# zbMATH — the first resource for mathematics

A smooth pseudo-gradient for the Lagrangian action functional. (English) Zbl 1185.37145
The authors consider the action functional of a Lagrangian system with quadratic growth in the velocity on the natural Hilbert manifold of absolutely continuous trajectories with square integrable derivative. In this setting, it is known that the action is $$C^1$$ and twice Gateaux differentiable but in general not twice Frechet differentiable [see V. Benci, J. Differ. Equations 63, 135–161 (1986; Zbl 0605.58034)].
The regularity of the action functional becomes a touchy business when dealing with Morse Theory, where having a $$C^2$$ functional would be desirable. In this paper, the authors are able to overcome this difficulty by showing that if the critical points of the action are non-degenerate, there exists a smooth Morse-Smale vector field for which the action is a Lyapunov function. This provides the construction of the Morse complex and gives several multiplicity results.

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
Full Text:
##### References:
 [1] Abbondandolo, The homology of path spaces and Floer homology with conormal boundary conditions Fixed Point Theory, Appl pp 263– (2008) · Zbl 1171.53344 [2] Lasry, A remark on regularization in Hilbert spaces Israel, Math pp 55– (1986) · Zbl 0631.49018 [3] Abbondandolo, Lectures on the Morse complex for infinite dimen - sional manifolds editors Methods in Nonlinear Analysis and in Symplectic Topology Montreal pages, Theoretic pp 1– (2006) [4] Duc, Morse - Palais lemma for nonsmooth func - tionals on normed spaces, Amer Math Soc pp 135– (2007) [5] Mazzucchelli, The Lagrangian Conley Conjecture arXiv math ( to appear in Comment, Math Helv pp 0810– (2008) [6] Li, Splitting theorem Hopf theorem and jumping nonlinear problems, Funct Anal pp 221– (2005) · Zbl 1129.35392 [7] Abbondandolo, On the Floer homology of cotangent bundles Pure, Appl Math pp 59– (2006) · Zbl 1084.53074 [8] Abbondandolo, On the global stable manifold, Math pp 177– (2006) · Zbl 1105.37018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.