A Counterexample to the “Hot Spots” Conjecture on Nested Fractals

Abstract

Although the “hot spots” conjecture was proved to be false on some classical domains, the problem still generates a lot of interests on identifying the domains that the conjecture hold. The question can also be asked on fractal sets that admit Laplacians. It is known that the conjecture holds on the Sierpinski gasket and its variants. In this note, we show surprisingly that the “hot spots” conjecture fails on the hexagasket, a typical nested fractal set. The technique we use is the spectral decimation method of eigenvalues of Laplacian on fractals.

Appendix: Laplacian and Normal Derivative on Hexagasket

In this appendix, we derive the definition of Laplacian and normal derivative of the hexagasket, in Definitions 2.1, 2.2, following the general method in [14].

For any finite set U and V, we let $\ell (U)=\{f:\; f \text{ maps } U \text{ into } \mathbb {R} \}$, denote

$$\begin{aligned} L(U,V)=\{A:\; A \text{ is } \text{ a } \text{ linear } \text{ map } \text{ from } \ell (U) \text{ to } \ell (V)\}, \end{aligned}$$

and $L(U)=L(U,U)$. It is clear that for each $A\in L(U,V)$, there exists a unique matrix, denoted by $\{A_{pq}\}_{p\in V,q\in U}$, such that for any $f\in \ell (U)$,

$$\begin{aligned} (Af)(p)=\sum _{q\in U} A_{pq} f(q), \quad \forall \ p\in V. \end{aligned}$$

We will use the same notation for both linear map and the corresponding matrix. We define $\mathcal {H}(V)$ to be the collection of symmetric and irreducible matrix D in L(V) satisfying following two conditions: (i) $D_{pp}<0$ and $\sum _{q\in V} D_{pq}=0$ for all $p\in V$; (ii) $D_{pq}\ge 0$ if $p\not =q$.

Let K be a connected p.c.f. self-similar set with contractive maps $\{F_i\}_{i=1}^{N}$ on ${\mathbb {R}}^d$, and let $V_0$ be the boundary of K. Given $D\in \mathcal {H}(V_0)$ and $\mathbf {r}=(r_1, r_2, \ldots , r_N)$, where $r_i>0$ for all i, we define $H_m$ for any $m>0$ by

$$\begin{aligned} H_m=\sum _{|\omega |=m}r_\omega ^{-1}R^t_{\omega }DR_\omega , \end{aligned}$$

where $R_\omega \in L(V_m,V_0)$ is defined by $R_\omega f=f\circ F_\omega$, and $r_\omega =r_{\omega _1},\ldots , r_{\omega _m}$ for $\omega =(\omega _1,\ldots ,\omega _m) \in \Sigma ^m.$ Decompose $H_1$ into

$$\begin{aligned} H_1= \left[ \begin{array}{ccc} T &{} J^{t} \\ J &{} X \\ \end{array} \right] , \end{aligned}$$

where $T\in L(V_0), J\in L(V_0, V_1{\setminus } V_0)$ and $X \in L(V_1{\setminus } V_0)$. As Sect. 3.1 in [14], we call $(D,\mathbf {r})$ a harmonic structure if

$$\begin{aligned} T-J^t X^{-1}J=D. \end{aligned}$$

Given a continuous function u on K. We call u harmonic if $(H_mu)|_{V_m{\setminus } V_0}=0$ for all $m\ge 1$, and call u a piecewise harmonic spline of level m if $u\circ F_\omega$ is harmonic for all $\omega \in \Sigma ^m.$

Let $\mu$ be the standard self-similar measure on K [i.e., $\mu (K)=1$ and $\mu (\cdot )=\sum _{i=1}^N \frac{1}{N}\mu (F_i^{-1}(\cdot ))$], the following definitions of Laplacian and normal derivative are given in Sect. 6 in [13].

Definition 4.5

Assume that $(D,\mathbf {r})$ is a harmonic structure on K. For u a continuous function on K, we write $\Delta u=f$ if there exists a continuous function f on K such that

$$\begin{aligned} \lim _{m\rightarrow \infty }\max _{x\in V_m \backslash V_0}\left| \left( \int _{K} \psi _x^{(m)} \, d\mu \right) ^{-1}(H_mu)(x)-f(x)\right| =0, \end{aligned}$$

where $\psi _x^{(m)}$ is the piecewise harmonic spline of level m satisfying $\psi _x^{(m)}(y)=\delta _{xy}$ for all $y\in V_m$.

Definition 4.6

Assume that $(D,\mathbf {r})$ is a harmonic structure on K. Given a continuous function u on K. For $q\in V_0$, we define the Neumann derivative of u at q to be (if the limit exists)

$$\begin{aligned} \partial _n u(q) = -\lim _{m\rightarrow \infty } (H_mu)(q). \end{aligned}$$

Now for the hexagasket, let

$$\begin{aligned} D= \left( \begin{array}{ccc} -2 &{} 1 &{}1 \\ 1 &{} -2 &{}1 \\ 1&{}1&{}-2 \end{array} \right) , \quad \mathbf {r}=\left( r,r,r,r,r,r\right) , \end{aligned}$$

where $r>0$. It is easy to check that $(D,\mathbf {r})$ is a harmonic structure if and only if $r=\frac{3}{7}$. Thus, throughout the paper, we always choose $r=\frac{3}{7}$ so that

$$\begin{aligned} (H_m)u(x)=\Big (\frac{7}{3}\Big )^m\sum _{y\sim _mx}(u(y)-u(x)). \end{aligned}$$

Let $\mu$ be the standard self-similar measure on the hexagasket. Similar to Sect. 2.2 in [23], we obtain that $\int _{K} \psi _x^{(m)} \, \mathrm{d}\mu = {\deg _m(x)}/{6^{m+1}}$. As by definition, $\Delta _mu(x)=\frac{1}{\deg _m(x)}\sum _{y\sim _mx}(u(y)-u(x))$ for $x\in V_m{\setminus } V_0$, we have for $x\in V_*{\setminus } V_0,$

$$\begin{aligned} \Delta u(x) = \lim _{m\rightarrow \infty }\left( \frac{7}{3}\right) ^m \left( \int _{K} \psi _x^{(m)} \, \mathrm{d}\mu \right) ^{-1} \sum _{y\sim _mx}(u(y)-u(x)) =6 \lim _{m\rightarrow \infty }14^m \Delta _mu(x), \end{aligned}$$

and for $q_i \in V_0$,

$$\begin{aligned} \partial _nu(q_i)=\lim _{m\rightarrow \infty }\left( \frac{7}{3}\right) ^m\sum _{y\sim _m q_i}(u(q_i)-u(y)). \end{aligned}$$