COMMUNICATIONS ON doi:10.3934/cpaa.2016.15.287 PURE AND APPLIED ANALYSIS Volume 15, Number 1, January 2016 pp. 287–297
THE “HOT SPOTS” CONJECTURE ON HIGHER DIMENSIONAL SIERPINSKI GASKETS
School of Mathematical Science, Zhejiang University Hangzhou, 310027, China
(Communicated by Camil Muscalu)
Abstract. In this paper, using spectral decimation, we prove that the “hot spots” conjecture holds on higher dimensional Sierpinski gaskets.
1. Introduction. The “hot spots” conjecture was posed by J. Rauch at a confer- ence in 1974. Informally speaking, it was stated in [3] as follows: Suppose that D is an open connected bounded subset of Rd and u(t, x) is the solution of the heat equation in D with the Neumann boundary condition. Then for “most” initial conditions, if zt is a point at which the function x → u(t, x) attains its maximum, then the distance from zt to the boundary of D tends to zero as t tends to ∞. In other words, the “hot spots” move towards the boundary. Formally, there are several versions of the hot spots conjecture. See [3] for details. In this paper, we will use the following version: every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian attains its maximum and minimum on the boundary. The hot spots conjecture holds in many typical domains in Euclidean space, especially for certain convex planar domains and lip domains. For examples, please see [1,3,9]. On the other hand, Burdzy and Werner [5] and Burdzy [4] constructed interesting planar domains such that the hot spots conjecture fails. The underlying spaces in above works are domains in Euclidean space. Recently, Ruan [13] and Ruan and Zheng [14] proved that the conjecture holds on the classical Sierpinski gasket (SG2 for short) and the level-3 Sierpinski gasket (SG3 for short), where the Laplacian was introduced by Kigami for p.c.f. self-similar fractals [10, 11]. In this paper, we prove that the hot spots conjecture holds on higher dimensional Sierpinski gaskets. This generalizes the result on the classical Sierpinski gasket in [13].
1.1. Definitions and notations. In this subsection, we recall definitions and basic notations in [7, 10, 12, 15, 17].
2000 Mathematics Subject Classification. Primary: 28A80, 47A75; Secondary: 39A70, 47B39. Key words and phrases. Neumann Laplacian, “hot spots” conjecture, higher dimensional Sier- pinski gaskets, eigenvalue, eigenfunction, spectral decimation. The work is supported in part by NSFC grant 11271327, ZJNSFC grant LR14A010001 and the Fundamental Research Funds for the Central Universities of China. ∗ Corresponding author: Huo-Jun Ruan.
287 288 XIAO-HUI LI AND HUO-JUN RUAN
N−1 Let N ≥ 2 be a given positive integer. For qi ∈ R , i = 1, 2,...,N, such that −−→ q1qj, j = 2,...,N, are linearly independent, we define 1 F (x) = (x + q ), x ∈ N−1. i 2 i R Then there exists a unique nonempty compact subset KN ⊂ RN−1 satisfying KN = SN N N i=1 Fi(K ). We call K the (N −1)-dimensional Sierpinski gasket. For example, K3 is the classical Sierpinski gasket. m Let Σ = {1, 2,...,N} and Σ = {ω1ω2 ··· ωm| ωj ∈ Σ for all 1 ≤ j ≤ m} for ∗ S∞ m ∗ every positive integer m. Define Σ = m=1 Σ . For each ω = ω1 ··· ωm ∈ Σ , we call ω a word with length |ω| := m and define
Fω = Fω1 ◦ · · · ◦ Fωm . S Define V0 = {q1, q2, . . . , qN } and Vm = |ω|=m Fω(V0) for every positive integer m S∞ m. Fω(V0) is called a cell of Vm for every ω ∈ Σ . Define V∗ = m=1 Vm. It is N clear that K is the closure of V∗. Now we can define graphs Γm with vertex set Vm for all non-negative integers m. We define Γ0 to be the complete graph with edge relation ∼0 on the vertex set V0, i.e., qi ∼0 qj for all distinct elements qi, qj ∈ V0. For m ≥ 1, we define Γm to be the graph on the vertex set Vm with edge relation ∼m: x ∼m y if and only if x 6= y m and there is an ω ∈ Σ such that x, y ∈ Fω(V0). Definition 1.1. For any continuous function u on KN , we define the graph Lapla- cian ∆m for positive integers m by X ∆mu(x) = (u(y) − u(x)), x ∈ Vm \ V0.
