Bounce law at the corners of convex billiards

Let C be a convex subset of R-n. Given any elastic shock solution x((.)) of the differential inclusion x (t) + N-C(x(t)) There Exists 0, t > 0, the bounce of the trajectory at a regular point of the boundary of C follows the Descartes law. The aim of the paper is to exhibit the bounce law at the comers of the boundary. For that purpose, we define a sequence (C,,) of regular sets tending to C as epsilon --> 0, then we consider the approximate differential inclusion x(epsilon)(t) + N-Cepsilon,(x(epsilon)(t)) There Exists 0, and finally we pass to the limit when t; --> 0. For approximate sets defined by C-epsilon = C + epsilonB (where B is the unit euclidean ball of R-n), we recover the bounce law associated with the Moreau-Yosida regularization. (C) 2004 Elsevier Ltd. All rights reserved.