The $L^p$ Dirichlet Problem for the Stokes System on Lipschitz Domains

Abstract

We study the $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed $p>2$, we show that a reverse Hölder condition with exponent $p$ is sufficient for the solvability of the Dirichlet problem with boundary data in $L^p_N(\partialΩ,\rn{d})$. Then we obtain a much simpler condition which implies the reverse Hölder condition. Finally, we establish the solvability ofthe $L^p$ Dirichlet problem for $d\ge 4$ and $2-\epsilon <p<\frac{2(d-1)}{d-3}+\epsilon$.

To appear in Indiana Univ. Math. Journal. Updated to fix formatting issues