Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations Yehuda Pinchover Department of Mathematics, Technion 32000 Haifa, ISRAEL E-mail address:
[email protected]
Abstract In this paper we discuss some new results concerning perturbation theory for second order elliptic partial differential equations related to positivity properties of such equations. We continue the study of some different notions of “small” perturbations and discuss their relations to comparisons of Green’s functions, refined maximum and antimaximum principles, ground state, and the decay of eigenfunctions. In particular, we show that if V is a positive function which is a semismall perturbation of a subcritical Schr¨odinger operator H defined on a domain Ω ⊂ Rd , and {φk }k≥0 are the (Dirichlet) eigenfunctions of the equation Hu = λV u, then for any k ≥ 0, the function φk /φ0 is bounded and has a continuous extension up to the Martin boundary of the pair (Ω, H), where φ0 is the ground state of H with a principal eigenvalue λ0 .
1
Introduction
In a recent paper [24], M. Murata has introduced the following notion of semismall perturbation of a second order linear elliptic operator. 0
1991 Mathematics subject classification. Primary 35B05, 35B50, 35J10; Secondary 31C35, 35J15. Keywords and phrases. maximum and anti-maximum principle, Green function, Martin boundary, Schr¨ odinger operators.
1
Definition 1.1 Let P be an elliptic operator of the form P (x, ∂x ) = −
d X
aij (x)∂i ∂j +
i,j=1
d X
bi (x)∂i + c(x)
(1.1)
i=1
which is defined on a domain Ω ⊂ IRd and which is subcritical in Ω (that is, ∞ P admits a positive minimal Green function GΩ P (x, y)). Let {Ωk }k=1 be an α exhaustion of Ω, and x0 be some fixed point in Ω. Let W ∈ C (Ω). We say that W is a semismall perturbation of P in Ω if (
lim
k→∞
Z
sup
y∈Ω\Ωk Ω\Ωk
Ω GΩ P (x0 , z)|W (z)|GP (z, y) dz GΩ P (x0 , y)
)
=0.
(1.2)
Semismall perturbation is a refinement of the definition of small perturbation (see Definition 2.5) which was introduced by the author in [27] (see Section 2 for more precise assumptions and details). It turns out that these notions are useful tools in the study of positive solutions. For example, many positivity properties of the operator P in Ω, like the behavior of the Green function, the structure of the cone CP (Ω) which is the set of all positive solutions of the equation P u = 0 in Ω and the Martin boundary are stable under small or semismall perturbation. The main aim of the paper is to discuss some new results concerning perturbation theory for second order linear elliptic partial differential equations related to positivity properties of such equations. We pursue the study of some different notions of perturbations which were introduced in [24] and discuss their relations to comparisons of Green’s functions, refined maximum and anti-maximum principles, positivity of the ground state energy, and the decay of eigenfunctions. In Section 4, we discuss an extension of the classical generalized maximum principle which holds in relatively compact subdomains to a refined (generalized) maximum principle which holds in any domain. Recall that if CP (Ω) 6= ∅, then the generalized maximum principle asserts that for any subdomain Ω1 ⊂⊂ Ω a function u 6= 0 which satisfies P u ≥ 0 in Ω1 and u ≥ 0 on ∂Ω1 is strictly positive in Ω1 . Roughly speaking, our maximum principle is of Phragm´en-Lindel¨ of type and states that if P u ≥ 0 in Ω, P is subcritical in Ω and u is “bounded” near infinity in Ω, then u ≥ 0 (see Theorem 4.4). As a consequence, we prove an existence and uniqueness theorem for solutions of the nonhomogeneous equation (see Theorem 4.6). A quite recent topic in the theory of second order elliptic operators is the so-called anti-maximum principle discovered by Ph. Cl´ement, and 2
L. A. Peletier [13] (see also [4, 9, and the references therein]). This interesting phenomenon occurs above the generalized principal eigenvalue λ0 , where λ0 = λ0 (1, P, Ω) := sup{λ ∈ IR : CP −λ (Ω) 6= ∅} (recall that the generalized maximum principle holds true below λ0 ). Let uλ be a solution of the equation (P − λ)u = f ≥ 0
in Ω.
(1.3)
The anti-maximum principle reads roughly that under some “smallness” conditions on f and uλ in a neighborhood of infinity in Ω, there exists ² > 0 which may depend on f such that uλ < 0 for all λ ∈ (λ0 , λ0 + ²). In Theorem 5.3, we prove a refined anti-maximum principle which generalizes the above results. Our approach demonstrates that the anti-maximum principle follows actually from the perturbation theory of positive solutions. More precisely, we study the behavior of the principal eigencurve of a certain two-parameter eigenvalue problem and prove the anti-maximum principle using a fixed point argument. Note that uλ is a negative solution of Equation (1.3) if and only if wλ,t := −tuλ is a positive solution of the linear homogeneous equation (P − λ + tf /wλ,t )u = 0
in Ω,
where t > 0. Another application of the perturbation theory of positive solutions is in the study of the asymptotic behavior of solutions of the equation (P +V )u = 0 which “decay” in a neighborhood of infinity in Ω. An important example is of course, the study of the asymptotic behavior of L2 -solutions of the Schr¨odinger equation (see for example [17, and the references therein]). We assume that P is a general subcritical operator, V is a semismall perturbation of the operator P ∗ in Ω and |w| ≤ Cφ satisfies the equation (P +V )u = 0 in a neighborhood of infinity in Ω, where φ is a positive solution of minimal growth in a neighborhood of infinity in Ω of the equation P u = 0. We prove that w/φ has a continuous extension up to the Martin boundary of the pair (Ω, P ∗ ). Consequently, we obtain some new results concerning the asymptotic behavior of the Dirichlet eigenfunctions of a Schr¨ odinger operator H. In Theorem 6.3, we prove that if V is a positive function which is a semismall perturbation of a subcritical Schr¨ odinger operator H defined on a domain Ω ⊂ Rd , and {φk }k≥0 are the (Dirichlet) eigenfunctions of the 3
equation Hu = λV u, then for any k ≥ 0, the function φk /φ0 is bounded and has a continuous extension up to the Martin boundary of the pair (Ω, H), where φ0 is the ground state of H with a principal eigenvalue λ0 . Most of the results concerning precise asymptotic behaviors of eigenfunctions of Schr¨odinger operators are concerned with either pointwise upper bounds which hold for any eigenfunction, or pointwise lower bounds for eigenfunctions which are positive outside a compact set. For oscillatory eigenfunctions there are some lower bounds for the spherically averaged L2 eigenfunctions in the special case of Schr¨ odinger operators defined on IRd with radially symmetric potentials (see [17, and the references therein]). In Lemma 6.5, we prove such a lower bound which generalizes [17, Theorem 1.4]. The plan of the paper is the following. In Section 2, we collect some terminology, introduce some notions of perturbations related to small and semismall perturbations and discuss some results which we need in this paper. In Section 3, we discuss some necessary and sufficient conditions, connected with the above notions of perturbations, such that λ0 > 0. We prove in Section 4 the refined maximum principle (Theorem 4.4) and the existence and uniqueness theorem for the nonhomogeneous equation (Theorem 4.6). In Section 5, we prove the anti-maximum principle while in Section 6 we discuss our results concerning the asymptotic behavior of eigenfunctions. In section 7, we discuss relationships between different notions of perturbations provided that a certain quasi-metric inequality (related to 3G-inequalities) holds true in Ω. Finally, in Section 8, we present some examples which exemplify some of the phenomena discussed in the paper.
2
Preliminaries
In this section we collect some terminology and results which we need in this paper. Let P be a linear, second order, elliptic operator defined in a subdomain Ω of a noncompact, connected, C 3 -smooth Riemannian manifold X of dimension d. Here P is an elliptic operator with real H¨older continuous coefficients which in any coordinate system (U ; x1 , . . . , xd ) has the form P (x, ∂x ) = −
d X
aij (x)∂i ∂j +
i,j=1
d X i=1
where ∂i = ∂/∂xi . 4
bi (x)∂i + c(x),
(2.1)
We assume that for every x ∈ Ω the real quadratic form d X
aij (x)ξi ξj , ξ = (ξ1 , . . . , ξd ) ∈ IRd
(2.2)
i,j=1
is positive definite. We denote the cone of all positive (classical) solutions of the elliptic equation P u = 0 in Ω by CP (Ω). In case that the coefficients of P are smooth enough, we denote by P ∗ the formal adjoint of P . Let {Ωk }∞ k=1 be an exhaustion of Ω, i.e. a sequence of smooth, relatively compact domains such that Ω1 6= ∅, Ωk ⊂ Ωk+1 and ∪∞ k=1 Ωk = Ω. Assume that CP (Ω) 6= ∅. Then for every k ≥ 1 the Dirichlet Green function k GΩ P (x, y) exists and is positive. By the generalized maximum principle, Ωk {GP (x, y)}∞ k=1 is an increasing sequence which, by the Harnack inequality, converges uniformly in any compact subdomain of Ω either to GΩ P (x, y) , the positive minimal Green function of P in Ω and P is said to be a subcritical operator in Ω , or to infinity and in this case P is critical in Ω. The operator P is said to be supercritical in Ω if CP (Ω) = ∅ [21, 26, 33]. It follows that P is critical (resp. subcritical) in Ω if and only if P ∗ is critical (resp. subcritical) in Ω. Furthermore, if P is critical in Ω, then CP (Ω) is a one-dimensional cone and any positive supersolution of the equation P u = 0 in Ω is a solution. In this case φ ∈ CP (Ω) is called a ground state of P in Ω [1]. We fix a reference point x0 ∈ Ω1 . From time to time, we consider the convex set KP (Ω) := {u ∈ CP (Ω) | u(x0 ) = 1} of all normalized positive solutions. M Ω In the subcritical case, we denote by ΩM P , ∂P Ω and KP (x, ξ) the Martin compactification, the Martin boundary and the Martin function with a pole ξ ∈ ∂PM Ω of the pair (Ω, P ) (for the definitions and basic properties see [21, 24, 33, and the references therein]). Remark 2.1 We would like to point out that the criticality theory is also valid for the class of weak solutions of elliptic equations in divergence form and also for the class of strong solutions of strongly elliptic equations with locally bounded coefficients. Nevertheless, for the sake of clarity, we prefer to present our results only for the class of classical solutions.
