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

TL;DR

This paper proves directional differentiability of minimal and maximal solution maps for elliptic quasi-variational inequalities of obstacle type, characterizing derivatives as limits of solutions to related variational inequalities, with applications to thermoforming.

Contribution

It establishes the directional differentiability of solution maps for elliptic quasi-variational inequalities and characterizes derivatives as limits of variational inequality solutions, providing new analytical tools.

Findings

Solution maps are directionally differentiable with respect to the forcing term.

Derivatives can be characterized as limits of solutions to variational inequalities.

Application demonstrated in thermoforming process modeling.

Abstract

In this 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. Along 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. We conclude the paper with some examples and an application to thermoforming.