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THE “HOT SPOTS” CONJECTURE on HIGHER DIMENSIONAL SIERPINSKI GASKETS Xiao-Hui Li and Huo-Jun Ruan (Communicated by Camil Musc

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THE “HOT SPOTS” CONJECTURE on HIGHER DIMENSIONAL SIERPINSKI GASKETS Xiao-Hui Li and Huo-Jun Ruan (Communicated by Camil Musc 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 Xiao-Hui Li and Huo-Jun Ruan∗ 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 1. 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 2 R , i = 1; 2;:::;N, such that −−! q1qj, j = 2;:::;N, are linearly independent, we define 1 F (x) = (x + q ); x 2 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 = f1; 2;:::;Ng and Σ = f!1!2 ··· !mj !j 2 Σ for all 1 ≤ j ≤ mg for ∗ S1 m ∗ every positive integer m. Define Σ = m=1 Σ . For each ! = !1 ··· !m 2 Σ , we call ! a word with length j!j := m and define F! = F!1 ◦ · · · ◦ F!m : S Define V0 = fq1; q2; : : : ; qN g and Vm = j!j=m F!(V0) for every positive integer m S1 m. F!(V0) is called a cell of Vm for every ! 2 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 2 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 ! 2 Σ such that x; y 2 F!(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 2 Vm n 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 x dµ ∆mu(x); x 2 V∗ n V0; m!1 KN −m X @nu(x) = lim r (u(x) − u(y)); x 2 V0; m!1 y∼mx (m) provided the limits exist, where r is the renormalization factor of energy, x is the (m) piecewise harmonic spline of level m satisfying x (y) = δxy for all y 2 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 x 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 2 dom∆ N with ∆u = f on K n V0 if N (N + 2)m∆ u(x) 2 m converges uniformly to f on V∗ n V0 as m goes to infinity. N Definition 1.3. The normal derivative at a point qi 2 V0 of a function u on K is defined to be m N + 2 X @nu(qi) = lim (u(qi) − u(y)) m!1 N y∼mqi THE \HOT SPOTS" CONJECTURE 289 if the limit exists. Definition 1.4. A function u 2 dom∆ is called an eigenfunction of the Neumann Laplacian with eigenvalue λ if − ∆u = λu on V∗ n 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 fλ gi≥1 is 1. 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 2 EF2, we have min u(p) ≤ u(x) ≤ max u(p); 8x 2 KN : p2V0 p2V0 By using spectral decimation in [7, 15] (see Section2 for details), we know that N−1 EF2 is a linear space with a base fuigi=1 , where N − 1 1 u (q ) = ; u (q ) = − ; 8j 6= i: i i N i j N Let m m m xm = u1(F1 (q2)); ym = u1(F2 (q1)); zm = u1(F2 (q3)); 8m ≥ 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); 8m 2 Z : (3) p2V0 p2V0 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 u1jVn to u1jVn+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 2 Vm n V0; 2 P (u (x) − u (y)) = λ u (x); x 2 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 ! = !1 ··· !m 2 Σ . We know that C := F!(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 2 Son(C). We say p is a brother of p if p 2 Son(C) and p ∼m+1 p. Denote Bp the set of all brothers of p and define Cp = Son(C) n (fpg [ 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 ±(x) be inverse functions of Φ, i.e., s ! N + 2 4x (N + 2)2 (x) = 1 ± 1 − ; x 2 − 1; : ± 2 (N + 2)2 4 Theorem 2.1 (Spectral decimation theorem I, [15]).
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