Particle Dynamics with Elastic Collision at the Boundary: Existence and Partial Uniqueness of Solutions

We consider the dynamics of point particles which are confined to a bounded, possibly nonconvex domain $\Omega$. Collisions with the boundary are described as purely elastic collisions. This turns the description of the particle dynamics into a coupled system of second order ODEs with discontinuous right-hand side. The main contribution of this paper is to develop a precise solution concept for this particle system, and to prove existence of solutions. In this proof we construct a solution by passing to the limit in an auxiliary problem based on the Yosida approximation. In addition to existence of solutions, we establish a partial uniqueness theorem, and show by means of a counterexample that uniqueness of solutions cannot hold in general.
Comment: 21 pages