On the principal eigenvalue of elliptic operators in R N and applications

Transcription

1 On the principal eigenvalue of elliptic operators in R N and applications Henri Berestycki a and Luca Rossi a, a EHESS, CAMS, 54 Boulevard Raspail, F Paris, France Università La Sapienza Roma I, Dipartimento di Matematica, Piazzale Aldo Moro 2, I Roma, Italy Astract Two generalizations of the notion of principal eigenvalue for elliptic operators in R N are examined in this paper. We prove several results comparing these two eigenvalues in various settings: general operators in dimension one; self-adjoint operators; and limit periodic operators. These results apply to questions of existence and uniqueness for some semi-linear prolems in all of space. We also indicate several outstanding open prolems and formulate some conjectures. 1 Introduction The principal eigenvalue is a asic notion associated with an elliptic operator. For instance, the study of semi-linear elliptic prolems in ounded domains often involves the principal eigenvalue of an associated linear operator. To motivate the results of the present paper, let us first recall some classical properties of a class of semi-linear elliptic prolems in ounded domains. Let L e a linear elliptic operator acting on functions defined on a ounded and smooth domain Ω R N : Lu = a ij (x) ij u + i (x) i u + c(x) u (here and throughout the paper, the convention is adopted for summation on repeated indices). Consider the Dirichlet prolem: { Lu = g(x, u), x Ω u = 0 on Ω (1.1) This paper has een completed while the first author was visiting the Department of Mathematics, University of Chicago whose hospitality is gratefully acknowledged. 1

2 We are interested in positive solutions of (1.1). Assume that g is a C 1 function such that: and: g(x, s) < g s(x, 0)s, s > 0, x Ω M > 0 such that g(x, s) + c(x)s 0, s M Then, existence of positive solutions of (1.1) is determined y the principal eigenvalue µ 1 of the prolem linearized aout u = 0: { L ϕ gs (x, 0) ϕ = µ 1 ϕ in Ω ϕ = 0 on Ω (1.2) Recall that µ 1 is characterized y the existence of an associated eigenfunction ϕ > 0 of (1.2). It is known indeed that (1.1) has a positive solution if and only if µ 1 > 0 (see e.g. [2]). Under the additional assumption that s g(x, s)/s is decreasing, one further otains a uniqueness result [2]. Prolems of the type (1.1) arise in several contexts, in particular in population dynamics. In many of the applications, the prolem is set in an unounded domain, often in R N. Clearly, extensions to unounded domains of the previous result, as well as others of the same type, require one to understand the generalizations and properties of the notion of principal eigenvalue of elliptic operators in unounded domains. In Section 2, we indicate some new results aout such a semi-linear prolem, extending the result for (1.1). Another example of use of principal eigenvalue is the characterization of the existence of the Green function for linear periodic operators (see Agmon [1]). We refer to [18] and to its iliography for details on the suject. Moreover, the principal eigenvalue of an elliptic operator has een shown to play an important role in some questions in ranching processes (see Englander and Pinsky [9], Pinsky [19]). Very recently, the principal eigenvalue of an elliptic operator in R N is eing introduced in the context of economic models [11]. Some definitions of the notion of principal eigenvalue in unounded domains have emerged in the works of Agmon [1], Berestycki, Nirenerg and Varadhan [6], Pinsky [18] and others. With a view to applications to semi-linear equations, in particular two definitions have een used in [3], [4], [5]. We will recall these definitions later in this section. In this paper, we examine these definitions and further investigate their properties. In particular, we are interested in understanding when the two definitions coincide or for which classes of operators does one or the other inequality hold. We also further explain the choice of definition. This work grew out of our previous article with François Hamel [5] which already addressed some of these issues. We review the relevant results from [5] in Section 3. Let us now recall the definitions. In the sequel, we define the class of elliptic operators (in non-divergence form) as the elliptic operators L with Lu = a ij (x) ij u + i (x) i u + c(x)u, in R N. 2

3 Self-adjoint elliptic operators L, are defined y: Lu = i (a ij (x) j u) + c(x)u, in R N, (we use the usual summation convention on repeated indices). Throughout the paper, (a ij ) ij will denote a N N symmetric matrix field such that x, ξ R N, a ξ 2 a ij (x)ξ i ξ j a ξ 2, (1.3) where a and a are two positive constants, ( i ) i will denote a N dimensional vector field and c a real valued function. We always assume that there exists 0 < α 1 such that: in the case of general operators, and a ij, i, c C 0,α (R N ), (1.4) a ij C 1,α (R N ), c C 0,α (R N ), (1.5) in the self-adjoint case. By C k,α (R N ), we mean the class of functions φ C k (R N ) such that φ and the derivatives of φ up to the order k are ounded and are uniformly Hölder continuous with exponent α. Notice that every self-adjoint operator satisfying (1.5) can e viewed as a particular case of general elliptic operator satisfying (1.4). It is well known that any elliptic operator L as defined aove admits a unique principal eigenvalue, either in ounded smooth domains associated with Dirichlet oundary conditions, as well as in R N provided that its coefficients are periodic in each variale. This principal eigenvalue is the ottom of the spectrum of L in the appropriate function space, and it admits an associated positive principal eigenfunction. This result follows from the Krein Rutman theory and from compactness arguments (see [14] and [13]). In this paper, we examine some properties of two different generalizations of the principal eigenvalue in unounded domains. The first one, originally introduced in [6], reads: Definition 1.1 Let L e a general elliptic operator defined in a domain Ω R N. We set λ 1 ( L, Ω) := sup{λ φ C 2 (Ω) C 1 loc(ω), φ > 0 and (L + λ)φ 0 in Ω}, (1.6) Here, Cloc 1 (Ω) denotes the set of functions φ C1 (Ω) for which φ and φ can e extended y continuity on Ω, ut which are not necessarily ounded. Berestycki, Nirenerg and Varadhan showed that this is a natural generalization of the principal eigenvalue. Indeed, if Ω is ounded and smooth, then λ 1 ( L, Ω) coincides with the principal eigenvalue of L in Ω, with Dirichlet oundary conditions. As we will see later, the eigenvalue λ 1 does not suffice to completely descrie the properties of semi-linear equations in the whole space, in contradistinction with the Dirichlet principal eigenvalue in ounded domains for prolem (1.1). For this, we also require another generalization, whose definition is similar to that of λ 1. This generalization has een introduced in [3], [5] and reads: Definition 1.2 Let L e a general elliptic operator defined in a domain Ω R N. We set λ 1( L, Ω) := inf{λ φ C 2 (Ω) C 1 loc(ω) W 2, (Ω), φ > 0 and (L + λ)φ 0 in Ω, φ = 0 on Ω, if Ω }. (1.7) 3