y∼mx Following the work on the classical Sierpinski gasket (see Sections 2.2, 4.2 in [17] for details), we can define Laplacian and normal derivative on KN by Z −1 −m (m) ∆u(x) = lim r ψx dµ ∆mu(x), x ∈ V∗ \ V0, m→∞ KN −m X ∂nu(x) = lim r (u(x) − u(y)), x ∈ V0, m→∞ y∼mx (m) provided the limits exist, where r is the renormalization factor of energy, ψx is the (m) piecewise harmonic spline of level m satisfying ψx (y) = δxy for all y ∈ Vm, and µ is the standard self-similar measure on KN . Similarly as Section 2.2 in [17], we can R (m) 2 easily obtain that KN ψx dµ = N m+1 . On the other hand, from Proposition 4.1 N in [7], we know that r = N+2 . Thus, we have following definitions. Definition 1.2. Let f be a continuous function on KN . We say that u ∈ dom∆ N with ∆u = f on K \ V0 if N (N + 2)m∆ u(x) 2 m converges uniformly to f on V∗ \ V0 as m goes to infinity. N Definition 1.3. The normal derivative at a point qi ∈ V0 of a function u on K is defined to be m N + 2 X ∂nu(qi) = lim (u(qi) − u(y)) m→∞ N y∼mqi THE “HOT SPOTS” CONJECTURE 289 if the limit exists. Definition 1.4. A function u ∈ dom∆ is called an eigenfunction of the Neumann Laplacian with eigenvalue λ if
− ∆u = λu on V∗ \ V0, and ∂nu = 0 on V0. (1) 1.2. Statement of main results. It is well-known that we can list all Neumann eigenvalues of ∆ as follows: λ(1) ≤ λ(2) ≤ λ(3) ≤ · · · λ(n) ≤ · · · , where all Neumann eigenvalues are non-negative and of finite multiplicity, and the (i) only accumulation point of {λ }i≥1 is ∞. See e.g. Section 4.1 in [12]. A func- tion u is called an eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian if it is a Neumann eigenfunction with eigenvalue λ(2). Let EF2 be the set of all Neumann eigenfunctions corresponding to the eigenvalue λ(2) of ∆. Our main result of the paper is: N Theorem 1.5. The hot spots conjecture holds on K . That is, for every u ∈ EF2, we have min u(p) ≤ u(x) ≤ max u(p), ∀x ∈ KN . p∈V0 p∈V0 By using spectral decimation in [7, 15] (see Section2 for details), we know that N−1 EF2 is a linear space with a base {ui}i=1 , where N − 1 1 u (q ) = , u (q ) = − , ∀j 6= i. i i N i j N Let m m m xm = u1(F1 (q2)), ym = u1(F2 (q1)), zm = u1(F2 (q3)), ∀m ≥ 1. (2) In Section4, we prove that the following lemma holds on KN . Lemma 1.6. The hot spots conjecture holds if + xm ≤ max u1(p) and ym ≥ zm ≥ min u1(p), ∀m ∈ Z . (3) p∈V0 p∈V0 In section5, we prove that the inequality (3) holds on KN . Combining this with the above lemma, we know that Theorem 1.5 holds. We remark that the second author and Zheng [13, 14] also proved that the hot spots conjecture holds on SG2 and SG3 by showing that Lemma 1.6 and the inequal- ity (3) hold on these two sets. However, in [13, 14], the explicit expressions from u1|Vn to u1|Vn+1 are well-known, while in this paper, we obtain explicit expression by solving linear equations in Section 3.
