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 domainΩ⊂Rn\Omega \subset \mathbf {R}^n,n≥2n\geq 2, with boundary that is decomposed as∂Ω=D∪N\partial \Omega =D\cup N, withDDandNNdisjoint. We letΛ\Lambdadenote the boundary ofDD(relative to∂Ω\partial \Omega) and impose conditions on the dimension and shape ofΛ\Lambdaand the setsNNandDD. Under these geometric criteria, we show that there existsp0>1p_0>1depending on the domainΩ\Omegasuch that forppin the interval(1,p0)(1,p_0), the mixed problem with Neumann data in the spaceLp(N)L^p(N)and Dirichlet data in the Sobolev spaceW1,p(D)W^{1, p}(D)has a unique solution with the non-tangential maximal function of the gradient of the solution inLp(∂Ω)L^p(\partial \Omega ). We also obtain results forp=1p=1when the Dirichlet and Neumann data come from Hardy spaces, and a result when the boundary data comes from weighted Sobolev spaces.