Electron. J. Diff. Eqns.,
Vol. 2007(2007), No. 144, pp. 18.
Nonsymmetric elliptic operators on bounded Lipschitz
domains in the plane
David J. Rule
Abstract:
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 [4] 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 [4] 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; nonsymmetric 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
The King's Buildings,
Mayfield Road
Edinburgh, EH9 3JZ, UK
email: rule@uchicago.edu 
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