2. Spectral decimation on KN . Spectral decimation studied by Shima and Fukushima [7, 15, 16] is one basic tool for describing the properties of eigenfunctions and eigenvalues of Dirichlet Laplacian and Neumann Laplacian on p.c.f. self-similar sets; see e.g. [2,6,8]. Let m be a nonnegative integer and um a function on Vm and λm a real num- ber. We call um a discrete Neumann eigenfunction and λm a discrete Neumann eigenvalue of ∆m if ( −∆mum(x) = λmum(x), x ∈ Vm \ V0, 2 P (u (x) − u (y)) = λ u (x), x ∈ V . y∼mx m m m m 0 290 XIAO-HUI LI AND HUO-JUN RUAN
p 1 Bp ={ } p Cp ={ }
p 4 p 2
p 3
4 Figure 1. Bp and Cp in K
∗ Given ω = ω1 ··· ωm ∈ Σ . We know that C := Fω(V0) is a cell of Vm. Assume that the vertices of C are pi, i = 1,...,N. Define 1 Son(C) = (p + p ): i > j . 2 i j 0 0 0 Given p ∈ Son(C). We say p is a brother of p if p ∈ Son(C) and p ∼m+1 p. Denote Bp the set of all brothers of p and define
Cp = Son(C) \ ({p} ∪ Bp). It is easy to check that N(N − 1) (N − 2)(N − 3) #(Son(C)) = , #(B ) = 2N − 4, #(C ) = . (4) 2 p p 2 See Figure1 for an example. Let Φ(x) = x(N + 2 − x) and ψ±(x) be inverse functions of Φ, i.e., s ! N + 2 4x (N + 2)2 ψ (x) = 1 ± 1 − , x ∈ − ∞, . ± 2 (N + 2)2 4
Theorem 2.1 (Spectral decimation theorem I, [15]). Given a positive integer m. Suppose that λm 6∈ {2,N + 2, 2N} and λm−1 = Φ(λm). (i). If u is a discrete Neumann eigenfunction of ∆m−1 with eigenvalue λm−1, then there exists a unique extension ue on Vm such that ue is a discrete Neumann eigenfunction of ∆m with eigenvalue λm. Furthermore, ue take values on Vm by the N following method: Let {pi}i=1 be vertices of a Vm−1 cell. For every distinct elements 1 pi, pj in Vm−1, let p = 2 (pi + pj). Then N 1 X u(p) = (u(p ) + u(p ) + u(p )) e N + 2 i j k k=1 X X − λmaN ue(p) − λmbN ue(r) − λmcN ue(r), (5) r∈Bp r∈Cp
N+6 3 1 where aN = − 2N(N+2) , bN = − 2N(N+2) , cN = − N(N+2) . (ii). Conversely, if u is a discrete Neumann eigenfunction of ∆m with eigenvalue
λm, then u|Vm−1 is a discrete Neumann eigenfunction of ∆m−1 with eigenvalue λm−1. THE “HOT SPOTS” CONJECTURE 291
(iii). If λm is a discrete Neumann eigenvalue of ∆m, then the multiplicity of λm of ∆m equals that of λm−1 of ∆m−1. Theorem 2.2 (Spectral decimation theorem II, [7]). (i). Given a nonnegative integer m0, let u be a discrete Neumann eigenfunction of ∆m0 with eigenvalue λm0 .
Assume that {λm}m≥m0 is an infinite sequence related by λm−1 = Φ(λm), with all but a finite number of λm = ψ−(λm−1). If we define
N m λ = lim (N + 2) λm (6) 2 m→∞ and extend u to V∗ by successively using (5), then u is a Neumann eigenfunction of ∆ with eigenvalue λ. (ii). Every Neumann eigenvalue and its corresponding Neumann eigenfunctions of ∆ can be obtained by the process described in (i).
(iii). Let {λm}m≥m0 and λ be defined as in (i). Then the multiplicity of λ of ∆ equals that of λm0 of ∆m0 .
