The $L^p$ Regularity Problem on Lipschitz Domains

Abstract

This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1<p<\infty$, we prove that the solvability of the $L^p$ regularity problem is equivalent to that of the $L^{p^{\prime }}$ Dirichlet problem. For higher order elliptic equations and systems, we show that if $p>2$, the solvability of the $L^p$ regularity problem is equivalent to a weak reverse Hölder condition with exponent $p$.

To appear in Trans. of AMS. Updated to fix formatting issues