5
Subcriticality is a stable property in the following sense. If P is subcritical in Ω and V ∈ C0α (Ω), then there exists ² > 0 such that P − µV is subcritical for all |µ| < ² [21, 26]. On the other hand, if P is critical in Ω and V ∈ C α (Ω) is a nonzero, nonnegative function, then for any ² > 0 the operator P + ²V is subcritical and P − ²V is supercritical in Ω. We associate to any subdomain Ω ⊆ X an exhaustion {Ωn }∞ n=1 and a ∞ sequence {χn (x)}n=1 of smooth cutoff functions in Ω such that χn (x) ≡ 1 in Ωn , χn (x) ≡ 0 in Ω \ Ωn+1 , and 0 ≤ χn (x) ≤ 1 in Ω. Let W ∈ C α (Ω), 0 < α ≤ 1. We denote Wn (x) = χn (x)W (x) and Wn∗ (x) = W (x) − Wn (x). For every k ≥ 1, we denote Ω∗k = Ω \ Ωk and for every k > k0 we denote by ◦
Ωk,k0 the “annulus” Ωk \ Ωk0 . Let Q ⊂ Ω, Q denotes the interior of the set Q. We say that a function f ∈ C(Ω) vanishes at infinity of Ω if for every ² > 0 there exists N ∈ IN such that |f (x)| < ² for all x ∈ Ω∗N . Let f, g ∈ C(Ω), we use the notation f ≈ g on Ω if there exists a positive constant C such that C −1 g(x) ≤ f (x) ≤ Cg(x) for all x ∈ Ω . We denote by f + (resp. f − ) the positive (resp. negative) part of a function f . So, f = f + − f − . By 1, we denote the constant function on Ω taking at any point of x ∈ Ω the value 1. The next notion was introduced by S. Agmon in [1]. Definition 2.2 Let P be an elliptic operator defined in a domain Ω ⊆ X. A function u ∈ C(Ω∗n ) is said to be a positive solution of the operator P of minimal growth in a neighborhood of infinity in Ω if u satisfies the following two conditions: (i) The function u is a positive solution of the equation P u = 0 in Ω∗n ; (ii) If v is a continuous function on Ω∗k for some k > n which is a positive solution of the equation P u = 0 in Ω∗k , and u ≤ v on ∂Ωk , then u ≤ v on Ω∗k . Remark 2.3 If P is subcritical in Ω, then GΩ P (x, x0 ) is a positive solution of the equation P u = 0 of minimal growth in a neighborhood of infinity in Ω. On the other hand, if P is critical in Ω, then the ground state is a positive (global) solution of the equation P u = 0 in Ω which has minimal growth in a neighborhood of infinity in Ω. 6
Let V ∈ C α (Ω) and denote the generalized principal eigenvalue by λ0 = λ0 (V, P, Ω) := sup{λ ∈ IR : CP −λV (Ω) 6= ∅}. Set also λ∞ = λ∞ (V, P, Ω) := sup{λ ∈ IR : ∃K ⊂⊂ Ω s.t. CP −λV (Ω \ K) 6= ∅}. Clearly, −∞ ≤ λ0 ≤ λ∞ ≤ ∞. Moreover, if V is a positive function and P is a symmetric operator in L2 (Ω, V dx) with a domain C0∞ (Ω) which has a selfadjoint realization P˜ on L2 (Ω, V dx), then λ0 = inf σ(P˜ ) and λ∞ = inf σess (P˜ ), the bottom of the spectrum and the essential spectrum of P˜ respectively [1]. In the sequel, we shall use also the notations σdis (P˜ ), σp (H) for the discrete and the point spectrum of P˜ , respectively. Set S = S+ ∪ S0 , where S+ = S+ (V, P, Ω) = {t ∈ IR | P − tV is subcritical in Ω},
(2.3)
S0 = S0 (V, P, Ω) = {t ∈ IR | P − tV is critical in Ω}.
(2.4)
It is known [28] that S is a closed interval and S0 ⊂ ∂S. Definition 2.4 Let Pi , i = 1, 2 be two subcritical operators in Ω ⊆ X . Ω We say that the Green functions GΩ P1 (x, y) and GP2 (x, y) are equivalent Ω Ω (resp. semi-equivalent) if GP1 ≈ GP2 on Ω × Ω \ {(x, x) | x ∈ Ω} (resp. Ω GΩ P1 (·, y0 ) ≈ GP2 (·, y0 ) on Ω \ {y0 } for some fixed y0 ∈ Ω). We use the notation n
o
Ω E+ = E+ (V, P, Ω) := t ∈ IR | GΩ P −tV ≈ GP on Ω × Ω \ {(x, x) | x ∈ Ω} ,
n
o
Ω sE+ = sE+ (V, P, Ω) := t ∈ IR | GΩ P −tV (·, y0 ) ≈ GP (·, y0 ) on Ω \ {y0 } .
Many papers deal with sufficient conditions, in terms of proximity near infinity in Ω between two given subcritical operators, which imply that the Green functions are equivalent (see, [6, 21, 24, 26, 27, 29, and the references therein]). The following notion was introduced in [27] and is closely related to this problem. Definition 2.5 Let P be a subcritical operator in Ω ⊆ X and let W ∈ C α (Ω) . We say that W is a small perturbation of P in Ω if (
lim
k→∞
Z
sup
x,y∈Ω∗k Ω∗k
Ω GΩ P (x, z)|W (z)|GP (z, y) dz GΩ P (x, y)
7
)
= 0.
(2.5)
The following refinements of Definition 2.5 were recently introduced by M. Murata [24]. Definition 2.6 Let P be a subcritical operator in Ω and let W ∈ C α (Ω). (i) We say that W is a semismall perturbation of P in Ω if (
lim
k→∞
Z
sup
y∈Ω∗k Ω∗k
Ω GΩ P (x0 , z)|W (z)|GP (z, y) dz GΩ P (x0 , y)
)
=0.
(2.6)
(ii) We say that W is a G-bounded perturbation (resp. G-semibounded perturbation) of P in Ω if there exists a positive constant C such that Z Ω
Ω GΩ P (x, z)|W (z)|GP (z, y) dz ≤ C . GΩ P (x, y)
(2.7)
for all x, y ∈ Ω (resp. for some fixed x ∈ Ω and all y ∈ Ω \ {x}). (iii) We say that W is a H-bounded perturbation (resp. H-semibounded perturbation) of P in Ω if there exists a positive constant C such that Z Ω
GΩ P (x, z)|W (z)|u(z) dz ≤ C . u(x)
(2.8)
for all x ∈ Ω (resp. for some fixed x ∈ Ω) and all u ∈ CP (Ω). (iv) We say that W is a H-integrable perturbation of P in Ω if Z Ω
GΩ P (x, z)|W (z)|u(z) dz < ∞ .
(2.9)
for all x ∈ Ω and all u ∈ CP (Ω). Remarks 2.7 (i) A small perturbation is semismall [24]. Clearly, a G(resp. H-) bounded perturbation is G- (resp. H-) semibounded. Moreover, H-semibounded perturbation is H-integrable. Note that if dim CP (Ω) < ∞, then V is H-semibounded perturbation if and only if it is H-integrable. It is also known that if W is G-(resp. semi) bounded perturbation, then W is H-(resp. semi) bounded perturbation [24, 26]. In Example 8.7, we show a potential which is H-semibounded perturbation but neither H-bounded nor G-semibounded. We do not know of any example of a semismall (resp. G-semibounded) perturbation which is not a small (resp. G-bounded) perturbation. We are also not aware of any example of a H-bounded (resp. H-integrable) perturbation which is not G-bounded (resp. H-semibounded). 8
(ii) It is known [24, 27, 28] that if the operator P is subcritical and W is a small (resp. semismall) perturbation of P (resp. P ∗ ) in Ω, then W is a G-bounded (resp. G-semibounded) perturbation of P (resp. P ∗ ) in Ω and the Green functions of P and P − tW are (resp. semi-) equivalent for all t ∈ S+ . Moreover, ∂S = S0 . In other words, E+ = S+ (Resp. sE+ = S+ ). Assume now that W is a semismall perturbation of P ∗ in Ω and let λ ∈ S0 . So, P − λW is critical in Ω. Denote by φ0 the ground state of P − λW . Then φ0 ≈ GΩ P (·, x0 ) [24]. (iii) If W is a G-bounded (resp. G-semibounded) perturbation of P Ω (resp. P ∗ ) in Ω, then GΩ P (x, y) and GP +tW (x, y) are equivalent (resp. semiequivalent) provided that |t| is small enough [24, 26, 27] (for a further discussion see Example 8.6). Ω On the other hand, if GΩ P (x, y) and GP +W (x, y) are equivalent (resp. semi-equivalent) and W has a definite sign, then W is a G-bounded (resp. G-semibounded) perturbation of P (resp. P ∗ ) in Ω. In this case, the set E+ (resp. sE+ ) is an open half line which is contained in S+ (see, [28, Corollary 3.6]). (iv) If W is a semismall perturbation of the operator P , and P + W is M subcritical in Ω, then the Martin spaces ΩM P and ΩP +W are homeomorphic and in particular, CP (Ω) and CP +W (Ω) are affine homeomorphic [24]. On the other hand, if the corresponding Green functions are equivalent, then it is only known that the cones CP (Ω) and CP +W (Ω) are homeomorphic [26]. Ω (v) Suppose that GΩ P (x, y) and GP −|W | (x, y) are equivalent (resp. semiequivalent). Using the resolvent equation it follows that the best (resp. Ω semi-) equivalence constants of GΩ P (x, y) and GP ±|Wn∗ | (x, y) tend to 1 as n → ∞ if and only if W is a (resp. semi-) small perturbation of P (resp. P ∗ ) in Ω. On the other hand, if W is a G-(resp. semi-) bounded perturbation of P (resp. P ∗ ) in Ω and K ∈ C α (Ω) is an arbitrary function which vanishes at infinity of Ω, then clearly the function K(x)W (x) is a (resp. semi-) small perturbation of the operator P (resp. P ∗ ) in Ω. (vi) A notion related to G- and H-boundedness in the self-adjoint case is intrinsic ultracontractivity (IU) (see [24]). Recall that if P is a formally self-adjoint operator such that the corresponding semigroup is IU in Ω, then 1 is a G-bounded perturbation of P (see [24]). On the other hand, Ba˜ nuelos and Davis [7] gave an example of an operator P and domain Ω such that 1 is a G-bounded perturbation of the Laplacian but the semigroup is not IU. See also Example 8.8 and Remark 6.4 (ii) for the relations of IU to small
9
and semismall perturbations. (vii) By the Martin boundary theory (or by remark (i)), it follows easily that if V is a G-semibounded perturbation of the operator P in Ω then KP (Ω) is a bounded set in L1 (Ω, |V (x)|GΩ P (x0 , x)dx). Moreover, by the Riesz decomposition theorem, the cone of all normalized positive supersolutions is bounded in L1 (Ω, |V (x)|GΩ P (x0 , x)dx). Ω In particular, if 1/GP (x0 , x) is a G-semibounded perturbation of the operator P in Ω then every positive solution (resp. supersolution) is integrable and the cone of all normalized positive solutions (resp. supersolutions) is a bounded set in L1 (Ω) (see also [2, 15]). Remark 2.8 In [6], A. Ancona investigates the question of the equivalence of the Green functions of two uniformly elliptic, weakly coercive operators with uniformly bounded coefficients defined on a complete Riemannian manifold. The perturbation is allowed to be not only in the zero order term but also in the higher order terms. It is proved that the Green functions and the Martin boundaries of such two operators are equivalent provided that the “distance near infinity” between the operators is sufficiently small. Moreover, the equivalence constant tends to 1 if the supports of the perturbations are concentrated at infinity in Ω. Therefore, zero order perturbations of Ancona’s type gives us a huge and almost optimal class of examples of small perturbations. Note that unlike the notion of small and semismall perturbations the definition of Ancona’s distance does not depend explicitly on the behavior of the Green functions at infinity but on the difference between the coefficients of the operators (see also [2, 21, 24, 27, 29, and the references therein]). The discussion above demonstrates that the above notions of perturbations are closely related to the equivalence of the Green functions and the Martin boundaries. Nevertheless, it turns out that some other positivity properties in the criticality theory are preserved under a wider class of perturbations, namely, weak perturbations (see [32]). Definition 2.9 Let P be a subcritical operator in Ω ⊆ X. A function W ∈ C α (Ω) is said to be a weak perturbation of the operator P in Ω if the following condition holds true. (∗) For every η ∈ IR there exists N ∈ IN such that the operator P − ηWn∗ (x) is subcritical in Ω for any n ≥ N .
