On the Laplace operator and on the vector potential problems in the half‐space: an approach using weighted spaces

The purpose of the present paper is twofold. The first object is to study the Laplace equation with inhomogeneous Dirichlet and Neumann boundary conditions in the half‐space of ?^N. The behaviour of solutions at infinity is described by means of a family of weighted Sobolev spaces. A class of existence, uniqueness and regularity results are obtained. The second purpose is to investigate some properties of grad, div and curl operators in order to treat curl–div systems of the form curl w = u, div w = 0 and problems related to vector potentials and Helmholtz decomposition.Copyright © 2003 John Wiley & Sons, Ltd.