A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions


In this paper, we study the multiplicity problem for Euler–Lagrange orbits that satisfy the conormal boundary conditions for a suitable class of reversible Lagrangian functions on compact manifolds. Such a class contains, e.g. the energy function of reversible Finsler metrics that satisfy a convexity condition on the boundary.

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  1. 1.

    Two curves \(\gamma _1,\gamma _2: [0,1] \rightarrow {\overline{\Omega }}\) are considered distinct if \(\gamma _1([0,1]) \ne \gamma _2([0,1])\).

  2. 2.

    There is a standard construction of metrics for which a given closed embedded submanifold of a differentiable manifold is totally geodesic. Such metrics are constructed in a tubular neighbourhood first, using a normal bundle construction, and then extended using a partition of unity argument.


  1. 1.

    Abbondandolo, A.: Lectures on the free period lagrangian action functional. J. Fixed Point Theory Appl. 13, 397–430 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Abbondandolo, A., Schwarzy, M.: A smooth pseudo-gradient for the Lagrangian action functional. Adv. Nonlinear Stud. 9(4) (2009)

  3. 3.

    Asselle, L.: On the existence of orbits satisfying periodic or conormal boundary conditions for Euler–Lagrange flows. PhD thesis, Ruhr Universität Bochum, Fakultät für Mathematik (2015)

  4. 4.

    Asselle, L.: On the existence of euler-lagrange orbits satisfying the conormal boundary conditions. J. Funct. Anal. 271, 3513–3553 (2016)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bos, W.: Kritische sehnen auf riemannschen elementarraumstücken. Math. Ann. 151(5), 431–451 (1963)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Canino, A.: Periodic solutions of lagrangian systems on manifolds with boundary. Nonlinear Anal. Theory Methods Appl. 16, 567–586 (1991)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Degiovanni, M., Marzocchi, M.: A critical point theory for nonsmooth functional. Ann. Mat. Pura Appl. 167(1), 73–100 (1994)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Giambò, R., Giannoni, F., Piccione, P.: Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk. Nonlinear Anal. Theory Methods Appl. (2010)

  9. 9.

    Giambò, R., Giannoni, F., Piccione, P.: Multiple brake orbits in \(m\)-dimensional disks. Calc. Var. 54, 2253–2580 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Giambò, R., Giannoni, F., Piccione, P.: Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles. Calc. Var. Partial Differ. Equ. 57, 117 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

    Google Scholar 

  12. 12.

    Liu, C.: Index Theory in Nonlinear Analysis. Springer Singapore, Singapore (2019)

    Google Scholar 

  13. 13.

    Marino, A., Scolozzi, D.: Geodetiche con ostacolo. Boll. UMI B (6) 2(1), 1–31 (1983)

    MATH  Google Scholar 

  14. 14.

    Palais, R.S.: Lusternik–Schnirelman theory on Banach manifolds. Topology 5(2), 115–132 (1966)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. Math. 108(3), 507–518 (1978)

    MathSciNet  Article  Google Scholar 

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I wish to thank the referee for his/her useful suggestions that improved the paper.

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Correspondence to Dario Corona.

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Corona, D. A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions. J. Fixed Point Theory Appl. 22, 60 (2020). https://doi.org/10.1007/s11784-020-00795-4

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  • Lagrange’s equations
  • variational methods
  • holonomic systems

Mathematics Subject Classification

  • 70H03
  • 70G75
  • 70F20