10
A function W ∈ C α (Ω) is said to be a weak perturbation of a critical operator P in Ω if there exists a nonzero, nonnegative function V ∈ C0α (Ω) such that the function W is a weak perturbation of the subcritical operator P + V in Ω. Remarks 2.10 (i) If W is a weak perturbation of critical or subcritical operator P in Ω, then ∂S = S0 and λ∞ = ∞ [32] (see also Theorem 6.3). (ii) Using the results in [24, 27], it follows that if W is a small or even semismall perturbation of an operator P in Ω, then |W | is a weak perturbation of P in Ω. (iii) One can use integral estimates on the number of the negative eigenvalues of subcritical Schr¨odinger operators to show that a perturbation is weak. For example, let d ≥ 3. By the Cwikel-Lieb-Rozenblum bound (see, [10]), if W ∈ Ld/2 (IRd ), then |W | is a weak perturbation of −∆ in IRd . Note that the function W (x) = (1 + |x|)−2 is not a weak perturbation of −∆ in IRd , d ≥ 3. On the other hand, for any ² > 0 the function W (x) = (1+|x|)−(2+²) is a small perturbation of −∆ in IRd , d ≥ 3 [21, 26, 29]. We conclude this section with the following remark. Remark 2.11 Let P be a subcritical operator in Ω and assume that V > 0. Then V is a perturbation of one of the types considered throughout this section if and only if 1 is such a perturbation of the operator V −1 P . Therefore, when considering perturbations by a positive function, we assume from time to time that the perturbation is by the function 1 and we leave to the reader to formulate the results for a general positive V .
3
Necessary and sufficient conditions for λ0 > 0
In this section we assume that P is subcritical and discuss some necessary and sufficient conditions, related to H-boundedness, such that λ0 > 0. Lemma 3.1 Suppose that P is a subcritical operator in Ω and V ∈ C α (Ω) is a nonzero, nonnegative function such that λ0 = λ0 (V, P, Ω) > 0. (i) Z Ω
Ω GΩ P (x, z)V (z)GP (z, y) dz < ∞.
(3.1)
(ii) Define the sequence ( (n) GΩ P (x, y)
:=
Ω
G (x, y) RP Ω (n−1) Ω dz Ω GP (x, z)V (z)GP (z, y) 11
if n = 0 if n ≥ 1.
P∞
Then for every 0 < ² < λ0 the Neumann series verges to GΩ P −²V (x, y) in the compact open topology.
n=0 ²
n GΩ (x, y)(n) P
con-
Proof of (i) and (ii): Let 0 < ² < δ < λ0 . Then it follows from the proof of Theorem 2.1 in [28] that GΩ P −δV (x, y) satisfies the resolvent equation Z Ω GΩ P −δV (x, y) = GP (x, y) + δ
Ω
Ω GΩ P −δV (x, z)V (z)GP (z, y) dz
(3.2)
and therefore, Z Ω
Z Ω GΩ P (x, z)V (z)GP (z, y) dz ≤
Ω
Ω GΩ P −δV (x, z)V (z)GP (z, y) dz ≤
δ −1 GΩ P −δV (x, y).
(3.3)
(n)
Thus, (3.1) is proved and by induction, GΩ is well defined for n ≥ 0 P (x, y) and satisfies (n) GΩ ≤ δ −n GΩ (3.4) P (x, y) P −δV (x, y). P
(n)
n Ω Hence, for every 0 < ² < δ the Neumann series ∞ converges n=0 ² GP (x, y) Ω to GP −²V (x, y) in the compact open topology and consequently, also in 2,α Cloc (Ω × Ω \ {(x, x) | x ∈ Ω}).
Remark 3.2 Condition (3.1) is not sufficient for λ0 (V, P, Ω) > 0 (see example 8.7). Lemma 3.3 Suppose that V is a H-bounded perturbation of a subcritical operator P in Ω. ◦ Then S 6= ∅. In particular, λ0 (V, P, Ω) > 0. Furthermore, there exists 0 < ²0 < λ0 such that for any u ∈ CP (Ω) and |²| < ²0 there exists u² ∈ CP −²V (Ω) which satisfies C²−1 u ≤ u² ≤ C² u ,
(3.5)
where C² is a positive constant. In particular, the mapping u 7→ u² is one to one from the set of the extreme points of KP (Ω) into CP −²V (Ω). Assume further that V is a nonzero, nonnegative function. Then the sets Iu := {² ∈ IR
|
∃ u² ∈ CP −²V (Ω) s.t. u² ≈ u, and
Z
Ω
GΩ P −²V (x, z)V 12
(z)u² (z) dz ≤ C² u²
¾
(3.6)
and Ju := {² < 0 | ∃ u² ∈ CP −²V (Ω) s.t. u² ≈ u, } are open. Proof: Define the sequence (
u
(n)
u(x) if n = 0 Ω (n−1) (z) dz if n ≥ 1. Ω GP (x, z)V (z)u
R
(x) :=
By induction, u(n) (x) are well defined and |u(n) (x)| ≤ C n u(x) for all x ∈ Ω and n ≥ 0. Moreover, by dominated convergence, P u(n) = V u(n−1) if n ≥ 1. P n (n) converges to a Consequently, for any |²| < (2C)−1 the series ∞ n=0 ² u positive solution u² of the equation (P − ²V )u = 0 in Ω which satisfies C²−1 u ≤ u² ≤ C² u, where C² is a positive constant. Note that for 0 < ² < C −1 the above series still converges to a positive solution u² of the equation (P − ²V )u = 0 in Ω. Thus, λ0 (V, P, Ω) > C −1 > 0 . Clearly, u² satisfies the integral equation Z
u² (x) = u(x) + ²
Ω
GΩ P (x, z)V (z)u² (z) dz.
(3.7)
Suppose now that V is a nonzero, nonnegative function. Lemma 3.1 implies that for sufficiently small ² > 0 the function u² satisfies the equation Z
u² (x) = u(x) + ²
Ω
GΩ P −²V (x, z)V (z)u(z) dz.
(3.8)
Recall that for sufficiently small ² > 0 we have that u² ≈ u. Hence, for sufficiently small ² > 0 there exists a positive constant c² such that Z Ω
GΩ P −²V (x, z)V (z)u² (z) dz ≤ c² u² (x).
(3.9)
On the other hand, if ² < 0 then by [25, Lemma 4.2], Z
−²
Ω
GΩ P −²V (x, z)V (z)u(z) dz ≤ u(x)
(3.10)
for every u ∈ CP (Ω). Hence, if u² ≈ u then there exists a positive constant c² such that Z Ω
GΩ P −²V (x, z)V (z)u² (z) dz ≤ c² u² (x), 13
(3.11)
also for ² < 0 sufficiently small. It follows that the set Iu is open. Now, suppose that ² < 0 and u² ∈ CP −²V (Ω), u² ≈ u. Then by (3.10), R Ω G Ω P −²V (x, z)u² (z) dz ≤ C² u² (x) and it follows that that Ju is open. It will be interesting to know if, under the conditions of the Lemma and V ≥ 0, the set Iu , or the set Ju is an open half line. Recall that the set (resp. sE+ ) E+ is an open half line provided that V ≥ 0 is a G-(resp. semi) bounded perturbation. Lemma 3.3 should be compared with the following proposition Proposition 3.4 Suppose that V is a nonzero, nonnegative function such that V is a H-integrable perturbation of a subcritical operator P in Ω and let u ∈ CP (Ω). Then for any ² < 0 there exists u² ∈ CP −²V (Ω) which satisfies 0 < u² ≤ u and the resolvent equation Z
u² (x) = u(x) + ²
Ω
GΩ P −²V (x, z)V (z)u(z) dz.
(3.12)
Proof: The proof is similar to the proof of [14, Theorem 2.1]. Let uk be the solution of the Dirichlet problem (P − ²V )v = 0 v = u
in Ωk , on ∂Ωk .
By the generalized maximum principle, 0 < uk ≤ u and {uk } is a decreasing sequence. Moreover, Z
uk (x) = u(x) + ²
Ω
k GΩ P −²V (x, z)V (z)u(z) dz.
(3.13)
Therefore, the sequence {uk } converges to 0 ≤ u² ≤ u which is a nonnegative solution of the equation (P − ²V )u = 0 in Ω and by the Lebesgue monotone convergence theorem, u² satisfies Equation (3.12). It remains to prove that u² > 0. Clearly, Z
w(x) := −²
Ω
GΩ P (x, z)V (z)u(z) dz ≥ u(x) − u² (x).
