The study of boundary value problems on the domains of Sierpinski gasket (SG) was initiated by [1]. Since then, two natural choices have been considered, namely the upper part of SG cut by a horizontal line (cf. [1, 2]) and half Sierpinski gasket constructed by cutting SG with a vertical line in the middle (cf. [3]). For more related works see, for example, [4–8]. This work is strongly motivated by [3].

In this work, we will introduce a new class of domains on level-n Sierpinski gasket and prove the exact form of the solution to the boundary value problems on these domains. Note that these domains are new examples of non-p.c.f. (postcritically finite) type fractals (can also be viewed as fractafold in [9, 10]) where harmonic functions can be well defined.

We follow [11, 12] by recalling that the fractal *K* is the invariant set for a finite iterated function systems (IFS) of contractive similarities in the Euclidean space \(\mathbb{R}^{2}\). We denote the mappings \(\{ F_{i} \} _{i=0,\ldots.N-1}\) for some positive integer *N*. Then *K* is the unique nonempty compact set satisfying

$$ K=\underset{i=0}{\overset{N-1}{\bigcup}}F_{i}(K). $$

(1.1)

For \(m\ge1\), we define the space of words of length *m* by

$$W_{m}^{N}=\{0,1,2,\ldots,N-1\}^{m}= \bigl\{ w_{1}w_{2}\ldots w_{m}:w_{i}\in \{ 0,1,2,\ldots,N-1 \} \bigr\} . $$

\(w\in W_{m}^{N}\) is called a word of length *m* with symbols \(\{ 0,1,2,\ldots,N-1\}\). We also set \(W_{*}^{N}=\bigcup_{m\ge0}W_{m}^{N}\) and denote the length of \(w\in W_{*}^{N}\) by \(\vert w \vert \).

Recall that *K* is called *postcritically finite* (p.c.f.) if *K* is connected and there exists a finite set \(V_{0}\subseteq K\) called the *boundary* such that

$$ F_{w}K\cap F_{w'}K\subseteq F_{w}V_{0} \cap F_{w'}V_{0}\quad\text{for }w\neq w' \text{ with } \vert w \vert = \bigl\vert w' \bigr\vert , $$

(1.2)

with the intersection disjoint from \(V_{0}\). Set \(V_{0}= \{ q_{0},q_{1},\ldots,q_{N_{0}} \} \) for \(N_{0}< N\). We require that each boundary point is the fixed point of one of the mappings \(\{ F_{i} \} \) and that

$$ F_{i}(q_{i})=q_{i}\quad\text{for } 0\leq i\le N_{0}. $$

(1.3)

The standard SG is the unique nonempty compact set *K* satisfying (1.1) with the boundary set \(V_{0}= \{ q_{0},q_{1},q_{2} \} \), where the contractive mappings \(\{ F_{i} \} _{i=0,1,2}\) are given by

$$F_{i}(x)=\frac{1}{2}(x-q_{i-1})+q_{i-1}. $$

Similarly, the *level-3 Sierpinski gasket*
\(\mathrm{SG}_{3}\) is the unique nonempty compact set *K* satisfying (1.1) with the boundary set \(V_{0}= \{ q_{0},q_{1},q_{2} \} \), where \(\{ F_{i} \} _{i=0\ldots,5}\) are given by

$$ F_{i}(x)=\frac{1}{3}(x-q_{i})+q_{i}. $$

(1.4)

Here \(q_{3}=\frac{q_{1}+q_{2}}{2}\), \(q_{4}=\frac{q_{0}+q_{2}}{2}\), \(q_{5}=\frac{q_{0}+q_{1}}{2}\). See Figure 2 for an illustration.

As above, we can define *level-n Sierpinski gasket* in a similar way.

Inspired by [3] we will construct a new class of domains in the following statement.

### Description of the general domains

Let \(K=\mathrm{SG}_{n}\) and \(\tilde{K}=\frac{1}{n}K\), that is, shrinking *K*
*n* times. Denote by *Ẽ* the compact triangular domain with boundary set \(\{q_{0},q_{1},q_{2}\}\) which is constructed by gluing finite copies of *K̃* at boundary (see Example 1.3 below). Assume that the compact triangular domain \(\widetilde{E}_{1}\) with boundary \(\{q_{0},q_{1}',q_{2}'\}\) (as a part of *Ẽ*) satisfies \(\widetilde{E}_{1}=F_{0}(\widetilde{E})\), where \(F_{0}(x):=q_{0}+\frac{x-q_{0}}{L}\) for some constant *L* (see Figure 1(a)). Pick some point *q̂* such that the contractive map

$$\widehat{F}(x):=\widehat{q}+\frac{x-\widehat{q}}{L} $$

satisfies \(\widehat{F}(q_{1})=q_{0}\). Set \(E=\widetilde{E}\backslash\{q_{1},q_{2}\}\), \(y_{0}=q_{1}\), \(y_{m}=\widehat{F}^{m}(q_{1})\), \(x_{m}=\widehat{F}^{m-1}(q_{2})\), \(Y_{m}:=\widehat{F}^{m-1}(E)\) for integer \(m>0\) (see Figure 1(b)). Let \(X=\bigcup_{m=1}^{\infty}\{x_{m}\}\). Set

$$\partial Y_{m}= \{ y_{m-1},y_{m},x_{m} \}, \qquad \overline {Y}_{m}=Y_{m}\cup\partial Y_{m}. $$

For each \(\overline {Y}_{m}\) define mapping \(F_{m}\) as

$$ F_{m}(z)=y_{m}+\frac{z-y_{m}}{L}. $$

(1.5)

Then by assumption \(\overline {Y}_{m}\) is self-similar with respect to \(F_{m}\). Define

$$ \Omega:=\underset{m=0}{\overset{\infty}{\bigcup}}Y_{m}, \qquad \overline {\Omega}:=\underset{m=0}{\overset{\infty}{\bigcup}} \overline {Y}_{m}. $$

(1.6)

We say that the set Ω is a DCPB *(domain of countable-point boundary)* with boundary \(\partial\Omega:=X\cup{y_{0}}\cup\hat{q}\) for \(K=\mathrm{SG}_{n}\).

### Remark 1.1

In application, we need the constant \(L=n^{k}\) with \(k>0\) to ensure that *Ẽ* is self-similar with respect to the map \(F_{0}\), and thus \(F_{m} (\overline {Y}_{m})\) is exactly a copy of \(\overline {Y}_{m+1}\). This property is useful in constructing harmonic functions on these domains in a sequel. The description of those domains will be justified by the examples below.

### Examples

### Example 1.2

For \(K=\mathrm{SG}_{3}\), let \(\widetilde{E}=F_{1}(K)\) be the compact triangular domain with boundary set \(\{q_{1},p_{1},p_{6}\}\) (see Figure 2). Let \(E=\widetilde{E}/\{q_{1},p_{1}\}\), set \(y_{0}=q_{1}\), \(y_{m}=F_{5}(q_{1})\), \(x_{m}=F_{5}^{m-1}(p_{1})\) and \(Y_{m}=F_{5}^{m-1}(E)\) for all positive integers *m*, where \(F_{5}\) is as defined in (1.4). Set \(\widehat{F}=F_{5}\), \(\hat{q}=q_{5}\). Then Ω (green part) can be well established as in (1.6) .

### Example 1.3

For \(K=\mathrm{SG}_{3}\), \(\widetilde{E}=F_{1}(K)\cup F_{3}(K)\cup F_{5}(K)\) is the compact triangular domain with boundary set \(\{ q_{1},p_{2},p_{5}\}\) (see Figure 2). Let \(E=\widetilde{E}/\{q_{1},p_{2}\}\), set \(y_{0}=q_{1}\), \(y_{m}=F_{0}^{m}(q_{1})\), \(x_{m}=F_{0}^{m-1}(p_{2})\) and \(Y_{m}=F_{0}^{m-1}(E)\) for all positive integers *m*. Let \(X=\bigcup_{m=1}^{\infty}\{x_{m}\}\). Setting \(\widehat{F}=F_{0}\), \(\hat{q}=q_{0}\), we obtain Ω (gray part) as a DCPB by (1.6).

### Example 1.4

For \(K=\mathrm{SG}_{2}\), \(\widetilde{E}=F_{1}(K)\) is the compact triangular domain with boundary points \(\{q_{1},p_{2},p_{0}\}\) (see Figure 3). Let \(q_{4}=\frac{2p_{2}+q_{0}}{3}\) and define the contractive mapping

$$ F_{4}(x)=\frac{x-q_{4}}{4}+q_{4}. $$

(1.7)

Let \(E=\widetilde{E}/\{q_{1},p_{0}\}\), set \(y_{0}=q_{1}\), \(y_{m}=F_{4}^{m}(q_{1})\), \(x_{m}=F_{4}^{m-1}(p_{0})\) and \(Y_{m}=F_{4}^{m-1}(E)\) for all \(m>0\). Let \(X=\bigcup_{m=1}^{\infty}\{x_{m}\}\). Set \(\widehat {F}=F_{4}\), \(\hat{q}=q_{4}\). Now we can define the desired domain Ω by (1.6). Note that \(F_{4}\) is not one of the contractive mappings for standard SG.

In the following section, we construct a solution to the boundary value problem using harmonic extension algorithm. Denote by \(C(U)\) the space of all continuous functions on some set *U*. We will see that the space of \(C(\Omega)\)-solutions to the boundary value problem is one-dimensional, but in general, the solution blows up at *q̂*. We show that if the boundary data on *X* converges, there exists a unique \(C(\overline {\Omega})\)-solution.