An evolutionary vector-valued variational inequality and Lagrange multiplier
Davide Azevedo, Lisa Santos · Journal of Mathematical Analysis and Applications · 2025
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 | v | ≤ 1 . We show that we can write the variational inequality as a system of equations on the unknowns ( λ , u ) , where λ is a (unique) Lagrange multiplier belonging to L p and u solves the variational inequality. Given data ( f n , u n 0 ) converging to ( f , u 0 ) in L ∞ ( Q T ) × H 0 1 ( Ω ) , we prove the convergence of the solutions ( λ n , u n ) of the Lagrange multiplier problem to the solution of the limit problem, when we let n → ∞ .