Bounce law at the corners of convex billiards

✍ Scribed by Alexandre Cabot

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 279 KB
• Volume: 57
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2004.03.008

✦ Synopsis

Let C be a convex subset of R n . Given any elastic shock solution x(•) of the di erential inclusion

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 deÿne a sequence (C ) of regular sets tending to C as → 0, then we consider the approximate di erential inclusion x (t) + NC (x (t)) 0, and ÿnally we pass to the limit when → 0. For approximate sets deÿned by C = C + B (where B is the unit euclidean ball of R n ), we recover the bounce law associated with the Moreau-Yosida regularization.