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

12 pages