BGD domains in p.c.f. self-similar sets I: boundary value problems for harmonic functions

We study the boundary value problems for harmonic functions on open connected subsets of post-critically finite (p.c.f.) self-similar sets, on which the Laplacian is defined through a strongly recurrent self-similar local regular Dirichlet form. For a p.c.f. self-similar set K, we prove that for any open connected subset Ω⊂ K whose "geometric" boundary is a graph-directed self-similar set, there exists a finite number of matrices called flux transfer matrices whose products generate the hitting probability from a point in Ω to the "resistance" boundary ∂ Ω. The harmonic functions on Ω can be expressed by integrating functions on ∂ Ω against the probability measures. Furthermore, we obtain a two-sided estimate of the energy of a harmonic function in terms of its values on ∂ Ω.