p.c.f.自相似集中的BGD域I：调和函数的边值问题
BGD domains in p.c.f. self-similar sets I: boundary value problems for harmonic functions
AI总结：
本文研究p.c.f.自相似集上调和函数的边值问题，证明当开连通子集的几何边界为图导向自相似集时，可用有限个通量转移矩阵的乘积表示击中概率，并给出调和函数用边界值积分表示及能量双边估计。
摘要：
我们研究后临界有限（p.c.f.）自相似集的开连通子集上调和函数的边值问题，其中拉普拉斯算子通过强递归自相似局部正则狄利克雷型定义。对于一个p.c.f.自相似集K，我们证明：对于K的任意开连通子集Ω，只要其“几何”边界是图导向自相似集，就存在有限多个称为“通量转移矩阵”的矩阵，这些矩阵的乘积生成从Ω中一点到“电阻”边界∂Ω的击中概率。Ω上的调和函数可以通过对∂Ω上的函数关于这些概率测度进行积分来表示。进一步地，我们得到了调和函数的能量关于其在∂Ω上取值的双边估计。
英文摘要：
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 $Ω\subset K$ whose "geometric" boundary is a graph-directed self-similar set, there exists a finite number of matrices called $\textit{flux transfer matrices}$ whose products generate the hitting probability from a point in $Ω$ to the "resistance" boundary $\partial Ω$. The harmonic functions on $Ω$ can be expressed by integrating functions on $\partial Ω$ against the probability measures. Furthermore, we obtain a two-sided estimate of the energy of a harmonic function in terms of its values on $\partial Ω$.
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