4 Several other generalizations are possile, starting from Definition 1.1 and playing on the space of functions or the inf and sup inequalities. We will explain why Definition 1.2 is relevant. If L is periodic (in the sense that its coefficients are periodic in each variale, with same period) then λ 1 ( L, R N ) λ 1( L, R N ), as is shown y taking φ equal to a positive periodic principal eigenfunction in (1.6) and (1.7). More generally, if there exists a ounded positive eigenfunction ϕ, then λ 1 λ 1. But in general, if the operator L is not self-adjoint, equality need not hold etween λ 1 and λ 1, even if L is periodic (see Section 3). It is then natural to ask what are the relations etween λ 1 and λ 1 in the general case. In Section 3, we review a list of statements, most of them given in [5], which answer this question in some particular cases. In Section 4, we state our new main results as well as some prolems which are still open. In Section 5, we motivate our choice of taking (1.6) and (1.7) as generalizations of the principal eigenvalue. The last three sections are dedicated to the proofs of our main results. 2 Positive solutions of semi-linear elliptic prolems in R N Let us precisely descrie how the eigenvalues λ 1 and λ 1 are involved in the study of the following class of nonlinear prolems: a ij (x) ij u(x) i (x) i u(x) = f(x, u(x)), in R N. (2.8) This type of prolem arises in particular in iology and in population dynamics. Here and in the sequel, the function f(x, s) : R N R R is assumed to e in C 0,α (R N ) with respect to the variale x, locally uniformly in s R, and to e locally lipschitz-continuous in the variale s, uniformly in x R N. Furthermore, we always assume that f satisfies: x R N, f(x, 0) = 0, δ > 0 such that s f(x, s) elongs to C 1 ([0, δ]), uniformly in x R N, f s (x, 0) C 0,α (R N ). We will always denote with L 0 the linearized operator around the solution u 0 associated to the equation (2.8), that is: L 0 u = a ij (x) ij u + i (x) i u + f s (x, 0)u, in R N. In [3] it is proved that, under suitale assumptions on f, if L 0 is self-adjoint and the functions a ij and x f(s, x) are periodic (in each variale) with same period, then (2.8) admits a unique positive ounded solution if and only if the periodic principal eigenvalue of L 0 is less than or equal to zero (see Theorems 2.1 and 2.4 in [3]). This result has een extended in [5] to non-periodic, non-self-adjoint operators, y using λ 1 ( L 0, R N ) and λ 1( L 0, R N ) instead of the periodic principal eigenvalue of L 0. The assumptions required are: M > 0, x R N, s M, f(x, s) 0, (2.9) 4

5 The existence result of [5] is: x R N, s 0, f(x, s) f s (x, 0)s. (2.10) Theorem 2.1 Let L 0 e the linearized operator around zero associated to equation (2.8). 1) If (2.9) holds and either λ 1 ( L 0, R N ) < 0 or λ 1( L 0, R N ) < 0, then there exists at least one positive ounded solution of (2.8). 2) If (2.10) holds and λ 1( L 0, R N ) > 0, then there is no nonnegative ounded solution of (2.8), other than the trivial one u 0. Theorem 2.1 follows essentially from Definitions 1.1, 1.2 and a characterization of λ 1 (see Theorem 5.1 and Propositions 6.1, 6.5 in [5] for details). In [9], Engländer and Pinsky proved a similar existence result for a class of solution of minimal growth (which they define there) for nonlinearities of the type f(x, u) = (x)u a(x)u 2, with inf a > 0 (see also [8], [19]). Since the theorem involves oth λ 1 and λ 1, one does not have a simple necessary and sufficient condition. This is one of the motivations to investigate the properties of these two generalized eigenvalues. In particular, it is useful to determine conditions which yield equality etween them or at least that yield an ordering. From the results we prove in this paper we can deal in particular with the case that the operator is self-adjoint limit periodic. The notion of limit periodic operator is defined precisely elow in section 4.2. Essentially, it means that the operator is the uniform limit (in the sense of coefficients) of a sequence of periodic operators. In this case, we still have a condition, extending that in theorem 2.1, which is nearly necessary and sufficient. Theorem 2.2 If L 0 is a self-adjoint limit periodic operator, then we have: 1) If (2.9) holds and λ 1 ( L 0, R N ) < 0, then there exists at least one positive ounded solution of (2.8). If, in addition, (2.11) elow holds, then such a solution is unique. 2) If (2.10) holds and λ 1 ( L 0, R N ) > 0, then there is no nonnegative ounded solution of (2.8), other than the trivial one u 0. The same result holds in dimension N=1 if L 0 is an aritrary self adjoint operator. The case of equality: λ 1 ( L 0, R N ) = 0 is open. For uniqueness, in unounded domains, one needs to replace the classical assumption that s f(x, s)/s is decreasing y the following one: 0 < s 1 < s 2, inf x R N ( f(x, s1 ) s 1 f(x, s 2) s 2 ) > 0. (2.11) The uniqueness result of [5] is more delicate and involves the principal eigenvalue of some limit operators defined there. It ecomes simpler to state in the case the coefficients in (2.8) are almost periodic, in the sense of the following definition: Definition 2.3 A function g : R N R is said to e almost periodic (a.p.) if from any sequence (x n ) n N in R N one can extract a susequence (x nk ) k N such that g(x nk +x) converges uniformly in x R N. 5

