WIAS Preprint No. 2758, (2020)

On the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities


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

2010 Mathematics Subject Classification

  • 47J20 49J40 49J52 49J50


  • Quasi-variational inequality, obstacle problem, directional differentiability, minimal and maximal solutions, ordered solutions




In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasi-variational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities.

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