The Palais-Smale condition on contact type energy levels for convex Lagrangian systems

@article{Contreras2003ThePC,
  title={The Palais-Smale condition on contact type energy levels for convex Lagrangian systems},
  author={Gonzalo Contreras},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2003},
  volume={27},
  pages={321-395}
}
  • G. Contreras
  • Published 2003
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
We prove that for a uniformly convex Lagrangian system L on a compact manifold M, almost all energy levels contain a periodic orbit. We also prove that below Mañé's critical value of the lift of the Lagrangian to the universal cover, cu(L), almost all energy levels have conjugate points.We in addition prove that if an energy level is of contact type, projects onto M and $$M\ne{\mathbb T}^2$$, then the free time action functional of L+k satisfies the Palais-Smale condition. 

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References

SHOWING 1-10 OF 86 REFERENCES
The Palais-Smale Condition and Mañé's Critical Values
Abstract. Let $ {\Bbb L} $ be a convex superlinear autonomous Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [23]. We define energyExpand
Convex Hamiltonians without conjugate points
We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that theExpand
On minimizing measures of the action of autonomous Lagrangians
We apply J Mather's theory (1991) on minimizing measures to the case of positive definite autonomous Lagrangians L:TM to R. We show that the minimal action function beta (h)=min( integral Ld mu modExpand
Lagrangian flows: The dynamics of globally minimizing orbits-II
Define the critical levelc(L) of a convex superlinear LagragianL as the infimum of thek ∈ ℝsuch that the LagragianL+k has minimizers with fixed endpoints and free time interval. We provide proofs forExpand
Lagrangian flows: The dynamics of globally minimizing orbits
The objective of this note is to present some results, to be proved in a forthcoming paper, about certain special solutions of the Euler-Lagrange equations on closed manifolds. Our main resultsExpand
Action potential and weak KAM solutions
Abstract. For convex superlinear lagrangians on a compact manifold M we characterize the Peierls barrier and the weak KAM solutions of the Hamilton-Jacobi equation, as defined by A. Fathi [9], inExpand
Lagrangian Graphs, Minimizing Measures and Mañé's Critical Values
Abstract. Let $\Bbb L$ be a convex superlinear Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mañé in [M3]. We show that the critical valueExpand
Critical values of autonomous Lagrangian systems
Abstract. Let M be a closed manifold and $ L : TM \rightarrow \bf {R} $ a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Mañé in [5]. Let cu(L) denote theExpand
Periodic orbits for exact magnetic flows on surfaces
We show that any exact magnetic flow on a closed surface has periodic orbits in all energy levels. Moreover, we give homological and homotopical properties of these periodic orbits in terms of theExpand
Hamiltonian dynamics on convex symplectic manifolds
We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish an explicit isomorphism between the Floer homology and the Morse homology of such aExpand
...
1
2
3
4
5
...