A uniqueness result for quasilinear elliptic equations with measures as data

In this paper, Ω is a bounded domain in R (N ≥ 2), with a Lipschitz continuous boundary. The unit normal to ∂Ω outward to Ω is denoted by n. We denote by x·y the usual Euclidean product of two vectors (x, y) ∈ R × R ; the associated Euclidean norm is written |.|. The Lebesgue measure of a measurable subset E in R is denoted by |E|; σ is the Lebesgue measure on ∂Ω (i.e. the (N−1)-dimensional Hausdorff measure). Γd and Γf are measurable subsets of ∂Ω such that ∂Ω = Γd ∪ Γf and σ(Γd ∩ Γf ) = 0. For q ∈ [1,+∞], we denote by q′ the conjugate exponent of q (i.e. q′ = q/(q − 1)). W (Ω) is the usual Sobolev space, endowed with the norm ||u||W 1,q(Ω) = ||u||Lq(Ω) + || |∇u| ||Lq(Ω). W 1,q Γd (Ω) is the space of functions of W (Ω) which have a null trace on Γd. When q = 2, we write H Γd(Ω) instead of W 1,q Γd (Ω). The space of the traces of functions in H Γd(Ω) is denoted by H 1/2 Γd (Ω) and it is endowed with the norm