Directional differentiability for elliptic quasi-variational inequalities of obstacle type

TL;DR

This paper investigates the directional differentiability of solutions to obstacle-type quasi-variational inequalities, extending classical results to cases with multiple solutions and applying the theory to thermoforming with numerical validation.

Contribution

It extends Mignot's classical differentiability results to quasi-variational inequalities with multiple solutions, providing new selection procedures and estimates.

Findings

Established directional differentiability under broader conditions.

Developed solution selection procedures for multiple solutions.

Applied theory to thermoforming with numerical experiments.

Abstract

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.