Potential Theory on Lipschitz Domains in Riemannian Manifolds: the Case of Dini Metric Tensors
نویسندگان
چکیده
We study the applicability of the the method of layer potentials, in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz subdomains of Riemannian manifolds, in the case when metric tensor gjkdxj ⊗ dxk has low regularity. Under the assumption that |gjk(x) − gjk(y)| ≤ C ω(|x − y|), where the modulus of continuity ω satisfies a Dini type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with L boundary data, for sharp ranges of p’s and with optimal non-tangential maximal function estimates. 1Partially supported by NSF grants DMS-9870018 and DMS-0139801 2Partially supported by NSF grant DMS-9877077 1
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