6 Theorem 2.4 (Theorem 1.5 in [5]) Assume that the functions a ij, i and f s (, 0) are a.p. If (2.11) holds and λ 1 ( L 0, R N ) < 0, then (2.8) admits at most one nonnegative ounded solution, esides the trivial one: u 0. Theorems 2.1 and 2.4 essentially contain the results in the periodic self-adjoint framework (which hold under the same assumptions (2.9), (2.10) and (2.11)). In that case, in fact, λ 1 ( L 0, R N ) and λ 1( L 0, R N ) coincide with the periodic principal eigenvalue of L 0 (see Proposition 3.3 elow) and then the only case which is not covered is when the periodic principal eigenvalue is equal to zero. 3 Some properties of the generalized principal eigenvalues λ 1 and λ 1 in R N In this section, unless otherwise specified, L denotes a general elliptic operator. When we say that L is periodic, we mean that there exist N positive constants l 1,, l N such that x R N, i {1,, N}, a ij (x + l i e i ) = a ij (x), i (x + l i e i ) = i (x), c(x + l i e i ) = c(x), where (e 1,, e N ) is the canonical ase of R N. The following are some of the known results concerning λ 1 and λ 1. Actually, in some statements of [5], the coefficients of L were in C 0,α (R N ) L (R N ) and the test functions φ in the definition of λ 1 were taken in C 2 (R N ) W 1, (R N ) instead of C 2 (R N ) W 2, (R N ). However, one can check that the following results - as well as Theorem can e proved arguing exactly as in the proofs of the corresponding results in [5]. Proposition 3.1 ([6] and Proposition 4.2 in [5]) Let Ω e a general domain in R N (Ω n ) n N e a sequence of nonempty open sets such that Ω n Ω n+1, Ω n = Ω. n N and Then λ 1 ( L, Ω n ) λ 1 ( L, Ω) as n +. Proposition 3.1 yields that λ 1 ( L, R N ) < +. Furthermore, taking φ 1 as a test function in (1.6), we see that λ 1 ( L, R N ) c. Thus, λ 1 is always a well defined real numer. In the case of L periodic, the periodic principal eigenvalue of L is defined as the unique constant such that there exists a positive periodic ϕ C 2 (R N ) satisfying (L + λ)ϕ = 0 in R N. Its existence follows from the Krein Rutman theory. Proposition 3.2 (Proposition 6.3 in [5]) If L is periodic, then its periodic principal eigenvalue λ p coincides with λ 1( L, R N ). 6

7 It is known that, in the general non-self-adjoint case, λ 1 λ 1. Indeed, as an example, consider the one dimensional operator Lu = u + u, which is periodic with aritrary positive period. Then it is easily seen that λ 1( L, R) = 0 < 1 4 = λ 1( L, R N ). In fact, since ϕ 1 satisfies Lϕ = 0, it follows that the periodic principal eigenvalue of L is 0 and then, y Proposition 3.2, λ 1( L, R) = 0. On the other hand, for any R > 0, the function ( π ) ϕ R (x) := cos 2R x e 1 2 x satisfies: Lϕ R = ( 1 + π2 )ϕ 4 4R 2 R, which shows that ϕ R is a principal eigenfunction of L in ( R, R), under Dirichlet oundary conditions. Therefore, y Proposition 3.1, λ 1 ( L, R) = lim R ( π2 4R 2 ) = 1 4 > λ 1( L, R). Proposition 3.3 (Proposition 6.6 in [5]) If the elliptic operator L is self-adjoint and periodic, then λ 1 ( L, R N ) = λ 1( L, R N ) = λ p, where λ p is the periodic principal eigenvalue of L. For the sequel of this paper it is useful to recall the proof of the last statement. Proof of Proposition 3.3. First, from Proposition 3.2 one knows that λ p = λ 1( L, R N ). Now, let ϕ p e a positive periodic principal eigenfunction of L in R N. Taking φ = ϕ p in (1.6), it is straightforward to see that λ 1 ( L, R N ) λ p. To show the reverse inequality, consider a family of cutoff functions (χ R ) R 1 in C 2 (R N ), uniformly ounded in W 2, (R N ), such that 0 χ R 1, suppχ R and χ R = 1 in 1. Fix R > 1 and call λ R the principal eigenvalue of L in. It is otained y the following variational formula: λ R = min ( aij (x) i v j v c(x)v 2) v 2 ; v H0(B 1 R ), v 0. (3.12) Taking v = ϕ p χ R as a test function in (3.12), and calling C R = \ 1, we find: ( L(ϕp B λ R R χ R ) ) ϕ p χ R = λ p 1 ϕ 2 p ( L(ϕp C R χ R ) ) ϕ p χ R ϕ 2 pχ 2 R ϕ 2 pχ 2 R = λ p λ p C R ϕ 2 pχ 2 R + ( L(ϕp C R χ R ) ) ϕ p χ R. ϕ 2 pχ 2 R Since min ϕ p > 0, it follows that there exists K > 0, independent of R, such that ϕ 2 pχ 2 R ϕ 2 p K(R 1) N. 1 7

