Bounce law at the corners of convex billiards

Abstract

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))∋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 corners of the boundary. For that purpose, we define a sequence ( C ε ) of regular sets tending to C as ε→0, then we consider the approximate differential inclusion x ̈ ε(t)+N C ε (x ε(t))∋0 , and finally we pass to the limit when ε→0. For approximate sets defined by C ε=C+ε B (where B is the unit euclidean ball of R n ), we recover the bounce law associated with the Moreau–Yosida regularization.