A multidirectional Dirichlet problem

  • Gregory C. Verchota
  • Andrew L. Vogel


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 Lp spaces p > 2 − δ are extended to compact polyhedral domains of ℝn. 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.

Math Subject Classifications

35J25 31B25 

Key Words and Phrases

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


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Copyright information

© Mathematica Josephina, Inc. 2003

Authors and Affiliations

  1. 1.Department of MathematicsSyracuse UniversitySyracuse

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