WIAS Preprint No. 2492, (2018)

Directional differentiability for elliptic quasi-variational inequalities of obstacle type


  • Alphonse, Amal
    ORCID: 0000-0001-7616-3293
  • 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, state constraint, conical derivative, directional differentiability, thermoforming




The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments.

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