The $L^p$ Regularity Problem on Lipschitz Domains

This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1<p<\mathrm{\infty }$, we prove that the solvability of the $L^p$ regularity problem is equivalent to that of the $L^{p^{\mathrm{\prime }}}$ 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 H"older condition with exponent $p$.