8 Consequently, R N 1 λ R λ p + K (R 1), N where K is a positive constant independent of R. Letting R go to infinity and using Proposition 3.1, we get: λ 1 ( L, R N ) λ p and therefore, λ 1 ( L, R N ) = λ p. The next result is an extension of the previous proposition. It is still aout periodic operators, ut which are not necessarily self-adjoint. A gradient type assumption on the first order coefficients is required. Theorem 3.4 (Theorem 6.8 in [5]) Consider the operator Lu := i (a ij (x) j u) + i (x) i u + c(x)u, x R N, where a ij, i, c are periodic in x, with the same period (l 1,, l N ), the matrix field A(x) = (a ij (x)) 1 i,j N is in C 1,α (R N ), elliptic and symmetric, the vector field = ( 1,, q N ) C 1,α (R N ) and c C 0,α (R N ). Assume that there is a function B C 2,α (R N ) such that a ij j B = i for all i = 1,..., N and assume that the vector field A 1 has zero average in the periodicity cell C = (0, l 1 ) (0, l N ). Then λ 1 ( L, R N ) = λ p = λ 1( L, R N ), where λ p is the periodic principal eigenvalue of L in R N. Next, the natural question is to ask what happens when we drop the periodicity assumption. Up to now, the only availale result had een otained in [5] in the case of dimension one. It stated: Proposition 3.5 (Proposition 6.11 in [5]) Let L e a self-adjoint operator in dimension one. Then λ 1 ( L, R) λ 1( L, R). This type of result will e extended elow. 4 Main results and open prolems The goal of this paper is to further explore these properties. We will examine three main classes: self-adjoint operators in low dimension, limit periodic operators and general operators in dimension one. We seek to identify classes of operators for which either equality or an inequality etween λ 1 and λ 1 hold. 4.1 Self-adjoint case Our first result is an extension of the comparison result of Proposition 3.5 to dimensions N = 2, 3 in the self-adjoint framework. Theorem 4.1 Let L e a self-adjoint elliptic operator in R N, with N 3. Then λ 1 ( L, R N ) λ 1( L, R N ). The assumption N 3 in Theorem 4.1 seems to e only technical, as well as it was for the assumption N = 1 in Proposition 3.5. This is why we elieve that the aove result holds in any dimension N. But the prolem is open at the moment. 8

9 4.2 Limit periodic operators Next, we examine the class of limit periodic operators which extends that of periodic operators. In a sense, this class is intermediate etween periodic and a.p. Here is the definition: Definition 4.2 1) We say that a general elliptic operator L is general limit periodic if there exists a sequence of general elliptic periodic operators L n u := a n ij ij u n i i u c n u such that a n ij a ij, n i i and c n c in C 0,α (R N ) as n goes to infinity. 2) We say that a self-adjoint elliptic operator L is self-adjoint limit periodic if there exists a sequence of self-adjoint elliptic periodic operators L n u := i (a n ij j u) c n u such that a n ij a ij in C 1,α (R N ) and c n c in C 0,α (R N ) as n goes to infinity. Clearly, if all the coefficients of the operators L n in Definition 4.2 have the same period (l 1,, l N ), then L is periodic too. It is immediate to show that the coefficients of a limit periodic operator are in particular a.p. in the sense of Definition 2.3. One of the results we otain is: Theorem 4.3 Let L e a general limit periodic operator. Then λ 1( L, R N ) λ 1 ( L, R N ). The other result otained concerns self-adjoint limit periodic operators. It extends Proposition 3.3. Theorem 4.4 Let L e a self-adjoint limit periodic operator. Then λ 1 ( L, R N ) = λ 1( L, R N ). In the proofs of Theorems 4.3 and 4.4, we make use of the Shauder interior estimates and the Harnack inequality. One can find a treatment of these results in [10], or one can consult [15], [16] and [21] for the original proofs of the Harnack inequality. Going ack to the nonlinear prolem, owing to Theorem 4.4, the existence and uniqueness results in the limit periodic case can e expressed in terms of λ 1 (or, equivalently, λ 1) only, which is the statement of Theorem The case of dimension N = 1 Our last result estalishes a comparison etween λ 1 and λ 1 for general elliptic operators in dimension one: Theorem 4.5 Let L e a general elliptic operator in dimension one. Then λ 1( L, R) λ 1 ( L, R). Notice that, y Theorems 4.3 and 4.5, if L 0 is limit periodic or N = 1, then we can state Theorem 2.1 without mentioning λ 1. Hence, only the sign of λ 1 is involved in the existence result. 9

10 4.4 Open prolems The notions of generalized principal eigenvalue raise several questions which still need an answer. Some of them are: Open prolem 4.6 Does (2.8) admit positive ounded solutions (even in the self-adjoint case) if λ 1( L 0, R N ) = 0? Open prolem 4.7 Is it true that λ 1( L, R N ) λ 1 ( L, R N ), for any general elliptic operator L and any dimension N? Conjecture 4.8 If L is a self-adjoint elliptic operator, then λ 1 ( L, R N ) λ 1( L, R N ) in any dimension N. Note that should the answers to oth 4.7 and 4.8 e positive, then we would have λ 1 = λ 1 in the self-adjoint case, in aritrary dimension. 5 Different definitions of the generalized principal eigenvalue In this section, we present various definitions which one could consider as generalizations of the principal eigenvalue in the whole space. Then, we explain the choice of (1.6) and (1.7) as the most relevant extensions. Here, L will always denote a general elliptic operator (satisfying (1.3) and (1.4)). The quantity λ 1 given y (1.6) is often called the generalized principal eigenvalue. It is considered the natural generalization of the principal eigenvalue ecause, as already mentioned, it coincides with the Dirichlet principal eigenvalue in ounded smooth domains. Also, the sign of λ 1 determines the existence or non-existence of a Green function for the operator (see Theorem 3.2 in [17]). The constant λ 1 has een introduced, more recently, in [3]. If Ω is ounded and smooth, then λ 1( L, Ω) = λ 1 ( L, Ω). Moreover, as we have seen in Proposition 3.2, in the periodic case λ 1 coincides with the periodic principal eigenvalue. The quantity λ 1 is the largest constant λ for which (L+λ) admits a positive susolution. The definition of λ 1 is ased on that of λ 1, with two changes: first, we take susolutions instead of supersolutions (and we replace the sup with the inf); second, we take test functions in W 2,. If we introduce only one of these changes, we otain the following definitions: µ 1 ( L, Ω) := sup{λ φ C 2 (Ω) C 1 (Ω) W 2, (Ω), φ > 0 and (L + λ)φ 0 in Ω, or φ = 0 on Ω, if Ω }, (5.13) µ 1( L, Ω) := inf{λ φ C 2 (Ω) C 1 loc(ω), φ > 0 and (L + λ)φ 0 in Ω}, (5.14) The quantity µ 1 is not interesting for us ecause, as is shown y Remark 6.2 in [5], if we replace λ 1 with µ 1, then the necessary condition given y Theorem 2.1 part 2) fails to hold. For completeness, we include this oservation here. 10

