ(x, y) dy > nvn (x), x ∈ [0, L]. c(y) 0 λp ≥

inf

In particular, vn0 > 0 and hence for large n and for positive constants c0 and c1 , we have Z L n 0 κ(x, y)vn (y)dy. (4.3) c0 vn (x) − vn (x) > c1 2 0 Take a small  > 0 such that κ(x, y) > 0 if |x − y| ≤  and x, y ∈ [0, L]. Observe that for x ∈ [0, L], we have Z L Z L κ(x, y)vn (y)dy ≥ κ(x, y)vn (y)dy 0



Z ≥ vn ()

L

κ(x, y)dy ≥ κ0 vn (), 

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FANG LI, JEROME COVILLE AND XUEFENG WANG

where κ0 is a positive constant. Combining this with (4.3), we have that for x ∈ [0, L], n c1 κ0 vn () vn0 (x) − vn (x) > , 2c0 c0   0 n c1 κ0 vn () − 2cn x e− 2c0 x vn (x) ≥ e 0 , c0 which yields that  n 2c1 κ0 vn () 2cn x  vn (x) ≥ e 0 1 − e− 2c0 x . n Taking x =  and sending n → ∞, we reach a contradiction. We have shown that λp is a finite number. Theorem 4.1. Suppose that (A3) holds. Then (i) λp defined by (4.1) or (4.2) is an eigenvalue of L3 : Y → X, with a corresponding eigenfunction φp > 0 on (0, L]. (ii) λp is the only eigenvalue which corresponds to an eigenfunction positive on (0, L]. (iii) The spectrum of L3 consists of isolated eigenvalues of finite algebraic multiplicity; the algebraic multiplicity of λp is equal to 1. (iv) All eigenvalues λ of L3 satisfy Re λ ≥ λp . Theorem 4.2. Assume that (A3) holds. Then λp > 0 if and only if L3 satisfies the maximum principle, i.e., if u ∈ C 1 ([0, L]), u(0) ≥ 0 and L3 u ≥ 0 on [0, L], then u ≥ 0 on [0, L]. Theorem 4.3. Suppose that (A3) holds. Then λp =

inf

sup

u∈Y,u>0 on (0,L) x∈(0,L)

L3 u(x) . u(x)

The proofs of these theorems use the following property. Lemma 4.1 (Touching Lemma). Assume that (A3) holds. Suppose that there exists u ∈ C 1 ([0, L]) such that u(0) ≥ 0, u > 0 on (0, L] and L3 u ≥ 0 on [0, L]. If there exists v ∈ C 1 ([0, L]) satisfying v(0) ≥ 0 and L3 v ≥ 0 on [0, L], then v does not change sign on [0, L]. Moreover, if v < 0 somewhere in (0, L], then L3 u = 0 = L3 v and v is a constant multiple of u. The result holds for L3 − λ with any λ ∈ R. Proof. We just need to discuss the case when v < 0 somewhere in (0, L]. Let δ¯ = sup{δ > 0 | u + δv > 0 on (0, L]}. ¯ ≥ 0 on [0, L], either (u + δv)(x ¯ Then 0 < δ¯ < ∞, u + δv 0 ) = 0 for some x0 ∈ (0, L] 0 ¯ ¯ ¯ or (u + δv)(0) = 0 = (u + δv) (0) and (u + δv)(x) > 0 for x ∈ (0, L]. The latter case cannot occur because Z L
0 ¯ ¯ ¯ c(x)(u + δv)(x) + a(x)(u + δv)(x) ≥ κ(x, y)(u + δv)(y)dy 0

cannot hold at x = 0; in the former case, taking x = x0 , we are led to Z L ¯ κ(x0 , y)(u + δv)(y)dy = 0. 0

¯ 6≡ 0 and we take x0 to be a boundary point of This cannot be true if u + δv ¯ ¯ ≡ 0. {x ∈ (0, L] | (u + δv)(x) > 0}. Thus u + δv

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Proof of Theorem 4.1. For any λ ∈ C, L3 − λ : Y → X is a compact perturbation of 0 T − λ : u ∈ Y → (a(x) − λ)u + (c(x)u(x)) ∈ X. By the existence and uniqueness theorem for linear ODE, T − λ is one-to-one and onto. Thus L3 − λ is Fredholm of index zero. In particular, L3 has no essential spectrum. By [14, Page 243], the spectrum of L3 consists of isolated eigenvalues with finite algebraic multiplicity. This proves the first part of (iii). Claim 1. For any λ < λp , λ 6∈ σ(L3 ). Suppose otherwise. Then there exists real eigenfunction u ∈ Y corresponding to λ, i.e., (L3 − λ)u(x) = 0, x ∈ [0, L]. Since λ < λp , by the definition of λp , there exists uλ ∈ Y such that uλ > 0 on (0, L] and (L3 − λ)uλ (x) > 0, x ∈ [0, L]. By Touching Lemma (Lemma 4.1), the last inequality is impossible. This completes the proof of Claim 1. According to Claim 1, for any λ < λp and  ∈ (0, 1), there exists unique uλ ∈ Y such that (L3 − λ)uλ (x) = , x ∈ [0, L], i.e., Z L 0 − κ(x, y)uλ (y)dy + (a(x) − λ)uλ (x) + (c(x)uλ (x)) = . (4.4) 0

