Skip to main content

Boundary value problems on part of a level-n Sierpinski gasket


We study the boundary value problems for the Laplacian on a sequence of domains constructed by cutting level-n Sierpinski gaskets properly. Under proper assumptions on these domains, we manage to give an explicit Poisson integral formula to obtain a series of solutions subject to the boundary data. In particular, it is proved that there exists a unique solution continuous on the closure of the domain for a given sequence of convergent boundary values.

1 Introduction

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, [48]. 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

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

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

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

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

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

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

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

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.

1.1 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 the compact triangular domain with boundary set \(\{q_{0},q_{1},q_{2}\}\) which is constructed by gluing finite copies of 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 ) 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 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

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

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

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

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 \(K=\mathrm{SG}_{n}\).

Figure 1
figure 1

General domain.

Remark 1.1

In application, we need the constant \(L=n^{k}\) with \(k>0\) to ensure that 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.

1.2 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) .

Figure 2
figure 2

\(\pmb{\mathrm{SG}_{3}}\) .

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

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

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.

Figure 3
figure 3

Domain in \(\pmb{\mathrm{SG}_{2}}\) .

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 . We show that if the boundary data on X converges, there exists a unique \(C(\overline {\Omega})\)-solution.

2 Main results

The Laplacian on the standard SG was first constructed as a generator of a stochastic process by Goldstein [13] and Kusuoka [14]. Kigami [15, 16] developed an analytical version of the Laplacian for SG, and then generalized it to any p.c.f. self-similar set (see [11], Definition 3.7.1, p.108).

We now study the boundary value problems on Ω as a DCPB defined in Section 1.1:

$$ \textstyle\begin{cases} \triangle u=0 & \text{on }\Omega,\\ u(y_{0})=a_{0}, \quad u(x_{m})=a_{m} & \text{on }\partial\Omega, \end{cases} $$

where denotes the Kigami’s Laplacian for \(K=\mathrm{SG}_{n}\) with respect to the standard self-similar measure, \(u:\overline {\Omega}\to\mathbb{R}\) is the unknown, and \(\{ a_{m} \} _{m=0}^{\infty}\) is the boundary data. Note that the Laplacian here is well defined for all cells \(Y_{m}\), hence the whole Ω by recalling that every cell \(\overline {Y}_{m}\) of Ω can be viewed as a part of \(K=\mathrm{SG}_{n}\) or gluing several copies of it.

Harmonic extension algorithm is the simplest tool for constructing harmonic functions subject to boundary value problems on SG n . In fact, we can apply this algorithm infinitely many times and obtain a function harmonic on SG n .

Using this, we will give an explicit solution to (BVP) based on the following assumption.

Assumption 0

Let Ω be a DCPB for \(K=\mathrm{SG}_{n}\). For each cell \(Y_{m}\) with boundary set \(\partial \overline {Y}_{m}=\{y_{m-1}, y_{m}, x_{m}\}\), if some function u is harmonic on \(Y_{m}\) and satisfies that

$$ u(y_{m})=c_{1},\qquad u(y_{m-1})=c_{2},\qquad u(x_{m})=c_{3} $$

for some real constants \(c_{1}, c_{2}, c_{3}\), then

$$\begin{aligned} \begin{bmatrix} u(y_{m})\\ u(y_{m-1}')\\u(x_{m}') \end{bmatrix} = M_{0} \begin{bmatrix} u(y_{m})\\ u(y_{m-1})\\u(x_{m}) \end{bmatrix} = M_{0} \begin{bmatrix} c_{1}\\ c_{2}\\c_{3} \end{bmatrix} , \quad M_{0}= \begin{bmatrix} 1 & 0 & 0\\ \theta_{1} & \theta_{2} & \theta_{3}\\\theta_{1} & \theta _{3} & \theta_{2} \end{bmatrix} \end{aligned}$$

for some positive constants \(\theta_{1}, \theta_{2}, \theta_{3}\) satisfying that

$$ \theta_{1}+\theta_{2}+\theta_{3}=1, $$

where \(y_{m-1}'=F_{m}(y_{m-1})\), \(x_{m}'=F_{m}(x_{m})\) with \(F_{m}\) given by (1.5).

Note that this assumption can be easily verified by harmonic extension algorithm. In Example 1.3, we have (see Figure 4)

$$M_{0}= \begin{bmatrix} 1 & 0 & 0\\ \frac{42}{75} & \frac{19}{75} & \frac{14}{75}\\\frac {42}{75} & \frac{14}{75} & \frac{19}{75} \end{bmatrix} . $$
Figure 4
figure 4

Values for Assumption  0 in Example  1.3 .