Let Λm be the set of all discrete Neumann eigenvalues of ∆m. It is easy to calculate that Λ0 = {0, 2N} while the multiplicity of 0 and 2N are 1 and N − 1, respectively. Furthermore, every eigenfunction corresponding to the eigenvalue 0 is a constant function on V0, while every eigenfunction u corresponding to the eigenvalue 2N satisfies X u(x) = 0.
x∈V0 By using the spectral decimation (Proposition 3.3 in [15]), we have Λ1 = {0,N, 2N} and
Λm+1 \{N + 2, 2N} = {ψ±(z)| z ∈ Λm}\{2,N + 2}, m = 1, 2,... In the sequel we will always define
λ1 = N, λm+1 = ψ−(λm), ∀m ≥ 1. (7)
It is easy to check that 0 < λ2 < 1 < λ1. Thus 0 < λm < 1 for all m ≥ 2 since ψ− (N+2)2 is increasing on (0, 4 ]. N m Theorem 2.3. Let {λm}m≥1 be defined as in (7) and λ = 2 limm→∞(N + 2) λm. Then λ is the second-smallest Neumann eigenvalue of ∆. Furthermore, the multi- plicity of λ equals N − 1. Proof. We can use the same method in the proof of Theorem 3.3 in [14]. Thus we omit the details.
N Recall that EF2 is the set of all Neumann eigenfunctions on K corresponding (2) to the eigenvalue λ of ∆. Let u ∈ EF2. In case that N ≥ 3, we have λ1 6∈
{2,N + 2, 2N}. Thus, from Theorem 2.1, u|V0 is a discrete Neumann eigenfunction of ∆0 with eigenvalue λ0 = Φ(λ1) = 2N. Hence u satisfies N X u(qi) = 0, i=1 and the values of u on Vm+1 \ Vm, ∀m ≥ 0, can be recurrently determined by (5). Let ui ∈ EF2, 1 ≤ i ≤ N, be defined as N − 1 1 u (q ) = , u (q ) = − , ∀j 6= i. (8) i i N i j N 292 XIAO-HUI LI AND HUO-JUN RUAN
N−1 PN−1 Then it is easy to see that {ui}i=1 is a basis of EF2 and uN = − i=1 ui. In case that N = 2, we have λ1 = 2 ∈ {2,N +2, 2N}. However, Theorem 2.3 still holds. Let u ∈ EF2. It is easy to check that u(q1) + u(q2) = 0 and u(F1(q2)) = 0. Thus the multiplicity of λ1 of ∆1 is 1. Let u1 be defined by
1 1 u (q ) = , u (q ) = − , u (F (q )) = 0, (9) 1 1 2 1 2 2 1 1 2 and the values of u1 on Vm+1 \ Vm, ∀m ≥ 1, are recurrently determined by (5). Then {u1} is a basis of EF2.
3. Solution of equations (5). We can obtain explicit expression from (5) as fol- lows.
N Lemma 3.1. Let {pj}j=1 be the vertices of a cell C of Vm−1, where m ≥ 1. If N 1 u gives values on {pj}j=1 and ue is extension to { 2 (pi + pj)}i,j by (5) with λm 6∈ {2,N + 2}, then for every distinct elements i, j ∈ {1,...,N},
1 X u (p + p ) = α(λ )[u(p ) + u(p )] + β(λ ) u(p ), (10) e 2 i j m i j m k k6=i,j
4−λ 2 where α(λ) = (2−λ)(N+2−λ) , β(λ) = (2−λ)(N+2−λ) .
1 Proof. Notice that (5) are linear equations of {ue( 2 (pi + pj))}i6=j. Thus, by the N symmetry of K , it suffices to show that if u(p1) = 1 and u(pj) = 0 for all j > 1, then
1 1 u (p + p ) = α(λ ), ∀j > 1, and u (p + p ) = β(λ ), ∀j > i > 1. e 2 1 j m e 2 i j m
1
x x
x 0 0 y y y 0
Figure 2. values of ue on C and Son(C), where N = 4 THE “HOT SPOTS” CONJECTURE 293
1 1 Denote ue( 2 (p1 + p2)) by x and ue( 2 (p2 + p3)) by y. See Figure2. From (5) and the symmetry of KN , we have 1 (N − 2)(N − 3) x = · 2 − λ a x + b (N − 2)x + (N − 2)y + c · y N + 2 m N N N 2 2 2x (N − 2)y = + λ + , N + 2 m N + 2 2(N + 2) 1 y = · 1 − λ a y + b 2x + (2N − 6)y N + 2 m N N (N − 3)(N − 4) + c (N − 3)x + y N 2 1 x Ny = + λ + . N + 2 m N + 2 2(N + 2)
By direct calculation, we can obtain that x = α(λm) and y = β(λm) if λm 6∈ {2,N + 2}. Thus the lemma holds. We remark that in case that N = 2, the equation (10) is reduced to 1 1 ue (pi + pj) = α(λm)[u(pi) + u(pj)] = [u1(pi) + u1(pj)]. (11) 2 2 − λm 4. Proof of Lemma 1.6.
