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 $\mathit{v}$ subject to the constraint $\mathrm{\mid }\mathit{v}\mathrm{\mid }\le 1$. We show that we can write the variational inequality as a system of equations on the unknowns $(\lambda ,\mathit{u})$, where $\lambda$ is a (unique) Lagrange multiplier belonging to $L^p$ and $\mathit{u}$ solves the variational inequality. Given data $({\mathit{f}}_n,{\mathit{u}}_{n0})$ converging to $(\mathit{f},{\mathit{u}}_0)$ in ${\mathit{L}}^{\mathrm{\infty }}(Q_T)\times H_0^1(\mathrm{\Omega })$, we prove the convergence of the solutions $({\lambda }_n,{\mathit{u}}_n)$ of the Lagrange multiplier problem to the solution of the limit problem, when we let $n\to \mathrm{\infty }$.