Periodic solutions with prescribed minimal period of vortex type problem in domains
We consider Hamiltonian systems with two degrees of freedom of point vortex type \[ κ_j \dot{z}_j = J \nabla_{z_j} H_Ω(z_1,z_2), \quad j=1,2, \] for $z_1,z_2$ in a domain $Ω\subset\mathbb{R}^2$. In the classical point vortex context the Hamiltonian $H_Ω$ is of the form \[ H_Ω(z_1,z_2) = -\frac{κ_1 κ_2}π \log |z_1-z_2| - 2κ_1 κ_2g(z_1,z_2) - κ_1^2 h(z_1) - κ_2^2 h(z_2), \] where $g:Ω\timesΩ\to\mathbb{R}$ is the regular part of a hydrodynamic Green function in $Ω$, $h:Ω\to\mathbb{R}$ is the Robin function: $h(z)=g(z,z)$, and $κ_1$, $κ_2$ 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 Poincaré-Birkhoff theorem due to Fonda and Ureña.