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

Abstract

We consider divergence form elliptic operators $L = mathop{ m div} A abla$ 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 ${X = (x,t) : phi(mathbf{e}cdot X) < mathbf{e}^perpcdot X }$, 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.

Keywords

• T(b) Theorem
• layer potentials
• L^p Neumann problem
• L^p regularity problem
• non-symmetric elliptic equations