Morse–Palais lemma for nonsmooth functionals on normed spaces

Duc, Duong Minh; Hung, Tran Vinh; Nguyen, Khai T.

doi:10.1090/s0002-9939-06-08662-x

Cited by 10 publications

(18 citation statements)

References 9 publications

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“…Our proof needs the following parameterized version of the Morse-Palais lemma due to Duc-Hung-Khai (Theorem 1.1 in [7]), whose proof can be obtained by almost repeating the proof in [7] (cf. Appendix A of [20] for details).…”

Section: A Splitting Lemma For C -Functionalsmentioning

confidence: 99%

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

2009

Journal of Functional Analysis

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In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713-756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C 1 -Hamiltonian systems on the cotangent bundle of a C 3 -smooth compact manifold M without boundary, of a time 1-periodic C 2 -smooth Hamiltonian H : R × T * M → R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H (−t, q, −p) = H (t, q, p) for any (t, q, p) ∈ R×T * M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T * M. If M is C 5 -smooth and dim M > 1, H is of C 4 class and independent of time t, then for any τ > 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H , L : R × T M → R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.

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Section: A Splitting Lemma For C -Functionalsmentioning

confidence: 99%

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

2009

Journal of Functional Analysis

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“…So the functionals are assumed to be at least C 2 . Based on the proof ideas of the Morse-Palais lemma due to Duc-Hung-Khai [9] and some techniques from [11], [17], [18] we find a new method to establish the splitting theorems for nonsmooth functionals on Hilbert spaces in [13], [14]. We shall follow the notations therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Step 3 of the proof of Theorem 2.1 (given in Appendix A of [13,14]) we can get the desired conclusion from Lemma 2.3 of [9].…”

Section: As Inmentioning

confidence: 93%

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The splitting lemmas for nonsmooth functional on Hilbert spaces II. The case at infinity.

Lu¹

2016

TMNA

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We generalize the Bartsch-Li's splitting lemma at infinity for C 2 -functionals in [2] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Different from the previous flow methods our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [9] with some techniques from [11], [17], [18]. A simple application is also presented.

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“…These existence results could be proved also by determining the critical groups of the critical points of S L by some weak version of the Morse Lemma (see [17] and [12] for two abstract statements of this kind, and [9] for their use in the study of the Finsler energy functional), or by replacing the space H 1 ([0, 1], M ) by finite dimensional spaces of continuous piecewise extremals of the Euler-Lagrange equation (see [19] for the case where L is the Finsler energy, or [18] for more general Lagrangian functions). However, our main motivation for constructing the Morse complex for the action functional associated to a Lagrangian L which has quadratic growth in the velocities comes from the study of Floer homology of cotangent bundles and its relationship with string topology.…”

Section: Introductionmentioning

confidence: 98%

A Smooth Pseudo-Gradient for the Lagrangian Action Functional

Abbondandolo

Schwarz

2009

Advanced Nonlinear Studies

137

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We study the action functional associated to a smooth Lagrangian function on the tangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H 1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is sufficient to associate a Morse complex to the Lagrangian action functional.

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Morse–Palais lemma for nonsmooth functionals on normed spaces

Cited by 10 publications

References 9 publications

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

The splitting lemmas for nonsmooth functional on Hilbert spaces II. The case at infinity.

A Smooth Pseudo-Gradient for the Lagrangian Action Functional

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