No abstract

Abstract. Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function d C is continuously differentiable everywhere on an open "tube" of uniform thickness around C. Here a corresponding local theory is developed for the property of d C being continuously differentiable outside of C on some neighborhood of a point x ∈ C. This is shown to be equivalent to the prox-regularity of C at x, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of d 2 C being locally of class C 1+ or such that d 2 C + σ| · | 2 is convex around x for some σ > 0. Prox-regularity of C at x corresponds further to the normal cone mapping N C having a hypomonotone truncation around x, and leads to a formula for P C by way of N C . The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.

This paper is devoted to the study of differential inclusions, particularly discontinuous perturbed sweeping processes in the infinite-dimensional setting. On the one hand, the sets involved are assumed to be prox-regular and to have a variation given by a function which is of bounded variation and right continuous. On the other hand, the perturbation satisfies a linear growth condition with respect to a fixed compact subset. Finally, the case where the sets move in an absolutely continuous way is recovered as a consequence.