The”Hot Spots”Conjecture on Homogeneous Hierarchical Gaskets

Xiaofen Qiu

Division of Fundamental Education,Shanghai Industry and Commerce Foreign Language College,Shanghai 201399,China and School of Mathematical Science,Zhejiang University,Hangzhou 310027,Zhejiang,China

Abstract.In this paper,using spectral decimation,we prove that the”hot spots”conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e.,every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian(introduced by Kigami)attains its maximum and minimum on the boundary.

Key Words:Neumann Laplacian,”hot spots”conjecture,homogeneous hierarchical gasket,spectral decimation,analysis on fractals.

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 Rdand u(t,x)is the solution of the heat equation in D with the Neumann boundary condition.Then for”most”initial conditions,if ztis a point at which the function x→u(t,x)attains its maximum,then the distance from ztto 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.

Figure 1:HH(b),where b=(2,3,3,2,···).

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(SG2for short),the level-3 Sierpinski gasket(SG3for 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+1times 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 bmequals 2 or 3 for each m.See Fig.1 for an example.

Notice that SG2and SG3are 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).

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 R2as follows:

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 R2as follows:

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

First we define a sequence of graphswith verticesand V?=The initial graph Γ0is 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 SG2construction.If bm=3,then each cell splits into six cells of level m,adding seven vertices in Vm,connected exactly as in the SG3construction.See Fig.2.For x,y∈Vm,we useto denote that x and y is connected in Γm.

Figure 2:Building block for SG2 and SG3.

Definition 2.1.For any continuous function u on HH(b),we define the graph Laplacian?mfor 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

converges uniformly to f on V?V0as m goes to infinity,where m=m2+m3and m2is the cardinality of the set{j≤m:bj=2}and m3is the cardinality of the set{j≤m:bj=3}.

Definition 2.2.The normal derivative at p∈V0of a function u on HH(b)is defined to be

if the limit exists,where m2and m3are defined as in Definition 2.1.

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

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 SG3can also be found in[2]and[10].

Let m be a nonnegative integer and uma function on Vmand λma real number.We call uma discrete N-eigenfunction and λma discrete N-eigenvalue on Vmif

We denote by Λmthe set of all discrete N-eigenvalue of ?m.

Define

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

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

and similarly for the other vertices if bm=3.

(ii).Conversely,if u is a discrete N-eigenfunction of ?mwith eigenvalue λm,thenis a discrete N-eigenfunction of ?m?1with eigenvalue λm?1.

(iii).If λm∈Λm,then the multiplicity of λmon ?mequals that of λm?1on ?m?1.

Theorem 3.2(Spectral decimation theorem II,[6,20]).(i).Let m0≥0,let u be a discrete Neigenfunction of ?m0with eigenvalue λm0.Assume that{λm}m≥m0is an infinite sequence related bywith all but a finite number ofIf 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).

Figure 3:The functionon one cell of Vm

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.Letbe defined as in(3.7)and(3.8).DefineThen 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 λmdefined in the lemma is the smallest element in Λm{0}for all m≥1.We will show this by induction.

In case that m=1 and b1=2,we can directly compute all N-eigenvaluesof ?1from(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.

Assume that λkis the second-smallest N-eigenvalues of ?kfor some positive integer k.Set m=k+1.Let τ0 is an N-eigenvalues of ?k+1.In case thatwe havefrom(3.9).In case thatfrom Spectral decimation theorem,there existsand i∈{2,3}such thatFrom the inductive assumption,we haveSince Ri(where i∈{2,3})is strictly increasing in(0,1],we know that.Thusis the second-smallest N-eigenvalue of ?k+1.

By induction,λ is the second-smallest N-eigenvalues of ?.Since the multiplicity of λ1of ?1equals 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≥1are defined as in(3.7),(3.8)and Theorem 3.3.

Let EF2be 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 u1and u2to be functions in EF2such thatandare functions as in Fig.4(or Fig.5).It is easy to check thatandare N-eigenfunctions corresponding to the eigenvalue λ1of ?1.Thus u1and u2are well-defined by Spectral decimation theorem.Furthermore,it is easy to see that u1and u2are linearly independent,and so{u1,u2}is a base of EF2.In the sequel of the paper,we will always use u1and u2to refer to these functions.

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>ηmfor all m≥2.

Figure 4:The functions u1 and u2 on V1 in case that b1=2.

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

Then

Figure 5:The functions u1 and u2 on V1 in case that b1=3.

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.

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

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

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≥1be defined as in Lemmas 4.1,4.2,andbe defined as(2.1),(2.2).Defineifif 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.

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

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

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

By definition of x1and y1,we know that(4.6c)holds for m=1.Thus it suffices to show that(4.6c)holds for m≥2.

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

The following lemma plays an essential role in our proof.

Lemma 4.4.Let{xm,ym,zm}m≥1be defined as in Lemmas 4.1 and 4.2.Let p1,p2,p3be three distinct vertices of one cell of Vmwhere m≥1.Assume that u1(p1)≤u1(p2)≤u1(p3).Then

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,p3be three distinct vertices of one cell C of Vk+1.Then there exists a unique cell C'of Vkwhich contains C.Letbe three distinct vertices of C'.

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

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>ηmfor all m≥2 andfor all m≥1.By inductive assumption,

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

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.u1attains its maximum and minimum on V0.

Define

Noticing that u2is a rotation of u1,and ?(u1+u2)is a symmetry of u1,we know that u2and ?(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,c2such that u=c1u1+c2u2.By(4.10),we have f+g+h=1 so thatandIt 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.