An evolutionary vector-valued variational inequality and Lagrange multiplier

Abstract

We prove existence and uniqueness of solution of an evolutionary vector-valued variational inequality defined in the convex set of vector valued functions v subject to the constraint $\left|v\right|\le 1$. We show that we can write the variational inequality as a system of equations on the unknowns $(\lambda ,u)$, where λ is a (unique) Lagrange multiplier belonging to $L^p$ and u solves the variational inequality. Given data $(f_n,u_{n0})$ converging to $(f,u_0)$ in $L^{\infty }\left(Q_T\right)\times H_0^1\left(\Omega \right)$, we prove the convergence of the solutions $({\lambda }_n,u_n)$ of the Lagrange multiplier problem to the solution of the limit problem, when we let $n\to \infty$.