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

Abstract:

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