Electron. J. Diff. Eqns., Vol. 2007(2007), No. 144, pp. 1-8.

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

David J. Rule

Abstract:
We consider divergence form elliptic operators $L = \hbox{div} A\nabla$ 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 $\{X = (x,t) : \phi(\hbox{e}\cdot X) < \hbox{e}^\perp\cdot X \}$, for a fixed unit vector $\hbox{e} = (e_1,e_2)$ and $\hbox{e}^\perp = (-e_2,e_1)$. This is proved in [4] only in the case $\hbox{e} = (1,0)$. 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.

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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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