11 Remark 5.1 Consider the equation u (x)u = 0 in R. We show that, for opportunely chosen, there exists a positive function φ C 2 (R) W 2, (R) such that (L 0 + 1)φ 0 in R (L 0 u = u +(x)u in this case). Therefore, µ 1 ( L 0, R) 1, ut all the functions u identically equal to a positive constant solve u (x)u = 0. The function φ is defined in [ 1, 1] y: φ(x) = 2 x 2. For x ( 1, 1), we have that (L 0 +1)φ = 2 2(x)x+φ 2(x)x. Hence, it is sufficient to take (x) 0 for x 0 and (x) 0 for x 0 to otain: (L 0 + 1)φ 0 in ( 1, 1). Then set { e x if x 2 φ(x) := e x if x 2. For x < 2, (L 0 + 1)φ = e x (2 + (x)) and, for x > 2, (L 0 + 1)φ = e x (2 (x)). Hence, if 2 for x < 2 and (x) > 2 for x > 2, we find: (L 0 + 1)φ 0 in (, 2) (2, + ). Clearly, it is possile to define φ in ( 2, 1) (1, 2) in such a way that inf [ 2, 1] φ > 0, sup [1,2] φ < 0 and φ C 2 (R) W 2, (R). Consequently, taking < M in ( 2, 1) and > M in (1, 2), with M > 0 large enough, we get: (L 0 + 1)φ 0 in R. Neither is the definition (5.14) much meaningful as, in general, µ 1 =. This is seen next. Remark 5.2 Consider the following family of functions: k > 0, φ k (x) := e kx v, x R N, where v is an aritrary unit vector in R N. Straightforward computation yields: Lφ k = a ij (x)v i v j k 2 φ k ((x) v)kφ k c(x)φ k ( a k 2 + k + c )φ k, where a is given y (1.3) and (x) = ( 1 (x),, N (x)). Therefore, for every λ R there exists k > 0 large enough such that (L+λ)φ k 0 and then the quantity µ 1( L, R) defined y (5.14) is equal to. On the contrary, we have: Remark 5.3 The quantity λ 1( L, R N ) given y (1.7) satisfies: c λ 1( L, R N ) c. In fact, taking φ 1 in (1.7), we see that λ 1( L, R N ) c. For the other inequality, consider λ R and φ C 2 (R N ) W 2, (R N ) such that: φ > 0 and (L+λ)φ 0. Let M e the supremum of φ and (x n ) n N e a maximizing sequence for φ. For n N, define: x R N, θ n (x) := φ(x) (M φ(x n )) x x n 2. Arguing similarly to the proof of Lemma 7.2 elow, one can see that every function θ n has a local maximum in a point y n B 1 (x n ). Furthermore, lim n φ(y n ) = lim n φ(x n ) = M. We have: 0 (L + λ)φ(y n ) 2(φ(x n ) M)a ii (y n ) + 2(φ(x n ) M) i (y n )(y n x n ) i (c(y n ) + λ)φ(y n ). It follows that λ lim inf n c(y n) c. This shows that λ 1( L, R N ) c. 11

12 6 Self-adjoint operators in dimension N 3 The proof of Theorem 4.1 consists in a not so immediate adaptation of the proof of Proposition 3.3. It makes use of the following oservation, which holds in any dimension N. Lemma 6.1 Let φ C 2 (R N ) e a nonnegative function. Let Λ(x) e the largest eigenvalue of the matrix ( ij φ(x)) ij and assume that Λ := sup x R N Λ(x) < +. Then we have: x R N, φ(x) 2 2Λφ(x). (6.15) Proof. First, if Λ 0 then, for every i = 1,, N, ii φ 0. This shows that φ is concave in every direction x i and then, since it is nonnegative, is constant. In particular, (6.15) holds. Consider the case Λ > 0. The Taylor expansion of φ at the point x R N gives: y R N, φ(y) = φ(x) + φ(x)(y x) ijφ(z)(y x) i (y x) j, where z is a point lying on the segment connecting x and y. Hence, 0 φ(y) φ(x) + φ(x)(y x) Λ y x 2. If we take in particular y = x φ(x)/λ we otain: and the statement is proved. 0 φ(x) φ(x) 2 2Λ Note that, if φ is a positive function in W 2, (R N ), then Lemma 6.1 yields that its gradient is controlled y the square root of φ. Actually, this is the reason why in (1.7) we take test functions in W 2, (R N ). Proof of Theorem 4.1. Let λ R N e such that there exists a positive function φ C 2 (R N ) W 2, (R N ) satisfying: (L + λ)φ 0. We would like to proceed as in the proof of Proposition 3.3, with ϕ p replaced y φ, and otain λ 1 ( L, R N ) λ. This is not possile ecause, in general, φ is not ounded from elow away from zero. Lemma 6.1 allows to overcome this difficulty. Consider in fact the same type of cutoff functions (χ R ) R 1 as in Proposition 3.3 and call λ R the principal eigenvalue of L in with Dirichlet oundary conditions. The representation formula (3.12), yields for R 1: ] λ R [ aij (x) i (φχ R ) j (φχ R ) c(x)φ 2 χ 2 R φ 2 χ 2 R. 12

