EIGENVALUES OF THE LAPLACE-BELTRAMI OPERATOR UNDER THE HOMOGENEOUS NEUMANN ON A ZONAL THE UNIT

. We consider the eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a spherical domain. Especially, we investigate the case when the domain is a large zonal one and letting the zone larger so that the zone covers the whole sphere as a limit. We discuss the behavior of eigenvalues according to the rate of expansion of the zone.

1. Introduction. In this paper we study the eigenvalue problem    Λ n u + λu = 0 in Ω ε,η ⊂ S n ⊂ R n+1 , ∂ n u = 0 on ∂Ω ε,η (1.1) where Λ n is the Laplace-Beltrami operator on the unit sphere S n , n ≥ 2, the domain x 2 i = 1, cos(π − ε) < x n+1 < cos η} is the zonal domain with the geodesic distance from the North Pole (0, 0, . . . , 0, 1) ∈ R n+1 between η > 0 and π − ε ( η > 0 and ε > 0 are small), and ∂ n denotes the derivative in the direction of the outer normal. In the precedent work by Bandle, Kabeya and Ninomiya [1], we studied the case when η = 0 (large spherical cap, or one hole case) and showed the asymptotic behavior of the eigenvalues as ε → 0. We also studied the nonlinear problem there from the bifurcation-theoretic point of view.
Here, we consider the large zonal domain on the unit sphere and enlarge the domain so that the domain covers the whole sphere.
Concerning the relation between the perturbation of domains and the behavior of eigenvalues, Ozawa [11,12,13] was one of the pioneers, who studied the dependence of the eigenvalues of the Laplacian with the homogeneous Dirichlet boundary condition, under small perturbations of the domain. Also, he assumed that the eigenvalue of the unperturbed problem is simple, which does not always hold, and which is not the case for our spherical domain.
Extensions to manifolds are found in Courtois [3], see also the references cited therein. In contrast to the Dirichlet case very little is known for the behavior of eigenvalues with the Neumann boundary conditions under the domain perturbation. Some results for simple eigenvalues in planar domains with holes are found in Lanza de Cristoforis [6]. Also several interesting properties of eigenvalues of the Laplacian under the Neumann condition on a domain in the Euclidean space are discussed in Chapter 1 in Ni [9] based on Ni and Wang [10]. The authors stress that the domain monotonicity property of the eigenvalues does not necessarily hold.
The asymptotic behavior of eigenvalues of −Λ n under the Robin condition with η = 0, ε > 0 when n = 2, 3, was discussed in Kabeya, Kawakami, Kosaka and Ninomiya [5]. The difference of the asymptotic behavior of the eigenvalues subject to the boundary condition is observed. More precisely, the asymptotic behavior of those under the Dirichlet condition or under the Neumann condition is different from the Robin case. In fact, for example, the three dimensional case, the asymptotic behavior as ε → +0 of the k-th eigenvalue whose eigenfunction depends only on x 4 = cos θ is as follows: for the Dirichlet boundary condition λ = k(k + 2) + C * 2 ε 2 + o(ε 2 ) for the Robin boundary condition for the Neumann boundary condition with C * i (i = 1, 2, 3) being nonzero constant. The Dirichlet case is due to Macdonald [7] and the Neumann case is due to [1].
We shall treat the problem (1.1) as a perturbation in and η of the problem Λ n u + λu = 0 in S n . In the whole sphere case, all the eigenvalues are completely understood. That is, the eigenvalues of Λ n Y + σY = 0 in S n are σ k,n := k(k + n − 1), k = 0, 1, 2, . . . and the corresponding eigenfunctions are the regular spherical harmonics Y (k,n) of degree k. The multiplicity of σ k,n is d(k, n) := (2k + n − 1) (k + n − 2)! (n − 1)!k! , which can be found in Müller [8] or in Shimakura [14]. The corresponding eigenfunctions Y (k,n) are expressed in terms of the associated Legendre polynomials as it will be seen in Section 2 (see also [8]). The eigenfunction corresponding to k = 0 is the constant. We may always assume that η is at most of ε order since the equation Λ n u+λu = 0 is unchanged under the transformation x n → −x n .
