Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains

We study the homogeneous elliptic systems of order $2\mathrm{\ell }$ with real constant coefficients on Lipschitz domains in $R^n$, $n\ge 4$. For any fixed $p>2$, we show that a reverse H"older condition with exponent $p$ is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in $L^p$. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the $L^p$ Dirichlet problem for $n\ge 4$ and $2-\text{e}<p<\frac{2(n-1)}{n-3}+\text{e}$. The range of $p$ is known to be sharp if $\mathrm{\ell }\ge 2$ and $4\le n\le 2\mathrm{\ell }+1$. For the polyharmonic equation, the sharp range of $p$ is also found in the case $n=6$, 7 if $\mathrm{\ell }=2$, and $n=2\mathrm{\ell }+2$ if $\mathrm{\ell }\ge 3$.