On generalized solution concepts for nonsmooth evolution arising in the differential sensitivity analysis of parabolic variational inequalities

This talk is concerned with directional differentiability results for solution operators of parabolic evolution variational inequalities. We discuss the (in large parts still open) problem of characterizing the directional derivatives of the solution map of a parabolic variational inequality by means of a suitably defined auxiliary problem. It is shown that the main difficulty that arises in this context is that the appearing directional derivatives are discontinuous as functions of time. This implies that standard weak formulations for evolutionary problems without any additional selection criteria are not able to provide a unique characterization. We present new results that show how this problem can be overcome for sweeping processes in the sense of Moreau by means of Kurzweil-Stieltjes integration theory and a vectorial version of Stampacchia’s lemma.

The talk concludes with a discussion of open questions and directions for future research.