Lemma 4.1. The hot spots conjecture holds if u1 attains its maximum and mini- mum on V0.
Proof. Assume that u1 attains its maximum and minimum on V0. Define fi = 1 ui + N , 1 ≤ i ≤ N. It follows from (8) that every fi is a rotation of f1 for 2 ≤ i ≤ N. Thus every fi has same maximum and minimum as f1. Noticing that f1(V0) = {0, 1}, we have 0 ≤ fi ≤ 1 for all i. Now, let u be a Neumann eigenfunction with respect to the second-smallest N−1 Neumann eigenvalue of ∆. Since {ui}i=1 is a basis of EF2, there exist constants PN−1 PN−1 ci, 1 ≤ i ≤ N − 1, such that u = i=1 ciui. Noticing that uN = − i=1 ui, we PN 1 PN have i=1 fi = 1 so that ui = fi − N i=1 fi for 1 ≤ i ≤ N − 1. It follows that N−1 N X X u = ciui = difi, i=1 i=1 1 PN−1 1 PN−1 where di = ci − N j=1 cj for 1 ≤ i ≤ N − 1 and dN = − N j=1 cj. Notice that PN 0 ≤ fi ≤ 1 for all 1 ≤ i ≤ N, i=1 fi = 1 and {u(qi)| i = 1, 2,...,N} = {di| i = 1, 2,...,N}. Hence
min{u(qi)| i = 1, 2,...,N} ≤ u(x) ≤ max{u(qi)| i = 1, 2,...,N} for all x ∈ KN by convexity. Recall that 4 − λ 2 α(λ) = , β(λ) = . (2 − λ)(N + 2 − λ) (2 − λ)(N + 2 − λ)
In the sequel, we define αm = α(λm) and βm = β(λm) for all m ≥ 1, where {λm}m≥1 is defined by (7). For every m ≥ 2, we have 0 < λm < 1 so that αm ≥ βm. By spectral decimation and Lemma 3.1, we have following two lemmas. 294 XIAO-HUI LI AND HUO-JUN RUAN
N−2 Lemma 4.2. Let {xm}m≥1 be defined as in (2). Then x1 = 2N , and for m ≥ 1, N − 1 x = α + (α + (N − 2)β )x . (12) m+1 N m+1 m+1 m+1 m Proof. It follows from (9) and (11) that the lemma holds for N = 2. Thus we can assume that N ≥ 3. We will show that (12) holds for every m ≥ 0, where 1 m x0 = u1(q2) = − N . Given a nonnegative integer m. We denote F1 (V0) by C and m N−1 denote F1 (qi) by pi, i = 1, ··· N. Then u1(p1) = u1(q1) = N . By the symmetry 1 of u1, we have u1(pi) = xm for all i 6= 1. Notice that xm+1 = u1( 2 (p1 + p2)). Thus by spectral decimation and Lemma 3.1, we can see that (12) holds for m. N−4 1 1 Now, combining (12) with α1 = 2(N−2) , β1 = − N−2 and x0 = − N , we can N−2 obtain that x1 = 2N .
N−2 Lemma 4.3. Let {ym}m≥1 and {zm}m≥1 be defined as in (2). Then y1 = 2N , z1 = 1 − N , and for m ≥ 1, 1 y = − α + α y + (N − 2)β z , (13) m+1 N m+1 m+1 m m+1 m 1 z = − α + β y + (α + (N − 3)β )z . (14) m+1 N m+1 m+1 m m+1 m+1 m
N−2 Proof. It is clear that y1 = 2N and (13) holds for N = 2. Thus we can assume that N ≥ 3. First we will show that (13) and (14) hold for all m ≥ 0, where y0 = N−1 1 u1(q1) = N and z0 = u1(q3) = − N . Given a nonnegative integer m. We denote m m 1 F2 (V0) by C and denote F2 (qi) by pi, i = 1,...,N. Then u1(p2) = u1(q2) = − N and u1(p1) = ym. By the symmetry of u1, we have u1(pi) = zm for all i 6= 1, 2. 1 1 Notice that ym+1 = u1( 2 (p1 + p2)) and zm+1 = u1( 2 (p2 + p3)). Thus by spectral decimation and Lemma 3.1, we can see that (13) and (14) holds for m. N−4 1 N−1 1 From α1 = 2(N−2) , β1 = − N−2 , y0 = N and z0 = − N , we can obtain that N−2 1 y1 = 2N and z1 = − N .
