Solutions with minimal period for Hamiltonian systems in a potential well

Abstract

Let $\mathrm{U}\in C^2(\Omega )$, where $\Omega$ is a bounded set in $R^N$. Suppose that $\mathrm{U}(x)$ tends to $+\infty$ as $x$ tends to $\partial \Omega$. Our main results concern the existence of periodic solutions of $-\ddot{x}={\mathrm{U}}^{\prime }(x)$ having a prescribed number $\mathrm{T}$ as minimal period. The results are also generalized to first order Hamiltonian systems.

Résumé

Soit $\mathrm{U}\in C^2(\Omega )$, où $\Omega$ est un ouvert donné de $R^N$. On suppose que $\mathrm{U}(x)\to +\infty$ quand $x\to \partial \Omega$. On montre l’existence de solutions périodiques de $\ddot{x}+{\mathrm{U}}^{\prime }(x)=0$, de période minimale prescrite. On étend ces résultats aux systèmes hamiltoniens du premier ordre.