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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors


Authors: Marius Mitrea and Michael Taylor
Journal: Trans. Amer. Math. Soc. 355 (2003), 1961-1985
MSC (2000): Primary 31C12, 35J25, 45E05
Published electronically: November 14, 2002
MathSciNet review: 1953534
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Abstract: We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor $g_{jk} dx_j\otimes dx_k$ has low regularity. Under the assumption that

\begin{displaymath}\vert g_{jk}(x)-g_{jk}(y)\vert\leq C\,\omega(\vert x-y\vert),\end{displaymath}

where the modulus of continuity $\omega$ satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with $L^p$ boundary data, for sharp ranges of $p$'s and with optimal nontangential maximal function estimates.


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Additional Information

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: marius@math.missouri.edu

Michael Taylor
Affiliation: Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599
Email: met@math.unc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03150-1
PII: S 0002-9947(02)03150-1
Received by editor(s): April 24, 2002
Received by editor(s) in revised form: July 4, 2002
Published electronically: November 14, 2002
Additional Notes: The first author was partially supported by NSF grants DMS-9870018 and DMS-0139801
The second author was partially supported by NSF grant DMS-9877077
Article copyright: © Copyright 2002 American Mathematical Society