# American Institute of Mathematical Sciences

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

## Boundary value problems for harmonic functions on domains in Sierpinski gaskets

 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.

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:

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##### References:
Upper and lower domains in $\mathcal{SG}$ and $\mathcal{SG}_3$
Half domains in $\mathcal{SG}$ and $\mathcal{SG}_3$
$\Gamma_1,\Gamma_2,\Gamma_3$ of $\mathcal{SG}_3$
Harmonic extension algorithm of $\mathcal{SG}_3$
The values of $h_a$ on $V_1\cap\bar{\Omega}$
Simple sets $O_1,O_2$
The values of $v$ on $O_2$
The upper domain
The relationship between $\Omega_{\lambda_0}$ and $\Omega_{\lambda_1}$
The upper domain $\Omega_{0.39}$. The shaded regions are $F^\lambda_5\Omega_{\lambda_1}$, $F^\lambda_{54}\Omega_{\lambda_2}$, $F^\lambda_{543}\Omega_{\lambda_3}$
Boundary values of $h^{(1)}$, $h^{(2)}$
Two typical domain $\Omega_\lambda^-$'s. $\lambda$ is (or not) a dyadic rational
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
$h_1+h_2$ and $h_1-h_2$ in two cases
$V_m^\lambda$ and some conductances. ($\frac{5}{8}<\lambda<\frac{3}{4},m = 3$)
The half domain of $\mathcal{SG}_4$
The vertices on a half $\mathcal{SG}$
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