An evolutionary vector-valued variational inequality and Lagrange multiplier

We prove existence and uniqueness of solution of an evolutionary vector-valued variational inequality defined in the convex set of vector valued functions $\boldsymbol v$ subject to the constraint $|\boldsymbol v|\le1$. We show that we can write the variational inequality as a system of equations on the unknowns $(\lambda,\boldsymbol u)$, where $\lambda$ is a (unique) Lagrange multiplier belonging to $L^p$ and $\boldsymbol u$ solves the variational inequality. Given data $(\boldsymbol f_n,\boldsymbol u_{n0})$ converging to $(\boldsymbol f,\boldsymbol u_0)$ in $\boldsymbol L^\infty(Q_T)\times H^1_0(\Omega)$, we prove the convergence of the solutions $(\lambda_n,\boldsymbol u_n)$ of the Lagrange multiplier problem to the solution of the limit problem, when we let $n\rightarrow \infty$.