We set, for Assumption 0,

$$\begin{gathered} \Theta_{0}=\theta_{2}+\theta_{3},\qquad \Theta_{1}=2(2-\theta_{1}), \\ T_{+}=\frac{\Theta_{1}+K}{2},\qquad T_{-}=\frac{\Theta_{1}-K}{2},\qquad K=\sqrt { \Theta_{1}^{2}-4\Theta_{0}}. \end{gathered} $$

Theorem 2.1

For every choice of the convergent boundary data \(\{a_{m}\}\) for some Ω as a DCPB (defined in Section  1.1) satisfying Assumption  0, there exists a one-dimensional space of \(C(\Omega)\) solutions to the (BVP). For each real constant λ, there exists a unique solution to the (BVP) \(u_{\lambda}\) such that \(u_{\lambda}(y_{1})=\lambda\) and that \(u_{\lambda}(x_{m})=a_{m}\) for \(m\ge1\). Furthermore, for \(m\ge2\)

$$ u_{\lambda}(y_{m})=K^{-1} \bigl\{ T_{+}^{m} \phi^{+}_{m}(\lambda)-T_{-}^{m}\phi ^{-}_{m}(\lambda) \bigr\} , $$


$$\begin{gathered} \phi^{+}_{m}(\lambda)=\lambda-T_{+}^{-1}\Theta_{0}(a_{0}+a_{1})-( \Theta _{0}+T_{+})\overset{m}{\underset{k=2}{\sum}}T_{+}^{-k}a_{k}, \\ \phi^{-}_{m}(\lambda)=\lambda-T_{-}^{-1}\Theta_{0}(a_{0}+a_{1})-( \Theta _{0}+T_{-})\overset{m}{\underset{k=2}{\sum}}T_{-}^{-k}a_{k}. \end{gathered} $$


For fixed \(m\ge2\), let u be a continuous piecewise harmonic function for (BVP). In view of Assumption 0, it is easy to adapt the argument for [3], proof of Lemma 2.1, to obtain that \(\triangle u(y_{m})=0\) holds if and only if

$$ u(y_{m})=\Theta_{1} u(y_{m-1})- \Theta_{0}u(y_{m-2})-a_{m}-\Theta_{0}a_{m-1}. $$

The rest is trivial algebra as in [3], proof of Theorem 2.2. □

The theorem below can be obtained by following the argument in [3], proof of Theorem 2.4, Corollary 2.5. We include a brief proof for the readers’ convenience.

Theorem 2.2

If \(a_{m}\to A\) as \(m\to\infty\) for some constant A, there exists a unique solution to (BVP) \(u\in C(\overline {\Omega})\) which satisfies that

$$ u(y_{1})=T_{+}^{-1}\Theta_{0}(a_{0}+a_{1})+( \Theta_{0}+T_{+})\overset{\infty }{\underset{k=2}{\sum }}T_{+}^{-k}a_{k}, $$

and for \(m\ge2\)

$$\begin{aligned} u(y_{m})&=K^{-1} (\Theta_{0}+T_{+} ) \Bigl\{ \overset{\infty }{\underset{k=1}{\sum}}T_{+}^{-k}a_{m+k}-T_{-}^{m} \overset{\infty}{\underset {k=2}{\sum}}T_{+}^{-k}a_{k} \Bigr\} \\ &\quad {}+T_{-}^{m} \Bigl\{ a_{0}+a_{1}+K^{-1}( \Theta_{0}+T_{-})\overset{m}{\underset {k=2}{\sum }}T_{-}^{-k}a_{k} \Bigr\} . \end{aligned}$$


We first prove the theorem for the case \(A=0\).

Substituting (2.6) into (2.4) yields (2.7).

By using the triangle inequality, we have

$$\begin{aligned} \bigl\vert u(y_{m}) \bigr\vert &\leq K^{-1} ( \Theta_{0}+T_{+} ) \Bigl\{ \overset{\infty}{\underset{k=1}{\sum }}T_{+}^{-k} \vert a_{m+k} \vert -T_{-}^{m}\overset{\infty}{\underset{k=2}{\sum }}T_{+}^{-k} \vert a_{k} \vert \Bigr\} \\ &\quad {}+T_{-}^{m} \Bigl\{ \vert a_{0} \vert + \vert a_{1} \vert +K^{-1}(\Theta_{0}+T_{-}) \overset{m}{\underset{k=2}{\sum }}T_{-}^{-k} \vert a_{k} \vert \Bigr\} . \end{aligned}$$

From this and \(a_{m} \to0\) we can easily see that \(u(y_{m})\to0\). Thus, by the definition of BVP,

$$\underset{m\to\infty}{\lim}u(x_{m}) = \underset{m\to\infty}{ \lim}u(y_{m}) = 0. $$

It follows by [3], Lemma 2.3, that \(u\in C(\overline{\Omega})\). Since harmonic functions that are continuous to the boundary satisfy the maximum principle [17], we obtain the uniqueness by the standard argument for linear differential equations satisfying the maximum principle.

