WIAS Preprint No. 3093, (2024)

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


  • Alphonse, Amal
    ORCID: 0000-0001-7616-3293
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Rautenberg, Carlos N.
    ORCID: 0000-0001-9497-9296
  • Wachsmuth, Gerd

2020 Mathematics Subject Classification

  • 49J40 47J20 49J52 49J21 49K21


  • Quasi-variational inequality, optimal control, stationary system, ordered solution, directional differentiability


Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau?Yosida-type penalisation for the QVI wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result.

Download Documents