### Abstract

B.E.J. Dahlberg's theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace's equation with data taken in L ^{p} spaces p > 2 - δ are extended to compact polyhedral domains of ℝ ^{n}. Consequently, for q < 2 + δ, Dahlberg's reverse Hölder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.

Original language | English (US) |
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Pages (from-to) | 495-520 |

Number of pages | 26 |

Journal | Journal of Geometric Analysis |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 2003 |

### Keywords

- Complexes
- Curtis-Zeeman manifold
- Lipschitz
- Mazur manifold
- NTA
- PL
- polyhedron
- reverse Hölder

### ASJC Scopus subject areas

- Geometry and Topology