The mixed problem for the Laplacian in Lipschitz domains

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.Comment: Version 5 includes a correction to one step of the main proof. Since the paper appeared long ago, this submission includes the complete paper, followed by a short section that gives the correction to one step in the proo

Pressure distribution in a reservoir affected by capillarity and hydrodynamic drive: Griffin Field, North West Shelf, Australia

The effects of capillarity in a multilayered reservoir with flow in the aquifer beneath have characteristic signatures on pressure-elevation plots. Such signatures are observed for the Griffin and Scindian/Chinook fields of the Carnarvon Basin North West Shelf of Australia. The Griffin and Scindian/Chinook fields have a highly permeable lower part to the reservoir, a less permeable upper part, and a low permeability top seal. In the Griffin Field there is a systematic tilt of the free-water level in the direction of groundwater flow. Where the oil-water contact occurs in the less permeable part of the reservoir, it lies above the free-water level due to capillarity. In addition to these observable hydrodynamic and capillary effects on hydrocarbon distribution, the multi-well pressure analysis shows that the gas-oil contacts in the Scindian/Chinook fields occur at different elevations. For both the Griffin and Scindian/Chinook fields the oil pressure gradients from each well are non-coincident despite continuous oil saturation, and the difference is not attributable to data error. Furthermore, the shift in oil pressure gradient has a geographical pattern seemingly linked to the hydrodynamics of the aquifer. The simplest explanation for all the observed pressure trends is an oil leg that is still in the process of equilibrating with the prevailing hydrodynamic regime