July  1997, 3(3): 439-450. doi: 10.3934/dcds.1997.3.439

Periodic orbits on Riemannian manifolds with convex boundary

1. 

Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 BARI, Italy

Received  November 1996 Published  April 1997

We look for $T$-periodic solutions on a convex Riemannian manifold $\mathcal{M}$ of the differential equation

$D_s\dot x(s) + \nabla V_x(x(s),s) = 0$

where $D_s\dot x(s)$ is the covariant derivative of $\dot x(s)$, $V$ is a $\mathcal{C}^2$ real function on $\mathcal{M}\times \mathbf{R}$, $T$-periodic in $s$. The manifold is allowed to be noncompact and to have boundary, so the action integral associated to the equation does not satisfy the Palais-Smale compactness condition. We overcome this problem under a assumption on the sectional curvature of $\mathcal{M}$ which allows to control the Morse index of the critical points of $f$ at "infinity". If $\mathcal{M}$ has a "rich" topology it is proved that there exist infinitely many periodic solutions.

Citation: Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439
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