The $L^p$ Dirichlet Problem for Elliptic Systems on Lipschitz Domains

Abstract

We develop a new approach to the $L^p$ Dirichlet problem via $L^2$ estimates and reverse Holder inequalities. We apply this approach to second order elliptic systems and the polyharmonic equation on a bounded Lipschitz domain $\Omega$ in $R^n$. For $n\ge 4$ and $2-\epsilon <p<2(n-1)/(n-3)+\epsilon$, we establish the solvability of the Dirichlet problem with boundary value data in $L^p(\partial \Omega )$. In the case of the polyharmonic equation ${\Delta }^{\ell }u=0$ with $\ell \ge 2$, the range of $p$ is sharp if $4\le n\le 2\ell +1$.