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Quasi-variational inequalities (QVIs) of obstacle type in many cases havemultiple solutions that can be ordered. We study a multitude of properties ofthe operator mapping the source term to the minimal or maximal solution of suchQVIs. We prove that the solution maps are locally Lipschitz continuous anddirectionally differentiable and show existence of optimal controls forproblems that incorporate these maps as the control-to-state operator. We alsoconsider a Moreau--Yosida-type penalisation for the QVI wherein we show that itis possible to approximate the minimal and maximal solutions by sequences ofminimal and maximal solutions (respectively) of certain PDEs, which have asimpler structure and offer a convenient characterisation in particular forcomputation. For solution mappings of these penalised problems, we prove anumber of properties including Lipschitz and differential stability. Making useof the penalised equations, we derive (in the limit) C-stationarity conditionsfor the control problem, in addition to the Bouligand stationarity we get fromthe differentiability result.

Minimal and maximal solution maps of elliptic QVIs: penalisation, Lipschitz stability, differentiability and optimal control