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

DOI:
https://doi.org/10.1090/S0002-9947-02-03150-1

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 has low regularity. Under the assumption that

where the modulus of continuity satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with boundary data, for sharp ranges of '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:
https://doi.org/10.1090/S0002-9947-02-03150-1

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