On the Solvability of a Variational Inequality Problem and Application to a Problem of Two Membranes

The purpose of this work is to give a continuous convex function, for which we can characterize the subdifferential, in order to reformulate a variational inequality problem: find u = (u1,u2) ∈ K such that for all v = (v1,v2) ∈ K, ∫ Ω∇u1∇(v1 −u1)+ ∫ Ω∇u2∇(v2−u2)+(f ,v−u)≥ 0 as a system of independent equations, where f belongs to L2(Ω)×L2(Ω) and K = {v ∈H1 0(Ω)×H 0(Ω) : v1 ≥ v2 a.e. in Ω}. 2000 Mathematics Subject Classification. Primary 35J85.