Periodic solutions with prescribed minimal period of vortex type problem in domains

Abstract

We consider Hamiltonian systems with two degrees of freedom of point vortex type \[ \kappa_j \dot{z}_j = J \nabla_{z_j} H_\Omega(z_1,z_2), \quad j=1,2, \] for $z_1,z_2$ in a domain $\Omega\subset\mathbb{R}^2$. In the classical point vortex context the Hamiltonian $H_\Omega$ is of the form \[ H_\Omega(z_1,z_2) = -\frac{\kappa_1 \kappa_2}{\pi} \log |z_1-z_2| - 2\kappa_1 \kappa_2g(z_1,z_2) - \kappa_1^2 h(z_1) - \kappa_2^2 h(z_2), \] where $g:\Omega\times\Omega\to\mathbb{R}$ is the regular part of a hydrodynamic Green function in $\Omega$, $h:\Omega\to\mathbb{R}$ is the Robin function: $h(z)=g(z,z)$, and $\kappa_1$, $\kappa_2$ are the vortex strengths. We prove the existence of infinitely many periodic solutions with prescribed minimal period that are superpositions of a slow motion of the center of vorticity close to a star-shaped level line of $h$ and of a fast rotation of the two vortices around their center of vorticity. The proofs are based on a recent higher dimensional version of the Poincar\'e-Birkhoff theorem due to Fonda and Ure\~na.