(ii): If η = κε + o(ε) as ε → 0 with some κ > 0, then there holds λ k,ε,η,m − k(k + n − 1) = C k,m,n,κ ε nm + o(ε nm ) as ε → 0 with some constant C k,m,n and with C k,m,n,κ . Here n m = max{2m + n − 2, n}. Remark 1. The constants C k,m,n and C k,m,n,κ can be computed explicitly and C k,0,n = C k,1,n (cf. Sections 4, 5 and 6). Moreover, we see the following: (i): If η is of small order of ε, then the first approximation does not depend on η. In fact, C k,m,n is the same as that obtained for the large spherical cap as in [1]. Only when η is of the same order of ε, the dependence appears. (ii): The multiplicity of λ k,ε,η,m is d(m, n − 1), which is the same as that with η = 0 in [1]. (iii): If η is "exponentially small", that is, η is of the same order as ε r with some r(≥ 2) ∈ N, then in a higher order expansion, we will see that the difference in some order of ε would appear (see also Remark 2). (iv): If η is of exp(−1/ε) oder or even smaller than this, then we will not see the difference of the expansion of the eigenvalues. Asymptotically, we cannot hear the difference of sound of the zonal drum if one of the holes radius is exponentially small to the other. (v): Similarly, we can treat the asymptotic behavior of the eigenvalues under the Robin condition. However, more complicated calculations are necessary and we do not do that here and postpone to a future work. Similar difference as in [5] would be found.
This paper is organized as follows: In the next section, Section 2, we give the exact form of eigenfunctions. We give a strategy to a proof of Theorem 1.1 in Section 3. An actual proof is given in Sections 4, 5 and 6 depdending on the dimension. Formulas on the associated Legendre functions, the Gauss hypergeometric functions, and the Gamma function and di-Gamma function are listed in Section 7 as appendix.
By the Weierstrass polynomial approximation theorem (see also Titchmarsh [15,16]), any eigenfunction to (1.1) is obtained by separation of variables. We are therefore looking for solutions of (1.1) of the form v = U 1 (t)Y (ζ). Y is an eigenfunction of −Λ n−1 on S n−1 corresponding to the eigenvalue k(k + n − 2) for any k ∈ N. Recalling the ingredients of [1], we see that the eigenfunctions Y of −Λ n−1 on S n−1 are expressed as Y =P k2+(n−2)/2 ν (cos θ 1 ) (sin θ 1 ) (n−2)/2 n−2 j=1P k n−j+1 +(j−1)/2 , if µ is a half integer. For the latter use, we set In the polar coordinate, the boundary condition yields d dθ Then P(t) is a solution to and the boundary condition (2.1) yields in terms of t as where τ η = cos η ∼ 1 − η 2 /2, t := cos(π − ) ∼ −1 + 2 /2 and we have used the notation := d/dt. Since the associated Legendre functions P µ ν (t) and Q µ ν (t) are fundamental solutions of the associated Legendre equation (see, e.g., Section 9.5, p.318 of Beals and Wong [2]), we see that with m being a natural number. In the following, we fix m. For a half-integer µ, by (61) in p. 230 in Hobson [4], we see that Thus the eigenfunction P has the form P(cos θ) = C 1 P µ ν (cos θ) + C 2 P −µ ν (cos θ) P(cos θ) = C 1 P µ ν (cos θ) + C 2 Q µ ν (cos θ) and we determine the relation between C 1 and C 2 then we do the behavior of ν from (2.2). In the case of a spherical cap, we use one of P µ ν or Q µ ν since one of them always has a singularity at t = 1 (at the North Pole). In this case, we need both of them to represent the eigenfunctions. According to the boundary conditions, ν is determined and hence the eigenvalues are done. In the following sections, we investigate the behavior of ν and give a proof of Theorem 1.1.
3. Basic strategy to a proof of Theorem 1.1. In this section we discuss the behavior of the eigenvalues as η → 0 and ε → 0. We compute the eigenvalues by carefully analyzing the associated Legendre functions introduced in Section 2.
Since the existence of eigenvalues is ensured by Lemma 3.1, we concentrate on the asymptotic behavior of ν as η → 0 and as ε → 0.
First we note that for odd n. For n even, we have Using the recursion formula (7.3) in Section 7 for odd n, we have Similarly, from (7.4) in Section 7, for even n we have In the following three sections, we first deduce the relation between C 1 and C 2 and then determine the behavior of eigenvalues by investigating the behavior of ν as η and ε go to zero.

4.
Proof of Theorem 1.1: The odd dimensional case. In the proof, we use the symbol f (x) g(x) in the sense that at least two leading terms are the same. That is, suppose that functions f (x) and g(x) defined for x > 0 are expanded near where a 1 , a 2 , p, q are constants. In such a case, we write is regarded as the second approximation of f (x) near x = 0. Hereafter, we do not use the symbol like o(x p ). First, we calculate the ratio between C 1 and C 2 from (3.1). We use the expression of P µ ν in terms of the Gauss hypergeometric function, which can be found in p. 319 Since η is close to 0, we may regard cos η = 1 − η 2 /2 and put µ = m + (n − 2)/2. Then we have and similarly, we get As for the negative super suffix, we have Hence we obtain Simplifying the relation above, we have Using the relation (7.6) in Section 7, we have Hence we see that and we see that For the moment, we do not need the exact value, we write We recall that we have defined µ as µ = m + (n − 2)/2 and denote n m := max{2m + n − 2, n}.
In the odd dimensional case, however, the Legendre function P µ ν (t) has a singularity at t = −1 and we need to use the conversion formula (7.2) in Section 7 and the relation between the associated Legendre function and the Gauss hypergeometric functions (4.1) to analyze the behavior of P µ ν near the singularity. In terms of the Gauss hypergeometric functions, the right-hand side of (4.4), which is denoted byA, is expressed as According to the conversion formulas (7.2) in Section 7, we see from the right-hand side of (4.4) that .
Also, from the left-hand side of (4.4), we have by using the conversion formula again. As there hold for any small ε > 0, we may replace the terms which have cos ε and cos(π − ε) by those as above mentioned ones. Since (4.5) and (4.6) express asymtotically the same value, we have the following relation: Hence we obtain (4.7) We multiply both sides of (4.7) by (ε 2 /4) −µ/2 , we have . (4.8) As before we recall ν = k + (n − 2)/2 + ν η,ε with ν η, → 0 as η, → 0 and µ = m + (n − 2)/2. Then we see that We note the fact that the Gamma function has a pole (see Lemma 7.1) of order one for m ≤ k and that the following holds Thus, the terms having Γ(m − ν), Γ(−µ − ν − 1) in the left-hand side of (4.8) converges to 0 as ε → 0. On the other hand, as we have seen in [1], only the case when the Γ(µ − ν) and ε −2µ are of the same order is admissible. If η → 0 in such a way as η/ε → 0 as ε → 0, then the left-hand side of (4.8) converges to 0 when m ≥ 1, while the right-hand side of (4.8) is of oder ε 2 when m = 0 as in Subsection 3.1 in [1]. Even in this case, if η/ε → 0 as ε → 0, then the left-hand side is of smaller order of ε 2 . Thus, we conclude that ν ε has the same order as shown in Subsection 3.1 of [1] and the leading term depends only on ε and is determined as The constant c k,m,n is calculated as in Theorem 1.1 in [1] by letting the left-hand side of (4.8) be zero and we have Remark 2. Since the Gauss hypergeometric functions are expanded in power series, we can expand (4.5) and (4.6) more. If η = o(ε 2 ), then we will see that ν η,ε = c k,m,n ε max{2m+n−2,n} +c k,m,n ε max{2m+n−2,n}+2 + o(ε max{2m+n−2,n}+2 ) The constantc k,m,n is determined in the similar way. Thus in principle, we can prove (iii) and (iv) in Remark 1.
