  We consider the mixed boundary value problem or Zaremba's problem for the
Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on
part of the boundary and Neumann data on the remainder of the boundary. We
assume that the boundary between the sets where we specify Dirichlet and
Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p
and the Dirichlet data is in the Sobolev space of functions having one
derivative in L^p for some p near 1. Under these conditions, there is a unique
solution to the mixed problem with the non-tangential maximal function of the
gradient of the solution in L^p of the boundary. We also obtain results with
data from Hardy spaces when p=1.
