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 Ω ⊂ Rn, n ≥ 2, with boundary that is decomposed as ∂Ω = D ∪ N , D and N disjoint. We let Λ denote the boundary of D (relative to ∂Ω) and impose conditions on the dimension and shape of Λ and the sets N and D. Under these geometric criteria, we show that there exists p0 > 1 depending on the domain Ω such that for p in the interval (1, p0), the mixed problem with Neumann data in the space Lp(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 Lp(∂Ω). 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.