Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 144, pp. 1-8.

Title: Non-symmetric elliptic operators on bounded Lipschitz 
       domains in the plane

Author: David J. Rule (The Univ. of Edinburgh, UK)

Abstract: 
 We consider divergence form elliptic operators
 $L = \mathop{\rm div}  A\nabla$ in  $\mathbb{R}^2$ with a coefficient
 matrix $A = A(x)$ of bounded measurable functions independent of the
 $t$-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 $L^p$ Neumann
 and regularity problems are solvable for $1 < p < p_0$ for some
 $p_0$ in domains of the form $\{(x,t) : \phi(x) < t\}$, where $\phi$
 is a Lipschitz function. The exponent $p_0$ depends only on the
 ellipticity constants and the Lipschitz constant of $\phi$.
 The principal modification of the argument for the original
 result is to prove the boundedness of the layer potentials on domains
 of the form
 $\{X = (x,t) : \phi(\mathbf{e}\cdot X) < \mathbf{e}^\perp\cdot X \}$,
 for a fixed unit vector $\mathbf{e} = (e_1,e_2)$ and
 $\mathbf{e}^\perp = (-e_2,e_1)$. This is proved in [4] only in
 the case $\mathbf{e} = (1,0)$. 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; L^p Neumann problem;
           L^p regularity problem; non-symmetric elliptic equations.