### Asymptotically critical points and multiple elastic bounce trajectories

DOI: http://dx.doi.org/10.12775/TMNA.2007.034

#### 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.

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.

#### Keywords

Elastic bounce trajectories; asymptotically critical points; Lusternik-Schnirelmann category; $\varphi$-convex functions; $\nabla$-theorems

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