Asymptotically critical points and multiple elastic bounce trajectories

Abstract

We study multiplicity of elastic bounce trajectories (e.b.t.'s) with fixed end points $A$ and $B$ on a nonconvex "billiard table" $\Omega$. As well known, in general, such trajectories might not exist at all. Assuming the existence of a "bounce free" trajectory $\gamma_0$ in $\Omega$ joining $A$ and $B$ we prove the existence of multiple families of e.b.t.'s $\gamma_{\lambda}$ bifurcating from $\gamma_0$ as a suitable parameter $\lambda$ varies. Here $\lambda$ appears in the dynamics equation as a multiplier of the potential term.

We use a variational approach and look for solutions as the critical points of the standard Lagrange integrals on the space $X(A,B)$ of curves joining $A$ and $B$. Moreover, we adopt an approximation scheme to obtain the elastic response of the walls as the limit of a sequence of repulsive potentials fields which vanish inside $\Omega$ and get stronger and stronger outside. To overcome the inherent difficulty of distinct solutions for the approximating problems covering to a single solutions to the limit one, we use the notion of "asymptotically critical points" (a.c.p.'s) for a sequence of functional. Such a notion behaves much better than the simpler one of "limit of critical points" and allows to prove multliplicity theorems in a quite natural way.

A remarkable feature of this framework is that, to obtain the e.b.t.'s as a.c.p.'s for the approximating Lagrange integrals, we are lead to consider the $L^2$ metric on $X(A,B)$. So we need to introduce a nonsmooth version of the definition of a.c.p. and prove nonsmooth versions of the multliplicity theorems, in particular of the "$\nabla$-theorems" used for the bifurcation result. To this aim we use several results from the theory of $\varphi$-convex functions.