1 Introduction

The”hot spots”conjecture was posed by J.Rauch at a conference 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[1,3,11].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. Since we can do analysis on fractals(see[12,13,21]),it is natural to ask whether the conjecture holds for p.c.f.fractals.Recently,there are some works on this topic.On the one hand,Ruan[17],Ruan and Zheng[18],Li and Ruan[15]proved that the conjecture hold on the Sierpinski gasket(SG2 for short),the level-3 Sierpinski gasket(SG3 for short)and higher dimensional Sierpinski gaskets.On the other hand,Lau,Li and Ruan[14]proved that the conjecture does not hold on the hexagasket.The basic tool used in these paper is spectral decimation.

The above fractals studied are all p.c.f. self-similar. Thus it is interesting to ask whether the conjecture holds for non p.c.f. self-similar fractals. In this paper,we will consider homogeneous hierarchical gaskets,which were introduced by Hambly[8,9].These gaskets are non p.c.f..Fortunately,they admit spectral decimation so that we can use similar method to prove that the conjecture holds on these gaskets.

Roughly speaking,the subdivision scheme for homogeneous hierarchical gaskets is a variant of the one for the usual Sierpinski gasket and constructed level by level.Each cell of level m is contained in a triangle,and that triangle is split into triangles of sides 1/bm+1 times the side of the original triangle,where bm+1 ∈{2,3,···}.If bm+1=2,we will have the cell of level m+1 as the same construction of SG2,if bm+1=3,we will have the cell of level m+1 as the same construction as SG3.The resulting gasket is denoted by HH(b)for b=(b1,b2,···).In this paper,we will restrict that bm equals 2 or 3 for each m.See Fig.1 for an example.

Notice that SG2 and SG3 are typical p.c.f.self-similar sets,while generally HH(b)is not a self-similar set.Meanwhile,the Dirichlet Laplacian and the Neumann Laplacian of these gaskets have already been discussed by Drenning and Strichartz[6].Thus,it is natural to ask whether the hot spots conjecture holds on certain homogeneous hierarchical gaskets.

The rest of the paper is organized as follows.Basic concepts are recalled in Section 2.Spectral decimation on HH(b)are described in Section 3.In Section 4,we prove that the”hot spots”conjecture holds on HH(b).

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2 Preliminaries

In this section,we recall some basic notations in[6,13,21].

Let qi,i=1,2,3,be non-collinear points in R2.Define functions Si,i=1,···,3,on R2 as follows:

converges uniformly to f on V∗V0 as m goes to infinity,where m=m2+m3 and m2 is the cardinality of the set{j≤m:bj=2}and m3 is the cardinality of the set{j≤m:bj=3}.

The Sierpinski gasket is the attractor of the iterated function system.

Let qi,i=1,2,3,be non-collinear points in R2.Define functions Fi,i=1,···,6,on R2 as follows:

For we define

First we define a sequence of graphs with vertices and V∗=The initial graph Γ0 is just the complete graph onthe vertices of a triangle which is considered as the boundary of HH(b).At stage m of the construction of HH(b),all the cells of level m−1 lie in triangles whose vertices make up Vm−1.If bm=2,then each cell of level m−1 splits into three cells of level m,adding three new vertices to Vm,connected exactly as in the SG2 construction.If bm=3,then each cell splits into six cells of level m,adding seven vertices in Vm,connected exactly as in the SG3 construction.See Fig.2.For x,y∈Vm,we use to denote that x and y is connected in Γm.

Definition 2.1.For any continuous function u on HH(b),we define the graph Laplacian∆m for positive integers m by

where degx is the cardinality of the setLet f be a continuous function on HH(b).We say that u∈dom∆with ∆u=f if

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Definition 2.2.The normal derivative at p∈V0 of a function u on HH(b)is defined to be

if the limit exists,where m2 and m3 are defined as in Definition 2.1.

Let EF2 be the set of all N-eigenfunctions on HH(b)corresponding to the eigenvalue λ of ∆.In case that b1=2(or b1=3),we define u1 and u2 to be functions in EF2 such that and are functions as in Fig.4(or Fig.5).It is easy to check thatandare N-eigenfunctions corresponding to the eigenvalue λ1 of ∆1.Thus u1 and u2 are well-defined by Spectral decimation theorem.Furthermore,it is easy to see that u1 and u2 are linearly independent,and so{u1,u2}is a base of EF2.In the sequel of the paper,we will always use u1 and u2 to refer to these functions.

For simplicity,we call λ an N-eigenvalue and u an N-eigenfunction if(2.4)holds.