(3.14)
Since w is the minimal positive solution of the equation P w = −²V u in Ω it follows that inf x∈Ω w(x)/u(x) = 0. Hence, (3.14) implies that supx∈Ω u² (x)/u(x) ≥ 1 and u² > 0.
14
Remarks 3.5 (i) There exists a perturbation W which is H-semibounded and λ0 (W, P, Ω) = 0 (see Example 8.7). On the other hand, there exists a perturbation W which is not H-integrable such that λ0 (W, P, Ω) > 0 (see (iv) below and Example 8.5). (ii) A necessary and sufficient condition for λ0 (V, P, Ω) > 0 for the case V ≥ 0 is given by the variational formula ½
(λ0 )
−1
¾
Z
= inf sup (φ(x)) φ>0 x∈Ω
−1 Ω
GΩ P (x, z)V
(z)φ(z) dz
(3.15)
(see Theorem 4.4 in [25]). Note that the first part of Lemma 3.3 clearly follows from (3.15). Furthermore, it follows from (3.15) that if there exists a positive constant C such that Z Ω
GΩ P (x, z)V (z) dz ≤ C
(3.16)
for all x ∈ Ω, then 0 < C −1 < λ0 . (iii) Another necessary and sufficient condition for λ0 (V, P, Ω) > 0, where V ≥ 0 and P is a formal self-adjoint subcritical operator in Ω is given in [25, Theorem 5.1]. Namely, λ0 (V, P, Ω) > 0 if and only if the Birman-Schwinger operator T is bounded on L2 (Ω). Recall that the BirmanSchwinger operator is the integral operator whose kernel t is given by 1/2 t(x, y) = V 1/2 (x)GΩ (y) . P (x, y)V
Moreover, kT k = λ−1 0 . Consider the quadratic form induced by T and the test functions V 1/2 (x)χK (x), where K ⊂⊂ Ω and χK is the characteristic function of the set K. It follows that λ0 (V, P, Ω) > 0 if and only if (R R
sup K⊂⊂Ω
K K
)
GΩ y)V (x)V (y) dx dy P (x, R 0 see [20]. (iv) Let P = −∆, Ω = IRd , d ≥ 3 and let 0 ≤ W ≤ C(1 + |x|)−2 such that R 2−d dy = ∞. So, W is not a H-integrable perturbation and IRd W (y)(1 + |y|) P − ²W is a Fuchsian type elliptic operator [30]. Then any u² ∈ CP −²W (IRd ) with 0 < ² ≤ λ0 satisfies limx→∞ u² (x) = 0 [35, Theorem 5.1] and u² does not satisfy (3.5). The following lemma gives a necessary condition, related to H-boundedness, for λ0 > 0 in the case where V ≥ 0. 15
Lemma 3.6 Let V be a nonzero, nonnegative function and assume that λ0 (V, P, Ω) > 0. Then there exists a positive supersolution v of the equation P u = 0 in Ω and C > 0 such that Z Ω
GΩ P (x, z)V (z)v(z) dz ≤ C . v(x)
(3.17)
for all x ∈ Ω. Proof: It follows from the variational formula (3.15) that if λ0 > 0 , then there exist a positive function u and C > 0 such that Z Ω
Set v(x) :=
4
R
Ω Ω GP (x, z)V
GΩ P (x, z)V (z)u(z) dz ≤ C . u(x)
(3.18)
(z)u(z) dz. Then v is the desired supersolution.
A refined maximum principle, minimal growth and bounded perturbation
The purpose of this section is to extend the classical generalized maximum principle to a refined (generalized) maximum principle which holds in any domain. Roughly speaking, our maximum principle is of Phragm´en-Lindel¨ of type and states that if P u ≥ 0 in Ω, P is subcritical in Ω and u is “bounded” near infinity in Ω, then u ≥ 0 (see Theorem 4.4). As a consequence, we prove an existence and uniqueness theorem for solutions of the nonhomogeneous equation (see Theorem 4.6). We first prove some simple lemmas. Lemma 4.1 Let P be a critical or subcritical operator in Ω and let φ be a positive solution of the equation P u = 0 in Ω∗1 which has a minimal growth in a neighborhood of infinity in Ω. Suppose that v ∈ C(Ω∗2 ) is a solution of the equation P u = 0 in Ω∗2 such that v = 0 in ∂Ω2 . Assume that there exists a positive constant C such that |v(x)| ≤ Cφ(x) for all x ∈ Ω∗2 . Then v = 0. Proof: Assume that v(x1 ) > 0 for some x1 in Ω∗2 . Define ²0 = sup{² > 0 | φ(x) − ²v(x) ≥ 0 for all x ∈ Ω∗2 } . Clearly, 0 < ²0 < ∞. Let w = φ − ²0 v. Since w is positive on ∂Ω2 it follows that w is a positive solution of the equation P u = 0 in Ω∗2 . But, φ is a 16
positive solution of the operator P in Ω∗1 which has minimal growth in a neighborhood of infinity in Ω and therefore, there exists a positive constant ²0 0 < γ < 1 such that w(x) ≥ γφ(x) for all x ∈ Ω∗2 . Hence, φ − (1−γ) v ≥ 0 in ∗ Ω2 which contradicts the maximality of ²0 . Lemma 4.2 Let P and φ be as in Lemma 4.1. Suppose that v ∈ C(Ω∗2 ) is a solution of the equation P u = 0 in Ω∗2 such that v ≥ 0, v 6= 0 on ∂Ω2 . Assume that there exists a positive constant C such that |v(x)| ≤ Cφ(x) for all x ∈ Ω∗2 . Then v > 0 in Ω∗2 . Proof: Denote by wk , k > 2 the solutions of the following Dirichlet problems Pu = 0
in Ωk,2 ,
u = v
on ∂Ω2 ,
u = 0
on ∂Ωk,2 \ ∂Ω2 .
By the generalized maximum principle, the sequence {wk }k>2 is a nondecreasing locally bounded sequence of nonnegative solutions which converges to a function w ≤ C1 φ. Recall that v 6= 0 on ∂Ω2 . Therefore, w is a positive solution of the equation P u = 0 of minimal growth in a neighborhood of infinity in Ω (see [30, Lemma 3.1]). Consequently, the function w−v satisfies all the conditions of Lemma 4.1 and therefore, v = w > 0 in Ω∗2 . Lemma 4.3 Let P and φ be as in Lemma 4.1. Let f ∈ C α (Ω∗2 ) be a nonnegative function. Suppose that v ∈ C(Ω∗2 ) is a solution of the equation P u = f in Ω∗2 such that v ≥ 0 on ∂Ω2 . Assume further that there exists a positive constant C such that for all x ∈ Ω∗2 .
|v(x)| ≤ Cφ(x)
(4.1)
Then v ≥ 0. Moreover, Z
v(x) = h(x) +
Ω∗
Ω∗2
GP 2 (x, y)f (y)dy,
(4.2)
where h ∈ C(Ω∗2 ) is either zero, or a positive solution of the equation P u = 0 in Ω∗2 of minimal growth in a neighborhood of infinity in Ω which satisfies the boundary condition h = v on ∂Ω2 . In particular, Z Ω∗2
Ω∗
GP 2 (x, y)f (y)dy < Cφ(x). 17
Proof: Since |v| = v + + v − it follows that v ± ≤ Cφ. Let wk,± , k > 2 be the nonnegative solutions of the following Dirichlet problems Pu = f±
in Ωk,2 ,
u = v
±
on ∂Ω2 ,
u = v
±
on ∂Ωk,2 \ ∂Ω2 .
By the generalized maximum principle, 0 ≤ wk,− ≤ Cφ and wk,+ ≥ 0. Since wk,+ − wk,− = v
(4.3)
it follows that wk,+ ≤ 2Cφ. By a standard elliptic argument, the sequence {wk,− }k>2 contains a subsequence {wkl ,− } converging to a function w− which is a nonnegative solution of the equation P u = 0 in Ω∗2 and satisfies the boundary condition w− = 0 on ∂Ω2 . Moreover, w− ≤ Cφ. Lemma 4.1 implies that w− = 0. Since this is true for any converging subsequence, Equation (4.3) implies now that the sequence {wk,+ } converges to v. Hence, v ≥ 0. Note that Z
wk,+ (x) = hk (x) +
Ω
Ωk,2
GP k,2 (x, y)f (y)dy,
where hk satisfies Pu = 0
in Ωk,2 ,
u = v
on ∂Ωk,2 .
Since 0 ≤ v ≤ Cφ, we deduce that the sequence {hk } converges either to zero, or to a positive solution of the equation P u = 0 in Ω∗2 of minimal growth in a neighborhood of infinity in Ω which satisfies the boundary condition h = v on ∂Ω2 . R Ω On the other hand, the sequence { Ωk,2 GP k,2 (x, y)f (y)dy} is a nondecreasing locally bounded sequence of nonnegative solutions of the equation P u = f . By the Lebesgue monotone convergence theorem, this sequence R Ω∗ converges to Ω∗ GP 2 (x, y)f (y)dy. 2
We now establish the refined maximum principle. Theorem 4.4 Let P be a subcritical operator in Ω and let φ be a positive solution of the equation P u = 0 in Ω∗1 which has a minimal growth in a neighborhood of infinity in Ω. Suppose that P v = f ≥ 0 in Ω, where f ∈ C α (Ω) and |v| ≤ Cφ in Ω∗1 . Then v ≥ 0 in Ω. 18
Proof: Suppose that there exists x1 ∈ Ω such that v(x1 ) > 0. Then there exists a ball B² = B(x1 , ²) ⊂ Ω such that v > 0 in B² . Lemma 4.3 implies that v ≥ 0 in B²∗ := Ω \ B² and therefore, v > 0 in Ω. So, we may assume that that v ≤ 0 in Ω and suppose that there exists x1 ∈ Ω such that v(x1 ) < 0. Then there exists a ball B² = B(x1 , ²) ⊂ Ω such that v < 0 in B² . Without loss of generality, we may assume that B² ⊂ Ω1 Let u²,k be the solution of the following Dirichlet problem Pu = 0
in Ωk \ B² ,
u = 0
on ∂B² ,
u = v
on ∂Ωk .
