International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 9, Pages 603-608
doi:10.1155/S0161171201004823

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: 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∈H01(Ω)×H01(Ω):v1≥v2  a.e. in Ω}.