November  2021, 41(11): 5329-5357. doi: 10.3934/dcds.2021079

A multiplicity result for orthogonal geodesic chords in Finsler disks

Università di Camerino, Scuola di Scienze e Tecnologie, Camerino (MC), Italy

* Corresponding author: Dario Corona

Received  August 2020 Revised  March 2021 Published  November 2021 Early access  May 2021

In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a $ N $-dimensional disk. Under a suitable assumption, which is weaker than convexity, we prove that, if the Finsler metric is reversible, then there are at least $ N $ orthogonal Finsler geodesic chords that are geometrically distinct. If the reversibility assumption does not hold, then there are at least two orthogonal Finsler geodesic chords with different values of the energy functional.

Citation: Dario Corona. A multiplicity result for orthogonal geodesic chords in Finsler disks. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5329-5357. doi: 10.3934/dcds.2021079
References:
[1]

A. Abbondandolo and A. Figalli, High action orbits for Tonelli Langrangians and superlinear Hamiltonians on compact configuration spaces, J. Differential Equations, 234 (2007), 626-653.  doi: 10.1016/j.jde.2006.10.015.

[2]

L. Asselle, On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Funct. Anal., 271 (2016), 3513-3553.  doi: 10.1016/j.jfa.2016.08.023.

[3]

R. BartoloE. CaponioA. V. Germinario and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calc. Var. Partial Differential Equations, 40 (2011), 335-356.  doi: 10.1007/s00526-010-0343-1.

[4]

A. Canino, Periodic solutions of Lagrangian systems on manifolds with boundary, Nonlinear Anal., 16 (1991), 567-586.  doi: 10.1016/0362-546X(91)90029-Z.

[5]

E. CaponioM. Á. Javaloyes and A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, Math. Ann., 351 (2011), 365-392.  doi: 10.1007/s00208-010-0602-7.

[6]

D. Corona, A multiplicity result for Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Fixed Point Theory Appl., 22 (2020), 60, 32 pp. doi: 10.1007/s11784-020-00795-4.

[7]

D. Corona and F. Giannoni, Brake orbits for Hamiltonian systems of classical type via Finsler geodesics, (to appear)

[8]

R. Giambò, F. Giannoni and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds, Adv. Differential Equations, 10 (2005), 931-960. arXiv: math/0410391

[9]

R. GiambòF. Giannoni and P. Piccione, On the multiplicity of orthogonal geodesies in Riemannian manifold with concave boundary. Applications to brake orbits and homoclinics, Adv. Nonlinear Stud., 9 (2009), 763-782.  doi: 10.1515/ans-2009-0409.

[10]

R. GiambòF. Giannoni and P. Piccione, Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and Homeomorphic to the $N$-dimensional disk, Nonlinear Anal., 73 (2010), 290-337.  doi: 10.1016/j.na.2010.03.019.

[11]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits and Homoclinics in Riemannian manifolds, Arch. Ration. Mech. Anal., 200 (2011), 691-724.  doi: 10.1007/s00205-010-0371-1.

[12]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits in $m$-dimensional disks, Calc. Var. Partial Differential Equations, 54 (2015), 2553-2580.  doi: 10.1007/s00526-015-0875-5.

[13]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 117, 26 pp. doi: 10.1007/s00526-018-1394-y.

[14]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits, arXiv: 2002.09687

[15]

F. Giannoni and P. Majer, On the effect of the domain on the number of orthogonal geodesic chords, Differential Geom. Appl., 7 (1997), 341-364.  doi: 10.1016/S0926-2245(96)00055-1.

[16]

F. Giannoni and A. Masiello, On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal., 101 (1991), 340-369.  doi: 10.1016/0022-1236(91)90162-X.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[18]

W. B. Gordon, The existence of geodesics joining two given points, J. Differential Geometry, 9 (1974), 443-450.  doi: 10.4310/jdg/1214432420.

[19]

C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math., 67 (2014), 1563-1604.  doi: 10.1002/cpa.21525.

[20]

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

[21]

F. Mercuri, The critical points theory for the closed geodesics problem, Math. Z., 156 (1977), 231-245.  doi: 10.1007/BF01214411.

[22]

D. Scolozzi, A result of local uniqueness for geodesics on manifolds with boundary, Boll. Un. Mat. Ital., 5-B (1986), 309–327. doi: 11587/118622.

[23]

H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z., 51 (1948), 197-216.  doi: 10.1007/BF01291002.