3 Spectral decimation on HH(b)

The main tool to prove the hot spots conjecture on p.c.f.self-similar fractals is the spectral decimation,which was studied in[7,16,19,20].Drenning and Strichartz[6]pointed out that we can also use this method to analyze all Neumann(or Dirichlet)eigenvalues and eigenfunctions.Relative discussions on Laplacian and spectral decimation on SG3 can also be found in[2]and[10].

Let m be a nonnegative integer and um a function on Vm and λm a real number.We call um a discrete N-eigenfunction and λm a discrete N-eigenvalue on Vm if

We denote by Λm the set of all discrete N-eigenvalue of ∆m.

and similarly for the other vertices if bm=2,and

The level-3 Sierpinski gasket is the attractor of the iterated function system

For ,we define

Theorem 3.1(Spectral decimation theorem I,see[6,20]).Let m>0,we assume that λm−1=and if bm=2,and if bm=3.(i).If u is a discrete N-eigenfunction of ∆m−1 with eigenvalue λm−1,then there exists a unique extension on Vm such that is a discrete N-eigenfunction of ∆m with eigenvalue λm.Furthermore,take values on Vm in one Vm−1 cell shown in Fig.3 with

Define

and similarly for the other vertices if bm=3.

(ii).Conversely,if u is a discrete N-eigenfunction of ∆m with eigenvalue λm,then is a discrete N-eigenfunction of ∆m−1 with eigenvalue λm−1.

(iii).If λm ∈Λm,then the multiplicity of λm on ∆m equals that of λm−1 on ∆m−1.

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Theorem 3.2(Spectral decimation theorem II,[6,20]).(i).Let m0 ≥0,let u be a discrete Neigenfunction of ∆m0 with eigenvalue λm0.Assume that{λm}m≥m0 is an infinite sequence related by with all but a finite number of If we define

and extend u to V∗by successively using(3.4)and(3.5),then u is an N-eigenfunction of ∆with eigenvalue λ.

(ii). Every N-eigenvalue and its corresponding N-eigenfunctions of ∆can be obtained by the process described in(i).

(iii).Let and λ be defined as in(i).Then the multiplicity of λ of ∆equals that of λm0 of ∆m0.

Define

For each m≥2,we inductively define

Using the similar method and results in[17,18],it is easy to see that 0<λm<1 for all m≥2,and for all positive integer m,we have

Theorem 3.3.Let be defined as in(3.7)and(3.8).Define Then λ is the second-smallest N-eigenvalue of ∆.Furthermore,the multiplicity of λ of ∆equals 2.

Proof.It is clear that 0 is the smallest N-eigenvalues of ∆with multiplicity 1.Thus,in order to prove the lemma,it suffices to prove that λm defined in the lemma is the smallest element in Λm{0}for all m≥1.We will show this by induction.

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In case that m=1 and b1=2,we can directly compute all N-eigenvaluesof ∆1 from(3.1).We can obtain that Λ1={6,3,0,6,3,6}. Furthermore,the multiplicities for eigenvalues 0,3,6 are 1,2,3,respectively.Thus 3 is the second-smallest N-eigenvalues of ∆1,while the multiplicity of 3 is 2.

In case that m=1 and b1=3,we can directly compute all N-eigenvalues of ∆1 from(3.1).We can obtain thatFurthermore,the multiplicities for eigenvalues are 1,2,1,2,4,respectively. Thus is the secondsmallest N-eigenvalues of ∆1,while the multiplicity of

Assume that λk is the second-smallest N-eigenvalues of ∆k for some positive integer k.Set m=k+1.Let τ0 is an N-eigenvalues of ∆k+1.In case thatwe have from(3.9).In case that from Spectral decimation theorem,there exists and i∈{2,3}such that From the inductive assumption,we have Since Ri(where i∈{2,3})is strictly increasing in(0,1],we know that .Thus is the second-smallest N-eigenvalue of ∆k+1.

By induction,λ is the second-smallest N-eigenvalues of ∆.Since the multiplicity of λ1 of ∆1 equals 2,we obtain from the Spectral decimation theorem that the multiplicity of λ of ∆is also 2.

4 Proof of the main result

In this section,we always assume that λ and{λm}m≥1 are defined as in(3.7),(3.8)and Theorem 3.3.

Definition 2.3.A function u ∈dom∆is called an eigenfunction of Neumann Laplacian with eigenvalue λ if

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In the sequel,we define

for all m≥1,where functions ζ,η,α,β,γ are defined by(3.3a)and(3.3b).

Recall that 0<λm<1 for all m≥2.From above equalities,ζm>ηm for all m≥2.

