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Transactions of the American Mathematical Society

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Dirichlet regularity of subanalytic domains


Author: Tobias Kaiser
Journal: Trans. Amer. Math. Soc. 360 (2008), 6573-6594
MSC (2000): Primary 31B25, 32B20; Secondary 03C64
DOI: https://doi.org/10.1090/S0002-9947-08-04609-6
Published electronically: July 22, 2008
MathSciNet review: 2434300
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Abstract: Let $ {\Omega}$ be a bounded and subanalytic domain in $ {{\mathbb{R}}^n}$, $ {n\, \geq \, 2}$. We show that the set of boundary points of $ {\Omega}$ which are regular with respect to the Dirichlet problem is again subanalytic. Moreover, we give sharp upper bounds for the dimension of the set of irregular boundary points. This enables us to decide whether the domain has a classical Green function. In dimensions 2 and 3, this is the case, given some mild and necessary conditions on the topology of the domain.


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Additional Information

Tobias Kaiser
Affiliation: Naturwissenschaftliche Fakultät-Mathematik, University of Regensburg, Universitätsstr. 31, 93040 Regensburg, Germany
Email: tobias.kaiser@mathematik.uni-regensburg.de

DOI: https://doi.org/10.1090/S0002-9947-08-04609-6
Received by editor(s): March 23, 2006
Received by editor(s) in revised form: February 5, 2007
Published electronically: July 22, 2008
Additional Notes: This research was supported by DFG-Projekt KN202/5-1
Article copyright: © Copyright 2008 American Mathematical Society