[24]

Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing, 2001. doi: 10.1142/9789812811622.

[25]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.  doi: 10.2307/1971185.

show all references

References:
[1]

A. Abbondandolo and A. Figalli, High action orbits for Tonelli Langrangians and superlinear Hamiltonians on compact configuration spaces, J. Differential Equations, 234 (2007), 626-653.  doi: 10.1016/j.jde.2006.10.015.

[2]

L. Asselle, On the existence of Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Funct. Anal., 271 (2016), 3513-3553.  doi: 10.1016/j.jfa.2016.08.023.

[3]

R. BartoloE. CaponioA. V. Germinario and M. Sánchez, Convex domains of Finsler and Riemannian manifolds, Calc. Var. Partial Differential Equations, 40 (2011), 335-356.  doi: 10.1007/s00526-010-0343-1.

[4]

A. Canino, Periodic solutions of Lagrangian systems on manifolds with boundary, Nonlinear Anal., 16 (1991), 567-586.  doi: 10.1016/0362-546X(91)90029-Z.

[5]

E. CaponioM. Á. Javaloyes and A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, Math. Ann., 351 (2011), 365-392.  doi: 10.1007/s00208-010-0602-7.

[6]

D. Corona, A multiplicity result for Euler-Lagrange orbits satisfying the conormal boundary conditions, J. Fixed Point Theory Appl., 22 (2020), 60, 32 pp. doi: 10.1007/s11784-020-00795-4.

[7]

D. Corona and F. Giannoni, Brake orbits for Hamiltonian systems of classical type via Finsler geodesics, (to appear)

[8]

R. Giambò, F. Giannoni and P. Piccione, Orthogonal geodesic chords, brake orbits and homoclinic orbits in Riemannian manifolds, Adv. Differential Equations, 10 (2005), 931-960. arXiv: math/0410391

[9]

R. GiambòF. Giannoni and P. Piccione, On the multiplicity of orthogonal geodesies in Riemannian manifold with concave boundary. Applications to brake orbits and homoclinics, Adv. Nonlinear Stud., 9 (2009), 763-782.  doi: 10.1515/ans-2009-0409.

[10]

R. GiambòF. Giannoni and P. Piccione, Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and Homeomorphic to the $N$-dimensional disk, Nonlinear Anal., 73 (2010), 290-337.  doi: 10.1016/j.na.2010.03.019.

[11]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits and Homoclinics in Riemannian manifolds, Arch. Ration. Mech. Anal., 200 (2011), 691-724.  doi: 10.1007/s00205-010-0371-1.

[12]

R. GiambòF. Giannoni and P. Piccione, Multiple brake orbits in $m$-dimensional disks, Calc. Var. Partial Differential Equations, 54 (2015), 2553-2580.  doi: 10.1007/s00526-015-0875-5.

[13]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords in nonconvex Riemannian disks using obstacles, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 117, 26 pp. doi: 10.1007/s00526-018-1394-y.

[14]

R. Giambò, F. Giannoni and P. Piccione, Multiple orthogonal geodesic chords and a proof of Seifert's conjecture on brake orbits, arXiv: 2002.09687

[15]

F. Giannoni and P. Majer, On the effect of the domain on the number of orthogonal geodesic chords, Differential Geom. Appl., 7 (1997), 341-364.  doi: 10.1016/S0926-2245(96)00055-1.

[16]

F. Giannoni and A. Masiello, On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal., 101 (1991), 340-369.  doi: 10.1016/0022-1236(91)90162-X.

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[18]

W. B. Gordon, The existence of geodesics joining two given points, J. Differential Geometry, 9 (1974), 443-450.  doi: 10.4310/jdg/1214432420.

[19]

C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math., 67 (2014), 1563-1604.  doi: 10.1002/cpa.21525.

[20]

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

[21]

F. Mercuri, The critical points theory for the closed geodesics problem, Math. Z., 156 (1977), 231-245.  doi: 10.1007/BF01214411.

[22]

D. Scolozzi, A result of local uniqueness for geodesics on manifolds with boundary, Boll. Un. Mat. Ital., 5-B (1986), 309–327. doi: 11587/118622.

[23]

H. Seifert, Periodische Bewegungen mechanischer Systeme, Math. Z., 51 (1948), 197-216.  doi: 10.1007/BF01291002.

[24]

Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing, 2001. doi: 10.1142/9789812811622.

[25]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.  doi: 10.2307/1971185.

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