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

摘要: This paper continues the study of mixed problem for Laplacian. We consider a bounded Lipschitz domain $\Omega\subset \reals^n$, $n\geq2$, with boundary that is decomposed as $\partial\Omega=D\cup N$, $D$ and $N$ disjoint. let $\Lambda$ denote (relative to $\partial\Omega$) impose conditions on dimension shape sets $D$. Under these geometric criteria, we show there exists $p_0>1$ depending $\Omega$ such $p$ in interval $(1,p_0)$, Neumann data space $L^p(N)$ Dirichlet Sobolev $W^ {1,p}(D) $ has unique solution non-tangential maximal function gradient $L^p(\partial\Omega)$. also obtain results $p=1$ when comes from Hardy spaces, result weighted spaces.