Since 0 ≤ −u²,k ≤ CGΩ P (·, x1 ) it follows from Lemma 4.1 that limk→∞ u²,k = 0. Consider also the Dirichlet problem Pu = 0
in Ωk \ B² ,
u = v
on ∂B² ,
u = 0
on ∂Ωk ,
and denote its negative solution by v²,k . Set, v² := limk→∞ v²,k . Clearly, 0 < −v² ≤ C² GΩ P (·, x1 ). Using the well known asymptotic behavior of GΩ (·, x ) near the pole x1 it follows that there exists C > 0 such that for 1 P ² > 0 small enough 0 < −v² ≤ C²d−2 GΩ P (·, x1 ) if d ≥ 3 and 0 < −v² ≤ −1 Ω −C(log ²) GP (·, x1 ) if d = 2. Therefore, lim²→0 v² = 0. Finally, let w²,k be the solution of the Dirichlet problem Pu = f
in Ωk \ B² ,
u = 0
on ∂B² ,
u = 0
on ∂Ωk .
Then w²,k ≥ 0. On the other hand, v = u²,k + v²,k + w²,k . Letting first k → ∞ and then ² → 0 in Equation (4.4), we obtain that µ
v = lim
²→0
¶
lim w²,k
k→∞
≥0
contradicting the hypothesis that v ≤ 0, v 6= 0. In the critical case we have 19
(4.4)
Proposition 4.5 Let P be a critical operator in Ω and let φ0 be a ground state of the operator P in Ω. Suppose that P u ≥ 0 in Ω and that for some C > O, u ≥ −Cφ0 in Ω∗1 . Then u = C1 φ0 , where C1 is a real constant. Proof: Choose a positive constant C such that the function v = u + Cφ0 is positive in Ω. It is well known that in the critical case any positive supersolution of the equation P u = 0 in Ω is a ground state. Thus, v = γφ0 and hence u = C1 φ0 . In the following theorem we prove existence, uniqueness and integral representation for solutions of the nonhomogeneous equation (compare with [5]). Theorem 4.6 Let P be a subcritical operator in Ω and let φ be a positive solution of the equation P u = 0 in Ω∗1 which has a minimal growth in a neighborhood of infinity in Ω. (i) Let f ∈ C α (Ω), 0 < α ≤ 1 be a function such that Z Ω
GΩ P (x, y)|f (y)|dy ≤ Cφ(x)
(4.5)
for all x ∈ Ω∗2 . Then there exists a unique solution u of the equation P u = f in Ω which satisfies |u| ≤ C1 φ for some positive constant C1 . Moreover, Z
u(x) =
Ω
GΩ P (x, y)f (y)dy.
(ii) Suppose that f ∈ C α (Ω), 0 < α ≤ 1 and f ≥ 0. Then f satisfies estimate (4.5) if and only if there exists a solution u of the equation P u = f in Ω which satisfies |u| ≤ Cφ for some positive constant C. In this case, u is the minimal nonnegative solution of the equation P v = f in Ω. Proof: (i) Let
Z
un (x) =
Ωn
n GΩ P (x, y)f (y)dy.
By the Lebesgue dominated convergence theorem and standard elliptic argument, Z u(x) = lim un (x) = n→∞
Ω
GΩ P (x, y)f (y)dy
is a solution of P u = f in Ω which satisfies |u| ≤ Cφ. The uniqueness follows from Theorem 4.4.
20
(ii) Suppose that u is a solution of the equation P u = f ≥ 0 in Ω which satisfies |u| ≤ Cφ for some positive constant C. Theorem 4.4 implies that u ≥ 0. Consider again the sequence Z
un (x) =
Ωn
n GΩ P (x, y)f (y)dy.
Clearly, 0 ≤ un ≤ u ≤ Cφ in Ωn and therefore, Z
Z
0 ≤ w(x) :=
Ω
GΩ P (x, y)f (y)dy = lim
n→∞ Ωn
n GΩ P (x, y)f (y)dy ≤ u(x) ≤ Cφ(x).
Part (i) implies now that w = u ≤ Cφ. We now reformulate Theorem 4.6 in terms of G-semibounded perturbations. Corollary 4.7 Let P and φ be as in Theorem 4.6. Assume further that φ ∈ C(Ω) is a positive function. Let f ∈ C α (Ω), 0 < α ≤ 1 be a function such that f /φ is a G-semibounded perturbation of the operator P ∗ in Ω. Then there exists a unique solution u of the equation P u = f in Ω which satisfies |u| ≤ Cφ for some positive constant C. Moreover, if f ≥ 0 and there exists a solution u of the equation P u = f in Ω which satisfies |u| ≤ Cφ, then f /φ is a G-semibounded perturbation. Remarks 4.8 (i) The results of this section continuous to hold if we suppose that f ∈ C(Ω) and we consider strong solutions. This follows by a standard approximation argument. We shall use this observation in the next section. (ii) For related maximum principles see [4, 8, 34].
5
Anti-maximum principle and semismall perturbations
In Section 4, we proved a refined maximum principle. Roughly speaking, it states that if (P − λ)uλ ≥ 0 in Ω, where λ ≤ λ0 , λ 6∈ S0 and uλ is “bounded” near infinity in Ω, then u ≥ 0. The main aim of this section is to prove a refined anti-maximum principle for solutions uλ of the equation (P − λ)u = f ≥ 0
21
in Ω,
λ0 < λ.
(5.1)
The anti-maximum principle (Theorem 5.3) reads roughly that under some “smallness” conditions on f and uλ in a neighborhood of infinity in Ω, there exists ² > 0 which may depend on f such that uλ < 0 for all λ ∈ (λ0 , λ0 + ²). Throughout this section we assume that P is subcritical in Ω and 1 is a semismall perturbation of P ∗ in Ω. Denote by λ0 = λ0 (1, P, Ω) and φ0 the generalized principal eigenvalue and the corresponding normalized ground state of the operator P in Ω. So, (P −λ0 )φ0 = 0 in Ω, λ0 > 0 and φ0 (x0 ) = 1. We denote by B the ordered Banach space B = {u ∈ C(Ω) | |u(x)| ≤ cφ0 (x) for some c > 0 and all x ∈ Ω} equipped with the norm kukB = inf{c > 0 | |u(x)| ≤ cφ0 (x) ∀ x ∈ Ω}. The ordering on B is the natural pointwise ordering of functions. Theorem 5.1 Let P be a subcritical operator in Ω and assume that 1 is a semismall perturbation of the operator P ∗ . Consider the integral operator Z
Gf (x) =
Ω
GΩ P (x, y)f (y)dy.
Then G is a compact operator on B. Denote the spectrum of G by σ(G). Then λ−1 0 ∈ σ(G) is the principal eigenvalue of G with a principal eigenfunction φ0 . Proof: Let {fn } be a bounded sequence in B. Since 1 is a semismall perturbation of P ∗ , it follows that the function un (x) := Gfn (x) is well defined and satisfies Z |un (x)| ≤ GΩ P (x, y)|fn (y)|dy ≤ Cφ0 (x), Ω
where C is a positive constant independent of n. Moreover, by Theorem 4.6 (and Remark 4.8) un is the unique function in B which is a solution of the equation P u = fn in Ω. By standard elliptic argument, it follows that the sequence un admits a subsequence which converge in the compact open topology to a function u. For any given ² > 0 there exists K such that for any k ≥ K and n, m ∈ IN Z
Z Ω∗k
GΩ P (x, y)|fn (y) − fm (y)|dy ≤ 2C
22
Ω∗k
GΩ P (x, y)φ0 (y)dy < ²φ0 (x) (5.2)
for all x ∈ Ω∗k . On the other hand, by the local uniform convergence, there exists N² ∈ IN such that |un − um | ≤ ²φ0 in ΩK for all n, m > N² . Consequently, by the generalized maximum principle, Estimate (5.2) and Lemma 4.3, we infer that |un − um | ≤ 2²φ0 in Ω∗K for all n, m > N² . Thus, un → u in B. The last statement of the theorem follows from the Krein-Rutman theorem and Theorem 1.5 in [24]. Remark 5.2 Note that if P is subcritical in Ω and 1 is a G-semibounded perturbation of P ∗ , then G is a bounded operator on B. Our main result of this section is the following anti-maximum principle which holds for semismall perturbations. Theorem 5.3 Let P be a subcritical operator in Ω and assume that 1 is a semismall perturbation of the operator P ∗ in Ω. Denote by λ0 = λ0 (1, P, Ω) and φ0 the generalized principal eigenvalue and the corresponding normalized ground state of the operator P . Let f ∈ B ∩C α (Ω) be a nonzero, nonnegative function. Then there exists ² > 0 (which may depend on f ) such that for every λ ∈ (λ0 , λ0 + ²) if (P − λ)u = f and u ∈ B, then u ≤ −C2 φ0 < 0, where C2 is a positive constant depending on f and λ. Proof: By the compactness of G, there exists ²0 > 0 such that ((λ0 + ²0 )−1 , (λ0 )−1 ) ∩ σ(G) = ∅. Therefore, it is enough to show that there exists ² ≤ ²0 such that the equation (P − λ)u = f admits a negative solution vλ ∈ B, |vλ | ≈ φ0 for every λ ∈ (λ0 , λ0 + ²) . Set wλ = −vλ . Then we need to prove the existence of a positive solution of the equation (P + f /wλ − λ)wλ = 0 in Ω such that wλ ≈ φ0 . For λ0 < λ ≤ λ0 + 1 consider the set Bλ := {u | u(x0 ) = 1, u ≈ φ0 , and ∃ V ∈ C α (Ω), 0 ≤ V ≤ 1, s. t. (P − λ + V )u = 0} . (5.3) Claim: Bλ is a nonempty convex compact set in the compact open topology. Proof of the claim: Bλ is nonempty: Set Vλ = λ − λ0 . Clearly, 0 ≤ Vλ ≤ 1 and φ0 satisfies the equation (P − λ + Vλ )φ0 = 0 and therefore, φ0 ∈ Bλ . Bλ is convex: Let ui ∈ Bλ , satisfy the equations (P − λ + Vi )ui = 0 in Ω, where 0 ≤ Vi ≤ 1, i = 1, 2, and let 0 < t < 1. Denote ut := tu1 +(1−t)u2 and q := tu1 /ut . Note that 0 < q < 1. It can be easily checked that ut satisfies the equation (P − λ + Vt )u = 0, where 0 ≤ Vt := qV1 + (1 − q)V2 ≤ 1. 23
Bλ is compact: Since 1 is a weak perturbation there exists k ≥ 0 such that the operator P − λ0 − 1 is subcritical in Ω∗k . Fix y1 ∈ Ωk,k+1 . By the Harnack inequality and the generalized maximum principle, there exists a positive constant C such that for any u ∈ Bλ Ω∗
Ω∗
C −1 GP k−λ0 +1 (x, y1 ) ≤ u(x) ≤ CGP k−λ0 −1 (x, y1 )
in Ω∗k+1 .
