Abstract
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.
Highlights
Quasi-variational inequalities (QVIs) are versatile models that are used to describe many different phenomena from fields as varied as physics, biology, finance and economics
QVIs are more complicated than variational inequalities (VIs) because solutions are sought in a constraint set which depends on the solution itself
This extends to the parabolic setting our previous work [2] where we provided a differentiability result for solution mappings associated to elliptic QVIs
Summary
Quasi-variational inequalities (QVIs) are versatile models that are used to describe many different phenomena from fields as varied as physics, biology, finance and economics. We focus in this work on parabolic QVIs with constraint sets of obstacle type and we address the issues of existence of solutions and directional differentiability for the solution map taking the source term of the QVI into the set of solutions. – Existence of solutions to (1) via time-discretisation (Theorem 9): we show that solutions to (1) can be formulated as the limit of a sequence constructed from considering time-discretised elliptic QVI problems This result makes use of the theory of sub- and supersolutions and the Tartar–Birkhoff fixed point method. It should be noted that the differentiability result essentially gives a characterisation of the contingent derivative (a concept frequently used in set-valued analysis) of P (between appropriate spaces) in terms of a parabolic QVI; see Proposition 41 for details
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