Lemma 4.4. Assume that the inequality (3) holds. Let p1, p2, . . . , pN be distinct vertices of a cell of Vm, where m ≥ 1, with u1(p1) ≤ u1(p2) ≤ · · · ≤ u1(pN ). Then
N − 1 u (p ) ≤ u (p ) ≤ · · · ≤ u (p ) ≤ x , u (p ) ≤ , and (15) 1 1 1 2 1 N−1 m 1 N N 1 u (p ) ≥ − , u (p ) ≥ u (p ) ··· u (p ) ≥ z , u (p ) ≥ y . (16) 1 1 N 1 N−1 1 N−2 1 2 m 1 N m
Proof. We will prove the lemma by induction. From definitions of {xm, ym, zm}m≥1 and u1, we know that the lemma holds for m = 1. Assume that the lemma holds for m ≤ k, where k is a positive integer. Let m = k + 1 and p1, p2, . . . , pN be distinct vertices of a cell C of Vk+1. Then there 0 0 0 0 exists a unique cell C of Vk which contains C. Let p1, p2, . . . , pN be distinct vertices 0 N of C . By the construction of K , there exists a permutation (i1, i2, . . . , iN ) of 0 0 (1, 2,...,N) such that pi1 ∈ {p1, . . . , pN } and pi2 , . . . , piN ∈ Vk+1\Vk. Furthermore, 0 1 0 0 we can assume without loss of generality that pi1 = p1 and pij = 2 (p1 + pj) for 2 ≤ j ≤ N. THE “HOT SPOTS” CONJECTURE 295
N−1 From αk+1 > βk+1, N ≥ xk and using the inductive assumption, we have
0 0 X 0 u1(pij ) = αk+1[u1(p1) + u1(pj)] + βk+1 u(pk) k6=1,j N − 1 ≤ α + x + β · (N − 2)x = x k+1 N k k+1 k k+1 0 N−1 for all j ≥ 2. Combining this with u1(pi1 ) = u1(p1) ≤ N , we know that (15) holds for m = k + 1. By the inductive assumption, there exists j ∈ {1,...,N}, such that u (p0 ) ≥ 0 1 j0 1 yk. If j0 6= 1, then from αk+1 > βk+1, zk ≥ − N and using the inductive assumption, we have X u (p ) = α [u (p0 ) + u (p0 )] + β u(p0 ) 1 ij0 k+1 1 1 1 j0 k+1 k k6=1,j0 1 ≥ α − + y + β · (N − 2)z = y . k+1 N k k+1 k k+1 1 Similarly, from αk+1 > βk+1 and yk ≥ zk ≥ − N , we have
0 0 X 0 u1(pij ) = αk+1[u1(p1) + u1(pj)] + βk+1 u(pk) k6=1,j 1 ≥ α − + z + β · ((N − 3)z + y ) = z , ∀j 6= j . k+1 N k k+1 k k k+1 0 0 1 Combining this with u1(pi1 ) = u1(p1) ≥ − N , we know that (16) holds for m = k+1. 1 If j0 = 1, then from αk+1 > βk+1 and zk ≥ − N , we have
0 0 X 0 u1(pij ) = αk+1[u1(p1) + u1(pj)] + βk+1 u(pk) k6=1,j 1 ≥ α y − + β · (N − 2)z = y k+1 k N k+1 k k+1 0 1 for all j ≥ 2. Combining this with yk+1 ≥ zk+1 and u1(pi1 ) = u1(p1) ≥ yk ≥ − N , we know that (16) holds for m = k + 1.
