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

Abstract

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: findu=(u1,u2)∈Ksuch that for allv=(v1,v2)∈K,∫Ω∇u1∇(v1−u1)+∫Ω∇u2∇(v2−u2)+(f,v−u)≥0as a system of independent equations, wherefbelongs toL2(Ω)×L2(Ω)andK={v∈H01(Ω)×H01(Ω):v1≥v2  a.e. inΩ}.