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.