N ∗ Proof of Lemma 1.6. Noticing that u1 is uniformly continuous on K and V is N dense in K , it directly follows from the above lemma that u1 attains it maximum and minimum on V0 if minp∈V0 u1(p) ≤ xm, ym, zm ≤ maxp∈V0 u1(p) for all positive integers m. Combining this with Lemma 4.1, we know that Lemma 1.6 holds.
5. Proof of inequality (3).
Lemma 5.1. N − 1 λ x = 1 − m , ∀m ≥ 1. (17) m N 2(N − 1) N−1 As a result, 0 ≤ xm ≤ N for all m. Proof. We will prove the lemma by induction. It is easy to check that the lemma holds for m = 1 since λ1 = N. Assume that the lemma holds for m ≤ k, where k is 296 XIAO-HUI LI AND HUO-JUN RUAN a positive integer. Let m = k + 1. From the definition of αk+1, βk+1 and noticing that λk = λk+1(N + 2 − λk+1), we have N λ (N + 2 − λ ) x = α + (α + (N − 2)β ) 1 − k+1 k+1 N − 1 k+1 k+1 k+1 k+1 2(N − 1)
λk+1(N+2−λk+1) (4 − λk+1) + (2N − λk+1) 1 − 2(N−1) = (2 − λk+1)(N + 2 − λk+1) λ (2N−λ ) 2 − k+1 k+1 λ = 2(N−1) = 1 − k+1 2 − λk+1 2(N − 1) so that the lemma holds for m = k + 1.
1 Lemma 5.2. In case that N = 2, we have ym ≥ − N for every positive integer m. In case that N ≥ 3, we have (N − 1)ym + xm = −(N − 2) (N − 1)zm + xm , (18) λ 0 ≥ (N − 1)z + x ≥ − m , (19) m m 2N 1 y > z ≥ u (q ) = − (20) m m 1 2 N for every positive integer m.
Proof. If N = 2, we can obtain from (12) and (13) that ym+1 + xm+1 = αm+1(ym + 1 xm) for all m ≥ 1. Thus from x1 = y1 = 0, we have ym = −xm ≥ − 2 for all m ≥ 1. Now we assume that N ≥ 3. First we will prove (18) and (19) by induction. In case that m = 1, N − 1 N − 2 1 1 (N − 1)z + x = − + = − = − λ , 1 1 N 2N 2 2N 1 N − 2 N − 2 N − 2 N − 2 (N − 1)y + x = (N − 1) + = = λ 1 1 2N 2N 2 2N 1 so that (18) and (19) hold. Assume that (18) and (19) hold for m ≤ k, where k is a positive integer. Let m = k + 1. From Lemmas 4.2, 4.3 and the inductive assumption, we have (N − 1)zk+1 + xk+1 = αk+1 + (N − 3)βk+1 (N − 1)zk + xk + βk+1 (N − 1)yk + xk
=(αk+1 − βk+1)((N − 1)zk + xk), (21)
(N − 1)yk+1 + xk+1 =αk+1((N − 1)yk + xk) + (N − 2)βk+1((N − 1)zk + xk)
= − (N − 2)(αk+1 − βk+1)((N − 1)zk + xk) so that (18) holds for m = k+1. Furthermore, by (21) and the inductive assumption, 1 0 ≥ (N − 1)zk+1 + xk+1 = ((N − 1)zk + xk) (N + 2 − λk+1) 1 −λ (N + 2 − λ ) λ ≥ · k+1 k+1 = − k+1 N + 2 − λk+1 2N 2N so that (19) holds for m = k + 1. THE “HOT SPOTS” CONJECTURE 297
Now we will show that (20) holds for all positive integers m. From (12), (13) and y1 > z1, we have ym+1 − zm+1 = (αm+1 − βm+1)(ym − zm) > 0 for all m ≥ 1. λm From (19), we have |(N − 1)zm + xm| ≤ 2N . Combining this with (17), we have 1 λ 1 z ≥ − m + x = − , ∀m ≥ 1. m N − 1 2N m N This completes the proof of the lemma. It directly follows from Lemmas 4.2 and 4.3 that inequality (3) holds.
Acknowledgements. The second author wishes to thank Professor Strichartz for his helpful discussion and encouragements. The authors are also grateful to the referees for their valuable suggestions.
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