The dirichlet-conormal problem with homogeneous and inhomogeneous boundary conditions

We consider the mixed Dirichlet-conormal problem on irregular domains in R-d. Two types of regularity results will be discussed: the W-1,W- p regularity and a non-tangential maximal function estimate. The domain is assumed to be Reifenberg-flat, and the interfacial boundary is either Reifenberg-flat of co-dimension 2 or is locally sufficiently close to a Lipschitz function of m variables, where m = 1, ... , d - 2: For the non-tangential maximal function estimate, we also require the domain to be Lipschitz.