In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55-81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of R n . This paper extends this study to the resolution of a mixed exterior Laplace's problem. Here, we give existence, unicity and regularity results in L p 's theory with 1 < p < ∞, in weighted Sobolev spaces.

MSC: 35J05 35J25Keywords: Weighted Sobolev spaces Laplacian Dirichlet and Neumann boundary conditions Exterior problems Half-spaceThe purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted L p 's theory with 1 < p < ∞. This paper extends the studies done in [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator, an approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1) (1997) 55-81] with Dirichlet and Neumann conditions.

The purpose of this work is to solve the exterior Stokes problem in the half-space R n + . We study the existence and the uniqueness of generalized solutions in weighted L p theory with 1 < p < ∞. Moreover, we consider the case of strong solutions and very weak solutions. This paper extends the studies done in Alliot, Amrouche (Math. Methods Appl. 23:575-600, 2000) for an exterior Stokes problem in the whole space and in Amrouche, Bonzom (Exterior Problems in the Half-space, submitted) for the Laplace equation in the same geometry as here.