Therefore, there exists C1 > 0 such that C1−1 φ0 ≤ u ≤ C1 φ0
in Ω
(5.4)
for every u ∈ Bλ and λ0 < λ ≤ λ0 + 1. Now, let un ∈ Bλ . By the Harnack inequality, {un } is locally bounded. Using the Schauder estimates it follows that one can extract a subsequence 2,α (Ω) {unk } which converges in the compact open topology and also in Cloc −(P −λ)u
nk to a function u > 0, u(x0 ) = 1. By (5.4), u ≈ φ0 . Since Vnk = it unk α follows that Vnk → V in Cloc (Ω), where 0 ≤ V ≤ 1. Hence, u satisfies the equation (P − λ + V )u = 0 in Ω and u ∈ Bλ . Thus, the claim is proved.
Now, fix f ∈ B ∩ C α (Ω), f ≥ 0. By (5.4), V1 := C1−1 f /φ0 ≤ f /u ≤ V2 := C1 f /φ0
(5.5)
for every u ∈ Bλ and λ0 < λ ≤ λ0 + 1. Note that for every nonzero, nonnegative, bounded function V and any t > 0 there exists a unique number λV (t) > λ0 such that the operator P + tV − λV (t) is critical in Ω. We recall some known properties of the principal eigencurve {(t, λV (t)) | t > 0}. The function, λV (t) is an increasing concave function of t such that λV (t) & λ0 as t & 0 [28]. Furthermore, if V1 ≤ V ≤ V2 , then λV1 (t) ≤ λV (t) ≤ λV2 (t). (5.6) It follows that there exists λ0 < λ1 ≤ λ0 + 1 such that for every u ∈ Bλ with λ0 < λ ≤ λ1 there exist a unique positive number tλ and a unique normalized ground state v ∈ B of the equation (P − λ + tλ f /u)v = 0. We denote Tλ u := v. We claim that there exists ² > 0 (which may depend on f ) such that for every λ ∈ (λ0 , λ0 + ²) we have Tλ : Bλ −→ Bλ . 24
Using (5.5) and (5.6), we infer that tλ & 0 as λ & λ0 uniformly in {λ0 0 such that tλ f /u ≤ 1 for every λ ∈ (λ0 , λ0 +²) and u ∈ Bλ . Thus, Tλ : Bλ −→ Bλ . Clearly, Tλ is continuous. Fix λ ∈ (λ0 , λ0 + ²). Applying the Schauder-Tychonoff fixed point theorem to the operator Tλ , we conclude that there exist tλ > 0 and uλ ∈ Bλ such that uλ is a positive solution of the the equation (P −λ+tλ f /uλ )uλ = 0 in Ω. Consequently, the negative function wλ := −uλ /tλ solves the equation S
(P − λ)u = f and we obtain the desired solution. Remark 5.4 Let u be the negative solution of the equation (P − λ)u = f ≥ 0 guaranteed by Theorem 5.3. Suppose that there exists a portion Γ of ∂Ω such that Γ and the coefficients of the operator P − λ − f /u satisfy the conditions of Hopf’s lemma on Γ. Recall that v = −u is a positive solution of the equation (P − λ + f /v)v = 0 of minimal growth in a neighborhood of infinity in Ω. It follows that v = 0 on Γ [30, Lemma 3.2]. Thus, the classical Hopf lemma implies that ∂u/∂ν > 0 on Γ, where ∂u/∂ν is the outer normal derivative of u .
6
Small perturbation and the behavior of eigenfunctions
In this section we investigate the asymptotic behavior of a solution w of the equation (P + V )u = 0 in a neighborhood of infinity in Ω. We assume that P is a subcritical operator, V is a semismall perturbation of the operator P ∗ in Ω and |w| ≤ Cφ, where φ is a positive solution of minimal growth in a neighborhood of infinity in Ω of the equation P u = 0. We prove that w/φ has a continuous extension up to the Martin boundary ∂PM∗ Ω. In particular, we obtain some new results concerning the asymptotic behavior of the Dirichlet eigenfunctions of a Schr¨ odinger operator H, when 1 is a semismall perturbation of H. The developments of this section will rest on the following simple lemma. Lemma 6.1 Let P be a subcritical operator in Ω and let V be a semismall perturbation of the operator P ∗ . Assume that P + V is critical in Ω and has a continuous extension denote by φ0 its ground state. Then Ωφ0 (x) GP (x, x0 ) ψ0 up to the Martin boundary ∂PM∗ Ω. Moreover, ψ0 is positive on ∂PM∗ Ω. 25
Proof: Fix ξ ∈ ∂PM∗ Ω and let {xn } ⊂ Ω, xn → ξ in the Martin space ΩM P∗. φ0 (xn ) exists and depends only on We need to prove that the limn→∞ Ω GP (xn , x0 ) ξ. By [24], the ground state φ0 satisfies the equation Z
φ0 (x) =
Ω
GΩ P (x, z)V (z)φ0 (z) dz .
(6.1)
Substitute x = xn in Equation (6.1) and divide the equation by GΩ P (xn , x0 ). ∗ and V is a semismall perturbation it follows that (·, x ) in Ω Since φ0 ≈ GΩ 0 1 P the sequence ( )∞ GΩ P (xn , z)V (z)φ0 (z) GΩ P (xn , x0 ) n=1 is uniformly integrable. Thus, φ0 (xn ) lim = n→∞ GΩ (xn , x0 ) P
Z
GΩ P (xn , z)V (z)φ0 (z) dz = n→∞ GΩ Ω P (xn , x0 ) lim
Z
Ω
KPΩ∗ (z, ξ)V (z)φ0 (z) dz
(6.2)
∗ and the limit exists. Since φ0 ≈ GΩ P (·, x0 ) in Ω1 the limit is positive.
Lemma 6.2 (i) Let V be a semismall perturbation of a subcritical operator P ∗ in Ω. Let u and v be positive solutions of minimal growth in a neighborhood of infinity in Ω of the equations P u = 0 and (P + V )v = 0 respectively. Then v/u has a positive continuous extension up to the Martin boundary ∂PM∗ Ω. (ii) Suppose further that w is a solution of the equation (P + V )u = 0 in a neighborhood of infinity in Ω and there exists a positive constant C such that in this neighborhood |w| ≤ Cu. Then w/u has a continuous extension up to the Martin boundary ∂PM∗ Ω. Proof: (i) We may assume that u and v are positive solutions of the corresponding equations in Ω∗1 . Any positive solution h of the equation P u = 0 in Ω∗1 of minimal growth in a neighborhood of infinity in Ω can be extended to ˜ which is a ground state with eigenvalue zero of a critical a positive function h operator (P + W0 ) in Ω, where W0 ∈ C0α (Ω) (see [26, Theorem 4.6]). So, we may assume that u and v are ground states of the equations (P + W0 )u = 0 and (P + V1 )u = 0 respectively, where V1 = V + V0 and W0 , V0 ∈ C0α (Ω). 26
Clearly, W0 and V1 are semismall perturbations of P ∗ in Ω. By Lemma 6.1, u(x) and Ω v(x) have the appropriate continuous extension and GΩ (x, x ) GP (x, x0 ) 0 P therefore also v/u. (ii) Recall that if φ and ψ are positive solutions of minimal growth in a neighborhood of infinity in Ω of the equations P u = 0 and (P + V )u = 0 respectively and V is a semismall perturbation of P ∗ , then φ ≈ ψ in some neighborhood of infinity in Ω. Without loss of generality, we may assume that w is a real function. Using Lemma 4.1 it is easily seen that w = w+ −w− , where w+ (resp. w− ) is either a positive solution of minimal growth in a neighborhood of infinity in Ω of the equation (P + V )u = 0, or identically zero. By part (i), the functions w± /u have continuous extensions up to the Martin boundary ∂PM∗ Ω and therefore also w/u. The next theorem deals with the asymptotic behavior of eigenfunctions in the self adjoint case. Theorem 6.3 (i) Let P be a second order elliptic operator defined on a domain Ω and let V ∈ C α (Ω) be a positive function. Suppose that P is a symmetric, nonnegative operator on L2 (Ω, V (x)dx) with a domain C0∞ (Ω) and assume that V is a weak perturbation of the operator P in Ω. Assume that P admits a (Dirichlet) selfadjoint realization P˜ on L2 (Ω, V (x)dx). Then P˜ has a purely discrete nonnegative spectrum (that is, σess (P˜ ) = ∅). Moreover, σ(P˜ ) = σdis (P˜ ) = σp (P˜ ) = {λn }∞ n=0 , where limn→∞ λn = ∞. In particular, if λ0 > 0, then the natural embedding E : H −→ L2 (Ω, V (x)dx) is compact, where H is the completion of C0∞ (Ω) with respect to the inner product induced by the corresponding quadratic form. (ii) Assume further that P is subcritical and V is a semismall perturbation of the operator P in Ω. Let {φn }∞ n=0 be the set of the corresponding eigenfunctions (P φn = λn V φn ). Then for every n ≥ 1 there exists a positive constant Cn such that |φn (x)| ≤ Cn φ0 (x) . (6.3) Moreover, the function φn /φ0 has a continuous extension ψn up to the Martin boundary ∂PM Ω and ψn satisfies Z
−1
ψn (ξ) = (ψ0 (ξ)) R
λn
Ω
KPΩ (z, ξ)V (z)φn (z) dz =
λn Ω KPΩ (z, ξ)V (z)φn (z) dz R λ0 Ω KPΩ (z, ξ)V (z)φ0 (z) dz 27
(6.4)
for every ξ ∈ ∂PM Ω, where ψ0 is the corresponding continuous extension of φ0 (x) guaranteed by Lemma 6.1. GΩ P (x, x0 ) Proof: (i) Recall that in the self-adjoint case λ0 = inf σ(P˜ ), λ∞ = inf σess (P˜ ). Since V is a weak perturbation of the operator P in Ω it follows that for every λ > 0 there exists n ∈ IN such that P − λVn∗ is subcritical in Ω. Hence, λ∞ = ∞ and therefore, σess (P˜ ) is empty. Note that under the above assumptions λ0 > 0 if and only if P is subcritical in Ω. (ii) Without loss of generality, we may assume that φn is a real eigenfunction of the operator P˜ with an eigenvalue λn . There exists k0 ∈ IN such that λ0 (V, P, Ω∗k0 ) > λn . For every k > k0 denote by φn,k,± the solutions of the following Dirichlet problems (P − λn )u = 0 u =
φ± n
u = 0
in Ωk,k0 , on ∂Ωk0 , on ∂Ωk .
