C. R. Acad. Sci. Paris, t. 328, Shrie I, p. 221-226, 1999 Equations aux d&i&es partielles/farfial Differential Equations Espaces de Sobolev avec poids pour l’hquation de Laplace dans le demi-espace Tahar Zamine BOULMEZAOUD Laboratoire d’analyse numkrique, Universit6 Paris VI, 4, place .Junsieu, 75252 Paris redex 05, France (Requ et accept6 le 27 novembre 1998) RCsum6. On Ctudie I’tquation de Laplace dans le demi-espace en utilisant une famille d’espaces de Sobolev avec poids. Une classe complete de resultats d’existence, d’unicite et de regularitt est obtenue pour des conditions aux limites de type Dirichlet ou Neumann non homogenes. 0 Academic des ScienceslElsevier, Paris Weighted Sobolev spaces for the Laplace equation in the half-space Abstract. We deal in this Note with the Laplace equation in the half-space. The use of a special family of weighted Sobolev spaces as a framework is at the heart of our approach. A complete class of existence, uniqueness and regularity results is obtained for inhomogeneous Dirichlet and Neumann problems. 0 Academic des Sciences/Elsevier, Paris A bridged English Version The aim of this Note is to study the Laplace operator in the half-space using a family of weighted Sobolev spaces (WSS) as a framework for describing the behaviour of solutions at infinity. This family of WSS was suggested by Hanouzet [4] and is defined as follows: given an open set 52of R” (N > 2), a non-negative integer m and a real number Q, consider the space where P(T) = (1 +r’)i is the basic weight. Here, we are mainly concerned with the case in which R is the upper half-space WY = {x = (z’,zN); x’ E RN-l, XN E [w and ZN > O}. The space W~(iw~) is a Hilbert spacefor the norm (2). We denote by $;I(Ry ) the closure of ‘o(Rf’) in W:z(Wy) and by W:r(W$v) the dual space of W;‘(Ry ). For any integer Ic, the space de (resp. Ne) denotes the space of harmonic polynomials of degree less or equal to AZ and odd (resp. even) with respect to 2,~. For the most significant basic properties of these spaces, the reader can consult Hanouzet’s paper [4]. It is worth noting that an extension of the above definition of W:(0) to real values of Note pr6sebt6e par Philippe G. CIARLET. 0764~4442/99/03280221 0 AcadCmie des Sciences/Elsevier, Paris 221