BGD domains in p.c.f. self-similar sets I: boundary value problems for harmonic functions
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929483940888576 |
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| author | Gu, Qingsong Qiu, Hua |
| author_facet | Gu, Qingsong Qiu, Hua |
| contents | 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 $\partial Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | BGD domains in p.c.f. self-similar sets I: boundary value problems for harmonic functions Gu, Qingsong Qiu, Hua Functional Analysis Probability 28A80, 31E05 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 $\partial Ω$. |
| title | BGD domains in p.c.f. self-similar sets I: boundary value problems for harmonic functions |
| topic | Functional Analysis Probability 28A80, 31E05 |
| url | https://arxiv.org/abs/2409.01623 |