The sequence {φn,k,+ }k>k0 (resp. {φn,k,− }k>k0 ) is an increasing sequence of nonnegative solutions which converges to un,+ (resp. un,− ), where un,+ (resp. un,− ) is either a positive solution of the equation (P − λn V )u = 0 of − minimal growth in a neighborhood of infinity in Ω if φ+ n 6= 0 (resp. φn 6= 0) − on ∂Ωk0 , or un,+ = 0 (resp. un,− = 0) in case that φ+ n = 0 (resp. φn = 0) on ∂Ωk0 . By Lemma 6.2, un,± /φ0 have continuous extensions up to the Martin boundary ∂PM Ω. Hence (un,+ − un,− )/φ0 has such an extension. Moreover, |un,+ − un,− | ≤ Cφ0 for some C > 0. The conclusion of Theorem 6.3 will thus follow if we can show that v := φn − (un,+ − un,− ) = 0 on Ω∗k0 . We may assume that un,± is positive. Since un,± and φ0 are positive solutions of minimal growth in a neighborhood of infinity in Ω of the equations (P − λn V )u = 0 and (P − λ0 V )u = 0 respectively, and V is a semismall perturbation of P in Ω it follows that there exist positive constants such that un,± ≤ C± φ0 in Ω∗k0 +1 . In particular, un,± ∈ L2 (Ω∗k0 , V (x)dx). It follows that v ∈ L2 (Ω∗k0 , V dx) ∩ H(Ω∗k0 ). Since λ0 (V, P, Ω∗k0 ) > λn and v is a solution of the equation (P − λn V )u = 0 in Ω∗k0 , we infer that v = 0 and φn = un,+ − un,− on Ω∗k0 . It remains to prove (6.4). By Theorem 4.6, the eigenfunction φn satisfies the equation Z
φn (x) = λn
Ω
GΩ P (x, z)V (z)φn (z) dz. 28
(6.5)
Divide Equation (6.5) by φ0 and take the limit x → ξ to obtain (6.4). Remarks 6.4 (i) Let P , V and φ0 be as in Theorem 6.3. Suppose that w ∈ L2 (Ω∗k , V (x)dx), k ≥ 1 is a solution of the equation (P + W )u = 0 in Ω∗k , where W is a semismall perturbation of the operator P in Ω. Then by a similar argument as in the proof of part (ii) of Theorem 6.3, it follows that w/φ0 is bounded and has a continuous extension up to the Martin boundary ∂PM Ω. (ii) By [12], the operator P˜ is IU if and only if the pointwise eigenfunction estimate (6.3) holds true with Cn = ct exp(tλn )kφn k2 , for every t > 0 and n > 1. Here ct is a positive function of t which may be taken as the function such that k(x, y, t) ≤ ct φ0 (x)φ0 (y), where k(x, y, t) is the corresponding heat kernel. It follows that if P˜ is IU, then the pointwise eigenfunction estimate (6.3) holds true with Cn = inf t>0 {ct exp(tλn )}kφn k2 . (iii) M. Murata [22] proved part (ii) of Theorem 6.3 for the special case of bounded Lipschitz domains. See also [5, 17] for related partial results on the asymptotic behavior of eigenfunctions of Schr¨ odinger operators in IRd . The next result concerns the asymptotics of the spherically average of oscillatory eigenfunctions of radially symmetric Schr¨ odinger operators in IRd . Let 1 ≤ p < ∞ and f ∈ C(B(0, R)∗ ) for some R > 0. Using polar coordinates r = |x| and ω = x/|x|, we denote by µZ
kf˜(r)kp =
S d−1
|f (rω)|p dω
¶1/p
the Lp spherically average of f (rω) on the unit sphere S d−1 . Lemma 6.5 (i) Consider a subcritical Schr¨ odinger operator of the form H = −∆ + V (x) on IRd , d ≥ 2, where V (x) = v(|x|) is a radial potential. Assume that W (x) = w(|x|) is a positive radial function which is a semismall perturbation of H on IRd . Assume further that the function (1 + |x|)−2 is also a semismall perturbation of H on IRd . ˜ be the (Dirichlet) selfadjoint realization of the symmetric operLet H −1 ator W H on L2 (IRd , W (x)dx) and denote by {φn }n≥0 the corresponding eigenfunction. Then kφ˜n (r)kp lim = Cn,p , r→∞ φ0 (r) where Cn,p is a positive constant. 29
(ii) Let W1 be a semismall radial perturbation of the operator H in IRd . Let w be any solution of the equation (H + W1 )u = 0 in a neighborhood of infinity in IRd such that w ∈ L2 (B(0, R)∗ , W (x)dx) for some R > 0. Then kw(r)k ˜ limr→∞ φ0 (r) p = C, where C is a positive constant. M (IRd ) Proof: (i) By [21], for every ² > 0 the Martin boundary ∂H−(λ 0 −²)W is either one point, or homeomorphic to S d−1 . Thus, using Theorem 6.3, we infer that for any ω ∈ S d−1 the following limit |φn (rω)|p lim = |ψn (ω)|p r→∞ (φ0 (r))p p
exists. Moreover, |φ(φn0(rω)| (r))p is bounded near infinity and hence, by dominated convergence, we have that Z kφ˜n (r)kpp Z |φn (rω)|p lim = lim dω = |ψn (ω)|p dω = (Cn,p )p ≥ 0 . p r→∞ (φ0 (r))p S d−1 r→∞ (φ0 (r)) S d−1 (6.6) It remains to prove that Cn,p > 0. First, we consider the case p = 2. As in the proof of part (iii) of Theorem 1.4 in [17], we express φ := φn in a series of surface harmonics Y (k) (ω) r(d−1)/2 φ(rω) =
∞ X
ak fk (r)Y (k) (ω)
k=0
and 2 ˜ kφ(r)k 2 =
∞ X
a2k fk2 (r),
k=0
where fk > 0 for r > R and satisfies the differential equation n
o
−fk00 + [k(k + d − 2) + (d − 1)(d − 3)/4] r−2 + V − λn W fk = 0 . By Lemma 6.2, fk ≈ φ0 . Since for some k ≥ 0, ak 6= 0 it follows that ˜ kφ(r)k 2 /φ0 ≥ |ak |fk /φ0 ≥ |ak |Ck > 0. Thus, Cn,2 > 0. It follows from (6.6) that Cn,p = 0 if and only if ψn = 0 almost everywhere. Since Cn,2 > 0, it follows that Cn,p > 0 also for p 6= 2. (ii) By Remark 6.4 (i), w has a continuous extension up to the MarM tin boundary ∂H−(λ (IRd ). To finish the proof apply now the same 0 −²)W argument as in the proof of part (i) of the Lemma. Corollary 6.6 Let φn , n ≥ 0 and w be the functions as in Lemma 6.5. Under the assumptions of Lemma 6.5 the functions |φn (rω)|/kφ˜n (r)kp and |w(rω)|/kw(r)k ˜ p are bounded. 30
7
Quasi-metric inequality and perturbations of elliptic operators
In this section we consider a sufficient condition for a semismall (resp. Gsemibounded) perturbation W to be a small (resp. G-bounded) perturbation. It is given in terms of quasi-metric inequality, some times known also as 3G-inequality (See [11, 19]). Lemma 7.1 Let x0 , y0 ∈ Ω1 be some fixed reference points. Suppose that P is a subcritical operator in Ω and the function dΩ P (x, y) = d(x, y) =
Ω GΩ P (x, y0 )GP (x0 , y) Ω GP (x, y)
(7.1)
satisfies the quasi-metric inequality d(x, y) ≤ C(d(x, z) + d(z, y)).
(7.2)
for every x, y, z ∈ Ω∗1 . Then W ∈ C α (Ω) is a small (resp. G-bounded) perturbation of P in Ω if and only if W is a semismall (resp. G-semibounded) perturbation of both P and P ∗ . Proof: Inequality (7.2) is clearly equivalent to the following pointwise 3Ginequality Ω GΩ P (x, z)GP (z, y) ≤C GΩ P (x, y)
Ã
Ω Ω GΩ GΩ P (x, z)GP (z, y0 ) P (x0 , z)GP (z, y) + GΩ GΩ P (x, y0 ) P (x0 , y)
!
. (7.3)
Now, multiply (7.3) by |W | and integrate. Example 7.2 Let Ω be a bounded C 1,1 domain in IRd , d ≥ 3 and P a subcritical uniformly elliptic operator with bounded H¨older continuous coδ(x)δ(y) efficients. It is known that GΩ P (x, y) ≈ d1 (x,y) , where δ(x) = δΩ (x) = dist (x, ∂Ω) and d1 (x, y) = |x − y|d−2 (|x − y|2 + δ(x)2 + δ(y)2 )
(7.4)
satisfies the quasi-metric inequality (see [19]). Let x0 , y0 ∈ Ω1 . It is also well known that for C 1,1 domains, GΩ P (x0 , z) ≈ ∗ Ω δ(z) and GP (z, y0 ) ≈ δ(z) in Ω2 . Therefore, the function d satisfies the inequality (7.2). It follows from Lemma 7.1 that W is a small (resp. Gbounded) perturbation of P in Ω if W is a semismall (resp. G-semibounded) perturbation of P . 31
Example 7.3 Consider a uniformly elliptic operator P in a divergence form defined on IRd , d ≥ 3. So, we assume that P (x, ∂x ) = −
d X
∂i (aij (x)∂j ).
i,j=1 d
2−d . The function It is well known that GIR P (x, y) ≈ |x − y|
ρ(x, y) =
|x − y|d−2 |x|d−2 |y|d−2
is a quasi-metric in IRd (see Lemma A.1 in Appendix A). By Lemma 7.1, W is a small (resp. G-bounded) perturbation of P in IRd provided that W is a semismall (resp. G-semibounded) perturbation of P (see also Example 8.2). Remarks 7.4 (i) Note that the function d(x, y) defined by (7.1) is always a (nonnegative) bounded function on Ω∗1 × Ω∗1 (see [26]). (ii) Suppose that the function dΩ P defined by (7.1) is a quasi-metric and Ω Ω Ω that GP ≈ GP1 then obviously, dP1 is also a quasi-metric. In particular, it follows from [6, Theorem 9.1’] that if Ω is a C 1,1 bounded domain then dΩ P is a quasi-metric also for a subcritical uniformly elliptic operator P in divergence form with uniformly bounded coefficients in the sense of Ancona. In [19], N. Kalton and I. Verbitsky conjectured that dΩ P satisfies the quasimetric inequality for any bounded Lipschitz domain. (iii) Another useful pointwise 3G-inequality holds true for the Laplacian in a Jordan domain Ω ⊂ IR2 (see [11, Theorem 6.24]). It implies, for instance, that if W is an integrable function and the integral operator induced by ∞ the kernel GΩ P (x, y)|W (y)| is bounded on L (Ω), then W is a G-bounded perturbation.
