Sweeping process by prox-regular sets in Riemannian Hilbert manifolds

Frédéric Bernicot (1) , Juliette Venel (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 LAMAV - Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015

In this paper, we deal with sweeping processes on (possibly infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful notions (proximal normal cone, prox-regularity) already defined in the setting of a Hilbert space to the framework of such manifolds. Especially we introduce the concept of local prox-regularity of a closed subset in accordance with the geometrical features of the ambient manifold and we check that this regularity implies a property of hypomonotonicity for the proximal normal cone. Moreover we show that the metric projection onto a locally prox-regular set is single-valued in its neighborhood. Then under some assumptions, we prove the well-posedness of perturbed sweeping processes by locally prox-regular sets.

Mots clés: Differential inclusion; Sweeping process; Prox-regularity; Riemannian manifold

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie métrique [math.MG]

Frédéric Bernicot, Juliette Venel. Sweeping process by prox-regular sets in Riemannian Hilbert manifolds. Journal of Differential Equations, 2015, 259 (8), pp.4086-4121. ⟨10.1016/j.jde.2015.05.011⟩. ⟨hal-01110480⟩