For the case \(A\neq0\), we consider the modified BVP:

$$ \textstyle\begin{cases} \triangle u=0 & \text{on }\Omega,\\ u(y_{0})=a_{0} - A, \quad u(x_{m})=a_{m} - A & \text{on }\partial\Omega. \end{cases} $$

Noting that \(a_{m} - A \to0\), the rest of the proof can be done by using the result from the last part of the proof and the maximum principle. □

Remark 2.3

The results in [3], Section 2, reduce to a special case of our theorems above with parameters \(\theta_{0}=2/5\), \(\theta_{1}=2/5\), \(\theta_{2}=1/5\). Indeed, [3] proved many more results on half SG. Many of them are highly dependent on the fact that we can obtain SG from half SG by reflection, and thus the top point enjoys many more nice properties than our domains. We will touch on that elsewhere.


  1. Owen, J, Strichartz, RS: Boundary value problems for harmonic functions on a domain in the Sierpinski gasket. Indiana Univ. Math. J. 61(1), 319-335 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Guo, Z, Kogan, R, Qiu, H, Strichartz, RS: Boundary value problems for a family of domains in the Sierpinski gasket. Ill. J. Math. 58(2), 497-519 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Li, W, Strichartz, RS: Boundary value problems on a half Sierpinski gasket. J. Fractal Geom. 1(1), 1-43 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonanno, G, Bisci, GM, Radulescu, V: Existence results for gradient-type systems on the Sierpiński gasket. Chin. Ann. Math., Ser. B 34(2), 941-953 (2013)

    MATH  Google Scholar 

  5. Bonanno, G, Molica Bisci, G, Radulescu, V: Infinitely many solutions for a class of nonlinear elliptic problems on fractals. C. R. Math. Acad. Sci. Paris 350(3-4), 187-191 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonanno, G, Molica Bisci, G, Radulescu, V: Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket. ESAIM Control Optim. Calc. Var. 18(4), 941-953 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ferrara, M, Molica Bisci, G, Repovs, D: Existence results for nonlinear elliptic problems on fractal domains. Adv. Nonlinear Anal. 5(1), 75-84 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Molica Bisci, G, Radulescu, V: A characterization for elliptic problems on fractal sets. Proc. Am. Math. Soc. 143(7), 2959-2968 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Strichartz, RS: Fractafolds based on the Sierpiński gasket and their spectra. Trans. Am. Math. Soc. 355(10), 4019-4043 (2003)

    Article  MATH  Google Scholar 

  10. Strichartz, RS: Fractals in the large. Can. J. Math. 50(3), 638-657 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kigami, J: Analysis on Fractals. Cambridge Tracts in Mathematics, vol. 143. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  12. Strichartz, RS: Differential Equations on Fractals: A Tutorial. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  13. Goldstein, S: Random walks and diffusions on fractals. In: Percolation Theory and Ergodic Theory of Infinite Particle Systems (Minneapolis, Minn., 1984-1985). IMA Vol. Math. Appl., vol. 8, pp. 121-129. Springer, New York (1987)

    Google Scholar 

  14. Kusuoka, S: Dirichlet forms on fractals and products of random matrices. Publ. Res. Inst. Math. Sci. 25(4), 659-680 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kigami, J: A harmonic calculus on the Sierpiński spaces. Jpn. J. Appl. Math. 6(2), 259-290 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kigami, J: Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335(2), 721-755 (1993)

    MathSciNet  MATH  Google Scholar 

  17. Strichartz, RS: Some properties of Laplacians on fractals. J. Funct. Anal. 164(2), 181-208 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references


The author is indebted to Professor Robert S. Strichartz for inviting them to work on this topic. The author is also grateful to Professor Jiaxin Hu for helpful comments on a draft version of the present work. This work was supported by the National Natural Science Foundation of China (11371217).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Xuliang Li.

Additional information

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X. Boundary value problems on part of a level-n Sierpinski gasket. Bound Value Probl 2017, 47 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: