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浙江大学学报（理学版）
浙江大学学报（理学版）  2019, Vol. 46 Issue (6): 676-679    DOI: 10.3785/j.issn.1008-9497.2019.06.008
数学与计算机科学     
SG3左半定义域上的Dirichlet边值问题
蔡洁洁, 吴波
南京财经大学 应用数学学院，江苏南京 210023
The Dirichlet boundary value problem on a left sub-domain of the level-3 Sierpinski Gaskets
CAI Jiejie, WU Bo
School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing 210023, China
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摘要： 调和函数在SG3上的Dirichlet边值问题是分形分析领域的重要研究内容之一。考虑通过垂直切割自相似图形SG3，得到SG3上的特定定义域。对于边界值为Cantor集的新的自相似图形，试图探讨在该定义域上的性质。在研究SG3左半定义域上的Dirichlet边值问题的过程中，借助SG3上的二元有理点的调和函数值和正则导数求解格林函数。进一步，运用调和函数的有限能以及弱公式化等方法，最终得到了SG3左半定义域上的格林函数表达式及其相应的定理。
关键词： 调和函数;  正则导数;  有限能;  格林函数    
Abstract: The Dirichlet boundary value problem for harmonic function on SG3 is one of the important research topics in the field of fractal analysis. In this paper, the specific sub-domain of SG3 generated by vertical cut from self-similar set SG3 is addressed. We study the properties of this new self-similar structure which has the boundary of Cantor sets. To solve the Dirichlet boundary value problem on the left sub-domain of SG3, we introduce the normal derivative and the harmonic function of the dyadic points on SG3. Furthermore, by using the harmonic functions of finite energy and the weak formula, the explicit expression of the Green's function on the left sub-domain of SG3 and its corresponding theorem are obtained.
Key words: Harmonic function    normal derivative    finite energy    Green function
收稿日期: 2018-12-19 出版日期: 2019-11-25
CLC:  O175.3  
基金资助: 国家自然科学基金资助项目(11701270)； 江苏省高校自然科学基金项目(17KJB110003)； 江苏省政府留学奖学金项目.
通讯作者: ORCID:http：//orcid.org/0000-0002-1487-7255,E-mail：bowu8800@163.com.     E-mail: bowu8800@163.com
作者简介: 蔡洁洁(1992-),ORCID:http：//orcid.org/0000- 0002-7408-8725,女, 硕士研究生, 主要从事分形分析研究, E-mail:jiejieCai27@163.com.
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引用本文:

蔡洁洁, 吴波. SG3左半定义域上的Dirichlet边值问题[J]. 浙江大学学报（理学版）, 2019, 46(6): 676-679.

CAI Jiejie, WU Bo. The Dirichlet boundary value problem on a left sub-domain of the level-3 Sierpinski Gaskets. Journal of ZheJIang University(Science Edition), 2019, 46(6): 676-679.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2019.06.008    或    https://www.zjujournals.com/sci/CN/Y2019/V46/I6/676

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