The mixed problem in Lipschitz domains with general decompositions of the boundary

This paper continues the study of the mixed problem for the Laplacian. We consider a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, with boundary that is decomposed as $\mathrm{\partial }\mathrm{\Omega }=D\cup N$, $D$ and $N$ disjoint. We let $\mathrm{\Lambda }$ denote the boundary of $D$ (relative to $\mathrm{\partial }\mathrm{\Omega }$) and impose conditions on the dimension and shape of $\mathrm{\Lambda }$ and the sets $N$ and $D$. Under these geometric criteria, we show that there exists $p_0>1$ depending on the domain $\mathrm{\Omega }$ such that for $p$ in the interval $(1,p_0)$, the mixed problem with Neumann data in the space $L^p(N)$ and Dirichlet data in the Sobolev space $W^{1,p}(D)$ has a unique solution with the non-tangential maximal function of the gradient of the solution in $L^p(\mathrm{\partial }\mathrm{\Omega })$. We also obtain results for $p=1$ when the Dirichlet and Neumann data comes from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.