Existence, iteration procedures and directional differentiability for parabolic QVIs

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.Comment: 41 page

Stability of the solution set of quasi-variational inequalities and optimal control

For a class of quasi-variational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to {monotone} perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.Comment: 29 page

Developmental Correlations of the Deciduous and Permanent Teeth During the Human Fetal Period

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/67941/2/10.1177_00220345770560120501.pd

Recent trends and views on elliptic quasi-variational inequalities

We consider state-of-the-art methods, theoretical limitations, and open problems in elliptic Quasi-Variational Inequalities (QVIs). This involves the development of solution algorithms in function space, existence theory, and the study of optimization problems with QVI constraints. We address the range of applicability and theoretical limitations of fixed point and other popular solution algorithms, also based on the nature of the constraint, e.g., obstacle and gradient-type. For optimization problems with QVI constraints, we study novel formulations that capture the multivalued nature of the solution mapping to the QVI, and generalized differentiability concepts appropriate for such problems

Directional differentiability for elliptic quasi-variational inequalities of obstacle type

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

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

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.Comment: 13 page

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

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

Stability of the solution set of quasi-variational inequalities and optimal control

For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements to the solution set of the QVI with respect to monotonic perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed

Existence, iteration procedures and directional differentiability for parabolic QVIs

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator

Existence, iteration procedures and directional differentiability for parabolic QVIs

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator