  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.

Analysis of PDEs;Functional Analysis