Touching Lemma again yields that uλ ≥ 0 on [0, L], from which it follows 0

(a(x) − λ)uλ (x) + (c(x)uλ (x)) ≥ ,

x ∈ [0, L].

uλ

uλ

> 0 on (0, L]. to be zero in (0, L]. Thus This prohibits R Case 1. There exists  ∈ (0, 1), such that for any λ ∈ [λp − 1, λp ), Ω uλ (x)dx ≤ 1. Integrating (4.4) on [0, x], we have ! Z x Z L Z x  − κ(z, y)uλ (y)dy dz + (a(z) − λ)uλ (z)dz + c(x)uλ (x) = x, 0

0

0

from which we see that after passing to a subsequence of λ % λp , there exists u ∈ X such that uλ → u in X, and by (4.4), uλ → u in Y . Then u satisfies (4.4) with λ replaced by λp and u ≥ 0. As in the case of uλ , we can prove u > 0 on (0, L]. Now λp u (x) +  L3 u (x) = inf > λp .  u (x) x∈(0,L] x∈(0,L] u (x) inf

This contradicts the definition of λp . Thus Case 1 does not occur. RL Case 2. For any  ∈ (0, 1), there exists λ ∈ [λp −1, λp ) such that 0 uλ (x)dx > 1. Let u (x)  w (x) = R L λ , c = R L .   uλ (x)dx uλ (x)dx 0 0 RL Then 0 w (x)dx = 1 and Z L 0 − κ(x, y)w (y)dy + (a(x) − λ )w (x) + (c(x)w (x)) = c . 0

902

FANG LI, JEROME COVILLE AND XUEFENG WANG

Just like in Case 1, we can show that after passing to a subsequence of  → 0+ , there exist λ0 ∈ [λp − 1, λp ], w0 ∈ Y such that λ → λ0 , w → w0 in Y , w0 > 0 on (0, L] and Z L 0 − κ(x, y)w0 (y)dy + (a(x) − λ0 )w0 (x) + (c(x)w0 (x)) = 0. 0

λ0 < λp is ruled out by Claim 1. Then (λ0 , w0 ) is the eigenpair sought in (i). It is now clear that we can prove the rest of the theorem along the lines for the case of L1 ; we omit the details. The proof of Theorems 4.2 and 4.3 are easier than those of Theorems 2.3 and 2.4 (for L1 ), because of the existence of principal eigenfunction φp > 0 on (0, L]. We again omit the details. REFERENCES [1] X. Bai and F. Li, Optimization of species survival for logistic models with non-local dispersal, Nonlinear Anal. Real World Appl., 21 (2015), 53–62. [2] X. Bai and F. Li, Global dynamics of a competition model with nonlocal dispersal II: The full system, J. Differential Equations, 258 (2015), 2655–2685. [3] P. Bates and G. Zhao, Existence, uniqueness and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal, J. Math. Anal. Appl., 332 (2007), 428–440. [4] P. Bates and G. Zhao, Spectral convergence and turing patterns for nonlocal diffusion systems, preprints. [5] H. Berestycki, J. Coville and H.-H. Vo, Persistence criteria for populations with non-local dispersion, J. Math. Biol., 72 (2016), 1693–1745. [6] H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47–92. [7] J. Coville, On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators, J. Differential Equations, 249 (2010), 2921–2953. [8] J. Coville, Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., 26 (2013), 831–835. [9] J. Coville, Nonlocal refuge model with a partial control, Discrete Contin. Dyn. Syst., 35 (2015), 1421–1446. [10] J. Coville, J. Davila and S. Martinez, Pulsating fronts for nonlocal dispersion and KPP nonlinearity, Ann. I. H. Poincare – AN, 30 (2013), 179–223. [11] J. F. Crow and M. Kimura, An Introduction to Population Genetics Theory, Burgess Pub. Co., 1970. [12] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, Providence, 1998. [13] V. Huston, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483–517. [14] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. [15] F. Li, K. Nakashima and W.-M. Ni, Stability from the point of view of diffusion, relaxation and spatial inhomogeneity, Discrete Contin. Dyn. Syst., 20 (2008), 259–274. [16] F. Li, Y. Lou and Y. Wang, Global dynamics of a competition model with non-local dispersal I: the shadow system, J. Math. Anal. Appl., 412 (2014), 485–497. [17] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263–298. [18] W. Shen and X. Xie, On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications, Discrete Contin. Dyn. Syst., 35 (2015), 1665–1696. [19] D. B. Smith, A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications, J. Math. Anal. Appl., 418 (2014), 766–774. [20] L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267–1278.

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[21] J.-W. Sun, W.-T. Li and Z.-C. Wang, A nonlocal dispersal logistic equation with spatial degeneracy, Discrete Contin. Dyn. Syst., 35 (2015), 3217–3238. [22] J.-W. Sun, F.-Y. Yang and W.-T. Li, A nonlocal dispersal equation arising from a selectionmigration model in genetics, J. Differential Equations, 257 (2014), 1372–1402. [23] Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal., 118 (2015), 51–62.

Received June 2015; revised September 2015. E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]