### Abstract

The Neumann problem as formulated in Lipschitz domains with L ^{p} boundary data is solved for harmonic functions in any compact polyhedral domain of ℝ^{4} that has a connected 3-manifold boundary. Energy estimates on the boundary are derived from new polyhedral Rellich formulas together with a Whitney type decomposition of the polyhedron into similar Lipschitz domains. The classical layer potentials are thereby shown to be semi-Fredholm. To settle the onto question a method of continuity is devised that uses the classical 3-manifold theory of E. E. Moise in order to untwist the polyhedral boundary into a Lipschitz boundary. It is shown that this untwisting can be extended to include the interior of the domain in local neighborhoods of the boundary. In this way the flattening arguments of B. E. J. Dahlberg and C. E. Kenig for the H ^{1} _{at} Neumann problem can be extended to polyhedral domains in ℝ^{4}. A compact polyhedral domain in ℝ^{6} of M. L. Curtis and E. C. Zeeman, based on a construction of M. H. A. Newman, shows that the untwisting and flattening techniques used here are unavailable in general for higher dimensional boundary value problems in polyhedra.

Original language | English (US) |
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Pages (from-to) | 571-644 |

Number of pages | 74 |

Journal | Mathematische Annalen |

Volume | 335 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1 2006 |

### Keywords

- 3-manifold
- Atomic estimate
- Bi-Lipschitz
- Homology sphere
- Layer potentials
- Method of continuity
- Polyhedral Rellich
- Shelling

### ASJC Scopus subject areas

- Mathematics(all)