On the existence of connecting orbits for critical values of the energy

TL;DR

This paper investigates the existence of connecting orbits in a mechanical system with a smooth potential that vanishes on the boundary of a domain, focusing on orbits at zero energy level connecting boundary components, including cases with critical points and symmetry.

Contribution

It establishes conditions for the existence of connecting orbits at zero energy in systems with potentials vanishing on boundaries, including symmetric cases and those with critical points.

Findings

Existence of connecting orbits between boundary components at zero energy.

Results applicable to potentials with critical points on the boundary.

Analysis includes symmetric potential cases.

Abstract

We consider an open connected set $\Omega$ and a smooth potential $U$ which is positive in $\Omega$ and vanishes on $\partial\Omega$. We study the existence of orbits of the mechanical system \[ \ddot{u}=U_x(u), \] that connect different components of $\partial\Omega$ and lie on the zero level of the energy. We allow that $\partial\Omega$ contains a finite number of critical points of $U$. The case of symmetric potential is also considered.