Periodic Solutions of N-Vortex Type Hamiltonian Systems near the Domain Boundary

The paper deals with the existence of nonstationary collision-free periodic solutions of singular first order Hamiltonian systems of $N$-vortex type in a domain $\Omega\subset\mathbb R^2$. These are solutions $z(t)=(z_1(t),\dots,z_N(t)) \in \Omega^N$ of $\dot{z}_j(t)=J\nabla_{z_j} H(z(t)), j=1,\dots,N$, where the Hamiltonian $H$ has the form $H(z_1,\dots,z_N) = -\sum_{j,k=1, j\ne k}^N \frac{1}{2\pi}\log|z_j-z_k| -\sum_{j,k=1}^N g(z_j,z_k).$ Here $J = (\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})$ is the standard symplectic matrix in $\mathbb{R}^2$. The function $g:\Omega\times\Omega \to\mathbb{R}$ is required to be of class $\mathcal C^3$ and symmetric, the regular part of the Dirichlet Green's function being our model. The Hamiltonian is unbounded from above and below, and the associated action integral is not defined on an open subset of the space of periodic $H^{1/2}$ functions. Given a compact connected component $\Gamma\subset\partial\Omega$ of class $\mathcal C^3$ we are interested in peri...