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 Ω ⊂ R n \Omega \subset \mathbf {R}^n , n ≥ 2 n\geq 2 , with boundary that is decomposed as ∂ Ω = D ∪ N \partial \Omega =D\cup N , with D D and N N disjoint. We let Λ \Lambda denote the boundary of D D (relative to ∂ Ω \partial \Omega ) and impose conditions on the dimension and shape of Λ \Lambda and the sets N N and D D . Under these geometric criteria, we show that there exists p 0 > 1 p_0>1 depending on the domain Ω \Omega such that for p p in the interval ( 1 , p 0 ) (1,p_0) , the mixed problem with Neumann data in the space L p ( N ) L^p(N) and Dirichlet data in the Sobolev space W 1 , p ( D ) W^{1, p}(D) has a unique solution with the non-tangential maximal function of the gradient of the solution in L p ( ∂ Ω ) L^p(\partial \Omega ) . We also obtain results for p = 1 p=1 when the Dirichlet and Neumann data come from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.