Elliptic quasi-variational inequalities under a smallness assumption: Uniqueness, differential stability and optimal control

TL;DR

This paper investigates elliptic quasi-variational inequalities with a smallness condition on the moving set, establishing uniqueness, stability, and optimal control conditions under Lipschitz assumptions.

Contribution

It proves uniqueness and stability of solutions and derives optimality conditions for control problems involving these inequalities under a smallness assumption.

Findings

Unique solution under small Lipschitz constant

Lipschitz dependence of solution on data

Necessary optimality conditions for control

Abstract

We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.