The N -membranes problem with Neumann type boundary condition A. Azevedo, J. F. Rodrigues and L. Santos Abstract. We consider the problem of finding the equilibrium position of N membranes constrained not to pass through each other, under prescribed vo- lumic forces and boundary tensions. This model corresponds to solve vari- ationally a N -system for linear second order elliptic equations with sequen- tial constraints. We obtain interior and boundary Lewy-Stampacchia type inequalities for the respective solution and we establish the conditions for stability in measure of the interior contact zones of the membranes. 1. Introduction Let Ω be a bounded open subset of R d with Lipschitz boundary Γ. Denote by u =(u 1 ,...,u N ) the equilibrium displacements of N (N ≥ 2) elastic membranes, each one constrained not to pass through the others, subject to external volumic forces f =(f 1 ,...,f N ) and boundary tensions g =(g 1 ,...,g N ). The problem consists of minimizing the energy functional (1.1) E(u)=  Ω  1 2 ( a(u,u)+ cu · u) − f · u  +  Γ  1 2 bu · u − g · u  , in the convex set (1.2) K N =  v =(v 1 ,...,v N ) ∈  H 1 (Ω)  N : v 1 ≥···≥ v N a.e. in Ω  , where a(u,v)= N  k=1 a(u k ,v k ), with a(u,v)= a ij u xi v xj (using the summation con- vention for i,j =1,...,d) and u · v denotes the usual internal product between u and v. The N -membranes problem attached to rigid supports was considered in [3] for N linear coercive elliptic operators of second order and extended in [1] to Received by the editors 15.10.2005. 1991 Mathematics Subject Classification. Primary 35R35; Secondary 35J50. Key words and phrases. Variational inequalities, Lewy-Stampacchia inequalities, coincidence sets. This work was partially supported by FCT (Funda¸ c˜ ao para a Ciˆ encia e Tecnologia).