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.