Boundary value problems for harmonic functions on domains in Sierpinski gaskets

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February 2020, 19(2): 1147-1179. doi: 10.3934/cpaa.2020054

Boundary value problems for harmonic functions on domains in Sierpinski gaskets

Shiping Cao ^1, and Hua Qiu ^2,,

1.	Department of Mathematics, Cornell Univerisity, Ithaca 14853, USA
2.	Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, China

^* Corresponding author

Received May 2019 Revised July 2019 Published October 2019

Fund Project: The research of the second author was supported by Nature Science Foundation of China, Grant 11471157.

We study boundary value problems for harmonic functions on certain domains in the level-$ l $ Sierpinski gaskets $ \mathcal{SG}_l $($ l\geq 2 $) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of $ \mathcal{SG}_l $ and the upper and lower domains generated by horizontal cuts of $ \mathcal{SG}_l $ are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This paper settles several open problems raised in previous work.

Keywords: Boundary value problems, harmonic functions, fractal Laplacians, Sierpinski gasket, energy estimates.

Mathematics Subject Classification: Primary: 28A80; Secondary: 35J25.

Citation: Shiping Cao, Hua Qiu. Boundary value problems for harmonic functions on domains in Sierpinski gaskets. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1147-1179. doi: 10.3934/cpaa.2020054

References:

[1]	Z. Guo, R. Kogan, H. Qiu and R. S. Strichartz, Boundary value problems for a family of domains in the Sierpinski gasket, Illinois J. Math., 58 (2014), 497-519. Google Scholar
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[11]	J. Owen and R. S. Strichartz, Boundary value problems for harmonic functions on a domain in the Sierpinski gasket, Indiana Univ. Math. J., 61 (2012), 319-335. doi: 10.1512/iumj.2012.61.4539. Google Scholar
[12]	H. Qiu, Exact spectrum of the Laplacian on a domain in the Sierpinski gasket, J. Funct. Anal., 277 (2019), 806-888. doi: 10.1016/j.jfa.2018.08.018. Google Scholar
[13]	R. S. Strichartz, Some properties of Laplacians on fractals, J. Funct. Anal., 164 (1999), 181-208. doi: 10.1006/jfan.1999.3400. Google Scholar
[14]	R. S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83. doi: 10.1016/S0022-1236(02)00035-6. Google Scholar
[15]	R. S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton University Press, 2006. Google Scholar
[16]	R.S. Strichartz, "Graph paper" trace characterizations of functions of finite energy, J. Anal. Math., 128 (2016), 239-260. doi: 10.1007/s11854-016-0008-x. Google Scholar

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References:

[1]	Z. Guo, R. Kogan, H. Qiu and R. S. Strichartz, Boundary value problems for a family of domains in the Sierpinski gasket, Illinois J. Math., 58 (2014), 497-519. Google Scholar
[2]	M. Hino and T. Kumagai, A trace theorem for Dirichlet forms on fractals, J. Funct. Anal., 238 (2006), 578-611. doi: 10.1016/j.jfa.2006.05.012. Google Scholar
[3]	A. Jonsson, A trace theorem for the Dirichlet forms on the Sierpinski gasket, Math. Z., 250 (2005), 599-609. doi: 10.1007/s00209-005-0767-z. Google Scholar
[4]	J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan. J. Appl. Math., 6 (1989), 259-290. doi: 10.1007/BF03167882. Google Scholar
[5]	J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755. doi: 10.2307/2154402. Google Scholar
[6]	J. Kigami, Analysis on Fractals, Cambridge University Press, 2001. doi: 10.1017/CBO9780511470943. Google Scholar
[7]	J. Kigami, Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees, Adv. Math., 225 (2010), 2674-2730. doi: 10.1016/j.aim.2010.04.029. Google Scholar
[8]	J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012). doi: 10.1090/S0065-9266-2011-00632-5. Google Scholar
[9]	P. H. Li, N. Ryder, R. S. Strichartz and B. E. Ugurcan, Extensions and their minimizations on the Sierpinski gasket, Potential Anal., 41 (2014), 1167-1201. doi: 10.1007/s11118-014-9415-8. Google Scholar
[10]	W. Li and R. S. Strichartz, Boundary value problems on a half Sierpinski gasket, J. Fractal Geom., 1 (2014), 1-43. doi: 10.4171/JFG/1. Google Scholar
[11]	J. Owen and R. S. Strichartz, Boundary value problems for harmonic functions on a domain in the Sierpinski gasket, Indiana Univ. Math. J., 61 (2012), 319-335. doi: 10.1512/iumj.2012.61.4539. Google Scholar
[12]	H. Qiu, Exact spectrum of the Laplacian on a domain in the Sierpinski gasket, J. Funct. Anal., 277 (2019), 806-888. doi: 10.1016/j.jfa.2018.08.018. Google Scholar
[13]	R. S. Strichartz, Some properties of Laplacians on fractals, J. Funct. Anal., 164 (1999), 181-208. doi: 10.1006/jfan.1999.3400. Google Scholar
[14]	R. S. Strichartz, Function spaces on fractals, J. Funct. Anal., 198 (2003), 43-83. doi: 10.1016/S0022-1236(02)00035-6. Google Scholar
[15]	R. S. Strichartz, Differential Equations on Fractals: A Tutorial, Princeton University Press, 2006. Google Scholar
[16]	R.S. Strichartz, "Graph paper" trace characterizations of functions of finite energy, J. Anal. Math., 128 (2016), 239-260. doi: 10.1007/s11854-016-0008-x. Google Scholar

Figure 1. Upper and lower domains in $ \mathcal{SG} $ and $ \mathcal{SG}_3 $

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Figure 2. Half domains in $ \mathcal{SG} $ and $ \mathcal{SG}_3 $

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Figure 3. $ \Gamma_1,\Gamma_2,\Gamma_3 $ of $ \mathcal{SG}_3 $

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Figure 4. Harmonic extension algorithm of $ \mathcal{SG}_3 $

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Figure 5. The values of $ h_a $ on $ V_1\cap\bar{\Omega} $

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Figure 6. Simple sets $ O_1,O_2 $

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Figure 7. The values of $ v $ on $ O_2 $

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Figure 8. The upper domain

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Figure 9. The relationship between $ \Omega_{\lambda_0} $ and $ \Omega_{\lambda_1} $

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Figure 10. The upper domain $ \Omega_{0.39} $. The shaded regions are $ F^\lambda_5\Omega_{\lambda_1} $, $ F^\lambda_{54}\Omega_{\lambda_2} $, $ F^\lambda_{543}\Omega_{\lambda_3} $

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Figure 11. Boundary values of $ h^{(1)} $, $ h^{(2)} $

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Figure 12. Two typical domain $ \Omega_\lambda^- $'s. $ \lambda $ is (or not) a dyadic rational

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Figure 13. The relationship between $ \Omega_\lambda^- $ and $ \Omega_{S\lambda}^- $. $ e_1(\lambda) = 0 $ in the left one and $ e_1(\lambda) = 1 $ in the right one

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Figure 14. $ h_1+h_2 $ and $ h_1-h_2 $ in two cases

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Figure 15. $ V_m^\lambda $ and some conductances. ($ \frac{5}{8}<\lambda<\frac{3}{4},m = 3 $)

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Figure 16. The half domain of $ \mathcal{SG}_4 $

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Figure 17. The vertices on a half $ \mathcal{SG} $

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