Description

• Title:
  Asymptotically critical points and multiple elastic bounce trajectories
• Creator:
  Marino, Antonio;
  Saccon, Claudio
• Description:
  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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2007.12.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 30, No 2 (December 2007); 351 - 395;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2007.034/8859
• Subject:
  Elastic bounce trajectories;
  asymptotically critical points;
  Lusternik-Schnirelmann category;
  $\varphi$-convex functions;
  $\nabla$-theorems;
  58E05;
  58E30;
  73K05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2007.034
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other