13 Hence, since χ R = 1 on 1, we get: λ R λ C R [ 2aij (x)( i φ)( j χ R )φχ R + i (a ij (x) j χ R )φ 2 χ R ] φ 2 χ 2 R Our aim is to prove that, y appropriately choosing the cutoff functions (χ R ) R 1 we get: [ ] 2aij (x)( i φ)( j χ R )φχ R + i (a ij (x) j χ R )φ 2 χ R C lim sup R 0. (6.16) R φ 2 χ 2 R These conditions are: x \ 1/2, ( ) 1 χ R (x) = exp, x R x 1/2, χ R (x) e 1/2. By direct computation, we see that, for x \ 1/2, and ij χ R (x) = χ R (x) = x x (R x ) 2 exp ( 1 x R [( xi x j x δ ) ij ( x R) x ix j 3 x x ( x R) + x ] ( ) ix j 1 ( x R) 4 exp. 2 x 2 x R Consequently, using the usual summation convention, we have: x \ 1/2, ), i (a ij (x) j χ R ) [a C( x R)] ( x R) 4 exp. ( ) 1, x R where C only depends on N and the W 1, norm of the a ij (and not on R) and a is given y (1.3). Therefore, there exists h independent of R with 0 < h 1/2 and such that i (a ij (x) j χ R ) 0 in \ h. Since χ R > exp( h 1 ) in h, it is possile to chose C large enough, independent of R, such that i (a ij (x) j χ R ) C χ R in. On the other hand, owing to Lemma 6.1, we can find another constant C > 0, depending only on N, a ij L (R N ), φ W 2, (R N ) and χ R W 2, (R N ) (which does not depend on R), such that a ij (x)( i φ)( j χ R ) C φ 1/2 χ 1/2 R. Assume, y way of contradiction, that (6.16) does not hold. There exist then ε > 0 and R 0 1 such that, for R R 0, [ ] ε φ 2 χ 2 R 2aij (x)( i φ)( j χ R )φχ R + i (a ij (x) j χ R )φ 2 χ R C R ( ) C χ 2 Rφ 2 + 2C φ 3/2 χ 3/2 R. C R 13

14 Since φ and χ R are ounded, the aove inequalities yield the existence of a positive constant k such that, for R R 0, k φ 2 χ 2 R φ 3/2 χ 3/2 R. C R Notice that, since φ > 0, we can choose k > 0 in such a way that the aove inequality holds for any R 1. Using the Hölder inequality, with p = 4/3 and p = 4, we then otain: R 1, ( φ 2 χ 2 R k 1 C R φ 2 χ 2 R ) 3/4 C R 1/4 K 1 R N 1 4 ( C R φ 2 χ 2 R ) 3/4, ( 3/4. where K is another positive constant. For n N call α n := C φ2 n χn) 2 Since for n N we have that φ 2 χ 2 n = B n n 1 φ 2 + C j φ 2 χ 2 n C n n φ 2 χ 2 j, C j it follows that j=1 α n Kn 1 N 4 n j=1 j=1 α 4/3 j. (6.17) We claim that the sequence (α n ) n N grows faster than any power of n. This is in contradiction with the definition of α n, ecause ( ) 3/4 α n = φ 2 χ 2 R φ 2 L (R N ) C n 3/4 Hn 3 4 (N 1), C n for some positive constant H. To prove our claim, we use (6.17) recursively. At the first step we have: α n K 0 n β 0, where K 0 = Kα 4/3 1 and β 0 = 1 N. At the second step we get: 4 α n KK 4/3 0 n 1 N n 4 j=1 j 4 3 β 0. If β 0 > 3/4 (i.e. if N < 4) then n j=1 j 4 3 β 0 n 4 3 β0+1. Hence, there exists K 1 > 0 such that α n K 1 n β 1, where β 1 = 4β N. Proceeding at the 4 same way we find, after m steps, that α n K m n βm, where K m is a positive constant and β m = 4β 3 m N, provided that β 4 0,..., β m 1 > 3/4. If β m 1 > 3/4, we have that β m > β m 1 β m 1 > 3 (N 5). 4 Since β 0 > 3 (N 5) N < 4, 4 it follows that for N 3 the sequence (β m ) m N is strictly increasing. Thus, lim m β m = + if N 3, ecause the only fixed point of the sequence x m = 4x 3 m N is 3 (N 5) 4 4 which is lesser than β 0. Therefore, as n, a n goes to infinity faster than any polynomial in n. 14

