On the solvability of a variational inequality problem and application to a problem of two membranes (Q5939671)

Summary: 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 \(u = (u_1,u_2)\in K\) such that for all \(v = (v_1,v_2)\in K\), \(\int_\Omega\nabla u_1\nabla(v_1-u_1) + \int_\Omega\nabla u_2 \nabla (v_2-u_2)+(f,v - u)\geq 0\) as a system of independent equations, where \(f\) belongs to \(L^2(\Omega)\times L^2(\Omega)\) and \(K=\{v\in H^1_0(\Omega)\times H^1_0(\Omega) :v_1 \geq v_2\) a.e. in \(\Omega\)\}.