An evolutionary vector-valued variational inequality and Lagrange multiplier
Davide Azevedo, Lisa Santos
TL;DR
This work analyzes evolutionary vector-valued variational inequalities under a pointwise constraint $|oldsymbol u|\le 1$, establishing that the VI can be recast as a Lagrange multiplier system with $\lambda\in L^p(Q_T)$ and $(\lambda-1)(|\boldsymbol u|-1)=0$. An approximation scheme using a monotone operator $\widehat{k}_{\varepsilon\delta}$ yields a unique $\boldsymbol u_{\varepsilon\delta}$, and ε-independent a priori estimates allow passage to the limit $\varepsilon\to0$ to obtain a unique pair $(\lambda_\delta,\boldsymbol u_\delta)$ solving the Lagrange multiplier system $\partial_t\boldsymbol u_\delta-\Delta\boldsymbol u_\delta+\lambda_\delta\boldsymbol u_\delta=\boldsymbol f$ with $\boldsymbol u_\delta(0)=\boldsymbol u_0$ and boundary conditions, along with the complementarity conditions $\lambda_\delta\ge\delta$, $(\lambda_\delta-\delta)(|\boldsymbol u_\delta|-1)=0$. The paper then proves continuous dependence of $(\lambda_\delta,\boldsymbol u_\delta)$ on data, showing convergence for data sequences and providing stability estimates. These results extend VI theory to a vector-valued, constrained, evolutionary setting and connect with elastic-plastic torsion-type models via the multiplier formulation.
Abstract
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 $(λ,\boldsymbol u)$, where $λ$ 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 $\boldsymbol L^\infty(Q_T)\times H^1_0(Ω)$, we prove the convergence of the solutions $(λ_n,\boldsymbol u_n)$ of the Lagrange multiplier problem to the solution of the limit problem, when we let $n\rightarrow \infty$.