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Lemma 4.1.Define inductively as follows:,and for m≥1,

Then

Proof.We will prove the lemma by induction.It is easy to check 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.In case that bk+1=2,by inductive assumption,we have

In case that bk+1=3,we have

By induction,the lemma holds for m≥1.

Lemma 4.2.Define{xm}m≥1 and{ym}m≥1 inductively as follows:and andif b1=3,and for all m≥1,

Let{zm}m≥1 be the sequence defined as in Lemma 4.1.Then for all m≥1,we have

Proof.We will prove the lemma by induction. In case that b1=2,we have 2x1+z1=In case that b1=3,we haveThus the lemma holds for m=1.

Assume that the lemma holds for m≤k,where k is a positive integer.Let m=k+1.In case that bk+1=2,from(4.2),(4.4a),(4.4b)and using inductive assumption,we have

Thus From(4.1)and using inductive assumption,we have

In case that bk+1=3,we have

Thus 2xk+1+zk+1=−(2yk+1+zk+1).From(4.1)and using inductive assumption,we have

By induction,the lemma holds for m≥1.

Lemma 4.3.Let{xm,ym,zm}m≥1 be defined as in Lemmas 4.1,4.2,andbe defined as(2.1),(2.2).Define if if bm=3,Then for all m≥1,

Proof.We will prove the lemma by induction. From Figs.4 and 5,we know that the lemma holds for m=1.

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Assume that the lemma holds for m≤k,where k is a positive integer.Let m=k+1 and bk+1=2.From Spectral decimation theorem and inductive assumption,we have

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In the case that bk+1=3,we have

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Similarly,we can prove that other equalities in(4.6a)and(4.6b)also hold for m=k+1.

By definition of x1 and y1,we know that(4.6c)holds for m=1.Thus it suffices to show that(4.6c)holds for m≥2.

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Let{zm}m≥1 be defined as in Lemma 4.1.From Lemma 4.2,we haveSubstituting zm byand noticing that 0<λm<1 for m≥2,we obtain that

Similarly,we have From(4.5),we have 2xm+zm>0>2ym+zm so that xm>ym.Hence(4.6c)holds for m≥2.

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The following lemma plays an essential role in our proof.

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In case that bk+1=2,from Fig.3,we know that there exists a permutation(i1,i2,i3)of(1,2,3)such thatwhile

Proof.We will prove the lemma by induction. From Figs.4 and 5,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,p3 be three distinct vertices of one cell C of Vk+1.Then there exists a unique cell C'of Vk which contains C.Let be three distinct vertices of C'.

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Lemma 4.4.Let{xm,ym,zm}m≥1 be defined as in Lemmas 4.1 and 4.2.Let p1,p2,p3 be three distinct vertices of one cell of Vm where m≥1.Assume that u1(p1)≤u1(p2)≤u1(p3).Then

By inductive assumption,we have

Without loss of generality,we assume that u1(pi2)≤u1(pi3).By Spectral decimation theorem,

where(j1,j2,j3)and(1,2,3)are two distinct permutations of(1,2,3).Notice that ζm>ηm for all m≥2 and for all m≥1.By inductive assumption,

Combining this with(4.8a),we know that(4.7a)holds for m=k+1.

By Lemma 4.3,we know that for all m≥1.By inductive assumption,

Sinceis another permutation of(1,2,3)and

Combining this with(4.8b)and(4.9),we know that(4.7b)holds for m=k+1.

In case that bk+1=3,we know from the proof of Lemma 4.4 in[18]that(4.7a)and(4.7b)also holds for m=k+1.

It directly follows from the above lemma that we have:

Theorem 4.1.u1 attains its maximum and minimum on V0.

Define

Noticing that u2 is a rotation of u1,and −(u1+u2)is a symmetry of u1,we know that u2 and −(u1+u2)also attains its maximum and minimum on V0.Clearly,0≤f,g,h≤1.

Now we can show that the”hot spots”conjecture holds on HH(b).

Theorem 4.2.Every eigenfunction of the second-smallest eigenvalue of Neumann Laplacian on HH(b)attains its maximum and minimum on the boundary V0.

Proof.Let u be an N-eigenfunctions with respect to the second-smallest N-eigenvalue of∆.Since{u1,u2}is a base of EF2,there exist constants c1,c2 such that u=c1u1+c2u2.By(4.10),we have f+g+h=1 so that and It follows that

Notice that 0≤f,g,h≤1,f+g+h=1 andHence,

for all x∈HH(b).

Acknowledgements

The work is supported in part by NSFC grants Nos.11271327,11771391. The author wishes to thank Professor Huojun Ruan for his helpful suggestions.

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