David J. Rule
We consider divergence form elliptic operators in with a coefficient matrix of bounded measurable functions independent of the -direction. The aim of this note is to demonstrate how the proof of the main theorem in  can be modified to bounded Lipschitz domains. The original theorem states that the Neumann and regularity problems are solvable for for some in domains of the form , where is a Lipschitz function. The exponent depends only on the ellipticity constants and the Lipschitz constant of . The principal modification of the argument for the original result is to prove the boundedness of the layer potentials on domains of the form , for a fixed unit vector and . This is proved in  only in the case . A simple localisation argument then completes the proof.
Submitted October 10, 2007. Published October 30, 2007.
Math Subject Classifications: 35J25, 31A25.
Key Words: T(b) Theorem; layer potentials; Neumann problem; regularity problem; non-symmetric elliptic equations.
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| David J. Rule |
School of Mathematics and Maxwell Institute for Mathematical Sciences
The University of Edinburghm James Clerk Maxwell Building
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Edinburgh, EH9 3JZ, UK
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