WIAS Preprint No. 2592, (2019)

Existence, iteration procedures and directional differentiability for parabolic QVIs


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

2010 Mathematics Subject Classification

  • 47J20 49J40 49J52

2008 Physics and Astronomy Classification Scheme

  • 49J50


  • Quasi-variational inequality, obstacle problem, conical derivative, directional differentiability, parabolic




We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

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