Hence, the equality above is written as In view of the famous formula we see that (4.11) Thus, we get Substituting (4.2) for C 0 (m), we obtain c k,m,n,κ = (−1) When m = 0. We may assume that ν η,ε = c k,0,n,κ ε n (1 + o (1)). Dividing both side of (4.8) by ε n and taking the limit, we have .

5.
Proof of Theorem 1.1: In the case of the even dimensional case (except for n = 2 and m = 0). In this section, we put with ν ν,ε → 0 as ν → 0, ε → 0. If n is even, M is an integer and ν converges to also an integer. This causes difficulty. In this case, Q M ν (t) is expressed as Thus Q M ν (t) has singularities at t = ±1. As in [1], we effectively use the function U defined as below.
Let α, β be non-integer values and be a positive integer. We define the new function U (α, β, , x) as follows where ψ(z) is the psi (or di-Gamma) function, whose properties are listed in Lemmas 7.1 and 7.2 in Section 7, defined as provided α + β + 1 − is a non-positive integer.
Since η > 0 is close to 0, we have As we treat the even dimensional case, we substitute M = m + (n − 2)/2 and ν = k + (n − 2)/2 +ν with k ≥ m and withν → 0. In the following sections, we always substitute those expressions for M and ν, respectively. Now we evaluate the right-hand side of (5.5) B: and we see that Finally, we obtain and Next, we calculate We use (5.2) to express the terms by using the Gauss hypergeometric functions. We have (5.7) Here we set and Hence, (3.3) yields Hence we obtain Now, we consider (3.4). The left-hand side of (3.4) has been calculated in Subsection 3.2 in [1] and from that, we see (5.9) Finally, we need to calculate the right-hand side of (3.4). Using (5.1) and (5.2), we first write the right-hand side of (3.4) in terms of the Gauss hypergeometric functions. We have (ν + n 2 )(cos ε)Q M ν (cos(π − ε)) + (ν − M + 1)Q M ν+1 (cos(π − ε)) (5.10) ) .

(5.15)
Hence we see thatν = c k,0,n,κ ε n + o(ε n ) in view of (5.15). The leading terms in (5.15) is the terms of ε 4−n order and we have Thus, the coefficient c k,0,n,κ is a roof of the quadratic equation In either case, due to Lemma 3.1, c k,m,n,κ is uniquely determined (we do not write down its exact form as its is too complicated). The proof of Theorem 1.1 is complete except for n = 2 and m = 0.
All the cases have been examined and the proof of Theorem 1.1 is complete.
7. Appendix. We here collect fundamental properties of the Gauss hypergeometric functions F (a, b, c; z), a recurrence formula for the associated Legendre functions, Γ(z) and ψ(z). For the Gauss hypergeometric function, there is a conversion formula, which is useful for the analysis on an odd dimensional case. Let a, b, c be real numbers such that none of c, a + b + 1 − c and c + 1 − a − b is a non-positive integer. Then for x ∈ (−1, 1), there holds 7.3. Recurrence formula. The general recurrence formulas for P µ ν and Q µ ν with its derivative is as follows . Here, µ and ν are real numbers and t varies within (−1, 1). There holds (see, e.g., (162) in p. 289 in [4]) (1 − t 2 ) d dt P µ ν (t) = (ν + 1)tP µ ν (t) − (ν − µ + 1)P µ ν+1 (t).
By this definition, we see that Γ(z) has a pole of order 1 at z = 0, −1, −2, . . . and the local behavior around one of these poles. The definition of the psi function is The psi function has the following property.
ψ(z + 1) = ψ(z) + 1 z . (7.5) The limiting behavior of the Gamma function and the di-psi function as follows. Also, the following relations are often used in the determination of the coefficient of eigenvalues.