8
Examples
In this section we present some examples which exemplify some of the phenomena discussed in this paper. Examples 8.1-8.4 demonstrate that in some special cases some of the notions of perturbations coincide and one can characterize such perturbations by a natural and simple integral condition. Example 8.1 Let V (x) = v(|x|) be a radial function defined on IRd , d ≥ 3 such that |v(r)|r2 is nonincreasing. The following conditions are equivalent 32
• V is a small perturbation of −∆ in IRd . • V is a H-integrable perturbation of −∆ in IRd . •
R∞
|v(r)|r dr < ∞.
(see [29, Lemma 2.3].) Example 8.2 Consider again the Laplacian on IRd , d ≥ 3 (or more generally the operator P as in Example 7.3) and let V ∈ C α (IRd ). It follows from the proof of Theorem 3.1 of [29] and Example 7.3 that the following conditions are equivalent • V is a G-bounded perturbation of −∆ in IRd . • V is a G-semibounded perturbation of −∆ in IRd . • V is a H-bounded perturbation −∆ in IRd . • The integral operator induced by the kernel K(x, y) = |x−y|2−d |V (y)| is a bounded operator on L∞ (IRd ). Example 8.3 Let P = −∆ + 1 on IRd , d ≥ 3. Let V (x) = v(|x|) be a radial function such that |v| is monotonically decreasing. By Remark 2.7 (vii) and [16] it follows that following conditions are equivalent • V is a small perturbation of P in IRd . • V is a G-semibounded perturbation of P in IRd . •
R∞
|v(r)| dr < ∞
Example 8.4 Let Ω ⊂ IRd be a C 1,1 bounded domain. Denote by δ the distance function to the boundary of Ω. Let V (x) = v(δ(x)) such that |v(r)|r2 is monotonically increasing. . Using Remark 2.7 (vii) and [2, 6] it follows that the following conditions are equivalent • V is a small perturbation of −∆ in Ω. • V is a G-semibounded perturbation of −∆ in Ω. •
R
0 |v(r)|r dr
< ∞.
In the next example we present a perturbation which is weak but not Hintegrable. 33
Example 8.5 Let P = −∆ in IRd , d ≥ 3, and consider a nonnegative function W ∈ Ld/2 (IRd ) such that Z IRd
W (y)(1 + |y|)2−d dy = ∞ .
By the Cwikel-Lieb-Rozenblum inequality, W is a weak perturbation. On the other hand, by definition, such a function is not H-integrable and therefore, not H- or G-semibounded (and obviously, not semismall). We present now a counter example to Conjecture 3.7 of [28] which demonstrates that even for nonnegative W it may happen that the set E+ (resp. sE+ ) of Definition 2.4 is not empty but E+ (resp. sE+ ) 6= S+ . In particular, it follows that G-(resp. semi) boundedness does not imply (resp. semi) smallness (see Remarks 2.7). Example 8.6 Let W ∈ C0∞ (IRd ), d ≥ 3 be a nonzero, nonnegative function. Set e1 = (1, 0, . . . , 0). Without loss of generality, we may assume that λ0 (W, −∆, IRd ) = 1. Let {an } be a strictly increasing sequence tending to 1. It follows from the principle of localization of binding [31] that there exists an increasing sequence {αn }∞ n=1 , limn→∞ αn = ∞ such that for every N > 1 the operator −∆ −
N X
an W (x + αn e1 )
n=1
P
is subcritical in IRd . Consider the function V (x) = ∞ n=2 an W (x + αn e1 ). Clearly, the operator −∆ − V is subcritical in IRd , λ0 (V, −∆, IRd ) = 1 and S = S+ = (−∞, 1]. By considering subsequences {ank } and {αnk } and P taking V (x) = ∞ k=2 ank W (x + αnk e1 ), we may assume that Z
sup { d |x − y|2−d V (y)dy} < ∞ . d IR x∈IR Hence, V is by definition, a H-bounded perturbation. In fact, V is even a G-bounded perturbation of the Laplacian in IRd (see Example 8.2). Recall that S is always a closed set and since V is a nonnegative, Gbounded perturbation, the set E+ (resp. sE+ ) is an unbounded open interval (see Remark 2.7 (iii)). It follows that in our case (resp. sE+ ) E+ is not d IRd empty but (resp. sE+ ) E+ 6= S+ . In particular, GIR −∆ (x, 0) 6≈ G−∆−V (x, 0). Moreover, since λ∞ (V, −∆, IRd ) = λ0 (V, −∆, IRd ) = 1 it follows that V is not a weak perturbation and therefore, V is not a semismall perturbation. 34
Let us discuss briefly some further properties of the operator −∆ − V . A subcritical operator P defined on IRd , d ≥ 3 is called uniformly subcritical if there exists a positive constant C such that d
2−d GIR P (x, y) ≤ C|x − y|
for all x, y ∈ IRd , 0 < |x − y| ≤ 1
(see [26]). By Theorem 3.1 of [26], we infer that −∆ − V is not a uniformly subcritical operator. On the other hand, it follows from [30] that if the d sequence {αn }∞ n=1 is sparse enough, then dim C−∆−V (IR ) = 1. It is also clear that limx→∞ u(x) 6= ∞, where u is the unique normalized solution in R d C−∆−V (IR ). Moreover, by [25, Lemma 4.2], IRd |x − y|2−d V (y)u(y)dy ≤ u(x) but since λ0 (V, −∆ − V, IRd ) = 0, it follows from Lemma 3.3 that V is not a H-bounded perturbation of −∆ − V . Therefore, ½Z
sup d x∈IR
¾
d
GIR −∆−V (x, y)V (y)u(y)dy/u(x) = ∞ . d IR
It would be interesting to understand the asymptotic behavior of u and d GIR −∆−V (x, y0 ). P
Example 8.7 Let V (x) = ∞ n=1 nW (x + αn e1 ), where the function W and the sequence {αn } are as in Example 8.6. It follows that if {αn } is sparse enough, then V is still a H-semibounded perturbation of the Laplacian in IRd , d ≥ 3 but since λ0 (V, −∆, IRd ) = 0, this perturbation is not H-bounded and not G-semibounded. Example 8.8 By the Cranston-McConnel inequality, 1 is a G-bounded perturbation of the Laplacian in any finite area domain of IR2 (see [7]). Moreover, M. Murata kindly informed the author that using the results of [3] it follows that 1 is actually a small perturbation of the Laplacian in any finite area planer domain. Thus, the example of Ba˜ nuelos and Davis in [7] gives us a finite area domain Ω ⊂ IR2 such that 1 is a small perturbation of the Laplacian in Ω but the corresponding semigroup is not IU. We conclude this section with a short discussion on sufficient conditions which imply that 1 is a small perturbation of a Schr¨ odinger operator H = d −∆ + V defined on IR . In [23] sufficient conditions for a Schr¨ odinger operator H = −∆ + V defined on IRd to be IU are proved. Recall that IU implies that 1 is a
35
G-bounded perturbation. On the other hand, in [4, 5, 17], maximum, antimaximum principles and asymptotic behavior of eigenfunctions are discussed for a Schr¨odinger operator with a potential which grow fast at infinity. These results seems to be special cases of the results in sections 4, 5 and 6. Among other assumptions, it is assumed in [4, 5, 17, 23] that Z ∞
(inf{V (x) | |x| = r})−1/2 dr < ∞ .
Similar integrability condition appears also in the next example. Example 8.9 By [21], if V (x) = |x|2+² , ² > 0, then 1 is a small perturbation of H = −∆ + V on IRd . Note that 1 is not a small perturbation of the harmonic oscillator and (6.3) is false for this operator. Furthermore, it follows from the resultsR in [6] that if V (x) = v(|x|) ≥ 0, 0 v ≥ 0 and v 0 (t)/v(t) ≤ C for t ≥ t0 and ∞ v(t)−1/2 dt < ∞, then 1 is a small perturbation of H = −∆ + V . More generally, assume that V ≥ 1, |∇V | ≤ CV on IRd and that p Z ∞ max{ V (x) | |x| = t} 0
min{V (x) | |x| ≥ t}
dt < ∞,
then 1 is a small perturbation of −∆ + V in IRd . (Ancona, private communication).
A
Appendix
Lemma A.1 The function ρ˜(x, y) =
|x − y|d−2 |x|d−2 |y|d−2
is a quasi-metric on IRd \ {0}, d ≥ 3 Lemma A.1 follows easily from the following lemma Lemma A.2 Let (V, h·, ·i) be an inner product space. Denote the induced norm by k · k. The function ρ(x, y) =
kx − yk kxk kyk
defines a metric on V \ {0}. 36
Proof: A direct computation shows that ρ(x, y) =
x y kx − yk =k − k. 2 kxk kyk kxk kyk2
Using the triangle inequality with respect to the norm k · k and the triple x , y , z , we infer that ρ satisfies the triangle inequality for every kxk2 kyk2 kzk2 x, y, z ∈ V \ {0}. Remark A.3 A quasi-metric inequality deals only with three vectors and therefore, for vector spaces, it is actually an inequality on spaces of dimension not greater than 3. It is well known ([18, Lemma 1.4.3]) that for any d-dimensional normed space (V, | · |) one can find an Euclidean norm k · k such that √ |x| ≤ kxk ≤ d|x| . It follows that for any normed space (V, | · |) with dim V = d ≤ ∞ the function |x − y| ρ(x, y) = |x| |y| √ satisfies the quasi-metric inequality with a constant 1 ≤ C ≤ 3 3 if d ≥ 3. √ while 1 ≤ C ≤ 2 2 for d = 2. Note added in Proof. It turns out that the anti-maximum principle (Theorem 5.3) holds true also for every f ∈ B such that φ˜0 (f ) > 0, where φ˜0 ∈ B ∗ is the ground state of the adjoint operator P ∗ . The proof will appear somewhere else. Acknowledgment. The author expresses his gratitude to Professors A. Ancona, M. Murata, M. Solomyak for helpful conversations and to Professors Y. Benyamini and S. Reich for useful comments on quasi-metric inequalities on normed spaces. The author thanks Professors J. FleckingerPell´e and P. Tak´aˇc for providing him their preprint [4] which motivated him to generalize the anti-maximum principle. This research was supported by the Fund for the Promotion of Research at the Technion.
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