15 7 Limit periodic operators Throughout this section, we consider limit periodic elliptic operators L. Definition 4.2, we let either According to if L is a general operator, or L n u = a n ij(x) ij u + n i (x) i u + c n (x)u L n u = i (a n ij(x) j u) + c n (x)u if L is self-adjoint. We denote with λ n and ϕ n respectively the periodic principal eigenvalue and a positive periodic principal eigenfunction of L n in R N. Our results make use of the following Lemma. Lemma 7.1 The sequence (λ n ) n N is ounded and lim (L L n )ϕ n n ϕ n = 0. L (R N ) Proof. We can assume, without loss of generality, that the operators L, L n are general elliptic. Since the operators L n are periodic, from Proposition 3.2 and Remark 5.3 it follows that c n λ 1( L n, R N ) = λ n c n. Hence, the sequence (λ n ) n N is ounded ecause c n c in C 0,α (R N ). For all n N, the functions ϕ n satisfy (L n + λ n )ϕ n = 0. Then, using interior Schauder estimates, we can find a constant C n > 0 such that x R N, ϕ n C 2,α (B 1 (x)) C n ϕ n L (B 2 (x)), where the C n are controlled y λ n and a n ij C 0,α (R N ), n i C 0,α (R N ), cn C 0,α (R N ). We know that the λ n are ounded in n N, and the same is true for the C 0,α norms of a n ij, n i and c n ecause they converge in the C 0,α norm to a ij, i and c respectively. Thus, there exists a positive constant C such that C C n for every n N. Moreover, applying the Harnack inequality for the operators (L n + λ n ), we can find another positive constant C, which is again independent of n (and x), such that x R N, ϕ n L (B 2 (x)) C ϕ n (x). Therefore, sup x R N (L L n )ϕ n (x) ϕ n (x) ( a ij sup an x R N ij + i n i + c c n ) ϕ n C 2,α (B 1 (x)) ϕ n (x) CC ( a ij a n ij + i n i + c c n ), 15

16 which goes to zero as n goes to infinity. Proof of Theorem 4.3. For n N define: H n := (L L n )ϕ n ϕ n L (R N ). (7.18) By Lemma 7.1, we know that lim n H n = 0. Since (L + λ n )ϕ n H n ϕ n, it follows that (L + λ n H n )ϕ n 0 and (L + λ n + H n )ϕ n 0. Hence, using ϕ n as a test function in (1.6) and (1.7), we infer that λ 1 ( L, R N ) λ n H n and λ 1( L, R N ) λ n + H n, for every n N. The proof is complete ecause, passing to the lim inf and the lim sup as n goes to infinity in the aove inequalities, we get: λ 1( L, R N ) lim inf n λ n lim sup λ n λ 1 ( L, R N ). (7.19) n The proof of Theorem 4.4 is divided into two parts, the first one eing the next Lemma. Lemma 7.2 The sequence (λ n ) n N converges to λ 1( L, R N ) as n goes to infinity. Proof. Proceeding as in the proof of Theorem 4.3, we derive (7.19). So, we only need to show that lim sup n λ n λ 1( L, R N ). To this end, consider a constant λ λ 1( L, R N ) such that there exists a positive function φ C 2 (R N ) W 2, (R N ) satisfying (L + λ)φ 0. Fix n N and define ψ n := k n ϕ n φ, where k n is the positive constant (depending on n) such that inf ψ n = 0 (such a constant always exists - and it is unique - ecause ϕ n is ounded from elow away from zero and φ is ounded from aove). From the inequalities (L + λ)ψ n k n (L + λ)ϕ n = k n (L n L)ϕ n + k n (λ n λ)ϕ n and defining H n as in (7.18), we find that (L + λ)ψ n k n (λ n λ H n )ϕ n. (7.20) Since inf ψ n = 0, there exists a sequence (x m ) m N in R N such that lim m ψ n (x m ) = 0. For m N, define the functions: θ m (x) := ψ n (x) + ψ n (x m ) x x m 2, x R N. Since θ m (x m ) = ψ n (x m ) and θ m (x) ψ n (x m ) for x B 1 (x m ), for any m N, there exists y m B 1 (x m ) point of local minimum of θ m. Hence, and 0 = θ m (y m ) = ψ n (y m ) + 2ψ n (x m )(y m x m ) 0 ( ij θ(y m )) ij = ( ij ψ n (y m )) ij + 2ψ n (x m )I, where I denotes the N N identity matrix. Thanks to the ellipticity of L, we then get: (L + λ)ψ n (y m ) 2ψ n (x m )a ii (y m ) + 2ψ n (x m ) i (y m )(y m x m ) i (c(y m ) + λ)ψ n (y m ). (7.21) 16

17 Furthermore, since θ m (y m ) = ψ n (y m ) + ψ n (x m ) y m x m 2 θ m (x m ) = ψ n (x m ), we see that ψ n (y m ) ψ n (x m ). Consequently, taking the limit as m goes to infinity in (7.21), we derive: lim sup m (L + λ)ψ n (y m ) 0. Therefore, y (7.20), lim sup k n (λ n λ H n )ϕ n (y m ) 0, m which implies that λ n λ H n 0 ecause inf R N ϕ n > 0. Since y Lemma 7.1 we know that H n goes to zero as n goes to infinity, it follows that λ lim sup(λ n H n ) = lim sup λ n. n n Taking the infimum over all such constants λ we finally get: λ 1( L, R N ) lim sup n λ n. Proof of Theorem 4.4. Owing to Theorem 4.3, it only remains to show that λ 1 ( L, R N ) λ 1( L, R N ). To do this, we fix R > 1 and n N and proceed as in the proof of Proposition 3.3, replacing the test function ϕ p y ϕ n. We this get: λ 1 ( L, ) ( L(ϕn χ R ) ) ϕ n χ R = λ n ϕ 2 nχ 2 R = 1 ((λ n + L n L)ϕ n )ϕ n C R ( L(ϕn χ R ) ) ϕ n χ R ϕ 2 nχ 2 R 1 ((L L n )ϕ n )ϕ n + C R ( (L + λn )ϕ n χ R ) ϕn χ R ϕ 2 nχ 2 R Setting H n as in (7.18), we get: λ 1 ( L, ) λ n + H n 1 ϕ 2 n + K n C R, ϕ 2 nχ 2 R where C R denotes the measure of the set C R and K n is a positive constant (independent of R ecause the χ R are uniformly ounded in C 2 (R N )). Therefore, since min R N ϕ n > 0, there exists another constant K n > 0 such that λ 1 ( L, ) λ n + H n + K n R. Letting R go to infinity in the aove inequality and using Proposition 3.1, this shows that: λ 1 ( L, R N ) λ n + H n. By Lemmas 7.1 and 7.2, we know that H n 0 and λ n λ 1( L, R N ), as n goes to infinity. Thus, we conclude that λ 1 ( L, R N ) λ 1( L, R N ).. 17

18 8 The inequality λ 1 λ 1 in dimension N = 1 In this section, we are concerned with general elliptic operators in dimension one, that is operators of the type: Lu = a(x)u (x)u c(x)u, x R, with the usual regularity assumptions on a,, c. The elliptic condition ecomes: a a(x) a, for some constants 0 < a a. Proof of Theorem 4.5. Fix R > 0 and call λ R and ϕ R the principal eigenvalue and eigenfunction respectively of L in ( R, R), with Dirichlet oundary condition. Then define ψ R (x) := h k e k(x R), x R where h, k are two positive constants that will e chosen later. The function ψ R satisfies: ( (L + λ R )ψ R = a(x)k + (x) (c(x) + λ R ) 1 ) he k(x R). k There exists k 0 > 0 (independent of h) such that (L + λ R )ψ R < 0 in R for any choice of k k 0. Our aim is to connect smoothly the functions ϕ R and ψ R in order to otain a function φ R C 2 ([0, + )) W 2, ([0, + )) satisfying (L + λ R )φ R 0. To this end, we set: g R (x) := η(x R + δ) 3, with η, δ > 0 to e chosen. Since (L + λ R )g R = [ 6a(x) 3(x)(x R + δ) (c(x) + λ R )(x R + δ) 2 ]η(x R + δ), we can find a constant δ 0 > 0 such that (L + λ R )g R 0 in (R δ, R), for any choice of 0 < δ δ 0. Then we define: ϕ R (x), for 0 x R δ φ R (x) := ϕ R (x) + g R (x), for R δ < x R (8.22) ψ R (x), for x > R. It follows that, if k k 0 and δ δ 0, then the function ϕ R satisfies (L + λ R )φ R 0 in (0, R δ) (R δ, R) (R, + ). In order to ensure the C 2 regularity of φ R, we need to solve the following system in the variales h, k, η, δ: ηδ 3 = h k ϕ R (R) + 3ηδ2 = h ϕ R (R) + 6ηδ = hk. One can see that, if h < ϕ R (R) (notice that ϕ R (R) < 0 y the Hopf s lemma), the previous system ecomes, after some simple algera, γ(h) = ϕ R(R)δ δk = 3h ϕ R (R) h η = hk ϕ R (R), 6δ 18 (8.23)

19 where γ(h) := 3h 2 ϕ R (R) h + 2(h + ϕ R(R)). We want to show that there exists δ small enough such that the system (8.23) admits positive solutions δ, h δ, k δ, η δ satisfying: δ δ 0, h δ < ϕ R(R), k δ k 0. (8.24) Let 0 < δ 1 δ 0 e such that ϕ R (R) δ 1 < ϕ R (R). Thus, if δ δ 1, the first equation of (8.23) yields: γ(h) < ϕ R (R). Since γ(0) = 2ϕ R (R) and lim h ϕ R (R) γ(h) = +, there exists a constant 0 < h 1 < ϕ R (R) such that, for any choice of δ (0, δ 1), the first equation of (8.23) admits a solution h δ [h 1, ϕ R (R)). For δ (0, δ 1) and h = h δ, the second equation of (8.23) gives: 3h δ k δ = ϕ R (R) h δ 1 3h 1 δ ϕ R (R) h δ 1. (8.25) 1 Hence, for δ small enough, we have k δ k 0. Finally, y the last equation of (8.23), for δ (0, δ 1 ), we have η δ = h δk δ ϕ R (R) h 1k δ ϕ R (R), 6δ 6δ and then, since k δ satisfies (8.25), η δ > 0 for δ small enough. Therefore, there exist four positive constants h, k, η, δ solving (8.23) and satisfying (8.24). With this choices of h, k, η, δ, the function φ R is in C 2 ([0, + )) W 2, ([0, + )). Proceeding as aove, we can extend ϕ R (x) for x negative, and get a function φ R C 2 (R) W 2, (R) such that (L + λ R )φ R 0 in R. Using φ R as a test function in (1.7), we find that λ 1( L, R) λ R. Thus, passing to the limit as R goes to infinity, y Proposition 3.1, we derive λ 1( L, R) λ 1 ( L, R). The proof is therey complete. Remark 8.1 Using the same type of construction as in Theorem 4.5, one can prove that the inequality λ 1( L, R N ) λ 1 ( L, R N ) holds for any elliptic operator L which is rotationally invariant. Consider in fact an elliptic operator of type Lu = a( x ) u ( x ) x x u c( x )u, in RN, with (0) = 0 and with the usual ellipticity and regularity assumptions on the coefficients. For R > 0, let λ R and ϕ R respectively denote its Dirichlet principal eigenvalue and eigenfunction in. It is easy to see that, for any orthogonal matrix M, the function ϕ R (Mx) is again a Dirichlet positive eigenfunction of L in. Hence, y uniqueness of the principal eigenfunction up to multiplication, it follows that ϕ R (x) ϕ R (Mx), that is ϕ R is a radial function. Since for any radial function u = u( x ) the expression of Lu reads: ( Lu = a( x )u + ( x ) + N 1 a( x ) x ) u + c( x )u, we can proceed as in the one-dimensional case and uild a radial function φ R C 2 (R N ) W 2, (R N ) such that (L + λ R )φ R 0. Therefore, λ 1( L, R N ) λ R and then, passing to the limit as R, we otain the stated inequality etween λ 1 and λ 1 19

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