Boundary value problems on part of a level-n Sierpinski gasket
- Xuliang Li^{1}Email authorView ORCID ID profile
Received: 18 January 2017
Accepted: 28 March 2017
Published: 5 April 2017
Abstract
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.
Keywords
MSC
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, [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.
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
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.
1.2 Examples
Example 1.2
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
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.
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).
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
Theorem 2.1
Proof
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
Proof
We first prove the theorem for the case \(A=0\).
Substituting (2.6) into (2.4) yields (2.7).
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.
Declarations
Acknowledgements
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).
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Authors’ Affiliations
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