Directional differentiability for elliptic quasi-variational inequalities of obstacle type
- Alphonse, Amal
- Hintermüller, Michael
- Rautenberg, Carlos N.
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.
- Calc. Var. Partial Differ. Equ., 58 (2019), pp. 39/1--39/47, DOI 10.1007/s00526-018-1473-0 .